diff --git a/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean b/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean
index 897b98f92..ab3513da1 100644
--- a/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean
+++ b/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Basic.lean
@@ -166,9 +166,8 @@ decreasing at that time. -/
 lemma energy_not_conserved (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (t : Time)
     (h1 : S.EquationOfMotion xₜ) (hx : ContDiff ℝ ∞ xₜ) (hdx : ∂ₜ xₜ t ≠ 0) (hγ : 0 < S.γ) :
     ∂ₜ (S.energy xₜ) t < 0 := by
-  rw [energy_dissipation_rate S xₜ t h1 hx]
-  rw [neg_mul]
-  exact neg_neg_of_pos (mul_pos hγ (real_inner_self_pos.mpr hdx))
+  simpa [energy_dissipation_rate S xₜ t h1 hx, neg_mul] using
+    neg_neg_of_pos (mul_pos hγ (real_inner_self_pos.mpr hdx))
 
 /-!
 ## C. Newton's second law
@@ -206,8 +205,7 @@ lemma equationOfMotion_iff_newtons_2nd_law (xₜ : Time → EuclideanSpace ℝ (
   constructor
   · intro h t
     have h' :
-        S.m • ∂ₜ (∂ₜ xₜ) t + (S.γ • ∂ₜ xₜ t + S.k • xₜ t) = 0 := by
-      simpa [add_assoc] using h t
+        S.m • ∂ₜ (∂ₜ xₜ) t + (S.γ • ∂ₜ xₜ t + S.k • xₜ t) = 0 := by simpa [add_assoc] using h t
     have ha :
         S.m • ∂ₜ (∂ₜ xₜ) t = -(S.γ • ∂ₜ xₜ t + S.k • xₜ t) :=
       eq_neg_of_add_eq_zero_left h'
@@ -267,39 +265,33 @@ lemma discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq :
 
 /-- The decay rate is nonnegative. -/
 lemma decayRate_nonneg : 0 ≤ S.decayRate := by
-  rw [decayRate]
-  exact div_nonneg S.γ_nonneg (by nlinarith [S.m_pos])
+  simpa [decayRate] using div_nonneg S.γ_nonneg (by nlinarith [S.m_pos])
 
 /-- An undamped oscillator lies in the underdamped regime. -/
 lemma isUnderdamped_of_gamma_eq_zero (hγ : S.γ = 0) : S.IsUnderdamped := by
-  rw [IsUnderdamped, discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq S, decayRate]
-  rw [hγ]
+  rw [IsUnderdamped, discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq S, decayRate, hγ]
   ring_nf
   nlinarith [sq_pos_of_pos S.m_pos, sq_pos_of_pos S.ω_pos]
 
 /-- An underdamped system has decay rate less than the natural frequency. -/
 lemma isUnderdamped_decayRate (hS : S.IsUnderdamped) : S.decayRate < S.ω := by
-  rw [IsUnderdamped] at hS
-  rw [discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq] at hS
+  rw [IsUnderdamped, discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq] at hS
   have hm_sq_pos : 0 < 4 * S.m^2 := by
     have hsq : 0 < S.m^2 := sq_pos_of_pos S.m_pos
     nlinarith
-  have hsq : S.decayRate^2 < S.ω^2 := by
-    nlinarith
+  have hsq : S.decayRate^2 < S.ω^2 := by nlinarith
   nlinarith [S.decayRate_nonneg, S.ω_pos]
 
 /-- A critically damped system has decay rate equal to the natural frequency. -/
 lemma isCriticallyDamped_decayRate (hS : S.IsCriticallyDamped) : S.ω = S.decayRate := by
-  rw [IsCriticallyDamped] at hS
-  rw [discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq] at hS
+  rw [IsCriticallyDamped, discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq] at hS
   have hm_sq_ne_zero : 4 * S.m^2 ≠ 0 := by
     have hm_sq_pos : 0 < 4 * S.m^2 := by
       have hsq : 0 < S.m^2 := sq_pos_of_pos S.m_pos
       nlinarith
     exact ne_of_gt hm_sq_pos
   have hsq : S.decayRate^2 = S.ω^2 := by
-    have hsub : S.decayRate^2 - S.ω^2 = 0 := by
-      exact (mul_eq_zero.mp hS).resolve_left hm_sq_ne_zero
+    have hsub : S.decayRate^2 - S.ω^2 = 0 := (mul_eq_zero.mp hS).resolve_left hm_sq_ne_zero
     linarith
   nlinarith [S.decayRate_nonneg, S.ω_pos]
 
@@ -317,20 +309,17 @@ lemma k_eq_m_mul_ω_sq : S.k = S.m * S.ω^2 := by
 lemma k_eq_m_mul_decayRate_sq_of_criticallyDamped (hS : S.IsCriticallyDamped) :
     S.k = S.m * S.decayRate^2 := by
   have hωa : S.ω = S.decayRate := S.isCriticallyDamped_decayRate hS
-  have hωsq : S.decayRate ^ 2 = S.k / S.m := by
-    simpa [hωa] using S.ω_sq
+  have hωsq : S.decayRate ^ 2 = S.k / S.m := by simpa [hωa] using S.ω_sq
   field_simp [S.m_ne_zero] at hωsq
   nlinarith
 
 /-- An overdamped system has decay rate greater than the natural frequency. -/
 lemma isOverdamped_decayRate (hS : S.IsOverdamped) : S.ω < S.decayRate := by
-  rw [IsOverdamped] at hS
-  rw [discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq] at hS
+  rw [IsOverdamped, discriminant_eq_four_mul_m_sq_mul_decayRate_sq_sub_ω_sq] at hS
   have hm_sq_pos : 0 < 4 * S.m^2 := by
     have hsq : 0 < S.m^2 := sq_pos_of_pos S.m_pos
     nlinarith
-  have hsq : S.ω^2 < S.decayRate^2 := by
-    nlinarith
+  have hsq : S.ω^2 < S.decayRate^2 := by nlinarith
   nlinarith [S.decayRate_nonneg, S.ω_pos]
 
 /-- In the underdamped regime, the selected frequency uses the oscillation frequency. -/
@@ -439,8 +428,7 @@ for the corresponding undamped harmonic oscillator. -/
 lemma toUndamped_equationOfMotion (S : DampedHarmonicOscillator) (hS : S.IsUndamped)
     (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff ℝ ∞ xₜ) :
     S.EquationOfMotion xₜ ↔ (S.toUndamped hS).EquationOfMotion xₜ := by
-  have hγ : S.γ = 0 := by
-    simpa [IsUndamped] using hS
+  have hγ : S.γ = 0 := by simpa [IsUndamped] using hS
   rw [S.equationOfMotion_iff_newtons_2nd_law xₜ,
     (S.toUndamped hS).equationOfMotion_iff_newtons_2nd_law xₜ hx]
   constructor
@@ -454,8 +442,7 @@ lemma toUndamped_equationOfMotion (S : DampedHarmonicOscillator) (hS : S.IsUndam
     calc
       S.m • ∂ₜ (∂ₜ xₜ) t = (S.toUndamped hS).m • ∂ₜ (∂ₜ xₜ) t := rfl
       _ = HarmonicOscillator.force (S.toUndamped hS) (xₜ t) := h t
-      _ = force S xₜ t := by
-        simp [force, HarmonicOscillator.force_eq_linear, toUndamped, hγ]
+      _ = force S xₜ t := by simp [force, HarmonicOscillator.force_eq_linear, toUndamped, hγ]
 
 end DampedHarmonicOscillator
 
diff --git a/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean b/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean
index 96fbaccb1..31581a804 100644
--- a/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean
+++ b/Physlib/ClassicalMechanics/DampedHarmonicOscillator/Solution.lean
@@ -178,8 +178,7 @@ private lemma exp_decay_smul_velocity
     ∂ₜ (fun t : Time => exp (-a * t.val) • y t) =
       fun t : Time => exp (-a * t.val) • (∂ₜ y t - a • y t) := by
   funext t
-  rw [Time.deriv]
-  rw [fderiv_fun_smul (by fun_prop) (hy t)]
+  rw [Time.deriv, fderiv_fun_smul (by fun_prop) (hy t)]
   rw [fderiv_exp (by fun_prop), fderiv_fun_mul (by fun_prop) (by fun_prop)]
   simp only [add_apply, ContinuousLinearMap.smulRight_apply,
     fderiv_fun_neg, fderiv_fun_const, Pi.zero_apply, Time.fderiv_val,
@@ -197,11 +196,9 @@ private lemma exp_decay_smul_acceleration
         (μ • y t - (2 * a) • ∂ₜ y t + a^2 • y t) := by
   rw [exp_decay_smul_velocity a y hy]
   funext t
-  rw [Time.deriv]
-  rw [fderiv_fun_smul (by fun_prop) (by fun_prop)]
+  rw [Time.deriv, fderiv_fun_smul (by fun_prop) (by fun_prop)]
   rw [fderiv_exp (by fun_prop), fderiv_fun_mul (by fun_prop) (by fun_prop)]
-  rw [fderiv_fun_sub (hdy t) (by fun_prop)]
-  rw [fderiv_fun_const_smul (hy t)]
+  rw [fderiv_fun_sub (hdy t) (by fun_prop), fderiv_fun_const_smul (hy t)]
   have hy''_t := congrFun hy'' t
   rw [Time.deriv] at hy''_t
   simp only [add_apply, _root_.sub_apply,
@@ -219,9 +216,7 @@ private lemma exp_decay_smul_equationOfMotion
     (hγ : S.γ = 2 * S.m * a) (hk : S.k = S.m * (a^2 - μ)) :
     S.EquationOfMotion (fun t : Time => exp (-a * t.val) • y t) := by
   intro t
-  rw [exp_decay_smul_acceleration a μ y hy hdy hy'']
-  rw [exp_decay_smul_velocity a y hy]
-  rw [hγ, hk]
+  rw [exp_decay_smul_acceleration a μ y hy hdy hy'', exp_decay_smul_velocity a y hy, hγ, hk]
   simp [smul_add, smul_sub, smul_smul]
   module
 
diff --git a/Physlib/ClassicalMechanics/EulerLagrange.lean b/Physlib/ClassicalMechanics/EulerLagrange.lean
index fe0086452..2340d6825 100644
--- a/Physlib/ClassicalMechanics/EulerLagrange.lean
+++ b/Physlib/ClassicalMechanics/EulerLagrange.lean
@@ -36,8 +36,7 @@ lemma eulerLagrangeOp_eq (L : Time → X → X → ℝ) (q : Time → X) :
     - ∂ₜ (fun t' => gradient (L t' (q t') ·) (∂ₜ q t')) t := by rfl
 
 lemma eulerLagrangeOp_zero (q : Time → X) :
-    eulerLagrangeOp (fun _ _ _ => 0) q = fun _ => 0 := by
-  simp [eulerLagrangeOp_eq, Time.deriv_eq]
+    eulerLagrangeOp (fun _ _ _ => 0) q = fun _ => 0 := by simp [eulerLagrangeOp_eq, Time.deriv_eq]
 
 /- The variational derivative of `L t (q' t) (deriv q' t))` for a lagrangian `L`
   is equal to the `eulerLagrangeOp`. -/
diff --git a/Physlib/ClassicalMechanics/HamiltonsEquations.lean b/Physlib/ClassicalMechanics/HamiltonsEquations.lean
index 71d0ed6d0..f9309e0a9 100644
--- a/Physlib/ClassicalMechanics/HamiltonsEquations.lean
+++ b/Physlib/ClassicalMechanics/HamiltonsEquations.lean
@@ -43,8 +43,7 @@ noncomputable def hamiltonEqOp (H : Time → X → X → ℝ) (p : Time → X) (
 
 lemma hamiltonEqOp_eq (H : Time → X → X → ℝ) (p : Time → X) (q : Time → X) :
     hamiltonEqOp H p q = fun t => (∂ₜ q t + -gradient (fun x => H t x (q t)) (p t),
-      - ∂ₜ p t + -gradient (fun x => H t (p t) x) (q t)) := by
-  rfl
+      - ∂ₜ p t + -gradient (fun x => H t (p t) x) (q t)) := rfl
 
 lemma hamiltonEqOp_eq_zero_iff_hamiltons_equations (H : Time → X → X → ℝ)
     (p : Time → X) (q : Time → X) :
diff --git a/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean b/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean
index 347e2bc7e..b4dc0d179 100644
--- a/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean
+++ b/Physlib/ClassicalMechanics/HarmonicOscillator/Basic.lean
@@ -155,8 +155,7 @@ lemma ω_pos : 0 < S.ω := sqrt_pos.mpr (div_pos S.k_pos S.m_pos)
 
 /-- The square of the angular frequency of the classical harmonic oscillator is equal to `k/m`. -/
 lemma ω_sq : S.ω^2 = S.k / S.m := by
-  rw [ω, sq_sqrt]
-  exact div_nonneg (le_of_lt S.k_pos) (le_of_lt S.m_pos)
+  simpa only [ω] using sq_sqrt (div_nonneg (le_of_lt S.k_pos) (le_of_lt S.m_pos))
 
 /-- The angular frequency of the classical harmonic oscillator is not equal to zero. -/
 lemma ω_ne_zero : S.ω ≠ 0 := Ne.symm (ne_of_lt S.ω_pos)
@@ -263,8 +262,7 @@ lemma kineticEnergy_deriv (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : Con
   unfold kineticEnergy
   conv_lhs => simp only [Time.deriv, one_div, ringHom_apply]
   change (fderiv ℝ ((fun x => 2⁻¹ * S.m * ⟪x, x⟫_ℝ) ∘ (fun t => ∂ₜ xₜ t)) t) 1 = _
-  rw [fderiv_comp]
-  rw [fderiv_const_mul (by fun_prop)]
+  rw [fderiv_comp, fderiv_const_mul (by fun_prop)]
   simp only [ContinuousLinearMap.smul_comp, FunLike.coe_smul,
     ContinuousLinearMap.coe_comp, Pi.smul_apply, Function.comp_apply, smul_eq_mul]
   rw [fderiv_inner_apply]
@@ -281,8 +279,7 @@ lemma potentialEnergy_deriv (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : C
   unfold potentialEnergy
   conv_lhs => simp only [Time.deriv, one_div, smul_eq_mul]
   change (fderiv ℝ ((fun x => 2⁻¹ * (S.k * ⟪x, x⟫_ℝ)) ∘ (fun t => xₜ t)) t) 1 = _
-  rw [fderiv_comp]
-  rw [fderiv_const_mul (by fun_prop), fderiv_const_mul (by fun_prop)]
+  rw [fderiv_comp, fderiv_const_mul (by fun_prop), fderiv_const_mul (by fun_prop)]
   simp only [ContinuousLinearMap.smul_comp, FunLike.coe_smul,
     ContinuousLinearMap.coe_comp, Pi.smul_apply, Function.comp_apply, smul_eq_mul]
   rw [fderiv_inner_apply]
@@ -302,11 +299,9 @@ lemma energy_deriv (xₜ : Time → EuclideanSpace ℝ (Fin 1)) (hx : ContDiff 
     ∂ₜ (energy S xₜ) = fun t => ⟪∂ₜ xₜ t, S.m • ∂ₜ (∂ₜ xₜ) t + S.k • xₜ t⟫_ℝ := by
   unfold energy
   funext t
-  rw [Time.deriv_eq]
-  rw [fderiv_fun_add (by fun_prop) (by apply S.potentialEnergy_differentiable xₜ hx)]
+  rw [Time.deriv_eq, fderiv_fun_add (by fun_prop) (by apply S.potentialEnergy_differentiable xₜ hx)]
   simp only [_root_.add_apply]
-  rw [← Time.deriv_eq, ← Time.deriv_eq]
-  rw [potentialEnergy_deriv, kineticEnergy_deriv]
+  rw [← Time.deriv_eq, ← Time.deriv_eq, potentialEnergy_deriv, kineticEnergy_deriv]
   simp only
   rw [← inner_add_right]
   fun_prop
@@ -418,10 +413,8 @@ lemma gradient_const_mul_inner_self (c : ℝ) (x : EuclideanSpace ℝ (Fin 1)) :
           rw [fderiv_const_mul]; fun_prop
     _ = c • gradient (fun y : EuclideanSpace ℝ (Fin 1) => ⟪y, y⟫_ℝ) x := by
           simp only [gradient, map_smul]
-    _ = c • ((2 : ℝ) • x) := by
-          rw [gradient_inner_self]
-    _ = (2 * c) • x := by
-          rw [smul_smul, mul_comm]
+    _ = c • ((2 : ℝ) • x) := by rw [gradient_inner_self]
+    _ = (2 * c) • x := by rw [smul_smul, mul_comm]
 
 /-!
 
@@ -689,12 +682,9 @@ noncomputable def toCanonicalMomentum (t : Time) (x : EuclideanSpace ℝ (Fin 1)
     EuclideanSpace ℝ (Fin 1) ≃ₗ[ℝ] EuclideanSpace ℝ (Fin 1) where
   toFun v := gradient (S.lagrangian t x ·) v
   invFun p := (1 / S.m) • p
-  left_inv v := by
-    simp [gradient_lagrangian_velocity_eq]
-  right_inv p := by
-    simp [gradient_lagrangian_velocity_eq]
-  map_add' v1 v2 := by
-    simp [gradient_lagrangian_velocity_eq]
+  left_inv v := by simp [gradient_lagrangian_velocity_eq]
+  right_inv p := by simp [gradient_lagrangian_velocity_eq]
+  map_add' v1 v2 := by simp [gradient_lagrangian_velocity_eq]
   map_smul' c v := by
     simp [gradient_lagrangian_velocity_eq]
     module
diff --git a/Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/Basic.lean b/Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/Basic.lean
index 58b56268b..3313f7d12 100644
--- a/Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/Basic.lean
+++ b/Physlib/ClassicalMechanics/HarmonicOscillator/Geometric/Basic.lean
@@ -122,9 +122,7 @@ homeomorphism underlies the single global chart used for the smooth-manifold str
 def valHomeomorphism : ConfigurationSpace ≃ₜ EuclideanSpace ℝ (Fin 1) where
   toEquiv := valEquiv
   continuous_toFun := continuous_induced_dom
-  continuous_invFun := by
-    apply continuous_induced_rng.mpr
-    exact continuous_id
+  continuous_invFun := continuous_induced_rng.mpr continuous_id
 
 /-- Configuration space is Hausdorff, transported from `EuclideanSpace ℝ (Fin 1)` across the
 coordinate homeomorphism. -/
@@ -148,8 +146,7 @@ chart's self-transition, which is analytic, so chart compatibility is immediate.
 instance : ChartedSpace (EuclideanSpace ℝ (Fin 1)) ConfigurationSpace where
   atlas := { valHomeomorphism.toOpenPartialHomeomorph }
   chartAt _ := valHomeomorphism.toOpenPartialHomeomorph
-  mem_chart_source := by
-    simp
+  mem_chart_source := by simp
   chart_mem_atlas := by
     intro x
     simp
diff --git a/Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean b/Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean
index 459ab748c..ceded4b46 100644
--- a/Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean
+++ b/Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean
@@ -386,8 +386,7 @@ The trajectory for zero initial conditions is the zero function.
 
 /-- For zero initial conditions, the trajectory is zero. -/
 @[simp]
-lemma trajectory_zero : trajectory S 0 = fun _ => 0 := by
-  simp [trajectory_eq]
+lemma trajectory_zero : trajectory S 0 = fun _ => 0 := by simp [trajectory_eq]
 
 /-!
 
@@ -549,8 +548,7 @@ lemma trajectories_unique (IC : InitialConditions) (x : Time → EuclideanSpace
   have hy_deriv : ∂ₜ y = fun t => ∂ₜ x t - ∂ₜ (IC.trajectory S) t :=
     dsub x _ (hx.differentiable (by simp)) (hTraj.differentiable (by simp))
   have hy_deriv2 : ∂ₜ (∂ₜ y) = fun t => ∂ₜ (∂ₜ x) t - ∂ₜ (∂ₜ (IC.trajectory S)) t := by
-    rw [hy_deriv]
-    exact dsub _ _ (deriv_differentiable_of_contDiff _ hx)
+    simpa [hy_deriv] using dsub _ _ (deriv_differentiable_of_contDiff _ hx)
       (deriv_differentiable_of_contDiff _ hTraj)
   have hNewt_x := (S.equationOfMotion_iff_newtons_2nd_law x hx).1 hEOM
   have hNewt_traj := (S.equationOfMotion_iff_newtons_2nd_law (IC.trajectory S) hTraj).1
@@ -1000,8 +998,7 @@ lemma trajectory_velocity_eq_zero_iff_exists_int (IC : InitialConditions)
       linarith
     calc
       (t : ℝ) = (S.ω * t) / S.ω := by field_simp [hω]
-      _ = ((AmplitudePhase.fromInitialConditions S IC).φ + n * π) / S.ω := by
-        rw [ht]
+      _ = ((AmplitudePhase.fromInitialConditions S IC).φ + n * π) / S.ω := by rw [ht]
   · intro h
     obtain ⟨n, hn⟩ := h
     rw [Real.sin_eq_zero_iff]
@@ -1039,8 +1036,7 @@ lemma trajectory_velocity_eq_zero_iff_norm_eq_amplitude (IC : InitialConditions)
       ext i
       fin_cases i
       simp [hA]
-  rw [trajectory_velocity_eq_zero_iff_sin_eq_zero S IC hA t]
-  rw [trajectory_eq_cos]
+  rw [trajectory_velocity_eq_zero_iff_sin_eq_zero S IC hA t, trajectory_eq_cos]
   set A := (AmplitudePhase.fromInitialConditions S IC).A
   set θ := S.ω * t.val - (AmplitudePhase.fromInitialConditions S IC).φ
   show sin θ = 0 ↔ ‖EuclideanSpace.single 0 (A * cos θ)‖ = A
@@ -1055,8 +1051,7 @@ lemma trajectory_velocity_eq_zero_iff_norm_eq_amplitude (IC : InitialConditions)
     · simp [hcos, abs_of_pos hA_pos]
     · simp [hcos, abs_of_pos hA_pos]
   · intro hnorm
-    have hnorm' : |A * cos θ| = A := by
-      simpa using hnorm
+    have hnorm' : |A * cos θ| = A := by simpa using hnorm
     have hcos_abs : |cos θ| = 1 := by
       calc
         |cos θ| = |A * cos θ| / A := by
@@ -1118,8 +1113,7 @@ lemma trajectory_eq_zero_iff_exists_int (IC : InitialConditions)
       linarith
     calc
       (t : ℝ) = (S.ω * t) / S.ω := by field_simp [hω]
-      _ = ((AmplitudePhase.fromInitialConditions S IC).φ + (2 * n + 1) * π / 2) / S.ω := by
-        rw [ht]
+      _ = ((AmplitudePhase.fromInitialConditions S IC).φ + (2 * n + 1) * π / 2) / S.ω := by rw [ht]
   · intro h
     obtain ⟨n, hn⟩ := h
     rw [Real.cos_eq_zero_iff]
@@ -1182,8 +1176,7 @@ lemma trajectory_periodic (IC : InitialConditions) :
   have h : S.ω * (t.val + 2 * π / S.ω) = S.ω * t.val + 2 * π := by
     have := S.ω_ne_zero
     ring_nf; field_simp
-  rw [InitialConditions.trajectory, add_val, period_eq, h, cos_add_two_pi, sin_add_two_pi]
-  rfl
+  simp [InitialConditions.trajectory, add_val, period_eq, h, cos_add_two_pi, sin_add_two_pi]
 
 /-!
 
@@ -1217,18 +1210,14 @@ lemma return_time (IC : InitialConditions) (non_trivial : IC.x₀ ≠ 0 ∨ IC.v
   have zero_lt_det :  0 < det := by
    cases non_trivial with
    | inl hx =>
-    have  xx_gt_zero : 0 < xx  := by
-        apply real_inner_self_pos.mpr
-        exact hx
+    have  xx_gt_zero : 0 < xx  := real_inner_self_pos.mpr hx
     calc
       0 < xx * S.ω^2 := by bound
       _ ≤  ‖IC.v₀‖^2 +   xx * S.ω^2  := by bound
       _ = vv +   xx * S.ω^2 := by rw [← real_inner_self_eq_norm_sq IC.v₀]
       _ = det := by rfl
    | inr hv =>
-     have vv_gt_zero : 0 < vv := by
-        apply real_inner_self_pos.mpr
-        exact hv
+     have vv_gt_zero : 0 < vv := real_inner_self_pos.mpr hv
      calc
         0 <  vv := vv_gt_zero
         _ ≤ vv +   ‖IC.x₀‖^2 * S.ω^2 := by bound
diff --git a/Physlib/ClassicalMechanics/Lagrangian/TotalDerivativeEquivalence.lean b/Physlib/ClassicalMechanics/Lagrangian/TotalDerivativeEquivalence.lean
index 2c74e1d5b..718aa4fbe 100644
--- a/Physlib/ClassicalMechanics/Lagrangian/TotalDerivativeEquivalence.lean
+++ b/Physlib/ClassicalMechanics/Lagrangian/TotalDerivativeEquivalence.lean
@@ -97,8 +97,7 @@ lemma isTotalTimeDerivative_explicit {δL : Time → X → X → ℝ} :
     ∀ t q v, δL t q v = fderiv ℝ ↿F (t, q) ((1 : Time), v)) := by
   -- Preliminary construction: properties of the function t => (t, q t)
   let tq := fun (q : Time → X) t => (t, q t)
-  have h_tq_contDiff : ∀ (q : Time → X), ContDiff ℝ ∞ q -> ContDiff ℝ ∞ (tq q) := by
-    fun_prop
+  have h_tq_contDiff : ∀ (q : Time → X), ContDiff ℝ ∞ q -> ContDiff ℝ ∞ (tq q) := by fun_prop
   have h_tq_der :  ∀ (q : Time → X) t, ContDiff ℝ ∞ q -> ∂ₜ (tq q) t = (1, ∂ₜ q t) := by
     intro q t h_ContDiff_q
     ext
@@ -114,8 +113,7 @@ lemma isTotalTimeDerivative_explicit {δL : Time → X → X → ℝ} :
         · simp
           rfl
         · apply ContDiffAt.differentiableAt
-          · apply ContDiff.contDiffAt
-            exact h_tq_contDiff q h_ContDiff_q
+          · exact (h_tq_contDiff q h_ContDiff_q).contDiffAt
           · by_contra
             rcases this
     apply Eq.symm
@@ -123,11 +121,9 @@ lemma isTotalTimeDerivative_explicit {δL : Time → X → X → ℝ} :
        (1, ∂ₜ q t).2 = fderiv ℝ (fun t' => (tq q t').2) t 1 := by rfl
        _ = (∂ₜ (tq q) t).2 := by
         rw [fderiv.snd]
-        · simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.coe_snd']
-          rfl
+        · rfl
         · apply ContDiffAt.differentiableAt
-          · apply ContDiff.contDiffAt
-            exact h_tq_contDiff q h_ContDiff_q
+          · exact (h_tq_contDiff q h_ContDiff_q).contDiffAt
           · by_contra
             rcases this
   have h_F_tq_der : ∀ (q : Time → X) (F : Time → X → ℝ) t, (ContDiff ℝ ∞ ↿F) → (ContDiff ℝ ∞ q)  →
@@ -139,13 +135,11 @@ lemma isTotalTimeDerivative_explicit {δL : Time → X → X → ℝ} :
       rw [← Time.deriv_eq,h_tq_der]
       exact hq
     · apply ContDiffAt.differentiableAt
-      · apply ContDiff.contDiffAt
-        exact hF
+      · exact hF.contDiffAt
       · by_contra
         rcases this
     · apply ContDiffAt.differentiableAt
-      · apply ContDiff.contDiffAt
-        exact h_tq_contDiff q hq
+      · exact (h_tq_contDiff q hq).contDiffAt
       · by_contra
         rcases this
   -- beginning of the proof
@@ -338,12 +332,10 @@ lemma isTotalTimeDerivativeVelocity  [CompleteSpace X]
     have h0 :
         δL (0 : X) =
           fderiv ℝ ↿F ((0 : Time), (0 : X))
-            ((1 : Time), (0 : X)) := by
-      simpa using (hEq (0 : Time) (0 : X)
+            ((1 : Time), (0 : X)) := by simpa using (hEq (0 : Time) (0 : X)
         (0 : X))
     have : dF ((1 : Time), (0 : X)) =
-        δL (0 : X) := by
-      simpa [dF] using h0.symm
+        δL (0 : X) := by simpa [dF] using h0.symm
     simpa [hδL0] using this
 
   -- Induced continuous linear functional on velocity: v ↦ dF (0,v).
@@ -356,10 +348,8 @@ lemma isTotalTimeDerivativeVelocity  [CompleteSpace X]
     have hv :
         δL v =
           fderiv ℝ ↿F ((0 : Time), (0 : X))
-            ((1 : Time), v) := by
-      simpa using (hEq (0 : Time) (0 : X) v)
-    have hv' : δL v = dF ((1 : Time), v) := by
-      simpa [dF] using hv
+            ((1 : Time), v) := by simpa using (hEq (0 : Time) (0 : X) v)
+    have hv' : δL v = dF ((1 : Time), v) := by simpa [dF] using hv
     calc
       δL v = dF ((1 : Time), v) := hv'
      _ = dF (((0  : Time), v) + ((1 : Time), (0 : X))) := by simp only [Prod.mk_add_mk, zero_add,
@@ -367,10 +357,8 @@ lemma isTotalTimeDerivativeVelocity  [CompleteSpace X]
       _ = dF ((0 : Time), v) + dF ((1 : Time), (0 : X)) := by
         simpa using
           (dF.map_add ((0 : Time), v) ((1 : Time), (0 : X)))
-      _ = dF ((0 : Time), v) := by
-        simp [h_time]
-      _ = φ v := by
-        simp [φ]
+      _ = dF ((0 : Time), v) := by simp [h_time]
+      _ = φ v := by simp [φ]
 
   -- Frechet–Riesz: represent φ as inner product with some g.
   refine ⟨(InnerProductSpace.toDual ℝ (X)).symm φ, ?_⟩
@@ -381,8 +369,7 @@ lemma isTotalTimeDerivativeVelocity  [CompleteSpace X]
         (E := X) (x := v) (y := φ)]
   calc
     δL v = φ v := hφ v
-    _ = ⟪(InnerProductSpace.toDual ℝ (X)).symm φ, v⟫_ℝ := by
-      rw [hinner.symm]
+    _ = ⟪(InnerProductSpace.toDual ℝ (X)).symm φ, v⟫_ℝ := by rw [hinner.symm]
 
 end Lagrangian
 
diff --git a/Physlib/ClassicalMechanics/Mass/MassUnit.lean b/Physlib/ClassicalMechanics/Mass/MassUnit.lean
index 2c0599458..71b46fece 100644
--- a/Physlib/ClassicalMechanics/Mass/MassUnit.lean
+++ b/Physlib/ClassicalMechanics/Mass/MassUnit.lean
@@ -39,8 +39,7 @@ structure MassUnit where
 namespace MassUnit
 
 @[simp]
-lemma val_ne_zero (x : MassUnit) : x.val ≠ 0 := by
-  exact Ne.symm (ne_of_lt x.property)
+lemma val_ne_zero (x : MassUnit) : x.val ≠ 0 := Ne.symm (ne_of_lt x.property)
 
 lemma val_pos (x : MassUnit) : 0 < x.val := x.property
 
@@ -73,9 +72,7 @@ lemma div_pos (x y : MassUnit) : (0 : ℝ≥0) < x/ y := by
 
 @[simp]
 lemma div_self (x : MassUnit) :
-    x / x = (1 : ℝ≥0) := by
-  simp [div_eq_val, x.val_ne_zero]
-  rfl
+    x / x = (1 : ℝ≥0) := NNReal.eq (_root_.div_self x.val_ne_zero)
 
 lemma div_symm (x y : MassUnit) :
     x / y = (y / x)⁻¹ := NNReal.eq <| by
@@ -101,8 +98,7 @@ def scale (r : ℝ) (x : MassUnit) (hr : 0 < r := by norm_num) : MassUnit :=
 
 @[simp]
 lemma scale_div_self (x : MassUnit) (r : ℝ) (hr : 0 < r) :
-    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by
-  simp [scale, div_eq_val]
+    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by simp [scale, div_eq_val]
 
 @[simp]
 lemma self_div_scale (x : MassUnit) (r : ℝ) (hr : 0 < r) :
@@ -111,8 +107,7 @@ lemma self_div_scale (x : MassUnit) (r : ℝ) (hr : 0 < r) :
   field_simp
 
 @[simp]
-lemma scale_one (x : MassUnit) : scale 1 x = x := by
-  simp [scale]
+lemma scale_one (x : MassUnit) : scale 1 x = x := by simp [scale]
 
 @[simp]
 lemma scale_div_scale (x1 x2 : MassUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0 < r2) :
@@ -125,8 +120,7 @@ lemma scale_div_scale (x1 x2 : MassUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0 <
 @[simp]
 lemma scale_scale (x : MassUnit) (r1 r2 : ℝ) (hr1 : 0 < r1) (hr2 : 0 < r2) :
     scale r1 (scale r2 x hr2) hr1 = scale (r1 * r2) x (mul_pos hr1 hr2) := by
-  simp [scale]
-  ring
+  simp [scale, mul_assoc]
 
 /-!
 
diff --git a/Physlib/ClassicalMechanics/WaveEquation/Basic.lean b/Physlib/ClassicalMechanics/WaveEquation/Basic.lean
index 8b85f180a..e58867685 100644
--- a/Physlib/ClassicalMechanics/WaveEquation/Basic.lean
+++ b/Physlib/ClassicalMechanics/WaveEquation/Basic.lean
@@ -78,8 +78,7 @@ noncomputable def planeWave (f₀ : ℝ → EuclideanSpace ℝ (Fin d))
     fun t x => f₀ (⟪x, s.unit⟫_ℝ - c * t)
 
 lemma planeWave_eq {d f₀ c x} {s : Direction d} :
-    planeWave f₀ c s t x = f₀ (⟪x, s.unit⟫_ℝ - c * t) :=
-  rfl
+    planeWave f₀ c s t x = f₀ (⟪x, s.unit⟫_ℝ - c * t) := rfl
 
 /-!
 
@@ -138,8 +137,7 @@ lemma planeWave_time_deriv {d f₀ c x} {s : Direction d}
     FunLike.coe_smul, Pi.smul_apply, PiLp.neg_apply, PiLp.smul_apply, smul_eq_mul,
     neg_mul, neg_inj, mul_eq_mul_left_iff]
   left
-  simp
-  rfl
+  simp [planeWave]
   repeat fun_prop
 
 lemma planeWave_time_deriv_time_deriv {d f₀ c x} {s : Direction d}
@@ -181,8 +179,7 @@ lemma planeWave_space_deriv {d f₀ c} {s : Direction d}
   rw [fderiv_comp]
   simp only [ContinuousLinearMap.coe_comp, Function.comp_apply, fderiv_eq_smul_deriv,
     PiLp.smul_apply, smul_eq_mul, one_smul, Pi.smul_apply]
-  rw [fderiv_sub_const]
-  rw [fderiv_inner_apply]
+  rw [fderiv_sub_const, fderiv_inner_apply]
   simp only [fderiv_fun_const, Pi.zero_apply, _root_.zero_apply, inner_zero_right,
     fderiv_fun_id, ContinuousLinearMap.coe_id', id_eq, basis_inner, zero_add, mul_eq_mul_left_iff]
   left
@@ -199,9 +196,7 @@ lemma planeWave_apply_space_deriv {d f₀ c} {s : Direction d}
   rw [fderiv_comp]
   simp only [ContinuousLinearMap.fderiv, ContinuousLinearMap.coe_comp, Function.comp_apply,
     PiLp.proj_apply, fderiv_eq_smul_deriv, one_smul, Pi.smul_apply, smul_eq_mul]
-  rw [← Space.deriv_eq_fderiv_basis, planeWave_space_deriv]
-  rfl
-  · fun_prop
+  exact congrArg (fun y => y.ofLp j) (congrFun (planeWave_space_deriv h' i) x)
   · fun_prop
   · fun_prop
 
diff --git a/Physlib/ClassicalMechanics/WaveEquation/HarmonicWave.lean b/Physlib/ClassicalMechanics/WaveEquation/HarmonicWave.lean
index 96b9f2719..faaa61b72 100644
--- a/Physlib/ClassicalMechanics/WaveEquation/HarmonicWave.lean
+++ b/Physlib/ClassicalMechanics/WaveEquation/HarmonicWave.lean
@@ -69,8 +69,7 @@ lemma transverseHarmonicPlaneWave_eq_planeWave {c : ℝ} {k : WaveVector} {f₀x
   have normk: ‖k‖ = ω/c := by
     rw [hk]
     simp [← abs_div, hc_ge_zero, hω_ge_zero, le_of_lt]
-  rw [normk]
-  rw [mul_sub, inner_smul_right, real_inner_comm, ← mul_assoc]
+  rw [normk, mul_sub, inner_smul_right, real_inner_comm, ← mul_assoc]
   ring_nf
   simp [ne_of_gt, hc_ge_zero, hω_ge_zero, mul_comm ω, mul_assoc, basis_repr_inner_eq]
 
diff --git a/Physlib/CondensedMatter/TightBindingChain/Basic.lean b/Physlib/CondensedMatter/TightBindingChain/Basic.lean
index 60bd6a73d..df9bee055 100644
--- a/Physlib/CondensedMatter/TightBindingChain/Basic.lean
+++ b/Physlib/CondensedMatter/TightBindingChain/Basic.lean
@@ -386,8 +386,7 @@ lemma quantaWaveNumber_exp_N (n : ℕ) (k : T.QuantaWaveNumber) :
   field_simp
   ring_nf
   congr 1
-  rw [mul_comm]
-  rfl
+  exact mul_comm _ _
 
 lemma quantaWaveNumber_exp_sub_one (n : Fin T.N) (k : T.QuantaWaveNumber) :
     Complex.exp (Complex.I * k * (n - 1).val * T.a) =
diff --git a/Physlib/Electromagnetism/Charge/ChargeUnit.lean b/Physlib/Electromagnetism/Charge/ChargeUnit.lean
index 805677145..a6e0b7d98 100644
--- a/Physlib/Electromagnetism/Charge/ChargeUnit.lean
+++ b/Physlib/Electromagnetism/Charge/ChargeUnit.lean
@@ -44,8 +44,7 @@ structure ChargeUnit where
 namespace ChargeUnit
 
 @[simp]
-lemma val_ne_zero (x : ChargeUnit) : x.val ≠ 0 := by
-  exact Ne.symm (ne_of_lt x.property)
+lemma val_ne_zero (x : ChargeUnit) : x.val ≠ 0 := Ne.symm (ne_of_lt x.property)
 
 lemma val_pos (x : ChargeUnit) : 0 < x.val := x.property
 
@@ -78,9 +77,7 @@ lemma div_pos (x y : ChargeUnit) : (0 : ℝ≥0) < x/ y := by
 
 @[simp]
 lemma div_self (x : ChargeUnit) :
-    x / x = (1 : ℝ≥0) := by
-  simp [div_eq_val, x.val_ne_zero]
-  rfl
+    x / x = (1 : ℝ≥0) := NNReal.eq (_root_.div_self x.val_ne_zero)
 
 lemma div_symm (x y : ChargeUnit) :
     x / y = (y / x)⁻¹ := NNReal.eq <| by
@@ -106,8 +103,7 @@ def scale (r : ℝ) (x : ChargeUnit) (hr : 0 < r := by norm_num) : ChargeUnit :=
 
 @[simp]
 lemma scale_div_self (x : ChargeUnit) (r : ℝ) (hr : 0 < r) :
-    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by
-  simp [scale, div_eq_val]
+    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by simp [scale, div_eq_val]
 
 @[simp]
 lemma self_div_scale (x : ChargeUnit) (r : ℝ) (hr : 0 < r) :
@@ -116,8 +112,7 @@ lemma self_div_scale (x : ChargeUnit) (r : ℝ) (hr : 0 < r) :
   field_simp
 
 @[simp]
-lemma scale_one (x : ChargeUnit) : scale 1 x = x := by
-  simp [scale]
+lemma scale_one (x : ChargeUnit) : scale 1 x = x := by simp [scale]
 
 @[simp]
 lemma scale_div_scale (x1 x2 : ChargeUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0 < r2) :
@@ -130,8 +125,7 @@ lemma scale_div_scale (x1 x2 : ChargeUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0
 @[simp]
 lemma scale_scale (x : ChargeUnit) (r1 r2 : ℝ) (hr1 : 0 < r1) (hr2 : 0 < r2) :
     scale r1 (scale r2 x hr2) hr1 = scale (r1 * r2) x (mul_pos hr1 hr2) := by
-  simp [scale]
-  ring
+  simp [scale, mul_assoc]
 
 /-!
 
diff --git a/Physlib/Electromagnetism/Distributional/Basic.lean b/Physlib/Electromagnetism/Distributional/Basic.lean
index 2c3004bf6..49c214689 100644
--- a/Physlib/Electromagnetism/Distributional/Basic.lean
+++ b/Physlib/Electromagnetism/Distributional/Basic.lean
@@ -188,8 +188,7 @@ lemma toTensor_distTensorDeriv_basis_repr_apply {d} (A : DistElectromagneticPote
       (Fin.append ![Color.down] ![Color.up])) :
     (Tensor.basis _).repr (Tensorial.toTensor (distTensorDeriv A ε)) b =
     distDeriv (b 0) A ε (b 1) := by
-  rw [Tensorial.basis_toTensor_apply]
-  rw [Tensorial.basis_map_prod]
+  rw [Tensorial.basis_toTensor_apply, Tensorial.basis_map_prod]
   simp only [Nat.reduceSucc, Nat.reduceAdd, Basis.repr_reindex, Finsupp.mapDomain_equiv_apply,
     Equiv.symm_symm, Fin.isValue]
   rw [Lorentz.Vector.tensor_basis_map_eq_basis_reindex,
diff --git a/Physlib/Electromagnetism/Distributional/Dynamics/CurrentDensity.lean b/Physlib/Electromagnetism/Distributional/Dynamics/CurrentDensity.lean
index 38e1ce7e7..182f1fd8b 100644
--- a/Physlib/Electromagnetism/Distributional/Dynamics/CurrentDensity.lean
+++ b/Physlib/Electromagnetism/Distributional/Dynamics/CurrentDensity.lean
@@ -66,8 +66,7 @@ namespace DistLorentzCurrentDensity
 noncomputable def chargeDensity {d : ℕ} (c : SpeedOfLight) :
     (DistLorentzCurrentDensity d) →ₗ[ℝ] (Time × Space d) →d[ℝ] ℝ where
   toFun J := (1 / (c : ℝ)) • (Lorentz.Vector.temporalCLM d ∘L distTimeSlice c J)
-  map_add' J1 J2 := by
-    simp
+  map_add' J1 J2 := by simp
   map_smul' r J := by
     simp only [one_div, map_smul, ContinuousLinearMap.comp_smulₛₗ, RingHom.id_apply]
     rw [smul_comm]
@@ -83,10 +82,8 @@ noncomputable def chargeDensity {d : ℕ} (c : SpeedOfLight) :
 noncomputable def currentDensity (c : SpeedOfLight) :
     DistLorentzCurrentDensity d →ₗ[ℝ] (Time × Space d) →d[ℝ] EuclideanSpace ℝ (Fin d) where
   toFun J := Lorentz.Vector.spatialCLM d ∘L distTimeSlice c J
-  map_add' J1 J2 := by
-    simp
-  map_smul' r J := by
-    simp
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
 
 end DistLorentzCurrentDensity
 end Electromagnetism
diff --git a/Physlib/Electromagnetism/Distributional/Dynamics/Lagrangian.lean b/Physlib/Electromagnetism/Distributional/Dynamics/Lagrangian.lean
index e77b0b035..458cf24bd 100644
--- a/Physlib/Electromagnetism/Distributional/Dynamics/Lagrangian.lean
+++ b/Physlib/Electromagnetism/Distributional/Dynamics/Lagrangian.lean
@@ -78,8 +78,7 @@ noncomputable def gradFreeCurrentPotential {d} :
     DistLorentzCurrentDensity d →ₗ[ℝ] ((SpaceTime d) →d[ℝ] Lorentz.Vector d) where
   toFun J := {
     toFun ε := ∑ μ, (η μ μ • (J ε μ) • Lorentz.Vector.basis μ)
-    map_add' ε₁ ε₂ := by
-      simp [Finset.sum_add_distrib, add_smul]
+    map_add' ε₁ ε₂ := by simp [Finset.sum_add_distrib, add_smul]
     map_smul' r ε := by
       simp only [map_smul, apply_smul, smul_smul, Real.ringHom_apply, Finset.smul_sum]
       congr
diff --git a/Physlib/Electromagnetism/Distributional/FieldStrength.lean b/Physlib/Electromagnetism/Distributional/FieldStrength.lean
index 4ec6cdb0a..3fbcda8fe 100644
--- a/Physlib/Electromagnetism/Distributional/FieldStrength.lean
+++ b/Physlib/Electromagnetism/Distributional/FieldStrength.lean
@@ -128,8 +128,7 @@ lemma fieldStrengthAux_tensor_basis_eq_basis {d} (A : DistElectromagneticPotenti
     (Tensor.basis _).repr (Tensorial.toTensor (A.fieldStrengthAux ε)) b =
     (Lorentz.Vector.basis.tensorProduct Lorentz.Vector.basis).repr (A.fieldStrengthAux ε)
       (b 0, b 1) := by
-  rw [Tensorial.basis_toTensor_apply]
-  rw [Tensorial.basis_map_prod]
+  rw [Tensorial.basis_toTensor_apply, Tensorial.basis_map_prod]
   simp only [Nat.reduceSucc, Nat.reduceAdd, Basis.repr_reindex, Finsupp.mapDomain_equiv_apply,
     Equiv.symm_symm, Fin.isValue]
   rw [Lorentz.Vector.tensor_basis_map_eq_basis_reindex]
@@ -243,9 +242,8 @@ lemma fieldStrength_eq_basis {d} (A : DistElectromagneticPotential d)
     (ε : 𝓢(SpaceTime d, ℝ)) :
     A.fieldStrength ε = ∑ μ, ∑ ν,
       ((η μ μ * distDeriv μ A ε ν) - η ν ν * distDeriv ν A ε μ)
-      • Lorentz.Vector.basis μ ⊗ₜ[ℝ] Lorentz.Vector.basis ν := by
-  rw [fieldStrength]
-  exact fieldStrengthAux_eq_basis A ε
+      • Lorentz.Vector.basis μ ⊗ₜ[ℝ] Lorentz.Vector.basis ν :=
+  (fieldStrength_eq_fieldStrengthAux A ε).trans (fieldStrengthAux_eq_basis A ε)
 
 lemma fieldStrength_basis_repr_eq_single {d} {μν : (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)}
     (A : DistElectromagneticPotential d) (ε : 𝓢(SpaceTime d, ℝ)) :
diff --git a/Physlib/Electromagnetism/Dynamics/Basic.lean b/Physlib/Electromagnetism/Dynamics/Basic.lean
index 513934f8d..056ddb8b1 100644
--- a/Physlib/Electromagnetism/Dynamics/Basic.lean
+++ b/Physlib/Electromagnetism/Dynamics/Basic.lean
@@ -102,8 +102,7 @@ lemma c_sq : (𝓕.c : ℝ) ^ 2 = 1 / (𝓕.ε₀ * 𝓕.μ₀) := by
   · positivity
 
 @[simp]
-lemma c_abs : abs (𝓕.c : ℝ) = 𝓕.c := by
-  rw [abs_of_pos (SpeedOfLight.val_pos 𝓕.c)]
+lemma c_abs : abs (𝓕.c : ℝ) = 𝓕.c := by rw [abs_of_pos (SpeedOfLight.val_pos 𝓕.c)]
 
 end FreeSpace
 
diff --git a/Physlib/Electromagnetism/Dynamics/CurrentDensity.lean b/Physlib/Electromagnetism/Dynamics/CurrentDensity.lean
index 3869cfc86..616e8187c 100644
--- a/Physlib/Electromagnetism/Dynamics/CurrentDensity.lean
+++ b/Physlib/Electromagnetism/Dynamics/CurrentDensity.lean
@@ -102,20 +102,13 @@ lemma chargeDensity_zero {d : ℕ} {c : SpeedOfLight}:
 lemma chargeDensity_differentiable {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) : Differentiable ℝ ↿(J.chargeDensity c) := by
   rw [chargeDensity_eq_timeSlice]
-  apply timeSlice_differentiable
-  have h1 : ∀ i, Differentiable ℝ (fun x => J x i) := by
-    rw [SpaceTime.differentiable_vector]
-    exact hJ
-  apply Differentiable.fun_const_smul
-  exact h1 (Sum.inl 0)
+  exact timeSlice_differentiable c _
+    (Differentiable.fun_const_smul ((SpaceTime.differentiable_vector J).mpr hJ (Sum.inl 0)) _)
 
 lemma chargeDensity_differentiable_space {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) (t : Time) :
-    Differentiable ℝ (fun x => J.chargeDensity c t x) := by
-  change Differentiable ℝ (↿(J.chargeDensity c) ∘ fun x => (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact chargeDensity_differentiable hJ
-  · fun_prop
+    Differentiable ℝ (fun x => J.chargeDensity c t x) :=
+  (chargeDensity_differentiable hJ).comp (f := fun x => (t, x)) (by fun_prop)
 
 /-!
 
@@ -126,12 +119,8 @@ lemma chargeDensity_differentiable_space {d : ℕ} {c : SpeedOfLight} {J : Loren
 lemma chargeDensity_contDiff {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : ContDiff ℝ n J) : ContDiff ℝ n ↿(J.chargeDensity c) := by
   rw [chargeDensity_eq_timeSlice]
-  apply timeSlice_contDiff
-  have h1 : ∀ i, ContDiff ℝ n (fun x => J x i) := by
-    rw [SpaceTime.contDiff_vector]
-    exact hJ
-  apply ContDiff.const_smul
-  exact h1 (Sum.inl 0)
+  exact timeSlice_contDiff c _
+    (ContDiff.const_smul _ ((SpaceTime.contDiff_vector J).mpr hJ (Sum.inl 0)))
 
 /-!
 
@@ -156,9 +145,7 @@ lemma currentDensity_eq_timeSlice {d : ℕ} {J : LorentzCurrentDensity d} :
 
 @[simp]
 lemma currentDensity_zero {d : ℕ} {c : SpeedOfLight}:
-    currentDensity c (0 : LorentzCurrentDensity d) = 0 := by
-  simp [currentDensity_eq_timeSlice, timeSlice]
-  rfl
+    currentDensity c (0 : LorentzCurrentDensity d) = 0 := rfl
 /-!
 
 ### C.2. Differentiability of the current density
@@ -168,57 +155,38 @@ lemma currentDensity_zero {d : ℕ} {c : SpeedOfLight}:
 lemma currentDensity_differentiable {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) : Differentiable ℝ ↿(J.currentDensity c) := by
   rw [currentDensity_eq_timeSlice]
-  apply timeSlice_differentiable
-  have h1 : ∀ i, Differentiable ℝ (fun x => J x i) := by
-    rw [SpaceTime.differentiable_vector]
-    exact hJ
-  exact differentiable_euclidean.mpr fun i => h1 (Sum.inr i)
+  exact timeSlice_differentiable c _
+    (differentiable_euclidean.mpr fun i => (SpaceTime.differentiable_vector J).mpr hJ (Sum.inr i))
 
 lemma currentDensity_apply_differentiable {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) (i : Fin d) :
-    Differentiable ℝ ↿(fun t x => J.currentDensity c t x i) := by
-  change Differentiable ℝ (EuclideanSpace.proj i ∘ ↿(J.currentDensity c))
-  refine Differentiable.comp ?_ ?_
-  · exact ContinuousLinearMap.differentiable (𝕜 := ℝ) (EuclideanSpace.proj i)
-  · exact currentDensity_differentiable hJ
+    Differentiable ℝ ↿(fun t x => J.currentDensity c t x i) :=
+  (ContinuousLinearMap.differentiable (𝕜 := ℝ) (EuclideanSpace.proj i)).comp
+    (currentDensity_differentiable hJ)
 
 lemma currentDensity_differentiable_space {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) (t : Time) :
-    Differentiable ℝ (fun x => J.currentDensity c t x) := by
-  change Differentiable ℝ (↿(J.currentDensity c) ∘ fun x => (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact currentDensity_differentiable hJ
-  · fun_prop
+    Differentiable ℝ (fun x => J.currentDensity c t x) :=
+  (currentDensity_differentiable hJ).comp (f := fun x => (t, x)) (by fun_prop)
 
 lemma currentDensity_apply_differentiable_space {d : ℕ} {c : SpeedOfLight}
     {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) (t : Time) (i : Fin d) :
-    Differentiable ℝ (fun x => J.currentDensity c t x i) := by
-  change Differentiable ℝ (EuclideanSpace.proj i ∘ (↿(J.currentDensity c) ∘ fun x => (t, x)))
-  refine Differentiable.comp ?_ ?_
-  · exact ContinuousLinearMap.differentiable (𝕜 := ℝ) _
-  · apply Differentiable.comp ?_ ?_
-    · exact currentDensity_differentiable hJ
-    · fun_prop
+    Differentiable ℝ (fun x => J.currentDensity c t x i) :=
+  (ContinuousLinearMap.differentiable (𝕜 := ℝ) (EuclideanSpace.proj i)).comp
+    (currentDensity_differentiable_space hJ t)
 
 lemma currentDensity_differentiable_time {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) (x : Space d) :
-    Differentiable ℝ (fun t => J.currentDensity c t x) := by
-  change Differentiable ℝ (↿(J.currentDensity c) ∘ fun t => (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact currentDensity_differentiable hJ
-  · fun_prop
+    Differentiable ℝ (fun t => J.currentDensity c t x) :=
+  (currentDensity_differentiable hJ).comp (f := fun t => (t, x)) (by fun_prop)
 
 lemma currentDensity_apply_differentiable_time {d : ℕ} {c : SpeedOfLight}
     {J : LorentzCurrentDensity d}
     (hJ : Differentiable ℝ J) (x : Space d) (i : Fin d) :
-    Differentiable ℝ (fun t => J.currentDensity c t x i) := by
-  change Differentiable ℝ (EuclideanSpace.proj i ∘ (↿(J.currentDensity c) ∘ fun t => (t, x)))
-  refine Differentiable.comp ?_ ?_
-  · exact ContinuousLinearMap.differentiable (𝕜 := ℝ) _
-  · apply Differentiable.comp ?_ ?_
-    · exact currentDensity_differentiable hJ
-    · fun_prop
+    Differentiable ℝ (fun t => J.currentDensity c t x i) :=
+  (ContinuousLinearMap.differentiable (𝕜 := ℝ) (EuclideanSpace.proj i)).comp
+    (currentDensity_differentiable_time hJ x)
 
 /-!
 
@@ -229,11 +197,8 @@ lemma currentDensity_apply_differentiable_time {d : ℕ} {c : SpeedOfLight}
 lemma currentDensity_ContDiff {d : ℕ} {c : SpeedOfLight} {J : LorentzCurrentDensity d}
     (hJ : ContDiff ℝ n J) : ContDiff ℝ n ↿(J.currentDensity c) := by
   rw [currentDensity_eq_timeSlice]
-  apply timeSlice_contDiff
-  have h1 : ∀ i, ContDiff ℝ n (fun x => J x i) := by
-    rw [SpaceTime.contDiff_vector]
-    exact hJ
-  exact contDiff_euclidean.mpr fun i => h1 (Sum.inr i)
+  exact timeSlice_contDiff c _
+    (contDiff_euclidean.mpr fun i => (SpaceTime.contDiff_vector J).mpr hJ (Sum.inr i))
 
 end LorentzCurrentDensity
 
diff --git a/Physlib/Electromagnetism/Dynamics/IsExtrema.lean b/Physlib/Electromagnetism/Dynamics/IsExtrema.lean
index 5c5cee879..dff99ce3f 100644
--- a/Physlib/Electromagnetism/Dynamics/IsExtrema.lean
+++ b/Physlib/Electromagnetism/Dynamics/IsExtrema.lean
@@ -233,8 +233,7 @@ lemma isExtrema_iff_gauss_ampere_magneticFieldMatrix {d} {𝓕 : FreeSpace}
     IsExtrema 𝓕 A J ↔ ∀ t, ∀ x, (∇ ⬝ (A.electricField 𝓕.c t)) x = J.chargeDensity 𝓕.c t x / 𝓕.ε₀
     ∧ ∀ i, 𝓕.μ₀ * 𝓕.ε₀ * ∂ₜ (fun t => A.electricField 𝓕.c t x) t i =
     ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x - 𝓕.μ₀ * J.currentDensity 𝓕.c t x i := by
-  rw [isExtrema_iff_gradLagrangian]
-  rw [funext_iff]
+  rw [isExtrema_iff_gradLagrangian, funext_iff]
   conv_lhs =>
     enter [x]
     rw [gradLagrangian_eq_electricField_magneticField (𝓕 := 𝓕) A hA J hJ]
@@ -469,8 +468,7 @@ lemma time_deriv_time_deriv_electricField_of_isExtrema {A : ElectromagneticPoten
         exact (hEd j j).differentiableAt
       _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -
           1 / (𝓕.μ₀ * 𝓕.ε₀) * (∂[i] (fun x => (∇ ⬝ (A.electricField 𝓕.c t)) x) x) -
-          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by
-        rfl
+          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by rfl
       _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -
           1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * ∂[i] (J.chargeDensity 𝓕.c t ·) x -
           1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by
@@ -489,9 +487,8 @@ lemma time_deriv_time_deriv_electricField_of_isExtrema {A : ElectromagneticPoten
           rw [fderiv_const_mul]
           simp [← Space.deriv_eq_fderiv_basis]
           field_simp
-          apply Differentiable.differentiableAt
-          apply LorentzCurrentDensity.chargeDensity_differentiable_space
-          exact hJ.differentiable (by simp)
+          exact (LorentzCurrentDensity.chargeDensity_differentiable_space
+            (hJ.differentiable (by simp)) _).differentiableAt
         · exact hA
         · exact hJ
       _ = 𝓕.c ^ 2 * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -
diff --git a/Physlib/Electromagnetism/Dynamics/KineticTerm.lean b/Physlib/Electromagnetism/Dynamics/KineticTerm.lean
index b9a583038..ab5d6c2ee 100644
--- a/Physlib/Electromagnetism/Dynamics/KineticTerm.lean
+++ b/Physlib/Electromagnetism/Dynamics/KineticTerm.lean
@@ -121,8 +121,7 @@ lemma kineticTerm_eq_sum {d} {𝓕 : FreeSpace} (A : ElectromagneticPotential d)
       (Lorentz.CoVector.basis.tensorProduct Lorentz.Vector.basis).repr (A.toFieldStrength x) (μ, ν)
       * (Lorentz.CoVector.basis.tensorProduct Lorentz.Vector.basis).repr
         (A.toFieldStrength x) (μ', ν') := by
-  rw [kineticTerm]
-  rw [toField_eq_repr]
+  rw [kineticTerm, toField_eq_repr]
   rw [contrT_basis_repr_apply_eq_fin]
   conv_lhs =>
     enter [2, 2, μ]
@@ -480,8 +479,7 @@ lemma kineticTerm_add_time_mul_const {d} {𝓕 : FreeSpace} (A : Electromagnetic
     _ = -1 / (2 * 𝓕.μ₀) *
     ∑ μ, ∑ ν, (η μ μ * η ν ν * (∂_ μ A x + if μ = Sum.inl 0 then c else 0) ν ^ 2 -
           (∂_ μ A x + if μ = Sum.inl 0 then c else 0) ν *
-          (∂_ ν A x + if ν = Sum.inl 0 then c else 0) μ) := by
-      simp only [hdiff]
+          (∂_ ν A x + if ν = Sum.inl 0 then c else 0) μ) := by simp only [hdiff]
     _ = -1 / (2 * 𝓕.μ₀) *
       ∑ μ, ∑ ν, (η μ μ * η ν ν * (∂_ μ A x ν + if μ = Sum.inl 0 then c ν else 0) ^ 2 -
           (∂_ μ A x ν + if μ = Sum.inl 0 then c ν else 0) *
diff --git a/Physlib/Electromagnetism/Kinematics/EMPotential.lean b/Physlib/Electromagnetism/Kinematics/EMPotential.lean
index f15200014..522271a1a 100644
--- a/Physlib/Electromagnetism/Kinematics/EMPotential.lean
+++ b/Physlib/Electromagnetism/Kinematics/EMPotential.lean
@@ -231,9 +231,8 @@ noncomputable def ofStaticPotentials {d} (c : SpeedOfLight) (ϕ : Space d → 
 
 lemma ofStaticPotentials_eq_ofPotentials {d} (c : SpeedOfLight) (ϕ : Space d → ℝ)
     (A : Space d → EuclideanSpace ℝ (Fin d)) :
-    ofStaticPotentials c ϕ A = ofPotentials c (fun _ => ϕ) (fun _ => A) := by
-  rw [ofPotentials_eq_add]
-  rfl
+    ofStaticPotentials c ϕ A = ofPotentials c (fun _ => ϕ) (fun _ => A) :=
+  (ofPotentials_eq_add c (fun _ => ϕ) (fun _ => A)).symm
 
 open MeasureTheory Matrix Space InnerProductSpace Time in
 /-- The electromagnetic potential from an electric and a magnetic field.
@@ -281,7 +280,7 @@ noncomputable instance {d} :
     simp
   smul_add Λ A B := by
     ext x μ
-    simp [Lorentz.Vector.smul_add]
+    simp
 
 /-!
 
@@ -295,38 +294,27 @@ open ContDiff
 @[fun_prop]
 lemma differentiable_component {d : ℕ}
     (A : ElectromagneticPotential d) (hA : Differentiable ℝ A) (μ : Fin 1 ⊕ Fin d) :
-    Differentiable ℝ (fun x => A x μ) := by
-  revert μ
-  rw [SpaceTime.differentiable_vector]
-  exact hA
+    Differentiable ℝ (fun x => A x μ) :=
+  (SpaceTime.differentiable_vector A).mpr hA μ
 
 @[fun_prop]
 lemma differentiable_action {d} (Λ : LorentzGroup d) (A : ElectromagneticPotential d)
-    (hA : Differentiable ℝ A) : Differentiable ℝ (fun x => Λ • A (Λ⁻¹ • x)) := by
-  apply Differentiable.comp
-  · exact ContinuousLinearMap.differentiable (Lorentz.Vector.actionCLM Λ)
-  · apply Differentiable.comp
-    · exact hA
-    · exact ContinuousLinearMap.differentiable (Lorentz.Vector.actionCLM Λ⁻¹)
+    (hA : Differentiable ℝ A) : Differentiable ℝ (fun x => Λ • A (Λ⁻¹ • x)) :=
+  (ContinuousLinearMap.differentiable (Lorentz.Vector.actionCLM Λ)).comp
+    (hA.comp (ContinuousLinearMap.differentiable (Lorentz.Vector.actionCLM Λ⁻¹)))
 
 @[fun_prop]
 lemma contDiff_action {d} (Λ : LorentzGroup d) (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) : ContDiff ℝ n (fun x => Λ • A (Λ⁻¹ • x)) := by
-  apply ContDiff.comp
-  · exact ContinuousLinearMap.contDiff (Lorentz.Vector.actionCLM Λ)
-  · apply ContDiff.comp
-    · exact hA
-    · exact ContinuousLinearMap.contDiff (Lorentz.Vector.actionCLM Λ⁻¹)
+    (hA : ContDiff ℝ n A) : ContDiff ℝ n (fun x => Λ • A (Λ⁻¹ • x)) :=
+  (ContinuousLinearMap.contDiff (Lorentz.Vector.actionCLM Λ)).comp
+    (hA.comp (ContinuousLinearMap.contDiff (Lorentz.Vector.actionCLM Λ⁻¹)))
 
 @[fun_prop]
 lemma differentiable_deriv {d} {A : ElectromagneticPotential d}
     (hA : ContDiff ℝ 2 A) (μ ν : Fin 1 ⊕ Fin d) :
-    Differentiable ℝ (fun x => ∂_ μ A x ν) := by
-  have diff_partial (μ) :
-      ∀ ν, Differentiable ℝ fun x => (fderiv ℝ A x) (Lorentz.Vector.basis μ) ν := by
-    rw [SpaceTime.differentiable_vector]
-    fun_prop
-  exact diff_partial μ ν
+    Differentiable ℝ (fun x => ∂_ μ A x ν) :=
+  (SpaceTime.differentiable_vector (fun x => (fderiv ℝ A x) (Lorentz.Vector.basis μ))).mpr
+    (by fun_prop) ν
 
 @[fun_prop]
 lemma differentiable_deriv_of_smooth {d} {A : ElectromagneticPotential d}
@@ -337,12 +325,9 @@ lemma differentiable_deriv_of_smooth {d} {A : ElectromagneticPotential d}
 @[fun_prop]
 lemma contDiff_deriv {n} {d} {A : ElectromagneticPotential d}
     (hA : ContDiff ℝ (n + 1) A) (μ ν : Fin 1 ⊕ Fin d) :
-    ContDiff ℝ n (fun x => ∂_ μ A x ν) := by
-  have diff_partial (μ) :
-      ∀ ν, ContDiff ℝ n fun x => (fderiv ℝ A x) (Lorentz.Vector.basis μ) ν := by
-    rw [SpaceTime.contDiff_vector]
-    fun_prop
-  exact diff_partial μ ν
+    ContDiff ℝ n (fun x => ∂_ μ A x ν) :=
+  (SpaceTime.contDiff_vector (fun x => (fderiv ℝ A x) (Lorentz.Vector.basis μ))).mpr
+    (by fun_prop) ν
 
 TODO "Add results related to the differentiability of the
   derivative of the Electromagnetic potential."
@@ -576,8 +561,7 @@ as taking the derivative and then applying the Lorentz transformation to the res
 lemma deriv_equivariant {d} {x : SpaceTime d} (A : ElectromagneticPotential d)
     (Λ : LorentzGroup d)
     (hf : Differentiable ℝ A) : deriv (Λ • A) x = Λ • (deriv A (Λ⁻¹ • x)) := by
-  rw [deriv_eq_tensorDeriv, deriv_eq_tensorDeriv]
-  rw [action_val, tensorDeriv_equivariant]
+  rw [deriv_eq_tensorDeriv, deriv_eq_tensorDeriv, action_val, tensorDeriv_equivariant]
   all_goals fun_prop
 
 /-!
@@ -634,8 +618,7 @@ lemma toTensor_deriv_basis_repr_apply {d} (A : ElectromagneticPotential d)
       (Fin.append ![Color.down] ![Color.up])) :
     (Tensor.basis _).repr (Tensorial.toTensor (deriv A x)) b =
     ∂_ (b 0) A x (b 1) := by
-  rw [Tensorial.basis_toTensor_apply]
-  rw [Tensorial.basis_map_prod]
+  rw [Tensorial.basis_toTensor_apply, Tensorial.basis_map_prod]
   simp only [Nat.reduceSucc, Nat.reduceAdd, Basis.repr_reindex, Finsupp.mapDomain_equiv_apply,
     Equiv.symm_symm, Fin.isValue]
   rw [Lorentz.Vector.tensor_basis_map_eq_basis_reindex,
diff --git a/Physlib/Electromagnetism/Kinematics/FieldStrength.lean b/Physlib/Electromagnetism/Kinematics/FieldStrength.lean
index 5b53bb856..d6c84b18d 100644
--- a/Physlib/Electromagnetism/Kinematics/FieldStrength.lean
+++ b/Physlib/Electromagnetism/Kinematics/FieldStrength.lean
@@ -151,10 +151,9 @@ tensor in.
   the components extracted via `toField`. -/
 lemma toFieldStrength_eq_sum_basis_eval {d} {A : ElectromagneticPotential d} :
     A.toFieldStrength = fun x => ∑ μ, ∑ ν, toField {A.toFieldStrength x| [μ] [ν]}ᵀ •
-      Vector.basis μ ⊗ₜ[ℝ] Vector.basis ν := by
-  ext x
-  exact prod_eq_sum_eval Vector.basis_eq_map_tensor_basis
-      Vector.basis_eq_map_tensor_basis (A.toFieldStrength x)
+      Vector.basis μ ⊗ₜ[ℝ] Vector.basis ν :=
+  funext fun x => prod_eq_sum_eval Vector.basis_eq_map_tensor_basis
+    Vector.basis_eq_map_tensor_basis (A.toFieldStrength x)
 
 /-- The statement that `F = F^{μν} eᵤ ⊗ eᵥ` written explicitly, with
   the components given by `∑ κ, (η μ κ * ∂_ κ A x ν - η ν κ * ∂_ κ A x μ)`. -/
@@ -317,8 +316,7 @@ lemma toFieldStrength_tensor_basis_eq_basis {d} (A : ElectromagneticPotential d)
     (Tensor.basis _).repr (Tensorial.toTensor (toFieldStrength A x)) b =
     (Lorentz.Vector.basis.tensorProduct Lorentz.Vector.basis).repr (toFieldStrength A x)
       (b 0, b 1) := by
-  rw [Tensorial.basis_toTensor_apply]
-  rw [Tensorial.basis_map_prod]
+  rw [Tensorial.basis_toTensor_apply, Tensorial.basis_map_prod]
   simp only [Nat.reduceSucc, Nat.reduceAdd, Basis.repr_reindex, Finsupp.mapDomain_equiv_apply,
     Equiv.symm_symm, Fin.isValue]
   rw [Lorentz.Vector.tensor_basis_map_eq_basis_reindex]
@@ -477,19 +475,15 @@ lemma fieldStrengthMatrix_differentiable {d} {A : ElectromagneticPotential d}
 
 lemma fieldStrengthMatrix_differentiable_space {d} {A : ElectromagneticPotential d}
     {μν} (hA : ContDiff ℝ 2 A) (t : Time) {c : SpeedOfLight} :
-    Differentiable ℝ (fun x => A.fieldStrengthMatrix ((toTimeAndSpace c).symm (t, x)) μν) := by
-  change Differentiable ℝ ((A.fieldStrengthMatrix · μν) ∘ fun x => (toTimeAndSpace c).symm (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact fieldStrengthMatrix_differentiable hA
-  · fun_prop
+    Differentiable ℝ (fun x => A.fieldStrengthMatrix ((toTimeAndSpace c).symm (t, x)) μν) :=
+  (fieldStrengthMatrix_differentiable hA).comp (f := fun x => (toTimeAndSpace c).symm (t, x))
+    (by fun_prop)
 
 lemma fieldStrengthMatrix_differentiable_time {d} {A : ElectromagneticPotential d}
     {μν} (hA : ContDiff ℝ 2 A) (x : Space d) {c : SpeedOfLight} :
-    Differentiable ℝ (fun t => A.fieldStrengthMatrix ((toTimeAndSpace c).symm (t, x)) μν) := by
-  change Differentiable ℝ ((A.fieldStrengthMatrix · μν) ∘ fun t => (toTimeAndSpace c).symm (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact fieldStrengthMatrix_differentiable hA
-  · fun_prop
+    Differentiable ℝ (fun t => A.fieldStrengthMatrix ((toTimeAndSpace c).symm (t, x)) μν) :=
+  (fieldStrengthMatrix_differentiable hA).comp (f := fun t => (toTimeAndSpace c).symm (t, x))
+    (by fun_prop)
 
 lemma fieldStrengthMatrix_contDiff {d} {n : WithTop ℕ∞} {A : ElectromagneticPotential d}
     {μν} (hA : ContDiff ℝ (n + 1) A) :
@@ -520,9 +514,8 @@ lemma fieldStrengthMatrix_contDiff {d} {n : WithTop ℕ∞} {A : Electromagnetic
 
 lemma fieldStrengthMatrix_smooth {d} {A : ElectromagneticPotential d}
     (hA : ContDiff ℝ ∞ A) (μν) :
-    ContDiff ℝ ∞ (A.fieldStrengthMatrix · μν) := by
-  apply fieldStrengthMatrix_contDiff
-  simpa using hA
+    ContDiff ℝ ∞ (A.fieldStrengthMatrix · μν) :=
+  fieldStrengthMatrix_contDiff (μν := μν) (by simpa using hA)
 
 /-!
 
diff --git a/Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean b/Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean
index c3c1cc631..bd75d39ed 100644
--- a/Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean
+++ b/Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean
@@ -104,8 +104,7 @@ lemma ofGradient_apply_sum {d} (χ : SpaceTime d → ℝ) (x : SpaceTime d) (μ
   only the `κ = μ` term survives. -/
 lemma ofGradient_apply {d} (χ : SpaceTime d → ℝ) (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d) :
     ofGradient χ x μ = η μ μ * ∂_ μ χ x := by
-  rw [ofGradient_apply_sum]
-  rw [Finset.sum_eq_single μ]
+  rw [ofGradient_apply_sum, Finset.sum_eq_single μ]
   · intro κ _ hκ
     rw [minkowskiMatrix.off_diag_zero (Ne.symm hκ)]
     simp
diff --git a/Physlib/Electromagnetism/Kinematics/MagneticField.lean b/Physlib/Electromagnetism/Kinematics/MagneticField.lean
index 1d88db0dd..dfeeb0f5f 100644
--- a/Physlib/Electromagnetism/Kinematics/MagneticField.lean
+++ b/Physlib/Electromagnetism/Kinematics/MagneticField.lean
@@ -239,8 +239,7 @@ lemma fieldStrengthMatrix_inr_inr_eq_magneticFieldMatrix {c : SpeedOfLight}
     (A : ElectromagneticPotential d)
     (x : SpaceTime d) (i j : Fin d) :
     A.fieldStrengthMatrix x (Sum.inr i, Sum.inr j) =
-    A.magneticFieldMatrix c (x.time c) x.space (i, j) := by
-  simp [magneticFieldMatrix_eq]
+    A.magneticFieldMatrix c (x.time c) x.space (i, j) := by simp [magneticFieldMatrix_eq]
 
 /-!
 
diff --git a/Physlib/Electromagnetism/Kinematics/ScalarPotential.lean b/Physlib/Electromagnetism/Kinematics/ScalarPotential.lean
index ba274ba3d..945a1f6c6 100644
--- a/Physlib/Electromagnetism/Kinematics/ScalarPotential.lean
+++ b/Physlib/Electromagnetism/Kinematics/ScalarPotential.lean
@@ -92,8 +92,7 @@ lemma ofVectorPotential_scalarPotential {d} (c : SpeedOfLight)
 @[simp]
 lemma ofStaticVectorPotential_scalarPotential {d} (c : SpeedOfLight)
     (A : Space d → EuclideanSpace ℝ (Fin d)) :
-    (ofStaticVectorPotential c A).scalarPotential = 0 := by
-  simp [ofStaticVectorPotential]
+    (ofStaticVectorPotential c A).scalarPotential = 0 := by simp [ofStaticVectorPotential]
 
 @[simp]
 lemma ofPotentials_scalarPotential {d} (c : SpeedOfLight) (φ : Time → Space d → ℝ)
@@ -114,8 +113,7 @@ lemma ofElectromagneticField_scalarPotential (c : SpeedOfLight)
     (E : Time → Space → EuclideanSpace ℝ (Fin 3))
     (B : Time → Space → EuclideanSpace ℝ (Fin 3)) :
     (ofElectromagneticField c E B).scalarPotential c = fun t x =>
-    - ∫ u in (0 : ℝ)..1, ⟪E t (u • x), basis.repr x⟫_ℝ ∂(volume) := by
-  simp [ofElectromagneticField]
+    - ∫ u in (0 : ℝ)..1, ⟪E t (u • x), basis.repr x⟫_ℝ ∂(volume) := by simp [ofElectromagneticField]
 
 open MeasureTheory Matrix Space InnerProductSpace Time in
 lemma ofElectromagneticField_scalarPotential_eq_add_vectorPotential (c : SpeedOfLight)
@@ -154,27 +152,20 @@ lemma scalarPotential_contDiff {n} {d} (c : SpeedOfLight) (A : ElectromagneticPo
 @[fun_prop]
 lemma scalarPotential_contDiff_space {n} {d} (c : SpeedOfLight)
     (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) (t : Time) : ContDiff ℝ n (A.scalarPotential c t) := by
-  change ContDiff ℝ n (↿(A.scalarPotential c) ∘ fun x => (t, x))
-  refine ContDiff.comp ?_ ?_
-  · exact scalarPotential_contDiff c A hA
-  · fun_prop
+    (hA : ContDiff ℝ n A) (t : Time) : ContDiff ℝ n (A.scalarPotential c t) :=
+  (scalarPotential_contDiff c A hA).comp (f := fun x => (t, x)) (by fun_prop)
 
 open ContDiff
 
 @[fun_prop]
 lemma scalarPotential_contDiff_space_of_smooth {n : ℕ} {d} (c : SpeedOfLight)
     (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ ∞ A) (t : Time) : ContDiff ℝ n (A.scalarPotential c t) := by
-  apply scalarPotential_contDiff_space
-  exact hA.of_le (ENat.LEInfty.out)
+    (hA : ContDiff ℝ ∞ A) (t : Time) : ContDiff ℝ n (A.scalarPotential c t) :=
+  scalarPotential_contDiff_space c A (hA.of_le (ENat.LEInfty.out)) t
 
 lemma scalarPotential_contDiff_time {n} {d} (c : SpeedOfLight) (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) (x : Space d) : ContDiff ℝ n (A.scalarPotential c · x) := by
-  change ContDiff ℝ n (↿(A.scalarPotential c) ∘ fun t => (t, x))
-  refine ContDiff.comp ?_ ?_
-  · exact scalarPotential_contDiff c A hA
-  · fun_prop
+    (hA : ContDiff ℝ n A) (x : Space d) : ContDiff ℝ n (A.scalarPotential c · x) :=
+  (scalarPotential_contDiff c A hA).comp (f := fun t => (t, x)) (by fun_prop)
 
 /-!
 
@@ -196,18 +187,12 @@ lemma scalarPotential_differentiable {d} (c : SpeedOfLight) (A : Electromagnetic
   exact h1 (Sum.inl 0)
 
 lemma scalarPotential_differentiable_space {d} (c : SpeedOfLight) (A : ElectromagneticPotential d)
-    (hA : Differentiable ℝ A) (t : Time) : Differentiable ℝ (A.scalarPotential c t) := by
-  change Differentiable ℝ (↿(A.scalarPotential c) ∘ fun x => (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact scalarPotential_differentiable c A hA
-  · fun_prop
+    (hA : Differentiable ℝ A) (t : Time) : Differentiable ℝ (A.scalarPotential c t) :=
+  (scalarPotential_differentiable c A hA).comp (f := fun x => (t, x)) (by fun_prop)
 
 lemma scalarPotential_differentiable_time {d} (c : SpeedOfLight) (A : ElectromagneticPotential d)
-    (hA : Differentiable ℝ A) (x : Space d) : Differentiable ℝ (A.scalarPotential c · x) := by
-  change Differentiable ℝ (↿(A.scalarPotential c) ∘ fun t => (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact scalarPotential_differentiable c A hA
-  · fun_prop
+    (hA : Differentiable ℝ A) (x : Space d) : Differentiable ℝ (A.scalarPotential c · x) :=
+  (scalarPotential_differentiable c A hA).comp (f := fun t => (t, x)) (by fun_prop)
 
 end ElectromagneticPotential
 
diff --git a/Physlib/Electromagnetism/Kinematics/VectorPotential.lean b/Physlib/Electromagnetism/Kinematics/VectorPotential.lean
index b5edd33d0..a909c266a 100644
--- a/Physlib/Electromagnetism/Kinematics/VectorPotential.lean
+++ b/Physlib/Electromagnetism/Kinematics/VectorPotential.lean
@@ -73,9 +73,7 @@ noncomputable def vectorPotential {d} (c : SpeedOfLight := 1) (A : Electromagnet
 
 @[simp]
 lemma ofScalarPotential_vectorPotential {d} (c : SpeedOfLight)
-    (φ : Time → Space d → ℝ) : (ofScalarPotential c φ).vectorPotential c = 0 := by
-  simp only [vectorPotential, ofScalarPotential]
-  rfl
+    (φ : Time → Space d → ℝ) : (ofScalarPotential c φ).vectorPotential c = 0 := rfl
 
 @[simp]
 lemma ofStaticScalarPotential_vectorPotential {d} (c : SpeedOfLight)
@@ -197,9 +195,8 @@ lemma vectorPotential_inner_radial_eq_zero_ofElectromagneticField
     {c : SpeedOfLight} {E B : Time → Space 3 → EuclideanSpace ℝ (Fin 3)}
     {t : Time} {x : Space 3} {a : ℝ} (hB : Continuous ↿B) :
     ⟪(ofElectromagneticField c E B).vectorPotential c t (a • x), Space.basis.repr x⟫_ℝ = 0 := by
-  rw [real_inner_comm]
-  rw [PiLp.inner_apply]
-  have h1 (a b : ℝ) : ⟪a, b⟫_ℝ = b * a:= by rfl
+  rw [real_inner_comm, PiLp.inner_apply]
+  have h1 (a b : ℝ) : ⟪a, b⟫_ℝ = b * a := rfl
   simp only [Space.basis_repr_apply, ofElectromagneticField_vectorPotential_apply_eq_expand hB,
     Fin.isValue, Space.smul_apply, h1, Fin.sum_univ_three, zero_add, Fin.reduceAdd]
   ring
@@ -238,57 +235,40 @@ the smoothness of the electromagnetic potential.
 lemma vectorPotential_contDiff {n} {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
     (hA : ContDiff ℝ n A) : ContDiff ℝ n ↿(A.vectorPotential c) := by
   simp [vectorPotential]
-  apply timeSlice_contDiff
-  refine contDiff_euclidean.mpr ?_
-  have h1 : ∀ i, ContDiff ℝ n (fun x => A x i) := by
-    rw [SpaceTime.contDiff_vector]
-    exact hA
-  exact fun i => h1 (Sum.inr i)
+  exact timeSlice_contDiff c _ (contDiff_euclidean.mpr fun i =>
+    (SpaceTime.contDiff_vector A).mpr hA (Sum.inr i))
 
 open ContDiff
 @[fun_prop]
 lemma vectorPotential_contDiff_of_smooth {n : ℕ} {d} {c : SpeedOfLight}
     (A : ElectromagneticPotential d) (hA : ContDiff ℝ ∞ A) :
-    ContDiff ℝ n ↿(A.vectorPotential c) := by
-  apply vectorPotential_contDiff
-  exact hA.of_le (ENat.LEInfty.out)
+    ContDiff ℝ n ↿(A.vectorPotential c) :=
+  vectorPotential_contDiff A (hA.of_le (ENat.LEInfty.out))
 
 lemma vectorPotential_apply_contDiff {n} {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) (i : Fin d) : ContDiff ℝ n ↿(fun t x => A.vectorPotential c t x i) := by
-  change ContDiff ℝ n (EuclideanSpace.proj i ∘ ↿(A.vectorPotential c))
-  refine ContDiff.comp ?_ ?_
-  · exact ContinuousLinearMap.contDiff (𝕜 := ℝ) (n := n) (EuclideanSpace.proj i)
-  · exact vectorPotential_contDiff A hA
+    (hA : ContDiff ℝ n A) (i : Fin d) : ContDiff ℝ n ↿(fun t x => A.vectorPotential c t x i) :=
+  (ContinuousLinearMap.contDiff (𝕜 := ℝ) (n := n) (EuclideanSpace.proj i)).comp
+    (vectorPotential_contDiff A hA)
 
 lemma vectorPotential_comp_contDiff {n} {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) (i : Fin d) : ContDiff ℝ n ↿(fun t x => A.vectorPotential c t x i) := by
-  change ContDiff ℝ n (EuclideanSpace.proj i ∘ ↿(A.vectorPotential c))
-  refine ContDiff.comp ?_ ?_
-  · exact ContinuousLinearMap.contDiff (𝕜 := ℝ) (n := n) (EuclideanSpace.proj i)
-  · exact vectorPotential_contDiff A hA
+    (hA : ContDiff ℝ n A) (i : Fin d) : ContDiff ℝ n ↿(fun t x => A.vectorPotential c t x i) :=
+  (ContinuousLinearMap.contDiff (𝕜 := ℝ) (n := n) (EuclideanSpace.proj i)).comp
+    (vectorPotential_contDiff A hA)
 
 lemma vectorPotential_contDiff_space {n} {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) (t : Time) : ContDiff ℝ n (A.vectorPotential c t) := by
-  change ContDiff ℝ n (↿(A.vectorPotential c) ∘ fun x => (t, x))
-  refine ContDiff.comp ?_ ?_
-  · exact vectorPotential_contDiff A hA
-  · fun_prop
+    (hA : ContDiff ℝ n A) (t : Time) : ContDiff ℝ n (A.vectorPotential c t) :=
+  (vectorPotential_contDiff A hA).comp (f := fun x => (t, x)) (by fun_prop)
 
 lemma vectorPotential_apply_contDiff_space {n} {d} {c : SpeedOfLight}
     (A : ElectromagneticPotential d)
     (hA : ContDiff ℝ n A) (t : Time) (i : Fin d) :
-    ContDiff ℝ n (fun x => A.vectorPotential c t x i) := by
-  change ContDiff ℝ n (EuclideanSpace.proj i ∘ (↿(A.vectorPotential c) ∘ fun x => (t, x)))
-  refine ContDiff.comp ?_ ?_
-  · exact ContinuousLinearMap.contDiff (𝕜 := ℝ) (n := n) (EuclideanSpace.proj i)
-  · exact vectorPotential_contDiff_space A hA t
+    ContDiff ℝ n (fun x => A.vectorPotential c t x i) :=
+  (ContinuousLinearMap.contDiff (𝕜 := ℝ) (n := n) (EuclideanSpace.proj i)).comp
+    (vectorPotential_contDiff_space A hA t)
 
 lemma vectorPotential_contDiff_time {n} {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
-    (hA : ContDiff ℝ n A) (x : Space d) : ContDiff ℝ n (A.vectorPotential c · x) := by
-  change ContDiff ℝ n (↿(A.vectorPotential c) ∘ fun t => (t, x))
-  refine ContDiff.comp ?_ ?_
-  · exact vectorPotential_contDiff A hA
-  · fun_prop
+    (hA : ContDiff ℝ n A) (x : Space d) : ContDiff ℝ n (A.vectorPotential c · x) :=
+  (vectorPotential_contDiff A hA).comp (f := fun t => (t, x)) (by fun_prop)
 
 /-!
 
@@ -302,19 +282,12 @@ the differentiablity of the electromagnetic potential.
 lemma vectorPotential_differentiable {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
     (hA : Differentiable ℝ A) : Differentiable ℝ ↿(A.vectorPotential c) := by
   simp [vectorPotential]
-  apply timeSlice_differentiable
-  refine differentiable_euclidean.mpr ?_
-  have h1 : ∀ i, Differentiable ℝ (fun x => A x i) := by
-    rw [SpaceTime.differentiable_vector]
-    exact hA
-  exact fun i => h1 (Sum.inr i)
+  exact timeSlice_differentiable c _ (differentiable_euclidean.mpr fun i =>
+    (SpaceTime.differentiable_vector A).mpr hA (Sum.inr i))
 
 lemma vectorPotential_differentiable_time {d} {c : SpeedOfLight} (A : ElectromagneticPotential d)
-    (hA : Differentiable ℝ A) (x : Space d) : Differentiable ℝ (A.vectorPotential c · x) := by
-  change Differentiable ℝ (↿(A.vectorPotential c) ∘ fun t => (t, x))
-  refine Differentiable.comp ?_ ?_
-  · exact vectorPotential_differentiable A hA
-  · fun_prop
+    (hA : Differentiable ℝ A) (x : Space d) : Differentiable ℝ (A.vectorPotential c · x) :=
+  (vectorPotential_differentiable A hA).comp (f := fun t => (t, x)) (by fun_prop)
 
 end ElectromagneticPotential
 
diff --git a/Physlib/Electromagnetism/PointParticle/OneDimension.lean b/Physlib/Electromagnetism/PointParticle/OneDimension.lean
index 127faf8de..c03c213b7 100644
--- a/Physlib/Electromagnetism/PointParticle/OneDimension.lean
+++ b/Physlib/Electromagnetism/PointParticle/OneDimension.lean
@@ -201,8 +201,7 @@ lemma oneDimPointParticle_electricField_timeDeriv (𝓕 : FreeSpace) (q : ℝ) (
 -/
 
 lemma oneDimPointParticle_magneticFieldMatrix (q : ℝ) (r₀ : Space 1) :
-    (oneDimPointParticle 𝓕 q r₀).magneticFieldMatrix 𝓕.c = 0 := by
-  simp
+    (oneDimPointParticle 𝓕 q r₀).magneticFieldMatrix 𝓕.c = 0 := by simp
 
 /-!
 
diff --git a/Physlib/Electromagnetism/Vacuum/Constant.lean b/Physlib/Electromagnetism/Vacuum/Constant.lean
index 4bbfd58b1..1f4dc9cdd 100644
--- a/Physlib/Electromagnetism/Vacuum/Constant.lean
+++ b/Physlib/Electromagnetism/Vacuum/Constant.lean
@@ -167,8 +167,7 @@ lemma constantEB_vectorPotential_space_deriv {c : SpeedOfLight}
     {B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)} (t : Time) (x : Space d) (i j : Fin d) :
     Space.deriv i ((constantEB c E₀ B₀ B₀_antisymm).vectorPotential c t · j) x =
     (1 / 2) * B₀ (j, i) := by
-  rw [constantEB_vectorPotential]
-  rw [Space.deriv_eq]
+  rw [constantEB_vectorPotential, Space.deriv_eq]
   rw [fderiv_const_mul (by fun_prop)]
   rw [fderiv_fun_sum (by fun_prop)]
   simp only [one_div, FunLike.coe_smul, FunLike.coe_sum, Pi.smul_apply,
diff --git a/Physlib/Electromagnetism/Vacuum/HarmonicWave.lean b/Physlib/Electromagnetism/Vacuum/HarmonicWave.lean
index bc4fbe15d..255a29080 100644
--- a/Physlib/Electromagnetism/Vacuum/HarmonicWave.lean
+++ b/Physlib/Electromagnetism/Vacuum/HarmonicWave.lean
@@ -86,15 +86,12 @@ noncomputable def harmonicWaveX (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → 
 @[simp]
 lemma harmonicWaveX_inl_zero {d} (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → ℝ) (φ : Fin d → ℝ)
     (x : SpaceTime d.succ) :
-    harmonicWaveX 𝓕 k E₀ φ x (Sum.inl 0) = 0 := by
-  simp [harmonicWaveX]
+    harmonicWaveX 𝓕 k E₀ φ x (Sum.inl 0) = 0 := by simp [harmonicWaveX]
 
 @[simp]
 lemma harmonicWaveX_inr_zero {d} (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → ℝ) (φ : Fin d → ℝ)
     (x : SpaceTime d.succ) :
-    harmonicWaveX 𝓕 k E₀ φ x (Sum.inr 0) = 0 := by
-  simp [harmonicWaveX]
-  rfl
+    harmonicWaveX 𝓕 k E₀ φ x (Sum.inr 0) = 0 := rfl
 
 /-!
 
@@ -104,9 +101,8 @@ lemma harmonicWaveX_inr_zero {d} (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d →
 
 lemma harmonicWaveX_differentiable {d} (𝓕 : FreeSpace) (k : ℝ)
     (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) :
-    Differentiable ℝ (harmonicWaveX 𝓕 k E₀ φ) := by
-  rw [← Lorentz.Vector.differentiable_apply]
-  intro μ
+    Differentiable ℝ (harmonicWaveX 𝓕 k E₀ φ) :=
+  (Lorentz.Vector.differentiable_apply _).mp fun μ => by
   match μ with
   | Sum.inl 0 => simp
   | Sum.inr ⟨0, h⟩ => simp
@@ -123,9 +119,8 @@ lemma harmonicWaveX_differentiable {d} (𝓕 : FreeSpace) (k : ℝ)
 
 lemma harmonicWaveX_contDiff {d} (n : WithTop ℕ∞) (𝓕 : FreeSpace) (k : ℝ)
     (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) :
-    ContDiff ℝ n (harmonicWaveX 𝓕 k E₀ φ) := by
-  rw [← Lorentz.Vector.contDiff_apply]
-  intro μ
+    ContDiff ℝ n (harmonicWaveX 𝓕 k E₀ φ) :=
+  (Lorentz.Vector.contDiff_apply _).mp fun μ => by
   match μ with
   | Sum.inl 0 => simp [harmonicWaveX]; fun_prop
   | Sum.inr ⟨0, h⟩ => simp [harmonicWaveX]; fun_prop
@@ -166,9 +161,7 @@ lemma harmonicWaveX_scalarPotential_eq_zero {d} (𝓕 : FreeSpace) (k : ℝ)
 @[simp]
 lemma harmonicWaveX_vectorPotential_zero_eq_zero {d} (𝓕 : FreeSpace) (k : ℝ)
     (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) (t : Time) (x : Space d.succ) :
-    (harmonicWaveX 𝓕 k E₀ φ).vectorPotential 𝓕.c t x 0 = 0 := by
-  simp [harmonicWaveX, vectorPotential, SpaceTime.timeSlice]
-  rfl
+    (harmonicWaveX 𝓕 k E₀ φ).vectorPotential 𝓕.c t x 0 = 0 := rfl
 
 lemma harmonicWaveX_vectorPotential_succ {d} (𝓕 : FreeSpace) (k : ℝ)
     (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) (t : Time) (x : Space d.succ) (i : Fin d) :
@@ -604,8 +597,7 @@ lemma harmonicWaveX_polarization_ellipse {d} (𝓕 : FreeSpace) (k : ℝ) (hk :
   symm
   calc _
     _ = ∑ (i : Fin d), ∑ (j : Fin d), (cos (τ + φ i) ^ 2 + cos (τ + φ j) ^ 2
-        - 2 * cos (τ + φ i) * cos (τ + φ j) * cos (φ j - φ i)) := by
-      simp [← hij]
+        - 2 * cos (τ + φ i) * cos (τ + φ j) * cos (φ j - φ i)) := by simp [← hij]
     _ = 2 * ∑ (i : Fin d), ∑ (j : Fin d), cos (τ + φ j) ^ 2
         - 2 * ∑ (i : Fin d), ∑ (j : Fin d), cos (τ + φ i) * cos (τ + φ j) * cos (φ j - φ i) := by
       rw [two_mul]
diff --git a/Physlib/Electromagnetism/Vacuum/IsPlaneWave.lean b/Physlib/Electromagnetism/Vacuum/IsPlaneWave.lean
index eb0811902..761811058 100644
--- a/Physlib/Electromagnetism/Vacuum/IsPlaneWave.lean
+++ b/Physlib/Electromagnetism/Vacuum/IsPlaneWave.lean
@@ -93,9 +93,8 @@ lemma electricField_eq_electricFunction {d : ℕ} {𝓕 : FreeSpace}
     {A : ElectromagneticPotential d} {s : Direction d}
     (P : IsPlaneWave 𝓕 A s) (t : Time) (x : Space d) :
     A.electricField 𝓕.c t x =
-    P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c * t) := by
-  rw [Classical.choose_spec P.1]
-  rfl
+    P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c * t) :=
+  congrFun (congrFun (Classical.choose_spec P.1) t) x
 
 /-- The corresponding magnetic field function from `ℝ` to
   `Fin d × Fin d → ℝ` of a plane wave. -/
@@ -108,9 +107,8 @@ lemma magneticFieldMatrix_eq_magneticFunction {d : ℕ}
     {𝓕 : FreeSpace} {A : ElectromagneticPotential d} {s : Direction d}
     (P : IsPlaneWave 𝓕 A s) (t : Time) (x : Space d) :
     A.magneticFieldMatrix 𝓕.c t x =
-    P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c * t) := by
-  rw [Classical.choose_spec P.2 t x]
-  rfl
+    P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c * t) :=
+  Classical.choose_spec P.2 t x
 
 /-!
 
@@ -379,19 +377,11 @@ lemma time_deriv_magneticFieldMatrix_eq_electricField_mul_propogator {d : ℕ}
   rw [P.electricField_space_deriv_eq_time_deriv, P.electricField_space_deriv_eq_time_deriv]
   conv_rhs =>
     rw [Time.deriv_eq]
-    rw [fderiv_fun_sub (by
-      apply Differentiable.differentiableAt
-      apply Differentiable.const_mul
-      exact electricField_apply_differentiable_time hA _ _) (by
-        apply Differentiable.differentiableAt
-        apply Differentiable.const_mul
-        exact electricField_apply_differentiable_time hA _ _)]
-    rw [fderiv_const_mul (by
-        apply Differentiable.differentiableAt
-        exact electricField_apply_differentiable_time hA _ _)]
-    rw [fderiv_const_mul (by
-        apply Differentiable.differentiableAt
-        exact electricField_apply_differentiable_time hA _ _)]
+    rw [fderiv_fun_sub
+      ((electricField_apply_differentiable_time hA _ _).const_mul _).differentiableAt
+      ((electricField_apply_differentiable_time hA _ _).const_mul _).differentiableAt]
+    rw [fderiv_const_mul (electricField_apply_differentiable_time hA _ _).differentiableAt]
+    rw [fderiv_const_mul (electricField_apply_differentiable_time hA _ _).differentiableAt]
   simp [← Time.deriv_eq]
   field_simp
   ring
@@ -446,12 +436,8 @@ lemma magneticFieldMatrix_eq_propogator_cross_electricField {d : ℕ}
       s.unit i * A.electricField 𝓕.c t x j) + C := by
   apply Space.equal_up_to_const_of_deriv_eq
   · exact magneticFieldMatrix_differentiable A hA (i, j)
-  · apply Differentiable.const_mul
-    apply Differentiable.sub
-    · apply Differentiable.const_mul
-      exact electricField_apply_differentiable hA
-    · apply Differentiable.const_mul
-      exact electricField_apply_differentiable hA
+  · exact (Differentiable.sub ((electricField_apply_differentiable hA).const_mul _)
+      ((electricField_apply_differentiable hA).const_mul _)).const_mul _
   · intro t x
     rw [P.time_deriv_magneticFieldMatrix_eq_electricField_mul_propogator hA t x i j]
     congr
@@ -506,8 +492,7 @@ lemma time_deriv_electricField_eq_magneticFieldMatrix {d : ℕ}
     ring_nf
     rfl
     · intro k _
-      apply DifferentiableAt.mul_const
-      exact (hBd k).differentiableAt
+      exact (hBd k).differentiableAt.mul_const _
 
   rw [h1, Finset.mul_sum, Finset.mul_sum,← Finset.sum_neg_distrib]
   field_simp
@@ -523,8 +508,7 @@ lemma time_deriv_electricField_eq_magneticFieldMatrix {d : ℕ}
   simp [← Time.deriv_eq]
   · refine DifferentiableAt.fun_sum ?_
     intro k _
-    apply DifferentiableAt.mul_const
-    exact (hBd k).differentiableAt
+    exact (hBd k).differentiableAt.mul_const _
   · change ContDiff ℝ ∞ (fun _ => 0)
     fun_prop
   · exact electricField_differentiable_time (hA.of_le (ENat.LEInfty.out)) x
@@ -562,22 +546,16 @@ lemma space_deriv_electricField_eq_magneticFieldMatrix {d : ℕ}
   field_simp
   any_goals apply Differentiable.differentiableAt
   · exact fieldStrengthMatrix_differentiable_space hA2 t
-  · apply Differentiable.mul_const
-    exact fieldStrengthMatrix_differentiable_space hA2 t
+  · exact (fieldStrengthMatrix_differentiable_space hA2 t).mul_const _
   · exact fieldStrengthMatrix_differentiable_time hA2 x
   · intro i _
-    apply Differentiable.differentiableAt
-    apply Differentiable.const_mul
-    apply Differentiable.mul_const
-    exact fieldStrengthMatrix_differentiable_space hA2 t
+    exact
+      (((fieldStrengthMatrix_differentiable_space hA2 t).mul_const _).const_mul _).differentiableAt
   · intro i _
-    apply Differentiable.differentiableAt
-    apply Differentiable.mul_const
-    exact fieldStrengthMatrix_differentiable_time hA2 x
+    exact ((fieldStrengthMatrix_differentiable_time hA2 x).mul_const _).differentiableAt
   · apply Differentiable.fun_sum
     intro i _
-    apply Differentiable.mul_const
-    exact fieldStrengthMatrix_differentiable_time hA2 x
+    exact (fieldStrengthMatrix_differentiable_time hA2 x).mul_const _
 
 /-!
 
@@ -594,11 +572,8 @@ lemma electricField_eq_propogator_cross_magneticFieldMatrix {d : ℕ}
   have hA2 : ContDiff ℝ 2 A := hA.of_le ENat.LEInfty.out
   apply Space.equal_up_to_const_of_deriv_eq
   · exact electricField_apply_differentiable hA2
-  · apply Differentiable.const_mul
-    apply Differentiable.fun_sum
-    intro j _
-    apply Differentiable.mul_const
-    exact magneticFieldMatrix_differentiable A hA2 (i, j)
+  · exact ((Differentiable.fun_sum fun j _ =>
+      (magneticFieldMatrix_differentiable A hA2 (i, j)).mul_const _).const_mul _)
   · intro t x
     rw [P.time_deriv_electricField_eq_magneticFieldMatrix hA _ t x i]
     congr
diff --git a/Physlib/FluidDynamics/NavierStokes/Momentum.lean b/Physlib/FluidDynamics/NavierStokes/Momentum.lean
index 40928d4fd..32b0baa9d 100644
--- a/Physlib/FluidDynamics/NavierStokes/Momentum.lean
+++ b/Physlib/FluidDynamics/NavierStokes/Momentum.lean
@@ -135,8 +135,7 @@ lemma timeDeriv_smul_velocity (d : ℕ) (rhoAtPosition : Time → ℝ)
   change (fderiv ℝ (rhoAtPosition • velocityAtPosition) t) 1 =
     rhoAtPosition t • (fderiv ℝ velocityAtPosition t) 1 +
       (fderiv ℝ rhoAtPosition t) 1 • velocityAtPosition t
-  rw [fderiv_smul hRho hVelocity]
-  rfl
+  exact congrArg (fun L => L 1) (fderiv_smul hRho hVelocity)
 
 /-- Product rule for the time derivative of the momentum density `rho u`. -/
 lemma timeDeriv_momentumDensity (d : ℕ) (fluid : FluidState d)
@@ -200,8 +199,7 @@ lemma matrixDiv_momentumFlux (d : ℕ) (fluid : FluidState d)
               hVelocity x]
             simp [momentumDensity, mul_comm, mul_assoc]
     _ = (∑ j, ∂[j] (fun x' => momentumDensity d fluid t x' j) x) * fluid.velocity t x i +
-        fluid.rho t x * (∑ j, fluid.velocity t x j * ∂[j] (fluid.velocity t) x i) := by
-          ring
+        fluid.rho t x * (∑ j, fluid.velocity t x j * ∂[j] (fluid.velocity t) x i) := by ring
 
 /-- The algebraic bridge between conservative and convective momentum.
 
diff --git a/Physlib/Mathematics/Calculus/AdjFDeriv.lean b/Physlib/Mathematics/Calculus/AdjFDeriv.lean
index 16130eb11..622fc1936 100644
--- a/Physlib/Mathematics/Calculus/AdjFDeriv.lean
+++ b/Physlib/Mathematics/Calculus/AdjFDeriv.lean
@@ -130,28 +130,24 @@ lemma gradient_eq_adjFDeriv
     {f : U → 𝕜} {x : U} (hf : DifferentiableAt 𝕜 f x) :
     gradient f x = adjFDeriv 𝕜 f x 1 := by
   apply ext_inner_right 𝕜
-  unfold gradient
-  simp [hf.hasAdjFDerivAt.hasAdjoint_fderiv.adjoint_inner_left]
+  simp [gradient, hf.hasAdjFDerivAt.hasAdjoint_fderiv.adjoint_inner_left]
 
 attribute [fun_prop] HasAdjFDerivAt.differentiableAt
 
 lemma hasAdjFDerivAt_id (x : E) : HasAdjFDerivAt 𝕜 (fun x : E => x) (fun dx => dx) x where
   differentiableAt := by fun_prop
-  hasAdjoint_fderiv := by
-    simp; apply hasAdjoint_id
+  hasAdjoint_fderiv := by simp; apply hasAdjoint_id
 
 lemma adjFDeriv_id : adjFDeriv 𝕜 (fun x : E => x) = fun _ dx => dx := by
   funext x
   rw[HasAdjFDerivAt.adjFDeriv (hasAdjFDerivAt_id x)]
 
-lemma adjFDeriv_id' : adjFDeriv 𝕜 (id : E → E) = fun _ dx => dx := by
-  exact adjFDeriv_id
+lemma adjFDeriv_id' : adjFDeriv 𝕜 (id : E → E) = fun _ dx => dx := adjFDeriv_id
 
 lemma hasAdjFDerivAt_const (x : E) (y : F) :
     HasAdjFDerivAt 𝕜 (fun _ : E => y) (fun _ => 0) x where
   differentiableAt := by fun_prop
-  hasAdjoint_fderiv := by
-    simp; apply hasAdjoint_zero
+  hasAdjoint_fderiv := by simp; apply hasAdjoint_zero
 
 lemma adjFDeriv_const (y : F) : adjFDeriv 𝕜 (fun _ : E => y) = fun _ _ => 0 := by
   funext x
@@ -160,8 +156,7 @@ lemma adjFDeriv_const (y : F) : adjFDeriv 𝕜 (fun _ : E => y) = fun _ _ => 0 :
 lemma HasAdjFDerivAt.comp {f : F → G} {g : E → F} {f' g'} {x : E}
     (hf : HasAdjFDerivAt 𝕜 f f' (g x)) (hg : HasAdjFDerivAt 𝕜 g g' x) :
     HasAdjFDerivAt 𝕜 (fun x => f (g x)) (fun dz => g' (f' dz)) x where
-  differentiableAt := by
-    fun_prop
+  differentiableAt := by fun_prop
   hasAdjoint_fderiv := by
     simp (disch:=fun_prop) [fderiv_fun_comp]
     exact hf.hasAdjoint_fderiv.comp hg.hasAdjoint_fderiv
@@ -169,11 +164,8 @@ lemma HasAdjFDerivAt.comp {f : F → G} {g : E → F} {f' g'} {x : E}
 lemma adjFDeriv_comp [CompleteSpace E] [CompleteSpace F] [CompleteSpace G]
     {f : F → G} {g : E → F} {x : E}
     (hf : DifferentiableAt 𝕜 f (g x)) (hg : DifferentiableAt 𝕜 g x) :
-    adjFDeriv 𝕜 (fun x => f (g x)) x = fun dy => adjFDeriv 𝕜 g x (adjFDeriv 𝕜 f (g x) dy) := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAdjFDerivAt.comp
-  apply hf.hasAdjFDerivAt
-  apply hg.hasAdjFDerivAt
+    adjFDeriv 𝕜 (fun x => f (g x)) x = fun dy => adjFDeriv 𝕜 g x (adjFDeriv 𝕜 f (g x) dy) :=
+  (HasAdjFDerivAt.comp hf.hasAdjFDerivAt hg.hasAdjFDerivAt).adjFDeriv
 
 lemma HasAdjFDerivAt.prodMk {f : E → F} {g : E → G} {f' g'} {x : E}
     (hf : HasAdjFDerivAt 𝕜 f f' x) (hg : HasAdjFDerivAt 𝕜 g g' x) :
@@ -194,9 +186,8 @@ lemma HasAjdFDerivAt.fst {f : E → F×G} {f'} {x : E} (hf : HasAdjFDerivAt 𝕜
 
 lemma adjFDeriv_fst [CompleteSpace E] [CompleteSpace F] [CompleteSpace G]
     {f : E → F×G} {x : E} (hf : DifferentiableAt 𝕜 f x) :
-    adjFDeriv 𝕜 (fun x => (f x).fst) x = fun dy => adjFDeriv 𝕜 f x (dy, 0) := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAjdFDerivAt.fst hf.hasAdjFDerivAt
+    adjFDeriv 𝕜 (fun x => (f x).fst) x = fun dy => adjFDeriv 𝕜 f x (dy, 0) :=
+  (HasAjdFDerivAt.fst hf.hasAdjFDerivAt).adjFDeriv
 
 @[simp]
 lemma adjFDeriv_prod_fst [CompleteSpace E] [CompleteSpace F] {x : F × E} :
@@ -216,9 +207,8 @@ lemma HasAjdFDerivAt.snd {f : E → F×G} {f'} {x : E} (hf : HasAdjFDerivAt 𝕜
 
 lemma adjFDeriv_snd [CompleteSpace E] [CompleteSpace F] [CompleteSpace G]
     {f : E → F×G} {x : E} (hf : DifferentiableAt 𝕜 f x) :
-    adjFDeriv 𝕜 (fun x => (f x).snd) x = fun dy => adjFDeriv 𝕜 f x (0, dy) := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAjdFDerivAt.snd hf.hasAdjFDerivAt
+    adjFDeriv 𝕜 (fun x => (f x).snd) x = fun dy => adjFDeriv 𝕜 f x (0, dy) :=
+  (HasAjdFDerivAt.snd hf.hasAdjFDerivAt).adjFDeriv
 
 @[simp]
 lemma adjFDeriv_prod_snd [CompleteSpace E] [CompleteSpace F] {x : F × E} :
@@ -262,9 +252,8 @@ lemma HasAdjFDerivAt.neg {f : E → F} {f'} {x : E} (hf : HasAdjFDerivAt 𝕜 f
 
 lemma adjFDeriv_neg [CompleteSpace E] [CompleteSpace F]
     {f : E → F} {x : E} (hf : DifferentiableAt 𝕜 f x) :
-    adjFDeriv 𝕜 (fun x => - f x) x = fun dy => - adjFDeriv 𝕜 f x dy := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAdjFDerivAt.neg hf.hasAdjFDerivAt
+    adjFDeriv 𝕜 (fun x => - f x) x = fun dy => - adjFDeriv 𝕜 f x dy :=
+  (HasAdjFDerivAt.neg hf.hasAdjFDerivAt).adjFDeriv
 
 lemma HasAjdFDerivAt.add {f g : E → F} {f' g'} {x : E}
     (hf : HasAdjFDerivAt 𝕜 f f' x) (hg : HasAdjFDerivAt 𝕜 g g' x) :
@@ -277,11 +266,8 @@ lemma HasAjdFDerivAt.add {f g : E → F} {f' g'} {x : E}
 lemma adjFDeriv_add [CompleteSpace E] [CompleteSpace F]
     {f g : E → F} {x : E}
     (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
-    adjFDeriv 𝕜 (fun x => f x + g x) x = fun dy => adjFDeriv 𝕜 f x dy + adjFDeriv 𝕜 g x dy := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAjdFDerivAt.add
-  apply hf.hasAdjFDerivAt
-  apply hg.hasAdjFDerivAt
+    adjFDeriv 𝕜 (fun x => f x + g x) x = fun dy => adjFDeriv 𝕜 f x dy + adjFDeriv 𝕜 g x dy :=
+  (HasAjdFDerivAt.add hf.hasAdjFDerivAt hg.hasAdjFDerivAt).adjFDeriv
 
 lemma HasAdjFDerivAt.sub
     {f g : E → F} {f' g'} {x : E}
@@ -294,11 +280,8 @@ lemma HasAdjFDerivAt.sub
 
 lemma adjFDeriv_sub [CompleteSpace E] [CompleteSpace F] {f g : E → F} {x : E}
     (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
-    adjFDeriv 𝕜 (fun x => f x - g x) x = fun dy => adjFDeriv 𝕜 f x dy - adjFDeriv 𝕜 g x dy := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAdjFDerivAt.sub
-  apply hf.hasAdjFDerivAt
-  apply hg.hasAdjFDerivAt
+    adjFDeriv 𝕜 (fun x => f x - g x) x = fun dy => adjFDeriv 𝕜 f x dy - adjFDeriv 𝕜 g x dy :=
+  (HasAdjFDerivAt.sub hf.hasAdjFDerivAt hg.hasAdjFDerivAt).adjFDeriv
 
 open ComplexConjugate in
 lemma HasAdjFDerivAt.smul {f : E → F} {g : E → 𝕜} {f' g'}
@@ -317,11 +300,8 @@ lemma adjFDeriv_smul [CompleteSpace E] [CompleteSpace F]
     {f : E → F} {g : E → 𝕜} {x : E}
     (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
     adjFDeriv 𝕜 (fun x => g x • f x) x = fun dy => conj (g x) • adjFDeriv 𝕜 f x dy +
-                                                  adjFDeriv 𝕜 g x (conj (inner 𝕜 dy (f x))) := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAdjFDerivAt.smul
-  apply hf.hasAdjFDerivAt
-  apply hg.hasAdjFDerivAt
+                                                  adjFDeriv 𝕜 g x (conj (inner 𝕜 dy (f x))) :=
+  (HasAdjFDerivAt.smul hf.hasAdjFDerivAt hg.hasAdjFDerivAt).adjFDeriv
 
 set_option backward.isDefEq.respectTransparency false in
 open InnerProductSpace in
@@ -348,6 +328,5 @@ open InnerProductSpace in
 lemma adjFDeriv_inner {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [InnerProductSpace' ℝ E]
     (x : E × E) :
     adjFDeriv ℝ (fun (x : E × E) => ⟪x.1, x.2⟫_ℝ) x =
-      fun y => y • (x.2, x.1) := by
-  apply HasAdjFDerivAt.adjFDeriv
-  apply HasAdjFDerivAt.inner
+      fun y => y • (x.2, x.1) :=
+  (HasAdjFDerivAt.inner x).adjFDeriv
diff --git a/Physlib/Mathematics/Calculus/Divergence.lean b/Physlib/Mathematics/Calculus/Divergence.lean
index a538a0e6b..9c18cf0ee 100644
--- a/Physlib/Mathematics/Calculus/Divergence.lean
+++ b/Physlib/Mathematics/Calculus/Divergence.lean
@@ -39,10 +39,8 @@ lemma divergence_zero : divergence 𝕜 (fun _ : E => 0) = fun _ => 0 := by
 lemma divergence_eq_sum_fderiv {s : Finset E} (b : Basis s 𝕜 E) {f : E → E} :
     divergence 𝕜 f = fun x => ∑ i : s, b.repr (fderiv 𝕜 f x (b i)) i := by
   funext x
-  unfold divergence
-  rw[LinearMap.trace_eq_matrix_trace_of_finset (s:=s) _ b]
-  simp only [Matrix.trace, Matrix.diag, LinearMap.toMatrix_apply]
-  rfl
+  rw [divergence, LinearMap.trace_eq_matrix_trace_of_finset (s := s) _ b]
+  simp [Matrix.trace, Matrix.diag, LinearMap.toMatrix_apply]
 
 lemma divergence_eq_sum_fderiv' {ι} [Fintype ι] (b : Basis ι 𝕜 E) {f : E → E} :
     divergence 𝕜 f = fun x => ∑ i, b.repr (fderiv 𝕜 f x (b i)) i := by
@@ -79,8 +77,7 @@ lemma divergence_eq_space_div {d} (f : Space d → Space d)
     ext j
     simp only [Function.comp_apply]
     rw [Space.coordCLM_apply, Space.coord_apply]
-  rw [h1]
-  rw [fderiv_comp]
+  rw [h1, fderiv_comp]
   simp [Space.coordCLM_apply, Space.coord_apply]
   · fun_prop
   · exact h x
@@ -107,30 +104,22 @@ lemma divergence_add {f g : E → E} {x : E}
     (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
     divergence 𝕜 (fun x => f x + g x) x
     =
-    divergence 𝕜 f x + divergence 𝕜 g x := by
-  unfold divergence
-  simp [fderiv_fun_add hf hg]
+    divergence 𝕜 f x + divergence 𝕜 g x := by simp [divergence, fderiv_fun_add hf hg]
 
 lemma divergence_neg {f : E → E} {x : E} :
-    divergence 𝕜 (fun x => -f x) x = -divergence 𝕜 f x := by
-  unfold divergence
-  simp
+    divergence 𝕜 (fun x => -f x) x = -divergence 𝕜 f x := by simp [divergence]
 
 lemma divergence_sub {f g : E → E} {x : E}
     (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
     divergence 𝕜 (fun x => f x - g x) x
     =
-    divergence 𝕜 f x - divergence 𝕜 g x := by
-  unfold divergence
-  simp [fderiv_fun_sub hf hg]
+    divergence 𝕜 f x - divergence 𝕜 g x := by simp [divergence, fderiv_fun_sub hf hg]
 
 lemma divergence_const_smul {f : E → E} {x : E} {c : 𝕜}
     (hf : DifferentiableAt 𝕜 f x) :
     divergence 𝕜 (fun x => c • f x) x
     =
-    c * divergence 𝕜 f x := by
-  unfold divergence
-  simp [fderiv_fun_const_smul hf]
+    c * divergence 𝕜 f x := by simp [divergence, fderiv_fun_const_smul hf]
 
 open InnerProductSpace' in
 lemma divergence_smul [InnerProductSpace' 𝕜 E] {f : E → 𝕜} {g : E → E} {x : E}
diff --git a/Physlib/Mathematics/Calculus/ParametricIntegration.lean b/Physlib/Mathematics/Calculus/ParametricIntegration.lean
index 60dd12ddd..ab16da442 100644
--- a/Physlib/Mathematics/Calculus/ParametricIntegration.lean
+++ b/Physlib/Mathematics/Calculus/ParametricIntegration.lean
@@ -48,8 +48,7 @@ lemma hasFDerivAt_parametric_intervalIntegral_of_contDiff
     fun_prop
   · apply Continuous.intervalIntegrable
     fun_prop
-  · apply Continuous.aestronglyMeasurable
-    exact Continuous.uncurry_left x₀ (by fun_prop)
+  · exact (Continuous.uncurry_left x₀ (by fun_prop)).aestronglyMeasurable
   · filter_upwards with t h x hx
     exact ha.2 (Set.mk_mem_prod hx (Set.Ioc_subset_Icc_self (by simpa using h)))
   · exact intervalIntegrable_const
diff --git a/Physlib/Mathematics/Calculus/Wirtinger/Basic.lean b/Physlib/Mathematics/Calculus/Wirtinger/Basic.lean
index 62e780ebe..5ba498cf9 100644
--- a/Physlib/Mathematics/Calculus/Wirtinger/Basic.lean
+++ b/Physlib/Mathematics/Calculus/Wirtinger/Basic.lean
@@ -215,8 +215,7 @@ rule.
 
 /-- Constants have zero holomorphic directional Wirtinger derivative, `∂_v c = 0`. -/
 @[simp] lemma dWirtingerDir_const (c : ℂ) (v u : V) :
-    dWirtingerDir (fun _ : V => c) v u = 0 := by
-  simp [dWirtingerDir_apply, fderiv_const_apply]
+    dWirtingerDir (fun _ : V => c) v u = 0 := by simp [dWirtingerDir_apply, fderiv_const_apply]
 
 /-- Constants have zero anti-holomorphic directional Wirtinger derivative, `∂̄_v c = 0`. -/
 @[simp] lemma dWirtingerAntiDir_const (c : ℂ) (v u : V) :
@@ -395,8 +394,7 @@ lemma realLinear_apply_eq_wirtinger (L : ℂ →L[ℝ] ℂ) (w : ℂ) :
   calc
     L w = L ((w.re : ℝ) • (1 : ℂ) + (w.im : ℝ) • Complex.I) := by
               congr 1; apply Complex.ext <;> simp
-    _ = (w.re : ℝ) • L 1 + (w.im : ℝ) • L Complex.I := by
-          rw [map_add, map_smul, map_smul]
+    _ = (w.re : ℝ) • L 1 + (w.im : ℝ) • L Complex.I := by rw [map_add, map_smul, map_smul]
     _ = ((1 / 2 : ℂ) * (L 1 - Complex.I * L Complex.I)) * w
           + ((1 / 2 : ℂ) * (L 1 + Complex.I * L Complex.I)) * star w := by
       apply Complex.ext <;>
@@ -664,13 +662,11 @@ private lemma fderiv_weightedDirDeriv (hf' : DifferentiableAt ℝ (fderiv ℝ f)
 
 /-- A directional derivative is a `weightedDirDeriv`: anti-holomorphic with `c = i`. -/
 private lemma dWirtingerAntiDir_eq_weightedDirDeriv (w : V) :
-    (fun p => dWirtingerAntiDir f w p) = weightedDirDeriv f Complex.I w (Complex.I • w) :=
-  rfl
+    (fun p => dWirtingerAntiDir f w p) = weightedDirDeriv f Complex.I w (Complex.I • w) := rfl
 
 /-- A directional derivative is a `weightedDirDeriv`: holomorphic with `c = -i`. -/
 private lemma dWirtingerDir_eq_weightedDirDeriv (v : V) :
-    (fun p => dWirtingerDir f v p) = weightedDirDeriv f (-Complex.I) v (Complex.I • v) :=
-  rfl
+    (fun p => dWirtingerDir f v p) = weightedDirDeriv f (-Complex.I) v (Complex.I • v) := rfl
 
 /-- Differentiating the anti-holomorphic directional derivative lands on the second
 real Fréchet derivative in the two slots. -/
@@ -740,8 +736,7 @@ lemma differentiableAt_dWirtingerAntiDir (hf2 : ContDiffAt ℝ 2 f u) (w : V) :
 fields agreeing on a neighbourhood have equal derivative. -/
 lemma dWirtingerDir_congr_of_eventuallyEq {f₁ f₂ : V → ℂ} {u : V}
     (h : f₁ =ᶠ[nhds u] f₂) (v : V) :
-    dWirtingerDir f₁ v u = dWirtingerDir f₂ v u := by
-  simp only [dWirtingerDir_apply, h.fderiv_eq]
+    dWirtingerDir f₁ v u = dWirtingerDir f₂ v u := by simp only [dWirtingerDir_apply, h.fderiv_eq]
 
 /-- The anti-holomorphic directional derivative depends only on the field near the
 point; dual of `dWirtingerDir_congr_of_eventuallyEq`. -/
diff --git a/Physlib/Mathematics/DataStructures/Matrix/LieTrace.lean b/Physlib/Mathematics/DataStructures/Matrix/LieTrace.lean
index 5abed796b..9c8362545 100644
--- a/Physlib/Mathematics/DataStructures/Matrix/LieTrace.lean
+++ b/Physlib/Mathematics/DataStructures/Matrix/LieTrace.lean
@@ -38,8 +38,7 @@ instance [UniformSpace 𝕂] : UniformSpace (Matrix m n 𝕂) := by unfold Matri
 lemma tsum_eq_zero
     {β : Type*} [TopologicalSpace β] [AddCommMonoid β]
     {f : ℕ → β} (h : ∀ n, f n = 0) :
-    (∑' n, f n) = 0 := by
-  simp_all only [tsum_zero]
+    (∑' n, f n) = 0 := by simp_all only [tsum_zero]
 
 /-!
   ### The determinant of the matrix exponential
@@ -59,10 +58,9 @@ lemma matrix_tsum_apply
   have h_row_summable : Summable (fun n ↦ (f n) i) := by
     have h := Pi.summable.1 hf
     exact h i
-  have h₁ : ((∑' n, f n) : Matrix m m 𝕂) i = (∑' n, (f n) i) := by
-    exact tsum_apply hf
-  have h₂ : ((∑' n, (f n) i) : m → 𝕂) j = (∑' n, (f n) i j) := by
-    exact tsum_apply h_row_summable
+  have h₁ : ((∑' n, f n) : Matrix m m 𝕂) i = (∑' n, (f n) i) := tsum_apply hf
+  have h₂ : ((∑' n, (f n) i) : m → 𝕂) j = (∑' n, (f n) i j) :=
+    tsum_apply h_row_summable
   rw [h₁, h₂]
 
 variable [Fintype m] [LinearOrder m]
@@ -145,8 +143,7 @@ theorem diag_exp_of_blockTriangular_id
   funext i
   rw [NormedSpace.exp_eq_tsum 𝕂, diag_apply]
   simp_rw [matrix_tsum_apply (NormedSpace.expSeries_summable' A) i i]
-  rw [matrix_exp_series_diag_eq_scalar_series hA i]
-  rw [NormedSpace.exp_eq_tsum 𝕂]
+  rw [matrix_exp_series_diag_eq_scalar_series hA i, NormedSpace.exp_eq_tsum 𝕂]
 
 /-- Lie's trace formula for upper triangular matrices. -/
 lemma det_exp_of_blockTriangular_id {A : Matrix m m 𝕂} (hA : BlockTriangular A id) :
@@ -189,8 +186,7 @@ lemma exp_unitary_conj (A : Matrix m m 𝕂) (U : unitaryGroup m 𝕂) :
 
 lemma det_exp_unitary_conj (A : Matrix m m 𝕂) (U : unitaryGroup m 𝕂) :
     (NormedSpace.exp ((U : Matrix m m 𝕂) * A * star (U : Matrix m m 𝕂))).det =
-    (NormedSpace.exp A).det := by
-  rw [exp_unitary_conj, det_unitary_conj]
+    (NormedSpace.exp A).det := by rw [exp_unitary_conj, det_unitary_conj]
 
 /-- The determinant of the exponential of a matrix is the exponential of its trace.
 This is also known as **Lie's trace formula**. -/
@@ -242,8 +238,7 @@ lemma exp_map_algebraMap {n : Type*} [Fintype n] [DecidableEq n]
   letI : NormedAlgebra ℂ (Matrix n n ℂ) := Matrix.linftyOpNormedAlgebra
   letI : CompleteSpace (Matrix n n ℂ) := inferInstance
   simp only [exp_eq_tsum ℝ]
-  have hs : Summable (fun k => (k.factorial : ℝ)⁻¹ • A ^ k) := by
-    exact NormedSpace.expSeries_summable' A
+  have hs : Summable (fun k => (k.factorial : ℝ)⁻¹ • A ^ k) := NormedSpace.expSeries_summable' A
   erw [Matrix.map_tsum (algebraMap ℝ ℂ).toAddMonoidHom RCLike.continuous_ofReal hs]
   apply tsum_congr
   intro k
@@ -269,8 +264,7 @@ theorem det_exp_real {n : Type*} [Fintype n] [LinearOrder n]
   have h_det_comm : (algebraMap ℝ ℂ) ((NormedSpace.exp A).det) = (NormedSpace.exp A_ℂ).det := by
     rw [@RingHom.map_det]
     rw [← NormedSpace.exp_map_algebraMap]; rfl
-  rw [← h_det_comm] at h_complex
-  rw [h_trace_comm] at h_complex
+  rw [← h_det_comm, h_trace_comm] at h_complex
   have h_exp_comm : Complex.exp ((algebraMap ℝ ℂ) A.trace) =
       (algebraMap ℝ ℂ) (Real.exp A.trace) := by
     rw [Complex.coe_algebraMap, ← Complex.ofReal_exp]
diff --git a/Physlib/Mathematics/Distribution/Basic.lean b/Physlib/Mathematics/Distribution/Basic.lean
index 20029cc6c..31e927c95 100644
--- a/Physlib/Mathematics/Distribution/Basic.lean
+++ b/Physlib/Mathematics/Distribution/Basic.lean
@@ -148,11 +148,10 @@ def ofLinear (s : Finset (ℕ × ℕ)) (u : 𝓢(E, 𝕜) →ₗ[𝕜] F)
       (f := fun kn : ℕ × ℕ ↦ (⟨SchwartzMap.seminorm 𝕜 kn.1 kn.2 η, apply_nonneg _ _⟩ : ℝ≥0))
 
 @[simp] lemma ofLinear_apply (s : Finset (ℕ × ℕ)) (u : 𝓢(E, 𝕜) →ₗ[𝕜] F)
-    (hu : ∃ C : ℝ, 0 ≤ C ∧ ∀ η : 𝓢(E, 𝕜), ∃ (k : ℕ) (n : ℕ) (x : E), (k, n) ∈ s ∧
-      ‖u η‖ ≤ C * (‖x‖ ^ k * ‖iteratedFDeriv ℝ n η x‖))
-    (η : 𝓢(E, 𝕜)) :
-    ofLinear 𝕜 s u hu η = u η :=
-  rfl
+  (hu : ∃ C : ℝ, 0 ≤ C ∧ ∀ η : 𝓢(E, 𝕜), ∃ (k : ℕ) (n : ℕ) (x : E), (k, n) ∈ s ∧
+    ‖u η‖ ≤ C * (‖x‖ ^ k * ‖iteratedFDeriv ℝ n η x‖))
+  (η : 𝓢(E, 𝕜)) :
+    ofLinear 𝕜 s u hu η = u η := rfl
 
 end NormedSpace
 
@@ -275,8 +274,7 @@ def fourierTransform : (E →d[ℂ] F) →ₗ[ℂ] (E →d[ℂ] F) where
   map_smul' c u := by simp
 
 @[simp] lemma fourierTransform_apply (u : E →d[ℂ] F) (η : 𝓢(E, ℂ)) :
-    u.fourierTransform E F η = u (fourierTransformCLM ℂ η) :=
-  rfl
+    u.fourierTransform E F η = u (fourierTransformCLM ℂ η) := rfl
 
 end Complex
 
@@ -356,10 +354,8 @@ def const [hμ : Measure.HasTemperateGrowth (volume (α := E))] (c : F) : E →d
     · subst hc
       simp
     rw [MeasureTheory.integral_add]
-    · refine (integrable_smul_const hc).mpr ?_
-      exact integrable η1
-    · refine (integrable_smul_const hc).mpr ?_
-      exact integrable η2
+    · exact (integrable_smul_const hc).mpr (integrable η1)
+    · exact (integrable_smul_const hc).mpr (integrable η2)
   · intro a η
     simp only [smul_apply, RingHom.id_apply, smul_assoc]
     rw [MeasureTheory.integral_smul]
@@ -414,8 +410,7 @@ lemma fderivD_const [hμ : Measure.IsAddHaarMeasure (volume (α := E))]
     change Integrable (SchwartzMap.evalCLM (𝕜 := ℝ) E ℝ v (SchwartzMap.fderivCLM ℝ E ℝ η)) volume
     exact integrable ((SchwartzMap.evalCLM ℝ E ℝ v) ((fderivCLM ℝ) E ℝ η))
   · simp
-  · apply MeasureTheory.Integrable.smul_const
-    exact integrable η
+  · exact (integrable η).smul_const _
   · fun_prop
   · simp
 
@@ -478,8 +473,7 @@ def diracDelta (a : E) : E →d[𝕜] 𝕜 :=
     (BoundedContinuousFunction.evalCLM 𝕜 a).comp (toBoundedContinuousFunctionCLM 𝕜 E 𝕜)
 
 @[simp] lemma diracDelta_apply (a : E) (η : 𝓢(E, 𝕜)) :
-    diracDelta 𝕜 a η = η a :=
-  rfl
+    diracDelta 𝕜 a η = η a := rfl
 
 /-- Dirac delta in a given direction `v : F`. `diracDelta' 𝕜 a v` takes in a test function
 `η : 𝓢(E, 𝕜)` and outputs `η a • v`. Intuitively this is an infinitely intense vector field
@@ -488,8 +482,7 @@ def diracDelta' (a : E) (v : F) : E →d[𝕜] F :=
   ContinuousLinearMap.smulRight (diracDelta 𝕜 a) v
 
 @[simp] lemma diracDelta'_apply (a : E) (v : F) (η : 𝓢(E, 𝕜)) :
-    diracDelta' 𝕜 a v η = η a • v :=
-  rfl
+    diracDelta' 𝕜 a v η = η a • v := rfl
 
 end DiracDelta
 /-!
@@ -513,10 +506,8 @@ def heavisideStep (d : ℕ) : (EuclideanSpace ℝ (Fin d.succ)) →d[ℝ] ℝ :=
   · intro η1 η2
     simp only [Nat.succ_eq_add_one, add_apply]
     rw [MeasureTheory.integral_add]
-    · apply MeasureTheory.Integrable.restrict
-      exact integrable η1
-    · apply MeasureTheory.Integrable.restrict
-      exact integrable η2
+    · exact (integrable η1).restrict
+    · exact (integrable η2).restrict
   · intro a η
     simp only [smul_apply, RingHom.id_apply]
     rw [MeasureTheory.integral_smul]
@@ -546,8 +537,7 @@ def heavisideStep (d : ℕ) : (EuclideanSpace ℝ (Fin d.succ)) →d[ℝ] ℝ :=
 
 lemma heavisideStep_apply (d : ℕ) (η : 𝓢(EuclideanSpace ℝ (Fin d.succ), ℝ)) :
     heavisideStep d η = ∫ x in {x : EuclideanSpace ℝ (Fin d.succ) | 0 < x (Fin.last d)},
-      η x ∂MeasureTheory.volume := by
-  rfl
+      η x ∂MeasureTheory.volume := rfl
 
 end Distribution
 end Physlib
diff --git a/Physlib/Mathematics/Distribution/PowMul.lean b/Physlib/Mathematics/Distribution/PowMul.lean
index 2feed10a9..3a93cd17c 100644
--- a/Physlib/Mathematics/Distribution/PowMul.lean
+++ b/Physlib/Mathematics/Distribution/PowMul.lean
@@ -76,8 +76,7 @@ def powOneMul : 𝓢(ℝ, 𝕜) →L[𝕜] 𝓢(ℝ, 𝕜) := by
         · exact SchwartzMap.smooth (ψ) ⊤
         · exact right_eq_inf.mp rfl
       · exact ContinuousMultilinearMap.opNorm_nonneg _
-      · refine pow_nonneg ?_ k
-        exact norm_nonneg x
+      · exact pow_nonneg (norm_nonneg x) k
     conv_lhs =>
       enter [2, 2, i, 1, 2]
       change ‖iteratedFDeriv ℝ i RCLike.ofRealCLM x‖
diff --git a/Physlib/Mathematics/FDerivCurry.lean b/Physlib/Mathematics/FDerivCurry.lean
index 3dd64883e..8c1c15735 100644
--- a/Physlib/Mathematics/FDerivCurry.lean
+++ b/Physlib/Mathematics/FDerivCurry.lean
@@ -37,9 +37,7 @@ lemma fderiv_uncurry (f : X → Y → Z) (xy dxy : X × Y)
 lemma fderiv_curry_fst (f : X × Y → Z) (x : X) (y : Y)
     (h : DifferentiableAt 𝕜 f (x,y)) (dx : X) :
     fderiv 𝕜 (fun x' => Function.curry f x' y) x dx = fderiv 𝕜 f (x,y) (dx, 0) := by
-  have h1 : f = ↿(Function.curry f) := by
-    ext x
-    rfl
+  have h1 : f = ↿(Function.curry f) := rfl
   conv_rhs =>
     rw [h1]
   rw [fderiv_uncurry]
@@ -49,9 +47,7 @@ lemma fderiv_curry_fst (f : X × Y → Z) (x : X) (y : Y)
 lemma fderiv_curry_snd (f : X × Y → Z) (x : X) (y : Y)
     (h : DifferentiableAt 𝕜 f (x,y)) (dy : Y) :
     fderiv 𝕜 (Function.curry f x) y dy = fderiv 𝕜 (f) (x,y) (0, dy) := by
-  have h1 : f = ↿(Function.curry f) := by
-    ext x
-    rfl
+  have h1 : f = ↿(Function.curry f) := rfl
   conv_rhs =>
     rw [h1]
   rw [fderiv_uncurry]
@@ -95,16 +91,14 @@ lemma fderiv_wrt_prod_clm_comp (f : X × Y → Z) (hf : Differentiable 𝕜 f) :
 lemma fderiv_curry_clm_apply (f : X → Y →L[𝕜] Z) (y : Y) (x dx : X) (h : Differentiable 𝕜 f) :
     fderiv 𝕜 f x dx y
     =
-    fderiv 𝕜 (f · y) x dx := by
-  rw [fderiv_clm_apply] <;> first | simp | fun_prop
+    fderiv 𝕜 (f · y) x dx := by rw [fderiv_clm_apply] <;> first | simp | fun_prop
 
 /- Helper rw lemmas for proving differentiability conditions. -/
 lemma fderiv_uncurry_comp_fst (f : X → Y → Z) (y : Y) (hf : Differentiable 𝕜 (↿f)) :
     fderiv 𝕜 (fun x' => (↿f) (x', y))
     =
     fun x => (fderiv 𝕜 (↿f) ((·, y) x)).comp (fderiv 𝕜 (·, y) x) := by
-  have hl (y : Y) : (fun x' => (↿f) (x', y)) = ↿f ∘ (·, y) := by
-    rfl
+  have hl (y : Y) : (fun x' => (↿f) (x', y)) = ↿f ∘ (·, y) := rfl
   rw [hl]
   funext x
   rw [fderiv_comp]
@@ -115,8 +109,7 @@ lemma fderiv_uncurry_comp_snd (f : X → Y → Z) (x : X) (hf : Differentiable 
     fderiv 𝕜 (fun y' => (↿f) (x, y'))
     =
     fun y => (fderiv 𝕜 (↿f) ((x, ·) y)).comp (fderiv 𝕜 (x, ·) y) := by
-  have hl (x : X) : (fun y' => (↿f) (x, y')) = ↿f ∘ (x, ·) := by
-    rfl
+  have hl (x : X) : (fun y' => (↿f) (x, y')) = ↿f ∘ (x, ·) := rfl
   rw [hl]
   funext y
   rw [fderiv_comp]
@@ -128,10 +121,8 @@ lemma fderiv_curry_comp_fst (f : X → Y → Z) (x dx : X) (y : Y)
     (fderiv 𝕜 (fun x' => f x' y) x) dx
     =
     (fderiv 𝕜 (↿f) ((·, y) x)) ((fderiv 𝕜 (·, y) x) dx) := by
-  have hl (y : Y) : (fun x' => f x' y) = ↿f ∘ (·, y) := by
-    rfl
-  rw [hl]
-  rw [fderiv_comp]
+  have hl (y : Y) : (fun x' => f x' y) = ↿f ∘ (·, y) := rfl
+  rw [hl, fderiv_comp]
   simp only [ContinuousLinearMap.coe_comp, Function.comp_apply]
   · fun_prop
   · fun_prop
@@ -141,10 +132,8 @@ lemma fderiv_curry_comp_snd (f : X → Y → Z) (x : X) (y dy : Y)
     (fderiv 𝕜 (fun y' => f x y') y) dy
     =
     (fderiv 𝕜 (↿f) ((x, ·) y)) ((fderiv 𝕜 (x, ·) y) dy) := by
-  have hl (x : X) : (fun y' => f x y') = ↿f ∘ (x, ·) := by
-    rfl
-  rw [hl]
-  rw [fderiv_comp]
+  have hl (x : X) : (fun y' => f x y') = ↿f ∘ (x, ·) := rfl
+  rw [hl, fderiv_comp]
   simp only [ContinuousLinearMap.coe_comp, Function.comp_apply]
   · fun_prop
   · fun_prop
@@ -161,53 +150,41 @@ lemma fderiv_inl_snd_clm (x : X) (y : Y) :
 
 lemma function_differentiableAt_fst (f : X → Y → Z) (x : X) (y : Y) (hf : Differentiable 𝕜 (↿f)) :
     DifferentiableAt 𝕜 (fun x' => f x' y) x := by
-  have hl : (fun x' => f x' y) = ↿f ∘ (·, y) := by
-    funext x'
-    rfl
+  have hl : (fun x' => f x' y) = ↿f ∘ (·, y) := rfl
   rw [hl]
-  apply Differentiable.differentiableAt
-  apply Differentiable.comp <;> fun_prop
+  exact (Differentiable.comp (by fun_prop) (by fun_prop)).differentiableAt
 
 lemma function_differentiableAt_snd (f : X → Y → Z) (x : X) (y : Y) (hf : Differentiable 𝕜 (↿f)) :
     DifferentiableAt 𝕜 (fun y' => f x y') y := by
-  have hl : (fun y' => f x y') = ↿f ∘ (x, ·) := by
-    funext y'
-    rfl
+  have hl : (fun y' => f x y') = ↿f ∘ (x, ·) := rfl
   rw [hl]
-  apply Differentiable.differentiableAt
-  apply Differentiable.comp <;> fun_prop
+  exact (Differentiable.comp (by fun_prop) (by fun_prop)).differentiableAt
 
 @[fun_prop]
 lemma fderiv_uncurry_differentiable_fst (f : X → Y → Z) (y : Y) (hf : ContDiff 𝕜 2 ↿f) :
-    Differentiable 𝕜 (fderiv 𝕜 fun x' => (↿f) (x', y)) := by
-  fun_prop
+    Differentiable 𝕜 (fderiv 𝕜 fun x' => (↿f) (x', y)) := by fun_prop
 
 @[fun_prop]
 lemma fderiv_uncurry_differentiable_snd (f : X → Y → Z) (x : X) (hf : ContDiff 𝕜 2 ↿f) :
-    Differentiable 𝕜 (fderiv 𝕜 fun y' => (↿f) (x, y')) := by
-  fun_prop
+    Differentiable 𝕜 (fderiv 𝕜 fun y' => (↿f) (x, y')) := by fun_prop
 
 @[fun_prop]
 lemma fderiv_uncurry_differentiable_fst_comp_snd (f : X → Y → Z) (x : X) (hf : ContDiff 𝕜 2 ↿f) :
-    Differentiable 𝕜 (fun y' => fderiv 𝕜 (fun x' => (↿f) (x', y')) x) := by
-  fun_prop
+    Differentiable 𝕜 (fun y' => fderiv 𝕜 (fun x' => (↿f) (x', y')) x) := by fun_prop
 
 @[fun_prop]
 lemma fderiv_uncurry_differentiable_fst_comp_snd_apply (f : X → Y → Z) (x δx : X)
     (hf : ContDiff 𝕜 2 ↿f) :
-    Differentiable 𝕜 (fun y' => fderiv 𝕜 (fun x' => (↿f) (x', y')) x δx) := by
-  fun_prop
+    Differentiable 𝕜 (fun y' => fderiv 𝕜 (fun x' => (↿f) (x', y')) x δx) := by fun_prop
 
 @[fun_prop]
 lemma fderiv_uncurry_differentiable_snd_comp_fst (f : X → Y → Z) (y : Y) (hf : ContDiff 𝕜 2 ↿f) :
-    Differentiable 𝕜 (fun x' => fderiv 𝕜 (fun y' => (↿f) (x', y')) y) := by
-  fun_prop
+    Differentiable 𝕜 (fun x' => fderiv 𝕜 (fun y' => (↿f) (x', y')) y) := by fun_prop
 
 @[fun_prop]
 lemma fderiv_uncurry_differentiable_snd_comp_fst_apply (f : X → Y → Z) (y δy : Y)
     (hf : ContDiff 𝕜 2 ↿f) :
-    Differentiable 𝕜 (fun x' => fderiv 𝕜 (fun y' => (↿f) (x', y')) y δy) := by
-  fun_prop
+    Differentiable 𝕜 (fun x' => fderiv 𝕜 (fun y' => (↿f) (x', y')) y δy) := by fun_prop
 
 @[fun_prop]
 lemma fderiv_curry_differentiableAt_fst_comp_snd (f : X → Y → Z) (x dx : X) (y : Y)
diff --git a/Physlib/Mathematics/Fin.lean b/Physlib/Mathematics/Fin.lean
index e89b8d3a5..4404545d2 100644
--- a/Physlib/Mathematics/Fin.lean
+++ b/Physlib/Mathematics/Fin.lean
@@ -58,8 +58,7 @@ lemma predAboveI_eq_iff {i x : Fin n.succ.succ} (h : i ≠ x) (y : Fin n.succ) :
   · simp [← h]
 
 lemma predAboveI_lt {i x : Fin n.succ.succ} (h : x.val < i.val) :
-    predAboveI i x = ⟨x.val, by omega⟩ := by
-  simp [predAboveI, h]
+    predAboveI i x = ⟨x.val, by omega⟩ := by simp [predAboveI, h]
 
 lemma predAboveI_ge {i x : Fin n.succ.succ} (h : i.val < x.val) :
     predAboveI i x = ⟨x.val - 1, by omega⟩ := by
@@ -85,9 +84,8 @@ def finExtractOne {n : ℕ} (i : Fin (n + 1)) : Fin (n + 1) ≃ Fin 1 ⊕ Fin n
 
 @[simp]
 lemma finExtractOne_apply_eq {n : ℕ} (i : Fin n.succ) :
-    finExtractOne i i = Sum.inl 0 := by
-  rw [Equiv.apply_eq_iff_eq_symm_apply]
-  rfl
+    finExtractOne i i = Sum.inl 0 :=
+  (Equiv.apply_eq_iff_eq_symm_apply (finExtractOne i)).2 rfl
 
 lemma finExtractOne_symm_inr {n : ℕ} (i : Fin n.succ) :
     (finExtractOne i).symm ∘ Sum.inr = i.succAbove := by
@@ -102,9 +100,7 @@ lemma finExtractOne_symm_inr {n : ℕ} (i : Fin n.succ) :
   by_cases hi : x.1 < i.1
   · generalize_proofs hp
     have h1 : (finSumFinEquiv.symm (Fin.cast hp x)) =
-        Sum.inl ⟨x, hi⟩ := by
-      rw [← finSumFinEquiv_symm_apply_castAdd]
-      rfl
+        Sum.inl ⟨x, hi⟩ := finSumFinEquiv_symm_apply_castAdd (n := n - i.1) (⟨x, hi⟩ : Fin i.1)
     rw [h1]
     simp only [Nat.succ_eq_add_one, Equiv.sumAssoc_symm_apply_inr_inl, Sum.map_inl,
       Equiv.symm_trans_apply, Equiv.symm_symm, Equiv.sumComm_symm, Equiv.sumComm_apply,
@@ -140,15 +136,13 @@ lemma finExtractOne_symm_inr_apply {n : ℕ} (i : Fin n.succ) (x : Fin n) :
 
 @[simp]
 lemma finExtractOne_symm_inl_apply {n : ℕ} (i : Fin n.succ) :
-    (finExtractOne i).symm (Sum.inl 0) = i := by
-  rfl
+    (finExtractOne i).symm (Sum.inl 0) = i := rfl
 
 lemma finExtractOne_apply_neq {n : ℕ} (i j : Fin (n + 1 + 1)) (hij : i ≠ j) :
     finExtractOne i j = Sum.inr (predAboveI i j) := by
   symm
   apply (Equiv.symm_apply_eq _).mp ?_
-  simp only [Nat.succ_eq_add_one, finExtractOne_symm_inr_apply]
-  exact succsAbove_predAboveI hij
+  simpa only [Nat.succ_eq_add_one, finExtractOne_symm_inr_apply] using succsAbove_predAboveI hij
 
 /-- Given an equivalence `Fin n.succ.succ ≃ Fin n.succ.succ`, and an `i : Fin n.succ.succ`,
   the map `Fin n.succ → Fin n.succ` obtained by dropping `i` and it's image. -/
@@ -200,8 +194,7 @@ def finExtractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
 
 @[simp]
 lemma finExtractTwo_apply_fst {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
-    finExtractTwo i j i = Sum.inl (Sum.inl 0) := by
-  simp [finExtractTwo]
+    finExtractTwo i j i = Sum.inl (Sum.inl 0) := by simp [finExtractTwo]
 
 lemma finExtractTwo_symm_inr {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
     (finExtractTwo i j).symm ∘ Sum.inr = i.succAbove ∘ j.succAbove := by
@@ -210,13 +203,11 @@ lemma finExtractTwo_symm_inr {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
 
 @[simp]
 lemma finExtractTwo_symm_inr_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) (x : Fin n) :
-    (finExtractTwo i j).symm (Sum.inr x) = i.succAbove (j.succAbove x) := by
-  simp [finExtractTwo]
+    (finExtractTwo i j).symm (Sum.inr x) = i.succAbove (j.succAbove x) := by simp [finExtractTwo]
 
 @[simp]
 lemma finExtractTwo_symm_inl_inr_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
-    (finExtractTwo i j).symm (Sum.inl (Sum.inr 0)) = i.succAbove j := by
-  simp [finExtractTwo]
+    (finExtractTwo i j).symm (Sum.inl (Sum.inr 0)) = i.succAbove j := by simp [finExtractTwo]
 
 @[simp]
 lemma finExtractTwo_symm_inl_inl_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
@@ -224,8 +215,7 @@ lemma finExtractTwo_symm_inl_inl_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin
 
 @[simp]
 lemma finExtractTwo_apply_snd {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
-    finExtractTwo i j (i.succAbove j) = Sum.inl (Sum.inr 0) := by
-  simp [← Equiv.eq_symm_apply]
+    finExtractTwo i j (i.succAbove j) = Sum.inl (Sum.inr 0) := by simp [← Equiv.eq_symm_apply]
 
 /-- Takes two maps `Fin n → Fin n` and returns the equivalence they form. -/
 def finMapToEquiv (f1 : Fin n → Fin m) (f2 : Fin m → Fin n)
diff --git a/Physlib/Mathematics/Fin/Involutions.lean b/Physlib/Mathematics/Fin/Involutions.lean
index c65067e45..cd47697fb 100644
--- a/Physlib/Mathematics/Fin/Involutions.lean
+++ b/Physlib/Mathematics/Fin/Involutions.lean
@@ -155,7 +155,6 @@ lemma involutionCons_ext {n : ℕ} {f1 f2 : (f : {f : Fin n → Fin n // Functio
   obtain ⟨val, property⟩ := fst
   obtain ⟨val_1, property_1⟩ := snd
   obtain ⟨val_2, property_2⟩ := snd_1
-  simp_all only
   rfl
 
 /-- Given an involution of `Fin n`, the optional choice of an element in `Fin n` which
@@ -201,8 +200,7 @@ lemma involutionAddEquiv_cast {n : ℕ} {f1 f2 : {f : Fin n → Fin n // Functio
     involutionAddEquiv f1 = (Equiv.subtypeEquivRight (by rw [hf]; simp)).trans
       ((involutionAddEquiv f2).trans (Equiv.optionCongr (finCongr (by rw [hf])))) := by
   subst hf
-  rw [finCongr_refl, Equiv.optionCongr_refl]
-  rfl
+  simp [Equiv.subtypeEquivRight]
 
 lemma involutionAddEquiv_cast' {m : ℕ} {f1 f2 : {f : Fin m → Fin m // Function.Involutive f}}
     {N : ℕ} (hf : f1 = f2) (n : Option (Fin N))
@@ -341,8 +339,7 @@ def involutionNoFixedSetOne {n : ℕ} :
       simp [f.2.1 i.succ.succ]
     · intro i
       simp only [succ_eq_add_one, ne_eq, Function.comp_apply, f''', f'', f']
-      rw [Fin.pred_eq_iff_eq_succ, Fin.pred_eq_iff_eq_succ]
-      exact f.2.2.1 i.succ.succ
+      simpa [Fin.pred_eq_iff_eq_succ, Fin.pred_eq_iff_eq_succ] using f.2.2.1 i.succ.succ
   invFun f := by
     let f' := fun (i : Fin n.succ.succ)=>
       match i with
diff --git a/Physlib/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean b/Physlib/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
index a6bf51cf6..0d88c7f14 100644
--- a/Physlib/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
+++ b/Physlib/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean
@@ -120,9 +120,7 @@ lemma neg_weight_implies_neg_value {E : Type*} [AddCommGroup E] [Module ℝ E]
     intro h
     have : f v = f 0 := by rw [h]
     have : f (f.symm v_std) = f 0 := by rw [← this]
-    have : v_std = 0 := by
-      rw [← f.apply_symm_apply v_std]
-      exact Eq.trans this (map_zero f)
+    have : v_std = 0 := by simpa using this
     have : v_std i = 0 := by rw [this]; rfl
     simp only [↓reduceIte, one_ne_zero, v_std] at this
   have hq_neg : q v < 0 := by
@@ -195,8 +193,7 @@ def pseudoRiemannianMetricValToQuadraticForm
     (symm : ∀ (x : M) (v w : TangentSpace I x), (val x v) w = (val x w) v)
     (x : M) : QuadraticForm ℝ (TangentSpace I x) where
   toFun v := val x v v
-  toFun_smul a v := by
-    simp only [ContinuousLinearMap.map_smul, _root_.smul_apply, smul_smul]
+  toFun_smul a v := by simp only [ContinuousLinearMap.map_smul, _root_.smul_apply, smul_smul]
   exists_companion' :=
       ⟨LinearMap.mk₂ ℝ (fun v y => val x v y + val x y v)
         (fun v₁ v₂ y => by simp only [map_add, add_apply]; ring)
@@ -486,16 +483,14 @@ lemma sharp_flat_apply
 lemma apply_sharp_sharp
     (g : PseudoRiemannianMetric E H M n I) (x : M) (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
     g.val x (g.sharpL x ω₁) (g.sharpL x ω₂) = ω₁ (g.sharpL x ω₂) := by
-  rw [← flatL_apply g x (g.sharpL x ω₁)]
-  rw [flatL_apply_sharpL g x ω₁]
+  rw [← flatL_apply g x (g.sharpL x ω₁), flatL_apply_sharpL g x ω₁]
 
 /-- The metric evaluated at `v` and `sharp ω`. -/
 lemma apply_vec_sharp
     (g : PseudoRiemannianMetric E H M n I) (x : M) (v : TangentSpace I x)
     (ω : TangentSpace I x →L[ℝ] ℝ) :
     g.val x v (g.sharpL x ω) = ω v := by
-  rw [g.symm x v (g.sharpL x ω)]
-  rw [← flatL_apply g x (g.sharpL x ω)]
+  rw [g.symm x v (g.sharpL x ω), ← flatL_apply g x (g.sharpL x ω)]
   rw [flatL_apply_sharpL g x ω]
 
 end Sharp
@@ -518,8 +513,7 @@ noncomputable def cotangentMetricVal (g : PseudoRiemannianMetric E H M n I) (x :
 @[simp]
 lemma cotangentMetricVal_eq_apply_sharp (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
-  cotangentMetricVal g x ω₁ ω₂ = ω₁ (g.sharpL x ω₂) := by
-  rw [cotangentMetricVal, apply_sharp_sharp]
+  cotangentMetricVal g x ω₁ ω₂ = ω₁ (g.sharpL x ω₂) := by rw [cotangentMetricVal, apply_sharp_sharp]
 
 lemma cotangentMetricVal_symm (g : PseudoRiemannianMetric E H M n I) (x : M)
     (ω₁ ω₂ : TangentSpace I x →L[ℝ] ℝ) :
@@ -606,8 +600,7 @@ lemma cotangentMetricVal_nondegenerate (g : PseudoRiemannianMetric E H M n I) (x
   have h_forall : ∀ w : TangentSpace I x, ω w = 0 := by
     intro w
     let ω' : TangentSpace I x →L[ℝ] ℝ := g.flatL x w
-    have this : g.sharpL x ω' = w := by
-      simp only [ω', sharpL_apply_flatL]
+    have this : g.sharpL x ω' = w := by simp only [ω', sharpL_apply_flatL]
     have h_apply : cotangentMetricVal g x ω ω' = 0 := h ω'
     simp only [cotangentMetricVal_eq_apply_sharp] at h_apply
     rw [this] at h_apply
diff --git a/Physlib/Mathematics/Geometry/Metric/Riemannian/Defs.lean b/Physlib/Mathematics/Geometry/Metric/Riemannian/Defs.lean
index 0e46651d2..dec2bae22 100644
--- a/Physlib/Mathematics/Geometry/Metric/Riemannian/Defs.lean
+++ b/Physlib/Mathematics/Geometry/Metric/Riemannian/Defs.lean
@@ -92,9 +92,8 @@ lemma toQuadraticForm_posDef (g : RiemannianMetric I n M) (x : M) :
   λ v hv => g.pos_def x v hv
 
 lemma riemannian_metric_negDim_zero (g : RiemannianMetric I n M) (x : M) :
-    (g.toQuadraticForm x).negDim = 0 := by
-  apply QuadraticForm.rankNeg_eq_zero
-  exact g.toQuadraticForm_posDef x
+    (g.toQuadraticForm x).negDim = 0 :=
+  QuadraticForm.rankNeg_eq_zero (g.toQuadraticForm_posDef x)
 
 /-! ## InnerProductSpace structure from RiemannianMetric -/
 
@@ -111,8 +110,7 @@ noncomputable def tangentInnerCore (g : RiemannianMetric I n M) (x : M) :
     InnerProductSpace.Core ℝ (TangentSpace I x) where
   inner := λ v w => g.inner x v w
   conj_inner_symm := λ v w => by
-    simp only [inner_apply, conj_trivial]
-    exact g.toPseudoRiemannianMetric.symm x w v
+    simpa only [inner_apply, conj_trivial] using g.toPseudoRiemannianMetric.symm x w v
   re_inner_nonneg := λ v => by
     simp only [inner_apply, RCLike.re_to_real]
     by_cases hv : v = 0
@@ -121,8 +119,7 @@ noncomputable def tangentInnerCore (g : RiemannianMetric I n M) (x : M) :
   add_left := λ u v w => by
     simp only [inner_apply, map_add, add_apply]
   smul_left := λ r u v => by
-    simp only [inner_apply, map_smul, conj_trivial]
-    rfl
+    simp [inner_apply]
   definite := fun v (h_inner_zero : g.inner x v v = 0) => by
     by_contra h_v_ne_zero
     have h_pos : g.inner x v v > 0 := g.pos_def x v h_v_ne_zero
@@ -170,9 +167,7 @@ example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) :
 
 -- Example showing how to use the metric inner product space
 example (g : RiemannianMetric I n M) (x : M) (v w : TangentSpace I x) :
-    (TangentSpace.metricInnerProductSpace g x).inner v w = g.inner x v w := by
-  letI := TangentSpace.metricInnerProductSpace g x
-  rfl
+    (TangentSpace.metricInnerProductSpace g x).inner v w = g.inner x v w := rfl
 
 /-- Helper function to compute the norm on a tangent space from a Riemannian metric,
     using the underlying `NormedAddCommGroup` structure. -/
@@ -196,12 +191,7 @@ example (g : RiemannianMetric I n M) (x : M) (v : TangentSpace I x) : ℝ :=
 
 lemma norm_eq_norm_of_metricNormedAddCommGroup (g : RiemannianMetric I n M) (x : M)
     (v : TangentSpace I x) : norm g x v = @Norm.norm (TangentSpace I x)
-    (@NormedAddCommGroup.toNorm _ (TangentSpace.metricNormedAddCommGroup g x)) v := by
-  unfold norm
-  let normed_group := TangentSpace.metricNormedAddCommGroup g x
-  unfold TangentSpace.metricNormedAddCommGroup
-  simp only [inner_apply]
-  rfl
+    (@NormedAddCommGroup.toNorm _ (TangentSpace.metricNormedAddCommGroup g x)) v := rfl
 
 end InnerProductSpace
 
diff --git a/Physlib/Mathematics/InnerProductSpace/Adjoint.lean b/Physlib/Mathematics/InnerProductSpace/Adjoint.lean
index 9161b1fb2..ddfdfd957 100644
--- a/Physlib/Mathematics/InnerProductSpace/Adjoint.lean
+++ b/Physlib/Mathematics/InnerProductSpace/Adjoint.lean
@@ -49,8 +49,7 @@ def adjoint (f : E → F) :=
 
 lemma HasAdjoint.adjoint_inner_right {f : E → F} (hf : HasAdjoint 𝕜 f f') :
     ⟪x, f' y⟫ = ⟪f x, y⟫ := by
-  rw [← inner_conj_symm']
-  rw [hf.adjoint_inner_left]
+  rw [← inner_conj_symm', hf.adjoint_inner_left]
   rw [inner_conj_symm']
 
 open InnerProductSpace' in
@@ -67,8 +66,7 @@ lemma adjoint_eq_clm_adjoint [CompleteSpace E] [CompleteSpace F] (f : E →L[
   simp[_root_.adjoint,h,h.choose_spec.adjoint_inner_left]
 
 lemma HasAdjoint.adjoint_apply_zero {f : E → F} {f' : F → E}
-    (hf : HasAdjoint 𝕜 f f') : f' 0 = 0 := by
-  simpa using hf.adjoint_inner_left (f' 0) 0
+    (hf : HasAdjoint 𝕜 f f') : f' 0 = 0 := by simpa using hf.adjoint_inner_left (f' 0) 0
 
 lemma HasAdjoint.adjoint
     {f : E → F} {f' : F → E}
diff --git a/Physlib/Mathematics/InnerProductSpace/Basic.lean b/Physlib/Mathematics/InnerProductSpace/Basic.lean
index f6d0b35ec..034bbfbc6 100644
--- a/Physlib/Mathematics/InnerProductSpace/Basic.lean
+++ b/Physlib/Mathematics/InnerProductSpace/Basic.lean
@@ -512,9 +512,7 @@ instance {ι : Type*} [Fintype ι] : InnerProductSpace' 𝕜 (ι → E) where
           rw [hi]
           use i
           simp
-      have hj : ∀ j, ‖x j‖ ≤ ‖x i‖ := by
-        rw [← hi]
-        exact fun j => norm_le_pi_norm x j
+      have hj : ∀ j, ‖x j‖ ≤ ‖x i‖ := by simpa [← hi] using fun j => norm_le_pi_norm x j
       rw [hi]
       constructor
       · apply le_trans (h (x i)).1
@@ -576,10 +574,8 @@ lemma _root_.isBoundedBilinearMap_inner' :
       apply h1.trans
       apply le_of_eq
       congr
-      · rw [norm_withLp2_eq_norm2]
-        rfl
-      · rw [norm_withLp2_eq_norm2]
-        rfl
+      · exact norm_withLp2_eq_norm2 _
+      · exact norm_withLp2_eq_norm2 _
     · have key (z : E) : |‖z‖₂| ≤ √ d * ‖z‖ := by
         apply le_of_sq_le_sq
         · simp [@mul_pow]
diff --git a/Physlib/Mathematics/InnerProductSpace/Calculus.lean b/Physlib/Mathematics/InnerProductSpace/Calculus.lean
index e5b79dc10..2b3b92ff0 100644
--- a/Physlib/Mathematics/InnerProductSpace/Calculus.lean
+++ b/Physlib/Mathematics/InnerProductSpace/Calculus.lean
@@ -35,8 +35,7 @@ lemma fderiv_inner_apply'
     {f g : E → F} {x : E}
     (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : E) :
     fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ := by
-  rw [(hf.hasFDerivAt.inner' hg.hasFDerivAt).fderiv]
-  rfl
+  simpa [fderivInnerCLM'] using DFunLike.congr_fun (hf.hasFDerivAt.inner' hg.hasFDerivAt).fderiv y
 
 -- todo: move this
 lemma deriv_inner_apply'
@@ -49,6 +48,5 @@ lemma deriv_inner_apply'
 @[fun_prop]
 lemma DifferentiableAt.inner' {f g : E → F} {x}
     (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
-    DifferentiableAt ℝ (fun x => ⟪f x, g x⟫) x := by
-  apply HasFDerivAt.differentiableAt
-  exact hf.hasFDerivAt.inner' hg.hasFDerivAt
+    DifferentiableAt ℝ (fun x => ⟪f x, g x⟫) x :=
+  (hf.hasFDerivAt.inner' hg.hasFDerivAt).differentiableAt
diff --git a/Physlib/Mathematics/InnerProductSpace/Submodule.lean b/Physlib/Mathematics/InnerProductSpace/Submodule.lean
index afeb98227..d0189e440 100644
--- a/Physlib/Mathematics/InnerProductSpace/Submodule.lean
+++ b/Physlib/Mathematics/InnerProductSpace/Submodule.lean
@@ -32,13 +32,9 @@ variable
 /-- The submodule of `WithLp 2 (E × F)` defined by `M`. -/
 def submoduleToLp : Submodule ℂ (WithLp 2 (E × F)) where
   carrier := {x | x.ofLp ∈ M}
-  add_mem' := by
-    intro a b ha hb
-    exact Submodule.add_mem M ha hb
+  add_mem' := fun {_ _} ha hb => Submodule.add_mem M ha hb
   zero_mem' := Submodule.zero_mem M
-  smul_mem' := by
-    intro c x hx
-    exact Submodule.smul_mem M c hx
+  smul_mem' := fun c {_} hx => Submodule.smul_mem M c hx
 
 lemma mem_submodule_iff_mem_submoduleToLp : f ∈ M ↔ (WithLp.toLp 2 f) ∈ submoduleToLp M :=
   Eq.to_iff rfl
@@ -75,8 +71,7 @@ lemma submoduleToLp_closure :
 
 lemma mem_submodule_closure_iff_mem_submoduleToLp_closure :
     f ∈ M.topologicalClosure ↔ (WithLp.toLp 2 f) ∈ (submoduleToLp M).topologicalClosure := by
-  rw [← submoduleToLp_closure]
-  rfl
+  simpa [submoduleToLp_closure] using mem_submodule_iff_mem_submoduleToLp M.topologicalClosure f
 
 lemma mem_submodule_adjoint_iff_mem_submoduleToLp_orthogonal :
     g ∈ M.adjoint ↔ WithLp.toLp 2 (g.2, -g.1) ∈ (submoduleToLp M)ᗮ := by
@@ -84,9 +79,7 @@ lemma mem_submodule_adjoint_iff_mem_submoduleToLp_orthogonal :
   · rw [mem_orthogonal]
     intro u hu
     rw [mem_adjoint_iff] at h
-    have h' : inner ℂ u.snd g.1 = inner ℂ u.fst g.2 := by
-      rw [← sub_eq_zero]
-      exact h u.fst u.snd hu
+    have h' : inner ℂ u.snd g.1 = inner ℂ u.fst g.2 := by simpa [sub_eq_zero] using h u.fst u.snd hu
     simp [h']
   · rw [mem_adjoint_iff]
     intro a b hab
diff --git a/Physlib/Mathematics/KroneckerDelta.lean b/Physlib/Mathematics/KroneckerDelta.lean
index 36d2933db..2ddde0146 100644
--- a/Physlib/Mathematics/KroneckerDelta.lean
+++ b/Physlib/Mathematics/KroneckerDelta.lean
@@ -109,20 +109,17 @@ open Finset
 variable [AddCommMonoid M]
 
 @[simp]
-lemma sum_mul [Fintype α] (i j : α) : ∑ k : α, δ[i,k] * δ[k,j] = δ[i,j] := by
-  simp [kroneckerDelta]
+lemma sum_mul [Fintype α] (i j : α) : ∑ k : α, δ[i,k] * δ[k,j] = δ[i,j] := by simp [kroneckerDelta]
 
 @[simp]
 lemma sum_smul [Fintype α] (i : α) (f : α → M) : ∑ j : α, δ[i,j] • f j = f i := by
   simp [kroneckerDelta]
 
 lemma sum_sum_smul_eq_zero [Fintype α] {f : α → α → M} (hf : ∀ i : α, f i i = 0) :
-    ∑ i : α, ∑ j : α, δ[i,j] • f i j = 0 := by
-  simp [sum_smul, hf, sum_const_zero]
+    ∑ i : α, ∑ j : α, δ[i,j] • f i j = 0 := by simp [sum_smul, hf, sum_const_zero]
 
 lemma finset_sum_smul (s : Finset α) (i : α) (f : α → M) :
-    ∑ j ∈ s, δ[i,j] • f j = if i ∈ s then f i else 0 := by
-  simp [kroneckerDelta]
+    ∑ j ∈ s, δ[i,j] • f j = if i ∈ s then f i else 0 := by simp [kroneckerDelta]
 
 lemma finset_sum_sum_smul_eq_zero {s s' : Finset α} {f : α → α → M}
     (hf : ∀ i ∈ s ∩ s', f i i = 0) : ∑ i ∈ s, ∑ j ∈ s', δ[i,j] • f i j = 0 := by
diff --git a/Physlib/Mathematics/LinearMaps.lean b/Physlib/Mathematics/LinearMaps.lean
index 35fbdd5b8..879a5489d 100644
--- a/Physlib/Mathematics/LinearMaps.lean
+++ b/Physlib/Mathematics/LinearMaps.lean
@@ -87,24 +87,19 @@ def mk₂ (f : V × V → ℚ) (map_smul : ∀ a S T, f (a • S, T) = a * f (S,
   map_add' := fun S1 S2 => LinearMap.ext fun T => map_add S1 S2 T
   swap' := swap
 
-lemma map_smul₁ (f : BiLinearSymm V) (a : ℚ) (S T : V) : f (a • S) T = a * f S T := by
-  have h : f (a • S) = a • (f S) := by
-    exact f.map_smul a S
-  simp [h]
+lemma map_smul₁ (f : BiLinearSymm V) (a : ℚ) (S T : V) : f (a • S) T = a * f S T :=
+  (congrArg (fun g : V →ₗ[ℚ] ℚ => g T) (f.map_smul a S)).trans (by rfl)
 
 lemma swap (f : BiLinearSymm V) (S T : V) : f S T = f T S := f.swap' S T
 
 lemma map_smul₂ (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : f S (a • T) = a * f S T := by
   rw [f.swap, f.map_smul₁, f.swap]
 
-lemma map_add₁ (f : BiLinearSymm V) (S1 S2 T : V) : f (S1 + S2) T = f S1 T + f S2 T := by
-  have h : f (S1 + S2) = f S1 + f S2 := by
-    exact f.map_add S1 S2
-  simp [h]
+lemma map_add₁ (f : BiLinearSymm V) (S1 S2 T : V) : f (S1 + S2) T = f S1 T + f S2 T :=
+  (congrArg (fun g : V →ₗ[ℚ] ℚ => g T) (f.map_add S1 S2)).trans (by rfl)
 
 lemma map_add₂ (f : BiLinearSymm V) (S : V) (T1 T2 : V) :
-    f S (T1 + T2) = f S T1 + f S T2 := by
-  rw [f.swap, f.map_add₁, f.swap T1 S, f.swap T2 S]
+    f S (T1 + T2) = f S T1 + f S T2 := by rw [f.swap, f.map_add₁, f.swap T1 S, f.swap T2 S]
 
 /-- Fixing the second input vectors, the resulting linear map. -/
 def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
@@ -115,8 +110,7 @@ def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
 lemma toLinear₁_apply (f : BiLinearSymm V) (S T : V) : f S T = f.toLinear₁ T S := rfl
 
 lemma map_sum₁ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
-    f (∑ i, S i) T = ∑ i, f (S i) T := by
-  simp [f.toLinear₁_apply, map_sum]
+    f (∑ i, S i) T = ∑ i, f (S i) T := by simp [f.toLinear₁_apply, map_sum]
 
 lemma map_sum₂ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
     f T (∑ i, S i) = ∑ i, f T (S i) := map_sum (f T) S Finset.univ
@@ -211,47 +205,35 @@ lemma swap₃ (f : TriLinearSymm V) (S T L : V) : f S T L = f L T S := by
   rw [f.swap₁, f.swap₂, f.swap₁]
 
 lemma map_smul₁ (f : TriLinearSymm V) (a : ℚ) (S T L : V) :
-    f (a • S) T L = a * f S T L := by
-  have h : f (a • S) = a • (f S) := by
-    exact f.map_smul a S
-  simp [h]
+    f (a • S) T L = a * f S T L :=
+  (congrArg (fun g : V →ₗ[ℚ] V →ₗ[ℚ] ℚ => g T L) (f.map_smul a S)).trans rfl
 
 lemma map_smul₂ (f : TriLinearSymm V) (S : V) (a : ℚ) (T L : V) :
-    f S (a • T) L = a * f S T L := by
-  rw [f.swap₁, f.map_smul₁, f.swap₁]
+    f S (a • T) L = a * f S T L := by rw [f.swap₁, f.map_smul₁, f.swap₁]
 
 lemma map_smul₃ (f : TriLinearSymm V) (S T : V) (a : ℚ) (L : V) :
-    f S T (a • L) = a * f S T L := by
-  rw [f.swap₃, f.map_smul₁, f.swap₃]
+    f S T (a • L) = a * f S T L := by rw [f.swap₃, f.map_smul₁, f.swap₃]
 
 lemma map_add₁ (f : TriLinearSymm V) (S1 S2 T L : V) :
-    f (S1 + S2) T L = f S1 T L + f S2 T L := by
-  have h : f (S1 + S2) = f S1 + f S2 := by
-    exact f.map_add S1 S2
-  simp [h]
+    f (S1 + S2) T L = f S1 T L + f S2 T L :=
+  (congrArg (fun g : V →ₗ[ℚ] V →ₗ[ℚ] ℚ => g T L) (f.map_add S1 S2)).trans rfl
 
 lemma map_add₂ (f : TriLinearSymm V) (S T1 T2 L : V) :
-    f S (T1 + T2) L = f S T1 L + f S T2 L := by
-  rw [f.swap₁, f.map_add₁, f.swap₁ S T1, f.swap₁ S T2]
+    f S (T1 + T2) L = f S T1 L + f S T2 L := by rw [f.swap₁, f.map_add₁, f.swap₁ S T1, f.swap₁ S T2]
 
 lemma map_add₃ (f : TriLinearSymm V) (S T L1 L2 : V) :
-    f S T (L1 + L2) = f S T L1 + f S T L2 := by
-  rw [f.swap₃, f.map_add₁, f.swap₃, f.swap₃ L2 T S]
+    f S T (L1 + L2) = f S T L1 + f S T L2 := by rw [f.swap₃, f.map_add₁, f.swap₃, f.swap₃ L2 T S]
 
 /-- Fixing the second and third input vectors, the resulting linear map. -/
 def toLinear₁ (f : TriLinearSymm V) (T L : V) : V →ₗ[ℚ] ℚ where
   toFun S := f S T L
   map_add' S1 S2 := map_add₁ f S1 S2 T L
-  map_smul' a S := by
-    simp only [f.map_smul₁]
-    rfl
+  map_smul' a S := by simp [f.map_smul₁]
 
 lemma toLinear₁_apply (f : TriLinearSymm V) (S T L : V) : f S T L = f.toLinear₁ T L S := rfl
 
 lemma map_sum₁ {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) :
-    f (∑ i, S i) T L = ∑ i, f (S i) T L := by
-  rw [f.toLinear₁_apply, map_sum]
-  rfl
+    f (∑ i, S i) T L = ∑ i, f (S i) T L := by simp [toLinear₁_apply, map_sum]
 
 lemma map_sum₂ {n : ℕ} (f : TriLinearSymm V) (S : Fin n → V) (T : V) (L : V) :
     f T (∑ i, S i) L = ∑ i, f T (S i) L := by
diff --git a/Physlib/Mathematics/LinearPMap.lean b/Physlib/Mathematics/LinearPMap.lean
index 5c7bf3ee0..bc4879fd7 100644
--- a/Physlib/Mathematics/LinearPMap.lean
+++ b/Physlib/Mathematics/LinearPMap.lean
@@ -198,8 +198,7 @@ lemma compRestricted_domain : (g ∘ᵣ f).domain = (g.domain.comap f.toFun).map
   exact h.choose
 
 lemma mem_compRestricted_domain_iff {x : E} :
-    x ∈ (v ∘ᵣ u).domain ↔ ∃ h : x ∈ u.domain, u ⟨x, h⟩ ∈ v.domain := by
-  simp [compRestricted_domain]
+    x ∈ (v ∘ᵣ u).domain ↔ ∃ h : x ∈ u.domain, u ⟨x, h⟩ ∈ v.domain := by simp [compRestricted_domain]
 
 lemma mem_compRestricted_domain_iff' {x : E} :
     x ∈ (v ∘ᵣ u).domain ↔ ∃ y : u.domain, x = y ∧ ∃ y' : v.domain, u y = y' := by
diff --git a/Physlib/Mathematics/List.lean b/Physlib/Mathematics/List.lean
index d2c0fa4e4..f257eb8a7 100644
--- a/Physlib/Mathematics/List.lean
+++ b/Physlib/Mathematics/List.lean
@@ -23,8 +23,7 @@ variable {n : Nat}
 lemma takeWile_eraseIdx {I : Type} (P : I → Prop) [DecidablePred P] :
     (l : List I) → (i : ℕ) → (hi : ∀ (i j : Fin l.length), i < j → P (l.get j) → P (l.get i)) →
     List.takeWhile P (List.eraseIdx l i) = (List.takeWhile P l).eraseIdx i
-  | [], _, h => by
-    rfl
+  | [], _, h => rfl
   | a :: [], 0, h => by
     simp only [List.takeWhile, List.eraseIdx_zero]
     split <;> rfl
@@ -34,8 +33,7 @@ lemma takeWile_eraseIdx {I : Type} (P : I → Prop) [DecidablePred P] :
   | a :: b :: l, 0, h => by
     simp only [List.takeWhile, List.eraseIdx_zero]
     by_cases hPb : P b
-    · have hPa : P a := by
-        simpa using h ⟨0, by simp⟩ ⟨1, by simp⟩ (by simp) (by simpa using hPb)
+    · have hPa : P a := by simpa using h ⟨0, by simp⟩ ⟨1, by simp⟩ (by simp) (by simpa using hPb)
       simp [hPb, hPa]
     · simp [hPb]
       split <;> rfl
@@ -62,15 +60,13 @@ lemma dropWile_eraseIdx {I : Type} (P : I → Prop) [DecidablePred P] :
     by_cases hPa : P a <;> simp [hPa, List.eraseIdx_of_length_le]
   | a :: b :: l, 0, h => by
     by_cases hPb : P b
-    · have hPa : P a := by
-        simpa using h ⟨0, by simp⟩ ⟨1, by simp⟩ (by simp) (by simpa using hPb)
+    · have hPa : P a := by simpa using h ⟨0, by simp⟩ ⟨1, by simp⟩ (by simp) (by simpa using hPb)
       simp [hPb, hPa]
     · by_cases hPa : P a <;> simp [hPa, hPb]
   | a :: b :: l, Nat.succ n, h => by
     simp only [Nat.succ_eq_add_one, List.eraseIdx_cons_succ]
     by_cases hPb : P b
-    · have hPa : P a := by
-        simpa using h ⟨0, by simp⟩ ⟨1, by simp⟩ (by simp) (by simpa using hPb)
+    · have hPa : P a := by simpa using h ⟨0, by simp⟩ ⟨1, by simp⟩ (by simp) (by simpa using hPb)
       simp only [List.dropWhile, hPa, decide_true, List.takeWhile, hPb, List.length_cons,
         add_le_add_iff_right, Nat.reduceSubDiff]
       rw [dropWile_eraseIdx P (b :: l) n (fun i j hij hP => by
@@ -176,10 +172,9 @@ lemma orderedInsertPos_take_eq_orderedInsert {I : Type} (le1 : I → I → Prop)
     exact Nat.le_of_lt_succ (orderedInsertPos_lt_length le1 r r0)
   · intro n h1 h2
     simp only [List.get_eq_getElem, List.getElem_take]
-    rw [orderedInsert_get_lt le1 r r0 n]
-    rfl
-    simp only [List.length_take, lt_inf_iff] at h1
-    exact h1.1
+    exact (orderedInsert_get_lt le1 r r0 n (by
+      simp only [List.length_take, lt_inf_iff] at h1
+      exact h1.1)).symm
 
 lemma orderedInsertPos_drop_eq_orderedInsert {I : Type} (le1 : I → I → Prop) [DecidableRel le1]
     (r : List I) (r0 : I) :
@@ -187,8 +182,7 @@ lemma orderedInsertPos_drop_eq_orderedInsert {I : Type} (le1 : I → I → Prop)
     List.drop (orderedInsertPos le1 r r0).succ (List.orderedInsert le1 r0 r) := by
   conv_rhs => simp [orderedInsertPos, List.orderedInsert_eq_take_drop]
   have hr : r = List.takeWhile (fun b => !decide (le1 r0 b)) r ++
-      List.dropWhile (fun b => !decide (le1 r0 b)) r := by
-    exact Eq.symm (List.takeWhile_append_dropWhile)
+      List.dropWhile (fun b => !decide (le1 r0 b)) r := Eq.symm (List.takeWhile_append_dropWhile)
   conv_lhs =>
     rhs
     rw [hr]
@@ -446,9 +440,7 @@ lemma orderedInsert_eraseIdx_orderedInsertEquiv_fin_succ
     (hr : ∀ (i j : Fin r.length), i < j → ¬le1 r0 (r.get j) → ¬le1 r0 (r.get i)) :
     (List.orderedInsert le1 r0 r).eraseIdx (orderedInsertEquiv le1 r r0 n.succ) =
     (List.orderedInsert le1 r0 (r.eraseIdx n)) := by
-  have hn : n.succ = ⟨n.val + 1, by omega⟩ := by
-    rw [Fin.ext_iff]
-    rfl
+  have hn : n.succ = ⟨n.val + 1, by omega⟩ := Fin.ext rfl
   rw [hn]
   exact orderedInsert_eraseIdx_orderedInsertEquiv_succ le1 r r0 n.val _ hr
 
@@ -608,8 +600,7 @@ lemma insertionSortEquiv_order {α : Type} {r : α → α → Prop} [DecidableRe
   | some i => List.eraseIdx l i
 
 lemma eraseIdx_length' {I : Type} (l : List I) (i : Fin l.length) :
-    (List.eraseIdx l i).length = l.length - 1 := by
-  simp [List.length_eraseIdx]
+    (List.eraseIdx l i).length = l.length - 1 := by simp [List.length_eraseIdx]
 
 lemma eraseIdx_length {I : Type} (l : List I) (i : Fin l.length) :
     (List.eraseIdx l i).length + 1 = l.length := by
@@ -624,8 +615,7 @@ lemma eraseIdx_length_succ {I : Type} (l : List I) (i : Fin l.length) :
   omega
 
 lemma eraseIdx_cons_length {I : Type} (a : I) (l : List I) (i : Fin (a :: l).length) :
-    (List.eraseIdx (a :: l) i).length= l.length := by
-  simp [List.length_eraseIdx]
+    (List.eraseIdx (a :: l) i).length= l.length := by simp [List.length_eraseIdx]
 
 lemma eraseIdx_get {I : Type} (l : List I) (i : Fin l.length) :
     (List.eraseIdx l i).get = l.get ∘ (Fin.cast (eraseIdx_length l i)) ∘
diff --git a/Physlib/Mathematics/List/InsertIdx.lean b/Physlib/Mathematics/List/InsertIdx.lean
index b9f4ae4fc..044eb9302 100644
--- a/Physlib/Mathematics/List/InsertIdx.lean
+++ b/Physlib/Mathematics/List/InsertIdx.lean
@@ -42,8 +42,7 @@ lemma eraseIdx_sorted {I : Type} (le : I → I → Prop) :
     simp only [List.pairwise_cons] at h
     refine And.intro ?_ (eraseIdx_sorted le as n h.2)
     intro b hb
-    refine h.1 _ ?_
-    exact List.mem_of_mem_eraseIdx hb
+    exact h.1 _ (List.mem_of_mem_eraseIdx hb)
 
 lemma mem_eraseIdx_nodup {I : Type} (i : I) :
     (l : List I) → (n : ℕ) → (hn : n < l.length) → (h : List.Nodup l) →
diff --git a/Physlib/Mathematics/List/InsertionSort.lean b/Physlib/Mathematics/List/InsertionSort.lean
index 7d2b893f7..85485eebd 100644
--- a/Physlib/Mathematics/List/InsertionSort.lean
+++ b/Physlib/Mathematics/List/InsertionSort.lean
@@ -73,9 +73,7 @@ lemma insertionSortMin_lt_mem_insertionSortDropMinPos_of_lt {α : Type} (r : α
     trans (insertionSortDropMinPos r a l).get i
     simp only [Fin.getElem_fin, List.get_eq_getElem]
     simp only [insertionSortDropMinPos, List.length_cons, Nat.succ_eq_add_one, finCongr_apply]
-    rw [eraseIdx_get]
-    simp only [List.length_cons, Function.comp_apply, List.get_eq_getElem, Fin.val_cast]
-    rfl
+    exact congr_fun (eraseIdx_get (a :: l) (insertionSortMinPos r a l)) i
   erw [h1]
   simp only [List.length_cons, Nat.succ_eq_add_one, List.get_eq_getElem]
   apply insertionSortEquiv_order
@@ -84,16 +82,14 @@ lemma insertionSortMin_lt_mem_insertionSortDropMinPos_of_lt {α : Type} (r : α
   apply lt_of_eq_of_lt (insertionSortMinPos_insertionSortEquiv r a l)
   apply insertionSortEquiv_gt_zero_of_ne_insertionSortMinPos r a l
   simp only [List.length_cons, ne_eq, Fin.ext_iff, Fin.val_cast]
-  have hl : (insertionSortMinPos r a l).val = (insertionSortMinPosFin r a l).val := by
-    rfl
+  have hl : (insertionSortMinPos r a l).val = (insertionSortMinPosFin r a l).val := rfl
   simp only [hl, Nat.succ_eq_add_one, Fin.val_eq_val, ne_eq]
   exact Fin.succAbove_ne (insertionSortMinPosFin r a l) i
 
 lemma insertionSort_insertionSort {α : Type} (r : α → α → Prop) [DecidableRel r]
     [Std.Total r] [IsTrans α r] (l1 : List α) :
-    List.insertionSort r (List.insertionSort r l1) = List.insertionSort r l1 := by
-  apply List.Pairwise.insertionSort_eq
-  exact List.pairwise_insertionSort r l1
+    List.insertionSort r (List.insertionSort r l1) = List.insertionSort r l1 :=
+  List.Pairwise.insertionSort_eq (List.pairwise_insertionSort r l1)
 
 lemma orderedInsert_commute {α : Type} (r : α → α → Prop) [DecidableRel r]
     [Std.Total r] [IsTrans α r] (a b : α) (hr : ¬ r a b) : (l : List α) →
@@ -154,9 +150,8 @@ lemma insertionSort_append_insertionSort_append {α : Type} (r : α → α → P
 
 @[simp]
 lemma orderedInsert_length {α : Type} (r : α → α → Prop) [DecidableRel r] (a : α) (l : List α) :
-    (List.orderedInsert r a l).length = (a :: l).length := by
-  apply List.Perm.length_eq
-  exact List.perm_orderedInsert r a l
+    (List.orderedInsert r a l).length = (a :: l).length :=
+  List.Perm.length_eq (List.perm_orderedInsert r a l)
 
 lemma takeWhile_orderedInsert {α : Type} (r : α → α → Prop) [DecidableRel r]
     [Std.Total r] [IsTrans α r]
@@ -242,8 +237,7 @@ lemma insertionSortEquiv_commute {α : Type} (r : α → α → Prop) [Decidable
       funext c
       simp only [Bool.eq_self_and, Bool.not_eq_eq_eq_not, Bool.not_true, decide_eq_false_iff_not]
       intro hbc hac
-      refine hbc ?_
-      exact IsTrans.trans _ _ _ hrba hac
+      exact hbc (IsTrans.trans _ _ _ hrba hac)
   have ha1 : b1.1 ≤ a2.1 := by
     simp only [orderedInsertPos, decide_not, b1]
     rw [ht]
@@ -289,8 +283,7 @@ lemma insertionSortEquiv_orderedInsert_append {α : Type} (r : α → α → Pro
       rw [insertionSortEquiv_congr _ _ h1]
       simp
     · have h1 : (List.orderedInsert r a (b :: l1) ++ a2 :: l2) =
-        (b :: List.orderedInsert r a (l1) ++ a2 :: l2) := by
-        simp [h]
+        (b :: List.orderedInsert r a (l1) ++ a2 :: l2) := by simp [h]
       rw [insertionSortEquiv_congr _ _ h1]
       simp only [List.cons_append, List.length_cons,
         Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.cast_mk,
@@ -313,7 +306,6 @@ lemma insertionSortEquiv_orderedInsert_append {α : Type} (r : α → α → Pro
       simp only [List.insertionSort, List.length_cons, hl]
       conv_rhs =>
         erw [insertionSortEquiv_commute _ _ _ h _ _]
-      simp
       rfl
 
 set_option backward.isDefEq.respectTransparency false in
@@ -335,10 +327,8 @@ lemma insertionSortEquiv_insertionSort_append {α : Type} (r : α → α → Pro
       equivCons_succ]
     erw [ih]
     have hl : (List.insertionSort r (List.insertionSort r l1 ++ a :: l2)) =
-        (List.insertionSort r (l1 ++ a :: l2)) := by
-      exact insertionSort_insertionSort_append r l1 (a :: l2)
+        (List.insertionSort r (l1 ++ a :: l2)) := insertionSort_insertionSort_append r l1 (a :: l2)
     erw [orderedInsertEquiv_congr _ _ _ hl]
-    simp only [List.foldr_cons, finCongr_apply]
     rfl
 
 /-!
@@ -360,11 +350,9 @@ lemma orderedInsert_filter_of_pos {α : Type} (r : α → α → Prop) [Decidabl
     by_cases hpb : p b <;> by_cases hab : r a b
     · simp only [hab, ↓reduceIte, hpb, decide_true, List.filter_cons_of_pos,
       List.orderedInsert.eq_2]
-      rw [List.filter_cons_of_pos (by simp [h])]
-      rw [List.filter_cons_of_pos (by simp [hpb])]
+      rw [List.filter_cons_of_pos (by simp [h]), List.filter_cons_of_pos (by simp [hpb])]
     · simp only [hab, ↓reduceIte]
-      rw [List.filter_cons_of_pos (by simp [hpb])]
-      rw [List.filter_cons_of_pos (by simp [hpb])]
+      rw [List.filter_cons_of_pos (by simp [hpb]), List.filter_cons_of_pos (by simp [hpb])]
       simp only [List.orderedInsert, hab, ↓reduceIte, List.cons.injEq, true_and]
       simp only [List.pairwise_cons] at hl
       exact orderedInsert_filter_of_pos r a p h l hl.2
@@ -383,8 +371,7 @@ lemma orderedInsert_filter_of_pos {α : Type} (r : α → α → Prop) [Decidabl
         simp only [List.pairwise_cons] at hl
         apply hl.1
         have hlf : (List.filter (fun b => decide (p b)) l)[0] ∈
-            (List.filter (fun b => decide (p b)) l) := by
-          exact List.getElem_mem c
+            (List.filter (fun b => decide (p b)) l) := List.getElem_mem c
         simp only [List.mem_filter, decide_eq_true_eq] at hlf
         exact hlf.1
       rw [hl]
@@ -394,8 +381,7 @@ lemma orderedInsert_filter_of_pos {α : Type} (r : α → α → Prop) [Decidabl
       rw [hl]
       simp
     · simp only [hab, ↓reduceIte]
-      rw [List.filter_cons_of_neg (by simp [hpb])]
-      rw [List.filter_cons_of_neg (by simp [hpb])]
+      rw [List.filter_cons_of_neg (by simp [hpb]), List.filter_cons_of_neg (by simp [hpb])]
       simp only [List.pairwise_cons] at hl
       exact orderedInsert_filter_of_pos r a p h l hl.2
 
@@ -404,8 +390,7 @@ lemma orderedInsert_filter_of_neg {α : Type} (r : α → α → Prop) [Decidabl
     List.filter p (List.orderedInsert r a l) = (List.filter p l) := by
   rw [List.orderedInsert_eq_take_drop]
   simp only [decide_not, List.filter_append]
-  rw [List.filter_cons_of_neg]
-  rw [← List.filter_append]
+  rw [List.filter_cons_of_neg, ← List.filter_append]
   congr
   exact List.takeWhile_append_dropWhile
   simp [h]
@@ -423,8 +408,7 @@ lemma insertionSort_filter {α : Type} (r : α → α → Prop) [DecidableRel r]
       simp only [List.insertionSort_cons]
       rw [insertionSort_filter]
     · rw [orderedInsert_filter_of_neg r a p h]
-      rw [List.filter_cons_of_neg (by simpa using h)]
-      rw [insertionSort_filter]
+      rw [List.filter_cons_of_neg (by simpa using h), insertionSort_filter]
 
 lemma takeWhile_sorted_eq_filter {α : Type} (r : α → α → Prop) [DecidableRel r]
     [IsTrans α r] (a : α) : (l : List α) → (hl : l.Pairwise r) →
@@ -440,8 +424,7 @@ lemma takeWhile_sorted_eq_filter {α : Type} (r : α → α → Prop) [Decidable
       not_false_eq_true, List.takeWhile_cons_of_neg, List.filter_cons_of_neg, List.nil_eq,
       List.filter_eq_nil_iff, Bool.not_eq_eq_eq_not, decide_eq_false_iff_not]
       intro c hc
-      apply IsTrans.trans a b c hb
-      exact hl.1 c hc
+      exact IsTrans.trans a b c hb (hl.1 c hc)
 
 lemma dropWhile_sorted_eq_filter {α : Type} (r : α → α → Prop) [DecidableRel r]
     [IsTrans α r] (a : α) : (l : List α) → (hl : l.Pairwise r) →
@@ -460,8 +443,7 @@ lemma dropWhile_sorted_eq_filter {α : Type} (r : α → α → Prop) [Decidable
       rw [List.filter_eq_self]
       intro c hc
       simp only [decide_eq_true_eq]
-      apply IsTrans.trans a b c hb
-      exact hl.1 c hc
+      exact IsTrans.trans a b c hb (hl.1 c hc)
 
 lemma dropWhile_sorted_eq_filter_filter {α : Type} (r : α → α → Prop) [DecidableRel r]
     [IsTrans α r] (a : α) :(l : List α) → (hl : l.Pairwise r) →
@@ -493,8 +475,7 @@ lemma dropWhile_sorted_eq_filter_filter {α : Type} (r : α → α → Prop) [De
           simp only [Bool.and_eq_true, decide_eq_true_eq, not_and]
           intro hac hca
           apply hba
-          apply IsTrans.trans b c a _ hca
-          exact hl.1 c hc
+          exact IsTrans.trans b c a (hl.1 c hc) hca
         rw [h1]
         rw [dropWhile_sorted_eq_filter_filter]
         simp only [Bool.decide_and, h1, decide_not, List.nil_append]
diff --git a/Physlib/Mathematics/PiTensorProduct.lean b/Physlib/Mathematics/PiTensorProduct.lean
index a8bcc6cae..26cddfebc 100644
--- a/Physlib/Mathematics/PiTensorProduct.lean
+++ b/Physlib/Mathematics/PiTensorProduct.lean
@@ -199,8 +199,7 @@ lemma elimPureTensor_update_right (p : (i : ι1) → s1 i) (q : (i : ι2) → s2
     Function.update (elimPureTensor p q) (Sum.inr y) r := by
   funext x
   match x with
-  | Sum.inl x =>
-    rfl
+  | Sum.inl x => rfl
   | Sum.inr x =>
     change Function.update q y r x = _
     simp only [Function.update, Sum.inr.injEq, Sum.elim_inr]
@@ -224,8 +223,7 @@ lemma elimPureTensor_update_left (p : (i : ι1) → s1 i) (q : (i : ι2) → s2
       subst h
       rfl
     · rfl
-  | Sum.inr y =>
-    rfl
+  | Sum.inr y => rfl
 
 end
 
diff --git a/Physlib/Mathematics/RatComplexNum.lean b/Physlib/Mathematics/RatComplexNum.lean
index 49c19f846..d1f489885 100644
--- a/Physlib/Mathematics/RatComplexNum.lean
+++ b/Physlib/Mathematics/RatComplexNum.lean
@@ -36,14 +36,8 @@ lemma ext {x y : RatComplexNum} (h1 : x.1 = y.1) (h2 : x.2 = y.2) : x = y := by
 def equivToProd : RatComplexNum ≃ ℚ × ℚ where
   toFun := fun x => (x.1, x.2)
   invFun := fun x => ⟨x.1, x.2⟩
-  left_inv := by
-    intro x
-    cases x
-    rfl
-  right_inv := by
-    intro x
-    cases x
-    rfl
+  left_inv := fun ⟨_, _⟩ => rfl
+  right_inv := fun ⟨_, _⟩ => rfl
 
 instance : DecidableEq RatComplexNum := Equiv.decidableEq equivToProd
 
@@ -242,8 +236,7 @@ open Complex
 /-- The inclusion of `RatComplexNum` into the complex numbers. -/
 noncomputable def toComplexNum : RatComplexNum →+* ℂ where
   toFun := fun x => x.fst + x.snd * I
-  map_one' := by
-    simp
+  map_one' := by simp
   map_add' a b := by
     simp only [add_fst, Rat.cast_add, add_snd]
     ring
@@ -252,8 +245,7 @@ noncomputable def toComplexNum : RatComplexNum →+* ℂ where
     ring_nf
     simp only [I_sq, mul_neg, mul_one]
     ring
-  map_zero' := by
-    simp
+  map_zero' := by simp
 
 @[simp]
 lemma I_mul_toComplexNum (a : RatComplexNum) : I * toComplexNum a = toComplexNum (⟨0, 1⟩ * a) := by
@@ -264,8 +256,7 @@ lemma I_mul_toComplexNum (a : RatComplexNum) : I * toComplexNum a = toComplexNum
   ring
 
 lemma ofNat_mul_toComplexNum (n : ℕ) (a : RatComplexNum) :
-    n * toComplexNum a = toComplexNum (n * a) := by
-  simp only [map_mul, map_natCast]
+    n * toComplexNum a = toComplexNum (n * a) := by simp only [map_mul, map_natCast]
 
 lemma toComplexNum_injective : Function.Injective toComplexNum := by
   intro a b ha
diff --git a/Physlib/Mathematics/SO3/Basic.lean b/Physlib/Mathematics/SO3/Basic.lean
index 69b497e9c..4f74e3d7b 100644
--- a/Physlib/Mathematics/SO3/Basic.lean
+++ b/Physlib/Mathematics/SO3/Basic.lean
@@ -57,8 +57,7 @@ def toGL : SO(3) →* GL (Fin 3) ℝ where
   map_mul' _ _ := (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl
 
 lemma subtype_val_eq_toGL : (Subtype.val : SO3 → Matrix (Fin 3) (Fin 3) ℝ) =
-    Units.val ∘ toGL.toFun :=
-  rfl
+    Units.val ∘ toGL.toFun := rfl
 
 /-- The inclusion of `SO(3)` into `GL(3,ℝ)` is an injection. -/
 lemma toGL_injective : Function.Injective toGL := by
diff --git a/Physlib/Mathematics/SchurTriangulation.lean b/Physlib/Mathematics/SchurTriangulation.lean
index 0ec7fb8e3..fec1510ec 100644
--- a/Physlib/Mathematics/SchurTriangulation.lean
+++ b/Physlib/Mathematics/SchurTriangulation.lean
@@ -180,8 +180,7 @@ protected noncomputable def SchurTriangulationAux.of
         have hf {bi i' bj j'} (hi : e i = ⟨bi, i'⟩) (hj : e j = ⟨bj, j'⟩) :=
           calc toMatrixOrthonormal basis f i j
             _ = toMatrixOrthonormal bE f (e i) (e j) := by
-              rw [f.toMatrixOrthonormal_reindex]
-              rfl
+              exact congr_fun (congr_fun (f.toMatrixOrthonormal_reindex bE e.symm) i) j
             _ = ⟪bE (e i), f (bE (e j))⟫_𝕜 := f.toMatrixOrthonormal_apply_apply ..
             _ = ⟪(B bi i' : E), f (B bj j')⟫_𝕜 := by rw [hB, hB, hi, hj]
 
@@ -252,8 +251,7 @@ noncomputable def schurTriangulationAux :
   fun i j (hji : j < i) =>
     calc LinearMap.toMatrixOrthonormal b' f i j
       _ = LinearMap.toMatrixOrthonormal b f (e.symm i) (e.symm j) := by
-        rw [f.toMatrixOrthonormal_reindex]
-        rfl
+        exact congrFun (congrFun (f.toMatrixOrthonormal_reindex b e.toEquiv) i) j
       _ = 0 := hut (e.symm.lt_iff_lt.mpr hji)
 
 /-- The change of basis that induces the upper triangular form `A.schurTriangulation` of a matrix
diff --git a/Physlib/Mathematics/SpecialFunctions/PhysHermite.lean b/Physlib/Mathematics/SpecialFunctions/PhysHermite.lean
index 68d76cc46..e7104beed 100644
--- a/Physlib/Mathematics/SpecialFunctions/PhysHermite.lean
+++ b/Physlib/Mathematics/SpecialFunctions/PhysHermite.lean
@@ -211,9 +211,8 @@ lemma deriv_physHermite' (x : ℝ)
     (f : ℝ → ℝ) (hf : DifferentiableAt ℝ f x) (n : ℕ) :
     deriv (fun x => physHermite n (f x)) x
     = (2 * n * physHermite (n - 1) (f x)) * deriv f x := by
-  unfold deriv
-  rw [fderiv_physHermite (hf := by fun_prop)]
-  rfl
+  simpa only [deriv, FunLike.coe_smul, Pi.smul_apply, smul_eq_mul] using
+    congrArg (fun g : ℝ →L[ℝ] ℝ => g 1) (fderiv_physHermite x f hf n)
 
 lemma physHermite_parity: (n : ℕ) → (x : ℝ) →
     physHermite n (-x) = (-1)^n * physHermite n x
@@ -325,8 +324,7 @@ lemma integral_physHermite_mul_physHermite_eq_integral_deriv_inductive (n m : 
     ∫ x : ℝ, (physHermite n x * physHermite m x) * Real.exp (- x ^ 2) =
     (-1 : ℝ) ^ (m - p) * ∫ x : ℝ, (deriv^[p] (physHermite n) x *
     (deriv^[m - p] fun x => Real.exp (-x ^ 2)) x)
-  | 0, h => by
-    exact integral_physHermite_mul_physHermite_eq_integral_deriv_exp n m
+  | 0, h => integral_physHermite_mul_physHermite_eq_integral_deriv_exp n m
   | p + 1, h => by
     rw [integral_physHermite_mul_physHermite_eq_integral_deriv_inductive n m p (by omega)]
     have h1 : m - p = m - (p + 1) + 1 := by omega
@@ -401,8 +399,7 @@ theorem physHermite_norm (n : ℕ) :
     rw [aeval_C]
     simp
   rw [MeasureTheory.integral_const_mul]
-  have h1 : ∫ (x : ℝ), Real.exp (- x^2) = Real.sqrt (Real.pi) := by
-    simpa using integral_gaussian 1
+  have h1 : ∫ (x : ℝ), Real.exp (- x^2) = Real.sqrt (Real.pi) := by simpa using integral_gaussian 1
   rw [h1]
 
 lemma physHermite_norm_cons (n : ℕ) (c : ℝ) :
@@ -482,8 +479,7 @@ lemma cos_mem_physHermite_span_topologicalClosure (c : ℝ) :
   simp only [HasSum] at h1
   have h1 : Filter.Tendsto
       (fun s => fun y => ∑ x ∈ s, (-1) ^ x * (c * y) ^ (2 * x) / ((2 * x)! : ℝ))
-    Filter.atTop (nhds (fun x => Real.cos (c * x))) := by
-    exact tendsto_pi_nhds.mpr fun x => h1 (c * x)
+    Filter.atTop (nhds (fun x => Real.cos (c * x))) := tendsto_pi_nhds.mpr fun x => h1 (c * x)
   have h2 (z : Finset ℕ) : (fun y => ∑ x ∈ z, (-1) ^ x * (c * y) ^ (2 * x) / ↑(2 * x)!) ∈
       ↑(Submodule.span ℝ (Set.range (fun n => (physHermite n : ℝ → ℝ)))) := by
     have h0 : (fun y => ∑ x ∈ z, (-1) ^ x * (c * y) ^ (2 * x) / ↑(2 * x)!) =
diff --git a/Physlib/Mathematics/Trigonometry/Tanh.lean b/Physlib/Mathematics/Trigonometry/Tanh.lean
index 9f17baa83..377484ac2 100644
--- a/Physlib/Mathematics/Trigonometry/Tanh.lean
+++ b/Physlib/Mathematics/Trigonometry/Tanh.lean
@@ -110,11 +110,8 @@ lemma polynomial_tanh_bounded (P : Polynomial ℝ) :
     ∃ C : ℝ, ∀ x : ℝ, |P.eval (Real.tanh x)| ≤ C := by
   -- Since tanh maps to (-1, 1), it maps to [-1+ε, 1-ε] for any ε > 0
   -- But more directly, tanh maps to (-1, 1) ⊆ [-1, 1]
-  have h_range : ∀ x : ℝ, Real.tanh x ∈ Set.Icc (-1) 1 := by
-    intro x
-    constructor
-    · exact le_of_lt (neg_one_lt_tanh x)
-    · exact le_of_lt (tanh_lt_one x)
+  have h_range : ∀ x : ℝ, Real.tanh x ∈ Set.Icc (-1) 1 :=
+    fun x => ⟨le_of_lt (neg_one_lt_tanh x), le_of_lt (tanh_lt_one x)⟩
   -- Apply polynomial boundedness on [-1, 1]
   obtain ⟨M, hM⟩ := polynomial_bounded_on_interval P (-1) 1
   use M
@@ -146,10 +143,8 @@ lemma tanh_hasTemperateGrowth : Function.HasTemperateGrowth Real.tanh := by
     use C
     intro x
     have h_equiv : ‖iteratedFDeriv ℝ n Real.tanh x‖ = |iteratedDeriv n Real.tanh x| := by
-      rw [← iteratedFDerivWithin_univ]
-      rw [← iteratedDerivWithin_univ]
-      rw [← norm_eq_abs]
-      rw [norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin]
+      rw [← iteratedFDerivWithin_univ, ← iteratedDerivWithin_univ]
+      rw [← norm_eq_abs, norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin]
     rw [h_equiv]
     simp only [pow_zero, mul_one]
     exact hC x
@@ -209,8 +204,7 @@ lemma iteratedDeriv_tanh_const_mul (n : ℕ) (κ : ℝ) : ∀ x : ℝ,
 lemma tanh_const_mul_hasTemperateGrowth (κ : ℝ) :
     Function.HasTemperateGrowth (fun x => Real.tanh (κ * x)) := by
   constructor
-  · have h : (fun x => Real.tanh (κ * x)) = (Real.tanh ∘ (fun x => κ * x)) :=
-      rfl
+  · have h : (fun x => Real.tanh (κ * x)) = (Real.tanh ∘ (fun x => κ * x)) := rfl
     have h' : ContDiff ℝ ∞ (fun x => κ * x) := by
       have h'': (fun x : ℝ => κ * x) = fun x => κ • x := rfl
       rw [contDiff_infty, h'']
diff --git a/Physlib/Mathematics/VariationalCalculus/Basic.lean b/Physlib/Mathematics/VariationalCalculus/Basic.lean
index 7244f8f22..98846c383 100644
--- a/Physlib/Mathematics/VariationalCalculus/Basic.lean
+++ b/Physlib/Mathematics/VariationalCalculus/Basic.lean
@@ -245,5 +245,4 @@ lemma fundamental_theorem_of_variational_calculus {f : X → V}
   · intro x
     simp only [Pi.zero_apply]
     apply real_inner_self_nonneg'
-  · apply IsTestFunction.integrable
-    exact IsTestFunction.inner hf hf
+  · exact (IsTestFunction.inner hf hf).integrable μ
diff --git a/Physlib/Mathematics/VariationalCalculus/HasVarAdjDeriv.lean b/Physlib/Mathematics/VariationalCalculus/HasVarAdjDeriv.lean
index 0bcaef70b..9671fdf30 100644
--- a/Physlib/Mathematics/VariationalCalculus/HasVarAdjDeriv.lean
+++ b/Physlib/Mathematics/VariationalCalculus/HasVarAdjDeriv.lean
@@ -66,8 +66,7 @@ lemma apply_smooth_of_smooth {F : (X → U) → (X → V)} {F' : (X → V) → (
   fun_prop
 
 lemma apply_smooth_self {F : (X → U) → (X → V)} {F' : (X → V) → (X → U)} {u : X → U}
-    (h : HasVarAdjDerivAt F F' u) : ContDiff ℝ ∞ (F u) := by
-  exact h.apply_smooth_of_smooth (h.smooth_at)
+    (h : HasVarAdjDerivAt F F' u) : ContDiff ℝ ∞ (F u) := h.apply_smooth_of_smooth (h.smooth_at)
 
 lemma smooth_R {F : (X → U) → (X → V)} {F' : (X → V) → (X → U)} {u : X → U}
     (h : HasVarAdjDerivAt F F' u) {φ : ℝ → X → U} (hφ : ContDiff ℝ ∞ ↿φ) (x : X) :
@@ -190,10 +189,8 @@ lemma comp {F : (Y → V) → (Z → W)} {G : (X → U) → (Y → V)} {u : X 
     {F' G'} (hF : HasVarAdjDerivAt F F' (G u)) (hG : HasVarAdjDerivAt G G' u) :
     HasVarAdjDerivAt (fun u => F (G u)) (fun ψ => G' (F' ψ)) u where
   smooth_at := hG.smooth_at
-  diff := by
-    intro φ hφ
-    apply hF.diff (φ := fun t x => G (φ t) x)
-    exact hG.diff φ hφ
+  diff := fun φ hφ =>
+    hF.diff (φ := fun t x => G (φ t) x) (hG.diff φ hφ)
   linearize := by
     intro φ hφ x
     rw[hF.linearize (fun t x => G (φ t) x) (hG.diff φ hφ)]
@@ -408,8 +405,7 @@ lemma deriv' (u : ℝ → U) (hu : ContDiff ℝ ∞ u) :
       rw [← fderiv_apply_one_eq_deriv]
     rw [fderiv_swap]
     simp only [fderiv_eq_smul_deriv, one_smul]
-    · apply ContDiff.of_le hφ
-      exact ENat.LEInfty.out
+    · exact hφ.of_le ENat.LEInfty.out
     · exact differentiableAt_id
   adjoint := by
     apply HasVarAdjoint.congr_fun (G := (fun δu x => deriv (fun x' => δu x') x))
@@ -605,8 +601,7 @@ lemma mul
     · apply hG.diff; assumption
   linearize := by
     intro φ hφ x
-    rw [deriv_fun_mul, deriv_fun_mul]
-    rw [hF.linearize _ hφ, hG.linearize _ hφ]
+    rw [deriv_fun_mul, deriv_fun_mul, hF.linearize _ hφ, hG.linearize _ hφ]
     · simp
     · exact hF.differentiable_linear hφ x 0
     · exact hG.differentiable_linear hφ x 0
@@ -683,11 +678,9 @@ protected lemma fderiv (u : X → U) (dx : X) (hu : ContDiff ℝ ∞ u)
     simp only [FunLike.coe_smul, Pi.smul_apply]
     exact (hφ.differentiable (by simp)).differentiableAt
   · intro φ hφ x
-    rw [← fderiv_apply_one_eq_deriv]
-    rw [fderiv_swap]
+    rw [← fderiv_apply_one_eq_deriv, fderiv_swap]
     simp only [fderiv_eq_smul_deriv, one_smul]
-    · apply ContDiff.of_le hφ
-      exact ENat.LEInfty.out
+    · exact hφ.of_le ENat.LEInfty.out
   · exact hu
   · exact HasVarAdjoint.fderiv_apply
 
@@ -712,16 +705,12 @@ protected lemma gradient {d} (u : Space d → ℝ) (hu : ContDiff ℝ ∞ u) :
     simp only [Space.deriv]
     fun_prop
   · intro φ1 φ2 h1 h2
-    rw [Space.gradient_eq_grad]
-    rw [Space.grad_add, Space.grad_eq_gradient, Space.grad_eq_gradient]
-    simp
+    rw [Space.gradient_eq_grad, Space.grad_add, Space.grad_eq_gradient, Space.grad_eq_gradient]
     rfl
     · exact h1.differentiable (by simp)
     · exact h2.differentiable (by simp)
   · intro c φ hφ
-    rw [Space.gradient_eq_grad]
-    rw [Space.grad_smul, Space.grad_eq_gradient]
-    simp
+    rw [Space.gradient_eq_grad, Space.grad_smul, Space.grad_eq_gradient]
     rfl
     exact hφ.differentiable (by simp)
   · intro φ hφ x
@@ -733,11 +722,9 @@ protected lemma gradient {d} (u : Space d → ℝ) (hu : ContDiff ℝ ∞ u) :
     rw [deriv_smul_const]
     congr
     simp [Space.deriv]
-    rw [← fderiv_apply_one_eq_deriv]
-    rw [fderiv_swap]
+    rw [← fderiv_apply_one_eq_deriv, fderiv_swap]
     simp only [fderiv_eq_smul_deriv, smul_eq_mul, one_mul]
-    · apply ContDiff.of_le hφ
-      exact ENat.LEInfty.out
+    · exact hφ.of_le ENat.LEInfty.out
     · simp [Space.deriv]
       apply Differentiable.differentiableAt
       apply fderiv_uncurry_differentiable_snd_comp_fst_apply
@@ -781,8 +768,7 @@ protected lemma grad {d} (u : Space d → ℝ) (hu : ContDiff ℝ ∞ u) :
     rw [← fderiv_apply_one_eq_deriv]
     rw [fderiv_swap]
     simp only [fderiv_eq_smul_deriv, smul_eq_mul, one_mul]
-    · apply ContDiff.of_le hφ
-      exact ENat.LEInfty.out
+    · exact hφ.of_le ENat.LEInfty.out
     · simp [Space.deriv]
       apply Differentiable.differentiableAt
       apply fderiv_uncurry_differentiable_snd_comp_fst_apply
@@ -835,8 +821,7 @@ lemma div {d} (u : Space d → EuclideanSpace ℝ (Fin d)) (hu : ContDiff ℝ 
       exact hφ.differentiable (by simp)
     · apply ContDiff.comp (g := EuclideanSpace.proj i)
       · fun_prop
-      · apply ContDiff.of_le hφ
-        exact ENat.LEInfty.out
+      · exact hφ.of_le ENat.LEInfty.out
     · intro i _
       apply Differentiable.differentiableAt
       simp [Space.deriv]
diff --git a/Physlib/Mathematics/VariationalCalculus/HasVarAdjoint.lean b/Physlib/Mathematics/VariationalCalculus/HasVarAdjoint.lean
index dabe8bf76..ab984557b 100644
--- a/Physlib/Mathematics/VariationalCalculus/HasVarAdjoint.lean
+++ b/Physlib/Mathematics/VariationalCalculus/HasVarAdjoint.lean
@@ -99,13 +99,9 @@ lemma comp {F : (Y → V) → (Z → W)} {G : (X → U) → (Y → V)} {F' G'}
 lemma congr_fun {F G : (X → U) → (Y → V)} {F' : (Y → V) → (X → U)}
     (h : HasVarAdjoint G F') (h' : ∀ φ, IsTestFunction φ → F φ = G φ) :
     HasVarAdjoint F F' where
-  test_fun_preserving φ hφ := by
-    rw[h' _ hφ]
-    exact h.test_fun_preserving φ hφ
+  test_fun_preserving φ hφ := by simpa [h' φ hφ] using h.test_fun_preserving φ hφ
   test_fun_preserving' φ hφ := h.test_fun_preserving' φ hφ
-  adjoint φ ψ hφ hψ := by
-    rw [h' φ hφ]
-    exact h.adjoint φ ψ hφ hψ
+  adjoint φ ψ hφ hψ := by simpa [h' φ hφ] using h.adjoint φ ψ hφ hψ
   ext' := h.ext'
 
 lemma of_eq {F : (X → U) → (Y → V)} {F' G' : (Y → V) → (X → U)}
@@ -113,12 +109,8 @@ lemma of_eq {F : (X → U) → (Y → V)} {F' G' : (Y → V) → (X → U)}
     (hlin : IsLocalizedFunctionTransform G') :
     HasVarAdjoint F G' where
   test_fun_preserving φ hφ := hF'.test_fun_preserving φ hφ
-  test_fun_preserving' φ hφ := by
-    rw [← h φ hφ]
-    exact hF'.test_fun_preserving' φ hφ
-  adjoint φ ψ hφ hψ := by
-    rw [← h ψ hψ]
-    exact hF'.adjoint φ ψ hφ hψ
+  test_fun_preserving' φ hφ := by simpa [← h φ hφ] using hF'.test_fun_preserving' φ hφ
+  adjoint φ ψ hφ hψ := by simpa [← h ψ hψ] using hF'.adjoint φ ψ hφ hψ
   ext' := hlin
 
 /-- Variational adjoint is unique only when applied to test functions. -/
@@ -129,8 +121,7 @@ lemma unique_on_test_functions {F : (X → U) → (Y → V)} {F' G' : (Y → V)
   obtain ⟨F_preserve_test, F'_preserve_test, F'_adjoint⟩ := hF'
   obtain ⟨F_preserve_test, G'_preserve_test, G'_adjoint⟩ := hG'
   intro φ hφ
-  rw [← zero_add (G' φ)]
-  rw [← sub_eq_iff_eq_add]
+  rw [← zero_add (G' φ), ← sub_eq_iff_eq_add]
   change (F' - G') φ = 0
   apply fundamental_theorem_of_variational_calculus (@volume X _)
   · simp
@@ -405,9 +396,7 @@ lemma clm_apply
     apply HasAdjoint.congr_adj
     apply ContinuousLinearMap.hasAdjoint
     funext y; simp[adjoint_eq_clm_adjoint]
-  ext' := by
-    intro K cK
-    exact ⟨K, cK, by intro _ _ hφ _ _; simp_all⟩
+  ext' := fun K cK => ⟨K, cK, by intro _ _ hφ _ _; simp_all⟩
   -- ext := IsLocalizedFunctionTransform.clm_apply _
 
 protected lemma deriv :
@@ -454,14 +443,12 @@ lemma fderiv_apply {dx}
     apply IsLocalizedFunctionTransform.neg
     apply IsLocalizedFunctionTransform.fderiv
   adjoint φ ψ hφ hψ := by
-    rw [← sub_eq_zero]
-    rw [← integral_sub]
+    rw [← sub_eq_zero, ← integral_sub]
     · trans ∫ (a : X), fderiv ℝ (fun a => ⟪φ a, ψ a⟫_ℝ) a dx
       · congr
         funext a
         simp only [inner_neg_right', sub_neg_eq_add]
-        rw [fderiv_inner_apply']
-        rw [add_comm]
+        rw [fderiv_inner_apply', add_comm]
         · exact hφ.differentiable a
         · exact hψ.differentiable a
       · have h1 := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable (f := fun a => 1)
@@ -695,9 +682,7 @@ lemma fst {F'} {F : (X → U) → (X → W×V)}
     HasVarAdjoint
       (fun φ x => (F φ x).1)
       (fun φ x => F' (fun x' => (φ x', 0)) x) where
-  test_fun_preserving _ hφ := by
-    apply IsTestFunction.prod_fst
-    exact hF.test_fun_preserving _ hφ
+  test_fun_preserving _ hφ := IsTestFunction.prod_fst (hF.test_fun_preserving _ hφ)
   test_fun_preserving' y hφ := by
     apply hF.test_fun_preserving'
     fun_prop
@@ -721,9 +706,7 @@ lemma snd {F'} {F : (X → U) → (X → W×V)}
     HasVarAdjoint
       (fun φ x => (F φ x).2)
       (fun φ x => F' (fun x' => (0, φ x')) x) where
-  test_fun_preserving _ hφ := by
-    apply IsTestFunction.prod_snd
-    exact hF.test_fun_preserving _ hφ
+  test_fun_preserving _ hφ := IsTestFunction.prod_snd (hF.test_fun_preserving _ hφ)
   test_fun_preserving' y hφ := by
     apply hF.test_fun_preserving' _
     fun_prop
diff --git a/Physlib/Mathematics/VariationalCalculus/HasVarGradient.lean b/Physlib/Mathematics/VariationalCalculus/HasVarGradient.lean
index 4698332a5..170644aae 100644
--- a/Physlib/Mathematics/VariationalCalculus/HasVarGradient.lean
+++ b/Physlib/Mathematics/VariationalCalculus/HasVarGradient.lean
@@ -72,8 +72,7 @@ lemma HasVarGradientAt.add (F F' : (X → U) → (X → ℝ))
   obtain ⟨F1,hF1,eq1⟩ := h
   obtain ⟨F2,hF2,eq2⟩ := h'
   apply HasVarGradientAt.intro (F1 + F2)
-  · apply hF1.add (V := ℝ)
-    exact hF2
+  · exact HasVarAdjDerivAt.add (V := ℝ) F F' F1 F2 u hF1 hF2
   · simp
     rw [eq1, eq2]
 
diff --git a/Physlib/Mathematics/VariationalCalculus/IsTestFunction.lean b/Physlib/Mathematics/VariationalCalculus/IsTestFunction.lean
index dd3e9fbcb..748b65919 100644
--- a/Physlib/Mathematics/VariationalCalculus/IsTestFunction.lean
+++ b/Physlib/Mathematics/VariationalCalculus/IsTestFunction.lean
@@ -85,10 +85,9 @@ lemma IsTestFunction.pi {ι} [Fintype ι] {φ : X → ι → U} (hφ : ∀ i, Is
     refine exists_compact_iff_hasCompactSupport.mp
       ⟨⋃ i, K i, isCompact_iUnion (fun i => (hK i).1), fun x hx => ?_⟩
     simp at hx
-    conv_lhs =>
-      enter [i]
-      rw [(hK i).2 x (hx i)]
-    rfl
+    apply funext
+    intro i
+    exact (hK i).2 x (hx i)
 
 lemma IsTestFunction.space_component {φ : X → Space d} (hφ : IsTestFunction φ) :
     IsTestFunction (fun x => φ x i) where
@@ -208,8 +207,7 @@ lemma IsTestFunction.linearMap_comp {f : X → V} (hf : IsTestFunction f)
 lemma IsTestFunction.family_linearMap_comp {f : X → V} (hf : IsTestFunction f)
     {g : X → V →L[ℝ] U} (hg : ContDiff ℝ ∞ g) :
     IsTestFunction (fun x => g x (f x)) where
-  smooth := by
-    fun_prop
+  smooth := by fun_prop
   supp := by
     have hf' := hf.supp
     rw [← exists_compact_iff_hasCompactSupport] at hf' ⊢
@@ -232,9 +230,7 @@ lemma IsTestFunction.of_fderiv {f : X → U} (hf : IsTestFunction f) :
     · fun_prop
     · fun_prop
     · exact Preorder.le_refl (∞ + 1)
-  supp := by
-    apply HasCompactSupport.fderiv
-    exact hf.supp
+  supp := HasCompactSupport.fderiv ℝ hf.supp
 
 @[fun_prop]
 lemma IsTestFunction.fderiv_apply {f : X → U} (hf : IsTestFunction f) (δx : X) :
@@ -245,9 +241,7 @@ lemma IsTestFunction.fderiv_apply {f : X → U} (hf : IsTestFunction f) (δx : X
     · fun_prop
     · fun_prop
     · exact Preorder.le_refl (∞ + 1)
-  supp := by
-    apply HasCompactSupport.fderiv_apply
-    exact hf.supp
+  supp := HasCompactSupport.fderiv_apply ℝ hf.supp δx
 
 open InnerProductSpace' in
 @[fun_prop]
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/Basic.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/Basic.lean
index e7c308607..ebaea736e 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/Basic.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/Basic.lean
@@ -33,10 +33,7 @@ namespace SMνCharges
 variable {n : ℕ}
 
 lemma sum_one  [AddCommMonoid M] (f : Fin (SMνSpecies 1).numberCharges → M) :
-    ∑ i, f i = f ⟨0, by simp⟩ := by
-  change  ∑ (i : Fin 1), f i = _
-  simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton]
-  rfl
+    ∑ i, f i = f ⟨0, by simp⟩ := Fin.sum_univ_one f
 
 /-- An equivalence between `(SMνCharges n).charges` and `(Fin 6 → Fin n → ℚ)`
 splitting the charges into species. -/
@@ -55,8 +52,7 @@ lemma charges_eq_toSpecies_eq (S T : (SMνCharges n).Charges) :
     S = T ↔ ∀ i, toSpecies i S = toSpecies i T := by
   refine Iff.intro (fun h => ?_) (fun h => ?_)
   · exact fun i => congrArg (⇑(toSpecies i)) h
-  · apply toSpeciesEquiv.injective
-    exact funext (fun i => h i)
+  · exact toSpeciesEquiv.injective (funext (fun i => h i))
 
 lemma toSMSpecies_toSpecies_inv (i : Fin 6) (f : Fin 6 → Fin n → ℚ) :
     (toSpecies i) (toSpeciesEquiv.symm f) = f i := by
@@ -258,8 +254,7 @@ def quadBiLin : BiLinearSymm (SMνCharges n).Charges := BiLinearSymm.mk₂
 lemma quadBiLin_decomp (S T : (SMνCharges n).Charges) :
     quadBiLin S T = ∑ i, Q S i * Q T i - 2 * ∑ i, U S i * U T i +
         ∑ i, D S i * D T i - ∑ i, L S i * L T i + ∑ i, E S i * E T i := by
-  rw [quadBiLin]
-  rw [BiLinearSymm.mk₂_toFun_apply]
+  rw [quadBiLin, BiLinearSymm.mk₂_toFun_apply]
   repeat rw [Finset.sum_add_distrib]
   repeat rw [← Finset.mul_sum]
   simp only [toSpecies_apply, Fin.isValue, neg_mul, one_mul, add_left_inj]
@@ -325,8 +320,7 @@ lemma cubeTriLin_decomp (S T R : (SMνCharges n).Charges) :
     cubeTriLin S T R = 6 * ∑ i, (Q S i * Q T i * Q R i) + 3 * ∑ i, (U S i * U T i * U R i) +
       3 * ∑ i, (D S i * D T i * D R i) + 2 * ∑ i, (L S i * L T i * L R i) +
       ∑ i, (E S i * E T i * E R i) + ∑ i, (N S i * N T i * N R i) := by
-  rw [cubeTriLin]
-  rw [TriLinearSymm.mk₃_toFun_apply_apply]
+  rw [cubeTriLin, TriLinearSymm.mk₃_toFun_apply_apply]
   repeat rw [Finset.sum_add_distrib]
   repeat rw [← Finset.mul_sum]
 
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/FamilyMaps.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/FamilyMaps.lean
index a9725b545..40d49aed1 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/FamilyMaps.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/FamilyMaps.lean
@@ -32,8 +32,7 @@ def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).Charges →ₗ[ℚ]
   map_smul' a S := by
     rw [charges_eq_toSpecies_eq]
     intro i
-    rw [map_smul, toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv, map_smul]
-    rfl
+    simp [toSMSpecies_toSpecies_inv]
 
 lemma chargesMapOfSpeciesMap_toSpecies {n m : ℕ}
     (f : (SMνSpecies n).Charges →ₗ[ℚ] (SMνSpecies m).Charges)
@@ -76,8 +75,7 @@ def speciesEmbed (m n : ℕ) :
       eq_ratCast, Rat.cast_eq_id, id_eq]
     by_cases hi : i.val < m
     · rw [dif_pos hi, dif_pos hi]
-    · rw [dif_neg hi, dif_neg hi]
-      exact Eq.symm (Rat.mul_zero a)
+    · simp [dif_neg hi]
 
 /-- The embedding of the `m`-family charges onto the `n`-family charges, with all
 other charges zero. -/
@@ -100,8 +98,7 @@ def familyUniversal (n : ℕ) : (SMνCharges 1).Charges →ₗ[ℚ] (SMνCharges
 
 lemma toSpecies_familyUniversal {n : ℕ} (j : Fin 6) (S : (SMνCharges 1).Charges)
     (i : Fin n) : toSpecies j (familyUniversal n S) i = toSpecies j S ⟨0, by simp⟩ := by
-  rw [familyUniversal, chargesMapOfSpeciesMap_toSpecies]
-  rfl
+  exact congrFun (chargesMapOfSpeciesMap_toSpecies (speciesFamilyUniversial n) S j) i
 
 set_option backward.isDefEq.respectTransparency false in
 lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).Charges) (j : Fin 6) :
@@ -130,8 +127,7 @@ lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).Charges)
     Fin.isValue, Nat.reduceMul]
   rw [Finset.mul_sum]
   refine Finset.sum_congr rfl (fun i _ => ?_)
-  erw [toSpecies_familyUniversal]
-  rfl
+  exact congrArg (fun x => x * T (finProdFinEquiv (j, i))) (toSpecies_familyUniversal j S i)
 
 set_option backward.isDefEq.respectTransparency false in
 lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).Charges)
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/BMinusL.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/BMinusL.lean
index a3f4f6a30..5bbf06c83 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/BMinusL.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/BMinusL.lean
@@ -79,8 +79,7 @@ lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
 
 lemma add_quad (S : (PlusU1 n).QuadSols) (a b : ℚ) :
     accQuad (a • S.val + b • (BL n).val) = 0 := by
-  rw [add_AFL_quad, quadSol S]
-  exact Rat.mul_zero (a ^ 2)
+  exact (add_AFL_quad S.1 a b).trans (mul_eq_zero_of_right (a ^ 2) (quadSol S))
 
 /-- The `QuadSol` obtained by adding $B-L$ to a `QuadSol`. -/
 def addQuad (S : (PlusU1 n).QuadSols) (a b : ℚ) : (PlusU1 n).QuadSols :=
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/Basic.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/Basic.lean
index 8fb0b8bcd..e55585d60 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/Basic.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/Basic.lean
@@ -66,8 +66,7 @@ lemma quadSol (S : (PlusU1 n).QuadSols) : accQuad S.val = 0 := by
   simp only [PlusU1_quadraticACCs] at hS
   exact hS ⟨0, by simp⟩
 
-lemma cubeSol (S : (PlusU1 n).Sols) : accCube S.val = 0 := by
-  exact S.cubicSol
+lemma cubeSol (S : (PlusU1 n).Sols) : accCube S.val = 0 := S.cubicSol
 
 /-- An element of `charges` which satisfies the linear ACCs
   gives us a element of `LinSols`. -/
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/HyperCharge.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/HyperCharge.lean
index a7be87e52..3e250acee 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/HyperCharge.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/HyperCharge.lean
@@ -76,8 +76,7 @@ lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
   simp
 
 lemma add_quad (S : (PlusU1 n).QuadSols) (a b : ℚ) :
-    accQuad (a • S.val + b • (Y n).val) = 0 := by
-  rw [add_AFL_quad, quadSol S]; simp
+    accQuad (a • S.val + b • (Y n).val) = 0 := by rw [add_AFL_quad, quadSol S]; simp
 
 /-- The `QuadSol` obtained by adding hypercharge to a `QuadSol`. -/
 def addQuad (S : (PlusU1 n).QuadSols) (a b : ℚ) : (PlusU1 n).QuadSols :=
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/PlaneNonSols.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/PlaneNonSols.lean
index 519735102..a286cc98e 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/PlaneNonSols.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/PlaneNonSols.lean
@@ -87,8 +87,7 @@ lemma Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) :
   rw [quadBiLin.map_sum₂, Fintype.sum_eq_single i]
   · rw [quadBiLin.map_smul₂]
   · intro k hij
-    rw [quadBiLin.map_smul₂, Bi_Bj_quad hij.symm]
-    exact Rat.mul_zero (f k)
+    simpa [quadBiLin.map_smul₂] using mul_eq_zero_of_right (f k) (Bi_Bj_quad hij.symm)
 
 /-- The coefficients of the quadratic equation in our basis. -/
 @[simp]
diff --git a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/QuadSolToSol.lean b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/QuadSolToSol.lean
index 594c080ce..cf52cc138 100644
--- a/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/QuadSolToSol.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/PlusU1/QuadSolToSol.lean
@@ -51,8 +51,7 @@ lemma BL_add_α₁_α₂_cube (S : (PlusU1 n).QuadSols) :
   ring
 
 lemma BL_add_α₁_α₂_AF (S : (PlusU1 n).Sols) :
-    BL.addQuad S.1 (α₁ S.1) (α₂ S.1) = (α₁ S.1) • S.1 := by
-  rw [α₂_AF, BL.addQuad_zero]
+    BL.addQuad S.1 (α₁ S.1) (α₂ S.1) = (α₁ S.1) • S.1 := by rw [α₂_AF, BL.addQuad_zero]
 
 /-- The construction of a `Sol` from a `QuadSol` in the generic case. -/
 def generic (S : (PlusU1 n).QuadSols) : (PlusU1 n).Sols :=
@@ -60,13 +59,10 @@ def generic (S : (PlusU1 n).QuadSols) : (PlusU1 n).Sols :=
 
 lemma generic_on_AF (S : (PlusU1 n).Sols) : generic S.1 = (α₁ S.1) • S := by
   apply ACCSystem.Sols.ext
-  change (BL.addQuad S.1 (α₁ S.1) (α₂ S.1)).val = _
-  rw [BL_add_α₁_α₂_AF]
-  rfl
+  exact congrArg (fun Q : (PlusU1 n).QuadSols => Q.val) (BL_add_α₁_α₂_AF S)
 
 lemma generic_on_AF_α₁_ne_zero (S : (PlusU1 n).Sols) (h : α₁ S.1 ≠ 0) :
-    (α₁ S.1)⁻¹ • generic S.1 = S := by
-  rw [generic_on_AF, smul_smul, inv_mul_cancel₀ h, one_smul]
+    (α₁ S.1)⁻¹ • generic S.1 = S := by rw [generic_on_AF, smul_smul, inv_mul_cancel₀ h, one_smul]
 
 /-- The construction of a `Sol` from a `QuadSol` in the case when `α₁ S = 0` and `α₂ S = 0`. -/
 def special (S : (PlusU1 n).QuadSols) (a b : ℚ) (h1 : α₁ S = 0) (h2 : α₂ S = 0) :
diff --git a/Physlib/Particles/BeyondTheStandardModel/TwoHDM/GramMatrix.lean b/Physlib/Particles/BeyondTheStandardModel/TwoHDM/GramMatrix.lean
index e6e319dba..6623c009b 100644
--- a/Physlib/Particles/BeyondTheStandardModel/TwoHDM/GramMatrix.lean
+++ b/Physlib/Particles/BeyondTheStandardModel/TwoHDM/GramMatrix.lean
@@ -99,18 +99,15 @@ lemma gaugeGroupI_exists_fst_eq {H : TwoHiggsDoublet} (h1 : H.Φ1 ≠ 0) :
   simp at h
   simp [h]
   have h_fst : (g • H.Φ2).ofLp 0 = ⟪H.Φ1, H.Φ2⟫_ℂ / ‖H.Φ1‖ := by
-    have h2 : ⟪H.Φ1, H.Φ2⟫_ℂ = ⟪g • H.Φ1, g • H.Φ2⟫_ℂ := by
-      simp
+    have h2 : ⟪H.Φ1, H.Φ2⟫_ℂ = ⟪g • H.Φ1, g • H.Φ2⟫_ℂ := by simp
     rw [h] at h2
     conv_rhs at h2 =>
       simp [PiLp.inner_apply]
     rw [h2]
-    have hx : (‖H.Φ1‖ : ℂ) ≠ 0 := by
-      simp_all
+    have hx : (‖H.Φ1‖ : ℂ) ≠ 0 := by simp_all
     field_simp
   apply And.intro h_fst
-  have hx : ‖g • H.Φ2‖ ^ 2 = ‖H.Φ2‖ ^ 2 := by
-    simp
+  have hx : ‖g • H.Φ2‖ ^ 2 = ‖H.Φ2‖ ^ 2 := by simp
   rw [PiLp.norm_sq_eq_of_L2] at hx
   simp at hx
   have hx0 : ‖(g • H.Φ2).ofLp 1‖ ^ 2 = ‖H.Φ2‖ ^ 2 - ‖(g • H.Φ2).ofLp 0‖ ^ 2 := by
@@ -184,8 +181,7 @@ lemma mem_orbit_gaugeGroupI_iff_gramMatrix (H1 H2 : TwoHiggsDoublet) :
     obtain ⟨g1, H1_Φ1, H1_Φ2⟩ := gaugeGroupI_exists_fst_eq_snd_eq (H := H1) Φ1_zero
     have Φ2_nezero : H2.Φ1 ≠ 0 := by
       intro hzero
-      have hnorm : ‖H1.Φ1‖ = ‖H2.Φ1‖ := by
-        rw [← eq_fst_norm_of_eq_gramMatrix h]
+      have hnorm : ‖H1.Φ1‖ = ‖H2.Φ1‖ := by rw [← eq_fst_norm_of_eq_gramMatrix h]
       simp [hzero] at hnorm
       simp [hnorm] at Φ1_zero
     obtain ⟨g2, H2_Φ1, H2_Φ2⟩ := gaugeGroupI_exists_fst_eq_snd_eq (H := H2) Φ2_nezero
@@ -236,8 +232,7 @@ lemma gramMatrix_surjective_det_tr (K : Matrix (Fin 2) (Fin 2) ℂ)
   have det_eq_abc : K.det = a * b - ‖c‖ ^ 2 := by
     simp [hK_explicit2]
     rw [Complex.mul_conj']
-  have tra_eq_abc : K.trace.re = a + b := by
-    simp [hK_explicit2]
+  have tra_eq_abc : K.trace.re = a + b := by simp [hK_explicit2]
   simp [det_eq_abc, ← Complex.ofReal_pow] at hKdet
   rw [tra_eq_abc] at hKtr
   rw [hK_explicit2]
@@ -255,8 +250,7 @@ lemma gramMatrix_surjective_det_tr (K : Matrix (Fin 2) (Fin 2) ℂ)
     simp [PiLp.norm_eq_of_L2, ← Complex.ofReal_pow]
     exact Real.sq_sqrt hb_nonneg
   /- The case when a ≠ 0. -/
-  have h1 : (√a : ℂ) ≠ 0 := by
-      simp_all
+  have h1 : (√a : ℂ) ≠ 0 := by simp_all
   use ⟨(!₂[√a, 0] : HiggsVec), !₂[conj c/ √a, √(a * b - ‖c‖ ^ 2) / √a]⟩
   ext i j
   fin_cases i <;> fin_cases j <;> simp [gramMatrix, PiLp.norm_eq_of_L2, ← Complex.ofReal_pow]
diff --git a/Physlib/Particles/FlavorPhysics/CKMMatrix/PhaseFreedom.lean b/Physlib/Particles/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
index 36dba588d..a454c85b4 100644
--- a/Physlib/Particles/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
+++ b/Physlib/Particles/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
@@ -97,8 +97,7 @@ lemma shift_tb_phase_zero {V : CKMMatrix} (h1 : t + b = - arg [V]tb) :
 
 lemma shift_cd_phase_pi {V : CKMMatrix} (h1 : c + d = Real.pi - arg [V]cd) :
     [phaseShiftApply V u c t d s b]cd = - VcdAbs ⟦V⟧ := by
-  rw [phaseShiftApply.cd]
-  rw [← norm_mul_exp_arg_mul_I [V]cd]
+  rw [phaseShiftApply.cd, ← norm_mul_exp_arg_mul_I [V]cd]
   rw [mul_comm, mul_assoc, ← exp_add]
   have h2 : ↑(arg [V]cd) * I + (↑c * I + ↑d * I) = ↑(arg [V]cd + (c + d)) * I := by
     simp only [Fin.isValue, ofReal_add]
diff --git a/Physlib/Particles/FlavorPhysics/CKMMatrix/Relations.lean b/Physlib/Particles/FlavorPhysics/CKMMatrix/Relations.lean
index 18358a9bc..f16cc2013 100644
--- a/Physlib/Particles/FlavorPhysics/CKMMatrix/Relations.lean
+++ b/Physlib/Particles/FlavorPhysics/CKMMatrix/Relations.lean
@@ -221,8 +221,7 @@ lemma conj_Vtb_mul_Vud {V : CKMMatrix} {τ : ℝ}
   simp only [neg_mul, exp_neg, Fin.isValue, ne_eq, exp_ne_zero, not_false_eq_true,
     mul_inv_cancel_left₀]
   have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]ud = [V]cs
-      * [V]ud * conj [V]ud - [V]us * ([V]cd * conj [V]ud) := by
-    ring
+      * [V]ud * conj [V]ud - [V]us * ([V]cd * conj [V]ud) := by ring
   rw [h2, V.Vcd_mul_conj_Vud]
   rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
   simp only [Fin.isValue, neg_mul]
@@ -236,8 +235,7 @@ lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
   simp only [neg_mul, exp_neg, Fin.isValue, ne_eq, exp_ne_zero, not_false_eq_true,
     mul_inv_cancel_left₀, neg_add_rev]
   have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]us = ([V]cs
-      * conj [V]us) * [V]ud - [V]us * [V]cd * conj [V]us := by
-    ring
+      * conj [V]us) * [V]ud - [V]us * [V]cd * conj [V]us := by ring
   rw [h2, V.Vcs_mul_conj_Vus]
   rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
   simp only [Fin.isValue, neg_mul]
diff --git a/Physlib/Particles/FlavorPhysics/CKMMatrix/Rows.lean b/Physlib/Particles/FlavorPhysics/CKMMatrix/Rows.lean
index 8c8998beb..db248587e 100644
--- a/Physlib/Particles/FlavorPhysics/CKMMatrix/Rows.lean
+++ b/Physlib/Particles/FlavorPhysics/CKMMatrix/Rows.lean
@@ -207,8 +207,7 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
       rw [← hg, smul_smul, inv_mul_cancel₀, one_smul]
       by_contra hn
       simp [hn] at h2
-    have hg2 : norm (g 0)⁻¹ = 1 := by
-      simp [h2]
+    have hg2 : norm (g 0)⁻¹ = 1 := by simp [h2]
     have hg22 : ∃ (τ : ℝ), (g 0)⁻¹ = Complex.exp (τ * I) := by
       rw [← norm_mul_exp_arg_mul_I (g 0)⁻¹, hg2]
       use arg (g 0)⁻¹
@@ -216,8 +215,7 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
     obtain ⟨τ, hτ⟩ := hg22
     use τ
     rw [hx, hτ]
-  · have hx : norm (g 0) = -1 := by
-      simp [← ofReal_inj, Fin.isValue, ofReal_neg, ofReal_one, h2]
+  · have hx : norm (g 0) = -1 := by simp [← ofReal_inj, Fin.isValue, ofReal_neg, ofReal_one, h2]
     have h3 := norm_nonneg (g 0)
     simp_all only [ofReal_neg, ofReal_one, Left.nonneg_neg_iff]
     have h4 : (0 : ℝ) < 1 := by norm_num
@@ -247,8 +245,7 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
     ofReal_eq_one] at h2
   cases' h2 with h2 h2
   swap
-  · have hx : norm (g 2) = -1 := by
-      simp [h2, ← ofReal_inj, Fin.isValue, ofReal_neg, ofReal_one]
+  · have hx : norm (g 2) = -1 := by simp [h2, ← ofReal_inj, Fin.isValue, ofReal_neg, ofReal_one]
     have h3 := norm_nonneg (g 2)
     simp_all only [ofReal_neg, ofReal_one, Left.nonneg_neg_iff]
     have h4 : (0 : ℝ) < 1 := by norm_num
@@ -257,8 +254,7 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
       rw [← hg, @smul_smul, inv_mul_cancel₀, one_smul]
       by_contra hn
       simp [hn] at h2
-    have hg2 : norm (g 2)⁻¹ = 1 := by
-      simp [h2]
+    have hg2 : norm (g 2)⁻¹ = 1 := by simp [h2]
     have hg22 : ∃ (τ : ℝ), (g 2)⁻¹ = Complex.exp (τ * I) := by
       rw [← norm_mul_exp_arg_mul_I (g 2)⁻¹]
       use arg (g 2)⁻¹
diff --git a/Physlib/Particles/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean b/Physlib/Particles/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
index 63a9f9b0e..6f7d04085 100644
--- a/Physlib/Particles/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
+++ b/Physlib/Particles/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
@@ -155,21 +155,17 @@ lemma S₂₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) :
   rw [S₂₃, if_pos ha]
 
 lemma S₂₃_of_Vub_ne_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
-    S₂₃ V = VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by
-  rw [S₂₃, if_neg ha]
+    S₂₃ V = VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by rw [S₂₃, if_neg ha]
 
 end sines
 
 section cosines
 
-lemma C₁₂_eq_ℂcos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₂ V) = C₁₂ V := by
-  simp [C₁₂]
+lemma C₁₂_eq_ℂcos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₂ V) = C₁₂ V := by simp [C₁₂]
 
-lemma C₁₃_eq_ℂcos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₃ V) = C₁₃ V := by
-  simp [C₁₃]
+lemma C₁₃_eq_ℂcos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₃ V) = C₁₃ V := by simp [C₁₃]
 
-lemma C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₂₃ V) = C₂₃ V := by
-  simp [C₂₃]
+lemma C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₂₃ V) = C₂₃ V := by simp [C₂₃]
 
 lemma complexAbs_cos_θ₁₂ (V : Quotient CKMMatrixSetoid) : norm (Complex.cos (θ₁₂ V)) =
     cos (θ₁₂ V) := by
@@ -541,8 +537,7 @@ lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
       rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
       rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃_of_Vub_one h1]
       simp only [Fin.isValue, ofReal_one, one_mul, ofReal_zero, mul_one, VcdAbs, zero_mul, sub_zero]
-      have h3 : (Real.cos (θ₂₃ ⟦V⟧) : ℂ) = √(1 - S₂₃ ⟦V⟧ ^ 2) := by
-        rw [θ₂₃, Real.cos_arcsin]
+      have h3 : (Real.cos (θ₂₃ ⟦V⟧) : ℂ) = √(1 - S₂₃ ⟦V⟧ ^ 2) := by rw [θ₂₃, Real.cos_arcsin]
       simp only [ofReal_cos] at h3
       rw [h3, S₂₃_of_Vub_eq_one h1, hV.2.2.2.2.2.2]
     · simp only [cRow, Fin.isValue, Fin.reduceFinMk, cons_val_two, Nat.succ_eq_add_one,
diff --git a/Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean b/Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean
index bfddec06f..56b4baa1d 100644
--- a/Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean
+++ b/Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean
@@ -33,10 +33,7 @@ variable {n : ℕ}
 
 lemma sum_SMSpecies_numberCharges_one {M} [AddCommMonoid M]
     (f : Fin (SMSpecies 1).numberCharges → M) :
-    ∑ i, f i = f ⟨0, by simp⟩ := by
-  change ∑ (i : Fin 1), f i = _
-  simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton]
-  rfl
+    ∑ i, f i = f ⟨0, by simp⟩ := Fin.sum_univ_one f
 
 /-- An equivalence between the set `(SMCharges n).charges` and the set
   `(Fin 5 → Fin n → ℚ)`. -/
diff --git a/Physlib/Particles/StandardModel/AnomalyCancellation/FamilyMaps.lean b/Physlib/Particles/StandardModel/AnomalyCancellation/FamilyMaps.lean
index a0730be03..62a33bcfa 100644
--- a/Physlib/Particles/StandardModel/AnomalyCancellation/FamilyMaps.lean
+++ b/Physlib/Particles/StandardModel/AnomalyCancellation/FamilyMaps.lean
@@ -36,8 +36,7 @@ def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMSpecies n).Charges →ₗ[ℚ] (S
     intro i
     rw [map_smul]
     rw [toSMSpecies_toSpecies_inv, toSMSpecies_toSpecies_inv]
-    rw [map_smul]
-    rfl
+    exact (f.comp (toSpecies i)).map_smul a S
 
 /-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
 @[simps!]
@@ -73,8 +72,7 @@ def speciesEmbed (m n : ℕ) :
     simp only [HSMul.hSMul, ACCSystemCharges.chargesModule_smul, eq_ratCast, Rat.cast_eq_id, id_eq]
     by_cases hi : i.val < m
     · rw [dif_pos hi, dif_pos hi]
-    · rw [dif_neg hi, dif_neg hi]
-      exact Eq.symm (Rat.mul_zero a)
+    · simp [dif_neg hi]
 
 /-- The embedding of the `m`-family charges onto the `n`-family charges, with all
 other charges zero. -/
diff --git a/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/Basic.lean b/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/Basic.lean
index d7da16d48..f26e674d9 100644
--- a/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/Basic.lean
+++ b/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/Basic.lean
@@ -31,9 +31,7 @@ def SMNoGrav (n : ℕ) : ACCSystem where
     | 0 => @accSU2 n
     | 1 => accSU3
   numberQuadratic := 0
-  quadraticACCs := by
-    intro i
-    exact Fin.elim0 i
+  quadraticACCs := fun i => Fin.elim0 i
   cubicACC := accCube
 
 namespace SMNoGrav
diff --git a/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/One/LinearParameterization.lean b/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/One/LinearParameterization.lean
index efb447d2f..aad78b182 100644
--- a/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/One/LinearParameterization.lean
+++ b/Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/One/LinearParameterization.lean
@@ -82,8 +82,7 @@ def asLinear (S : linearParameters) : (SMNoGrav 1).LinSols :=
       LinearMap.coe_mk, AddHom.coe_mk, speciesVal, asCharges, neg_add_rev]
     ring)
 
-lemma asLinear_val (S : linearParameters) : S.asLinear.val = S.asCharges := by
-  rfl
+lemma asLinear_val (S : linearParameters) : S.asLinear.val = S.asCharges := rfl
 
 set_option backward.isDefEq.respectTransparency false in
 lemma cubic (S : linearParameters) :
@@ -295,8 +294,7 @@ lemma cubic (S : linearParametersQENeqZero) :
   field_simp
   simp [hQ]
   ring_nf
-  have h1 : -216 - S.v ^ 3 * 216 - S.w ^ 3 * 216 = - 216 *(S.v ^3 + S.w ^3 +1) := by
-    ring
+  have h1 : -216 - S.v ^ 3 * 216 - S.w ^ 3 * 216 = - 216 *(S.v ^3 + S.w ^3 +1) := by ring
   rw [h1]
   simp_all
   exact add_eq_zero_iff_eq_neg
diff --git a/Physlib/Particles/StandardModel/Basic.lean b/Physlib/Particles/StandardModel/Basic.lean
index 18c72a2fe..42bbacfc6 100644
--- a/Physlib/Particles/StandardModel/Basic.lean
+++ b/Physlib/Particles/StandardModel/Basic.lean
@@ -141,8 +141,7 @@ noncomputable def gaugeGroupℤ₆SU3OfRoot (α : rootsOfUnity 6 ℂ) :
   let z : ℂ := ((α : ℂˣ) : ℂ)
   ⟨scalar (Fin 3) (z ^ 2), by
     rw [mem_specialUnitaryGroup_iff]
-    have hz : ‖z‖ = 1 := by
-      simpa [z] using Complex.norm_eq_one_of_mem_rootsOfUnity α.prop
+    have hz : ‖z‖ = 1 := by simpa [z] using Complex.norm_eq_one_of_mem_rootsOfUnity α.prop
     have hz2 : star (z ^ 2) * z ^ 2 = 1 := by
       rw [RCLike.star_def, Complex.conj_mul', Complex.norm_pow, hz]
       norm_num
@@ -152,8 +151,7 @@ noncomputable def gaugeGroupℤ₆SU3OfRoot (α : rootsOfUnity 6 ℂ) :
         Matrix.diagonal_mul_diagonal, Matrix.diagonal_eq_one]
       funext i
       simpa [Pi.star_def] using hz2
-    · have hα : z ^ 6 = 1 := by
-        simpa [z] using (mem_rootsOfUnity' 6 (α : ℂˣ)).mp α.prop
+    · have hα : z ^ 6 = 1 := by simpa [z] using (mem_rootsOfUnity' 6 (α : ℂˣ)).mp α.prop
       rw [Matrix.scalar_apply, Matrix.det_diagonal, Fin.prod_univ_three]
       calc
         z ^ 2 * z ^ 2 * z ^ 2 = z ^ 6 := by ring
@@ -172,8 +170,7 @@ noncomputable def gaugeGroupℤ₆SU2OfRoot (α : rootsOfUnity 6 ℂ) :
       star ((u ^ 3 : unitary ℂ) : ℂ) = 1
     rw [star_star]
     exact (u ^ 3 : unitary ℂ).prop.2
-  have hα : z ^ 6 = 1 := by
-    simpa [z] using (mem_rootsOfUnity' 6 (α : ℂˣ)).mp α.prop
+  have hα : z ^ 6 = 1 := by simpa [z] using (mem_rootsOfUnity' 6 (α : ℂˣ)).mp α.prop
   have hw2 : w ^ 2 = 1 := by
     calc
       w ^ 2 = star (z ^ 6) := by
diff --git a/Physlib/Particles/StandardModel/HiggsBoson/Basic.lean b/Physlib/Particles/StandardModel/HiggsBoson/Basic.lean
index 5a647eab3..47bdbd260 100644
--- a/Physlib/Particles/StandardModel/HiggsBoson/Basic.lean
+++ b/Physlib/Particles/StandardModel/HiggsBoson/Basic.lean
@@ -176,8 +176,7 @@ lemma gaugeGroupI_smul_eq_U1_mul_SU2 (g : StandardModel.GaugeGroupI) (φ : Higgs
 
 lemma gaugeGroupI_smul_eq_U1_smul_SU2 (g : StandardModel.GaugeGroupI) (φ : HiggsVec) :
     g • φ = (WithLp.toLp 2 <| (g.toU1 ^ 3 • g.toSU2.1) *ᵥ φ.ofLp) := by
-  rw [gaugeGroupI_smul_eq]
-  rw [Matrix.smul_mulVec]
+  rw [gaugeGroupI_smul_eq, Matrix.smul_mulVec]
 
 instance : MulAction StandardModel.GaugeGroupI HiggsVec where
   one_smul φ := by simp [gaugeGroupI_smul_eq]
@@ -224,8 +223,7 @@ lemma gaugeGroupI_smul_inner (g : StandardModel.GaugeGroupI) (φ ψ : HiggsVec)
     _ = (g.toSU2.1 *ᵥ (g.toU1 ^ 3 • ψ)) ⬝ᵥ star (g.toSU2.1 *ᵥ (g.toU1 ^ 3 • φ)) := by
       rw [gaugeGroupI_smul_eq_U1_mul_SU2, gaugeGroupI_smul_eq_U1_mul_SU2]
     _ = (g.toSU2.1 *ᵥ (g.toU1 ^ 3 • ψ)) ⬝ᵥ (star ((g.toU1 ^ 3 • φ)) ᵥ* star (g.toSU2.1)) := by
-      rw [star_mulVec]
-      rfl
+      exact congrArg (fun v => _ ⬝ᵥ v) (star_mulVec (g.toSU2.1) (g.toU1 ^ 3 • φ))
     _ = ((star (g.toSU2.1) * g.toSU2.1) *ᵥ (g.toU1 ^ 3 • ψ)) ⬝ᵥ star ((g.toU1 ^ 3 • φ)) := by
       rw [dotProduct_comm, ← Matrix.dotProduct_mulVec, dotProduct_comm, mulVec_mulVec]
       rfl
@@ -450,8 +448,7 @@ def toRealScalars : HiggsVec →ₗ[ℝ] (Fin 4 → ℝ) where
     fin_cases i <;> simp
 
 lemma toRealScalars_smul_real (a : ℝ) (φ : HiggsVec) :
-    toRealScalars (a • φ) = a • toRealScalars φ := by
-  simp [toRealScalars]
+    toRealScalars (a • φ) = a • toRealScalars φ := by simp [toRealScalars]
 
 lemma ofReal_toRealScalars (a : ℝ) :
     toRealScalars (ofReal a) = !₄[Real.sqrt a, 0, 0, 0] := by
@@ -778,8 +775,7 @@ lemma normSq_expand (φ : HiggsField) :
 
 -/
 
-lemma normSq_nonneg (φ : HiggsField) (x : SpaceTime) : 0 ≤ ‖φ‖_H^2 x := by
-  simp [normSq]
+lemma normSq_nonneg (φ : HiggsField) (x : SpaceTime) : 0 ≤ ‖φ‖_H^2 x := by simp [normSq]
 
 /-!
 
@@ -817,8 +813,7 @@ lemma normSq_smooth (φ : HiggsField) : ContMDiff 𝓘(ℝ, SpaceTime) 𝓘(ℝ,
 
 @[simp]
 lemma const_normSq (φ : HiggsVec) (x : SpaceTime) :
-    ‖const φ‖_H^2 x = ‖φ‖ ^ 2 := by
-  simp [normSq, const_apply]
+    ‖const φ‖_H^2 x = ‖φ‖ ^ 2 := by simp [normSq, const_apply]
 
 /-!
 
diff --git a/Physlib/Particles/StandardModel/HiggsBoson/Potential.lean b/Physlib/Particles/StandardModel/HiggsBoson/Potential.lean
index d0dc88789..8da1593e5 100644
--- a/Physlib/Particles/StandardModel/HiggsBoson/Potential.lean
+++ b/Physlib/Particles/StandardModel/HiggsBoson/Potential.lean
@@ -89,8 +89,7 @@ lemma 𝓵_neg : P.neg.𝓵 = - P.𝓵 := by rfl
 -/
 
 @[simp]
-lemma toFun_zero (x : SpaceTime) : P.toFun 0 x = 0 := by
-  simp [toFun]
+lemma toFun_zero (x : SpaceTime) : P.toFun 0 x = 0 := by simp [toFun]
 
 lemma complete_square (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
     P.toFun φ x = P.𝓵 * (‖φ‖_H^2 x - P.μ2 / (2 * P.𝓵)) ^ 2 - P.μ2 ^ 2 / (4 * P.𝓵) := by
diff --git a/Physlib/Particles/StandardModel/Representations.lean b/Physlib/Particles/StandardModel/Representations.lean
index 90b7fd5d5..5b81c6cec 100644
--- a/Physlib/Particles/StandardModel/Representations.lean
+++ b/Physlib/Particles/StandardModel/Representations.lean
@@ -41,8 +41,7 @@ noncomputable def repU1 : unitary ℂ →* unitaryGroup (Fin 2) ℂ where
   toFun g := repU1Map g
   map_mul' g h := by
     simp only [repU1Map, Submonoid.mk_mul_mk, mul_smul_one, smul_smul, mul_comm, ← mul_pow]
-  map_one' := by
-    simp only [repU1Map, one_pow, one_smul, Submonoid.mk_eq_one]
+  map_one' := by simp only [repU1Map, one_pow, one_smul, Submonoid.mk_eq_one]
 
 /-- The fundamental representation of SU(2) as a homomorphism to `unitaryGroup (Fin 2) ℂ`. -/
 @[simps!]
diff --git a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/B3.lean b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/B3.lean
index 2cba2d097..407e88b8e 100644
--- a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/B3.lean
+++ b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/B3.lean
@@ -61,8 +61,7 @@ def B₃ : MSSMACC.Sols :=
     (by with_unfolding_all rfl) (by with_unfolding_all rfl) (by with_unfolding_all rfl)
     (by with_unfolding_all rfl) (by with_unfolding_all rfl)
 
-lemma B₃_val : B₃.val = B₃AsCharge := by
-  rfl
+lemma B₃_val : B₃.val = B₃AsCharge := rfl
 
 set_option backward.isDefEq.respectTransparency false in
 lemma doublePoint_B₃_B₃ (R : MSSMACC.LinSols) : cubeTriLin B₃.val B₃.val R.val = 0 := by
diff --git a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Basic.lean b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Basic.lean
index 340c8f26d..ce7c618a1 100644
--- a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Basic.lean
+++ b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Basic.lean
@@ -31,10 +31,7 @@ namespace MSSMCharges
 
 lemma sum_MSSMSpecies_numberCharges_eq_expand [AddCommMonoid M]
     (f : Fin MSSMSpecies.numberCharges → M) :
-    ∑ i, f i = f ⟨0, by simp⟩ + f ⟨1, by simp⟩ + f ⟨2, by simp⟩ := by
-  change ∑ (i : Fin 3), f i = f ⟨0, by simp⟩ + f ⟨1, by simp⟩ + f ⟨2, by simp⟩
-  simp only [Fin.sum_univ_three]
-  rfl
+    ∑ i, f i = f ⟨0, by simp⟩ + f ⟨1, by simp⟩ + f ⟨2, by simp⟩ := Fin.sum_univ_three f
 
 /-- An equivalence between `MSSMCharges.charges` and the space of maps
 `(Fin 18 ⊕ Fin 2 → ℚ)`. The first 18 factors corresponds to the SM fermions, while the last two
@@ -125,12 +122,10 @@ lemma charges_eq_toSpecies_eq (S T : MSSMCharges.Charges) :
       | 1 => exact h.2.2
 
 lemma Hd_toSpecies_inv (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) :
-    Hd (toSpecies.symm f) = f.2 0 := by
-  rfl
+    Hd (toSpecies.symm f) = f.2 0 := rfl
 
 lemma Hu_toSpecies_inv (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) :
-    Hu (toSpecies.symm f) = f.2 1 := by
-  rfl
+    Hu (toSpecies.symm f) = f.2 1 := rfl
 
 end MSSMCharges
 
@@ -455,8 +450,7 @@ def AnomalyFreeMk (S : MSSMACC.Charges) (hg : accGrav S = 0)
 lemma AnomalyFreeMk_val (S : MSSMACC.Charges) (hg : accGrav S = 0)
     (hsu2 : accSU2 S = 0) (hsu3 : accSU3 S = 0) (hyy : accYY S = 0)
     (hquad : accQuad S = 0) (hcube : accCube S = 0) :
-    (AnomalyFreeMk S hg hsu2 hsu3 hyy hquad hcube).val = S := by
-  rfl
+    (AnomalyFreeMk S hg hsu2 hsu3 hyy hquad hcube).val = S := rfl
 
 /-- A `QuadSol` from a `LinSol` satisfying the quadratic ACC. -/
 @[simp]
@@ -483,8 +477,7 @@ def AnomalyFreeMk'' (S : MSSMACC.QuadSols) (hcube : accCube S.val = 0) : MSSMACC
 
 lemma AnomalyFreeMk''_val (S : MSSMACC.QuadSols)
     (hcube : accCube S.val = 0) :
-    (AnomalyFreeMk'' S hcube).val = S.val := by
-  rfl
+    (AnomalyFreeMk'' S hcube).val = S.val := rfl
 
 set_option backward.isDefEq.respectTransparency false in
 /-- The dot product on the vector space of charges. -/
diff --git a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/Basic.lean b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/Basic.lean
index 2c038fff9..475c8ce0b 100644
--- a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/Basic.lean
+++ b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/Basic.lean
@@ -58,23 +58,20 @@ def proj (T : MSSMACC.LinSols) : MSSMACC.AnomalyFreePerp :=
 lemma proj_val (T : MSSMACC.LinSols) :
     (proj T).val = (dot B₃.val T.val - dot Y₃.val T.val) • Y₃.val +
     (dot Y₃.val T.val - 2 * dot B₃.val T.val) • B₃.val +
-    dot Y₃.val B₃.val • T.val := by
-  rfl
+    dot Y₃.val B₃.val • T.val := rfl
 
 lemma Y₃_plus_B₃_plus_proj (T : MSSMACC.LinSols) (a b c : ℚ) :
     a • Y₃.val + b • B₃.val + c • (proj T).val =
     (a + c * (dot B₃.val T.val - dot Y₃.val T.val)) • Y₃.val
     + (b + c * (dot Y₃.val T.val - 2 * dot B₃.val T.val)) • B₃.val
     + (dot Y₃.val B₃.val * c) • T.val:= by
-  rw [proj_val]
-  rw [DistribMulAction.smul_add, DistribMulAction.smul_add]
+  rw [proj_val, DistribMulAction.smul_add, DistribMulAction.smul_add]
   module
 
 set_option backward.isDefEq.respectTransparency false in
 lemma quad_Y₃_proj (T : MSSMACC.LinSols) :
     quadBiLin Y₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin Y₃.val T.val := by
-  rw [proj_val]
-  rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
+  rw [proj_val, quadBiLin.map_add₂, quadBiLin.map_add₂]
   rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
   rw [show quadBiLin Y₃.val B₃.val = 0 by with_unfolding_all rfl]
   rw [show quadBiLin Y₃.val Y₃.val = 0 by with_unfolding_all rfl]
@@ -83,8 +80,7 @@ lemma quad_Y₃_proj (T : MSSMACC.LinSols) :
 set_option backward.isDefEq.respectTransparency false in
 lemma quad_B₃_proj (T : MSSMACC.LinSols) :
     quadBiLin B₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin B₃.val T.val := by
-  rw [proj_val]
-  rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
+  rw [proj_val, quadBiLin.map_add₂, quadBiLin.map_add₂]
   rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
   rw [show quadBiLin B₃.val Y₃.val = 0 by with_unfolding_all rfl]
   rw [show quadBiLin B₃.val B₃.val = 0 by with_unfolding_all rfl]
@@ -95,8 +91,7 @@ lemma quad_self_proj (T : MSSMACC.Sols) :
     quadBiLin T.val (proj T.1.1).val =
     (dot B₃.val T.val - dot Y₃.val T.val) * quadBiLin Y₃.val T.val +
     (dot Y₃.val T.val - 2 * dot B₃.val T.val) * quadBiLin B₃.val T.val := by
-  rw [proj_val]
-  rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
+  rw [proj_val, quadBiLin.map_add₂, quadBiLin.map_add₂]
   rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
   rw [← quadBiLin.toHomogeneousQuad_apply]
   rw [← accQuad]
@@ -119,8 +114,7 @@ set_option backward.isDefEq.respectTransparency false in
 lemma cube_proj_proj_Y₃ (T : MSSMACC.LinSols) :
     cubeTriLin (proj T).val (proj T).val Y₃.val =
     (dot Y₃.val B₃.val)^2 * cubeTriLin T.val T.val Y₃.val := by
-  rw [proj_val]
-  rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
+  rw [proj_val, cubeTriLin.map_add₁, cubeTriLin.map_add₂]
   conv_lhs =>
     enter [1, 1]
     change ((cubeTriLin (lineY₃B₃ _ _).val) (lineY₃B₃ _ _).val) _
@@ -145,8 +139,7 @@ set_option backward.isDefEq.respectTransparency false in
 lemma cube_proj_proj_B₃ (T : MSSMACC.LinSols) :
     cubeTriLin (proj T).val (proj T).val B₃.val =
     (dot Y₃.val B₃.val)^2 * cubeTriLin T.val T.val B₃.val := by
-  rw [proj_val]
-  rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
+  rw [proj_val, cubeTriLin.map_add₁, cubeTriLin.map_add₂]
   rw [← lineY₃B₃_val, lineY₃B₃_doublePoint, lineY₃B₃_val]
   rw [cubeTriLin.map_add₂, cubeTriLin.swap₂, cubeTriLin.map_add₁, cubeTriLin.map_smul₁,
     cubeTriLin.map_smul₃, doublePoint_Y₃_B₃]
@@ -164,8 +157,7 @@ lemma cube_proj_proj_self (T : MSSMACC.Sols) :
     2 * dot Y₃.val B₃.val *
     ((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin T.val T.val Y₃.val +
     (dot Y₃.val T.val - 2 * dot B₃.val T.val) * cubeTriLin T.val T.val B₃.val) := by
-  rw [proj_val]
-  rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
+  rw [proj_val, cubeTriLin.map_add₁, cubeTriLin.map_add₂]
   rw [← lineY₃B₃_val, lineY₃B₃_doublePoint, lineY₃B₃_val]
   repeat rw [cubeTriLin.map_add₁]
   repeat rw [cubeTriLin.map_smul₁]
diff --git a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/PlaneWithY3B3.lean b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/PlaneWithY3B3.lean
index dae9178f8..9d36dbf14 100644
--- a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/PlaneWithY3B3.lean
+++ b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/PlaneWithY3B3.lean
@@ -30,8 +30,7 @@ def planeY₃B₃ (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) : MSSMACC.LinSols
   a • Y₃.1.1 + b • B₃.1.1 + c • R.1
 
 lemma planeY₃B₃_val (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
-    (planeY₃B₃ R a b c).val = a • Y₃.val + b • B₃.val + c • R.val := by
-  rfl
+    (planeY₃B₃ R a b c).val = a • Y₃.val + b • B₃.val + c • R.val := rfl
 
 lemma planeY₃B₃_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
     planeY₃B₃ R (d * a) (d * b) (d * c) = d • planeY₃B₃ R a b c := by
@@ -42,8 +41,7 @@ lemma planeY₃B₃_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
   rw [smul_smul, smul_smul, smul_smul]
 
 lemma planeY₃B₃_eq (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (h : a = a' ∧ b = b' ∧ c = c') :
-    (planeY₃B₃ R a b c) = (planeY₃B₃ R a' b' c') := by
-  rw [h.1, h.2.1, h.2.2]
+    (planeY₃B₃ R a b c) = (planeY₃B₃ R a' b' c') := by rw [h.1, h.2.1, h.2.2]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R.val ≠ 0)
@@ -67,8 +65,7 @@ set_option backward.isDefEq.respectTransparency false in
 lemma planeY₃B₃_quad (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
     accQuad (planeY₃B₃ R a b c).val = c * (2 * a * quadBiLin Y₃.val R.val
     + 2 * b * quadBiLin B₃.val R.val + c * quadBiLin R.val R.val) := by
-  rw [planeY₃B₃_val]
-  rw [accQuad, BiLinearSymm.toHomogeneousQuad_add]
+  rw [planeY₃B₃_val, accQuad, BiLinearSymm.toHomogeneousQuad_add]
   rw [← lineY₃B₃Charges_val, ← accQuad]
   rw [lineY₃B₃Charges_quad]
   rw [lineY₃B₃Charges_val, accQuad]
@@ -83,8 +80,7 @@ lemma planeY₃B₃_cubic (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
     accCube (planeY₃B₃ R a b c).val = c ^ 2 *
     (3 * a * cubeTriLin R.val R.val Y₃.val
     + 3 * b * cubeTriLin R.val R.val B₃.val + c * cubeTriLin R.val R.val R.val) := by
-  rw [planeY₃B₃_val]
-  rw [accCube, TriLinearSymm.toCubic_add, ← accCube]
+  rw [planeY₃B₃_val, accCube, TriLinearSymm.toCubic_add, ← accCube]
   rw [← lineY₃B₃Charges_val]
   rw [lineY₃B₃Charges_cubic]
   rw [TriLinearSymm.map_smul₃, lineY₃B₃Charges_val, ← lineY₃B₃_val]
@@ -105,8 +101,7 @@ def lineQuadAFL (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) : MSSMACC.LinSols
 
 lemma lineQuadAFL_quad (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) :
     accQuad (lineQuadAFL R c1 c2 c3).val = 0 := by
-  rw [lineQuadAFL, planeY₃B₃_quad]
-  rw [mul_eq_zero]
+  rw [lineQuadAFL, planeY₃B₃_quad, mul_eq_zero]
   apply Or.inr
   ring
 
@@ -118,8 +113,7 @@ lemma lineQuad_val (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) :
     (lineQuad R c1 c2 c3).val = (planeY₃B₃ R
     (c2 * quadBiLin R.val R.val - 2 * c3 * quadBiLin B₃.val R.val)
     (2 * c3 * quadBiLin Y₃.val R.val - c1 * quadBiLin R.val R.val)
-    (2 * c1 * quadBiLin B₃.val R.val - 2 * c2 * quadBiLin Y₃.val R.val)).val := by
-  rfl
+    (2 * c1 * quadBiLin B₃.val R.val - 2 * c2 * quadBiLin Y₃.val R.val)).val := rfl
 
 lemma lineQuad_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
     lineQuad R (d * a) (d * b) (d * c) = d • lineQuad R a b c := by
@@ -148,8 +142,7 @@ lemma lineQuad_cube (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ : ℚ) :
     accCube (lineQuad R c₁ c₂ c₃).val =
     - 4 * (c₁ * quadBiLin B₃.val R.val - c₂ * quadBiLin Y₃.val R.val) ^ 2 *
     (α₁ R * c₁ + α₂ R * c₂ + α₃ R * c₃) := by
-  rw [lineQuad_val]
-  rw [planeY₃B₃_cubic, α₁, α₂, α₃]
+  rw [lineQuad_val, planeY₃B₃_cubic, α₁, α₂, α₃]
   ring
 
 /-- The line in the plane spanned by `Y₃`, `B₃` and `R` which is in the cubic. -/
@@ -196,19 +189,14 @@ lemma α₂_proj (T : MSSMACC.Sols) : α₂ (proj T.1.1) =
 
 lemma α₁_proj (T : MSSMACC.Sols) : α₁ (proj T.1.1) =
     - α₃ (proj T.1.1) * (dot B₃.val T.val - dot Y₃.val T.val) := by
-  rw [α₃_proj, α₁]
-  rw [cube_proj_proj_B₃, quad_B₃_proj, quad_proj, cube_proj]
+  rw [α₃_proj, α₁, cube_proj_proj_B₃, quad_B₃_proj, quad_proj, cube_proj]
   ring
 
 lemma α₁_proj_zero (T : MSSMACC.Sols) (h1 : α₃ (proj T.1.1) = 0) :
-    α₁ (proj T.1.1) = 0 := by
-  rw [α₁_proj, h1]
-  exact mul_eq_zero_of_left rfl ((dot B₃.val) T.val - (dot Y₃.val) T.val)
+    α₁ (proj T.1.1) = 0 := by simp [α₁_proj, h1]
 
 lemma α₂_proj_zero (T : MSSMACC.Sols) (h1 : α₃ (proj T.1.1) = 0) :
-    α₂ (proj T.1.1) = 0 := by
-  rw [α₂_proj, h1]
-  exact mul_eq_zero_of_left rfl ((dot Y₃.val) T.val - 2 * (dot B₃.val) T.val)
+    α₂ (proj T.1.1) = 0 := by simp [α₂_proj, h1]
 
 end proj
 
diff --git a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/ToSols.lean b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/ToSols.lean
index f1d2b3469..be63be629 100644
--- a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/ToSols.lean
+++ b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/OrthogY3B3/ToSols.lean
@@ -70,8 +70,7 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
     simp only [mul_eq_zero, OfNat.ofNat_ne_zero, false_or] at h
     rw [α₁_proj, α₂_proj, h]
     simp only [neg_zero, zero_mul, and_self]
-  · rw [h.2.2]
-    exact Rat.mul_zero ((dot Y₃.val) B₃.val)
+  · simp [h.2.2]
 
 /-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
 entirely in the quadratic surface. -/
@@ -152,8 +151,7 @@ lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
 
 lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
     InCubeSolProp R ↔ InCubeProp (proj R.1.1) := by
-  rw [InCubeSolProp, InCubeProp]
-  rw [cube_proj, cube_proj_proj_Y₃, cube_proj_proj_B₃]
+  rw [InCubeSolProp, InCubeProp, cube_proj, cube_proj_proj_Y₃, cube_proj_proj_B₃]
   refine Iff.intro (fun h => ?_) (fun h => ?_)
   · rw [h.1, h.2]
     simp only [mul_zero, add_zero, and_self]
@@ -199,8 +197,7 @@ def toSolNSQuad (R : MSSMACC.AnomalyFreePerp) : MSSMACC.QuadSols :=
 
 lemma toSolNSQuad_cube (R : MSSMACC.AnomalyFreePerp) :
     accCube (toSolNSQuad R).val = 0 := by
-  rw [toSolNSQuad]
-  rw [lineQuad_val]
+  rw [toSolNSQuad, lineQuad_val]
   rw [planeY₃B₃_cubic]
   ring
 
@@ -260,9 +257,7 @@ def inLineEqProj (T : InLineEqSol) : InLineEq × ℚ × ℚ × ℚ :=
 lemma inLineEqTo_smul (R : InLineEq) (c₁ c₂ c₃ d : ℚ) :
     inLineEqToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inLineEqToSol (R, c₁, c₂, c₃) := by
   apply ACCSystem.Sols.ext
-  change (lineQuad _ _ _ _).val = _
-  rw [lineQuad_smul]
-  rfl
+  exact congrArg (fun x => x.val) (lineQuad_smul R.val c₁ c₂ c₃ d)
 
 lemma inLineEqToSol_proj (T : InLineEqSol) : inLineEqToSol (inLineEqProj T) = T.val := by
   rw [inLineEqProj, inLineEqTo_smul]
@@ -292,9 +287,7 @@ def inQuadToSol : InQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a₁,
 lemma inQuadToSol_smul (R : InQuad) (c₁ c₂ c₃ d : ℚ) :
     inQuadToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inQuadToSol (R, c₁, c₂, c₃) := by
   apply ACCSystem.Sols.ext
-  change (lineCube _ _ _ _).val = _
-  rw [lineCube_smul]
-  rfl
+  exact congrArg (fun x => x.val) (lineCube_smul R.val.val c₁ c₂ c₃ d)
 
 /-- On elements of `inQuadSol` a right-inverse to `inQuadToSol`. -/
 def inQuadProj (T : InQuadSol) : InQuad × ℚ × ℚ × ℚ :=
@@ -332,9 +325,7 @@ def inQuadCubeToSol : InQuadCube × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R
 lemma inQuadCubeToSol_smul (R : InQuadCube) (c₁ c₂ c₃ d : ℚ) :
     inQuadCubeToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inQuadCubeToSol (R, c₁, c₂, c₃) := by
   apply ACCSystem.Sols.ext
-  change (planeY₃B₃ _ _ _ _).val = _
-  rw [planeY₃B₃_smul]
-  rfl
+  exact congrArg (fun x => x.val) (planeY₃B₃_smul R.val.val.val c₁ c₂ c₃ d)
 
 /-- On elements of `inQuadCubeSol` a right-inverse to `inQuadCubeToSol`. -/
 def inQuadCubeProj (T : InQuadCubeSol) : InQuadCube × ℚ × ℚ × ℚ :=
@@ -355,8 +346,7 @@ lemma inQuadCubeToSol_proj (T : InQuadCubeSol) :
   simp only [zero_smul, add_zero, zero_add]
   rw [← SemigroupAction.mul_smul, mul_comm, mul_inv_cancel₀]
   · exact MulAction.one_smul (T.1).val
-  · rw [show dot Y₃.val B₃.val = 108 by with_unfolding_all rfl]
-    exact Ne.symm (OfNat.zero_ne_ofNat 108)
+  · simp [show dot Y₃.val B₃.val = 108 by with_unfolding_all rfl]
 
 /-- A solution from an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ`. We will
 show that this map is a surjection. -/
diff --git a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Y3.lean b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Y3.lean
index 87a0f57c4..b3b6b522e 100644
--- a/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Y3.lean
+++ b/Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Y3.lean
@@ -61,8 +61,7 @@ def Y₃ : MSSMACC.Sols :=
     (by with_unfolding_all rfl) (by with_unfolding_all rfl) (by with_unfolding_all rfl)
     (by with_unfolding_all rfl) (by with_unfolding_all rfl) (by with_unfolding_all rfl)
 
-lemma Y₃_val : Y₃.val = Y₃AsCharge := by
-  rfl
+lemma Y₃_val : Y₃.val = Y₃AsCharge := rfl
 
 set_option backward.isDefEq.respectTransparency false in
 lemma doublePoint_Y₃_Y₃ (R : MSSMACC.LinSols) :
diff --git a/Physlib/Particles/SuperSymmetry/N1/Basic.lean b/Physlib/Particles/SuperSymmetry/N1/Basic.lean
index 30c27e517..a27b42f1f 100644
--- a/Physlib/Particles/SuperSymmetry/N1/Basic.lean
+++ b/Physlib/Particles/SuperSymmetry/N1/Basic.lean
@@ -413,13 +413,11 @@ def conjChiralCovector
 `toField`. -/
 lemma toField_conjScalar (t : (chiralTensor (ι := ι)).Tensor ![]) :
     (conjScalar t).toField = star t.toField := by
-  rw [conjScalar, toField_permT]
-  rw [toField_eq_repr, toField_eq_repr]
+  rw [conjScalar, toField_permT, toField_eq_repr, toField_eq_repr]
   change componentMap (S := (chiralTensor (ι := ι)).toTensorSpecies)
       ((chiralTensor (ι := ι)).bar ∘ ![]) ((chiralTensor (ι := ι)).conjT t) (fun j => Fin.elim0 j) =
     star ((basis (S := (chiralTensor (ι := ι)).toTensorSpecies) ![]).repr t (fun j => Fin.elim0 j))
-  rw [ConjTensorSpecies.componentMap_conjT (S := chiralTensor (ι := ι))]
-  rfl
+  exact ConjTensorSpecies.componentMap_conjT (S := chiralTensor (ι := ι)) t (fun j => Fin.elim0 j)
 
 /-- Component formula for the holomorphic covector conjugate: the `![I]` basis component of
 `conjChiralCovector t` is the complex conjugate of the `![I]` component of `t`. -/
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Basic.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Basic.lean
index 7241858a8..56408a140 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Basic.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Basic.lean
@@ -176,8 +176,7 @@ instance subsetDecidable [DecidableEq 𝓩] (x y : ChargeSpectrum 𝓩) : Decida
   instDecidableAnd
 
 lemma subset_def {x y : ChargeSpectrum 𝓩} : x ⊆ y ↔ x.qHd.toFinset ⊆ y.qHd.toFinset ∧
-    x.qHu.toFinset ⊆ y.qHu.toFinset ∧ x.Q5 ⊆ y.Q5 ∧ x.Q10 ⊆ y.Q10 := by
-  rfl
+    x.qHu.toFinset ⊆ y.qHu.toFinset ∧ x.Q5 ⊆ y.Q5 ∧ x.Q10 ⊆ y.Q10 := Iff.rfl
 
 @[simp, refl]
 lemma subset_refl (x : ChargeSpectrum 𝓩) : x ⊆ x := ⟨by rfl, by rfl, by rfl, by rfl⟩
@@ -208,8 +207,7 @@ instance emptyInst : EmptyCollection (ChargeSpectrum 𝓩) where
 lemma empty_eq : (∅ : ChargeSpectrum 𝓩) = ⟨none, none, {}, {}⟩ := rfl
 
 @[simp]
-lemma empty_subset (x : ChargeSpectrum 𝓩) : ∅ ⊆ x := by
-  simp [Subset, empty_eq]
+lemma empty_subset (x : ChargeSpectrum 𝓩) : ∅ ⊆ x := by simp [Subset, empty_eq]
 
 @[simp]
 lemma subset_of_empty_iff_empty {x : ChargeSpectrum 𝓩} :
@@ -219,20 +217,16 @@ lemma subset_of_empty_iff_empty {x : ChargeSpectrum 𝓩} :
   simp
 
 @[simp]
-lemma empty_qHd : (∅ : ChargeSpectrum 𝓩).qHd = none := by
-  simp [empty_eq]
+lemma empty_qHd : (∅ : ChargeSpectrum 𝓩).qHd = none := by simp [empty_eq]
 
 @[simp]
-lemma empty_qHu : (∅ : ChargeSpectrum 𝓩).qHu = none := by
-  simp [empty_eq]
+lemma empty_qHu : (∅ : ChargeSpectrum 𝓩).qHu = none := by simp [empty_eq]
 
 @[simp]
-lemma empty_Q5 : (∅ : ChargeSpectrum 𝓩).Q5 = ∅ := by
-  simp [empty_eq]
+lemma empty_Q5 : (∅ : ChargeSpectrum 𝓩).Q5 = ∅ := by simp [empty_eq]
 
 @[simp]
-lemma empty_Q10 : (∅ : ChargeSpectrum 𝓩).Q10 = ∅ := by
-  simp [empty_eq]
+lemma empty_Q10 : (∅ : ChargeSpectrum 𝓩).Q10 = ∅ := by simp [empty_eq]
 
 /-!
 
@@ -246,8 +240,7 @@ def card (x : ChargeSpectrum 𝓩) : Nat :=
   x.qHu.toFinset.card + x.qHd.toFinset.card + x.Q5.card + x.Q10.card
 
 @[simp]
-lemma card_empty : card (∅ : ChargeSpectrum 𝓩) = 0 := by
-  simp [card, empty_eq]
+lemma card_empty : card (∅ : ChargeSpectrum 𝓩) = 0 := by simp [card, empty_eq]
 
 lemma card_mono {x y : ChargeSpectrum 𝓩} (h : x ⊆ y) : card x ≤ card y := by
   have h1 := Finset.card_le_card h.1
@@ -300,13 +293,11 @@ lemma mem_powerset_iff {x y : ChargeSpectrum 𝓩} :
     x.qHd ∈ y.qHd.powerset ∧
     x.qHu ∈ y.qHu.powerset ∧
     x.Q5 ∈ y.Q5.powerset ∧
-    x.Q10 ∈ y.Q10.powerset := by
-  simp [powerset, Finset.mem_product, toProd]
+    x.Q10 ∈ y.Q10.powerset := by simp [powerset, Finset.mem_product, toProd]
 
 @[simp]
 lemma mem_powerset_iff_subset {x y : ChargeSpectrum 𝓩} :
-    x ∈ powerset y ↔ x ⊆ y := by
-  simp [mem_powerset_iff, subset_def]
+    x ∈ powerset y ↔ x ⊆ y := by simp [mem_powerset_iff, subset_def]
 
 lemma self_mem_powerset (x : ChargeSpectrum 𝓩) :
     x ∈ powerset x := by simp
@@ -321,12 +312,9 @@ lemma powerset_of_empty :
   simp
 
 lemma powerset_mono {x y : ChargeSpectrum 𝓩} :
-    powerset x ⊆ powerset y ↔ x ⊆ y := by
-  constructor
-  · intro h
-    exact mem_powerset_iff_subset.mp (h (self_mem_powerset x))
-  · intro h z hz
-    exact mem_powerset_iff_subset.mpr (subset_trans (mem_powerset_iff_subset.mp hz) h)
+    powerset x ⊆ powerset y ↔ x ⊆ y :=
+  ⟨fun h => mem_powerset_iff_subset.mp (h (self_mem_powerset x)),
+    fun h _ hz => mem_powerset_iff_subset.mpr (subset_trans (mem_powerset_iff_subset.mp hz) h)⟩
 
 lemma min_exists_inductive (S : Finset (ChargeSpectrum 𝓩)) (hS : S ≠ ∅) :
     (n : ℕ) → (hn : S.card = n) →
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Completions.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Completions.lean
index 20a2b9332..ec1e993d8 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Completions.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Completions.lean
@@ -87,8 +87,7 @@ The empty charge spectrum is not complete, since it has no charges present.
 -/
 
 @[simp]
-lemma not_isComplete_empty : ¬ IsComplete (∅ : ChargeSpectrum 𝓩) := by
-  simp [IsComplete]
+lemma not_isComplete_empty : ¬ IsComplete (∅ : ChargeSpectrum 𝓩) := by simp [IsComplete]
 
 /-!
 
@@ -210,8 +209,7 @@ lemma completions_eq_singleton_of_complete {S5 S10 : Finset 𝓩} (x : ChargeSpe
   case' neg => simp_all
   by_cases h4 : x.Q10 ≠ ∅
   case' neg => simp_all
-  simp_all
-  rfl
+  simp_all [toProd]
 
 /-!
 
@@ -294,7 +292,6 @@ lemma self_subset_mem_completions (S5 S10 : Finset 𝓩) (x y : ChargeSpectrum 
     (hy : y ∈ completions S5 S10 x) : x ⊆ y := by
   simp [mem_completions_iff] at hy
   rw [Subset]
-  dsimp [hasSubset]
   refine ⟨?_, ?_, ?_, ?_⟩
   · by_cases h : x.qHd.isSome
     · simp_all
@@ -353,7 +350,6 @@ lemma exist_completions_subset_of_complete (S5 S10 : Finset 𝓩) (x y : ChargeS
     · split_ifs with h3 <;> simp_all
     · split_ifs with h4 <;> simp_all
   · rw [Subset]
-    dsimp [hasSubset]
     refine ⟨by simp_all, by simp_all, ?_, ?_⟩
     · split_ifs with h3 <;> simp_all [Finset.singleton_subset_iff]
     · split_ifs with h4 <;> simp_all [Finset.singleton_subset_iff]
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Map.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Map.lean
index f2270bc7a..e6c6a8aff 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Map.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Map.lean
@@ -113,8 +113,7 @@ lemma map_empty (f : 𝓩 →+ 𝓩1) : map f (∅ : ChargeSpectrum 𝓩) = ∅
 -/
 
 lemma map_map (f : 𝓩 →+ 𝓩1) (g : 𝓩1 →+ 𝓩2) (x : ChargeSpectrum 𝓩) :
-    map g (map f x) = map (g.comp f) x := by
-  simp [map, Option.map_map, Finset.image_image]
+    map g (map f x) = map (g.comp f) x := by simp [map, Option.map_map, Finset.image_image]
 
 /-!
 
@@ -222,16 +221,13 @@ lemma mem_map_ofPotentialTerm_iff [DecidableEq 𝓩]
 lemma mem_map_ofPotentialTerm'_iff[DecidableEq 𝓩]
     (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (T : PotentialTerm) :
     i ∈ (ofPotentialTerm' (map f x) T) ↔ i ∈ (ofPotentialTerm' x T).map f := by
-  rw [← mem_ofPotentialTerm_iff_mem_ofPotentialTerm]
-  rw [mem_map_ofPotentialTerm_iff]
+  rw [← mem_ofPotentialTerm_iff_mem_ofPotentialTerm, mem_map_ofPotentialTerm_iff]
   simp only [Multiset.mem_map]
   constructor
   · intro ⟨a, h, h1⟩
-    refine ⟨a, ?_, h1⟩
-    exact mem_ofPotentialTerm_iff_mem_ofPotentialTerm.mp h
+    exact ⟨a, mem_ofPotentialTerm_iff_mem_ofPotentialTerm.mp h, h1⟩
   · intro ⟨a, h, h1⟩
-    refine ⟨a, ?_, h1⟩
-    exact mem_ofPotentialTerm_iff_mem_ofPotentialTerm.mpr h
+    exact ⟨a, mem_ofPotentialTerm_iff_mem_ofPotentialTerm.mpr h, h1⟩
 
 lemma map_ofPotentialTerm'_toFinset [DecidableEq 𝓩]
     (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (T : PotentialTerm) :
@@ -298,8 +294,7 @@ lemma not_isPhenoConstrained_of_map {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 
 
 omit [DecidableEq 𝓩] in
 lemma map_isComplete_iff {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} :
-    (map f x).IsComplete ↔ x.IsComplete := by
-  simp [IsComplete, map]
+    (map f x).IsComplete ↔ x.IsComplete := by simp [IsComplete, map]
 
 /-!
 
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimalSuperSet.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimalSuperSet.lean
index 15604132a..7ecd6a333 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimalSuperSet.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimalSuperSet.lean
@@ -126,8 +126,7 @@ two different forms of this result.
 
 @[simp]
 lemma self_not_mem_minimalSuperSet (S5 S10 : Finset 𝓩) (x : ChargeSpectrum 𝓩) :
-    x ∉ minimalSuperSet S5 S10 x := by
-  simp [minimalSuperSet]
+    x ∉ minimalSuperSet S5 S10 x := by simp [minimalSuperSet]
 
 lemma self_ne_mem_minimalSuperSet (S5 S10 : Finset 𝓩) (x y : ChargeSpectrum 𝓩)
     (hy : y ∈ minimalSuperSet S5 S10 x) : x ≠ y := by
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimallyAllowsTerm/Basic.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimallyAllowsTerm/Basic.lean
index b615cfc72..49484a2a5 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimallyAllowsTerm/Basic.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/MinimallyAllowsTerm/Basic.lean
@@ -114,8 +114,7 @@ lemma allowsTerm_of_has_minimallyAllowsTerm_subset
     (hx : ∃ y ∈ powerset x, y.MinimallyAllowsTerm T) : x.AllowsTerm T := by
   obtain ⟨y, hy⟩ := hx
   simp only [mem_powerset_iff_subset] at hy
-  apply allowsTerm_mono hy.1
-  exact allowsTerm_of_minimallyAllowsTerm hy.2
+  exact allowsTerm_mono hy.1 (allowsTerm_of_minimallyAllowsTerm hy.2)
 
 /-!
 
@@ -178,8 +177,7 @@ lemma minimallyAllowsTerm_iff_powerset_countP_eq_one :
       exact Eq.symm (Multiset.countP_eq_card_filter (fun y => y.AllowsTerm T) x.powerset.val)
     rw [Multiset.card_eq_one] at h1
     obtain ⟨a, ha⟩ := h1
-    have haMem : a ∈ Multiset.filter (fun y => y.AllowsTerm T) x.powerset.val := by
-      simp [ha]
+    have haMem : a ∈ Multiset.filter (fun y => y.AllowsTerm T) x.powerset.val := by simp [ha]
     simp at haMem
     have hxMem : x ∈ Multiset.filter (fun y => y.AllowsTerm T) x.powerset.val := by
       simpa using allowsTerm_mono haMem.1 haMem.2
@@ -241,8 +239,7 @@ lemma card_le_degree_of_minimallyAllowsTerm (h : x.MinimallyAllowsTerm T) :
     x.card ≤ T.degree := by
   obtain ⟨y, y_mem_power, y_card,y_present⟩ :=
     subset_card_le_degree_allowsTerm_of_allowsTerm (allowsTerm_of_minimallyAllowsTerm h)
-  have hy : y ∈ x.powerset.filter (fun y => y.AllowsTerm T) := by
-    simp_all
+  have hy : y ∈ x.powerset.filter (fun y => y.AllowsTerm T) := by simp_all
   rw [minimallyAllowsTerm_iff_powerset_filter_eq] at h
   rw [h] at hy
   simp at hy
@@ -272,8 +269,7 @@ lemma eq_allowsTermForm_of_minimallyAllowsTerm (h1 : x.MinimallyAllowsTerm T) :
   obtain ⟨a, b, c, h2, h3⟩ := allowsTermForm_subset_allowsTerm_of_allowsTerm
     (allowsTerm_of_minimallyAllowsTerm h1)
   use a, b, c
-  have hy : allowsTermForm a b c T ∈ x.powerset.filter (fun y => y.AllowsTerm T) := by
-    simp_all
+  have hy : allowsTermForm a b c T ∈ x.powerset.filter (fun y => y.AllowsTerm T) := by simp_all
   rw [minimallyAllowsTerm_iff_powerset_filter_eq] at h1
   rw [h1] at hy
   simp at hy
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/OfPotentialTerm.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/OfPotentialTerm.lean
index 49658a0a8..f642ddfe9 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/OfPotentialTerm.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/OfPotentialTerm.lean
@@ -381,10 +381,9 @@ which is not important for us.
 -/
 lemma mem_ofPotentialTerm_iff_mem_ofPotentialTerm [DecidableEq 𝓩]
     {T : PotentialTerm} {n : 𝓩} {y : ChargeSpectrum 𝓩} :
-    n ∈ y.ofPotentialTerm T ↔ n ∈ y.ofPotentialTerm' T := by
-  constructor
-  · exact fun h => ofPotentialTerm_subset_ofPotentialTerm' T h
-  · exact fun h => ofPotentialTerm'_subset_ofPotentialTerm T h
+    n ∈ y.ofPotentialTerm T ↔ n ∈ y.ofPotentialTerm' T :=
+  ⟨fun h => ofPotentialTerm_subset_ofPotentialTerm' T h,
+    fun h => ofPotentialTerm'_subset_ofPotentialTerm T h⟩
 
 /-!
 
@@ -399,8 +398,7 @@ lemma ofPotentialTerm'_mono [DecidableEq 𝓩] {x y : ChargeSpectrum 𝓩}
     (h : x ⊆ y) (T : PotentialTerm) :
     x.ofPotentialTerm' T ⊆ y.ofPotentialTerm' T := by
   intro i
-  rw [← mem_ofPotentialTerm_iff_mem_ofPotentialTerm, ← mem_ofPotentialTerm_iff_mem_ofPotentialTerm]
-  exact fun a => ofPotentialTerm_mono h T a
+  simpa [← mem_ofPotentialTerm_iff_mem_ofPotentialTerm] using fun a => ofPotentialTerm_mono h T a
 
 end ChargeSpectrum
 
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/PhenoConstrained.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/PhenoConstrained.lean
index bfd14fa6b..c8f96fb5e 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/PhenoConstrained.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/PhenoConstrained.lean
@@ -138,8 +138,7 @@ def phenoConstrainingChargesSP (x : ChargeSpectrum 𝓩) : Multiset 𝓩 :=
 
 @[simp]
 lemma phenoConstrainingChargesSP_empty :
-    phenoConstrainingChargesSP (∅ : ChargeSpectrum 𝓩) = ∅ := by
-  simp [phenoConstrainingChargesSP]
+    phenoConstrainingChargesSP (∅ : ChargeSpectrum 𝓩) = ∅ := by simp [phenoConstrainingChargesSP]
 
 /-!
 
@@ -183,8 +182,7 @@ def IsPhenoConstrainedQ5 [DecidableEq 𝓩] (x : ChargeSpectrum 𝓩) (q5 : 𝓩
 lemma isPhenoConstrainedQ5_iff [DecidableEq 𝓩] (x : ChargeSpectrum 𝓩) (q5 : 𝓩) :
     x.IsPhenoConstrainedQ5 q5 ↔
     x.AllowsTermQ5 q5 β ∨ x.AllowsTermQ5 q5 Λ ∨ x.AllowsTermQ5 q5 W4 ∨
-    x.AllowsTermQ5 q5 K1 ∨ x.AllowsTermQ5 q5 W1 := by
-  simp [IsPhenoConstrainedQ5, AllowsTermQ5]
+    x.AllowsTermQ5 q5 K1 ∨ x.AllowsTermQ5 q5 W1 := by simp [IsPhenoConstrainedQ5, AllowsTermQ5]
 
 /-!
 
diff --git a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Yukawa.lean b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Yukawa.lean
index 030ae5816..7c2b35b31 100644
--- a/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Yukawa.lean
+++ b/Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/Yukawa.lean
@@ -84,11 +84,9 @@ lemma ofYukawaTerms_subset_of_subset [DecidableEq 𝓩] {x y : ChargeSpectrum 
   intro hr
   rcases hr with hr | hr
   · left
-    apply ofPotentialTerm'_mono h
-    exact hr
+    exact ofPotentialTerm'_mono h topYukawa hr
   · right
-    apply ofPotentialTerm'_mono h
-    exact hr
+    exact ofPotentialTerm'_mono h bottomYukawa hr
 
 /-!
 
@@ -131,8 +129,7 @@ lemma ofYukawaTermsNSum_subset_of_subset [DecidableEq 𝓩] {x y : ChargeSpectru
     · exact ih hz1
     use z2
     simp_all only [and_true]
-    apply ofYukawaTerms_subset_of_subset h
-    exact hz2
+    exact ofYukawaTerms_subset_of_subset h hz2
 
 /-!
 
@@ -191,8 +188,7 @@ lemma yukawaGeneratesDangerousAtLevel_iff_toFinset (x : ChargeSpectrum 𝓩) (n
     ext i
     simp only [Finset.mem_inter, Multiset.mem_toFinset, Finset.notMem_empty, iff_false, not_and]
     intro h1 h2
-    have h3 : i ∈ (x.ofYukawaTermsNSum n) ∩ x.phenoConstrainingChargesSP := by
-      simpa using ⟨h1, h2⟩
+    have h3 : i ∈ (x.ofYukawaTermsNSum n) ∩ x.phenoConstrainingChargesSP := by simpa using ⟨h1, h2⟩
     simp_all
 
 /-!
diff --git a/Physlib/QFT/AnomalyCancellation/Basic.lean b/Physlib/QFT/AnomalyCancellation/Basic.lean
index d9b158e81..98957f780 100644
--- a/Physlib/QFT/AnomalyCancellation/Basic.lean
+++ b/Physlib/QFT/AnomalyCancellation/Basic.lean
@@ -383,9 +383,8 @@ def quadSolsIncl (χ : ACCSystemQuad) : χ.QuadSols →[ℚ] χ.Charges :=
 lemma quadSolsIncl_injective (χ : ACCSystemQuad) :
     Function.Injective χ.quadSolsIncl := by
   intro S T h
-  have h' : χ.quadSolsInclLinSols S = χ.quadSolsInclLinSols T := by
-    apply ACCSystemLinear.linSolsIncl_injective (χ := χ.toACCSystemLinear)
-    exact h
+  have h' : χ.quadSolsInclLinSols S = χ.quadSolsInclLinSols T :=
+    ACCSystemLinear.linSolsIncl_injective (χ := χ.toACCSystemLinear) h
   exact quadSolsInclLinSols_injective χ h'
 
 end ACCSystemQuad
diff --git a/Physlib/QFT/AnomalyCancellation/GroupActions.lean b/Physlib/QFT/AnomalyCancellation/GroupActions.lean
index 09f91bde5..fb85e12a6 100644
--- a/Physlib/QFT/AnomalyCancellation/GroupActions.lean
+++ b/Physlib/QFT/AnomalyCancellation/GroupActions.lean
@@ -60,14 +60,10 @@ def linSolRep {χ : ACCSystem} (G : ACCSystemGroupAction χ) :
   toFun := G.linSolMap
   map_mul' g1 g2 := by
     refine LinearMap.ext fun S ↦ ACCSystemLinear.LinSols.ext ?_
-    change (G.rep (g1 * g2)) S.val = _
-    rw [G.rep.map_mul]
-    rfl
+    exact LinearMap.congr_fun (G.rep.map_mul g1 g2) S.val
   map_one' := by
     refine LinearMap.ext fun S ↦ ACCSystemLinear.LinSols.ext ?_
-    change (G.rep.toFun 1) S.val = _
-    rw [G.rep.map_one']
-    rfl
+    exact LinearMap.congr_fun G.rep.map_one S.val
 
 /-- The representation on the charges and anomaly free solutions
 commutes with the inclusion. -/
@@ -82,14 +78,10 @@ instance quadSolAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) :
   smul f S := ⟨G.linSolRep f S.1, by simp [linSolRep_apply_apply_val, G.quadInvariant, S.quadSol]⟩
   mul_smul f1 f2 S := by
     apply ACCSystemQuad.QuadSols.ext
-    change (G.rep.toFun (f1 * f2)) S.val = _
-    rw [G.rep.map_mul']
-    rfl
+    exact LinearMap.congr_fun (G.rep.map_mul' f1 f2) S.val
   one_smul S := by
     apply ACCSystemQuad.QuadSols.ext
-    change (G.rep.toFun 1) S.val = _
-    rw [G.rep.map_one']
-    rfl
+    exact LinearMap.congr_fun G.rep.map_one S.val
 
 lemma linSolRep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
     (S : χ.QuadSols) : χ.quadSolsInclLinSols (G.quadSolAction.toFun _ _ S g) =
@@ -107,14 +99,10 @@ instance solAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) : MulAction G.
     rw [G.cubicInvariant, S.cubicSol]⟩
   mul_smul f1 f2 S := by
     apply ACCSystem.Sols.ext
-    change (G.rep.toFun (f1 * f2)) S.val = _
-    rw [G.rep.map_mul']
-    rfl
+    exact LinearMap.congr_fun (G.rep.map_mul' f1 f2) S.val
   one_smul S := by
     apply ACCSystem.Sols.ext
-    change (G.rep.toFun 1) S.val = _
-    rw [G.rep.map_one']
-    rfl
+    exact LinearMap.congr_fun G.rep.map_one S.val
 
 lemma quadSolAction_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
     (S : χ.Sols) : χ.solsInclQuadSols (G.solAction.toFun _ _ S g) =
diff --git a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
index 169c50c34..237a98cc2 100644
--- a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Basic.lean
@@ -155,19 +155,16 @@ def crPartF : 𝓕.FieldOp → 𝓕.FieldOpFreeAlgebra := fun φ =>
 
 @[simp]
 lemma crPartF_negAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    crPartF (FieldOp.inAsymp φ) = ofCrAnOpF ⟨FieldOp.inAsymp φ, ()⟩ := by
-  simp [crPartF]
+    crPartF (FieldOp.inAsymp φ) = ofCrAnOpF ⟨FieldOp.inAsymp φ, ()⟩ := by simp [crPartF]
 
 @[simp]
 lemma crPartF_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
     crPartF (FieldOp.position φ) =
-    ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by
-  simp [crPartF]
+    ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by simp [crPartF]
 
 @[simp]
 lemma crPartF_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    crPartF (FieldOp.outAsymp φ) = 0 := by
-  simp [crPartF]
+    crPartF (FieldOp.outAsymp φ) = 0 := by simp [crPartF]
 
 /-- The algebra map taking an element of the free-state algebra to
   the part of it in the creation and annihilation free algebra
@@ -180,19 +177,16 @@ def anPartF : 𝓕.FieldOp → 𝓕.FieldOpFreeAlgebra := fun φ =>
 
 @[simp]
 lemma anPartF_negAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    anPartF (FieldOp.inAsymp φ) = 0 := by
-  simp [anPartF]
+    anPartF (FieldOp.inAsymp φ) = 0 := by simp [anPartF]
 
 @[simp]
 lemma anPartF_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
     anPartF (FieldOp.position φ) =
-    ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by
-  simp [anPartF]
+    ofCrAnOpF ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by simp [anPartF]
 
 @[simp]
 lemma anPartF_posAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    anPartF (FieldOp.outAsymp φ) = ofCrAnOpF ⟨FieldOp.outAsymp φ, ()⟩ := by
-  simp [anPartF]
+    anPartF (FieldOp.outAsymp φ) = ofCrAnOpF ⟨FieldOp.outAsymp φ, ()⟩ := by simp [anPartF]
 
 lemma ofFieldOpF_eq_crPartF_add_anPartF (φ : 𝓕.FieldOp) :
     ofFieldOpF φ = crPartF φ + anPartF φ := by
diff --git a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Grading.lean b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Grading.lean
index 40f76ac7d..de54ee5f1 100644
--- a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Grading.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/Grading.lean
@@ -43,8 +43,7 @@ lemma ofCrAnListF_bosonic_or_fermionic (φs : List 𝓕.CrAnFieldOp) :
 
 lemma ofCrAnOpF_bosonic_or_fermionic (φ : 𝓕.CrAnFieldOp) :
     ofCrAnOpF φ ∈ statisticSubmodule bosonic ∨ ofCrAnOpF φ ∈ statisticSubmodule fermionic := by
-  rw [← ofCrAnListF_singleton]
-  exact ofCrAnListF_bosonic_or_fermionic [φ]
+  simpa [ofCrAnListF_singleton] using ofCrAnListF_bosonic_or_fermionic [φ]
 
 /-- The projection of an element of `FieldOpFreeAlgebra` onto it's bosonic part. -/
 def bosonicProjF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) bosonic :=
@@ -104,9 +103,8 @@ lemma bosonicProjF_of_bonosic_part
 @[simp]
 lemma bosonicProjF_of_fermionic_part
     (a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
-    bosonicProjF (a fermionic).1 = 0 := by
-  apply bosonicProjF_of_mem_fermionic
-  exact Submodule.coe_mem (a.toFun fermionic)
+    bosonicProjF (a fermionic).1 = 0 :=
+  bosonicProjF_of_mem_fermionic (a.toFun fermionic).1 (Submodule.coe_mem (a.toFun fermionic))
 
 /-- The projection of an element of `FieldOpFreeAlgebra` onto it's fermionic part. -/
 def fermionicProjF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] statisticSubmodule (𝓕 := 𝓕) fermionic :=
@@ -171,9 +169,8 @@ lemma fermionicProjF_of_mem_bosonic (a : 𝓕.FieldOpFreeAlgebra)
 @[simp]
 lemma fermionicProjF_of_bosonic_part
     (a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
-    fermionicProjF (a bosonic).1 = 0 := by
-  apply fermionicProjF_of_mem_bosonic
-  exact Submodule.coe_mem (a.toFun bosonic)
+    fermionicProjF (a bosonic).1 = 0 :=
+  fermionicProjF_of_mem_bosonic (a.toFun bosonic).1 (Submodule.coe_mem (a.toFun bosonic))
 
 @[simp]
 lemma fermionicProjF_of_fermionic_part
@@ -255,9 +252,7 @@ instance fieldOpFreeAlgebraGrade :
     simp only [statisticSubmodule]
     refine Submodule.mem_span.mpr fun p a => a ?_
     simp only [Set.mem_setOf_eq]
-    use []
-    simp only [ofCrAnListF_nil, ofList_empty, true_and]
-    rfl
+    exact ⟨[], rfl, rfl⟩
   mul_mem f1 f2 a1 a2 h1 h2 := by
     let p (a2 : 𝓕.FieldOpFreeAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
       a1 * a2 ∈ statisticSubmodule (f1 + f2)
@@ -282,8 +277,7 @@ instance fieldOpFreeAlgebraGrade :
         cases f1 <;> cases f2 <;> rfl
       · simp [p]
       · intro x y hx hy hx1 hx2
-        simp only [add_mul, p]
-        exact Submodule.add_mem _ hx1 hx2
+        simpa only [add_mul, p] using Submodule.add_mem _ hx1 hx2
       · intro c a hx h1
         simp only [Algebra.smul_mul_assoc, p]
         exact Submodule.smul_mem _ _ h1
@@ -350,17 +344,13 @@ lemma bosonicProjF_mul (a b : 𝓕.FieldOpFreeAlgebra) :
     right
     rw [bosonicProjF_of_mem_bosonic _
       (by
-      have h1 : bosonic = fermionic + fermionic := by
-        simp only [add_eq_mul, mul_self]
-        rfl
+      have h1 : bosonic = fermionic + fermionic := rfl
       conv_lhs => rw [h1]
       apply fieldOpFreeAlgebraGrade.mul_mem
       simp only [SetLike.coe_mem]
       simp)]
   simp only [ZeroMemClass.coe_zero, add_zero, zero_add]
-  · have h1 : bosonic = bosonic + bosonic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+  · have h1 : bosonic = bosonic + bosonic := rfl
     conv_lhs => rw [h1]
     apply fieldOpFreeAlgebraGrade.mul_mem
     simp only [SetLike.coe_mem]
@@ -378,9 +368,7 @@ lemma fermionicProjF_mul (a b : 𝓕.FieldOpFreeAlgebra) :
     left
     rw [fermionicProjF_of_mem_bosonic _
       (by
-      have h1 : bosonic = bosonic + bosonic := by
-        simp only [add_eq_mul, mul_self]
-        rfl
+      have h1 : bosonic = bosonic + bosonic := rfl
       conv_lhs => rw [h1]
       apply fieldOpFreeAlgebraGrade.mul_mem
       simp only [SetLike.coe_mem]
@@ -410,9 +398,7 @@ lemma fermionicProjF_mul (a b : 𝓕.FieldOpFreeAlgebra) :
     right
     rw [fermionicProjF_of_mem_bosonic _
       (by
-      have h1 : bosonic = fermionic + fermionic := by
-        simp only [add_eq_mul, mul_self]
-        rfl
+      have h1 : bosonic = fermionic + fermionic := rfl
       conv_lhs => rw [h1]
       apply fieldOpFreeAlgebraGrade.mul_mem
       simp only [SetLike.coe_mem]
diff --git a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
index 3cb794dd1..c9a25faab 100644
--- a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/NormalOrder.lean
@@ -115,17 +115,11 @@ lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
 
 lemma normalOrderF_normalOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) :
     𝓝ᶠ(a * b) = 𝓝ᶠ(a * 𝓝ᶠ(b)) := by
-  trans 𝓝ᶠ(a * b * 1)
-  · simp
-  · rw [normalOrderF_normalOrderF_mid]
-    simp
+  simpa using normalOrderF_normalOrderF_mid a b (1 : 𝓕.FieldOpFreeAlgebra)
 
 lemma normalOrderF_normalOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) :
     𝓝ᶠ(a * b) = 𝓝ᶠ(𝓝ᶠ(a) * b) := by
-  trans 𝓝ᶠ(1 * a * b)
-  · simp
-  · rw [normalOrderF_normalOrderF_mid]
-    simp
+  simpa using normalOrderF_normalOrderF_mid (1 : 𝓕.FieldOpFreeAlgebra) a b
 
 /-!
 
@@ -173,8 +167,7 @@ lemma normalOrderF_crPartF_mul (φ : 𝓕.FieldOp) (a : FieldOpFreeAlgebra 𝓕)
     crPartF φ * 𝓝ᶠ(a) := by
   match φ with
   | .inAsymp φ =>
-    rw [crPartF]
-    exact normalOrderF_create_mul ⟨FieldOp.inAsymp φ, ()⟩ rfl a
+    simpa [crPartF] using normalOrderF_create_mul ⟨FieldOp.inAsymp φ, ()⟩ rfl a
   | .position φ =>
     rw [crPartF]
     exact normalOrderF_create_mul _ rfl _
@@ -221,8 +214,7 @@ lemma normalOrderF_swap_create_annihilate_ofCrAnListF (φc φa : 𝓕.CrAnFieldO
   refine LinearMap.congr_fun (ofCrAnListFBasis.ext fun l ↦ ?_) a
   simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
     LinearMap.coe_comp, Function.comp_apply]
-  rw [normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF φc φa hφc hφa]
-  rfl
+  exact normalOrderF_swap_create_annihilate_ofCrAnListF_ofCrAnListF _ _ hφc hφa _ _
 
 lemma normalOrderF_swap_create_annihilate (φc φa : 𝓕.CrAnFieldOp)
     (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
@@ -267,23 +259,19 @@ lemma normalOrderF_swap_crPartF_anPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFr
   | .position φ, .position φ' =>
     simp only [crPartF_position, anPartF_position]
     rw [normalOrderF_swap_create_annihilate]
-    simp only [crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
-    rfl; rfl
+    all_goals rfl
   | .inAsymp φ, .outAsymp φ' =>
     simp only [crPartF_negAsymp, anPartF_posAsymp]
     rw [normalOrderF_swap_create_annihilate]
-    simp only [crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
-    rfl; rfl
+    all_goals rfl
   | .inAsymp φ, .position φ' =>
     simp only [crPartF_negAsymp, anPartF_position]
     rw [normalOrderF_swap_create_annihilate]
-    simp only [crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
-    rfl; rfl
+    all_goals rfl
   | .position φ, .outAsymp φ' =>
     simp only [crPartF_position, anPartF_posAsymp]
     rw [normalOrderF_swap_create_annihilate]
-    simp only [crAnStatistics, Function.comp_apply, crAnFieldOpToFieldOp_prod]
-    rfl; rfl
+    all_goals rfl
 
 /-!
 
@@ -296,8 +284,7 @@ Using the results from above.
 lemma normalOrderF_swap_anPartF_crPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
     𝓝ᶠ(a * (anPartF φ) * (crPartF φ') * b) =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝ᶠ(a * (crPartF φ') *
-      (anPartF φ) * b) := by
-  simp [normalOrderF_swap_crPartF_anPartF, smul_smul]
+      (anPartF φ) * b) := by simp [normalOrderF_swap_crPartF_anPartF, smul_smul]
 
 lemma normalOrderF_superCommuteF_crPartF_anPartF (φ φ' : 𝓕.FieldOp) (a b : FieldOpFreeAlgebra 𝓕) :
     𝓝ᶠ(a * superCommuteF
@@ -348,38 +335,32 @@ lemma normalOrderF_crPartF_mul_crPartF (φ φ' : 𝓕.FieldOp) :
     𝓝ᶠ(crPartF φ * crPartF φ') =
     crPartF φ * crPartF φ' := by
   rw [normalOrderF_crPartF_mul]
-  conv_lhs => rw [← mul_one (crPartF φ')]
-  rw [normalOrderF_crPartF_mul, normalOrderF_one]
-  simp
+  simpa [normalOrderF_one] using congrArg (fun x => crPartF φ * x)
+    (normalOrderF_crPartF_mul φ' (1 : 𝓕.FieldOpFreeAlgebra))
 
 @[simp]
 lemma normalOrderF_anPartF_mul_anPartF (φ φ' : 𝓕.FieldOp) :
     𝓝ᶠ(anPartF φ * anPartF φ') =
     anPartF φ * anPartF φ' := by
   rw [normalOrderF_mul_anPartF]
-  conv_lhs => rw [← one_mul (anPartF φ)]
-  rw [normalOrderF_mul_anPartF, normalOrderF_one]
-  simp
+  simpa [normalOrderF_one] using congrArg (fun x => x * anPartF φ')
+    (normalOrderF_mul_anPartF φ (1 : 𝓕.FieldOpFreeAlgebra))
 
 @[simp]
 lemma normalOrderF_crPartF_mul_anPartF (φ φ' : 𝓕.FieldOp) :
     𝓝ᶠ(crPartF φ * anPartF φ') =
     crPartF φ * anPartF φ' := by
   rw [normalOrderF_crPartF_mul]
-  conv_lhs => rw [← one_mul (anPartF φ')]
-  rw [normalOrderF_mul_anPartF, normalOrderF_one]
-  simp
+  simpa [normalOrderF_one] using congrArg (fun x => crPartF φ * x)
+    (normalOrderF_mul_anPartF φ' (1 : 𝓕.FieldOpFreeAlgebra))
 
 @[simp]
 lemma normalOrderF_anPartF_mul_crPartF (φ φ' : 𝓕.FieldOp) :
     𝓝ᶠ(anPartF φ * crPartF φ') =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
     (crPartF φ' * anPartF φ) := by
-  conv_lhs => rw [← one_mul (anPartF φ * crPartF φ')]
-  conv_lhs => rw [← mul_one (1 * (anPartF φ *
-    crPartF φ'))]
-  rw [← mul_assoc, normalOrderF_swap_anPartF_crPartF]
-  simp
+  simpa using normalOrderF_swap_anPartF_crPartF φ φ'
+    (1 : 𝓕.FieldOpFreeAlgebra) (1 : 𝓕.FieldOpFreeAlgebra)
 
 lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
     𝓝ᶠ(ofFieldOpF φ * ofFieldOpF φ') =
diff --git a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
index 84da10d1b..4b2c3d9ae 100644
--- a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/SuperCommute.lean
@@ -92,16 +92,14 @@ lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs
 lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
     [ofFieldOpF φ, ofFieldOpListF φs]ₛF = ofFieldOpF φ * ofFieldOpListF φs -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
-  rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList,
-    ofFieldOpListF_singleton]
-  simp
+  simpa [ofFieldOpListF_singleton] using
+    superCommuteF_ofFieldOpListF_ofFieldOpFsList [φ] φs
 
 lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
     [ofFieldOpListF φs, ofFieldOpF φ]ₛF = ofFieldOpListF φs * ofFieldOpF φ -
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
-  rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList,
-    ofFieldOpListF_singleton]
-  simp
+  simpa [ofFieldOpListF_singleton] using
+    superCommuteF_ofFieldOpListF_ofFieldOpFsList φs [φ]
 
 lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
     [anPartF φ, crPartF φ']ₛF = anPartF φ * crPartF φ' -
@@ -244,15 +242,13 @@ lemma superCommuteF_crPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
     [crPartF φ, ofFieldOpF φ']ₛF =
     crPartF φ * ofFieldOpF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * crPartF φ := by
-  rw [← ofFieldOpListF_singleton, superCommuteF_crPartF_ofFieldOpListF]
-  simp
+  simpa [ofFieldOpListF_singleton] using superCommuteF_crPartF_ofFieldOpListF φ [φ']
 
 lemma superCommuteF_anPartF_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
     [anPartF φ, ofFieldOpF φ']ₛF =
     anPartF φ * ofFieldOpF φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOpF φ' * anPartF φ := by
-  rw [← ofFieldOpListF_singleton, superCommuteF_anPartF_ofFieldOpListF]
-  simp
+  simpa [ofFieldOpListF_singleton] using superCommuteF_anPartF_ofFieldOpListF φ [φ']
 
 /-!
 
@@ -265,8 +261,7 @@ multiplication with a sign plus the super commutator.
 lemma ofCrAnListF_mul_ofCrAnListF_eq_superCommuteF (φs φs' : List 𝓕.CrAnFieldOp) :
     ofCrAnListF φs * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnListF φs
     + [ofCrAnListF φs, ofCrAnListF φs']ₛF := by
-  rw [superCommuteF_ofCrAnListF_ofCrAnListF]
-  simp [ofCrAnListF_append]
+  simp [superCommuteF_ofCrAnListF_ofCrAnListF, ofCrAnListF_append]
 
 lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.CrAnFieldOp) :
     ofCrAnOpF φ * ofCrAnListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnListF φs' * ofCrAnOpF φ
@@ -278,8 +273,7 @@ lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.F
     ofFieldOpListF φs * ofFieldOpListF φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
     + [ofFieldOpListF φs, ofFieldOpListF φs']ₛF := by
-  rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
-  simp
+  simp [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
 
 lemma ofFieldOpF_mul_ofFieldOpListF_eq_superCommuteF (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
     ofFieldOpF φ * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpF φ
@@ -296,37 +290,27 @@ lemma ofFieldOpListF_mul_ofFieldOpF_eq_superCommuteF (φs : List 𝓕.FieldOp) (
 lemma crPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     crPartF φ * anPartF φ' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * crPartF φ +
-    [crPartF φ, anPartF φ']ₛF := by
-  rw [superCommuteF_crPartF_anPartF]
-  simp
+    [crPartF φ, anPartF φ']ₛF := by simp [superCommuteF_crPartF_anPartF]
 
 lemma anPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     anPartF φ * crPartF φ' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') •
     crPartF φ' * anPartF φ +
-    [anPartF φ, crPartF φ']ₛF := by
-  rw [superCommuteF_anPartF_crPartF]
-  simp
+    [anPartF φ, crPartF φ']ₛF := by simp [superCommuteF_anPartF_crPartF]
 
 lemma crPartF_mul_crPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     crPartF φ * crPartF φ' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPartF φ' * crPartF φ +
-    [crPartF φ, crPartF φ']ₛF := by
-  rw [superCommuteF_crPartF_crPartF]
-  simp
+    [crPartF φ, crPartF φ']ₛF := by simp [superCommuteF_crPartF_crPartF]
 
 lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
     anPartF φ * anPartF φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPartF φ' * anPartF φ +
-    [anPartF φ, anPartF φ']ₛF := by
-  rw [superCommuteF_anPartF_anPartF]
-  simp
+    [anPartF φ, anPartF φ']ₛF := by simp [superCommuteF_anPartF_anPartF]
 
 lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp)
     (φs' : List 𝓕.FieldOp) : ofCrAnListF φs * ofFieldOpListF φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
-    + [ofCrAnListF φs, ofFieldOpListF φs']ₛF := by
-  rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
-  simp
+    + [ofCrAnListF φs, ofFieldOpListF φs']ₛF := by simp [superCommuteF_ofCrAnListF_ofFieldOpFsList]
 
 /-!
 
@@ -531,8 +515,7 @@ lemma superCommuteF_grade {a b : 𝓕.FieldOpFreeAlgebra} {f1 f2 : FieldStatisti
         rw [mul_comm]
     · simp [p]
     · intro x y hx hy hp1 hp2
-      simp only [add_eq_mul, map_add, LinearMap.add_apply, p]
-      exact Submodule.add_mem _ hp1 hp2
+      simpa only [add_eq_mul, map_add, LinearMap.add_apply, p] using Submodule.add_mem _ hp1 hp2
     · intro c x hx hp1
       simp only [add_eq_mul, map_smul, LinearMap.smul_apply, p]
       exact Submodule.smul_mem _ c hp1
@@ -663,15 +646,11 @@ lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisti
 
 lemma superCommuteF_bonsonic_symm {a b : 𝓕.FieldOpFreeAlgebra}
     (hb : b ∈ statisticSubmodule bosonic) :
-    [a, b]ₛF = - [b, a]ₛF := by
-  rw [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
-  simp
+    [a, b]ₛF = - [b, a]ₛF := by simp [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
 
 lemma bonsonic_superCommuteF_symm {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule bosonic) :
-    [a, b]ₛF = - [b, a]ₛF := by
-  rw [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
-  simp
+    [a, b]ₛF = - [b, a]ₛF := by simp [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
 
 lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
@@ -709,8 +688,7 @@ lemma superCommuteF_fermionic_fermionic {a b : 𝓕.FieldOpFreeAlgebra}
 lemma superCommuteF_fermionic_fermionic_symm {a b : 𝓕.FieldOpFreeAlgebra}
     (ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
     [a, b]ₛF = [b, a]ₛF := by
-  rw [superCommuteF_fermionic_fermionic ha hb]
-  rw [superCommuteF_fermionic_fermionic hb ha]
+  rw [superCommuteF_fermionic_fermionic ha hb, superCommuteF_fermionic_fermionic hb ha]
   abel
 
 lemma superCommuteF_expand_bosonicProjF_fermionicProjF (a b : 𝓕.FieldOpFreeAlgebra) :
@@ -731,33 +709,25 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_bosonic_or_fermionic (φs φs' : Lis
     [ofCrAnListF φs, ofCrAnListF φs']ₛF ∈ statisticSubmodule fermionic := by
   by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
   · left
-    have h : bosonic = bosonic + bosonic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+    have h : bosonic = bosonic + bosonic := rfl
     rw [h]
     apply superCommuteF_grade
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ h1
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ h2
   · right
-    have h : fermionic = bosonic + fermionic := by
-      simp only [add_eq_mul]
-      rfl
+    have h : fermionic = bosonic + fermionic := rfl
     rw [h]
     apply superCommuteF_grade
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ h1
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h2)
   · right
-    have h : fermionic = fermionic + bosonic := by
-      simp only [add_eq_mul]
-      rfl
+    have h : fermionic = fermionic + bosonic := rfl
     rw [h]
     apply superCommuteF_grade
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ h2
   · left
-    have h : bosonic = fermionic + fermionic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+    have h : bosonic = fermionic + fermionic := rfl
     rw [h]
     apply superCommuteF_grade
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ (by simpa using h1)
@@ -775,27 +745,19 @@ lemma superCommuteF_superCommuteF_ofCrAnOpF_bosonic_or_fermionic (φ1 φ2 φ3 :
   rcases superCommuteF_ofCrAnOpF_ofCrAnOpF_bosonic_or_fermionic φ2 φ3 with hs | hs
     <;> rcases ofCrAnOpF_bosonic_or_fermionic φ1 with h1 | h1
   · left
-    have h : bosonic = bosonic + bosonic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+    have h : bosonic = bosonic + bosonic := rfl
     rw [h]
     apply superCommuteF_grade h1 hs
   · right
-    have h : fermionic = fermionic + bosonic := by
-      simp only [add_eq_mul]
-      rfl
+    have h : fermionic = fermionic + bosonic := rfl
     rw [h]
     apply superCommuteF_grade h1 hs
   · right
-    have h : fermionic = bosonic + fermionic := by
-      simp only [add_eq_mul]
-      rfl
+    have h : fermionic = bosonic + fermionic := rfl
     rw [h]
     apply superCommuteF_grade h1 hs
   · left
-    have h : bosonic = fermionic + fermionic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+    have h : bosonic = fermionic + fermionic := rfl
     rw [h]
     apply superCommuteF_grade h1 hs
 
@@ -871,17 +833,13 @@ lemma statistic_ne_of_superCommuteF_fermionic {φs φs' : List 𝓕.CrAnFieldOp}
   by_contra hn
   refine h0 (eq_zero_of_bosonic_and_fermionic ?_ h)
   by_cases hc : (𝓕 |>ₛ φs) = bosonic
-  · have h1 : bosonic = bosonic + bosonic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+  · have h1 : bosonic = bosonic + bosonic := rfl
     rw [h1]
     apply superCommuteF_grade
     apply ofCrAnListF_mem_statisticSubmodule_of _ _ hc
     apply ofCrAnListF_mem_statisticSubmodule_of _ _
     rw [← hn, hc]
-  · have h1 : bosonic = fermionic + fermionic := by
-      simp only [add_eq_mul, mul_self]
-      rfl
+  · have h1 : bosonic = fermionic + fermionic := rfl
     rw [h1]
     apply superCommuteF_grade
     apply ofCrAnListF_mem_statisticSubmodule_of _ _
diff --git a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
index 8f07abd3c..0b4d44a7f 100644
--- a/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldOpFreeAlgebra/TimeOrder.lean
@@ -105,16 +105,10 @@ lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
     simp_all [pc]
 
 lemma timeOrderF_timeOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(a * 𝓣ᶠ(b)) := by
-  trans 𝓣ᶠ(a * b * 1)
-  · simp
-  · rw [timeOrderF_timeOrderF_mid]
-    simp
+  simpa using timeOrderF_timeOrderF_mid a b (1 : 𝓕.FieldOpFreeAlgebra)
 
 lemma timeOrderF_timeOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b) = 𝓣ᶠ(𝓣ᶠ(a) * b) := by
-  trans 𝓣ᶠ(1 * a * b)
-  · simp
-  · rw [timeOrderF_timeOrderF_mid]
-    simp
+  simpa using timeOrderF_timeOrderF_mid (1 : 𝓕.FieldOpFreeAlgebra) a b
 
 lemma timeOrderF_ofFieldOpListF (φs : List 𝓕.FieldOp) :
     𝓣ᶠ(ofFieldOpListF φs) = timeOrderSign φs • ofFieldOpListF (timeOrderList φs) := by
@@ -142,8 +136,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.FieldOp} (h : timeO
   rw [← ofFieldOpListF_singleton, ← ofFieldOpListF_singleton, ← ofFieldOpListF_append,
     timeOrderF_ofFieldOpListF]
   simp only [List.singleton_append]
-  rw [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
-  simp
+  simp [timeOrderSign_pair_ordered h, timeOrderList_pair_ordered h]
 
 lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.FieldOp} (h : ¬ timeOrderRel φ ψ) :
     𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • ofFieldOpF ψ * ofFieldOpF φ := by
@@ -156,8 +149,7 @@ lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered {φ ψ : 𝓕.FieldOp} (h : 
 lemma timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF {φ ψ : 𝓕.FieldOp}
     (h : ¬ timeOrderRel φ ψ) :
     𝓣ᶠ(ofFieldOpF φ * ofFieldOpF ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣ᶠ(ofFieldOpF ψ * ofFieldOpF φ) := by
-  rw [timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h]
-  rw [timeOrderF_ofFieldOpF_ofFieldOpF_ordered]
+  rw [timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered h, timeOrderF_ofFieldOpF_ofFieldOpF_ordered]
   simp only [Algebra.smul_mul_assoc]
   have hx := Std.Total.total (r := timeOrderRel) ψ φ
   simp_all
@@ -275,24 +267,19 @@ lemma timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_all_not_crAnTimeOrderRel
     simp
   simp_all only [imp_false, Decidable.not_not]
   by_cases h12 : ¬ crAnTimeOrderRel φ1 φ2
-  · have h13 : ¬ crAnTimeOrderRel φ1 φ3 := by
-      intro h13
-      apply h12
-      exact IsTrans.trans φ1 φ3 φ2 h13 h32
+  · have h13 : ¬ crAnTimeOrderRel φ1 φ3 :=
+      fun h13 => h12 (IsTrans.trans φ1 φ3 φ2 h13 h32)
     rw [timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel h12 h13]
   simp_all only [Decidable.not_not, forall_const]
   have h13 : crAnTimeOrderRel φ1 φ3 := IsTrans.trans φ1 φ2 φ3 h12 h23
   simp_all only [forall_const]
   by_cases h21 : ¬ crAnTimeOrderRel φ2 φ1
   · simp_all only [IsEmpty.forall_iff]
-    have h31 : ¬ crAnTimeOrderRel φ3 φ1 := by
-      intro h31
-      apply h21
-      exact IsTrans.trans φ2 φ3 φ1 h23 h31
+    have h31 : ¬ crAnTimeOrderRel φ3 φ1 :=
+      fun h31 => h21 (IsTrans.trans φ2 φ3 φ1 h23 h31)
     rw [timeOrderF_superCommuteF_ofCrAnOpF_superCommuteF_not_crAnTimeOrderRel' h21 h31]
   simp_all only [Decidable.not_not, forall_const]
-  refine False.elim (h ?_)
-  exact IsTrans.trans φ3 φ2 φ1 h32 h21
+  exact False.elim (h (IsTrans.trans φ3 φ2 φ1 h32 h21))
 
 lemma timeOrderF_superCommuteF_ofCrAnOpF_ofCrAnOpF_eq_time
     {φ ψ : 𝓕.CrAnFieldOp} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
diff --git a/Physlib/QFT/PerturbationTheory/FieldSpecification/CrAnSection.lean b/Physlib/QFT/PerturbationTheory/FieldSpecification/CrAnSection.lean
index 6067efcf6..92235ada7 100644
--- a/Physlib/QFT/PerturbationTheory/FieldSpecification/CrAnSection.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldSpecification/CrAnSection.lean
@@ -109,8 +109,7 @@ def nilEquiv : CrAnSection (𝓕 := 𝓕) [] ≃ Unit where
     have h2 := ψs.2
     simp only [List.map_eq_nil_iff] at h2
     simp [h2]
-  right_inv _ := by
-    simp
+  right_inv _ := by simp
 
 /-- The creation and annihilation sections for a singleton list is given by
   a choice of `𝓕.fieldOpToCreateAnnihilateType φ`. If `φ` is a asymptotic state
@@ -127,9 +126,7 @@ def singletonEquiv {φ : 𝓕.FieldOp} : CrAnSection [φ] ≃
     have h2 := ψs.tail.2
     simp only [List.tail_cons, List.map_eq_nil_iff] at h2
     simp [h2]
-  right_inv ψ := by
-    simp only [head]
-    rfl
+  right_inv ψ := rfl
 
 /-- An equivalence separating the head of a creation and annihilation section
   from the tail. -/
@@ -139,9 +136,7 @@ def consEquiv {φ : 𝓕.FieldOp} {φs : List 𝓕.FieldOp} : CrAnSection (φ ::
   invFun ψψs :=
     match ψψs with
     | (ψ, ψs) => cons ψ ψs
-  left_inv ψs := by
-    apply Subtype.ext
-    exact Eq.symm eq_head_cons_tail
+  left_inv ψs := Subtype.ext (Eq.symm eq_head_cons_tail)
   right_inv ψψs := by
     match ψψs with
     | (ψ, ψs) => rfl
@@ -156,14 +151,11 @@ instance fintype : (φs : List 𝓕.FieldOp) → Fintype (CrAnSection φs)
 
 @[simp]
 lemma card_nil_eq : Fintype.card (CrAnSection (𝓕 := 𝓕) []) = 1 := by
-  rw [Fintype.ofEquiv_card nilEquiv.symm]
-  simp
+  simpa using Fintype.ofEquiv_card nilEquiv.symm
 
 lemma card_cons_eq {φ : 𝓕.FieldOp} {φs : List 𝓕.FieldOp} :
     Fintype.card (CrAnSection (φ :: φs)) = Fintype.card (𝓕.fieldOpToCrAnType φ) *
-    Fintype.card (CrAnSection φs) := by
-  rw [Fintype.ofEquiv_card consEquiv.symm]
-  simp
+    Fintype.card (CrAnSection φs) := by simpa using Fintype.ofEquiv_card consEquiv.symm
 
 lemma card_eq_mul : {φs : List 𝓕.FieldOp} → Fintype.card (CrAnSection φs) =
     2 ^ (List.countP 𝓕.statesIsPosition φs)
@@ -171,20 +163,17 @@ lemma card_eq_mul : {φs : List 𝓕.FieldOp} → Fintype.card (CrAnSection φs)
     simp
   | FieldOp.position _ :: φs => by
       simp only [statesIsPosition, List.countP_cons_of_pos]
-      rw [card_cons_eq]
-      rw [card_eq_mul]
+      rw [card_cons_eq, card_eq_mul]
       simp only [fieldOpToCrAnType]
       erw [CreateAnnihilate.CreateAnnihilate_card_eq_two]
       ring
   | FieldOp.inAsymp x_ :: φs => by
       simp only [statesIsPosition, Bool.false_eq_true, not_false_eq_true, List.countP_cons_of_neg]
-      rw [card_cons_eq]
-      rw [card_eq_mul]
+      rw [card_cons_eq, card_eq_mul]
       simp [fieldOpToCrAnType]
   | FieldOp.outAsymp _ :: φs => by
       simp only [statesIsPosition, Bool.false_eq_true, not_false_eq_true, List.countP_cons_of_neg]
-      rw [card_cons_eq]
-      rw [card_eq_mul]
+      rw [card_cons_eq, card_eq_mul]
       simp [fieldOpToCrAnType]
 
 lemma card_perm_eq {φs φs' : List 𝓕.FieldOp} (h : φs.Perm φs') :
@@ -196,21 +185,17 @@ lemma card_perm_eq {φs φs' : List 𝓕.FieldOp} (h : φs.Perm φs') :
 @[simp]
 lemma sum_nil (f : CrAnSection (𝓕 := 𝓕) [] → M) [AddCommMonoid M] :
     ∑ (s : CrAnSection []), f s = f ⟨[], rfl⟩ := by
-  rw [← nilEquiv.symm.sum_comp]
-  simp only [Finset.univ_unique, PUnit.default_eq_unit, Finset.sum_singleton]
-  rfl
+  simpa [nilEquiv] using Eq.symm (Equiv.sum_comp nilEquiv.symm f)
 
 lemma sum_cons (f : CrAnSection (φ :: φs) → M) [AddCommMonoid M] :
     ∑ (s : CrAnSection (φ :: φs)), f s = ∑ (a : 𝓕.fieldOpToCrAnType φ),
     ∑ (s : CrAnSection φs), f (cons a s) := by
-  rw [← consEquiv.symm.sum_comp, Fintype.sum_prod_type]
-  rfl
+  simpa [Fintype.sum_prod_type, consEquiv] using Eq.symm (consEquiv.symm.sum_comp f)
 
 lemma sum_over_length {s : CrAnSection φs} (f : Fin s.1.length → M)
     [AddCommMonoid M] : ∑ (n : Fin s.1.length), f n =
-    ∑ (n : Fin φs.length), f (Fin.cast (length_eq s).symm n) := by
-  rw [← (finCongr (length_eq s)).sum_comp]
-  rfl
+    ∑ (n : Fin φs.length), f (Fin.cast (length_eq s).symm n) :=
+  Eq.symm (Equiv.sum_comp (finCongr (length_eq s)).symm f)
 
 /-- The equivalence between `CrAnSection φs` and
   `CrAnSection φs'` induced by an equality `φs = φs'`. -/
@@ -394,8 +379,7 @@ lemma eraseIdxEquiv_symm_eq_take_cons_drop {n : ℕ} (φs : List 𝓕.FieldOp) (
   simp only [congr_fst]
   have hn : (List.take n φs).length = n := by
     rw [@List.length_take]
-    simp only [inf_eq_left]
-    exact Nat.le_of_succ_le hn
+    simpa only [inf_eq_left] using Nat.le_of_succ_le hn
   rw [hn]
 
 @[simp]
@@ -423,8 +407,7 @@ lemma sum_eraseIdxEquiv (n : ℕ) (φs : List 𝓕.FieldOp) (hn : n < φs.length
     (f : CrAnSection φs → M) [AddCommMonoid M] : ∑ (s : CrAnSection φs), f s =
     ∑ (a : 𝓕.fieldOpToCrAnType φs[n]), ∑ (s : CrAnSection (φs.eraseIdx n)),
     f ((eraseIdxEquiv n φs hn).symm ⟨a, s⟩) := by
-  rw [← (eraseIdxEquiv n φs hn).symm.sum_comp]
-  rw [Fintype.sum_prod_type]
+  rw [← (eraseIdxEquiv n φs hn).symm.sum_comp, Fintype.sum_prod_type]
 
 end CrAnSection
 
diff --git a/Physlib/QFT/PerturbationTheory/FieldSpecification/Filters.lean b/Physlib/QFT/PerturbationTheory/FieldSpecification/Filters.lean
index 9af522575..02bc26ec2 100644
--- a/Physlib/QFT/PerturbationTheory/FieldSpecification/Filters.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldSpecification/Filters.lean
@@ -40,13 +40,11 @@ lemma createFilter_cons_annihilate {φ : 𝓕.CrAnFieldOp}
 
 lemma createFilter_append (φs φs' : List 𝓕.CrAnFieldOp) :
     createFilter (φs ++ φs') = createFilter φs ++ createFilter φs' := by
-  rw [createFilter, List.filter_append]
-  rfl
+  simp [createFilter, List.filter_append]
 
 lemma createFilter_singleton_create (φ : 𝓕.CrAnFieldOp)
     (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) :
-    createFilter [φ] = [φ] := by
-  simp [createFilter, hφ]
+    createFilter [φ] = [φ] := by simp [createFilter, hφ]
 
 lemma createFilter_singleton_annihilate (φ : 𝓕.CrAnFieldOp)
     (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) : createFilter [φ] = [] := by
@@ -76,17 +74,14 @@ lemma annihilateFilter_cons_annihilate {φ : 𝓕.CrAnFieldOp}
 
 lemma annihilateFilter_append (φs φs' : List 𝓕.CrAnFieldOp) :
     annihilateFilter (φs ++ φs') = annihilateFilter φs ++ annihilateFilter φs' := by
-  rw [annihilateFilter, List.filter_append]
-  rfl
+  simp [annihilateFilter, List.filter_append]
 
 lemma annihilateFilter_singleton_create (φ : 𝓕.CrAnFieldOp)
     (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) :
-    annihilateFilter [φ] = [] := by
-  simp [annihilateFilter, hφ]
+    annihilateFilter [φ] = [] := by simp [annihilateFilter, hφ]
 
 lemma annihilateFilter_singleton_annihilate (φ : 𝓕.CrAnFieldOp)
     (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) :
-    annihilateFilter [φ] = [φ] := by
-  simp [annihilateFilter, hφ]
+    annihilateFilter [φ] = [φ] := by simp [annihilateFilter, hφ]
 
 end FieldSpecification
diff --git a/Physlib/QFT/PerturbationTheory/FieldSpecification/NormalOrder.lean b/Physlib/QFT/PerturbationTheory/FieldSpecification/NormalOrder.lean
index 7ad2e23e2..d501ea7e9 100644
--- a/Physlib/QFT/PerturbationTheory/FieldSpecification/NormalOrder.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldSpecification/NormalOrder.lean
@@ -171,8 +171,7 @@ lemma normalOrderSign_swap_create_create (φc φc' : 𝓕.CrAnFieldOp)
     (hφc : 𝓕 |>ᶜ φc = CreateAnnihilate.create) (hφc' : 𝓕 |>ᶜ φc' = CreateAnnihilate.create) :
     (φs φs' : List 𝓕.CrAnFieldOp) →
     normalOrderSign (φs ++ φc :: φc' :: φs') = normalOrderSign (φs ++ φc' :: φc :: φs')
-  | [], φs' => by
-    exact normalOrderSign_swap_create_create_fst φc φc' hφc hφc' φs'
+  | [], φs' => normalOrderSign_swap_create_create_fst φc φc' hφc hφc' φs'
   | φ :: φs, φs' => by
     rw [normalOrderSign]
     dsimp only [List.cons_append, Wick.koszulSign]
@@ -205,8 +204,7 @@ lemma normalOrderSign_swap_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
     (hφa : 𝓕 |>ᶜ φa = CreateAnnihilate.annihilate)
     (hφa' : 𝓕 |>ᶜ φa' = CreateAnnihilate.annihilate) : (φs φs' : List 𝓕.CrAnFieldOp) →
     normalOrderSign (φs ++ φa :: φa' :: φs') = normalOrderSign (φs ++ φa' :: φa :: φs')
-  | [], φs' => by
-    exact normalOrderSign_swap_annihilate_annihilate_fst φa φa' hφa hφa' φs'
+  | [], φs' => normalOrderSign_swap_annihilate_annihilate_fst φa φa' hφa hφa' φs'
   | φ :: φs, φs' => by
     rw [normalOrderSign]
     dsimp only [List.cons_append, Wick.koszulSign]
@@ -214,11 +212,8 @@ lemma normalOrderSign_swap_annihilate_annihilate (φa φa' : 𝓕.CrAnFieldOp)
     rw [normalOrderSign_swap_annihilate_annihilate φa φa' hφa hφa']
     dsimp only [normalOrderSign, Wick.koszulSign]
     rw [← normalOrderSign]
-    simp only [mul_eq_mul_right_iff]
-    apply Or.inl
-    apply Wick.koszulSignInsert_eq_perm
-    refine List.Perm.append_left φs ?h.h.a
-    exact List.Perm.swap φa' φa φs'
+    simpa only [mul_eq_mul_right_iff] using Or.inl (Wick.koszulSignInsert_eq_perm _ _ _ _ _
+      (List.Perm.append_left φs (List.Perm.swap φa' φa φs')))
 open FieldStatistic
 
 /-!
@@ -238,13 +233,11 @@ def normalOrderList (φs : List 𝓕.CrAnFieldOp) : List 𝓕.CrAnFieldOp :=
   List.insertionSort 𝓕.normalOrderRel φs
 
 @[simp]
-lemma normalOrderList_nil : normalOrderList (𝓕 := 𝓕) [] = [] := by
-  simp [normalOrderList]
+lemma normalOrderList_nil : normalOrderList (𝓕 := 𝓕) [] = [] := by simp [normalOrderList]
 
 @[simp]
 lemma normalOrderList_statistics (φs : List 𝓕.CrAnFieldOp) :
-    (𝓕 |>ₛ (normalOrderList φs)) = 𝓕 |>ₛ φs := by
-  simp [normalOrderList]
+    (𝓕 |>ₛ (normalOrderList φs)) = 𝓕 |>ₛ φs := by simp [normalOrderList]
 
 lemma orderedInsert_create (φ : 𝓕.CrAnFieldOp)
     (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.create) :
@@ -363,8 +356,7 @@ lemma normalOrderSign_eraseIdx (φs : List 𝓕.CrAnFieldOp) (i : Fin φs.length
     normalOrderSign (φs.eraseIdx i) = normalOrderSign φs *
     𝓢(𝓕 |>ₛ (φs.get i), 𝓕 |>ₛ (φs.take i)) *
     𝓢(𝓕 |>ₛ (φs.get i), 𝓕 |>ₛ ((normalOrderList φs).take (normalOrderEquiv i))) := by
-  rw [normalOrderSign, Wick.koszulSign_eraseIdx, ← normalOrderSign]
-  rfl
+  exact Wick.koszulSign_eraseIdx 𝓕.crAnStatistics normalOrderRel φs i
 
 lemma orderedInsert_createFilter_append_annihilate (φ : 𝓕.CrAnFieldOp)
     (hφ : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate) : (φs φs' : List 𝓕.CrAnFieldOp) →
@@ -417,8 +409,7 @@ lemma normalOrderList_eq_createFilter_append_annihilateFilter : (φs : List 𝓕
       rw [List.filter_cons_of_neg]
       swap
       simp only [hφ, reduceCtorEq, decide_false, Bool.false_eq_true, not_false_eq_true]
-      rw [normalOrderList_eq_createFilter_append_annihilateFilter φs]
-      rfl
+      exact congrArg (List.cons φ) (normalOrderList_eq_createFilter_append_annihilateFilter φs)
     · simp only [normalOrderList, List.insertionSort_cons]
       rw [← normalOrderList]
       have hφ' : 𝓕 |>ᶜ φ = CreateAnnihilate.annihilate := by
diff --git a/Physlib/QFT/PerturbationTheory/FieldSpecification/TimeOrder.lean b/Physlib/QFT/PerturbationTheory/FieldSpecification/TimeOrder.lean
index 96b35b211..2144fc7a2 100644
--- a/Physlib/QFT/PerturbationTheory/FieldSpecification/TimeOrder.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldSpecification/TimeOrder.lean
@@ -135,9 +135,7 @@ def timeOrderSign (φs : List 𝓕.FieldOp) : ℂ :=
   Wick.koszulSign 𝓕.fieldOpStatistic 𝓕.timeOrderRel φs
 
 @[simp]
-lemma timeOrderSign_nil : timeOrderSign (𝓕 := 𝓕) [] = 1 := by
-  simp only [timeOrderSign]
-  rfl
+lemma timeOrderSign_nil : timeOrderSign (𝓕 := 𝓕) [] = 1 := rfl
 
 lemma timeOrderSign_pair_ordered {φ ψ : 𝓕.FieldOp} (h : timeOrderRel φ ψ) :
     timeOrderSign [φ, ψ] = 1 := by
@@ -156,7 +154,6 @@ lemma timerOrderSign_of_eraseMaxTimeField (φ : 𝓕.FieldOp) (φs : List 𝓕.F
     𝓢(𝓕 |>ₛ maxTimeField φ φs, 𝓕 |>ₛ (φ :: φs).take (maxTimeFieldPos φ φs)) := by
   rw [eraseMaxTimeField, insertionSortDropMinPos, timeOrderSign,
     Wick.koszulSign_eraseIdx_insertionSortMinPos]
-  rw [← timeOrderSign, ← maxTimeField]
   rfl
 
 /-- The time ordering of a list of states. A schematic example is:
@@ -178,8 +175,7 @@ lemma timeOrderList_pair_not_ordered {φ ψ : 𝓕.FieldOp} (h : ¬ timeOrderRel
   exact fun h' => False.elim (h h')
 
 @[simp]
-lemma timeOrderList_nil : timeOrderList (𝓕 := 𝓕) [] = [] := by
-  simp [timeOrderList]
+lemma timeOrderList_nil : timeOrderList (𝓕 := 𝓕) [] = [] := by simp [timeOrderList]
 
 lemma timeOrderList_eq_maxTimeField_timeOrderList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
     timeOrderList (φ :: φs) = maxTimeField φ φs :: timeOrderList (eraseMaxTimeField φ φs) := by
@@ -237,9 +233,7 @@ def crAnTimeOrderSign (φs : List 𝓕.CrAnFieldOp) : ℂ :=
   Wick.koszulSign 𝓕.crAnStatistics 𝓕.crAnTimeOrderRel φs
 
 @[simp]
-lemma crAnTimeOrderSign_nil : crAnTimeOrderSign (𝓕 := 𝓕) [] = 1 := by
-  simp only [crAnTimeOrderSign]
-  rfl
+lemma crAnTimeOrderSign_nil : crAnTimeOrderSign (𝓕 := 𝓕) [] = 1 := rfl
 
 lemma crAnTimeOrderSign_pair_ordered {φ ψ : 𝓕.CrAnFieldOp} (h : crAnTimeOrderRel φ ψ) :
     crAnTimeOrderSign [φ, ψ] = 1 := by
@@ -264,8 +258,7 @@ def crAnTimeOrderList (φs : List 𝓕.CrAnFieldOp) : List 𝓕.CrAnFieldOp :=
   List.insertionSort 𝓕.crAnTimeOrderRel φs
 
 @[simp]
-lemma crAnTimeOrderList_nil : crAnTimeOrderList (𝓕 := 𝓕) [] = [] := by
-  simp [crAnTimeOrderList]
+lemma crAnTimeOrderList_nil : crAnTimeOrderList (𝓕 := 𝓕) [] = [] := by simp [crAnTimeOrderList]
 
 lemma crAnTimeOrderList_pair_ordered {φ ψ : 𝓕.CrAnFieldOp} (h : crAnTimeOrderRel φ ψ) :
     crAnTimeOrderList [φ, ψ] = [φ, ψ] := by
@@ -416,8 +409,7 @@ lemma orderedInsert_crAnTimeOrderRel_crAnSection {φ : 𝓕.FieldOp} {ψ : 𝓕.
       rw [← h1.1] at hr
       simp only [crAnFieldOpToFieldOp] at hr
       simp only [hr, ↓reduceIte, List.map_cons, List.cons.injEq]
-      apply And.intro h1.1
-      exact orderedInsert_crAnTimeOrderRel_crAnSection h ⟨ψs, h1.2⟩
+      exact And.intro h1.1 (orderedInsert_crAnTimeOrderRel_crAnSection h ⟨ψs, h1.2⟩)
 
 lemma crAnTimeOrderList_crAnSection_is_crAnSection : {φs : List 𝓕.FieldOp} → (ψs : CrAnSection φs) →
     (crAnTimeOrderList ψs.1).map 𝓕.crAnFieldOpToFieldOp = timeOrderList φs
diff --git a/Physlib/QFT/PerturbationTheory/FieldStatistics/Basic.lean b/Physlib/QFT/PerturbationTheory/FieldStatistics/Basic.lean
index c28b361e7..6284299c7 100644
--- a/Physlib/QFT/PerturbationTheory/FieldStatistics/Basic.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldStatistics/Basic.lean
@@ -88,13 +88,11 @@ instance : Fintype FieldStatistic where
     intro c
     cases c
     · exact Finset.mem_insert_self bosonic {fermionic}
-    · refine Finset.insert_eq_self.mp ?_
-      exact rfl
+    · exact Finset.insert_eq_self.mp rfl
 
 @[simp]
-lemma fermionic_not_eq_bonsic : ¬ fermionic = bosonic := by
-  intro h
-  exact FieldStatistic.noConfusion h
+lemma fermionic_not_eq_bonsic : ¬ fermionic = bosonic :=
+  fun h => FieldStatistic.noConfusion h
 
 lemma bonsic_eq_fermionic_false : bosonic = fermionic ↔ false := by
   simp only [reduceCtorEq, Bool.false_eq_true]
@@ -203,8 +201,7 @@ lemma ofList_append_eq_mul (s : 𝓕 → FieldStatistic) (φs φs' : List 𝓕)
 
 lemma ofList_perm (s : 𝓕 → FieldStatistic) {l l' : List 𝓕} (h : l.Perm l') :
     ofList s l = ofList s l' := by
-  rw [ofList_eq_prod, ofList_eq_prod]
-  exact List.Perm.prod_eq (List.Perm.map s h)
+  simpa [ofList_eq_prod] using List.Perm.prod_eq (List.Perm.map s h)
 
 lemma ofList_orderedInsert (s : 𝓕 → FieldStatistic) (le1 : 𝓕 → 𝓕 → Prop) [DecidableRel le1]
     (φs : List 𝓕) (φ : 𝓕) : ofList s (List.orderedInsert le1 φ φs) = ofList s (φ :: φs) :=
@@ -238,18 +235,15 @@ lemma ofList_map_eq_finset_prod (s : 𝓕 → FieldStatistic) :
     simp only [List.length_cons, mul_ite, ite_mul, one_mul, mul_one]
     by_cases ha : a = i
     · simp only [ha, ↓reduceIte, mul_self, true_or]
-      rw [if_neg]
-      rfl
-      simp only [List.length_cons, List.nodup_cons] at hl
-      exact hl.1
-    · simp only [ha, ↓reduceIte, false_or]
-      rfl
+      exact if_neg (by
+        simp only [List.length_cons, List.nodup_cons] at hl
+        exact hl.1)
+    · simp only [ha, ↓reduceIte, false_or, Fin.getElem_fin]
     simp only [List.length_cons, List.nodup_cons] at hl
     exact hl.2
 
 lemma ofList_pair (s : 𝓕 → FieldStatistic) (φ1 φ2 : 𝓕) :
-    ofList s [φ1, φ2] = s φ1 * s φ2 := by
-  rw [ofList_cons_eq_mul, ofList_singleton]
+    ofList s [φ1, φ2] = s φ1 * s φ2 := by rw [ofList_cons_eq_mul, ofList_singleton]
 
 /-!
 
@@ -271,9 +265,7 @@ lemma ofList_take_eraseIdx (n : ℕ) (φs : List 𝓕) :
   rw [take_eraseIdx_same]
 
 lemma ofList_take_zero (φs : List 𝓕) :
-    ofList q (List.take 0 φs) = 1 := by
-  simp only [List.take_zero, ofList_empty]
-  rfl
+    ofList q (List.take 0 φs) = 1 := rfl
 
 lemma ofList_take_succ_cons (n : ℕ) (φ1 : 𝓕) (φs : List 𝓕) :
     ofList q ((φ1 :: φs).take (n + 1)) = q φ1 * ofList q (φs.take n) := by
@@ -310,8 +302,7 @@ instance : AddMonoid FieldStatistic where
   add_assoc a b c := by
     cases a <;> cases b <;> cases c <;> rfl
   nsmul_zero a := by
-    simp only [Finset.univ_eq_empty, Finset.prod_const, Finset.card_empty, pow_zero]
-    rfl
+    cases a <;> rfl
   nsmul_succ a n := by
     simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin]
     rfl
diff --git a/Physlib/QFT/PerturbationTheory/FieldStatistics/ExchangeSign.lean b/Physlib/QFT/PerturbationTheory/FieldStatistics/ExchangeSign.lean
index 025ffab39..bc0ce6c1c 100644
--- a/Physlib/QFT/PerturbationTheory/FieldStatistics/ExchangeSign.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldStatistics/ExchangeSign.lean
@@ -81,8 +81,7 @@ lemma bosonic_exchangeSign (a : FieldStatistic) : 𝓢(bosonic, a) = 1 := by
   rw [exchangeSign_symm, exchangeSign_bosonic]
 
 @[simp]
-lemma fermionic_exchangeSign_fermionic : 𝓢(fermionic, fermionic) = - 1 := by
-  rfl
+lemma fermionic_exchangeSign_fermionic : 𝓢(fermionic, fermionic) = - 1 := rfl
 
 lemma exchangeSign_eq_if (a b : FieldStatistic) :
     𝓢(a, b) = if a = fermionic ∧ b = fermionic then - 1 else 1 := by
@@ -98,8 +97,7 @@ lemma exchangeSign_mul_self_swap (a b : FieldStatistic) : 𝓢(a, b) * 𝓢(b, a
 
 lemma exchangeSign_ofList_cons (a : FieldStatistic)
     (s : 𝓕 → FieldStatistic) (φ : 𝓕) (φs : List 𝓕) :
-    𝓢(a, ofList s (φ :: φs)) = 𝓢(a, s φ) * 𝓢(a, ofList s φs) := by
-  rw [ofList_cons_eq_mul, map_mul]
+    𝓢(a, ofList s (φ :: φs)) = 𝓢(a, s φ) * 𝓢(a, ofList s φs) := by rw [ofList_cons_eq_mul, map_mul]
 
 /-- The exchange sign is a cocycle. -/
 lemma exchangeSign_cocycle (a b c : FieldStatistic) :
diff --git a/Physlib/QFT/PerturbationTheory/FieldStatistics/OfFinset.lean b/Physlib/QFT/PerturbationTheory/FieldStatistics/OfFinset.lean
index b86ef25e3..85f2f0f4d 100644
--- a/Physlib/QFT/PerturbationTheory/FieldStatistics/OfFinset.lean
+++ b/Physlib/QFT/PerturbationTheory/FieldStatistics/OfFinset.lean
@@ -32,8 +32,7 @@ lemma ofFinset_empty (q : 𝓕 → FieldStatistic) (f : Fin n → 𝓕) :
   rfl
 
 lemma ofFinset_singleton {n : ℕ} (q : 𝓕 → FieldStatistic) (f : Fin n → 𝓕) (i : Fin n) :
-    ofFinset q f {i} = q (f i) := by
-  simp [ofFinset]
+    ofFinset q f {i} = q (f i) := by simp [ofFinset]
 
 lemma ofFinset_finset_map {n m : ℕ}
     (q : 𝓕 → FieldStatistic) (i : Fin m → Fin n) (hi : Function.Injective i)
@@ -59,8 +58,7 @@ lemma ofFinset_insert (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finse
   simp only [ofFinset, Fin.getElem_fin]
   rw [← ofList_cons_eq_mul]
   have h1 : (φs[↑i] :: List.map φs.get (a.sort (fun x1 x2 => x1 ≤ x2)))
-      = List.map φs.get (i :: a.sort (fun x1 x2 => x1 ≤ x2)) := by
-      simp
+      = List.map φs.get (i :: a.sort (fun x1 x2 => x1 ≤ x2)) := by simp
   erw [h1]
   apply ofList_perm
   refine List.Perm.map φs.get ?_
@@ -74,8 +72,7 @@ lemma ofFinset_insert (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finse
 lemma ofFinset_erase (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length))
     (i : Fin φs.length) (h : i ∈ a) :
     ofFinset q φs.get (a.erase i) = (q φs[i]) * ofFinset q φs.get a := by
-  have ha : a = Insert.insert i (a.erase i) := by
-    exact Eq.symm (Finset.insert_erase h)
+  have ha : a = Insert.insert i (a.erase i) := Eq.symm (Finset.insert_erase h)
   conv_rhs => rw [ha]
   rw [ofFinset_insert]
   rw [← mul_assoc]
@@ -84,8 +81,7 @@ lemma ofFinset_erase (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset
 
 lemma ofFinset_eq_prod (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length)) :
     ofFinset q φs.get a = ∏ (i : Fin φs.length), if i ∈ a then (q φs[i]) else 1 := by
-  rw [ofFinset]
-  rw [ofList_map_eq_finset_prod]
+  rw [ofFinset, ofList_map_eq_finset_prod]
   congr
   funext i
   simp only [Finset.mem_sort, Fin.getElem_fin]
@@ -93,8 +89,7 @@ lemma ofFinset_eq_prod (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Fins
 
 lemma ofFinset_union (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a b : Finset (Fin φs.length)) :
     ofFinset q φs.get a * ofFinset q φs.get b = ofFinset q φs.get ((a ∪ b) \ (a ∩ b)) := by
-  rw [ofFinset_eq_prod, ofFinset_eq_prod, ofFinset_eq_prod]
-  rw [← Finset.prod_mul_distrib]
+  rw [ofFinset_eq_prod, ofFinset_eq_prod, ofFinset_eq_prod, ← Finset.prod_mul_distrib]
   congr
   funext x
   simp only [Fin.getElem_fin, mul_ite, ite_mul, mul_self, one_mul, mul_one,
@@ -108,8 +103,7 @@ lemma ofFinset_union (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a b : Fins
 lemma ofFinset_union_disjoint (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a b : Finset (Fin φs.length))
     (h : Disjoint a b) :
     ofFinset q φs.get a * ofFinset q φs.get b = ofFinset q φs.get (a ∪ b) := by
-  rw [ofFinset_union, Finset.disjoint_iff_inter_eq_empty.mp h]
-  simp
+  simpa [Finset.disjoint_iff_inter_eq_empty.mp h] using ofFinset_union q φs a b
 
 lemma ofFinset_filter_mul_neg (q : 𝓕 → FieldStatistic) (φs : List 𝓕) (a : Finset (Fin φs.length))
     (p : Fin φs.length → Prop) [DecidablePred p] :
diff --git a/Physlib/QFT/PerturbationTheory/Koszul/KoszulSign.lean b/Physlib/QFT/PerturbationTheory/Koszul/KoszulSign.lean
index 5de3325d7..c5e27524c 100644
--- a/Physlib/QFT/PerturbationTheory/Koszul/KoszulSign.lean
+++ b/Physlib/QFT/PerturbationTheory/Koszul/KoszulSign.lean
@@ -31,8 +31,7 @@ def koszulSign (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [Deci
 
 @[simp]
 lemma koszulSign_singleton (q : 𝓕 → FieldStatistic) (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] (φ : 𝓕) :
-    koszulSign q le [φ] = 1 := by
-  simp [koszulSign, koszulSignInsert]
+    koszulSign q le [φ] = 1 := by simp [koszulSign, koszulSignInsert]
 
 lemma koszulSign_mul_self (l : List 𝓕) : koszulSign q le l * koszulSign q le l = 1 := by
   induction l with
@@ -47,8 +46,7 @@ lemma koszulSign_mul_self (l : List 𝓕) : koszulSign q le l * koszulSign q le
 @[simp]
 lemma koszulSign_freeMonoid_of (φ : 𝓕) : koszulSign q le (FreeMonoid.of φ) = 1 := by
   change koszulSign q le [φ] = 1
-  simp only [koszulSign, mul_one]
-  rfl
+  simp [koszulSign, koszulSignInsert, mul_one]
 
 lemma koszulSignInsert_erase_boson {𝓕 : Type} (q : 𝓕 → FieldStatistic)
     (le : 𝓕 → 𝓕 → Prop) [DecidableRel le] (φ : 𝓕) :
@@ -116,12 +114,10 @@ lemma koszulSign_insertIdx [Std.Total le] [IsTrans 𝓕 le] (φ : 𝓕) :
       change 𝓢(q φ, ofList q ((List.insertionSort le (φ1 :: φs)).take
         (↑(orderedInsertPos le ((List.insertionSort le (φ1 :: φs))) φ))))
       rw [← koszulSignInsert_eq_exchangeSign_take q le]
-    rw [ofList_take_zero]
     simp
   | φ1 :: φs, n + 1, h => by
     conv_lhs =>
-      rw [List.insertIdx_succ_cons]
-      rw [koszulSign]
+      rw [List.insertIdx_succ_cons, koszulSign]
     rw [koszulSign_insertIdx _ _ _ (Nat.le_of_lt_succ h)]
     conv_rhs =>
       rhs
@@ -171,12 +167,10 @@ lemma koszulSign_insertIdx [Std.Total le] [IsTrans 𝓕 le] (φ : 𝓕) :
       simp only [Function.comp_apply, Equiv.symm_apply_apply, List.get_eq_getElem, ni]
       simp_all only [List.length_cons, add_le_add_iff_right, List.getElem_insertIdx_self]
     have hc1 (hninro : ni.castSucc < nro) : ¬ le φ1 φ := by
-      rw [← hns]
-      exact lt_orderedInsertPos_rel le φ1 rs ni hninro
+      simpa [← hns] using lt_orderedInsertPos_rel le φ1 rs ni hninro
     have hc2 (hninro : ¬ ni.castSucc < nro) : le φ1 φ := by
-      rw [← hns]
-      refine gt_orderedInsertPos_rel le φ1 rs ?_ ni hninro
-      exact List.pairwise_insertionSort le (List.insertIdx φs n φ)
+      simpa [← hns] using gt_orderedInsertPos_rel le φ1 rs
+        (List.pairwise_insertionSort le (List.insertIdx φs n φ)) ni hninro
     by_cases hn : ni.castSucc < nro
     · simp only [hn, ↓reduceIte, Fin.val_castSucc]
       rw [ofList_take_insertIdx_gt]
@@ -210,8 +204,7 @@ lemma koszulSign_eraseIdx [Std.Total le] [IsTrans 𝓕 le] (φs : List 𝓕) (n
     𝓢(q (φs.get n), ofList q (List.take (↑(insertionSortEquiv le φs n))
     (List.insertionSort le φs))) := by
   let φs' := φs.eraseIdx ↑n
-  have hφs : List.insertIdx φs' n (φs.get n) = φs := by
-    exact insertIdx_eraseIdx n.1 φs n.prop
+  have hφs : List.insertIdx φs' n (φs.get n) = φs := insertIdx_eraseIdx n.1 φs n.prop
   conv_rhs =>
     lhs
     lhs
@@ -241,10 +234,7 @@ lemma koszulSign_eraseIdx [Std.Total le] [IsTrans 𝓕 le] (φs : List 𝓕) (n
     rhs
     rw [exchangeSign_mul_self]
   simp only [Fin.getElem_fin, mul_one]
-  conv_rhs =>
-    rhs
-    rw [ofList_take_eraseIdx, exchangeSign_mul_self]
-  simp
+  simp [ofList_take_eraseIdx, exchangeSign_mul_self]
 
 lemma koszulSign_eraseIdx_insertionSortMinPos [Std.Total le] [IsTrans 𝓕 le] (φ : 𝓕) (φs : List 𝓕) :
     koszulSign q le ((φ :: φs).eraseIdx (insertionSortMinPos le φ φs)) = koszulSign q le (φ :: φs)
@@ -272,8 +262,7 @@ lemma koszulSign_swap_eq_rel_cons {ψ φ : 𝓕}
 lemma koszulSign_swap_eq_rel {ψ φ : 𝓕} (h1 : le φ ψ) (h2 : le ψ φ) : (φs φs' : List 𝓕) →
     koszulSign q le (φs ++ φ :: ψ :: φs') = koszulSign q le (φs ++ ψ :: φ :: φs')
   | [], φs' => by
-    simp only [List.nil_append]
-    exact koszulSign_swap_eq_rel_cons q le h1 h2 φs'
+    simpa only [List.nil_append] using koszulSign_swap_eq_rel_cons q le h1 h2 φs'
   | φ'' :: φs, φs' => by
     simp only [List.cons_append, koszulSign]
     rw [koszulSign_swap_eq_rel h1 h2]
@@ -292,8 +281,7 @@ lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le]
   congr
   simp only [koszulSignInsert, ite_mul, neg_mul]
   simp_all only [and_self, ite_true]
-  rw [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hq]
-  simp
+  simp [koszulSignInsert_eq_rel_eq_stat q le h1 h2 hq]
 
 lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
     (h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs' φs : List 𝓕) →
@@ -324,9 +312,8 @@ lemma koszulSign_of_sorted : (φs : List 𝓕)
 
 @[simp]
 lemma koszulSign_of_insertionSort [Std.Total le] [IsTrans 𝓕 le] (φs : List 𝓕) :
-    koszulSign q le (List.insertionSort le φs) = 1 := by
-  apply koszulSign_of_sorted
-  exact List.pairwise_insertionSort le φs
+    koszulSign q le (List.insertionSort le φs) = 1 :=
+  koszulSign_of_sorted q le _ (List.pairwise_insertionSort le φs)
 
 lemma koszulSign_of_append_eq_insertionSort_left [Std.Total le] [IsTrans 𝓕 le] :
     (φs φs' : List 𝓕) → koszulSign q le (φs ++ φs') =
@@ -354,15 +341,13 @@ lemma koszulSign_of_append_eq_insertionSort_left [Std.Total le] [IsTrans 𝓕 le
     congr 3
     · have h2 : (List.insertionSort le φs ++ φ :: φs') =
           List.insertIdx (List.insertionSort le φs ++ φs') φs.length φ := by
-        rw [← insertIdx_length_fst_append]
-        simp
+        simp [← insertIdx_length_fst_append]
       rw [insertionSortEquiv_congr _ _ h2.symm]
       simp only [Equiv.trans_apply, RelIso.coe_fn_toEquiv, Fin.castOrderIso_apply, Fin.cast_mk,
         Fin.val_cast]
       rw [insertionSortEquiv_insertionSort_append]
       simp only [finCongr_apply, Fin.val_cast]
-      rw [insertionSortEquiv_congr _ _ h1.symm]
-      simp
+      simp [insertionSortEquiv_congr _ _ h1.symm]
     · rw [insertIdx_length_fst_append]
       rw [show φs.length = (List.insertionSort le φs).length by simp]
       rw [insertIdx_length_fst_append]
@@ -379,10 +364,9 @@ lemma koszulSign_of_append_eq_insertionSort [Std.Total le] [IsTrans 𝓕 le] : (
     simp only [List.cons_append, koszulSign]
     rw [koszulSign_of_append_eq_insertionSort φs'' φs φs', ← mul_assoc]
     congr 2
-    apply koszulSignInsert_eq_perm
-    refine (List.perm_append_right_iff φs').mpr ?_
-    refine List.Perm.append_left φs'' ?_
-    exact List.Perm.symm (List.perm_insertionSort le φs)
+    exact koszulSignInsert_eq_perm q le _ _ _
+      ((List.perm_append_right_iff φs').mpr
+        (List.Perm.append_left φs'' (List.Perm.symm (List.perm_insertionSort le φs))))
 
 /-!
 
@@ -415,8 +399,7 @@ lemma koszulSign_perm_eq_append [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : L
     simp_all only [and_self, implies_true, nonempty_prop, forall_const, motive]
     refine (ih2 ?_)
     intro φ' hφ
-    refine h φ' ?_
-    exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφ
+    exact h φ' ((List.Perm.mem_iff (id (List.Perm.symm h1))).mp hφ)
 
 lemma koszulSign_perm_eq [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : List 𝓕) →
     (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) → (hp : φs.Perm φs') →
@@ -429,8 +412,7 @@ lemma koszulSign_perm_eq [IsTrans 𝓕 le] (φ : 𝓕) : (φs1 φs φs' φs2 : L
     have ih := koszulSign_perm_eq φ φs1 φs φs' φs2 h hp
     rw [ih]
     congr 1
-    apply koszulSignInsert_eq_perm
-    refine (List.perm_append_right_iff φs2).mpr ?_
-    exact List.Perm.append_left φs1 hp
+    exact koszulSignInsert_eq_perm q le _ _ _
+      ((List.perm_append_right_iff φs2).mpr (List.Perm.append_left φs1 hp))
 
 end Wick
diff --git a/Physlib/QFT/PerturbationTheory/Koszul/KoszulSignInsert.lean b/Physlib/QFT/PerturbationTheory/Koszul/KoszulSignInsert.lean
index 96add368d..7cf6a5af4 100644
--- a/Physlib/QFT/PerturbationTheory/Koszul/KoszulSignInsert.lean
+++ b/Physlib/QFT/PerturbationTheory/Koszul/KoszulSignInsert.lean
@@ -103,8 +103,7 @@ lemma koszulSignInsert_eq_filter (φ : 𝓕) : (φs : List 𝓕) →
 lemma koszulSignInsert_eq_cons [Std.Total le] (φ : 𝓕) (φs : List 𝓕) :
     koszulSignInsert q le φ φs = koszulSignInsert q le φ (φ :: φs) := by
   simp only [koszulSignInsert, and_self]
-  have h1 : le φ φ := by
-    simpa only [or_self] using Std.Total.total (r := le) φ φ
+  have h1 : le φ φ := (Std.Total.total (r := le) φ φ).elim id id
   simp [h1]
 
 lemma koszulSignInsert_eq_grade (φ : 𝓕) (φs : List 𝓕) :
@@ -148,15 +147,13 @@ lemma koszulSignInsert_eq_perm (φs φs' : List 𝓕) (φ : 𝓕) (h : φs.Perm
   simp only [decide_not, eq_iff_iff, and_congr_right_iff]
   intro h'
   have hg : ofList q (List.filter (fun i => !decide (le φ i)) φs) =
-      ofList q (List.filter (fun i => !decide (le φ i)) φs') := by
-    apply ofList_perm
-    exact List.Perm.filter (fun i => !decide (le φ i)) h
+      ofList q (List.filter (fun i => !decide (le φ i)) φs') :=
+    ofList_perm q (List.Perm.filter (fun i => !decide (le φ i)) h)
   rw [hg]
 
 lemma koszulSignInsert_eq_sort (φs : List 𝓕) (φ : 𝓕) :
-    koszulSignInsert q le φ φs = koszulSignInsert q le φ (List.insertionSort le φs) := by
-  apply koszulSignInsert_eq_perm
-  exact List.Perm.symm (List.perm_insertionSort le φs)
+    koszulSignInsert q le φ φs = koszulSignInsert q le φ (List.insertionSort le φs) :=
+  koszulSignInsert_eq_perm q le _ _ _ (List.Perm.symm (List.perm_insertionSort le φs))
 
 lemma koszulSignInsert_eq_exchangeSign_take [Std.Total le] [IsTrans 𝓕 le] (φ : 𝓕) (φs : List 𝓕) :
     koszulSignInsert q le φ φs = 𝓢(q φ, ofList q
@@ -168,8 +165,7 @@ lemma koszulSignInsert_eq_exchangeSign_take [Std.Total le] [IsTrans 𝓕 le] (φ
       (List.insertionSort le φs))) = if FieldStatistic.ofList q [φ] = fermionic ∧
       FieldStatistic.ofList q (List.take (↑(orderedInsertPos le (List.insertionSort le φs) φ))
       (List.insertionSort le φs)) = fermionic then - 1 else 1 := by
-    rw [exchangeSign_eq_if]
-    simp
+    simp [exchangeSign_eq_if]
   rw [hx]
   congr
   simp only [List.filter_filter, Bool.and_self]
@@ -209,9 +205,8 @@ lemma koszulSignInsert_eq_exchangeSign_take [Std.Total le] [IsTrans 𝓕 le] (φ
     · simp [orderedInsertPos]
 
 lemma koszulSignInsert_insertIdx (i j : 𝓕) (r : List 𝓕) (n : ℕ) (hn : n ≤ r.length) :
-    koszulSignInsert q le j (List.insertIdx r n i) = koszulSignInsert q le j (i :: r) := by
-  apply koszulSignInsert_eq_perm
-  exact List.perm_insertIdx i r hn
+    koszulSignInsert q le j (List.insertIdx r n i) = koszulSignInsert q le j (i :: r) :=
+  koszulSignInsert_eq_perm q le _ _ _ (List.perm_insertIdx i r hn)
 
 /-- The difference in `koszulSignInsert` on inserting `r0` into `r` compared to
   into `r1 :: r` for any `r`. -/
@@ -236,8 +231,7 @@ lemma koszulSignCons_eq_exchangeSign (φ0 φ1 : 𝓕) : koszulSignCons q le φ0
 
 lemma koszulSignInsert_cons (r0 r1 : 𝓕) (r : List 𝓕) :
     koszulSignInsert q le r0 (r1 :: r) = (koszulSignCons q le r0 r1) *
-    koszulSignInsert q le r0 r := by
-  simp [koszulSignInsert, koszulSignCons]
+    koszulSignInsert q le r0 r := by simp [koszulSignInsert, koszulSignCons]
 
 lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀ b ∈ φs, le φ0 b) →
     koszulSignInsert q le φ0 φs = 1
@@ -263,8 +257,7 @@ lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le]
     · simp only [hr, ↓reduceIte]
       have h1' : le ψ φ' := by
         apply IsTrans.trans ψ φ φ' h2 hr
-      simp only [h1', ↓reduceIte]
-      exact koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs
+      simpa only [h1', ↓reduceIte] using koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs
     · have hψφ' : ¬ le ψ φ' := by
         intro hψφ'
         apply hr
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/Basic.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/Basic.lean
index 813ef0739..35eec32e3 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/Basic.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/Basic.lean
@@ -72,9 +72,8 @@ instance : Coe (𝓕.FieldOpFreeAlgebra) (𝓕.WickAlgebra) :=
 instance : Setoid (FieldOpFreeAlgebra 𝓕) := (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).ringCon.toSetoid
 
 lemma equiv_iff_sub_mem_ideal (x y : FieldOpFreeAlgebra 𝓕) :
-    x ≈ y ↔ x - y ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
-  rw [← TwoSidedIdeal.rel_iff]
-  rfl
+    x ≈ y ↔ x - y ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet :=
+  (TwoSidedIdeal.span 𝓕.fieldOpIdealSet).rel_iff x y
 
 lemma equiv_iff_exists_add (x y : FieldOpFreeAlgebra 𝓕) :
     x ≈ y ↔ ∃ a, x = y + a ∧ a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
@@ -380,8 +379,7 @@ lemma ofFieldOpList_append (φs ψs : List 𝓕.FieldOp) :
   simp
 
 lemma ofFieldOpList_cons (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
-    ofFieldOpList (φ :: φs) = ofFieldOp φ * ofFieldOpList φs := by
-  aesop
+    ofFieldOpList (φ :: φs) = ofFieldOp φ * ofFieldOpList φs := by aesop
 
 lemma ofFieldOpList_singleton (φ : 𝓕.FieldOp) :
     ofFieldOpList [φ] = ofFieldOp φ := by
@@ -418,8 +416,7 @@ lemma ofCrAnList_append (φs ψs : List 𝓕.CrAnFieldOp) :
   simp
 
 lemma ofCrAnList_singleton (φ : 𝓕.CrAnFieldOp) :
-    ofCrAnList [φ] = ofCrAnOp φ := by
-  simp only [ofCrAnList, ofCrAnOp, ofCrAnListF_singleton]
+    ofCrAnList [φ] = ofCrAnOp φ := by simp only [ofCrAnList, ofCrAnOp, ofCrAnListF_singleton]
 
 lemma ofFieldOpList_eq_sum (φs : List 𝓕.FieldOp) :
     ofFieldOpList φs = ∑ s : CrAnSection φs, ofCrAnList s.1 := by
@@ -438,19 +435,16 @@ lemma anPart_eq_ι_anPartF (φ : 𝓕.FieldOp) : anPart φ = ι (anPartF φ) :=
 
 @[simp]
 lemma anPart_inAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    anPart (FieldOp.inAsymp φ) = 0 := by
-  simp [anPart, anPartF]
+    anPart (FieldOp.inAsymp φ) = 0 := by simp [anPart, anPartF]
 
 @[simp]
 lemma anPart_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
     anPart (FieldOp.position φ) =
-    ofCrAnOp ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by
-  simp [anPart, ofCrAnOp]
+    ofCrAnOp ⟨FieldOp.position φ, CreateAnnihilate.annihilate⟩ := by simp [anPart, ofCrAnOp]
 
 @[simp]
 lemma anPart_outAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    anPart (FieldOp.outAsymp φ) = ofCrAnOp ⟨FieldOp.outAsymp φ, ()⟩ := by
-  simp [anPart, ofCrAnOp]
+    anPart (FieldOp.outAsymp φ) = ofCrAnOp ⟨FieldOp.outAsymp φ, ()⟩ := by simp [anPart, ofCrAnOp]
 
 /-- For a field specification `𝓕`, and an element `φ` of `𝓕.FieldOp`, the
   creation part of `𝓕.FieldOp` as an element of `𝓕.WickAlgebra`.
@@ -464,19 +458,16 @@ lemma crPart_eq_ι_crPartF (φ : 𝓕.FieldOp) : crPart φ = ι (crPartF φ) :=
 
 @[simp]
 lemma crPart_inAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    crPart (FieldOp.inAsymp φ) = ofCrAnOp ⟨FieldOp.inAsymp φ, ()⟩ := by
-  simp [crPart, ofCrAnOp]
+    crPart (FieldOp.inAsymp φ) = ofCrAnOp ⟨FieldOp.inAsymp φ, ()⟩ := by simp [crPart, ofCrAnOp]
 
 @[simp]
 lemma crPart_position (φ : (Σ f, 𝓕.PositionLabel f) × SpaceTime) :
     crPart (FieldOp.position φ) =
-    ofCrAnOp ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by
-  simp [crPart, ofCrAnOp]
+    ofCrAnOp ⟨FieldOp.position φ, CreateAnnihilate.create⟩ := by simp [crPart, ofCrAnOp]
 
 @[simp]
 lemma crPart_outAsymp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
-    crPart (FieldOp.outAsymp φ) = 0 := by
-  simp [crPart]
+    crPart (FieldOp.outAsymp φ) = 0 := by simp [crPart]
 
 /-- For field specification `𝓕`, and an element `φ` of `𝓕.FieldOp` the following relation holds:
 
@@ -486,18 +477,15 @@ That is, every field operator splits into its creation part plus its annihilatio
 -/
 lemma ofFieldOp_eq_crPart_add_anPart (φ : 𝓕.FieldOp) :
     ofFieldOp φ = crPart φ + anPart φ := by
-  rw [ofFieldOp, crPart, anPart, ofFieldOpF_eq_crPartF_add_anPartF]
-  simp [map_add]
+  simp [ofFieldOp, crPart, anPart, ofFieldOpF_eq_crPartF_add_anPartF, map_add]
 
 lemma anPart_outAsymp_eq_ofFieldOp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
     anPart (FieldOp.outAsymp φ) = ofFieldOp (FieldOp.outAsymp φ) := by
-  rw [ofFieldOp_eq_crPart_add_anPart]
-  simp
+  simp [ofFieldOp_eq_crPart_add_anPart]
 
 lemma crPart_inAsymp_eq_ofFieldOp (φ : (Σ f, 𝓕.AsymptoticLabel f) × Momentum) :
     crPart (FieldOp.inAsymp φ) = ofFieldOp (FieldOp.inAsymp φ) := by
-  rw [ofFieldOp_eq_crPart_add_anPart]
-  simp
+  simp [ofFieldOp_eq_crPart_add_anPart]
 
 end WickAlgebra
 end FieldSpecification
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/Grading.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/Grading.lean
index 077cdd961..de43bef79 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/Grading.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/Grading.lean
@@ -44,8 +44,7 @@ lemma mem_bosonic_of_mem_free_bosonic (a : 𝓕.FieldOpFreeAlgebra)
     simp only [Set.mem_setOf_eq] at hx
     obtain ⟨φs, rfl, h⟩ := hx
     simp [p]
-    apply ofCrAnList_mem_statSubmodule_of_eq
-    exact h
+    exact ofCrAnList_mem_statSubmodule_of_eq φs bosonic h
   · simp only [map_zero, p]
     exact Submodule.zero_mem (statSubmodule bosonic)
   · intro x y hx hy hpx hpy
@@ -65,8 +64,7 @@ lemma mem_fermionic_of_mem_free_fermionic (a : 𝓕.FieldOpFreeAlgebra)
     simp only [Set.mem_setOf_eq] at hx
     obtain ⟨φs, rfl, h⟩ := hx
     simp [p]
-    apply ofCrAnList_mem_statSubmodule_of_eq
-    exact h
+    exact ofCrAnList_mem_statSubmodule_of_eq φs fermionic h
   · simp only [map_zero, p]
     exact Submodule.zero_mem (statSubmodule fermionic)
   · intro x y hx hy hpx hpy
@@ -126,8 +124,7 @@ def bosonicProj : 𝓕.WickAlgebra →ₗ[ℂ] statSubmodule (𝓕 := 𝓕) boso
     obtain ⟨x, hx⟩ := ι_surjective x
     obtain ⟨y, hy⟩ := ι_surjective y
     subst hx hy
-    rw [← map_add, ι_apply, ι_apply, ι_apply]
-    rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
+    rw [← map_add, ι_apply, ι_apply, ι_apply, Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
     simp
   map_smul' c y := by
     obtain ⟨y, hy⟩ := ι_surjective y
@@ -172,8 +169,7 @@ def fermionicProj : 𝓕.WickAlgebra →ₗ[ℂ] statSubmodule (𝓕 := 𝓕) fe
     obtain ⟨x, hx⟩ := ι_surjective x
     obtain ⟨y, hy⟩ := ι_surjective y
     subst hx hy
-    rw [← map_add, ι_apply, ι_apply, ι_apply]
-    rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
+    rw [← map_add, ι_apply, ι_apply, ι_apply, Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
     simp
   map_smul' c y := by
     obtain ⟨y, hy⟩ := ι_surjective y
@@ -209,10 +205,8 @@ lemma bosonicProj_mem_bosonic (a : 𝓕.WickAlgebra) (ha : a ∈ statSubmodule .
     simp only [p]
     apply Subtype.ext
     simp only
-    rw [ofCrAnList]
-    rw [bosonicProj_eq_bosonicProjFree]
-    rw [bosonicProjFree_eq_ι_bosonicProjF]
-    rw [bosonicProjF_of_mem_bosonic]
+    rw [ofCrAnList, bosonicProj_eq_bosonicProjFree]
+    rw [bosonicProjFree_eq_ι_bosonicProjF, bosonicProjF_of_mem_bosonic]
     exact ofCrAnListF_mem_statisticSubmodule_of _ _ h
   · simp only [map_zero, p]
     rfl
@@ -233,9 +227,7 @@ lemma fermionicProj_mem_fermionic (a : 𝓕.WickAlgebra) (ha : a ∈ statSubmodu
     simp only [p]
     apply Subtype.ext
     simp only
-    rw [ofCrAnList]
-    rw [fermionicProj_eq_fermionicProjFree]
-    rw [fermionicProjFree_eq_ι_fermionicProjF]
+    rw [ofCrAnList, fermionicProj_eq_fermionicProjFree, fermionicProjFree_eq_ι_fermionicProjF]
     rw [fermionicProjF_of_mem_fermionic]
     exact ofCrAnListF_mem_statisticSubmodule_of _ _ h
   · simp only [map_zero, p]
@@ -291,15 +283,13 @@ lemma eq_zero_of_bosonic_and_fermionic {a : 𝓕.WickAlgebra}
 
 @[simp]
 lemma bosonicProj_fermionicProj_eq_zero (a : 𝓕.WickAlgebra) :
-    bosonicProj (fermionicProj a).1 = 0 := by
-  apply bosonicProj_mem_fermionic
-  exact Submodule.coe_mem (fermionicProj a)
+    bosonicProj (fermionicProj a).1 = 0 :=
+  bosonicProj_mem_fermionic (fermionicProj a).1 (Submodule.coe_mem (fermionicProj a))
 
 @[simp]
 lemma fermionicProj_bosonicProj_eq_zero (a : 𝓕.WickAlgebra) :
-    fermionicProj (bosonicProj a).1 = 0 := by
-  apply fermionicProj_mem_bosonic
-  exact Submodule.coe_mem (bosonicProj a)
+    fermionicProj (bosonicProj a).1 = 0 :=
+  fermionicProj_mem_bosonic (bosonicProj a).1 (Submodule.coe_mem (bosonicProj a))
 
 @[simp]
 lemma bosonicProj_bosonicProj_eq_bosonicProj (a : 𝓕.WickAlgebra) :
@@ -320,16 +310,14 @@ lemma bosonicProj_of_bosonic_part
 @[simp]
 lemma bosonicProj_of_fermionic_part
     (a : DirectSum FieldStatistic (fun i => (statSubmodule (𝓕 := 𝓕) i))) :
-    bosonicProj (a fermionic).1 = 0 := by
-  apply bosonicProj_mem_fermionic
-  exact Submodule.coe_mem (a.toFun fermionic)
+    bosonicProj (a fermionic).1 = 0 :=
+  bosonicProj_mem_fermionic (a.toFun fermionic).1 (Submodule.coe_mem (a.toFun fermionic))
 
 @[simp]
 lemma fermionicProj_of_bosonic_part
     (a : DirectSum FieldStatistic (fun i => (statSubmodule (𝓕 := 𝓕) i))) :
-    fermionicProj (a bosonic).1 = 0 := by
-  apply fermionicProj_mem_bosonic
-  exact Submodule.coe_mem (a.toFun bosonic)
+    fermionicProj (a bosonic).1 = 0 :=
+  fermionicProj_mem_bosonic (a.toFun bosonic).1 (Submodule.coe_mem (a.toFun bosonic))
 
 @[simp]
 lemma fermionicProj_of_fermionic_part
@@ -400,9 +388,7 @@ instance WickAlgebraGrade : GradedAlgebra (A := 𝓕.WickAlgebra) statSubmodule
     simp only [statSubmodule]
     refine Submodule.mem_span.mpr fun p a => a ?_
     simp only [Set.mem_setOf_eq]
-    use []
-    simp only [ofCrAnList, ofCrAnListF_nil, map_one, ofList_empty, true_and]
-    rfl
+    exact ⟨[], rfl, rfl⟩
   mul_mem f1 f2 a1 a2 h1 h2 := by
     let p (a2 : 𝓕.WickAlgebra) (hx : a2 ∈ statSubmodule f2) : Prop :=
       a1 * a2 ∈ statSubmodule (f1 + f2)
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Basic.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Basic.lean
index 81412c81b..70229a01b 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Basic.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Basic.lean
@@ -67,8 +67,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero
     exact ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero φa φa' φs l
   change (ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
     mulLinearMap ((ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛF))) a = 0
-  rw [hf]
-  simp
+  simp [hf]
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul (φa φa' : 𝓕.CrAnFieldOp)
     (a b : 𝓕.FieldOpFreeAlgebra) :
@@ -83,21 +82,17 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul (φa φa' : 𝓕.CrAnF
     simp only [mulLinearMap, FieldOpFreeAlgebra.ofListBasis_eq_ofList, LinearMap.coe_comp,
       Function.comp_apply, LinearMap.flip_apply, LinearMap.coe_mk, AddHom.coe_mk,
       AlgHom.toLinearMap_apply, LinearMap.zero_apply]
-    rw [← mul_assoc]
-    exact ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero φa φa' _ _
-  rw [hf]
-  simp
+    simpa [← mul_assoc] using ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero φa φa' _ _
+  simp [hf]
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul (φa : 𝓕.CrAnFieldOp)
     (φs : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
     ι 𝓝ᶠ(a * [ofCrAnOpF φa, ofCrAnListF φs]ₛF * b) = 0 := by
   rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum]
-  rw [Finset.mul_sum, Finset.sum_mul]
-  rw [map_sum, map_sum]
+  rw [Finset.mul_sum, Finset.sum_mul, map_sum, map_sum]
   apply Fintype.sum_eq_zero
   intro n
-  rw [← mul_assoc, ← mul_assoc]
-  rw [mul_assoc _ _ b, ofCrAnListF_singleton]
+  rw [← mul_assoc, ← mul_assoc, mul_assoc _ _ b, ofCrAnListF_singleton]
   rw [ι_normalOrderF_superCommuteF_ofCrAnOpF_eq_zero_mul]
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul (φa : 𝓕.CrAnFieldOp)
@@ -106,8 +101,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul (φa : 
   rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF_symm, ofCrAnListF_singleton]
   simp only [ofList_singleton, Algebra.mul_smul_comm, Algebra.smul_mul_assoc,
     map_smul]
-  rw [ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul]
-  simp
+  simp [ι_normalOrderF_superCommuteF_ofCrAnOpF_ofCrAnListF_eq_zero_mul]
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul
     (φs φs' : List 𝓕.CrAnFieldOp) (a b : 𝓕.FieldOpFreeAlgebra) :
@@ -116,8 +110,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul
   rw [map_sum, map_sum]
   apply Fintype.sum_eq_zero
   intro n
-  rw [← mul_assoc, ← mul_assoc]
-  rw [mul_assoc _ _ b]
+  rw [← mul_assoc, ← mul_assoc, mul_assoc _ _ b]
   rw [ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnOpF_eq_zero_mul]
 
 lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul
@@ -134,8 +127,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul
       LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
       LinearMap.zero_apply]
     rw [ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul]
-  rw [hf]
-  simp
+  simp [hf]
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero_mul
@@ -150,32 +142,26 @@ lemma ι_normalOrderF_superCommuteF_eq_zero_mul
       LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
       LinearMap.zero_apply]
     rw [ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul]
-  rw [hf]
-  simp
+  simp [hf]
 
 @[simp]
 lemma ι_normalOrder_superCommuteF_eq_zero_mul_right (b c d : 𝓕.FieldOpFreeAlgebra) :
     ι 𝓝ᶠ([d, c]ₛF * b) = 0 := by
-  rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 b c d]
-  simp
+  simpa using ι_normalOrderF_superCommuteF_eq_zero_mul (1 : 𝓕.FieldOpFreeAlgebra) b c d
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero_mul_left (a c d : 𝓕.FieldOpFreeAlgebra) :
     ι 𝓝ᶠ(a * [d, c]ₛF) = 0 := by
-  rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a 1 c d]
-  simp
+  simpa using ι_normalOrderF_superCommuteF_eq_zero_mul a (1 : 𝓕.FieldOpFreeAlgebra) c d
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero_mul_mul_right (a b1 b2 c d: 𝓕.FieldOpFreeAlgebra) :
     ι 𝓝ᶠ(a * [d, c]ₛF * b1 * b2) = 0 := by
-  rw [← ι_normalOrderF_superCommuteF_eq_zero_mul a (b1 * b2) c d]
-  congr 2
-  noncomm_ring
+  simpa [mul_assoc] using ι_normalOrderF_superCommuteF_eq_zero_mul a (b1 * b2) c d
 
 @[simp]
 lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ([d, c]ₛF) = 0 := by
-  rw [← ι_normalOrderF_superCommuteF_eq_zero_mul 1 1 c d]
-  simp
+  simpa using ι_normalOrderF_superCommuteF_eq_zero_mul (1 : 𝓕.FieldOpFreeAlgebra) 1 c d
 
 /-!
 
@@ -236,8 +222,7 @@ noncomputable def normalOrder : WickAlgebra 𝓕 →ₗ[ℂ] WickAlgebra 𝓕 wh
   map_add' x y := by
     obtain ⟨x, rfl⟩ := ι_surjective x
     obtain ⟨y, rfl⟩ := ι_surjective y
-    rw [← map_add, ι_apply, ι_apply, ι_apply]
-    rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
+    rw [← map_add, ι_apply, ι_apply, ι_apply, Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
     simp
   map_smul' c y := by
     obtain ⟨y, rfl⟩ := ι_surjective y
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Lemmas.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Lemmas.lean
index c7d41c4c1..d33355862 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Lemmas.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/NormalOrder/Lemmas.lean
@@ -34,29 +34,21 @@ lemma normalOrder_eq_ι_normalOrderF (a : 𝓕.FieldOpFreeAlgebra) :
 
 lemma normalOrder_ofCrAnList (φs : List 𝓕.CrAnFieldOp) :
     𝓝(ofCrAnList φs) = normalOrderSign φs • ofCrAnList (normalOrderList φs) := by
-  rw [ofCrAnList, normalOrder_eq_ι_normalOrderF, normalOrderF_ofCrAnListF]
-  rfl
+  simp only [ofCrAnList, normalOrder_eq_ι_normalOrderF, normalOrderF_ofCrAnListF, map_smul]
 
 @[simp]
 lemma normalOrder_one_eq_one : normalOrder (𝓕 := 𝓕) 1 = 1 := by
-  have h1 : 1 = ofCrAnList (𝓕 := 𝓕) [] := by simp [ofCrAnList]
-  rw [h1]
-  rw [normalOrder_ofCrAnList]
-  simp
+  simpa [ofCrAnList, normalOrderSign_nil, normalOrderList_nil, one_smul] using
+    normalOrder_ofCrAnList (𝓕 := 𝓕) []
 
 @[simp]
 lemma normalOrder_ofFieldOpList_nil : normalOrder (𝓕 := 𝓕) (ofFieldOpList []) = 1 := by
-  rw [ofFieldOpList]
-  rw [normalOrder_eq_ι_normalOrderF]
-  simp only [ofFieldOpListF_nil]
-  change normalOrder (𝓕 := 𝓕) 1 = _
-  simp
+  simp [ofFieldOpList, ofFieldOpListF_nil]
 
 @[simp]
 lemma normalOrder_ofCrAnList_nil : normalOrder (𝓕 := 𝓕) (ofCrAnList []) = 1 := by
-  rw [normalOrder_ofCrAnList]
-  simp only [normalOrderSign_nil, normalOrderList_nil, ofCrAnList_nil]
-  module
+  simpa only [normalOrderSign_nil, normalOrderList_nil, ofCrAnList_nil, one_smul] using
+    normalOrder_ofCrAnList (𝓕 := 𝓕) []
 
 lemma ofCrAnList_eq_normalOrder (φs : List 𝓕.CrAnFieldOp) :
     ofCrAnList (normalOrderList φs) = normalOrderSign φs • 𝓝(ofCrAnList φs) := by
@@ -71,34 +63,16 @@ lemma normalOrder_normalOrder_mid (a b c : 𝓕.WickAlgebra) :
   rw [normalOrder_eq_ι_normalOrderF]
   simp only [← map_mul]
   rw [normalOrder_eq_ι_normalOrderF]
-  rw [normalOrderF_normalOrderF_mid]
-  rfl
+  exact congrArg ι (normalOrderF_normalOrderF_mid a b c)
 
 lemma normalOrder_normalOrder_left (a b : 𝓕.WickAlgebra) :
-    𝓝(a * b) = 𝓝(𝓝(a) * b) := by
-  obtain ⟨a, rfl⟩ := ι_surjective a
-  obtain ⟨b, rfl⟩ := ι_surjective b
-  rw [normalOrder_eq_ι_normalOrderF]
-  simp only [← map_mul]
-  rw [normalOrder_eq_ι_normalOrderF]
-  rw [normalOrderF_normalOrderF_left]
-  rfl
+    𝓝(a * b) = 𝓝(𝓝(a) * b) := by simpa using normalOrder_normalOrder_mid (1 : 𝓕.WickAlgebra) a b
 
 lemma normalOrder_normalOrder_right (a b : 𝓕.WickAlgebra) :
-    𝓝(a * b) = 𝓝(a * 𝓝(b)) := by
-  obtain ⟨a, rfl⟩ := ι_surjective a
-  obtain ⟨b, rfl⟩ := ι_surjective b
-  rw [normalOrder_eq_ι_normalOrderF]
-  simp only [← map_mul]
-  rw [normalOrder_eq_ι_normalOrderF]
-  rw [normalOrderF_normalOrderF_right]
-  rfl
+    𝓝(a * b) = 𝓝(a * 𝓝(b)) := by simpa using normalOrder_normalOrder_mid a b (1 : 𝓕.WickAlgebra)
 
 lemma normalOrder_normalOrder (a : 𝓕.WickAlgebra) : 𝓝(𝓝(a)) = 𝓝(a) := by
-  trans 𝓝(𝓝(a) * 1)
-  · simp
-  · rw [← normalOrder_normalOrder_left]
-    simp
+  simpa using (normalOrder_normalOrder_left a (1 : 𝓕.WickAlgebra)).symm
 
 /-!
 
@@ -108,14 +82,12 @@ lemma normalOrder_normalOrder (a : 𝓕.WickAlgebra) : 𝓝(𝓝(a)) = 𝓝(a) :
 lemma normalOrder_mul_anPart (φ : 𝓕.FieldOp) (a : 𝓕.WickAlgebra) :
     𝓝(a * anPart φ) = 𝓝(a) * anPart φ := by
   obtain ⟨a, rfl⟩ := ι_surjective a
-  rw [anPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_mul_anPartF]
-  rfl
+  exact congrArg ι (normalOrderF_mul_anPartF _ _)
 
 lemma crPart_mul_normalOrder (φ : 𝓕.FieldOp) (a : 𝓕.WickAlgebra) :
     𝓝(crPart φ * a) = crPart φ * 𝓝(a) := by
   obtain ⟨a, rfl⟩ := ι_surjective a
-  rw [crPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_crPartF_mul]
-  rfl
+  exact congrArg ι (normalOrderF_crPartF_mul _ _)
 
 /-!
 
@@ -130,8 +102,7 @@ lemma normalOrder_superCommute_eq_zero (a b : 𝓕.WickAlgebra) :
     𝓝([a, b]ₛ) = 0 := by
   obtain ⟨a, rfl⟩ := ι_surjective a
   obtain ⟨b, rfl⟩ := ι_surjective b
-  rw [superCommute_eq_ι_superCommuteF, normalOrder_eq_ι_normalOrderF]
-  simp
+  simp [superCommute_eq_ι_superCommuteF, normalOrder_eq_ι_normalOrderF]
 
 @[simp]
 lemma normalOrder_superCommute_left_eq_zero (a b c: 𝓕.WickAlgebra) :
@@ -139,8 +110,7 @@ lemma normalOrder_superCommute_left_eq_zero (a b c: 𝓕.WickAlgebra) :
   obtain ⟨a, rfl⟩ := ι_surjective a
   obtain ⟨b, rfl⟩ := ι_surjective b
   obtain ⟨c, rfl⟩ := ι_surjective c
-  rw [superCommute_eq_ι_superCommuteF, ← map_mul, normalOrder_eq_ι_normalOrderF]
-  simp
+  simp [superCommute_eq_ι_superCommuteF, ← map_mul, normalOrder_eq_ι_normalOrderF]
 
 @[simp]
 lemma normalOrder_superCommute_right_eq_zero (a b c: 𝓕.WickAlgebra) :
@@ -148,8 +118,7 @@ lemma normalOrder_superCommute_right_eq_zero (a b c: 𝓕.WickAlgebra) :
   obtain ⟨a, rfl⟩ := ι_surjective a
   obtain ⟨b, rfl⟩ := ι_surjective b
   obtain ⟨c, rfl⟩ := ι_surjective c
-  rw [superCommute_eq_ι_superCommuteF, ← map_mul, normalOrder_eq_ι_normalOrderF]
-  simp
+  simp [superCommute_eq_ι_superCommuteF, ← map_mul, normalOrder_eq_ι_normalOrderF]
 
 @[simp]
 lemma normalOrder_superCommute_mid_eq_zero (a b c d : 𝓕.WickAlgebra) :
@@ -158,8 +127,7 @@ lemma normalOrder_superCommute_mid_eq_zero (a b c d : 𝓕.WickAlgebra) :
   obtain ⟨b, rfl⟩ := ι_surjective b
   obtain ⟨c, rfl⟩ := ι_surjective c
   obtain ⟨d, rfl⟩ := ι_surjective d
-  rw [superCommute_eq_ι_superCommuteF, ← map_mul, ← map_mul, normalOrder_eq_ι_normalOrderF]
-  simp
+  simp [superCommute_eq_ι_superCommuteF, ← map_mul, ← map_mul, normalOrder_eq_ι_normalOrderF]
 
 /-!
 
@@ -192,12 +160,10 @@ lemma normalOrder_anPart_ofFieldOpList_swap (φ : 𝓕.FieldOp) (φ' : List 𝓕
     module
   | .position φ =>
     simp only [anPart_position]
-    rw [normalOrder_ofCrAnOp_ofFieldOpList_swap]
-    rfl
+    exact normalOrder_ofCrAnOp_ofFieldOpList_swap _ _
   | .outAsymp φ =>
     simp only [anPart_outAsymp]
-    rw [normalOrder_ofCrAnOp_ofFieldOpList_swap]
-    rfl
+    exact normalOrder_ofCrAnOp_ofFieldOpList_swap _ _
 
 lemma normalOrder_ofFieldOpList_anPart_swap (φ : 𝓕.FieldOp) (φ' : List 𝓕.FieldOp) :
     𝓝(ofFieldOpList φ' * anPart φ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • 𝓝(anPart φ * ofFieldOpList φ') := by
@@ -215,9 +181,7 @@ lemma anPart_mul_normalOrder_ofFieldOpList_eq_superCommute (φ : 𝓕.FieldOp)
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • 𝓝(ofFieldOpList φs' * anPart φ) +
     [anPart φ, 𝓝(ofFieldOpList φs')]ₛ := by
   rw [anPart, ofFieldOpList, normalOrder_eq_ι_normalOrderF, ← map_mul]
-  rw [anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF]
-  simp only [map_add, map_smul]
-  rfl
+  exact congrArg ι (anPartF_mul_normalOrderF_ofFieldOpListF_eq_superCommuteF _ _)
 
 /-!
 
@@ -261,8 +225,7 @@ lemma ofCrAnOp_superCommute_normalOrder_ofCrAnList_sum (φ : 𝓕.CrAnFieldOp)
       𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ ((normalOrderList φs).take (normalOrderEquiv n))))
       * 𝓢(𝓕 |>ₛ (φs.get n), 𝓕 |>ₛ (φs.take n))
     · ring_nf
-      rw [hs]
-      rfl
+      simp [hs]
     · simp [hs]
   · erw [superCommute_diff_statistic hs]
     simp only [zero_mul, smul_zero]
@@ -330,9 +293,7 @@ Within a proto-operator algebra we have that
 lemma anPart_mul_normalOrder_ofFieldOpList_eq_superCommute_reorder (φ : 𝓕.FieldOp)
     (φs : List 𝓕.FieldOp) : anPart φ * 𝓝(ofFieldOpList φs) =
     𝓝(anPart φ * ofFieldOpList φs) + [anPart φ, 𝓝(ofFieldOpList φs)]ₛ := by
-  rw [anPart_mul_normalOrder_ofFieldOpList_eq_superCommute]
-  simp only [add_left_inj]
-  rw [normalOrder_anPart_ofFieldOpList_swap]
+  rw [anPart_mul_normalOrder_ofFieldOpList_eq_superCommute, normalOrder_anPart_ofFieldOpList_swap]
 
 /--
 Within a proto-operator algebra we have that
@@ -438,15 +399,13 @@ lemma normalOrder_crPart_mul_anPart (φ φ' : 𝓕.FieldOp) :
 @[simp]
 lemma normalOrder_anPart_mul_crPart (φ φ' : 𝓕.FieldOp) :
     𝓝(anPart φ * crPart φ') = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
-  rw [anPart, crPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_anPartF_mul_crPartF]
-  simp
+  simp [anPart, crPart, ← map_mul, normalOrder_eq_ι_normalOrderF, normalOrderF_anPartF_mul_crPartF]
 
 lemma normalOrder_ofFieldOp_mul_ofFieldOp (φ φ' : 𝓕.FieldOp) : 𝓝(ofFieldOp φ * ofFieldOp φ') =
     crPart φ * crPart φ' + 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • (crPart φ' * anPart φ) +
     crPart φ * anPart φ' + anPart φ * anPart φ' := by
-  rw [ofFieldOp, ofFieldOp, ← map_mul, normalOrder_eq_ι_normalOrderF,
-    normalOrderF_ofFieldOpF_mul_ofFieldOpF]
-  rfl
+  rw [ofFieldOp, ofFieldOp, ← map_mul, normalOrder_eq_ι_normalOrderF]
+  simp [normalOrderF_ofFieldOpF_mul_ofFieldOpF, crPart, anPart]
 
 end WickAlgebra
 end FieldSpecification
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/StaticWickTheorem.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/StaticWickTheorem.lean
index 3261bf4f4..5a7f83d96 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/StaticWickTheorem.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/StaticWickTheorem.lean
@@ -68,8 +68,7 @@ theorem static_wick_theorem : (φs : List 𝓕.FieldOp) →
     rw [Finset.mul_sum]
     apply Finset.sum_congr rfl
     intro c _
-    rw [mul_staticWickTerm_eq_sum]
-    rfl
+    simp [mul_staticWickTerm_eq_sum]
 
 end WickAlgebra
 end FieldSpecification
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/SuperCommute.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/SuperCommute.lean
index 34ced8b47..171a599ee 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/SuperCommute.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/SuperCommute.lean
@@ -41,9 +41,8 @@ lemma ι_superCommuteF_eq_zero_of_ι_left_zero (a b : 𝓕.FieldOpFreeAlgebra) (
 -/
 
 lemma ι_superCommuteF_right_zero_of_mem_ideal (a b : 𝓕.FieldOpFreeAlgebra)
-    (h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛF = 0 := by
-  apply ι_superCommuteF_eq_zero_of_ι_right_zero
-  exact (ι_eq_zero_iff_mem_ideal b).mpr h
+    (h : b ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι [a, b]ₛF = 0 :=
+  ι_superCommuteF_eq_zero_of_ι_right_zero a b ((ι_eq_zero_iff_mem_ideal b).mpr h)
 
 lemma ι_superCommuteF_eq_of_equiv_right (a b1 b2 : 𝓕.FieldOpFreeAlgebra) (h : b1 ≈ b2) :
     ι [a, b1]ₛF = ι [a, b2]ₛF := by
@@ -60,8 +59,7 @@ noncomputable def superCommuteRight (a : 𝓕.FieldOpFreeAlgebra) :
     obtain ⟨x, hx⟩ := ι_surjective x
     obtain ⟨y, hy⟩ := ι_surjective y
     subst hx hy
-    rw [← map_add, ι_apply, ι_apply, ι_apply]
-    rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
+    rw [← map_add, ι_apply, ι_apply, ι_apply, Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
     simp
   map_smul' c y := by
     obtain ⟨y, hy⟩ := ι_surjective y
@@ -156,13 +154,11 @@ lemma superCommute_annihilate_annihilate {φ φ' : 𝓕.CrAnFieldOp}
 
 lemma superCommute_diff_statistic {φ φ' : 𝓕.CrAnFieldOp} (h : (𝓕 |>ₛ φ) ≠ 𝓕 |>ₛ φ') :
     [ofCrAnOp φ, ofCrAnOp φ']ₛ = 0 := by
-  rw [ofCrAnOp, ofCrAnOp]
-  rw [superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_diff_statistic h]
+  rw [ofCrAnOp, ofCrAnOp, superCommute_eq_ι_superCommuteF, ι_superCommuteF_of_diff_statistic h]
 
 lemma superCommute_ofCrAnOp_ofFieldOp_diff_stat_zero (φ : 𝓕.CrAnFieldOp) (ψ : 𝓕.FieldOp)
     (h : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)) : [ofCrAnOp φ, ofFieldOp ψ]ₛ = 0 := by
-  rw [ofFieldOp_eq_sum, map_sum]
-  rw [Finset.sum_eq_zero]
+  rw [ofFieldOp_eq_sum, map_sum, Finset.sum_eq_zero]
   intro x hx
   apply superCommute_diff_statistic
   simpa [crAnStatistics] using h
@@ -186,10 +182,8 @@ lemma superCommute_ofCrAnOp_ofCrAnOp_mem_center (φ φ' : 𝓕.CrAnFieldOp) :
 
 lemma superCommute_ofCrAnOp_ofCrAnOp_commute (φ φ' : 𝓕.CrAnFieldOp)
     (a : WickAlgebra 𝓕) :
-    a * [ofCrAnOp φ, ofCrAnOp φ']ₛ = [ofCrAnOp φ, ofCrAnOp φ']ₛ * a := by
-  have h1 := superCommute_ofCrAnOp_ofCrAnOp_mem_center φ φ'
-  rw [@Subalgebra.mem_center_iff] at h1
-  exact h1 a
+    a * [ofCrAnOp φ, ofCrAnOp φ']ₛ = [ofCrAnOp φ, ofCrAnOp φ']ₛ * a :=
+  (Subalgebra.mem_center_iff.mp (superCommute_ofCrAnOp_ofCrAnOp_mem_center φ φ')) a
 
 lemma superCommute_ofCrAnOp_ofFieldOp_mem_center (φ : 𝓕.CrAnFieldOp) (φ' : 𝓕.FieldOp) :
     [ofCrAnOp φ, ofFieldOp φ']ₛ ∈ Subalgebra.center ℂ (WickAlgebra 𝓕) := by
@@ -201,10 +195,8 @@ lemma superCommute_ofCrAnOp_ofFieldOp_mem_center (φ : 𝓕.CrAnFieldOp) (φ' :
 
 lemma superCommute_ofCrAnOp_ofFieldOp_commute (φ : 𝓕.CrAnFieldOp) (φ' : 𝓕.FieldOp)
     (a : WickAlgebra 𝓕) : a * [ofCrAnOp φ, ofFieldOp φ']ₛ =
-    [ofCrAnOp φ, ofFieldOp φ']ₛ * a := by
-  have h1 := superCommute_ofCrAnOp_ofFieldOp_mem_center φ φ'
-  rw [@Subalgebra.mem_center_iff] at h1
-  exact h1 a
+    [ofCrAnOp φ, ofFieldOp φ']ₛ * a :=
+  (Subalgebra.mem_center_iff.mp (superCommute_ofCrAnOp_ofFieldOp_mem_center φ φ')) a
 
 lemma superCommute_anPart_ofFieldOp_mem_center (φ φ' : 𝓕.FieldOp) :
     [anPart φ, ofFieldOp φ']ₛ ∈ Subalgebra.center ℂ (WickAlgebra 𝓕) := by
@@ -226,59 +218,51 @@ lemma superCommute_anPart_ofFieldOp_mem_center (φ φ' : 𝓕.FieldOp) :
 lemma superCommute_ofCrAnList_ofCrAnList (φs φs' : List 𝓕.CrAnFieldOp) :
     [ofCrAnList φs, ofCrAnList φs']ₛ =
     ofCrAnList (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList (φs' ++ φs) := by
-  rw [ofCrAnList_eq_ι_ofCrAnListF, ofCrAnList_eq_ι_ofCrAnListF]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnListF_ofCrAnListF]
-  rfl
+  simp [ofCrAnList_eq_ι_ofCrAnListF, superCommute_eq_ι_superCommuteF,
+    superCommuteF_ofCrAnListF_ofCrAnListF]
 
 lemma superCommute_ofCrAnOp_ofCrAnOp (φ φ' : 𝓕.CrAnFieldOp) :
     [ofCrAnOp φ, ofCrAnOp φ']ₛ = ofCrAnOp φ * ofCrAnOp φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofCrAnOp φ' * ofCrAnOp φ := by
-  rw [ofCrAnOp, ofCrAnOp]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnOpF_ofCrAnOpF]
-  rfl
+  simp [ofCrAnOp, superCommute_eq_ι_superCommuteF,
+    superCommuteF_ofCrAnOpF_ofCrAnOpF]
 
 lemma superCommute_ofCrAnList_ofFieldOpList (φcas : List 𝓕.CrAnFieldOp)
     (φs : List 𝓕.FieldOp) :
     [ofCrAnList φcas, ofFieldOpList φs]ₛ = ofCrAnList φcas * ofFieldOpList φs -
     𝓢(𝓕 |>ₛ φcas, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofCrAnList φcas := by
-  rw [ofCrAnList, ofFieldOpList]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofCrAnListF_ofFieldOpFsList]
-  rfl
+  simp [ofCrAnList, ofFieldOpList, superCommute_eq_ι_superCommuteF,
+    superCommuteF_ofCrAnListF_ofFieldOpFsList]
 
 lemma superCommute_ofFieldOpList_ofFieldOpList (φs φs' : List 𝓕.FieldOp) :
     [ofFieldOpList φs, ofFieldOpList φs']ₛ = ofFieldOpList φs * ofFieldOpList φs' -
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs := by
-  rw [ofFieldOpList, ofFieldOpList]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofFieldOpFsList]
-  rfl
+  simp [ofFieldOpList, superCommute_eq_ι_superCommuteF,
+    superCommuteF_ofFieldOpListF_ofFieldOpFsList]
 
 lemma superCommute_ofFieldOp_ofFieldOpList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
     [ofFieldOp φ, ofFieldOpList φs]ₛ = ofFieldOp φ * ofFieldOpList φs -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * ofFieldOp φ := by
-  rw [ofFieldOp, ofFieldOpList]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpF_ofFieldOpFsList]
-  rfl
+  simpa [ofFieldOpList_singleton] using
+    superCommute_ofFieldOpList_ofFieldOpList (𝓕 := 𝓕) [φ] φs
 
 lemma superCommute_ofFieldOpList_ofFieldOp (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
     [ofFieldOpList φs, ofFieldOp φ]ₛ = ofFieldOpList φs * ofFieldOp φ -
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs := by
-  rw [ofFieldOpList, ofFieldOp]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_ofFieldOpListF_ofFieldOpF]
-  rfl
+  simpa [ofFieldOpList_singleton] using
+    superCommute_ofFieldOpList_ofFieldOpList (𝓕 := 𝓕) φs [φ]
 
 lemma superCommute_anPart_crPart (φ φ' : 𝓕.FieldOp) :
     [anPart φ, crPart φ']ₛ = anPart φ * crPart φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ := by
-  rw [anPart, crPart]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_crPartF]
-  rfl
+  simp [anPart, crPart, superCommute_eq_ι_superCommuteF,
+    superCommuteF_anPartF_crPartF]
 
 lemma superCommute_crPart_anPart (φ φ' : 𝓕.FieldOp) :
     [crPart φ, anPart φ']ₛ = crPart φ * anPart φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ := by
-  rw [anPart, crPart]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_anPartF]
-  rfl
+  simp [anPart, crPart, superCommute_eq_ι_superCommuteF,
+    superCommuteF_crPartF_anPartF]
 
 @[simp]
 lemma superCommute_crPart_crPart (φ φ' : 𝓕.FieldOp) : [crPart φ, crPart φ']ₛ = 0 := by
@@ -323,30 +307,26 @@ lemma superCommute_anPart_anPart (φ φ' : 𝓕.FieldOp) : [anPart φ, anPart φ
 lemma superCommute_crPart_ofFieldOpList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
     [crPart φ, ofFieldOpList φs]ₛ = crPart φ * ofFieldOpList φs -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * crPart φ := by
-  rw [crPart, ofFieldOpList]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofFieldOpListF]
-  rfl
+  simp [crPart, ofFieldOpList, superCommute_eq_ι_superCommuteF,
+    superCommuteF_crPartF_ofFieldOpListF]
 
 lemma superCommute_anPart_ofFieldOpList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
     [anPart φ, ofFieldOpList φs]ₛ = anPart φ * ofFieldOpList φs -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpList φs * anPart φ := by
-  rw [anPart, ofFieldOpList]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofFieldOpListF]
-  rfl
+  simp [anPart, ofFieldOpList, superCommute_eq_ι_superCommuteF,
+    superCommuteF_anPartF_ofFieldOpListF]
 
 lemma superCommute_crPart_ofFieldOp (φ φ' : 𝓕.FieldOp) :
     [crPart φ, ofFieldOp φ']ₛ = crPart φ * ofFieldOp φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * crPart φ := by
-  rw [crPart, ofFieldOp]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_crPartF_ofFieldOpF]
-  rfl
+  simp [crPart, ofFieldOp, superCommute_eq_ι_superCommuteF,
+    superCommuteF_crPartF_ofFieldOpF]
 
 lemma superCommute_anPart_ofFieldOp (φ φ' : 𝓕.FieldOp) :
     [anPart φ, ofFieldOp φ']ₛ = anPart φ * ofFieldOp φ' -
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * anPart φ := by
-  rw [anPart, ofFieldOp]
-  rw [superCommute_eq_ι_superCommuteF, superCommuteF_anPartF_ofFieldOpF]
-  rfl
+  simp [anPart, ofFieldOp, superCommute_eq_ι_superCommuteF,
+    superCommuteF_anPartF_ofFieldOpF]
 
 /-!
 
@@ -361,60 +341,46 @@ lemma ofCrAnList_mul_ofCrAnList_eq_superCommute (φs φs' : List 𝓕.CrAnFieldO
     ofCrAnList φs * ofCrAnList φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnList φs
     + [ofCrAnList φs, ofCrAnList φs']ₛ := by
-  rw [superCommute_ofCrAnList_ofCrAnList]
-  simp [ofCrAnList_append]
+  simp [superCommute_ofCrAnList_ofCrAnList, ofCrAnList_append]
 
 lemma ofCrAnOp_mul_ofCrAnList_eq_superCommute (φ : 𝓕.CrAnFieldOp)
     (φs' : List 𝓕.CrAnFieldOp) : ofCrAnOp φ * ofCrAnList φs' =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofCrAnList φs' * ofCrAnOp φ
     + [ofCrAnOp φ, ofCrAnList φs']ₛ := by
-  rw [← ofCrAnList_singleton, ofCrAnList_mul_ofCrAnList_eq_superCommute]
-  simp
+  simpa [ofCrAnList_singleton] using ofCrAnList_mul_ofCrAnList_eq_superCommute [φ] φs'
 
 lemma ofFieldOpList_mul_ofFieldOpList_eq_superCommute (φs φs' : List 𝓕.FieldOp) :
     ofFieldOpList φs * ofFieldOpList φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOpList φs
-    + [ofFieldOpList φs, ofFieldOpList φs']ₛ := by
-  rw [superCommute_ofFieldOpList_ofFieldOpList]
-  simp
+    + [ofFieldOpList φs, ofFieldOpList φs']ₛ := by simp [superCommute_ofFieldOpList_ofFieldOpList]
 
 lemma ofFieldOp_mul_ofFieldOpList_eq_superCommute (φ : 𝓕.FieldOp) (φs' : List 𝓕.FieldOp) :
     ofFieldOp φ * ofFieldOpList φs' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofFieldOp φ
     + [ofFieldOp φ, ofFieldOpList φs']ₛ := by
-  rw [superCommute_ofFieldOp_ofFieldOpList]
-  simp
+  simpa [ofFieldOpList_singleton] using ofFieldOpList_mul_ofFieldOpList_eq_superCommute [φ] φs'
 
 lemma ofFieldOp_mul_ofFieldOp_eq_superCommute (φ φ' : 𝓕.FieldOp) :
     ofFieldOp φ * ofFieldOp φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • ofFieldOp φ' * ofFieldOp φ
     + [ofFieldOp φ, ofFieldOp φ']ₛ := by
-  rw [← ofFieldOpList_singleton, ← ofFieldOpList_singleton]
-  rw [ofFieldOpList_mul_ofFieldOpList_eq_superCommute, ofFieldOpList_singleton]
-  simp
+  simpa [ofFieldOpList_singleton] using ofFieldOpList_mul_ofFieldOpList_eq_superCommute [φ] [φ']
 
 lemma ofFieldOpList_mul_ofFieldOp_eq_superCommute (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
     ofFieldOpList φs * ofFieldOp φ = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOp φ * ofFieldOpList φs
     + [ofFieldOpList φs, ofFieldOp φ]ₛ := by
-  rw [superCommute_ofFieldOpList_ofFieldOp]
-  simp
+  simpa [ofFieldOpList_singleton] using ofFieldOpList_mul_ofFieldOpList_eq_superCommute φs [φ]
 
 lemma ofCrAnList_mul_ofFieldOpList_eq_superCommute (φs : List 𝓕.CrAnFieldOp)
     (φs' : List 𝓕.FieldOp) : ofCrAnList φs * ofFieldOpList φs' =
     𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpList φs' * ofCrAnList φs
-    + [ofCrAnList φs, ofFieldOpList φs']ₛ := by
-  rw [superCommute_ofCrAnList_ofFieldOpList]
-  simp
+    + [ofCrAnList φs, ofFieldOpList φs']ₛ := by simp [superCommute_ofCrAnList_ofFieldOpList]
 
 lemma crPart_mul_anPart_eq_superCommute (φ φ' : 𝓕.FieldOp) :
     crPart φ * anPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • anPart φ' * crPart φ
-    + [crPart φ, anPart φ']ₛ := by
-  rw [superCommute_crPart_anPart]
-  simp
+    + [crPart φ, anPart φ']ₛ := by simp [superCommute_crPart_anPart]
 
 lemma anPart_mul_crPart_eq_superCommute (φ φ' : 𝓕.FieldOp) :
     anPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * anPart φ
-    + [anPart φ, crPart φ']ₛ := by
-  rw [superCommute_anPart_crPart]
-  simp
+    + [anPart φ, crPart φ']ₛ := by simp [superCommute_anPart_crPart]
 
 lemma crPart_mul_crPart_swap (φ φ' : 𝓕.FieldOp) :
     crPart φ * crPart φ' = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ') • crPart φ' * crPart φ := by
@@ -446,9 +412,8 @@ lemma superCommute_ofCrAnList_ofCrAnList_symm (φs φs' : List 𝓕.CrAnFieldOp)
 lemma superCommute_ofCrAnOp_ofCrAnOp_symm (φ φ' : 𝓕.CrAnFieldOp) :
     [ofCrAnOp φ, ofCrAnOp φ']ₛ =
     (- 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φ')) • [ofCrAnOp φ', ofCrAnOp φ]ₛ := by
-  rw [ofCrAnOp, ofCrAnOp, superCommute_eq_ι_superCommuteF,
-    superCommuteF_ofCrAnOpF_ofCrAnOpF_symm]
-  rfl
+  simpa [ofCrAnList_singleton] using
+    superCommute_ofCrAnList_ofCrAnList_symm (𝓕 := 𝓕) [φ] [φ']
 
 /-!
 
@@ -464,8 +429,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_eq_sum (φs φs' : List 𝓕.CrAnFieldO
   conv_lhs =>
     rw [ofCrAnList, ofCrAnList, superCommute_eq_ι_superCommuteF,
       superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum]
-  rw [map_sum]
-  rfl
+  exact map_sum _ _ _
 
 lemma superCommute_ofCrAnOp_ofCrAnList_eq_sum (φ : 𝓕.CrAnFieldOp)
     (φs' : List 𝓕.CrAnFieldOp) : [ofCrAnOp φ, ofCrAnList φs']ₛ =
@@ -477,8 +441,7 @@ lemma superCommute_ofCrAnOp_ofCrAnList_eq_sum (φ : 𝓕.CrAnFieldOp)
   funext n
   simp only [ofList_singleton, List.get_eq_getElem, Algebra.smul_mul_assoc]
   congr 1
-  rw [ofCrAnList_singleton, superCommute_ofCrAnOp_ofCrAnOp_commute]
-  rw [mul_assoc, ← ofCrAnList_append]
+  rw [ofCrAnList_singleton, superCommute_ofCrAnOp_ofCrAnOp_commute, mul_assoc, ← ofCrAnList_append]
   congr
   exact Eq.symm (List.eraseIdx_eq_take_drop_succ φs' ↑n)
 
@@ -490,8 +453,7 @@ lemma superCommute_ofCrAnList_ofFieldOpList_eq_sum (φs : List 𝓕.CrAnFieldOp)
   conv_lhs =>
     rw [ofCrAnList, ofFieldOpList, superCommute_eq_ι_superCommuteF,
       superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum]
-  rw [map_sum]
-  rfl
+  exact map_sum _ _ _
 
 lemma superCommute_ofCrAnOp_ofFieldOpList_eq_sum (φ : 𝓕.CrAnFieldOp) (φs' : List 𝓕.FieldOp) :
     [ofCrAnOp φ, ofFieldOpList φs']ₛ =
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeContraction.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeContraction.lean
index 091fa274b..571e33eb7 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeContraction.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeContraction.lean
@@ -45,18 +45,15 @@ lemma timeContract_of_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : timeOrderRel φ 
     rw [ofFieldOp_eq_crPart_add_anPart]
     rw [map_add, superCommute_anPart_anPart, superCommute_anPart_crPart]
   simp only [timeContract, Algebra.smul_mul_assoc, add_zero]
-  rw [timeOrder_ofFieldOp_ofFieldOp_ordered h]
-  rw [normalOrder_ofFieldOp_mul_ofFieldOp]
+  rw [timeOrder_ofFieldOp_ofFieldOp_ordered h, normalOrder_ofFieldOp_mul_ofFieldOp]
   rw [ofFieldOp_eq_crPart_add_anPart, ofFieldOp_eq_crPart_add_anPart]
   simp only [mul_add, add_mul]
   abel_nf
 
 lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderRel φ ψ) :
     timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • timeContract ψ φ := by
-  rw [timeContract_eq_smul]
-  rw [normalOrder_ofFieldOp_ofFieldOp_swap]
-  rw [timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder h]
-  rw [timeContract_eq_smul]
+  rw [timeContract_eq_smul, normalOrder_ofFieldOp_ofFieldOp_swap,
+    timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder h, timeContract_eq_smul]
   simp only [smul_add]
   rw [smul_smul, smul_smul, mul_comm]
 
@@ -66,8 +63,7 @@ lemma timeContract_of_not_timeOrderRel (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderR
   `timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • [anPart ψ, ofFieldOp φ]ₛ`. -/
 lemma timeContract_of_not_timeOrderRel_expand (φ ψ : 𝓕.FieldOp) (h : ¬ timeOrderRel φ ψ) :
     timeContract φ ψ = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • [anPart ψ, ofFieldOp φ]ₛ := by
-  rw [timeContract_of_not_timeOrderRel _ _ h]
-  rw [timeContract_of_timeOrderRel _ _ _]
+  rw [timeContract_of_not_timeOrderRel _ _ h, timeContract_of_timeOrderRel _ _ _]
   have h1 := Std.Total.total (r := 𝓕.timeOrderRel) φ ψ
   simp_all
 
@@ -146,10 +142,7 @@ lemma timeOrder_timeContract_eq_time_mid {φ ψ : 𝓕.FieldOp}
 lemma timeOrder_timeContract_eq_time_left {φ ψ : 𝓕.FieldOp}
     (h1 : timeOrderRel φ ψ) (h2 : timeOrderRel ψ φ) (b : 𝓕.WickAlgebra) :
     𝓣(timeContract φ ψ * b) = timeContract φ ψ * 𝓣(b) := by
-  trans 𝓣(1 * timeContract φ ψ * b)
-  simp only [one_mul]
-  rw [timeOrder_timeContract_eq_time_mid h1 h2]
-  simp
+  simpa using timeOrder_timeContract_eq_time_mid h1 h2 (1 : 𝓕.WickAlgebra) b
 
 lemma timeOrder_timeContract_ne_time {φ ψ : 𝓕.FieldOp}
     (h1 : ¬ (timeOrderRel φ ψ ∧ timeOrderRel ψ φ)) :
@@ -197,9 +190,7 @@ lemma timeOrder_timeContract_ne_time {φ ψ : 𝓕.FieldOp}
   This prevents Feynman diagrams where incoming vertices are
   connected to incoming vertices. -/
 lemma timeContract_inAsymp_inAsymp (φ ψ : ((f : Field 𝓕) × AsymptoticLabel 𝓕 f) × Momentum) :
-    timeContract (.inAsymp φ) (.inAsymp ψ) = 0 := by
-  rw [timeContract_eq_superCommute]
-  simp
+    timeContract (.inAsymp φ) (.inAsymp ψ) = 0 := by simp [timeContract_eq_superCommute]
 
 /-- The time contraction of an outgoing asymptotic field with
   another outgoing asymptotic field is zero.
@@ -208,8 +199,7 @@ lemma timeContract_inAsymp_inAsymp (φ ψ : ((f : Field 𝓕) × AsymptoticLabel
   connected to outgoing vertices. -/
 lemma timeContract_outAsymp_outAsymp (φ ψ : ((f : Field 𝓕) × AsymptoticLabel 𝓕 f) × Momentum) :
     timeContract (.outAsymp φ) (.outAsymp ψ) = 0 := by
-  rw [timeContract_eq_superCommute, ← anPart_outAsymp_eq_ofFieldOp, ← anPart_outAsymp_eq_ofFieldOp]
-  simp [- anPart_outAsymp]
+  simp [timeContract_eq_superCommute, ← anPart_outAsymp_eq_ofFieldOp, - anPart_outAsymp]
 
 end WickAlgebra
 
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeOrder.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeOrder.lean
index 06137c996..6dbd24ff9 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeOrder.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/TimeOrder.lean
@@ -71,9 +71,8 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
     simp only [decide_not,
       Bool.decide_and, ofCrAnListF_append, map_smul, map_mul, l1, l2, mul_assoc]
     congr 1
-    have hp : List.Perm [φ1, φ3, φ2] [φ1, φ2, φ3] := by
-      refine List.Perm.cons φ1 ?_
-      exact List.Perm.swap φ2 φ3 []
+    have hp : List.Perm [φ1, φ3, φ2] [φ1, φ2, φ3] :=
+      List.Perm.cons φ1 (List.Perm.swap φ2 φ3 [])
     rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
     · simp
     · intro φ4 hφ4
@@ -82,10 +81,9 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
       all_goals
         subst hφ4
         simp_all
-  have hp231 : List.Perm [φ2, φ3, φ1] [φ1, φ2, φ3] := by
-      refine List.Perm.trans (l₂ := [φ2, φ1, φ3]) ?_ ?_
-      refine List.Perm.cons φ2 (List.Perm.swap φ1 φ3 [])
-      exact List.Perm.swap φ1 φ2 [φ3]
+  have hp231 : List.Perm [φ2, φ3, φ1] [φ1, φ2, φ3] :=
+      List.Perm.trans (List.Perm.cons φ2 (List.Perm.swap φ1 φ3 []))
+        (List.Perm.swap φ1 φ2 [φ3])
   have h231 : ι 𝓣ᶠ(ofCrAnListF (φs1 ++ φ2 :: φ3 :: φ1 :: φs2)) =
       crAnTimeOrderSign (φs1 ++ φ1 :: φ2 :: φ3 :: φs2)
       • (ι (ofCrAnListF l1) * ι (ofCrAnListF [φ2, φ3, φ1]) * ι (ofCrAnListF l2)) := by
@@ -114,9 +112,8 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
     simp only [decide_not,
       Bool.decide_and, ofCrAnListF_append, map_smul, map_mul, l1, l2, mul_assoc]
     congr 1
-    have hp : List.Perm [φ3, φ2, φ1] [φ1, φ2, φ3] := by
-      refine List.Perm.trans ?_ hp231
-      exact List.Perm.swap φ2 φ3 [φ1]
+    have hp : List.Perm [φ3, φ2, φ1] [φ1, φ2, φ3] :=
+      List.Perm.trans (List.Perm.swap φ2 φ3 [φ1]) hp231
     rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
     · simp
     · intro φ4 hφ4
@@ -130,8 +127,7 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_eq_time_ofCrAnListF {φ1 φ2 φ3
     ι [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛF]ₛF *
     ι (ofCrAnListF l2)); swap
   · simp
-  rw [mul_assoc]
-  rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
+  rw [mul_assoc, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_singleton]
   rw [superCommuteF_ofCrAnListF_ofCrAnListF]
   simp only [List.singleton_append, ofList_singleton, map_sub, map_smul]
   rw [superCommuteF_ofCrAnListF_ofCrAnListF, superCommuteF_ofCrAnListF_ofCrAnListF]
@@ -275,8 +271,7 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}
               ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧
                 crAnTimeOrderRel c φ)) φs)) *
               ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧ ¬crAnTimeOrderRel c φ))
-                (List.insertionSort crAnTimeOrderRel (φs' ++ φs))))) := by
-            simp [mul_sub, sub_mul]
+                (List.insertionSort crAnTimeOrderRel (φs' ++ φs))))) := by simp [mul_sub, sub_mul]
         _ = crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs) •
             (ι (ofCrAnListF (List.takeWhile (fun c => ¬crAnTimeOrderRel φ c)
                 (List.insertionSort crAnTimeOrderRel (φs' ++ φs)))) *
@@ -304,14 +299,12 @@ lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}
       rw [← ofCrAnListF_append, ← ofCrAnListF_append, ← ofCrAnListF_append]
       have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
       simp [decide_not, Bool.decide_and, List.append_assoc, List.cons_append] at h1 ⊢
-      rw [← h1]
-      rw [← crAnTimeOrderList]
+      rw [← h1, ← crAnTimeOrderList]
       by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
       · rw [ι_superCommuteF_of_diff_statistic hq]
         simp
       · rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
-        rw [← ofCrAnListF_append]
-        rw [timeOrderF_ofCrAnListF]
+        rw [← ofCrAnListF_append, timeOrderF_ofCrAnListF]
         simp only [map_smul, Algebra.mul_smul_comm]
         exact hφψ
         exact hψφ
@@ -430,8 +423,7 @@ noncomputable def timeOrder : WickAlgebra 𝓕 →ₗ[ℂ] WickAlgebra 𝓕 wher
     obtain ⟨x, hx⟩ := ι_surjective x
     obtain ⟨y, hy⟩ := ι_surjective y
     subst hx hy
-    rw [← map_add, ι_apply, ι_apply, ι_apply]
-    rw [Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
+    rw [← map_add, ι_apply, ι_apply, ι_apply, Quotient.lift_mk, Quotient.lift_mk, Quotient.lift_mk]
     simp
   map_smul' c y := by
     obtain ⟨y, hy⟩ := ι_surjective y
@@ -467,7 +459,6 @@ lemma timeOrder_ofFieldOp_ofFieldOp_not_ordered_eq_timeOrder {φ ψ : 𝓕.Field
     𝓣(ofFieldOp φ * ofFieldOp ψ) = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ψ) • 𝓣(ofFieldOp ψ * ofFieldOp φ) := by
   rw [ofFieldOp, ofFieldOp, ← map_mul, timeOrder_eq_ι_timeOrderF,
     timeOrderF_ofFieldOpF_ofFieldOpF_not_ordered_eq_timeOrderF h]
-  simp only [map_smul]
   rfl
 
 lemma timeOrder_ofFieldOpList_nil : 𝓣(ofFieldOpList (𝓕 := 𝓕) []) = 1 := by
@@ -497,12 +488,10 @@ lemma timeOrder_superCommute_eq_time_mid {φ ψ : 𝓕.CrAnFieldOp}
     (hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.WickAlgebra) :
     𝓣(a * [ofCrAnOp φ, ofCrAnOp ψ]ₛ * b) =
     [ofCrAnOp φ, ofCrAnOp ψ]ₛ * 𝓣(a * b) := by
-  rw [ofCrAnOp, ofCrAnOp]
-  rw [superCommute_eq_ι_superCommuteF]
+  rw [ofCrAnOp, ofCrAnOp, superCommute_eq_ι_superCommuteF]
   obtain ⟨a, rfl⟩ := ι_surjective a
   obtain ⟨b, rfl⟩ := ι_surjective b
-  rw [← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF]
-  rw [ι_timeOrderF_superCommuteF_eq_time]
+  rw [← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF, ι_timeOrderF_superCommuteF_eq_time]
   rfl
   · simp_all
   · simp_all
@@ -519,9 +508,7 @@ lemma timeOrder_superCommute_eq_time_left {φ ψ : 𝓕.CrAnFieldOp}
 lemma timeOrder_superCommute_ne_time {φ ψ : 𝓕.CrAnFieldOp}
     (hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) :
     𝓣([ofCrAnOp φ, ofCrAnOp ψ]ₛ) = 0 := by
-  rw [ofCrAnOp, ofCrAnOp]
-  rw [superCommute_eq_ι_superCommuteF]
-  rw [timeOrder_eq_ι_timeOrderF]
+  rw [ofCrAnOp, ofCrAnOp, superCommute_eq_ι_superCommuteF, timeOrder_eq_ι_timeOrderF]
   trans ι (timeOrderF (1 * (superCommuteF (ofCrAnOpF φ)) (ofCrAnOpF ψ) * 1))
   simp only [one_mul, mul_one]
   rw [ι_timeOrderF_superCommuteF_ne_time]
@@ -557,26 +544,14 @@ lemma timeOrder_timeOrder_mid (a b c : 𝓕.WickAlgebra) :
   ← map_mul, ← map_mul, timeOrder_eq_ι_timeOrderF, timeOrderF_timeOrderF_mid]
 
 lemma timeOrder_timeOrder_left (b c : 𝓕.WickAlgebra) :
-    𝓣(b * c) = 𝓣(𝓣(b) * c) := by
-  trans 𝓣(1 * b * c)
-  simp only [one_mul]
-  rw [timeOrder_timeOrder_mid]
-  simp
+    𝓣(b * c) = 𝓣(𝓣(b) * c) := by simpa using timeOrder_timeOrder_mid (1 : 𝓕.WickAlgebra) b c
 
 lemma timeOrder_timeOrder_right (a b : 𝓕.WickAlgebra) :
-    𝓣(a * b) = 𝓣(a * 𝓣(b)) := by
-  trans 𝓣(a * b * 1)
-  simp only [mul_one]
-  rw [timeOrder_timeOrder_mid]
-  simp
+    𝓣(a * b) = 𝓣(a * 𝓣(b)) := by simpa using timeOrder_timeOrder_mid a b (1 : 𝓕.WickAlgebra)
 
 /-- Time ordering is a projection. -/
 lemma timeOrder_timeOrder (a : 𝓕.WickAlgebra) :
-    𝓣(𝓣(a)) = 𝓣(a) := by
-  trans 𝓣(𝓣(a) * 1)
-  · simp
-  · rw [← timeOrder_timeOrder_left]
-    simp
+    𝓣(𝓣(a)) = 𝓣(a) := by simpa using (timeOrder_timeOrder_left a (1 : 𝓕.WickAlgebra)).symm
 
 end WickAlgebra
 end FieldSpecification
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/Universality.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/Universality.lean
index 85ef247f5..cc5bf44eb 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/Universality.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/Universality.lean
@@ -68,8 +68,7 @@ def universalLift {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp
   commutes' r := by
     rw [Algebra.algebraMap_eq_smul_one r]
     rw [show r • 1 = ι (𝓕 := 𝓕) (r • 1) from rfl, universalLiftMap_ι]
-    simp only [map_smul, map_one]
-    exact Eq.symm (Algebra.algebraMap_eq_smul_one r)
+    simpa only [map_smul, map_one] using Eq.symm (Algebra.algebraMap_eq_smul_one r)
 
 @[simp]
 lemma universalLift_ι {A : Type} [Semiring A] [Algebra ℂ A] (f : 𝓕.CrAnFieldOp → A)
diff --git a/Physlib/QFT/PerturbationTheory/WickAlgebra/WickTerm.lean b/Physlib/QFT/PerturbationTheory/WickAlgebra/WickTerm.lean
index 947fd32ec..f90ee7c7b 100644
--- a/Physlib/QFT/PerturbationTheory/WickAlgebra/WickTerm.lean
+++ b/Physlib/QFT/PerturbationTheory/WickAlgebra/WickTerm.lean
@@ -40,9 +40,7 @@ def wickTerm {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : 
   corresponding to the empty set `∅` (the only Wick contraction of `[]`) is `1`. -/
 @[simp]
 lemma wickTerm_empty_nil :
-    wickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by
-  rw [wickTerm]
-  simp [sign_empty]
+    wickTerm (empty (n := ([] : List 𝓕.FieldOp).length)) = 1 := by simp [wickTerm, sign_empty]
 
 /--
 For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a Wick contraction `φsΛ` of `φs`, an element `φ` of
@@ -84,8 +82,7 @@ lemma wickTerm_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     · intro ha
       simp only [uncontracted, Finset.mem_filter, Finset.mem_univ, true_and, ha, and_true,
         forall_const]
-      have hx : φsΛ.getDual? a = none ↔ ¬ (φsΛ.getDual? a).isSome := by
-        simp
+      have hx : φsΛ.getDual? a = none ↔ ¬ (φsΛ.getDual? a).isSome := by simp
       rw [hx]
       simp only [Bool.not_eq_true, Bool.eq_false_or_eq_true_self, true_and]
       intro h1 h2
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Basic.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Basic.lean
index e9ae5dd39..c20a58d9c 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Basic.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Basic.lean
@@ -129,8 +129,7 @@ lemma congrLift_surjective {n m : ℕ} {c : WickContraction n} (h : n = m) :
   simp [Function.surjective_id]
 
 lemma congrLift_bijective {n m : ℕ} {c : WickContraction n} (h : n = m) :
-    Function.Bijective (c.congrLift h) := by
-  exact ⟨c.congrLift_injective h, c.congrLift_surjective h⟩
+    Function.Bijective (c.congrLift h) := ⟨c.congrLift_injective h, c.congrLift_surjective h⟩
 
 /-- Given a contracted pair in `c : WickContraction n` the contracted pair
   in `congr h c`. -/
@@ -203,8 +202,7 @@ lemma getDual?_get_self_mem (i : Fin n) (h : (c.getDual? i).isSome) :
 
 @[simp]
 lemma self_getDual?_get_mem (i : Fin n) (h : (c.getDual? i).isSome) :
-    {i, (c.getDual? i).get h} ∈ c.1 := by
-  rw [← getDual?_eq_some_iff_mem, Option.some_get]
+    {i, (c.getDual? i).get h} ∈ c.1 := by rw [← getDual?_eq_some_iff_mem, Option.some_get]
 
 lemma getDual?_eq_some_neq (i j : Fin n) (h : c.getDual? i = some j) :
     ¬ i = j := by
@@ -257,21 +255,17 @@ lemma getDual?_isSome_iff (i : Fin n) : (c.getDual? i).isSome ↔ ∃ (a : c.1),
       simpa using hy ▸ a.2
 
 lemma getDual?_isSome_of_mem (a : c.1) (i : a.1) : (c.getDual? i).isSome := by
-  rw [getDual?_isSome_iff]
-  exact ⟨⟨a.1, a.2⟩, Finset.coe_mem ..⟩
+  exact (getDual?_isSome_iff (c := c) i).2 ⟨a, Finset.coe_mem i⟩
 
 @[simp]
 lemma getDual?_getDual?_get_get (i : Fin n) (h : (c.getDual? i).isSome) :
-    c.getDual? ((c.getDual? i).get h) = some i := by
-  simp [getDual?_eq_some_iff_mem]
+    c.getDual? ((c.getDual? i).get h) = some i := by simp [getDual?_eq_some_iff_mem]
 
 lemma getDual?_getDual?_get_isSome (i : Fin n) (h : (c.getDual? i).isSome) :
-    (c.getDual? ((c.getDual? i).get h)).isSome := by
-  simp
+    (c.getDual? ((c.getDual? i).get h)).isSome := by simp
 
 lemma getDual?_getDual?_get_not_none (i : Fin n) (h : (c.getDual? i).isSome) :
-    ¬ (c.getDual? ((c.getDual? i).get h)) = none := by
-  simp
+    ¬ (c.getDual? ((c.getDual? i).get h)) = none := by simp
 
 /-!
 
@@ -368,13 +362,11 @@ lemma fstFieldOfContract_lt_sndFieldOfContract (c : WickContraction n) (a : c.1)
 
 @[simp]
 lemma fstFieldOfContract_mem (c : WickContraction n) (a : c.1) :
-    c.fstFieldOfContract a ∈ a.1 := by
-  simp [finset_eq_fstFieldOfContract_sndFieldOfContract]
+    c.fstFieldOfContract a ∈ a.1 := by simp [finset_eq_fstFieldOfContract_sndFieldOfContract]
 
 lemma fstFieldOfContract_getDual?_isSome (c : WickContraction n) (a : c.1) :
     (c.getDual? (c.fstFieldOfContract a)).isSome := by
-  rw [getDual?_isSome_iff]
-  exact ⟨a, fstFieldOfContract_mem ..⟩
+  exact (getDual?_isSome_iff (c := c) _).2 ⟨a, fstFieldOfContract_mem ..⟩
 
 @[simp]
 lemma fstFieldOfContract_getDual? (c : WickContraction n) (a : c.1) :
@@ -383,13 +375,11 @@ lemma fstFieldOfContract_getDual? (c : WickContraction n) (a : c.1) :
 
 @[simp]
 lemma sndFieldOfContract_mem (c : WickContraction n) (a : c.1) :
-    c.sndFieldOfContract a ∈ a.1 := by
-  simp [finset_eq_fstFieldOfContract_sndFieldOfContract]
+    c.sndFieldOfContract a ∈ a.1 := by simp [finset_eq_fstFieldOfContract_sndFieldOfContract]
 
 lemma sndFieldOfContract_getDual?_isSome (c : WickContraction n) (a : c.1) :
     (c.getDual? (c.sndFieldOfContract a)).isSome := by
-  rw [getDual?_isSome_iff]
-  exact ⟨a, sndFieldOfContract_mem ..⟩
+  exact (getDual?_isSome_iff (c := c) _).2 ⟨a, sndFieldOfContract_mem ..⟩
 
 @[simp]
 lemma sndFieldOfContract_getDual? (c : WickContraction n) (a : c.1) :
@@ -520,9 +510,6 @@ def sigmaContractedEquiv : (a : c.1) × a ≃ {x : Fin n // (c.getDual? x).isSom
         not_false_eq_true]
       exact Subtype.ext (id (Eq.symm hc))
     · simp
-  right_inv := by
-    intro x
-    cases x
-    rfl
+  right_inv := fun ⟨_, _⟩ => rfl
 
 end WickContraction
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Card.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Card.lean
index 16adb3620..fc5fffdcb 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Card.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Card.lean
@@ -53,8 +53,7 @@ lemma wickContraction_zero_some_eq_sum :
       ext
       · simp [c.2.2.2]
       · rfl}
-  rw [Fintype.card_congr e1]
-  simp
+  simpa using Fintype.card_congr e1
 
 lemma finset_succAbove_succ_disjoint (a : Finset (Fin n)) (i : Fin n.succ) :
     Disjoint ((Finset.map (Fin.succEmb (n + 1))) ((Finset.map i.succAboveEmb) a)) {0, i.succ} := by
@@ -78,8 +77,7 @@ def consAddContract (i : Fin n.succ) (c : WickContraction n) :
     rcases h with h | h
     · obtain ⟨a, ha, rfl⟩ := h
       rw [Finset.mapEmbedding_apply, Finset.mapEmbedding_apply]
-      simp only [Finset.card_map]
-      exact c.2.1 a ha
+      simpa only [Finset.card_map] using c.2.1 a ha
     · subst h
       rw [@Finset.card_eq_two]
       use 0, i.succ
@@ -111,14 +109,12 @@ def consAddContract (i : Fin n.succ) (c : WickContraction n) :
 @[simp]
 lemma consAddContract_getDual?_zero (i : Fin n.succ) (c : WickContraction n) :
     (consAddContract i c).getDual? 0 = some i.succ := by
-  rw [getDual?_eq_some_iff_mem]
-  simp [consAddContract]
+  simp [getDual?_eq_some_iff_mem, consAddContract]
 
 @[simp]
 lemma consAddContract_getDual?_self_succ (i : Fin n.succ) (c : WickContraction n) :
     (consAddContract i c).getDual? i.succ = some 0 := by
-  rw [getDual?_eq_some_iff_mem]
-  simp [consAddContract, Finset.pair_comm]
+  simp [getDual?_eq_some_iff_mem, consAddContract, Finset.pair_comm]
 
 lemma mem_consAddContract_of_mem_iff (i : Fin n.succ) (c : WickContraction n) (a : Finset (Fin n)) :
     a ∈ c.1 ↔ (a.map i.succAboveEmb).map (Fin.succEmb n.succ) ∈ (consAddContract i c).1 := by
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Erase.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Erase.lean
index 97c576804..d61221e16 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Erase.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Erase.lean
@@ -63,8 +63,7 @@ lemma mem_erase_uncontracted_iff (c : WickContraction n.succ) (i : Fin n.succ) (
       simp only [Nat.succ_eq_add_one, Finset.disjoint_insert_right, Finset.mem_insert,
         Finset.mem_singleton, true_or, not_true_eq_false, Finset.disjoint_singleton_right, not_or,
         false_and, or_false] at hc
-      have hi : i ∈ ({i.succAbove j, i.succAbove k} : Finset (Fin n.succ)) := by
-        simp [← hc]
+      have hi : i ∈ ({i.succAbove j, i.succAbove k} : Finset (Fin n.succ)) := by simp [← hc]
       simp only [Nat.succ_eq_add_one, Finset.mem_insert, Finset.mem_singleton] at hi
       rcases hi with hi | hi
       · exact False.elim (Fin.succAbove_ne _ _ hi.symm)
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/ExtractEquiv.lean b/Physlib/QFT/PerturbationTheory/WickContraction/ExtractEquiv.lean
index 7b8c3db81..d01363ef5 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/ExtractEquiv.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/ExtractEquiv.lean
@@ -47,8 +47,7 @@ def extractEquiv (i : Fin n.succ) : WickContraction n.succ ≃
     (c : WickContraction n) × Option c.uncontracted where
   toFun := fun c => ⟨erase c i, getDualErase c i⟩
   invFun := fun ⟨c, j⟩ => insertAndContractNat c i j
-  left_inv f := by
-    simp
+  left_inv f := by simp
   right_inv f := by
     refine extractEquiv_equiv ?_ ?_
     simp only [insertAndContractNat_erase]
@@ -108,9 +107,7 @@ lemma sum_extractEquiv_congr [AddCommMonoid M] {n m : ℕ} (i : Fin n) (f : Wick
     ∑ c, f c = ∑ (c : WickContraction m), ∑ (k : Option c.uncontracted),
     f (congr h.symm ((extractEquiv (finCongr h i)).symm ⟨c, k⟩)) := by
   subst h
-  simp only [finCongr_refl, Equiv.refl_apply, congr_refl]
-  rw [← (extractEquiv i).symm.sum_comp]
-  rw [Finset.sum_sigma']
+  rw [← (extractEquiv i).symm.sum_comp, Finset.sum_sigma']
   rfl
 
 /-- For `n = 3` there are `4` possible Wick contractions:
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContract.lean b/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContract.lean
index 3a2ade00b..c724b3cb0 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContract.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContract.lean
@@ -140,17 +140,14 @@ lemma insertAndContract_some_getDual?_some_eq (φ : 𝓕.FieldOp) (φs : List 
     ((φsΛ ↩Λ φ i (some j)).getDual?
       (Fin.cast (insertIdx_length_fin φ φs i).symm (i.succAbove j)))
     = some (Fin.cast (insertIdx_length_fin φ φs i).symm i) := by
-  rw [getDual?_eq_some_iff_mem]
-  rw [@Finset.pair_comm]
-  rw [← getDual?_eq_some_iff_mem]
+  rw [getDual?_eq_some_iff_mem, @Finset.pair_comm, ← getDual?_eq_some_iff_mem]
   simp
 
 @[simp]
 lemma insertAndContract_none_succAbove_getDual?_eq_none_iff (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
     (φsΛ ↩Λ φ i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
-      (i.succAbove j)) = none ↔ φsΛ.getDual? j = none := by
-  simp [insertAndContract, getDual?_congr]
+      (i.succAbove j)) = none ↔ φsΛ.getDual? j = none := by simp [insertAndContract, getDual?_congr]
 
 @[simp]
 lemma insertAndContract_some_succAbove_getDual?_eq_option (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
@@ -352,14 +349,12 @@ lemma stat_ofFinset_of_insertAndContractLiftFinset (φ : 𝓕.FieldOp) (φs : Li
       fin_list_sorted_monotone_sorted (a.sort (fun x1 x2 => x1 ≤ x2)) ?hl
         (⇑(finCongr (Eq.symm (insertIdx_length_fin φ φs i))) ∘ i.succAbove) ?hf
     exact a.pairwise_sort (fun x1 x2 => x1 ≤ x2)
-    refine StrictMono.comp (fun ⦃a b⦄ a => a) ?hf.hf
-    exact Fin.strictMono_succAbove i
+    exact StrictMono.comp (fun ⦃a b⦄ a => a) (Fin.strictMono_succAbove i)
   have h2 : (List.map (⇑(finCongr (insertIdx_length_fin φ φs i).symm))
       (List.map i.succAbove (a.sort (fun x1 x2 => x1 ≤ x2)))).Nodup := by
     simp only [Nat.succ_eq_add_one, List.map_map]
     refine List.Nodup.map ?_ ?_
-    apply (Equiv.comp_injective _ (finCongr _)).mpr
-    exact Fin.succAbove_right_injective
+    exact (Equiv.comp_injective _ (finCongr _)).mpr Fin.succAbove_right_injective
     exact a.sort_nodup (fun x1 x2 => x1 ≤ x2)
   have h3 : (List.map (⇑(finCongr (insertIdx_length_fin φ φs i).symm))
       (List.map i.succAbove (a.sort (fun x1 x2 => x1 ≤ x2)))).toFinset
@@ -374,8 +369,7 @@ lemma stat_ofFinset_of_insertAndContractLiftFinset (φ : 𝓕.FieldOp) (φs : Li
       intro x hx
       refine Fin.ne_of_val_ne ?h.inl.h
       simp only [Fin.val_cast, ne_eq]
-      rw [Fin.val_eq_val]
-      exact Fin.succAbove_ne i x
+      exact fun h => Fin.succAbove_ne i x (Fin.ext h)
     · obtain ⟨k, hk⟩ := hk
       subst hk
       simp only [Nat.succ_eq_add_one]
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContractNat.lean b/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContractNat.lean
index bc4cf4f25..e6a45f574 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContractNat.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/InsertAndContractNat.lean
@@ -44,8 +44,7 @@ def insertAndContractNat (c : WickContraction n) (i : Fin n.succ) (j : Option (c
         forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, f]
       intro a ha
       rw [Finset.mapEmbedding_apply]
-      simp only [Finset.card_map]
-      exact c.2.1 a ha
+      simpa only [Finset.card_map] using c.2.1 a ha
     | some j =>
       simp only [Finset.mem_insert, forall_eq_or_imp]
       apply And.intro
@@ -77,8 +76,7 @@ def insertAndContractNat (c : WickContraction n) (i : Fin n.succ) (j : Option (c
       simp_all only [Finset.mem_insert, Nat.succ_eq_add_one]
       match ha, hb with
       | Or.inl ha, Or.inl hb =>
-        rw [ha, hb]
-        simp
+        simp [ha, hb]
       | Or.inl ha, Or.inr hb =>
         apply Or.inr
         subst ha
@@ -161,8 +159,7 @@ lemma self_mem_uncontracted_of_insertAndContractNat_none (c : WickContraction n)
 lemma self_not_mem_uncontracted_of_insertAndContractNat_some (c : WickContraction n)
     (i : Fin n.succ) (j : c.uncontracted) :
     i ∉ (insertAndContractNat c i (some j)).uncontracted := by
-  rw [mem_uncontracted_iff_not_contracted]
-  simp [insertAndContractNat]
+  simp [mem_uncontracted_iff_not_contracted, insertAndContractNat]
 
 lemma insertAndContractNat_succAbove_mem_uncontracted_iff (c : WickContraction n) (i : Fin n.succ)
     (j : Fin n) :
@@ -343,18 +340,15 @@ lemma insertAndContractNat_some_uncontracted (c : WickContraction n) (i : Fin n.
 set_option backward.isDefEq.respectTransparency false in
 lemma insertAndContractNat_none_getDual?_isNone (c : WickContraction n) (i : Fin n.succ) :
     ((insertAndContractNat c i none).getDual? i).isNone := by
-  have hi : i ∈ (insertAndContractNat c i none).uncontracted := by
-    simp
+  have hi : i ∈ (insertAndContractNat c i none).uncontracted := by simp
   simp only [Nat.succ_eq_add_one, uncontracted, Finset.mem_filter, Finset.mem_univ, true_and] at hi
-  rw [hi]
-  simp
+  simp [hi]
 
 set_option backward.isDefEq.respectTransparency false in
 @[simp]
 lemma insertAndContractNat_none_getDual?_eq_none (c : WickContraction n) (i : Fin n.succ) :
     (insertAndContractNat c i none).getDual? i = none := by
-  have hi : i ∈ (insertAndContractNat c i none).uncontracted := by
-    simp
+  have hi : i ∈ (insertAndContractNat c i none).uncontracted := by simp
   simp only [Nat.succ_eq_add_one, uncontracted, Finset.mem_filter, Finset.mem_univ, true_and] at hi
   rw [hi]
 
@@ -392,8 +386,7 @@ lemma insertAndContractNat_succAbove_getDual?_get (c : WickContraction n) (i : F
 lemma insertAndContractNat_some_getDual?_eq (c : WickContraction n) (i : Fin n.succ)
     (j : c.uncontracted) :
     (insertAndContractNat c i (some j)).getDual? i = some (i.succAbove j) := by
-  rw [getDual?_eq_some_iff_mem]
-  simp [insertAndContractNat]
+  simp [getDual?_eq_some_iff_mem, insertAndContractNat]
 
 lemma insertAndContractNat_some_getDual?_ne_none (c : WickContraction n) (i : Fin n.succ)
     (j : c.uncontracted) (k : Fin n) (hkj : k ≠ j.1) :
@@ -503,8 +496,7 @@ lemma insertAndContractNat_erase (c : WickContraction n) (i : Fin n.succ)
           intro x
           exact fun a => Fin.succAbove_ne i x
         refine False.elim (hin ?_)
-        rw [ha]
-        simp
+        simp [ha]
       · obtain ⟨a', ha', ha''⟩ := ha
         rw [Finset.mapEmbedding_apply] at ha''
         simp only [Finset.map_inj] at ha''
@@ -667,8 +659,7 @@ lemma insertAndContractNat_fstFieldOfContract (c : WickContraction n) (i : Fin n
   · simp only [Nat.succ_eq_add_one, insertLift, Finset.mem_map, Fin.succAboveEmb_apply]
     use (c.sndFieldOfContract a)
     simp
-  · refine Fin.succAbove_lt_succAbove_iff.mpr ?_
-    exact fstFieldOfContract_lt_sndFieldOfContract c a
+  · exact Fin.succAbove_lt_succAbove_iff.mpr (fstFieldOfContract_lt_sndFieldOfContract c a)
 
 @[simp]
 lemma insertAndContractNat_sndFieldOfContract (c : WickContraction n) (i : Fin n.succ)
@@ -683,8 +674,7 @@ lemma insertAndContractNat_sndFieldOfContract (c : WickContraction n) (i : Fin n
   · simp only [Nat.succ_eq_add_one, insertLift, Finset.mem_map, Fin.succAboveEmb_apply]
     use (c.sndFieldOfContract a)
     simp
-  · refine Fin.succAbove_lt_succAbove_iff.mpr ?_
-    exact fstFieldOfContract_lt_sndFieldOfContract c a
+  · exact Fin.succAbove_lt_succAbove_iff.mpr (fstFieldOfContract_lt_sndFieldOfContract c a)
 
 /-- Given a contracted pair for a Wick contraction `WickContraction n`, the
   corresponding contracted pair of a wick contraction `(c.insert i (some j))` formed
@@ -707,8 +697,7 @@ lemma insertLiftSome_injective {c : WickContraction n} (i : Fin n.succ) (j : c.u
     simp only [Finset.map_insert, Fin.succAboveEmb_apply, Finset.map_singleton] at hab
     have hi : i ∈ ({i.succAbove (c.fstFieldOfContract a),
         i.succAbove (c.sndFieldOfContract a)} : Finset (Fin (n + 1))) := by
-      rw [← hab]
-      simp
+      simp [← hab]
     simp only [Nat.succ_eq_add_one, Finset.mem_insert, Finset.mem_singleton] at hi
     rcases hi with hi | hi
     · exact False.elim (Fin.ne_succAbove _ _ hi)
@@ -719,8 +708,7 @@ lemma insertLiftSome_injective {c : WickContraction n} (i : Fin n.succ) (j : c.u
     simp only [Finset.map_insert, Fin.succAboveEmb_apply, Finset.map_singleton] at hab
     have hi : i ∈ ({i.succAbove (c.fstFieldOfContract a),
         i.succAbove (c.sndFieldOfContract a)} : Finset (Fin (n + 1))) := by
-      rw [hab]
-      simp
+      simp [hab]
     simp only [Nat.succ_eq_add_one, Finset.mem_insert, Finset.mem_singleton] at hi
     rcases hi with hi | hi
     · exact False.elim (Fin.ne_succAbove _ _ hi)
@@ -773,8 +761,7 @@ lemma insertAndContractNat_surjective_on_nodual (i : Fin n.succ)
     obtain ⟨a', ha', rfl⟩ := h
     exact ha'
   · intro h
-    have hi : i ∈ c.uncontracted := by
-      simpa [uncontracted] using hc
+    have hi : i ∈ c.uncontracted := by simpa [uncontracted] using hc
     rw [mem_uncontracted_iff_not_contracted] at hi
     obtain ⟨j, hj⟩ := (@Fin.exists_succAbove_eq_iff _ i (c.fstFieldOfContract ⟨a, h⟩)).mpr
       (by
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/IsFull.lean b/Physlib/QFT/PerturbationTheory/WickContraction/IsFull.lean
index 0e76f2f02..f5b061883 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/IsFull.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/IsFull.lean
@@ -42,10 +42,8 @@ lemma isFull_iff_equivInvolution_no_fixed_point :
 /-- The equivalence between full contractions and fixed-point free involutions. -/
 def isFullInvolutionEquiv : {c : WickContraction n // IsFull c} ≃
     {f : Fin n → Fin n // Function.Involutive f ∧ (∀ i, f i ≠ i)} where
-  toFun c := ⟨equivInvolution c.1, by
-    apply And.intro (equivInvolution c.1).2
-    rw [← isFull_iff_equivInvolution_no_fixed_point]
-    exact c.2⟩
+  toFun c := ⟨equivInvolution c.1,
+    ⟨(equivInvolution c.1).2, (isFull_iff_equivInvolution_no_fixed_point (c := c.1)).mp c.2⟩⟩
   invFun f := ⟨equivInvolution.symm ⟨f.1, f.2.1⟩, by
     rw [isFull_iff_equivInvolution_no_fixed_point]
     simpa using f.2.2⟩
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Join.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Join.lean
index 5540142ca..0d8942184 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Join.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Join.lean
@@ -41,8 +41,7 @@ def join {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
     · exact φsΛ.2.1 a ha
     · obtain ⟨a, ha, rfl⟩ := ha
       rw [Finset.mapEmbedding_apply]
-      simp only [Finset.card_map]
-      exact φsucΛ.2.1 a ha, by
+      simpa only [Finset.card_map] using φsucΛ.2.1 a ha, by
     intro a ha b hb
     simp only [Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
       RelEmbedding.coe_toEmbedding] at ha hb
@@ -52,13 +51,11 @@ def join {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
       right
       symm
       rw [Finset.mapEmbedding_apply]
-      apply uncontractedListEmd_finset_disjoint_left
-      exact ha
+      exact uncontractedListEmd_finset_disjoint_left b a ha
     · obtain ⟨a, ha, rfl⟩ := ha
       right
       rw [Finset.mapEmbedding_apply]
-      apply uncontractedListEmd_finset_disjoint_left
-      exact hb
+      exact uncontractedListEmd_finset_disjoint_left a b hb
     · obtain ⟨a, ha, rfl⟩ := ha
       obtain ⟨b, hb, rfl⟩ := hb
       simp only [EmbeddingLike.apply_eq_iff_eq]
@@ -113,8 +110,7 @@ lemma jointLiftLeft_disjoint_joinLiftRight {φs : List 𝓕.FieldOp} {φsΛ : Wi
     Disjoint (@joinLiftLeft _ _ _ φsucΛ a).1 (joinLiftRight b).1 := by
   simp only [joinLiftLeft, joinLiftRight]
   symm
-  apply uncontractedListEmd_finset_disjoint_left
-  exact a.2
+  exact uncontractedListEmd_finset_disjoint_left b.1 a.1 a.2
 
 lemma jointLiftLeft_ne_joinLiftRight {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
     {φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} (a : φsΛ.1) (b : φsucΛ.1) :
@@ -166,8 +162,7 @@ lemma joinLift_surjective {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs
   · rw [Finset.mapEmbedding_apply] at ha3
     use Sum.inr ⟨a2, ha3.1⟩
     simp only [joinLift, joinLiftRight]
-    refine Subtype.ext ?_
-    exact ha3.2
+    exact Subtype.ext ha3.2
 
 lemma joinLift_bijective {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
     {φsucΛ : WickContraction [φsΛ]ᵘᶜ.length} : Function.Bijective (@joinLift _ _ φsΛ φsucΛ) := by
@@ -208,8 +203,7 @@ lemma join_fstFieldOfContract_joinLiftRight {φs : List 𝓕.FieldOp} (φsΛ : W
   apply eq_fstFieldOfContract_of_mem _ _ _ (uncontractedListEmd (φsucΛ.sndFieldOfContract a))
   · simp [joinLiftRight]
   · simp [joinLiftRight]
-  · apply uncontractedListEmd_strictMono
-    exact fstFieldOfContract_lt_sndFieldOfContract φsucΛ a
+  · exact uncontractedListEmd_strictMono (fstFieldOfContract_lt_sndFieldOfContract φsucΛ a)
 
 @[simp]
 lemma join_sndFieldOfContract_joinLiftRight {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
@@ -219,8 +213,7 @@ lemma join_sndFieldOfContract_joinLiftRight {φs : List 𝓕.FieldOp} (φsΛ : W
   apply eq_sndFieldOfContract_of_mem _ _ (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
   · simp [joinLiftRight]
   · simp [joinLiftRight]
-  · apply uncontractedListEmd_strictMono
-    exact fstFieldOfContract_lt_sndFieldOfContract φsucΛ a
+  · exact uncontractedListEmd_strictMono (fstFieldOfContract_lt_sndFieldOfContract φsucΛ a)
 
 @[simp]
 lemma join_fstFieldOfContract_joinLiftLeft {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
@@ -347,8 +340,7 @@ lemma mem_join_uncontracted_of_mem_right_uncontracted {φs : List 𝓕.FieldOp}
   simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
     RelEmbedding.coe_toEmbedding] at hp
   rcases hp with hp | hp
-  · have hi : uncontractedListEmd i ∈ φsΛ.uncontracted := by
-      exact uncontractedListEmd_mem_uncontracted i
+  · have hi : uncontractedListEmd i ∈ φsΛ.uncontracted := uncontractedListEmd_mem_uncontracted i
     rw [mem_uncontracted_iff_not_contracted] at hi
     exact hi p hp
   · obtain ⟨p, hp, rfl⟩ := hp
@@ -362,8 +354,7 @@ lemma exists_mem_left_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.Fie
     (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i : Fin φs.length)
     (ha : i ∈ (join φsΛ φsucΛ).uncontracted) :
     i ∈ φsΛ.uncontracted := by
-  rw [@mem_uncontracted_iff_not_contracted]
-  rw [@mem_uncontracted_iff_not_contracted] at ha
+  rw [@mem_uncontracted_iff_not_contracted] at ha ⊢
   simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
     RelEmbedding.coe_toEmbedding] at ha
   intro p hp
@@ -378,8 +369,7 @@ lemma exists_mem_right_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.Fi
   obtain ⟨j, rfl⟩ := uncontractedListEmd_surjective_mem_uncontracted i hi'
   use j
   simp only [true_and]
-  rw [mem_uncontracted_iff_not_contracted] at hi
-  rw [mem_uncontracted_iff_not_contracted]
+  rw [mem_uncontracted_iff_not_contracted] at hi ⊢
   intro p hp
   have hip := hi (p.map uncontractedListEmd) (by
     simp only [join, Finset.le_eq_subset, Finset.mem_union, Finset.mem_map,
@@ -393,9 +383,7 @@ lemma exists_mem_right_uncontracted_of_mem_join_uncontracted {φs : List 𝓕.Fi
 lemma join_uncontractedList {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
     (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
     (join φsΛ φsucΛ).uncontractedList = List.map uncontractedListEmd φsucΛ.uncontractedList := by
-  rw [uncontractedList_eq_sort]
-  rw [uncontractedList_eq_sort]
-  rw [fin_finset_sort_map_monotone]
+  rw [uncontractedList_eq_sort, uncontractedList_eq_sort, fin_finset_sort_map_monotone]
   congr
   ext a
   simp only [Finset.mem_map]
@@ -599,19 +587,14 @@ lemma join_singleton_getDual?_left {φs : List 𝓕.FieldOp}
     {i j : Fin φs.length} (h : i < j)
     (φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
     (join (singleton h) φsucΛ).getDual? i = some j := by
-  rw [@getDual?_eq_some_iff_mem]
-  simp [singleton, join]
+  simp [getDual?_eq_some_iff_mem, singleton, join]
 
 @[simp]
 lemma join_singleton_getDual?_right {φs : List 𝓕.FieldOp}
     {i j : Fin φs.length} (h : i < j)
     (φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
     (join (singleton h) φsucΛ).getDual? j = some i := by
-  rw [@getDual?_eq_some_iff_mem]
-  simp only [join, singleton, Finset.le_eq_subset, Finset.mem_union, Finset.mem_singleton,
-    Finset.mem_map, RelEmbedding.coe_toEmbedding]
-  left
-  exact Finset.pair_comm j i
+  simp [getDual?_eq_some_iff_mem, join, singleton, Finset.pair_comm]
 
 lemma exists_contraction_pair_of_card_ge_zero {φs : List 𝓕.FieldOp}
     (φsΛ : WickContraction φs.length) (h : 0 < φsΛ.1.card) :
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Perm.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Perm.lean
index f7a9b9323..5da4ea233 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Perm.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Perm.lean
@@ -49,21 +49,18 @@ variable {φs : List 𝓕.FieldOp} {φsΛ φsΛ₁ φsΛ₂ φsΛ₃ : WickContr
 
 /-- The reflexivity of the `Perm` relation. -/
 @[refl]
-lemma refl : Perm φsΛ φsΛ := by
-  rw [Perm]
+lemma refl : Perm φsΛ φsΛ := rfl
 
 /-- The symmetry of the `Perm` relation. -/
 @[symm]
-lemma symm (h : Perm φsΛ₁ φsΛ₂) : Perm φsΛ₂ φsΛ₁ := by
-  rw [Perm] at h ⊢
-  exact h.symm
+lemma symm (h : Perm φsΛ₁ φsΛ₂) : Perm φsΛ₂ φsΛ₁ :=
+  show φsΛ₂.wickTerm = φsΛ₁.wickTerm from Eq.symm h
 
 /-- The transitivity of the `Perm` relation. -/
 @[trans]
 lemma trans (h12 : Perm φsΛ₁ φsΛ₂) (h23 : Perm φsΛ₂ φsΛ₃) :
-    Perm φsΛ₁ φsΛ₃ := by
-  rw [Perm] at h12 h23 ⊢
-  exact h12.trans h23
+    Perm φsΛ₁ φsΛ₃ :=
+  show φsΛ₁.wickTerm = φsΛ₃.wickTerm from Eq.trans h12 h23
 
 /-- If `Perm φsΛ₁ φsΛ₂` and both contractions are grading-compliant,
   then if `φsΛ₁` is a full Wick contraction, so is `φsΛ₂`. -/
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Basic.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Basic.lean
index c128c5c08..e343f09d3 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Basic.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Basic.lean
@@ -51,9 +51,7 @@ def sign (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) : ℂ :=
     𝓕 |>ₛ ⟨φs.get, φsΛ.signFinset (φsΛ.fstFieldOfContract a) (φsΛ.sndFieldOfContract a)⟩)
 
 lemma sign_empty (φs : List 𝓕.FieldOp) :
-    sign φs empty = 1 := by
-  rw [sign]
-  simp [empty]
+    sign φs empty = 1 := by simp [sign, empty]
 
 lemma sign_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') (φsΛ : WickContraction φs.length) :
     sign φs' (congr (by simp [h]) φsΛ) = sign φs φsΛ := by
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertNone.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertNone.lean
index 4bf3e5334..dd36be363 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertNone.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertNone.lean
@@ -58,8 +58,7 @@ lemma signFinset_insertAndContract_none (φ : 𝓕.FieldOp) (φs : List 𝓕.Fie
       · simp only [Nat.succ_eq_add_one, finCongr_apply, Finset.mem_insert, Fin.ext_iff,
         Fin.val_cast, or_iff_right_iff_imp]
         intro h
-        have h1 : i.succAbove k ≠ i := by
-          exact Fin.succAbove_ne i k
+        have h1 : i.succAbove k ≠ i := Fin.succAbove_ne i k
         omega
       · simp
     rw [h1]
@@ -95,8 +94,7 @@ def signInsertNone (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickCo
 lemma sign_insert_none_eq_signInsertNone_mul_sign (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
     (φsΛ ↩Λ φ i none).sign = (φsΛ.signInsertNone φ φs i) * φsΛ.sign := by
-  rw [sign]
-  rw [signInsertNone, sign, ← Finset.prod_mul_distrib]
+  rw [sign, signInsertNone, sign, ← Finset.prod_mul_distrib]
   rw [insertAndContract_none_prod_contractions]
   congr
   funext a
@@ -193,8 +191,7 @@ lemma signInsertNone_eq_filter_map (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     φsΛ.signInsertNone φ φs i =
     𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ((List.filter (fun x => (φsΛ.getDual? x).isSome ∧ i.succAbove x < i)
     (List.finRange φs.length)).map φs.get)) := by
-  rw [signInsertNone_eq_prod_getDual?_Some]
-  rw [FieldStatistic.ofList_map_eq_finset_prod]
+  rw [signInsertNone_eq_prod_getDual?_Some, FieldStatistic.ofList_map_eq_finset_prod]
   rw [map_prod]
   congr
   funext a
@@ -208,8 +205,7 @@ lemma signInsertNone_eq_filter_map (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     · simp only [map_one]
   · rename_i h
     simp [h]
-  · refine List.Nodup.filter _ ?_
-    exact List.nodup_finRange φs.length
+  · exact List.Nodup.filter _ (List.nodup_finRange φs.length)
   · exact hG
 
 /-- The following signs for a grading compliant Wick contraction are equal:
@@ -254,8 +250,7 @@ lemma sign_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
     (φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (hG : GradingCompliant φs φsΛ) :
     (φsΛ ↩Λ φ i none).sign = 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ ⟨φs.get, Finset.univ.filter
     (fun x => (φsΛ.getDual? x).isSome ∧ i.succAbove x < i)⟩) * φsΛ.sign := by
-  rw [sign_insert_none_eq_signInsertNone_mul_sign]
-  rw [signInsertNone_eq_filterset]
+  rw [sign_insert_none_eq_signInsertNone_mul_sign, signInsertNone_eq_filterset]
   exact hG
 
 /-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertSome.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertSome.lean
index c819bae03..8087002ef 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertSome.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/InsertSome.lean
@@ -365,8 +365,7 @@ lemma signInsertSomeProd_eq_prod_fin (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldO
           then 𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs[x.1])
           else 1
       else 1 := by
-  rw [signInsertSomeProd_eq_prod_prod]
-  rw [Finset.prod_sigma']
+  rw [signInsertSomeProd_eq_prod_prod, Finset.prod_sigma']
   erw [← φsΛ.sigmaContractedEquiv.symm.prod_comp]
   let e2 : Fin φs.length ≃ {x // (φsΛ.getDual? x).isSome} ⊕ {x // ¬ (φsΛ.getDual? x).isSome} := by
     exact (Equiv.sumCompl fun a => (φsΛ.getDual? a).isSome = true).symm
@@ -395,8 +394,7 @@ lemma signInsertSomeProd_eq_finset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
       ∀ (h : (φsΛ.getDual? x).isSome),
       x < j ∧ (i.succAbove x < i ∧ i < i.succAbove ((φsΛ.getDual? x).get h)
       ∨ j < ((φsΛ.getDual? x).get h) ∧ ¬ i.succAbove x < i)))⟩) := by
-  rw [signInsertSomeProd_eq_prod_fin]
-  rw [ofFinset_eq_prod]
+  rw [signInsertSomeProd_eq_prod_fin, ofFinset_eq_prod]
   rw [map_prod]
   congr
   funext x
@@ -442,8 +440,7 @@ lemma stat_signFinset_insert_some_self_fst
   𝓕 |>ₛ ⟨φs.get,
     (Finset.univ.filter (fun x => i < i.succAbove x ∧ x < j ∧ ((φsΛ.getDual? x = none) ∨
       ∀ (h : (φsΛ.getDual? x).isSome), i < i.succAbove ((φsΛ.getDual? x).get h))))⟩ := by
-  rw [get_eq_insertIdx_succAbove φ _ i]
-  rw [ofFinset_finset_map]
+  rw [get_eq_insertIdx_succAbove φ _ i, ofFinset_finset_map]
   swap
   refine
     (Equiv.comp_injective i.succAbove (finCongr (Eq.symm (insertIdx_length_fin φ φs i)))).mpr ?hi.a
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Join.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Join.lean
index bb834f7b1..8c2a39b06 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Join.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Sign/Join.lean
@@ -193,10 +193,8 @@ lemma join_singleton_sign_left {φs : List 𝓕.FieldOp}
     (φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
     (singleton h).sign = 𝓢(𝓕 |>ₛ φs[j],
     (𝓕 |>ₛ ⟨φs.get, (join (singleton h) φsucΛ).signFinset i j⟩)) * (joinSignLeftExtra h φsucΛ) := by
-  rw [singleton_sign_expand]
-  rw [join_singleton_left_signFinset_eq_filter h φsucΛ]
-  rw [map_mul]
-  rfl
+  simpa [joinSignLeftExtra, join_singleton_left_signFinset_eq_filter h φsucΛ, map_mul] using
+    singleton_sign_expand h
 
 lemma join_singleton_sign_right {φs : List 𝓕.FieldOp}
     {i j : Fin φs.length} (h : i < j)
@@ -205,9 +203,8 @@ lemma join_singleton_sign_right {φs : List 𝓕.FieldOp}
     (joinSignRightExtra h φsucΛ) *
     (∏ a, 𝓢(𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a], 𝓕|>ₛ ⟨φs.get,
       ((join (singleton h) φsucΛ).signFinset (uncontractedListEmd (φsucΛ.fstFieldOfContract a))
-        (uncontractedListEmd (φsucΛ.sndFieldOfContract a)))⟩)) := by
-  rw [sign_right_eq_prod_mul_prod]
-  rfl
+        (uncontractedListEmd (φsucΛ.sndFieldOfContract a)))⟩)) :=
+  sign_right_eq_prod_mul_prod (singleton h) φsucΛ
 
 lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.FieldOp}
     {i j : Fin φs.length} (h : i < j) (φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
@@ -246,8 +243,7 @@ lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.FieldOp}
       not_forall, not_lt]
     intro hxij h1 h2
     omega
-  · have hj1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j := by
-      omega
+  · have hj1 : uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j := by omega
     by_cases hi1 : ¬ i < uncontractedListEmd (φsucΛ.sndFieldOfContract a)
     · simp only [hi1, and_false, ↓reduceIte, Finset.notMem_empty, or_false]
       have hj2 : ¬ j < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
@@ -255,8 +251,7 @@ lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.FieldOp}
         not_or, not_forall, not_lt]
       intro hxij h1 h2
       omega
-    · have hi1 : i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by
-        omega
+    · have hi1 : i < uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by omega
       simp only [hj1, true_and, hi1, and_true]
       by_cases hi2 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i
       · simp only [hi2, and_false, ↓reduceIte, Finset.notMem_empty, or_self, iff_false, not_and,
@@ -338,9 +333,8 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.FieldOp}
   funext a
   have hjneqsnd := singleton_uncontractedEmd_ne_right h (φsucΛ.sndFieldOfContract a)
   have hl : uncontractedListEmd (φsucΛ.fstFieldOfContract a) <
-      uncontractedListEmd (φsucΛ.sndFieldOfContract a) := by
-    apply uncontractedListEmd_strictMono
-    exact fstFieldOfContract_lt_sndFieldOfContract φsucΛ a
+      uncontractedListEmd (φsucΛ.sndFieldOfContract a) :=
+    uncontractedListEmd_strictMono (fstFieldOfContract_lt_sndFieldOfContract φsucΛ a)
   by_cases hj1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j
   · have hi1 : ¬ uncontractedListEmd (φsucΛ.fstFieldOfContract a) < i := by omega
     simp [hj1, hi1]
@@ -410,16 +404,14 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
     obtain ⟨i, j, hij, φsucΛ', rfl, h1, h2, h3⟩ :=
       exists_join_singleton_of_card_ge_zero φsΛ (by simp [hn]) hc
     rw [join_assoc, join_sign_singleton hij h1, join_sign_singleton hij h1]
-    have hn : φsucΛ'.1.card = n := by
-      omega
+    have hn : φsucΛ'.1.card = n := by omega
     rw [join_sign_induction φsucΛ' (congr (by simp [join_uncontractedListGet]) φsucΛ) h2
       n hn]
     rw [mul_assoc]
     simp only [mul_eq_mul_left_iff]
     left
     left
-    apply sign_congr
-    exact join_uncontractedListGet (singleton hij) φsucΛ'
+    exact sign_congr (join_uncontractedListGet (singleton hij) φsucΛ') φsucΛ
 
 /-- For a list `φs` of `𝓕.FieldOp`, a grading compliant Wick contraction `φsΛ` of `φs`,
   and a Wick contraction `φsucΛ` of `[φsΛ]ᵘᶜ`, the following relation holds
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Singleton.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Singleton.lean
index 103fd57af..b3bdcd747 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Singleton.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Singleton.lean
@@ -40,12 +40,10 @@ def singleton {i j : Fin n} (hij : i < j) : WickContraction n :=
     simp_all⟩
 
 lemma mem_singleton {i j : Fin n} (hij : i < j) :
-    {i, j} ∈ (singleton hij).1 := by
-  simp [singleton]
+    {i, j} ∈ (singleton hij).1 := by simp [singleton]
 
 lemma mem_singleton_iff {i j : Fin n} (hij : i < j) {a : Finset (Fin n)} :
-    a ∈ (singleton hij).1 ↔ a = {i, j} := by
-  simp [singleton]
+    a ∈ (singleton hij).1 ↔ a = {i, j} := by simp [singleton]
 
 lemma of_singleton_eq {i j : Fin n} (hij : i < j) (a : (singleton hij).1) :
     a = ⟨{i, j}, mem_singleton hij⟩ := by
@@ -55,8 +53,7 @@ lemma of_singleton_eq {i j : Fin n} (hij : i < j) (a : (singleton hij).1) :
 
 lemma singleton_prod {φs : List 𝓕.FieldOp} {i j : Fin φs.length} (hij : i < j)
     (f : (singleton hij).1 → M) [CommMonoid M] :
-    ∏ a, f a = f ⟨{i,j}, mem_singleton hij⟩:= by
-  simp [singleton, of_singleton_eq]
+    ∏ a, f a = f ⟨{i,j}, mem_singleton hij⟩:= by simp [singleton, of_singleton_eq]
 
 @[simp]
 lemma singleton_fstFieldOfContract {i j : Fin n} (hij : i < j) :
@@ -81,8 +78,7 @@ lemma singleton_sign_expand {φs : List 𝓕.FieldOp} {i j : Fin φs.length} (hi
 
 lemma singleton_getDual?_eq_none_iff_neq {i j : Fin n} (hij : i < j) (a : Fin n) :
     (singleton hij).getDual? a = none ↔ (i ≠ a ∧ j ≠ a) := by
-  rw [getDual?_eq_none_iff_mem_uncontracted]
-  rw [mem_uncontracted_iff_not_contracted]
+  rw [getDual?_eq_none_iff_mem_uncontracted, mem_uncontracted_iff_not_contracted]
   simp only [singleton, Finset.mem_singleton, forall_eq, Finset.mem_insert, not_or, ne_eq]
   omega
 
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/SubContraction.lean b/Physlib/QFT/PerturbationTheory/WickContraction/SubContraction.lean
index e0b2db5cd..6d7a1285f 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/SubContraction.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/SubContraction.lean
@@ -33,8 +33,7 @@ def subContraction (S : Finset (Finset (Fin φs.length))) (ha : S ⊆ φsΛ.1) :
     exact φsΛ.2.2 i (ha hi) j (ha hj)⟩
 
 lemma mem_of_mem_subContraction {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
-    {a : Finset (Fin φs.length)} (ha : a ∈ (φsΛ.subContraction S hs).1) : a ∈ φsΛ.1 := by
-  exact hs ha
+    {a : Finset (Fin φs.length)} (ha : a ∈ (φsΛ.subContraction S hs).1) : a ∈ φsΛ.1 := hs ha
 
 /-- Given a Wick contraction `φsΛ`, and a subset `S` of `φsΛ.1`, the Wick contraction
   on the uncontracted list `[φsΛ.subContraction S ha]ᵘᶜ`
@@ -101,9 +100,7 @@ lemma mem_subContraction_or_quotContraction {S : Finset (Finset (Fin φs.length)
 @[simp]
 lemma subContraction_uncontractedList_get {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
     {a : Fin [subContraction S hs]ᵘᶜ.length} :
-    [subContraction S hs]ᵘᶜ[a] = φs[uncontractedListEmd a] := by
-  erw [← getElem_uncontractedListEmd]
-  rfl
+    [subContraction S hs]ᵘᶜ[a] = φs[uncontractedListEmd a] := (getElem_uncontractedListEmd a).symm
 
 @[simp]
 lemma subContraction_fstFieldOfContract {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/TimeCond.lean b/Physlib/QFT/PerturbationTheory/WickContraction/TimeCond.lean
index b50b356d8..42fe58d92 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/TimeCond.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/TimeCond.lean
@@ -48,8 +48,7 @@ lemma timeOrderRel_both_of_eqTimeOnly {i j : Fin φs.length} (h : {i, j} ∈ φs
   apply And.intro
   · exact timeOrderRel_of_eqTimeOnly_pair φsΛ h hc
   · apply timeOrderRel_of_eqTimeOnly_pair φsΛ _ hc
-    rw [@Finset.pair_comm]
-    exact h
+    simpa [@Finset.pair_comm] using h
 
 lemma eqTimeOnly_iff_forall_finset {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
     φsΛ.EqTimeOnly ↔ ∀ (a : φsΛ.1),
@@ -95,8 +94,7 @@ lemma eqTimeOnly_iff_forall_finset {φs : List 𝓕.FieldOp} (φsΛ : WickContra
 
 @[simp]
 lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly := by
-  rw [eqTimeOnly_iff_forall_finset]
-  simp [empty]
+  simp [eqTimeOnly_iff_forall_finset, empty]
 
 /-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` within
   which every contraction involves two `𝓕.FieldOp`s that have the same time, then
@@ -110,9 +108,8 @@ lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
   simp only [List.get_eq_getElem]
   rw [timeContract_of_timeOrderRel]
   apply timeOrderRel_of_eqTimeOnly_pair φsΛ
-  rw [← finset_eq_fstFieldOfContract_sndFieldOfContract]
-  exact a.2
-  exact h
+  · simp [← finset_eq_fstFieldOfContract_sndFieldOfContract, a.2]
+  · exact h
 
 lemma eqTimeOnly_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') (φsΛ : WickContraction φs.length) :
     (congr (by simp [h]) φsΛ).EqTimeOnly (φs := φs') ↔ φsΛ.EqTimeOnly := by
@@ -177,8 +174,7 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction {φs : List 𝓕.Fi
   | Nat.succ n, hn => by
     obtain ⟨i, j, hij, φsucΛ, rfl, h2, h3, h4⟩ :=
       exists_join_singleton_of_card_ge_zero φsΛ (by simp [hn]) hl
-    rw [join_timeContract]
-    rw [singleton_timeContract]
+    rw [join_timeContract, singleton_timeContract]
     simp only [Fin.getElem_fin, MulMemClass.coe_mul]
     trans timeOrder (a * WickAlgebra.timeContract φs[↑i] φs[↑j] * (φsucΛ.timeContract.1 * b))
     simp only [mul_assoc, Fin.getElem_fin]
@@ -205,10 +201,8 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_left {φs : List 𝓕.FieldOp}
     (φsΛ : WickContraction φs.length)
     (hl : φsΛ.EqTimeOnly) (b : 𝓕.WickAlgebra) :
     𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b) := by
-  trans 𝓣(1 * φsΛ.timeContract.1 * b)
-  simp only [one_mul]
-  rw [timeOrder_timeContract_mul_of_eqTimeOnly_mid φsΛ hl]
-  simp
+  simpa using
+    timeOrder_timeContract_mul_of_eqTimeOnly_mid φsΛ hl (1 : 𝓕.WickAlgebra) b
 
 set_option maxHeartbeats 400000 in
 lemma exists_join_singleton_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
@@ -240,11 +234,9 @@ lemma timeOrder_timeContract_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
     (φsΛ : WickContraction φs.length)
     (hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.timeContract.1) = 0 := by
   obtain ⟨i, j, hij, φsucΛ, rfl, hr⟩ := exists_join_singleton_of_not_eqTimeOnly φsΛ hl
-  rw [join_timeContract]
-  rw [singleton_timeContract]
+  rw [join_timeContract, singleton_timeContract]
   simp only [Fin.getElem_fin, MulMemClass.coe_mul]
-  rw [timeOrder_timeOrder_left]
-  rw [timeOrder_timeContract_ne_time]
+  rw [timeOrder_timeOrder_left, timeOrder_timeContract_ne_time]
   simp only [zero_mul, map_zero]
   simp_all only [Fin.getElem_fin, not_and]
   intro h
@@ -258,8 +250,7 @@ lemma timeOrder_staticContract_of_not_mem {φs : List 𝓕.FieldOp} (φsΛ : Wic
   obtain ⟨i, j, hij, φsucΛ, rfl, hr⟩ := exists_join_singleton_of_not_eqTimeOnly φsΛ hl
   rw [join_staticContract]
   simp only [MulMemClass.coe_mul]
-  rw [singleton_staticContract]
-  rw [timeOrder_timeOrder_left]
+  rw [singleton_staticContract, timeOrder_timeOrder_left]
   rw [timeOrder_superCommute_anPart_ofFieldOp_ne_time]
   simp only [zero_mul, map_zero]
   intro h
@@ -308,14 +299,11 @@ lemma haveEqTime_iff_finset {φs : List 𝓕.FieldOp} (φsΛ : WickContraction 
     use φsΛ.fstFieldOfContract ⟨a, h1⟩
     use φsΛ.sndFieldOfContract ⟨a, h1⟩
     simp_all only [and_true, true_and]
-    rw [← finset_eq_fstFieldOfContract_sndFieldOfContract]
-    exact h1
+    simpa [← finset_eq_fstFieldOfContract_sndFieldOfContract] using h1
 
 @[simp]
 lemma empty_not_haveEqTime {φs : List 𝓕.FieldOp} :
-    ¬ HaveEqTime (empty : WickContraction φs.length) := by
-  rw [haveEqTime_iff_finset]
-  simp [empty]
+    ¬ HaveEqTime (empty : WickContraction φs.length) := by simp [haveEqTime_iff_finset, empty]
 
 /-- Given a Wick contraction the subset of contracted pairs between equal time fields. -/
 def eqTimeContractSet {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
@@ -495,8 +483,7 @@ lemma join_haveEqTime_of_eqTimeOnly_nonEmpty {φs : List 𝓕.FieldOp} (φsΛ :
   use i, j
   simp_all only [ne_eq, true_or, true_and]
   apply h1 j i
-  rw [Finset.pair_comm]
-  exact h
+  simpa [Finset.pair_comm] using h
 
 lemma hasEqTimeEquiv_ext_sigma {φs : List 𝓕.FieldOp} {x1 x2 :
     Σ (φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty}),
@@ -536,8 +523,7 @@ def hasEqTimeEquiv (φs : List 𝓕.FieldOp) :
         apply Subtype.ext
         ext a
         simp only [subContraction]
-        rw [join_eqTimeContractSet]
-        rw [eqTimeContractSet_of_not_haveEqTime h2]
+        rw [join_eqTimeContractSet, eqTimeContractSet_of_not_haveEqTime h2]
         simp only [Finset.le_eq_subset, ne_eq, Finset.map_empty, Finset.union_empty]
         rw [eqTimeContractSet_of_mem_eqTimeOnly h1.1]
       refine hasEqTimeEquiv_ext_sigma ?_ ?_
@@ -547,11 +533,9 @@ def hasEqTimeEquiv (φs : List 𝓕.FieldOp) :
         apply Subtype.ext
         ext a
         simp only [quotContraction, Finset.mem_filter, Finset.mem_univ, true_and]
-        rw [mem_congr_iff]
-        rw [mem_join_right_iff]
+        rw [mem_congr_iff, mem_join_right_iff]
         simp only [ne_eq]
-        rw [uncontractedListEmd_congr hs]
-        rw [Finset.map_map]
+        rw [uncontractedListEmd_congr hs, Finset.map_map]
 
 lemma sum_haveEqTime (φs : List 𝓕.FieldOp)
     (f : WickContraction φs.length → M) [AddCommMonoid M]:
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/TimeContract.lean b/Physlib/QFT/PerturbationTheory/WickContraction/TimeContract.lean
index ec014fd96..a34f80727 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/TimeContract.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/TimeContract.lean
@@ -77,9 +77,7 @@ lemma timeContract_insertAndContract_some
 
 @[simp]
 lemma timeContract_empty (φs : List 𝓕.FieldOp) :
-    (@empty φs.length).timeContract = 1 := by
-  rw [timeContract, empty]
-  simp
+    (@empty φs.length).timeContract = 1 := by simp [timeContract, empty]
 
 open FieldStatistic
 
@@ -122,8 +120,7 @@ lemma timeContract_insert_some_of_lt
       congr
       rw [← List.map_take]
       congr
-      rw [take_uncontractedIndexEquiv_symm]
-      rw [filter_uncontractedList]
+      rw [take_uncontractedIndexEquiv_symm, filter_uncontractedList]
     rw [h1]
     simp only [exchangeSign_mul_self]
     · exact ht
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/Uncontracted.lean b/Physlib/QFT/PerturbationTheory/WickContraction/Uncontracted.lean
index 5f4a8ebf1..b8ce1a2d8 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/Uncontracted.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/Uncontracted.lean
@@ -30,8 +30,7 @@ lemma congr_uncontracted {n m : ℕ} (c : WickContraction n) (h : n = m) :
   simp
 
 lemma getDual?_eq_none_iff_mem_uncontracted (i : Fin n) :
-    c.getDual? i = none ↔ i ∈ c.uncontracted := by
-  simp [uncontracted]
+    c.getDual? i = none ↔ i ∈ c.uncontracted := by simp [uncontracted]
 
 /-- The equivalence of `Option c.uncontracted` for two propositionally equal Wick contractions. -/
 def uncontractedCongr {c c': WickContraction n} (h : c = c') :
@@ -40,8 +39,7 @@ def uncontractedCongr {c c': WickContraction n} (h : c = c') :
 
 @[simp]
 lemma uncontractedCongr_none {c c': WickContraction n} (h : c = c') :
-    (uncontractedCongr h) none = none := by
-  simp [uncontractedCongr]
+    (uncontractedCongr h) none = none := by simp [uncontractedCongr]
 
 @[simp]
 lemma uncontractedCongr_some {c c': WickContraction n} (h : c = c') (i : c.uncontracted) :
@@ -85,8 +83,7 @@ lemma getDual?_empty_eq_none (i : Fin n) : empty.getDual? i = none := by
   simpa [uncontracted] using mem_uncontracted_empty i
 
 @[simp]
-lemma uncontracted_empty {n : ℕ} : (@empty n).uncontracted = Finset.univ := by
-  simp [uncontracted]
+lemma uncontracted_empty {n : ℕ} : (@empty n).uncontracted = Finset.univ := by simp [uncontracted]
 
 lemma uncontracted_card_le (c : WickContraction n) : c.uncontracted.card ≤ n := by
   simp only [uncontracted]
@@ -103,9 +100,7 @@ lemma uncontracted_card_eq_iff (c : WickContraction n) :
     by_contra hn
     have hc' := exists_pair_of_not_eq_empty c hn
     obtain ⟨i, j, hij⟩ := hc'
-    have hci : c.getDual? i = some j := by
-      rw [@getDual?_eq_some_iff_mem]
-      exact hij
+    have hci : c.getDual? i = some j := by simpa [getDual?_eq_some_iff_mem] using hij
     simp_all
   · intro h
     subst h
diff --git a/Physlib/QFT/PerturbationTheory/WickContraction/UncontractedList.lean b/Physlib/QFT/PerturbationTheory/WickContraction/UncontractedList.lean
index 732cf0dfe..2613a8574 100644
--- a/Physlib/QFT/PerturbationTheory/WickContraction/UncontractedList.lean
+++ b/Physlib/QFT/PerturbationTheory/WickContraction/UncontractedList.lean
@@ -44,29 +44,21 @@ lemma fin_list_sorted_monotone_sorted {n m : ℕ} (l: List (Fin n)) (hl : l.Pair
       exact ih hl.2
 
 lemma fin_list_sorted_succAboveEmb_sorted (l: List (Fin n)) (hl : l.Pairwise (· ≤ ·))
-    (i : Fin n.succ) : ((List.map i.succAboveEmb l)).Pairwise (· ≤ ·) := by
-  apply fin_list_sorted_monotone_sorted
-  exact hl
-  simp only [Fin.coe_succAboveEmb]
-  exact Fin.strictMono_succAbove i
+    (i : Fin n.succ) : ((List.map i.succAboveEmb l)).Pairwise (· ≤ ·) :=
+  fin_list_sorted_monotone_sorted l hl i.succAboveEmb (Fin.strictMono_succAbove i)
 
 lemma fin_finset_sort_map_monotone {n m : ℕ} (a : Finset (Fin n)) (f : Fin n ↪ Fin m)
     (hf : StrictMono f) : (a.sort (· ≤ ·)).map f =
     ((a.map f).sort (· ≤ ·)) := by
-  have h1 : ((a.sort (· ≤ ·)).map f).Pairwise (· ≤ ·) := by
-    apply fin_list_sorted_monotone_sorted
-    exact a.pairwise_sort (fun x1 x2 => x1 ≤ x2)
-    exact hf
-  have h2 : ((a.sort (· ≤ ·)).map f).Nodup := by
-    refine (List.nodup_map_iff_inj_on ?_).mpr ?_
-    exact a.sort_nodup (fun x1 x2 => x1 ≤ x2)
-    intro a ha b hb hf
-    exact f.2 hf
+  have h1 : ((a.sort (· ≤ ·)).map f).Pairwise (· ≤ ·) :=
+    fin_list_sorted_monotone_sorted _ (a.pairwise_sort (fun x1 x2 => x1 ≤ x2)) f hf
+  have h2 : ((a.sort (· ≤ ·)).map f).Nodup :=
+    (List.nodup_map_iff_inj_on (a.sort_nodup (fun x1 x2 => x1 ≤ x2))).mpr
+      fun _ _ _ _ hf => f.2 hf
   have h3 : ((a.sort (· ≤ ·)).map f).toFinset = (a.map f) := by
     ext a
     simp
-  rw [← h3]
-  exact ((List.toFinset_sort (· ≤ ·) h2).mpr h1).symm
+  simpa [← h3] using ((List.toFinset_sort (· ≤ ·) h2).mpr h1).symm
 
 lemma fin_list_sorted_split :
     (l : List (Fin n)) → (hl : l.Pairwise (· ≤ ·)) → (i : ℕ) →
@@ -196,8 +188,7 @@ lemma orderedInsert_eq_insertIdx_of_fin_list_sorted (l : List (Fin n)) (hl : l.P
 def uncontractedList : List (Fin n) := List.filter (fun x => x ∈ c.uncontracted) (List.finRange n)
 
 lemma uncontractedList_mem_iff (i : Fin n) :
-    i ∈ c.uncontractedList ↔ i ∈ c.uncontracted := by
-  simp [uncontractedList]
+    i ∈ c.uncontractedList ↔ i ∈ c.uncontracted := by simp [uncontractedList]
 
 @[simp]
 lemma uncontractedList_empty : (empty (n := n)).uncontractedList = List.finRange n := by
@@ -220,23 +211,19 @@ lemma uncontractedList_sorted : List.Pairwise (· ≤ ·) c.uncontractedList :=
   rw [uncontractedList]
   apply List.Pairwise.filter
   rw [← List.ofFn_id]
-  refine List.sortedLE_iff_pairwise.mp ?_
-  exact Monotone.sortedLE_ofFn fun ⦃a b⦄ a => a
+  exact List.sortedLE_iff_pairwise.mp (Monotone.sortedLE_ofFn fun ⦃a b⦄ a => a)
 
 lemma uncontractedList_sorted_lt : List.Pairwise (· < ·) c.uncontractedList := by
   rw [uncontractedList]
   apply List.Pairwise.filter
-  rw [← List.ofFn_id]
-  exact List.pairwise_ofFn.mpr fun ⦃i j⦄ a => a
+  simp [← List.ofFn_id]
 
 lemma uncontractedList_nodup : c.uncontractedList.Nodup := by
   rw [uncontractedList]
-  refine List.Nodup.filter (fun x => decide (x ∈ c.uncontracted)) ?_
-  exact List.nodup_finRange n
+  exact List.Nodup.filter (fun x => decide (x ∈ c.uncontracted)) (List.nodup_finRange n)
 
 lemma uncontractedList_toFinset (c : WickContraction n) :
-    c.uncontractedList.toFinset = c.uncontracted := by
-  simp [uncontractedList]
+    c.uncontractedList.toFinset = c.uncontracted := by simp [uncontractedList]
 
 lemma uncontractedList_eq_sort (c : WickContraction n) :
     c.uncontractedList = c.uncontracted.sort (· ≤ ·) := by
@@ -247,24 +234,19 @@ lemma uncontractedList_eq_sort (c : WickContraction n) :
   · exact uncontractedList_sorted c
 
 lemma uncontractedList_length_eq_card (c : WickContraction n) :
-    c.uncontractedList.length = c.uncontracted.card := by
-  rw [uncontractedList_eq_sort]
-  exact Finset.length_sort fun x1 x2 => x1 ≤ x2
+    c.uncontractedList.length = c.uncontracted.card := by simp [uncontractedList_eq_sort]
 
 lemma filter_uncontractedList (c : WickContraction n) (p : Fin n → Prop) [DecidablePred p] :
     (c.uncontractedList.filter p) = (c.uncontracted.filter p).sort (· ≤ ·) := by
-  have h1 : (c.uncontractedList.filter p).Pairwise (· ≤ ·) := by
-    apply List.Pairwise.filter
-    exact uncontractedList_sorted c
-  have h2 : (c.uncontractedList.filter p).Nodup := by
-    refine List.Nodup.filter _ ?_
-    exact uncontractedList_nodup c
+  have h1 : (c.uncontractedList.filter p).Pairwise (· ≤ ·) :=
+    List.Pairwise.filter _ (uncontractedList_sorted c)
+  have h2 : (c.uncontractedList.filter p).Nodup :=
+    List.Nodup.filter _ (uncontractedList_nodup c)
   have h3 : (c.uncontractedList.filter p).toFinset = (c.uncontracted.filter p) := by
     ext a
     simp only [List.toFinset_filter, decide_eq_true_eq, Finset.mem_filter, List.mem_toFinset,
       and_congr_left_iff]
-    rw [uncontractedList_mem_iff]
-    simp
+    simp [uncontractedList_mem_iff]
   have hx := (List.toFinset_sort (· ≤ ·) h2).mpr h1
   rw [← hx, h3]
 
@@ -301,8 +283,7 @@ lemma uncontractedIndexEquiv_symm_eq_filter_length (k : c.uncontracted) :
   rw [fin_list_sorted_indexOf_mem]
   · simp
   · exact uncontractedList_sorted c
-  · rw [uncontractedList_mem_iff]
-    exact k.2
+  · simp [uncontractedList_mem_iff, k.2]
 
 lemma take_uncontractedIndexEquiv_symm (k : c.uncontracted) :
     c.uncontractedList.take (c.uncontractedIndexEquiv.symm k).val =
@@ -311,8 +292,7 @@ lemma take_uncontractedIndexEquiv_symm (k : c.uncontracted) :
   conv_lhs =>
     rhs
     rw [hl]
-  rw [uncontractedIndexEquiv_symm_eq_filter_length]
-  simp
+  simp [uncontractedIndexEquiv_symm_eq_filter_length]
 /-!
 
 ## Uncontracted List get
@@ -332,8 +312,7 @@ scoped[WickContraction] notation "[" φsΛ "]ᵘᶜ" => uncontractedListGet φs
 
 @[simp]
 lemma uncontractedListGet_empty {φs : List 𝓕.FieldOp} :
-    (empty (n := φs.length)).uncontractedListGet = φs := by
-  simp [uncontractedListGet]
+    (empty (n := φs.length)).uncontractedListGet = φs := by simp [uncontractedListGet]
 
 /-!
 
@@ -351,8 +330,7 @@ def uncontractedFieldOpEquiv (φs : List 𝓕.FieldOp) (φsΛ : WickContraction
 
 @[simp]
 lemma uncontractedFieldOpEquiv_none (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :
-    (uncontractedFieldOpEquiv φs φsΛ).toFun none = none := by
-  simp [uncontractedFieldOpEquiv]
+    (uncontractedFieldOpEquiv φs φsΛ).toFun none = none := by simp [uncontractedFieldOpEquiv]
 
 lemma uncontractedFieldOpEquiv_list_sum [AddCommMonoid α] (φs : List 𝓕.FieldOp)
     (φsΛ : WickContraction φs.length) (f : Option (Fin [φsΛ]ᵘᶜ.length) → α) :
@@ -380,17 +358,14 @@ lemma uncontractedListEmd_congr {φs : List 𝓕.FieldOp} {φsΛ φsΛ' : WickCo
 
 lemma uncontractedListEmd_toFun_eq_get (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :
     (uncontractedListEmd (φsΛ := φsΛ)).toFun =
-    φsΛ.uncontractedList.get ∘ (finCongr (by simp [uncontractedListGet])) := by
-  rfl
+    φsΛ.uncontractedList.get ∘ (finCongr (by simp [uncontractedListGet])) := rfl
 
 lemma uncontractedListEmd_strictMono {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
     {i j : Fin [φsΛ]ᵘᶜ.length} (h : i < j) : uncontractedListEmd i < uncontractedListEmd j := by
   simp only [uncontractedListEmd, uncontractedIndexEquiv, List.get_eq_getElem,
     Equiv.trans_toEmbedding, Function.Embedding.trans_apply, Equiv.coe_toEmbedding, finCongr_apply,
     Equiv.coe_fn_mk, Fin.val_cast, Function.Embedding.coe_subtype]
-  apply List.SortedLT.strictMono_get
-  exact φsΛ.uncontractedList_sorted_lt.sortedLT
-  exact h
+  exact List.SortedLT.strictMono_get φsΛ.uncontractedList_sorted_lt.sortedLT h
 
 lemma uncontractedListEmd_mem_uncontracted {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
     (i : Fin [φsΛ]ᵘᶜ.length) : uncontractedListEmd i ∈ φsΛ.uncontracted := by
@@ -452,9 +427,8 @@ lemma uncontractedListEmd_empty {φs : List 𝓕.FieldOp} :
 -/
 
 lemma uncontractedList_succAboveEmb_sorted (c : WickContraction n) (i : Fin n.succ) :
-    ((List.map i.succAboveEmb c.uncontractedList)).Pairwise (· ≤ ·) := by
-  apply fin_list_sorted_succAboveEmb_sorted
-  exact uncontractedList_sorted c
+    ((List.map i.succAboveEmb c.uncontractedList)).Pairwise (· ≤ ·) :=
+  fin_list_sorted_succAboveEmb_sorted c.uncontractedList (uncontractedList_sorted c) i
 
 lemma uncontractedList_succAboveEmb_nodup (c : WickContraction n) (i : Fin n.succ) :
     ((List.map i.succAboveEmb c.uncontractedList)).Nodup := by
@@ -487,8 +461,7 @@ lemma uncontractedList_succAbove_orderedInsert_toFinset (c : WickContraction n)
   ext a
   simp only [Nat.succ_eq_add_one, Fin.coe_succAboveEmb, List.mem_toFinset, List.mem_orderedInsert,
     List.mem_map, Finset.mem_insert, Finset.mem_map, Fin.succAboveEmb_apply]
-  rw [← uncontractedList_toFinset]
-  simp
+  simp [← uncontractedList_toFinset]
 
 lemma uncontractedList_succAbove_orderedInsert_eq_sort (c : WickContraction n) (i : Fin n.succ) :
     (List.orderedInsert (· ≤ ·) i (List.map i.succAboveEmb c.uncontractedList)) =
@@ -540,14 +513,12 @@ lemma uncontractedList_succAboveEmb_eraseIdx_toFinset (c : WickContraction n) (i
   exact uncontractedList_succAboveEmb_nodup c i
 
 lemma uncontractedList_succAboveEmb_eraseIdx_sorted (c : WickContraction n) (i : Fin n.succ)
-    (k: ℕ) : ((List.map i.succAboveEmb c.uncontractedList).eraseIdx k).Pairwise (· ≤ ·) := by
-  apply Physlib.List.eraseIdx_sorted
-  exact uncontractedList_succAboveEmb_sorted c i
+    (k: ℕ) : ((List.map i.succAboveEmb c.uncontractedList).eraseIdx k).Pairwise (· ≤ ·) :=
+  Physlib.List.eraseIdx_sorted (· ≤ ·) _ k (uncontractedList_succAboveEmb_sorted c i)
 
 lemma uncontractedList_succAboveEmb_eraseIdx_nodup (c : WickContraction n) (i : Fin n.succ) (k: ℕ) :
-    ((List.map i.succAboveEmb c.uncontractedList).eraseIdx k).Nodup := by
-  refine List.Nodup.eraseIdx k ?_
-  exact uncontractedList_succAboveEmb_nodup c i
+    ((List.map i.succAboveEmb c.uncontractedList).eraseIdx k).Nodup :=
+  List.Nodup.eraseIdx k (uncontractedList_succAboveEmb_nodup c i)
 
 lemma uncontractedList_succAboveEmb_eraseIdx_eq_sort (c : WickContraction n) (i : Fin n.succ)
     (k : ℕ) (hk : k < c.uncontractedList.length) :
@@ -563,8 +534,7 @@ lemma uncontractedList_succAboveEmb_eraseIdx_eq_sort (c : WickContraction n) (i
 lemma uncontractedList_extractEquiv_symm_some (c : WickContraction n) (i : Fin n.succ)
     (k : c.uncontracted) : ((extractEquiv i).symm ⟨c, some k⟩).uncontractedList =
     ((c.uncontractedList).map i.succAboveEmb).eraseIdx (c.uncontractedIndexEquiv.symm k) := by
-  rw [uncontractedList_eq_sort]
-  rw [uncontractedList_succAboveEmb_eraseIdx_eq_sort]
+  rw [uncontractedList_eq_sort, uncontractedList_succAboveEmb_eraseIdx_eq_sort]
   swap
   simp only [Fin.is_lt]
   congr
@@ -580,8 +550,7 @@ lemma uncontractedList_succAboveEmb_toFinset (c : WickContraction n) (i : Fin n.
   ext a
   simp only [Fin.coe_succAboveEmb, List.mem_toFinset, List.mem_map, Finset.mem_map,
     Fin.succAboveEmb_apply]
-  rw [← c.uncontractedList_toFinset]
-  simp
+  simp [← c.uncontractedList_toFinset]
 
 /-!
 
@@ -614,18 +583,14 @@ lemma take_uncontractedListOrderPos_eq_filter_sort (c : WickContraction n) (i :
     (c.uncontractedList.take (c.uncontractedListOrderPos i)) =
     (c.uncontracted.filter (fun x => x.1 < i.1)).sort (· ≤ ·) := by
   rw [take_uncontractedListOrderPos_eq_filter]
-  have h1 : (c.uncontractedList.filter (fun x => x.1 < i.1)).Pairwise (· ≤ ·) := by
-    apply List.Pairwise.filter
-    exact uncontractedList_sorted c
-  have h2 : (c.uncontractedList.filter (fun x => x.1 < i.1)).Nodup := by
-    refine List.Nodup.filter _ ?_
-    exact uncontractedList_nodup c
+  have h1 : (c.uncontractedList.filter (fun x => x.1 < i.1)).Pairwise (· ≤ ·) :=
+    List.Pairwise.filter _ (uncontractedList_sorted c)
+  have h2 : (c.uncontractedList.filter (fun x => x.1 < i.1)).Nodup :=
+    List.Nodup.filter _ (uncontractedList_nodup c)
   have h3 : (c.uncontractedList.filter (fun x => x.1 < i.1)).toFinset =
     (c.uncontracted.filter (fun x => x.1 < i.1)) := by
-    rw [uncontractedList_eq_sort]
-    simp
-  rw [← h3]
-  exact ((List.toFinset_sort (α := Fin n) (· ≤ ·) h2).mpr h1).symm
+    simp [uncontractedList_eq_sort]
+  simpa [← h3] using ((List.toFinset_sort (α := Fin n) (· ≤ ·) h2).mpr h1).symm
 
 lemma orderedInsert_succAboveEmb_uncontractedList_eq_insertIdx (c : WickContraction n)
     (i : Fin n.succ) :
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Basic.lean b/Physlib/QFT/QED/AnomalyCancellation/Basic.lean
index 593e4f8f8..3a9e802a6 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Basic.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Basic.lean
@@ -128,9 +128,7 @@ lemma pureU1_linear {n : ℕ} (S : (PureU1 n).LinSols) :
 
 /-- A solution to the pure U(1) accs satisfies the cubic ACCs. -/
 lemma pureU1_cube {n : ℕ} (S : (PureU1 n).Sols) :
-    ∑ i, (S.val i) ^ 3 = 0 := by
-  rw [← PureU1.accCube_explicit]
-  exact S.cubicSol
+    ∑ i, (S.val i) ^ 3 = 0 := (PureU1.accCube_explicit n S.val) ▸ S.cubicSol
 
 /-- The last charge of a solution to the linear ACCs is equal to the negation of the sum
   of the other charges. -/
@@ -151,8 +149,7 @@ lemma pureU1_anomalyFree_ext {n : ℕ} {S T : (PureU1 n.succ).LinSols}
   · obtain ⟨j, hj⟩ := hi
     subst hj
     exact h j
-  · rw [hi, pureU1_last, pureU1_last]
-    exact neg_inj.mpr (Finset.sum_congr rfl fun j _ => h j)
+  · simpa [hi, pureU1_last] using neg_inj.mpr (Finset.sum_congr rfl fun j _ => h j)
 
 namespace PureU1
 
@@ -164,8 +161,7 @@ lemma sum_of_charges {n : ℕ} (f : Fin k → (PureU1 n).Charges) (j : Fin n) :
   · rfl
   · rename_i k hl
     rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
-    erw [← hl (f ∘ Fin.castSucc)]
-    rfl
+    exact congrArg₂ (· + ·) (hl (f ∘ Fin.castSucc)) rfl
 
 /-- The `j`th charge of a sum of solutions to the linear ACC is equal to the sum of
   their `j`th charges. -/
@@ -175,7 +171,6 @@ lemma sum_of_anomaly_free_linear {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j
   · rfl
   · rename_i k hl
     rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
-    erw [← hl (f ∘ Fin.castSucc)]
-    rfl
+    exact congrArg₂ (· + ·) (hl (f ∘ Fin.castSucc)) rfl
 
 end PureU1
diff --git a/Physlib/QFT/QED/AnomalyCancellation/BasisLinear.lean b/Physlib/QFT/QED/AnomalyCancellation/BasisLinear.lean
index 4b9f954ae..ba277ced6 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/BasisLinear.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/BasisLinear.lean
@@ -33,8 +33,7 @@ def asCharges (j : Fin n) : (PureU1 n.succ).Charges :=
       else 0)
 
 lemma asCharges_eq_castSucc (j : Fin n) :
-    asCharges j (Fin.castSucc j) = 1 := by
-  simp [asCharges]
+    asCharges j (Fin.castSucc j) = 1 := by simp [asCharges]
 
 lemma asCharges_ne_castSucc {k j : Fin n} (h : k ≠ j) :
     asCharges k ⟨j, by simp⟩= 0 := by
@@ -51,8 +50,7 @@ def asLinSols (j : Fin n) : (PureU1 n.succ).LinSols :=
     | ⟨0, _⟩=>
     simp only [PureU1_linearACCs, accGrav,
       LinearMap.coe_mk, AddHom.coe_mk]
-    rw [Fin.sum_univ_castSucc]
-    rw [Finset.sum_eq_single j]
+    rw [Fin.sum_univ_castSucc, Finset.sum_eq_single j]
     · simp only [asCharges, ↓reduceIte]
       have hn : ¬ (Fin.last n = Fin.castSucc j) := Fin.ne_of_gt j.prop
       split
@@ -72,12 +70,8 @@ lemma sum_of_vectors {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
 noncomputable
 def coordinateMap : (PureU1 n.succ).LinSols ≃ₗ[ℚ] Fin n →₀ ℚ where
   toFun S := (Finsupp.linearEquivFunOnFinite ℚ ℚ (Fin n)).symm (S.1 ∘ Fin.castSucc)
-  map_add' S T := by
-    rw [← map_add]
-    rfl
-  map_smul' a S := by
-    rw [← map_smul]
-    rfl
+  map_add' S T := map_add _ _ _
+  map_smul' a S := (Finsupp.linearEquivFunOnFinite ℚ ℚ (Fin n)).symm.map_smul a (S.1 ∘ Fin.castSucc)
   invFun f := ∑ i : Fin n, f i • asLinSols i
   left_inv S := by
     simp only [Nat.succ_eq_add_one,
@@ -89,8 +83,7 @@ def coordinateMap : (PureU1 n.succ).LinSols ≃ₗ[ℚ] Fin n →₀ ℚ where
     rw [Finset.sum_eq_single j]
     · simp only [asCharges, ↓reduceIte, mul_one]
     · intro k _ hkj
-      erw [asCharges_ne_castSucc hkj]
-      exact Rat.mul_zero (S.val k.castSucc)
+      exact mul_eq_zero_of_right (S.val k.castSucc) (asCharges_ne_castSucc hkj)
     · simp
   right_inv f := by
     simp only
@@ -103,8 +96,7 @@ def coordinateMap : (PureU1 n.succ).LinSols ≃ₗ[ℚ] Fin n →₀ ℚ where
     rw [Finset.sum_eq_single j]
     · simp only [asCharges, ↓reduceIte, mul_one]
     · intro k _ hkj
-      erw [asCharges_ne_castSucc hkj]
-      exact Rat.mul_zero (f k)
+      exact mul_eq_zero_of_right (f k) (asCharges_ne_castSucc hkj)
     · simp
 
 /-- The basis of `LinSols`. -/
diff --git a/Physlib/QFT/QED/AnomalyCancellation/ConstAbs.lean b/Physlib/QFT/QED/AnomalyCancellation/ConstAbs.lean
index 13008404c..660d09e74 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/ConstAbs.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/ConstAbs.lean
@@ -41,8 +41,7 @@ lemma constAbs_perm (S : (PureU1 n).Charges) (M :(FamilyPermutations n).group) :
   exact h2
 
 lemma constAbs_sort {S : (PureU1 n).Charges} (CA : ConstAbs S) : ConstAbs (sort S) := by
-  rw [sort]
-  exact (constAbs_perm S _).mpr CA
+  simpa [sort] using (constAbs_perm S _).mpr CA
 
 /-- The condition for a set of charges to be `sorted`, and have `constAbs`-/
 def ConstAbsSorted (S : (PureU1 n).Charges) : Prop := ConstAbs S ∧ Sorted S
@@ -63,10 +62,8 @@ lemma lt_eq {k i : Fin n.succ} (hk : S k ≤ 0) (hik : i ≤ k) : S i = S k := b
   · linarith
 
 include hS in
-lemma val_le_zero {i : Fin n.succ} (hi : S i ≤ 0) : S i = S (0 : Fin n.succ) := by
-  symm
-  apply lt_eq hS hi
-  exact Fin.zero_le i
+lemma val_le_zero {i : Fin n.succ} (hi : S i ≤ 0) : S i = S (0 : Fin n.succ) :=
+  (lt_eq hS hi (Fin.zero_le i)).symm
 
 include hS in
 lemma gt_eq {k i: Fin n.succ} (hk : 0 ≤ S k) (hik : k ≤ i) : S i = S k := by
@@ -115,8 +112,7 @@ lemma boundary_succ {k : Fin n} (hk : Boundary S k) : S k.succ = - S (0 : Fin n.
   rw [opposite_signs_eq_neg hS (le_of_lt hk.left) (le_of_lt hk.right)] at hn
   linear_combination -(1 * hn)
 
-lemma boundary_split (k : Fin n) : k.succ.val + (n.succ - k.succ.val) = n.succ := by
-  omega
+lemma boundary_split (k : Fin n) : k.succ.val + (n.succ - k.succ.val) = n.succ := by omega
 
 set_option backward.isDefEq.respectTransparency false in
 lemma boundary_accGrav' (k : Fin n) : accGrav n.succ S =
@@ -131,8 +127,7 @@ set_option backward.isDefEq.respectTransparency false in
 include hS in
 lemma boundary_accGrav'' (k : Fin n) (hk : Boundary S k) :
     accGrav n.succ S = (2 * ↑↑k + 1 - ↑n) * S (0 : Fin n.succ) := by
-  rw [boundary_accGrav' k]
-  rw [Fin.sum_univ_add]
+  rw [boundary_accGrav' k, Fin.sum_univ_add]
   have hfst (i : Fin k.succ.val) :
       S (Fin.cast (boundary_split k) (Fin.castAdd (n.succ - k.succ.val) i)) = S k.castSucc := by
     apply lt_eq hS (le_of_lt hk.left) (Fin.is_le i)
@@ -207,9 +202,8 @@ lemma AFL_odd_zero {A : (PureU1 (2 * n + 1)).LinSols} (h : ConstAbsSorted A.val)
   exact (AFL_odd_noBoundary h hn) (AFL_hasBoundary h hn)
 
 theorem AFL_odd (A : (PureU1 (2 * n + 1)).LinSols) (h : ConstAbsSorted A.val) :
-    A = 0 := by
-  apply ACCSystemLinear.LinSols.ext
-  exact is_zero h (AFL_odd_zero h)
+    A = 0 :=
+  ACCSystemLinear.LinSols.ext (is_zero h (AFL_odd_zero h))
 
 lemma AFL_even_Boundary {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.val)
     (hA : A.val (0 : Fin (2 * n.succ)) ≠ 0) {k : Fin (2 * n + 1)} (hk : Boundary A.val k) :
@@ -232,16 +226,14 @@ lemma AFL_even_below' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.
   apply lt_eq h (le_of_lt hk.left)
   rw [Fin.le_def]
   simp only [Fin.val_cast, Fin.val_castAdd, mul_eq, Fin.val_castSucc]
-  rw [AFL_even_Boundary h hA hk]
-  exact Fin.is_le i
+  simpa [AFL_even_Boundary h hA hk] using Fin.is_le i
 
 lemma AFL_even_below (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.val)
     (i : Fin n.succ) :
     A.val (Fin.cast (split_equal n.succ) (Fin.castAdd n.succ i))
     = A.val (0 : Fin (2*n.succ)) := by
   by_cases hA : A.val (0 : Fin (2*n.succ)) = 0
-  · rw [is_zero h hA]
-    rfl
+  · exact (congrFun (is_zero h hA) _).trans hA.symm
   · exact AFL_even_below' h hA i
 
 lemma AFL_even_above' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.val)
@@ -253,8 +245,7 @@ lemma AFL_even_above' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.
   apply gt_eq h (le_of_lt hk.right)
   rw [Fin.le_def]
   simp only [mul_eq, Fin.val_succ, Fin.val_cast, Fin.val_natAdd]
-  rw [AFL_even_Boundary h hA hk]
-  exact Nat.le_add_right (n + 1) ↑i
+  simp [AFL_even_Boundary h hA hk]
 
 lemma AFL_even_above (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.val)
     (i : Fin n.succ) :
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Even/BasisLinear.lean b/Physlib/QFT/QED/AnomalyCancellation/Even/BasisLinear.lean
index aed41de11..45965bc82 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Even/BasisLinear.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Even/BasisLinear.lean
@@ -152,8 +152,7 @@ lemma sum_even (S : Fin (2 * n.succ) → ℚ) :
 
 -/
 
-lemma n_cond₂ (n : ℕ) : 1 + ((n + n) + 1) = 2 * n.succ := by
-  linarith
+lemma n_cond₂ (n : ℕ) : 1 + ((n + n) + 1) = 2 * n.succ := by linarith
 
 /-- The inclusion of `Fin n` into `Fin (1 + (n + n + 1))` via the first `n`,
   casted into `Fin (2 * n.succ)`. -/
@@ -298,8 +297,7 @@ lemma basis_on_evenSnd_self (j : Fin n.succ) : basisAsCharges j (evenSnd j) = -
   rw [basis_evenSnd_eq_neg_evenFst, basis_on_evenFst_self]
 
 lemma basis_on_evenSnd_other {k j : Fin n.succ} (h : k ≠ j) : basisAsCharges k (evenSnd j) = 0 := by
-  rw [basis_evenSnd_eq_neg_evenFst, basis_on_evenFst_other h]
-  rfl
+  simp [basis_evenSnd_eq_neg_evenFst, basis_on_evenFst_other h]
 
 /-!
 
@@ -358,11 +356,9 @@ lemma P_evenFst (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (evenFst j) = f
   rw [P, sum_of_charges]
   simp only [succ_eq_add_one, HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis_on_evenFst_self]
-    exact Rat.mul_one (f j)
+  · simp [basis_on_evenFst_self]
   · intro k _ hkj
-    rw [basis_on_evenFst_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis_on_evenFst_other hkj)
   · simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
 
 lemma P_evenSnd (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (evenSnd j) = - f j := by
@@ -407,8 +403,7 @@ lemma P_accCube (f : Fin n.succ → ℚ) : accCube (2 * n.succ) (P f) = 0 := by
 lemma P_zero (f : Fin n.succ → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
   intro i
   erw [← P_evenFst f]
-  rw [h]
-  rfl
+  exact congr_fun h (evenFst i)
 
 /-!
 
@@ -422,8 +417,7 @@ def P' (f : Fin n.succ → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i 
 lemma P'_val (f : Fin n.succ → ℚ) : (P' f).val = P f := by
   simp only [succ_eq_add_one, P', P]
   funext i
-  rw [sum_of_anomaly_free_linear, sum_of_charges]
-  rfl
+  exact (sum_of_anomaly_free_linear _ i).trans (sum_of_charges _ i).symm
 
 /-!
 
@@ -461,8 +455,7 @@ lemma vectorLikeEven_in_span (S : (PureU1 (2 * n.succ)).LinSols)
   erw [P'_val]
   apply ext_even
   · intro i
-    rw [P_evenFst]
-    rfl
+    exact P_evenFst f i
   · intro i
     rw [P_evenSnd]
     have ht := hS i
@@ -470,8 +463,7 @@ lemma vectorLikeEven_in_span (S : (PureU1 (2 * n.succ)).LinSols)
     have h : sort S.val (evenSnd i) = - sort S.val (evenFst i) := by
       rw [ht]
       ring
-    rw [h]
-    rfl
+    exact h.symm
 
 /-!
 
@@ -550,8 +542,7 @@ lemma basis!_on_evenShiftSnd_self (j : Fin n) : basis!AsCharges j (evenShiftSnd
 
 lemma basis!_on_evenShiftSnd_other {k j : Fin n} (h : k ≠ j) :
     basis!AsCharges k (evenShiftSnd j) = 0 := by
-  rw [basis!_evenShftSnd_eq_neg_evenShiftFst, basis!_on_evenShiftFst_other h]
-  rfl
+  simp [basis!_evenShftSnd_eq_neg_evenShiftFst, basis!_on_evenShiftFst_other h]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma basis!_on_evenShiftZero (j : Fin n) : basis!AsCharges j evenShiftZero = 0 := by
@@ -601,8 +592,7 @@ lemma basis!_linearACC (j : Fin n) : (accGrav (2 * n.succ)) (basis!AsCharges j)
 
 lemma basis!_accCube (j : Fin n) :
     accCube (2 * n.succ) (basis!AsCharges j) = 0 := by
-  rw [accCube_explicit, sum_evenShift]
-  rw [basis!_on_evenShiftLast, basis!_on_evenShiftZero]
+  rw [accCube_explicit, sum_evenShift, basis!_on_evenShiftLast, basis!_on_evenShiftZero]
   simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, add_zero, Function.comp_apply,
     zero_add]
   apply Finset.sum_eq_zero
@@ -643,22 +633,18 @@ lemma P!_evenShiftFst (f : Fin n → ℚ) (j : Fin n) : P! f (evenShiftFst j) =
   rw [P!, sum_of_charges]
   simp only [HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis!_on_evenShiftFst_self]
-    exact Rat.mul_one (f j)
+  · simp [basis!_on_evenShiftFst_self]
   · intro k _ hkj
-    rw [basis!_on_evenShiftFst_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis!_on_evenShiftFst_other hkj)
   · simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
 
 lemma P!_evenShiftSnd (f : Fin n → ℚ) (j : Fin n) : P! f (evenShiftSnd j) = - f j := by
   rw [P!, sum_of_charges]
   simp only [HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis!_on_evenShiftSnd_self]
-    exact mul_neg_one (f j)
+  · simp [basis!_on_evenShiftSnd_self]
   · intro k _ hkj
-    rw [basis!_on_evenShiftSnd_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis!_on_evenShiftSnd_other hkj)
   · simp
 
 lemma P!_evenShiftZero (f : Fin n → ℚ) : P! f (evenShiftZero) = 0 := by
@@ -693,8 +679,7 @@ lemma P!_accCube (f : Fin n → ℚ) : accCube (2 * n.succ) (P! f) = 0 := by
 lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
   intro i
   rw [← P!_evenShiftFst f]
-  rw [h]
-  rfl
+  exact congr_fun h (evenShiftFst i)
 
 /-!
 
@@ -719,8 +704,7 @@ def P!' (f : Fin n → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i • ba
 lemma P!'_val (f : Fin n → ℚ) : (P!' f).val = P! f := by
   simp only [succ_eq_add_one, P!', P!]
   funext i
-  rw [sum_of_anomaly_free_linear, sum_of_charges]
-  rfl
+  exact (sum_of_anomaly_free_linear _ i).trans (sum_of_charges _ i).symm
 
 /-!
 
@@ -917,8 +901,7 @@ def Pa' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n.succ)).LinSols :
     ∑ i, f i • basisa i
 
 lemma Pa'_P'_P!' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) :
-    Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := by
-  exact Fintype.sum_sum_type _
+    Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := Fintype.sum_sum_type _
 
 /-!
 
@@ -956,9 +939,7 @@ lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f =
   · rw [Pa', Pa'] at h
     have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by
       simp only [add_smul, neg_smul]
-      rw [Finset.sum_add_distrib]
-      rw [h]
-      rw [← Finset.sum_add_distrib]
+      rw [Finset.sum_add_distrib, h, ← Finset.sum_add_distrib]
       simp
     have h2 : ∀ i, (f i + (- f' i)) = 0 := by
       exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent n)
@@ -979,8 +960,7 @@ lemma Pa'_elim_eq_iff (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
 
 lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
     Pa g f = Pa g' f' ↔ g = g' ∧ f = f' := by
-  rw [← Pa'_elim_eq_iff, ← Sum.elim_eq_iff]
-  exact Pa'_eq _ _
+  simpa [← Pa'_elim_eq_iff, ← Sum.elim_eq_iff] using Pa'_eq _ _
 
 /-!
 
@@ -1036,10 +1016,8 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
       S'.val = P g' + P! f' ∧ P! f' = P! f +
       (S.val (evenShiftSnd j) - S.val (evenShiftFst j)) • basis!AsCharges j ∧ g' = g := by
   let X := P! f + (S.val (evenShiftSnd j) - S.val (evenShiftFst j)) • basis!AsCharges j
-  have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
-    apply Submodule.add_mem
-    exact (P!_in_span f)
-    exact (smul_basis!AsCharges_in_span S j)
+  have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) :=
+    Submodule.add_mem _ (P!_in_span f) (smul_basis!AsCharges_in_span S j)
   have hXsum := (Submodule.mem_span_range_iff_exists_fun ℚ).mp hX
   obtain ⟨f', hf'⟩ := hXsum
   use g
@@ -1047,9 +1025,7 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
   change P! f' = _ at hf'
   erw [hf']
   simp only [and_self, and_true, X]
-  rw [← add_assoc, ← h]
-  apply swap!_as_add at hS
-  exact hS
+  simpa [← add_assoc, ← h] using swap!_as_add j hS
 
 end VectorLikeEvenPlane
 
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Even/Parameterization.lean b/Physlib/QFT/QED/AnomalyCancellation/Even/Parameterization.lean
index 5c469ac08..db61c4261 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Even/Parameterization.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Even/Parameterization.lean
@@ -108,8 +108,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
     GenericCase S ∨ SpecialCase S := by
   obtain ⟨g, f, h⟩ := span_basis S.1.1
   have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
-    accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
-    exact ne_or_eq _ _
+    accCubeTriLinSymm (P g) (P g) (P! f) = 0 := ne_or_eq _ _
   rcases h1 with h1 | h1
   · exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
   · exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
@@ -121,8 +120,7 @@ theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
   rw [parameterization]
   apply ACCSystem.Sols.ext
   rw [parameterizationAsLinear_val]
-  rw [anomalyFree_param _ _ hS]
-  rw [neg_neg, ← smul_add, smul_smul, inv_mul_cancel₀, one_smul]
+  rw [anomalyFree_param _ _ hS, neg_neg, ← smul_add, smul_smul, inv_mul_cancel₀, one_smul]
   · exact hS
   · have h := h g f hS
     rw [anomalyFree_param _ _ hS] at h
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Odd/BasisLinear.lean b/Physlib/QFT/QED/AnomalyCancellation/Odd/BasisLinear.lean
index b1e743abf..2323ae1d6 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Odd/BasisLinear.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Odd/BasisLinear.lean
@@ -106,8 +106,7 @@ section theDeltas
 
 -/
 
-lemma odd_shift_eq (n : ℕ) : (1 + n) + n = 2 * n +1 := by
-  omega
+lemma odd_shift_eq (n : ℕ) : (1 + n) + n = 2 * n +1 := by omega
 
 /-- The inclusion of `Fin n` into `Fin ((n + 1) + n)` via the first `n`.
   This is then casted to `Fin (2 * n + 1)`. -/
@@ -131,12 +130,10 @@ lemma sum_odd (S : Fin (2 * n + 1) → ℚ) :
     · intro i
       simp only [mem_univ, Fin.symm_castOrderIso, RelIso.coe_fn_toEquiv]
     · exact fun _ _ => rfl
-  rw [h1]
-  rw [Fin.sum_univ_add, Fin.sum_univ_add]
+  rw [h1, Fin.sum_univ_add, Fin.sum_univ_add]
   simp only [univ_unique, Fin.default_eq_zero, Fin.isValue, sum_singleton, Function.comp_apply]
   nth_rewrite 2 [add_comm]
-  rw [add_assoc]
-  rw [Finset.sum_add_distrib]
+  rw [add_assoc, Finset.sum_add_distrib]
   rfl
 
 /-!
@@ -178,8 +175,7 @@ lemma sum_oddShift (S : Fin (2 * n + 1) → ℚ) :
 
 -/
 
-lemma odd_shift_shift_eq (n : ℕ) : ((1+n)+1) + n.succ = 2 * n.succ + 1 := by
-  omega
+lemma odd_shift_shift_eq (n : ℕ) : ((1+n)+1) + n.succ = 2 * n.succ + 1 := by omega
 
 /-- The element representing the first `1` in `Fin (1 + n + 1 + n.succ)` casted
   to `Fin (2 * n.succ + 1)`. -/
@@ -219,8 +215,7 @@ lemma oddShiftShiftFst_eq_oddFst_succ (j : Fin n) :
   exact Nat.add_comm 1 ↑j
 
 lemma oddShiftShiftFst_eq_oddShiftFst_castSucc (j : Fin n) :
-    oddShiftShiftFst j = oddShiftFst j.castSucc := by
-  rfl
+    oddShiftShiftFst j = oddShiftFst j.castSucc := rfl
 
 lemma oddShiftShiftMid_eq_oddMid : @oddShiftShiftMid n = oddMid := by
   rw [Fin.ext_iff]
@@ -228,17 +223,14 @@ lemma oddShiftShiftMid_eq_oddMid : @oddShiftShiftMid n = oddMid := by
     Fin.val_natAdd, Fin.val_eq_zero, add_zero, oddMid]
   exact Nat.add_comm 1 n
 
-lemma oddShiftShiftMid_eq_oddShiftFst_last : oddShiftShiftMid = oddShiftFst (Fin.last n) := by
-  rfl
+lemma oddShiftShiftMid_eq_oddShiftFst_last : oddShiftShiftMid = oddShiftFst (Fin.last n) := rfl
 
 lemma oddShiftShiftSnd_eq_oddSnd (j : Fin n.succ) : oddShiftShiftSnd j = oddSnd j := by
   rw [Fin.ext_iff]
   simp only [succ_eq_add_one, oddShiftShiftSnd, Fin.val_cast, Fin.val_natAdd, oddSnd, add_left_inj]
   exact Nat.add_comm 1 n
 
-lemma oddShiftShiftSnd_eq_oddShiftSnd (j : Fin n.succ) : oddShiftShiftSnd j = oddShiftSnd j := by
-  rw [Fin.ext_iff]
-  rfl
+lemma oddShiftShiftSnd_eq_oddShiftSnd (j : Fin n.succ) : oddShiftShiftSnd j = oddShiftSnd j := rfl
 
 lemma oddSnd_eq_oddShiftSnd (j : Fin n) : oddSnd j = oddShiftSnd j := by
   rw [Fin.ext_iff]
@@ -296,8 +288,7 @@ def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).Charges :=
 
 -/
 
-lemma basis_on_oddFst_self (j : Fin n) : basisAsCharges j (oddFst j) = 1 := by
-  simp [basisAsCharges]
+lemma basis_on_oddFst_self (j : Fin n) : basisAsCharges j (oddFst j) = 1 := by simp [basisAsCharges]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma basis_on_oddFst_other {k j : Fin n} (h : k ≠ j) :
@@ -349,8 +340,7 @@ lemma basis_on_oddSnd_self (j : Fin n) : basisAsCharges j (oddSnd j) = - 1 := by
   rw [basis_oddSnd_eq_minus_oddFst, basis_on_oddFst_self]
 
 lemma basis_on_oddSnd_other {k j : Fin n} (h : k ≠ j) : basisAsCharges k (oddSnd j) = 0 := by
-  rw [basis_oddSnd_eq_minus_oddFst, basis_on_oddFst_other h]
-  rfl
+  simp [basis_oddSnd_eq_minus_oddFst, basis_on_oddFst_other h]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma basis_on_oddMid (j : Fin n) : basisAsCharges j oddMid = 0 := by
@@ -412,22 +402,18 @@ lemma P_oddFst (f : Fin n → ℚ) (j : Fin n) : P f (oddFst j) = f j := by
   rw [P, sum_of_charges]
   simp only [HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis_on_oddFst_self]
-    exact Rat.mul_one (f j)
+  · simp [basis_on_oddFst_self]
   · intro k _ hkj
-    rw [basis_on_oddFst_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis_on_oddFst_other hkj)
   · simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
 
 lemma P_oddSnd (f : Fin n → ℚ) (j : Fin n) : P f (oddSnd j) = - f j := by
   rw [P, sum_of_charges]
   simp only [HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis_on_oddSnd_self]
-    exact mul_neg_one (f j)
+  · simp [basis_on_oddSnd_self]
   · intro k _ hkj
-    rw [basis_on_oddSnd_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis_on_oddSnd_other hkj)
   · simp
 
 lemma P_oddMid (f : Fin n → ℚ) : P f oddMid = 0 := by
@@ -463,8 +449,7 @@ lemma P_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P f) = 0 := by
 lemma P_zero (f : Fin n → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
   intro i
   erw [← P_oddFst f]
-  rw [h]
-  rfl
+  exact congr_fun h (oddFst i)
 
 /-- A point in the span of the first part of the basis. -/
 def P' (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basis i
@@ -472,8 +457,7 @@ def P' (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basi
 lemma P'_val (f : Fin n → ℚ) : (P' f).val = P f := by
   simp only [P', P]
   funext i
-  rw [sum_of_anomaly_free_linear, sum_of_charges]
-  rfl
+  exact (sum_of_anomaly_free_linear _ i).trans (sum_of_charges _ i).symm
 
 /-!
 
@@ -535,13 +519,12 @@ lemma basis!_on_oddShiftFst_other {k j : Fin n} (h : k ≠ j) :
     simp_all
     rw [Fin.ext_iff] at h
     simp_all
-  · split
-    · rename_i h1 h2
-      simp_all
-      rw [Fin.ext_iff] at h2
-      simp only [Fin.val_castAdd, Fin.val_natAdd] at h2
-      omega
-    rfl
+  · split <;> try rfl
+    rename_i h1 h2
+    simp_all
+    rw [Fin.ext_iff] at h2
+    simp only [Fin.val_castAdd, Fin.val_natAdd] at h2
+    omega
 
 set_option backward.isDefEq.respectTransparency false in
 lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)}
@@ -573,8 +556,7 @@ lemma basis!_on_oddShiftSnd_self (j : Fin n) : basis!AsCharges j (oddShiftSnd j)
 
 lemma basis!_on_oddShiftSnd_other {k j : Fin n} (h : k ≠ j) :
     basis!AsCharges k (oddShiftSnd j) = 0 := by
-  rw [basis!_oddShiftSnd_eq_minus_oddShiftFst, basis!_on_oddShiftFst_other h]
-  rfl
+  simp [basis!_oddShiftSnd_eq_minus_oddShiftFst, basis!_on_oddShiftFst_other h]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma basis!_on_oddShiftZero (j : Fin n) : basis!AsCharges j oddShiftZero = 0 := by
@@ -661,22 +643,18 @@ lemma P!_oddShiftFst (f : Fin n → ℚ) (j : Fin n) : P! f (oddShiftFst j) = f
   rw [P!, sum_of_charges]
   simp only [HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis!_on_oddShiftFst_self]
-    exact Rat.mul_one (f j)
+  · simp [basis!_on_oddShiftFst_self]
   · intro k _ hkj
-    rw [basis!_on_oddShiftFst_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis!_on_oddShiftFst_other hkj)
   · simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
 
 lemma P!_oddShiftSnd (f : Fin n → ℚ) (j : Fin n) : P! f (oddShiftSnd j) = - f j := by
   rw [P!, sum_of_charges]
   simp only [HSMul.hSMul, SMul.smul]
   rw [Finset.sum_eq_single j]
-  · rw [basis!_on_oddShiftSnd_self]
-    exact mul_neg_one (f j)
+  · simp [basis!_on_oddShiftSnd_self]
   · intro k _ hkj
-    rw [basis!_on_oddShiftSnd_other hkj]
-    exact Rat.mul_zero (f k)
+    exact mul_eq_zero_of_right (f k) (basis!_on_oddShiftSnd_other hkj)
   · simp
 
 lemma P!_oddShiftZero (f : Fin n → ℚ) : P! f oddShiftZero = 0 := by
@@ -712,8 +690,7 @@ lemma P!_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P! f) = 0 := by
 lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
   intro i
   rw [← P!_oddShiftFst f]
-  rw [h]
-  rfl
+  exact congr_fun h (oddShiftFst i)
 
 /-!
 
@@ -727,8 +704,7 @@ def P!' (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • bas
 lemma P!'_val (f : Fin n → ℚ) : (P!' f).val = P! f := by
   simp only [P!', P!]
   funext i
-  rw [sum_of_anomaly_free_linear, sum_of_charges]
-  rfl
+  exact (sum_of_anomaly_free_linear _ i).trans (sum_of_charges _ i).symm
 
 /-!
 
@@ -878,8 +854,7 @@ def Pa' (f : (Fin n) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n + 1)).LinSols :=
     ∑ i, f i • basisa i
 
 lemma Pa'_P'_P!' (f : (Fin n) ⊕ (Fin n) → ℚ) :
-    Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := by
-  exact Fintype.sum_sum_type _
+    Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := Fintype.sum_sum_type _
 
 /-!
 
@@ -942,9 +917,7 @@ lemma Pa'_elim_eq_iff (g g' : Fin n.succ → ℚ) (f f' : Fin n.succ → ℚ) :
 
 lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n.succ → ℚ) :
     Pa g f = Pa g' f' ↔ g = g' ∧ f = f' := by
-  rw [← Pa'_elim_eq_iff]
-  rw [← Sum.elim_eq_iff]
-  exact Pa'_eq _ _
+  simpa [← Pa'_elim_eq_iff, ← Sum.elim_eq_iff] using Pa'_eq _ _
 
 /-!
 
@@ -1011,10 +984,8 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
     apply SetLike.mem_of_subset
     apply Submodule.subset_span
     simp_all only [Set.mem_range, exists_apply_eq_apply]
-  have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
-    apply Submodule.add_mem
-    exact hf
-    exact hP
+  have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) :=
+    Submodule.add_mem _ hf hP
   have hXsum := (Submodule.mem_span_range_iff_exists_fun ℚ).mp hX
   obtain ⟨f', hf'⟩ := hXsum
   use g
@@ -1022,9 +993,7 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
   change P! f' = _ at hf'
   erw [hf']
   simp only [and_self, and_true, X]
-  rw [← add_assoc, ← hS1]
-  apply swap!_as_add at hS
-  exact hS
+  simpa [← add_assoc, ← hS1] using swap!_as_add j hS
 
 end VectorLikeOddPlane
 
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Odd/Parameterization.lean b/Physlib/QFT/QED/AnomalyCancellation/Odd/Parameterization.lean
index 567fbb4a6..38489f821 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Odd/Parameterization.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Odd/Parameterization.lean
@@ -46,9 +46,7 @@ set_option backward.isDefEq.respectTransparency false in
 lemma parameterizationCharge_cube (g f : Fin n → ℚ) (a : ℚ) :
     accCube (2 * n + 1) (parameterizationAsLinear g f a).val = 0 := by
   change accCubeTriLinSymm.toCubic _ = 0
-  rw [parameterizationAsLinear_val]
-  rw [HomogeneousCubic.map_smul]
-  rw [TriLinearSymm.toCubic_add]
+  rw [parameterizationAsLinear_val, HomogeneousCubic.map_smul, TriLinearSymm.toCubic_add]
   rw [HomogeneousCubic.map_smul, HomogeneousCubic.map_smul]
   erw [P_accCube g, P!_accCube f]
   rw [accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂,
@@ -110,8 +108,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
     GenericCase S ∨ SpecialCase S := by
   obtain ⟨g, f, h⟩ := span_basis S.1.1
   have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
-      accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
-    exact ne_or_eq _ _
+      accCubeTriLinSymm (P g) (P g) (P! f) = 0 := ne_or_eq _ _
   cases h1 <;> rename_i h1
   · exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
   · exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Permutations.lean b/Physlib/QFT/QED/AnomalyCancellation/Permutations.lean
index 221f1e0bc..4041e08bc 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Permutations.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Permutations.lean
@@ -80,13 +80,11 @@ def FamilyPermutations (n : ℕ) : ACCSystemGroupAction (PureU1 n) where
 
 lemma FamilyPermutations_charges_apply (S : (PureU1 n).Charges)
     (i : Fin n) (f : (FamilyPermutations n).group) :
-    ((FamilyPermutations n).rep f S) i = S (f.invFun i) := by
-  rfl
+    ((FamilyPermutations n).rep f S) i = S (f.invFun i) := rfl
 
 lemma FamilyPermutations_anomalyFreeLinear_apply (S : (PureU1 n).LinSols)
     (i : Fin n) (f : (FamilyPermutations n).group) :
-    ((FamilyPermutations n).linSolRep f S).val i = S.val (f.invFun i) := by
-  rfl
+    ((FamilyPermutations n).linSolRep f S).val i = S.val (f.invFun i) := rfl
 
 /-- A permutation of fermions which takes one ordered subset into another. -/
 noncomputable def permOfInjection (f g : Fin m ↪ Fin n) : (FamilyPermutations n).group :=
@@ -101,8 +99,7 @@ def permTwoInj : Fin 2 ↪ Fin n where
   toFun s := match s with
     | 0 => i
     | 1 => j
-  inj' s1 s2 := by
-    aesop
+  inj' s1 s2 := by aesop
 
 lemma permTwoInj_fst : i ∈ Set.range ⇑(permTwoInj hij) := by
   simp only [Set.mem_range]
@@ -158,8 +155,7 @@ def permThreeInj : Fin 3 ↪ Fin n where
     | 0 => i
     | 1 => j
     | 2 => k
-  inj' s1 s2 := by
-    aesop
+  inj' s1 s2 := by aesop
 
 lemma permThreeInj_fst : i ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
   simp only [Set.mem_range]
diff --git a/Physlib/QFT/QED/AnomalyCancellation/Sorts.lean b/Physlib/QFT/QED/AnomalyCancellation/Sorts.lean
index e28cd3136..5befe83f3 100644
--- a/Physlib/QFT/QED/AnomalyCancellation/Sorts.lean
+++ b/Physlib/QFT/QED/AnomalyCancellation/Sorts.lean
@@ -43,8 +43,7 @@ lemma sort_perm {n : ℕ} (S : (PureU1 n).Charges) (M :(FamilyPermutations n).gr
   @Tuple.comp_perm_comp_sort_eq_comp_sort n ℚ _ S M⁻¹
 
 lemma sort_apply {n : ℕ} (S : (PureU1 n).Charges) (j : Fin n) :
-    sort S j = S ((Tuple.sort S) j) := by
-  rfl
+    sort S j = S ((Tuple.sort S) j) := rfl
 
 lemma sort_zero {n : ℕ} (S : (PureU1 n).Charges) (hS : sort S = 0) : S = 0 := by
   funext i
@@ -55,8 +54,7 @@ lemma sort_zero {n : ℕ} (S : (PureU1 n).Charges) (hS : sort S = 0) : S = 0 :=
   have hi := hj ((Tuple.sort S).invFun i)
   rw [sort_apply] at hi
   simp only [Equiv.invFun_as_coe, Equiv.apply_symm_apply] at hi
-  rw [hi]
-  rfl
+  exact hi
 
 lemma sort_projection {n : ℕ} (S : (PureU1 n).Charges) : sort (sort S) = sort S :=
   sort_perm S (Tuple.sort S).symm
@@ -65,15 +63,13 @@ lemma sort_projection {n : ℕ} (S : (PureU1 n).Charges) : sort (sort S) = sort
 def sortAFL {n : ℕ} (S : (PureU1 n).LinSols) : (PureU1 n).LinSols :=
   ((FamilyPermutations n).linSolRep (Tuple.sort S.val).symm S)
 
-lemma sortAFL_val {n : ℕ} (S : (PureU1 n).LinSols) : (sortAFL S).val = sort S.val := by
-  rfl
+lemma sortAFL_val {n : ℕ} (S : (PureU1 n).LinSols) : (sortAFL S).val = sort S.val := rfl
 
 lemma sortAFL_zero {n : ℕ} (S : (PureU1 n).LinSols) (hS : sortAFL S = 0) : S = 0 := by
   apply ACCSystemLinear.LinSols.ext
   have h1 : sort S.val = 0 := by
     rw [← sortAFL_val]
-    rw [hS]
-    rfl
+    exact congrArg ACCSystemLinear.LinSols.val hS
   exact sort_zero S.val h1
 
 end PureU1
diff --git a/Physlib/QuantumMechanics/DDimensions/Basic.lean b/Physlib/QuantumMechanics/DDimensions/Basic.lean
index 3c7f22b3a..cad943ed4 100644
--- a/Physlib/QuantumMechanics/DDimensions/Basic.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Basic.lean
@@ -114,8 +114,7 @@ lemma potentialCLM_apply (h_HTG : Q.potential.HasTemperateGrowth) (ψ : 𝓢(Spa
 @[simp]
 lemma potentialCLM_apply_apply
     (h_HTG : Q.potential.HasTemperateGrowth) (ψ : 𝓢(Space Q.d, ℂ)) (x : Space Q.d) :
-    Q.potentialCLM ψ x = Q.potential x • ψ x := by
-  rw [potentialCLM_apply h_HTG]
+    Q.potentialCLM ψ x = Q.potential x • ψ x := by rw [potentialCLM_apply h_HTG]
 
 /-- The potential operator as a self-adjoint, unbounded multiplication operator
   with domain `{ψ ∈ Q.HS | Q.potential • ψ ∈ Q.HS}`. -/
diff --git a/Physlib/QuantumMechanics/DDimensions/Hydrogen/Basic.lean b/Physlib/QuantumMechanics/DDimensions/Hydrogen/Basic.lean
index 9396e875d..d5d2fa985 100644
--- a/Physlib/QuantumMechanics/DDimensions/Hydrogen/Basic.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Hydrogen/Basic.lean
@@ -51,8 +51,7 @@ lemma potential_eq : H.potential = fun x ↦ -H.k * ‖x‖⁻¹ := H.coulomb_po
 
 @[fun_prop]
 lemma potential_AESM : AEStronglyMeasurable H.potential := by
-  rw [potential_eq]
-  exact AEMeasurable.aestronglyMeasurable (by fun_prop)
+  simpa [potential_eq] using AEMeasurable.aestronglyMeasurable (by fun_prop)
 
 @[fun_prop]
 lemma potential_AEM : AEMeasurable H.potential := H.potential_AESM.aemeasurable
diff --git a/Physlib/QuantumMechanics/DDimensions/Hydrogen/LaplaceRungeLenzVector.lean b/Physlib/QuantumMechanics/DDimensions/Hydrogen/LaplaceRungeLenzVector.lean
index 44d43347b..546cfd16f 100644
--- a/Physlib/QuantumMechanics/DDimensions/Hydrogen/LaplaceRungeLenzVector.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Hydrogen/LaplaceRungeLenzVector.lean
@@ -448,8 +448,7 @@ private lemma sum_Lprx (d : ℕ) (ε : ℝˣ) :
     calc
       _ = 𝐋 i j ∘L (𝐩 j ∘L 𝐱 i - 𝐩 i ∘L 𝐱 j) := by
         simp [angularMomentumOperator_antisymm j i, comp_assoc, sub_eq_add_neg]
-      _ = 𝐋 i j ∘L 𝐋 i j := by
-        simp [momentum_comp_position_eq, symm j i, angularMomentumOperator]
+      _ = 𝐋 i j ∘L 𝐋 i j := by simp [momentum_comp_position_eq, symm j i, angularMomentumOperator]
   rw [← angularMomentumSqr_comp_radiusRegPow_commute, angularMomentumOperatorSqr]
 
 private lemma sum_rxpL (d : ℕ) (ε : ℝˣ) :
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/Covariance.lean b/Physlib/QuantumMechanics/DDimensions/Operators/Covariance.lean
index aa5fb91c5..39ddee379 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/Covariance.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/Covariance.lean
@@ -62,8 +62,7 @@ def covariance : ℝ :=
 /-- Covariance, unfolded to the real part of the centered inner product. -/
 lemma covariance_eq_re_inner_centered :
     covariance A B ψ hψB =
-      (⟪centered A ψ, centered B ⟨ψ, hψB⟩⟫_ℂ).re :=
-  rfl
+      (⟪centered A ψ, centered B ⟨ψ, hψB⟩⟫_ℂ).re := rfl
 
 /-- Swapping the two observables does not change the covariance. -/
 lemma covariance_comm :
@@ -84,8 +83,7 @@ lemma covariance_eq_re_symm_centered :
       ((⟪centered A ψ, centered B ⟨ψ, hψB⟩⟫_ℂ +
         ⟪centered B ⟨ψ, hψB⟩, centered A ψ⟫_ℂ).re) / 2 := by
   let z : ℂ := ⟪centered A ψ, centered B ⟨ψ, hψB⟩⟫_ℂ
-  have hz : ⟪centered B ⟨ψ, hψB⟩, centered A ψ⟫_ℂ = star z := by
-    simp [z, inner_conj_symm]
+  have hz : ⟪centered B ⟨ψ, hψB⟩, centered A ψ⟫_ℂ = star z := by simp [z, inner_conj_symm]
   rw [covariance_eq_re_inner_centered, hz]
   change z.re = ((z + star z).re) / 2
   simp only [Complex.add_re, Complex.star_def, Complex.conj_re, add_self_div_two]
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/Momentum.lean b/Physlib/QuantumMechanics/DDimensions/Operators/Momentum.lean
index cd6fadb33..82531220e 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/Momentum.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/Momentum.lean
@@ -69,8 +69,7 @@ notation "𝐩[" d' "]" => momentumCLM (d := d')
 lemma momentumCLM_apply_fun (ψ : 𝓢(Space d, ℂ)) : 𝐩 i ψ = (-I * ℏ) • ∂[i] ψ := rfl
 
 @[simp]
-lemma momentumCLM_apply (ψ : 𝓢(Space d, ℂ)) (x : Space d) : 𝐩 i ψ x = -I * ℏ * ∂[i] ψ x :=
-  rfl
+lemma momentumCLM_apply (ψ : 𝓢(Space d, ℂ)) (x : Space d) : 𝐩 i ψ x = -I * ℏ * ∂[i] ψ x := rfl
 
 /-!
 
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/Multiplication.lean b/Physlib/QuantumMechanics/DDimensions/Operators/Multiplication.lean
index 67daf3b36..f56a9d797 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/Multiplication.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/Multiplication.lean
@@ -294,8 +294,7 @@ lemma mulOperator_adjoint_eq_conj {f : Space d → ℂ} (hf : AEStronglyMeasurab
 
 lemma mulOperator_isSelfAdjoint_ofReal
     {f : Space d → ℂ} (hf : AEStronglyMeasurable f) (hf' : conj ∘ f = f) :
-    IsSelfAdjoint (𝓜 f) := by
-  rw [isSelfAdjoint_def, mulOperator_adjoint_eq_conj hf, hf']
+    IsSelfAdjoint (𝓜 f) := by rw [isSelfAdjoint_def, mulOperator_adjoint_eq_conj hf, hf']
 
 /-!
 ## D. Closable & unbounded
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/Position.lean b/Physlib/QuantumMechanics/DDimensions/Operators/Position.lean
index faf088297..01ebe70b8 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/Position.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/Position.lean
@@ -110,14 +110,12 @@ notation "𝐫₀[" d' "]" => radiusRegPowCLM (d := d')
 lemma radiusRegPowCLM_apply_fun {d : ℕ} (ε : ℝˣ) (s : ℝ) (ψ : 𝓢(Space d, ℂ)) :
     𝐫₀ ε s ψ = fun x ↦ (‖x‖ ^ 2 + ε ^ 2) ^ (s / 2) • ψ x := by
   ext x
-  dsimp [radiusRegPowCLM]
   refine smulLeftCLM_apply_apply ?_ ψ x
   exact HasTemperateGrowth.comp (by fun_prop) (normRegularizedPow_hasTemperateGrowth d ε s)
 
 @[simp]
 lemma radiusRegPowCLM_apply {d : ℕ} (ε : ℝˣ) (s : ℝ) (ψ : 𝓢(Space d, ℂ)) (x : Space d) :
-    𝐫₀ ε s ψ x = (‖x‖ ^ 2 + ε ^ 2) ^ (s / 2) • ψ x := by
-  rw [radiusRegPowCLM_apply_fun]
+    𝐫₀ ε s ψ x = (‖x‖ ^ 2 + ε ^ 2) ^ (s / 2) • ψ x := by rw [radiusRegPowCLM_apply_fun]
 
 @[simp]
 lemma radiusRegPowCLM_comp_eq {d : ℕ} (ε : ℝˣ) (s t : ℝ) :
@@ -158,8 +156,7 @@ lemma radiusPowLM_apply_fun {d : ℕ} (s : ℝ) (ψ : 𝓢(Space d, ℂ)) :
 
 @[simp]
 lemma radiusPowLM_apply {d : ℕ} (s : ℝ) (ψ : 𝓢(Space d, ℂ)) (x : Space d) :
-    𝐫 s ψ x = ‖x‖ ^ s • ψ x := by
-  rw [radiusPowLM_apply_fun]
+    𝐫 s ψ x = ‖x‖ ^ s • ψ x := by rw [radiusPowLM_apply_fun]
 
 /-- `x ↦ ‖x‖ˢψ(x)` is smooth away from `x = 0`. -/
 @[fun_prop]
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean b/Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean
index d10a099b4..38d41d1f2 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean
@@ -150,11 +150,10 @@ lemma isLowerBound_closure
 def regularityDomain (T : H →ₗ.[ℂ] H) : Set ℂ := {z : ℂ | ∃ c > 0, IsLowerBound T z c}
 
 @[simp]
-lemma regularityDomain_neg (T : H →ₗ.[ℂ] H) : (-T).regularityDomain = -T.regularityDomain := by
-  ext z
-  constructor
-  · exact fun ⟨c, hc, h_bound⟩ ↦ ⟨c, hc, neg_neg T ▸ isLowerBound_neg h_bound⟩
-  · exact fun ⟨c, hc, h_bound⟩ ↦ ⟨c, hc, neg_neg z ▸ isLowerBound_neg h_bound⟩
+lemma regularityDomain_neg (T : H →ₗ.[ℂ] H) : (-T).regularityDomain = -T.regularityDomain :=
+  Set.ext fun z =>
+    ⟨fun ⟨c, hc, h_bound⟩ ↦ ⟨c, hc, neg_neg T ▸ isLowerBound_neg h_bound⟩,
+      fun ⟨c, hc, h_bound⟩ ↦ ⟨c, hc, neg_neg z ▸ isLowerBound_neg h_bound⟩⟩
 
 @[simp]
 lemma regularityDomain_smul (T : H →ₗ.[ℂ] H) {w : ℂ} (hw : w ≠ 0) :
@@ -166,8 +165,7 @@ lemma regularityDomain_smul (T : H →ₗ.[ℂ] H) {w : ℂ} (hw : w ≠ 0) :
     · refine ⟨‖w‖⁻¹ * c, by positivity, fun x ↦ ?_⟩
       rw [mul_assoc]
       apply (inv_mul_le_iff₀ <| norm_pos_iff.mpr hw).mpr
-      rw [← norm_smul, smul_sub, smul_smul, mul_inv_cancel_left₀ hw]
-      exact h_bound x
+      simpa [← norm_smul, smul_sub, smul_smul, mul_inv_cancel_left₀ hw] using h_bound x
     · simp [hw]
   · intro ⟨u, ⟨c, hc, h_bound⟩, huz⟩
     refine ⟨‖w‖ * c, by positivity, fun x ↦ ?_⟩
@@ -658,8 +656,7 @@ def resolventSet (T : H →ₗ.[ℂ] H) : Set ℂ :=
 scoped notation "ρ" => resolventSet
 
 lemma resolventSet_eq (T : H →ₗ.[ℂ] H) :
-    ρ T = {z | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤ ∧ Continuous (𝑅 T z)} :=
-  rfl
+    ρ T = {z | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤ ∧ Continuous (𝑅 T z)} := rfl
 
 lemma mem_resolventSet_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :
     z ∈ ρ T ↔ (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤ ∧ Continuous (𝑅 T z) :=
@@ -783,8 +780,7 @@ def residualSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ :=
 scoped notation "σʳ" => residualSpectrum
 
 lemma residualSpectrum_eq (T : H →ₗ.[ℂ] H) :
-    σʳ T = {z | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range ≠ ⊤ ∧ Continuous (𝑅 T z)} :=
-  rfl
+    σʳ T = {z | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range ≠ ⊤ ∧ Continuous (𝑅 T z)} := rfl
 
 lemma mem_residualSpectrum_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :
     z ∈ σʳ T ↔ (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range ≠ ⊤ ∧ Continuous (𝑅 T z) :=
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/ExpectedValue.lean b/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/ExpectedValue.lean
index e5dc6f9b0..e1f4e8459 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/ExpectedValue.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/ExpectedValue.lean
@@ -40,8 +40,7 @@ variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
 
 private lemma conj_inner_apply_self_eq_of_isSymmetric (T : H →ₗ.[ℂ] H) (hT : T.IsSymmetric)
     (ψ : T.domain) :
-    (starRingEnd ℂ) ⟪(ψ : H), T ψ⟫_ℂ = ⟪(ψ : H), T ψ⟫_ℂ := by
-  simpa [inner_conj_symm] using hT ψ ψ
+    (starRingEnd ℂ) ⟪(ψ : H), T ψ⟫_ℂ = ⟪(ψ : H), T ψ⟫_ℂ := by simpa [inner_conj_symm] using hT ψ ψ
 
 /-- Expectation value `re ⟪ψ, Tψ⟫_ℂ` for `ψ ∈ T.domain`.
 
@@ -52,8 +51,7 @@ def expectedValue (T : H →ₗ.[ℂ] H) (ψ : T.domain) : ℝ :=
 
 /-- The expectation value, unfolded as a real part. -/
 lemma expectedValue_eq_re_inner (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    expectedValue T ψ = (⟪(ψ : H), T ψ⟫_ℂ).re :=
-  rfl
+    expectedValue T ψ = (⟪(ψ : H), T ψ⟫_ℂ).re := rfl
 
 /-- If `T` is symmetric, `⟪ψ, Tψ⟫_ℂ` is the expectation value, coerced to `ℂ`. -/
 lemma expectedValue_eq_inner (T : H →ₗ.[ℂ] H) (hT : T.IsSymmetric) (ψ : T.domain) :
@@ -74,13 +72,11 @@ def centered (T : H →ₗ.[ℂ] H) (ψ : T.domain) : H :=
 
 /-- The centered vector, unfolded to its raw expression. -/
 lemma centered_eq (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    centered T ψ = T ψ - (expectedValue T ψ : ℂ) • (ψ : H) :=
-  rfl
+    centered T ψ = T ψ - (expectedValue T ψ : ℂ) • (ψ : H) := rfl
 
 /-- A centered vector vanishes exactly when `Tψ = ⟨T⟩_ψ ψ`. -/
 lemma centered_eq_zero_iff (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    centered T ψ = 0 ↔ T ψ = (expectedValue T ψ : ℂ) • (ψ : H) := by
-  rw [centered_eq, sub_eq_zero]
+    centered T ψ = 0 ↔ T ψ = (expectedValue T ψ : ℂ) • (ψ : H) := by rw [centered_eq, sub_eq_zero]
 
 /-- For a unit vector and symmetric `T`, the centered vector is orthogonal to the state. -/
 lemma inner_state_centered_eq_zero (T : H →ₗ.[ℂ] H) (hT : T.IsSymmetric)
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/Variance.lean b/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/Variance.lean
index 8cbf85f9f..4916268b4 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/Variance.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/StateObservables/Variance.lean
@@ -52,14 +52,12 @@ def variance (T : H →ₗ.[ℂ] H) (ψ : T.domain) : ℝ :=
 
 /-- The variance is the squared norm of the centered vector. -/
 lemma variance_eq_centered_norm_sq (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    variance T ψ = ‖centered T ψ‖ ^ 2 :=
-  rfl
+    variance T ψ = ‖centered T ψ‖ ^ 2 := rfl
 
 /-- `variance` with `centered` unfolded to `Tψ - ⟨T⟩_ψ • ψ`. -/
 lemma variance_eq_norm_sub_sq (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
     variance T ψ =
-      ‖T ψ - (expectedValue T ψ : ℂ) • (ψ : H)‖ ^ 2 :=
-  rfl
+      ‖T ψ - (expectedValue T ψ : ℂ) • (ψ : H)‖ ^ 2 := rfl
 
 /-- For symmetric `T` and `‖ψ‖ = 1`, variance equals `‖Tψ‖ ^ 2 - ⟨T⟩_ψ ^ 2`. -/
 lemma variance_eq_norm_sq_sub_expectedValue_sq (T : H →ₗ.[ℂ] H)
@@ -67,10 +65,8 @@ lemma variance_eq_norm_sq_sub_expectedValue_sq (T : H →ₗ.[ℂ] H)
     variance T ψ = ‖T ψ‖ ^ 2 - expectedValue T ψ ^ 2 := by
   let μ := expectedValue T ψ
   let a : H := T ψ
-  have hμ_right : ⟪(ψ : H), a⟫_ℂ = (μ : ℂ) := by
-    simpa [a, μ] using expectedValue_eq_inner T hT ψ
-  have hμ_left : ⟪a, (ψ : H)⟫_ℂ = (μ : ℂ) := by
-    simpa [inner_conj_symm] using congrArg star hμ_right
+  have hμ_right : ⟪(ψ : H), a⟫_ℂ = (μ : ℂ) := by simpa [a, μ] using expectedValue_eq_inner T hT ψ
+  have hμ_left : ⟪a, (ψ : H)⟫_ℂ = (μ : ℂ) := by simpa [inner_conj_symm] using congrArg star hμ_right
   have h_re_inner_centered : (⟪a, (μ : ℂ) • (ψ : H)⟫_ℂ).re = μ ^ 2 := by
     rw [inner_smul_right, hμ_left]
     simp [μ]
@@ -88,15 +84,11 @@ lemma variance_eq_norm_sq_sub_expectedValue_sq (T : H →ₗ.[ℂ] H)
 
 /-- Variance is nonnegative. -/
 lemma variance_nonneg (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    0 ≤ variance T ψ := by
-  rw [variance_eq_centered_norm_sq]
-  exact sq_nonneg _
+    0 ≤ variance T ψ := by simp [variance_eq_centered_norm_sq]
 
 /-- Zero variance is the same as a zero centered vector. -/
 lemma variance_eq_zero_iff_centered_eq_zero (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    variance T ψ = 0 ↔ centered T ψ = 0 := by
-  rw [variance_eq_centered_norm_sq]
-  exact sq_eq_zero_iff.trans norm_eq_zero
+    variance T ψ = 0 ↔ centered T ψ = 0 := by simp [variance_eq_centered_norm_sq]
 
 /-- Zero variance is the same as `Tψ = ⟨T⟩_ψ ψ`. -/
 lemma variance_eq_zero_iff (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
@@ -126,20 +118,17 @@ def standardDeviation (T : H →ₗ.[ℂ] H) (ψ : T.domain) : ℝ :=
 
 /-- The standard deviation, unfolded to the square root of the variance. -/
 lemma standardDeviation_eq_sqrt_variance (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    standardDeviation T ψ = Real.sqrt (variance T ψ) :=
-  rfl
+    standardDeviation T ψ = Real.sqrt (variance T ψ) := rfl
 
 /-- Standard deviation is nonnegative. -/
 lemma standardDeviation_nonneg (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
-    0 ≤ standardDeviation T ψ := by
-  rw [standardDeviation_eq_sqrt_variance]
-  exact Real.sqrt_nonneg _
+    0 ≤ standardDeviation T ψ :=
+  Real.sqrt_nonneg _
 
 @[simp]
 lemma standardDeviation_sq (T : H →ₗ.[ℂ] H) (ψ : T.domain) :
     standardDeviation T ψ ^ 2 = variance T ψ := by
-  rw [standardDeviation_eq_sqrt_variance, Real.sq_sqrt]
-  exact variance_nonneg T ψ
+  simpa [standardDeviation_eq_sqrt_variance] using Real.sq_sqrt (variance_nonneg T ψ)
 
 /-- Zero standard deviation is the same as a zero centered vector. -/
 lemma standardDeviation_eq_zero_iff_centered_eq_zero (T : H →ₗ.[ℂ] H)
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean b/Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean
index 0da5aa699..9096d2c80 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean
@@ -366,8 +366,7 @@ lemma adjoint_add_le_add_adjoint [CompleteSpace H]
 
 lemma adjoint_sub_le_sub_adjoint [CompleteSpace H]
     (U₁ U₂ : H →ₗ.[ℂ] H') (h₁₂ : (U₁ - U₂).HasDenseDomain) : U₁† - U₂† ≤ (U₁ - U₂)† := by
-  simp only [sub_eq_add_neg, ← adjoint_neg]
-  exact adjoint_add_le_add_adjoint U₁ (-U₂) h₁₂
+  simpa only [sub_eq_add_neg, ← adjoint_neg] using adjoint_add_le_add_adjoint U₁ (-U₂) h₁₂
 
 lemma adjoint_compRestricted_le_compRestricted_adjoint [CompleteSpace H] [CompleteSpace H']
     (hV : V.HasDenseDomain) (hVU : (V ∘ᵣ U).HasDenseDomain) : U† ∘ᵣ V† ≤ (V ∘ᵣ U)† := by
@@ -690,9 +689,8 @@ lemma IsSelfAdjoint.isEssentiallySelfAdjoint [CompleteSpace H] (h : IsSelfAdjoin
   isEssentiallySelfAdjoint_def.mpr (h.isClosed.closure_eq.symm ▸ h)
 
 @[aesop safe apply]
-lemma IsSelfAdjoint.adjoint [CompleteSpace H] (h : IsSelfAdjoint T) : IsSelfAdjoint T† := by
-  apply isSelfAdjoint_def.mp at h
-  exact h.symm ▸ h
+lemma IsSelfAdjoint.adjoint [CompleteSpace H] (h : IsSelfAdjoint T) : IsSelfAdjoint T† :=
+  isSelfAdjoint_def.mpr ((isSelfAdjoint_def.mp h).symm ▸ isSelfAdjoint_def.mp h)
 
 @[aesop safe apply]
 lemma IsSelfAdjoint.smul [CompleteSpace H]
diff --git a/Physlib/QuantumMechanics/DDimensions/Operators/Uncertainty.lean b/Physlib/QuantumMechanics/DDimensions/Operators/Uncertainty.lean
index 01f7556fb..aad85eae0 100644
--- a/Physlib/QuantumMechanics/DDimensions/Operators/Uncertainty.lean
+++ b/Physlib/QuantumMechanics/DDimensions/Operators/Uncertainty.lean
@@ -115,18 +115,14 @@ lemma raw_commutator_eq_of_symmetric
     (h_raw : ⟪(ψ : H), A ⟨B ⟨ψ, hψB⟩, hAB⟩ - B ⟨A ψ, hBA⟩⟫_ℂ = Complex.I * c) :
     ⟪A ψ, B ⟨ψ, hψB⟩⟫_ℂ - ⟪B ⟨ψ, hψB⟩, A ψ⟫_ℂ = Complex.I * c := by
   have ha_pairing :
-      ⟪A ψ, B ⟨ψ, hψB⟩⟫_ℂ = ⟪(ψ : H), A ⟨B ⟨ψ, hψB⟩, hAB⟩⟫_ℂ := by
-    exact hA ψ ⟨B ⟨ψ, hψB⟩, hAB⟩
+      ⟪A ψ, B ⟨ψ, hψB⟩⟫_ℂ = ⟪(ψ : H), A ⟨B ⟨ψ, hψB⟩, hAB⟩⟫_ℂ := hA ψ ⟨B ⟨ψ, hψB⟩, hAB⟩
   have hb_pairing :
-      ⟪B ⟨ψ, hψB⟩, A ψ⟫_ℂ = ⟪(ψ : H), B ⟨A ψ, hBA⟩⟫_ℂ := by
-    exact hB ⟨ψ, hψB⟩ ⟨A ψ, hBA⟩
+      ⟪B ⟨ψ, hψB⟩, A ψ⟫_ℂ = ⟪(ψ : H), B ⟨A ψ, hBA⟩⟫_ℂ := hB ⟨ψ, hψB⟩ ⟨A ψ, hBA⟩
   calc
     ⟪A ψ, B ⟨ψ, hψB⟩⟫_ℂ - ⟪B ⟨ψ, hψB⟩, A ψ⟫_ℂ =
       ⟪(ψ : H), A ⟨B ⟨ψ, hψB⟩, hAB⟩⟫_ℂ -
-        ⟪(ψ : H), B ⟨A ψ, hBA⟩⟫_ℂ := by
-          rw [ha_pairing, hb_pairing]
-    _ = ⟪(ψ : H), A ⟨B ⟨ψ, hψB⟩, hAB⟩ - B ⟨A ψ, hBA⟩⟫_ℂ := by
-          rw [inner_sub_right]
+        ⟪(ψ : H), B ⟨A ψ, hBA⟩⟫_ℂ := by rw [ha_pairing, hb_pairing]
+    _ = ⟪(ψ : H), A ⟨B ⟨ψ, hψB⟩, hAB⟩ - B ⟨A ψ, hBA⟩⟫_ℂ := by rw [inner_sub_right]
     _ = Complex.I * c := h_raw
 
 /-- The scalar commutator of the centered vectors of `A` and `B` in the state `ψ`. -/
@@ -245,27 +241,22 @@ lemma inner_centered_commutator_of_raw_commutator :
   let b : H := B ⟨ψ, hψB⟩
   let μa : ℝ := expectedValue A ψ
   let μb : ℝ := expectedValue B ⟨ψ, hψB⟩
-  have hμa_right : ⟪(ψ : H), a⟫_ℂ = (μa : ℂ) := by
-    simpa [a, μa] using expectedValue_eq_inner A hA ψ
+  have hμa_right : ⟪(ψ : H), a⟫_ℂ = (μa : ℂ) := by simpa [a, μa] using expectedValue_eq_inner A hA ψ
   have hμa_left : ⟪a, (ψ : H)⟫_ℂ = (μa : ℂ) := by
-    have h_symm : ⟪a, (ψ : H)⟫_ℂ = ⟪(ψ : H), a⟫_ℂ := by
-      simpa [a] using hA ψ ψ
+    have h_symm : ⟪a, (ψ : H)⟫_ℂ = ⟪(ψ : H), a⟫_ℂ := by simpa [a] using hA ψ ψ
     simpa [h_symm] using hμa_right
   have hμb_right : ⟪(ψ : H), b⟫_ℂ = (μb : ℂ) := by
     simpa [b, μb] using expectedValue_eq_inner B hB ⟨ψ, hψB⟩
   have hμb_left : ⟪b, (ψ : H)⟫_ℂ = (μb : ℂ) := by
-    have h_symm : ⟪b, (ψ : H)⟫_ℂ = ⟪(ψ : H), b⟫_ℂ := by
-      simpa [b] using hB ⟨ψ, hψB⟩ ⟨ψ, hψB⟩
+    have h_symm : ⟪b, (ψ : H)⟫_ℂ = ⟪(ψ : H), b⟫_ℂ := by simpa [b] using hB ⟨ψ, hψB⟩ ⟨ψ, hψB⟩
     simpa [h_symm] using hμb_right
   calc
     centeredCommutatorExpectation A B ψ hψB =
       ⟪centered A ψ, centered B ⟨ψ, hψB⟩⟫_ℂ -
-        ⟪centered B ⟨ψ, hψB⟩, centered A ψ⟫_ℂ := by
-          rfl
+        ⟪centered B ⟨ψ, hψB⟩, centered A ψ⟫_ℂ := rfl
     _ =
       ⟪a - (μa : ℂ) • (ψ : H), b - (μb : ℂ) • (ψ : H)⟫_ℂ -
-        ⟪b - (μb : ℂ) • (ψ : H), a - (μa : ℂ) • (ψ : H)⟫_ℂ := by
-          rfl
+        ⟪b - (μb : ℂ) • (ψ : H), a - (μa : ℂ) • (ψ : H)⟫_ℂ := rfl
     _ = ⟪a, b⟫_ℂ - ⟪b, a⟫_ℂ :=
       sub_expectation_commutator_eq_raw (ψ : H) a b μa μb
         hμa_right hμa_left hμb_right hμb_left hψ_norm
diff --git a/Physlib/QuantumMechanics/DDimensions/SpaceDHilbertSpace/PolyBddSchwartzSubmodule.lean b/Physlib/QuantumMechanics/DDimensions/SpaceDHilbertSpace/PolyBddSchwartzSubmodule.lean
index 6ac8e891a..c8b15c11a 100644
--- a/Physlib/QuantumMechanics/DDimensions/SpaceDHilbertSpace/PolyBddSchwartzSubmodule.lean
+++ b/Physlib/QuantumMechanics/DDimensions/SpaceDHilbertSpace/PolyBddSchwartzSubmodule.lean
@@ -77,8 +77,7 @@ def PolyBddSchwartzMap (d : ℕ) (a : ℕ∞) : Submodule ℂ 𝓢(Space d, ℂ)
     obtain ⟨C₂, hC₂_pos, hC₂⟩ := hg k hk
     refine ⟨C₁ + C₂, by positivity, fun x ↦ ?_⟩
     refine le_trans ?_ (add_le_add (hC₁ x) (hC₂ x))
-    rw [← mul_add]
-    exact mul_le_mul_of_nonneg_left (norm_add_le (f x) (g x)) (by positivity)
+    simpa [← mul_add] using mul_le_mul_of_nonneg_left (norm_add_le (f x) (g x)) (by positivity)
   zero_mem' := fun _ _ ↦ ⟨1, by simp⟩
   smul_mem' := by
     intro c f hf k hk
@@ -119,8 +118,7 @@ instance {d : ℕ} {a : ℕ∞} : CoeFun (PolyBddSchwartzMap d a) (fun _ ↦ Spa
 
 @[simp]
 lemma toFun_eq_coe {d : ℕ} {a : ℕ∞} (f : PolyBddSchwartzMap d a) (x : Space d) :
-    f.val.toFun x = f.val x :=
-  rfl
+    f.val.toFun x = f.val x := rfl
 
 lemma polyBddSchwartzEquiv_symm_apply_coe {d : ℕ} {a : ℕ∞}
     {ψ : SchwartzSubmodule d} (hψ : ↑ψ ∈ PolyBddSchwartzSubmodule d a) :
diff --git a/Physlib/QuantumMechanics/OneDimension/GeneralPotential/Basic.lean b/Physlib/QuantumMechanics/OneDimension/GeneralPotential/Basic.lean
index 6c2b716a9..231b6ccff 100644
--- a/Physlib/QuantumMechanics/OneDimension/GeneralPotential/Basic.lean
+++ b/Physlib/QuantumMechanics/OneDimension/GeneralPotential/Basic.lean
@@ -22,11 +22,9 @@ namespace OneDimension
 open HilbertSpace Constants
 
 private lemma fun_add {α : Type*} (f g : α → ℂ) :
-  (fun x ↦ f x) + (fun x ↦ g x) = fun x ↦ f x + g x := by
-  rfl
+  (fun x ↦ f x) + (fun x ↦ g x) = fun x ↦ f x + g x := rfl
 
-private lemma fun_smul (a1: ℂ) (f : ℝ → ℂ) : (a1 • fun x ↦ f x) = (fun x ↦ a1*(f x)) := by
-    rfl
+private lemma fun_smul (a1: ℂ) (f : ℝ → ℂ) : (a1 • fun x ↦ f x) = (fun x ↦ a1*(f x)) := by rfl
 
 lemma momentumOperator_linear (a1 a2 : ℂ) (ψ1 ψ2 : ℝ → ℂ)
     (hψ1_x: Differentiable ℝ ψ1) (hψ2_x: Differentiable ℝ ψ2) :
@@ -102,8 +100,7 @@ noncomputable def schrodingerOperator (ψ : ℝ → ℂ) : ℝ → ℂ :=
   fun x ↦ 1 / (2 * Q.m) * (momentumOperator (momentumOperator ψ) x) + (Q.V x) * ψ x
 
 private lemma eval_add (f g : ℝ → ℂ) :
-    (f + g) x = f x + g x :=
-  rfl
+    (f + g) x = f x + g x := rfl
 
 lemma schrodingerOperator_linear (a1 a2 : ℂ) (ψ1 ψ2 : ℝ → ℂ)
     (hψ1_x: Differentiable ℝ ψ1) (hψ2_x: Differentiable ℝ ψ2)
diff --git a/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Basic.lean b/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Basic.lean
index 26f3c20e4..1dbba481e 100644
--- a/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Basic.lean
+++ b/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Basic.lean
@@ -38,8 +38,7 @@ lemma Complex.ofReal_hasDerivAt : HasDerivAt Complex.ofReal 1 x := by
   let f1 : ℂ → ℂ := id
   change HasDerivAt (f1 ∘ Complex.ofReal) 1 x
   apply HasDerivAt.comp_ofReal
-  simp only [f1]
-  exact hasDerivAt_id _
+  simpa only [f1] using hasDerivAt_id _
 
 @[simp]
 lemma Complex.deriv_ofReal : deriv Complex.ofReal x = 1 := by
@@ -131,9 +130,7 @@ lemma one_over_ξ : 1/Q.ξ = √(Q.m * Q.ω / ℏ) := by
   rw [← Real.sqrt_inv]
   field_simp
 
-lemma ξ_inverse : Q.ξ⁻¹ = √(Q.m * Q.ω / ℏ) := by
-  rw [inv_eq_one_div]
-  exact one_over_ξ Q
+lemma ξ_inverse : Q.ξ⁻¹ = √(Q.m * Q.ω / ℏ) := by simpa only [inv_eq_one_div] using one_over_ξ Q
 
 lemma one_over_ξ_sq : (1/Q.ξ)^2 = Q.m * Q.ω / ℏ := by
   rw [one_over_ξ]
diff --git a/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean b/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
index 8d2e465d2..42180c32f 100644
--- a/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
+++ b/Physlib/QuantumMechanics/OneDimension/HarmonicOscillator/Completeness.lean
@@ -108,9 +108,7 @@ lemma mul_polynomial_integrable (f : ℝ → ℂ) (hf : MemHS f) (P : Polynomial
     funext x
     simp only [Complex.ofReal_exp, Complex.ofReal_div, Complex.ofReal_neg, Complex.ofReal_pow,
       Complex.ofReal_mul, Complex.ofReal_ofNat, Complex.real_smul]
-  rw [hf']
-  apply MeasureTheory.Integrable.fun_smul
-  exact Q.mul_physHermite_integrable f hf i
+  simpa [hf'] using (Q.mul_physHermite_integrable f hf i).fun_smul (a i)
 
 lemma mul_power_integrable (f : ℝ → ℂ) (hf : MemHS f) (r : ℕ) :
     MeasureTheory.Integrable (fun x => x ^ r * (f x * Real.exp (- x^2 / (2 * Q.ξ^2)))) volume := by
@@ -143,8 +141,7 @@ lemma mul_power_integrable (f : ℝ → ℂ) (hf : MemHS f) (r : ℕ) :
 lemma orthogonal_eigenfunction_of_mem_orthogonal (f : ℝ → ℂ) (hf : MemHS f)
     (hOrth : ∀ n : ℕ, ⟪HilbertSpace.mk (Q.eigenfunction_memHS n), HilbertSpace.mk hf⟫_ℂ = 0)
     (n : ℕ) : ∫ (x : ℝ), Q.eigenfunction n x * f x = 0 := by
-  rw [← hOrth n]
-  rw [inner_mk_mk]
+  rw [← hOrth n, inner_mk_mk]
   simp
 
 local notation "m" => Q.m
diff --git a/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Basic.lean b/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Basic.lean
index 67623e9d4..b7aa49ef4 100644
--- a/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Basic.lean
+++ b/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Basic.lean
@@ -152,12 +152,10 @@ lemma mk_smul {f : ℝ → ℂ} {c : ℂ} {hf : MemHS f} :
     mk (memHS_smul (c := c) hf) = c • mk hf := rfl
 
 lemma mk_eq_iff {f g : ℝ → ℂ} {hf : MemHS f} {hg : MemHS g} :
-    mk hf = mk hg ↔ f =ᵐ[volume] g := by
-  simp [mk]
+    mk hf = mk hg ↔ f =ᵐ[volume] g := by simp [mk]
 
 lemma ext_iff {f g : HilbertSpace} :
-    f = g ↔ (f : ℝ → ℂ) =ᶠ[ae volume] (g : ℝ → ℂ) := by
-  exact Lp.ext_iff
+    f = g ↔ (f : ℝ → ℂ) =ᶠ[ae volume] (g : ℝ → ℂ) := Lp.ext_iff
 
 end HilbertSpace
 end
diff --git a/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Gaussians.lean b/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Gaussians.lean
index e855bf752..a8215732a 100644
--- a/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Gaussians.lean
+++ b/Physlib/QuantumMechanics/OneDimension/HilbertSpace/Gaussians.lean
@@ -72,9 +72,8 @@ lemma exp_mul_gaussian_integrable (b c : ℝ) (hb : 0 < b) :
     field_simp
     ring
   rw [h1]
-  apply MeasureTheory.Integrable.const_mul
-  apply Integrable.comp_sub_right (f := (fun x => Real.exp (- b * x ^ 2)))
-  exact integrable_exp_neg_mul_sq hb
+  exact (Integrable.comp_sub_right (f := (fun x => Real.exp (- b * x ^ 2)))
+    (integrable_exp_neg_mul_sq hb) (c / (2 * b))).const_mul _
 
 lemma exp_abs_mul_gaussian_integrable (b c : ℝ) (hb : 0 < b) :
     MeasureTheory.Integrable (fun x => Real.exp (|c * x|) * Real.exp (- b * x ^ 2)) := by
diff --git a/Physlib/QuantumMechanics/OneDimension/HilbertSpace/SchwartzSubmodule.lean b/Physlib/QuantumMechanics/OneDimension/HilbertSpace/SchwartzSubmodule.lean
index 85891c6f9..51b3e4335 100644
--- a/Physlib/QuantumMechanics/OneDimension/HilbertSpace/SchwartzSubmodule.lean
+++ b/Physlib/QuantumMechanics/OneDimension/HilbertSpace/SchwartzSubmodule.lean
@@ -48,8 +48,7 @@ lemma schwartzIncl_inner (ψ1 ψ2 : 𝓢(ℝ, ℂ)) :
   filter_upwards [h1, h2] with _ h1 h2
   rw [← h1, ← h2]
   simp only [RCLike.inner_apply]
-  rw [mul_comm]
-  rfl
+  exact mul_comm _ _
 
 end HilbertSpace
 end
diff --git a/Physlib/QuantumMechanics/OneDimension/Operators/Momentum.lean b/Physlib/QuantumMechanics/OneDimension/Operators/Momentum.lean
index 448334e5c..eaf4d2488 100644
--- a/Physlib/QuantumMechanics/OneDimension/Operators/Momentum.lean
+++ b/Physlib/QuantumMechanics/OneDimension/Operators/Momentum.lean
@@ -42,8 +42,7 @@ open HilbertSpace SchwartzMap
 def momentumOperator (ψ : ℝ → ℂ) : ℝ → ℂ := fun x ↦ - Complex.I * ℏ * deriv ψ x
 
 lemma momentumOperator_eq_smul (ψ : ℝ → ℂ) :
-    momentumOperator ψ = fun x => (- Complex.I * ℏ) • deriv ψ x := by
-  rfl
+    momentumOperator ψ = fun x => (- Complex.I * ℏ) • deriv ψ x := rfl
 
 @[fun_prop]
 lemma continuous_momentumOperator (ψ : ℝ → ℂ) (hψ : ContDiff ℝ 1 ψ) :
@@ -75,16 +74,12 @@ lemma momentumOperator_add {ψ1 ψ2 : ℝ → ℂ}
   `𝓢(ℝ, ℂ)` to itself, such that `ψ` is taken to `fun x => - I ℏ * ψ' x`. -/
 def momentumOperatorSchwartz : 𝓢(ℝ, ℂ) →L[ℂ] 𝓢(ℝ, ℂ) where
   toFun ψ := (- Complex.I * ℏ) • SchwartzMap.derivCLM ℂ ℂ ψ
-  map_add' ψ1 ψ2 := by
-    simp only [neg_mul, map_add, smul_add, neg_smul]
-  map_smul' a ψ := by
-    simp only [map_smul, RingHom.id_apply, smul_comm (-Complex.I * ℏ) a]
+  map_add' ψ1 ψ2 := by simp only [neg_mul, map_add, smul_add, neg_smul]
+  map_smul' a ψ := by simp only [map_smul, RingHom.id_apply, smul_comm (-Complex.I * ℏ) a]
   cont := by fun_prop
 
 lemma momentumOperatorSchwartz_apply (ψ : 𝓢(ℝ, ℂ))
-    (x : ℝ) : (momentumOperatorSchwartz ψ) x = (- Complex.I * ℏ) * (deriv ψ x) := by
-  rw [momentumOperatorSchwartz]
-  rfl
+    (x : ℝ) : (momentumOperatorSchwartz ψ) x = (- Complex.I * ℏ) * (deriv ψ x) := rfl
 
 /-- The unbounded momentum operator, whose domain is Schwartz maps. -/
 def momentumOperatorUnbounded : UnboundedOperator schwartzIncl schwartzIncl_injective :=
@@ -99,8 +94,7 @@ def momentumOperatorUnbounded : UnboundedOperator schwartzIncl schwartzIncl_inje
 lemma planeWaveFunctional_generalized_eigenvector_momentumOperatorUnbounded (k : ℝ) :
     momentumOperatorUnbounded.IsGeneralizedEigenvector
       (planewaveFunctional k) (2 * Real.pi * ℏ * k) := by
-  dsimp [momentumOperatorUnbounded]
-  rw [UnboundedOperator.isGeneralizedEigenvector_ofSelfCLM_iff]
+  rw [momentumOperatorUnbounded, UnboundedOperator.isGeneralizedEigenvector_ofSelfCLM_iff]
   intro ψ
   trans (-((Complex.I * ↑↑ℏ) •
       (SchwartzMap.fourierTransformCLM ℂ) ((SchwartzMap.derivCLM ℂ ℂ) ψ) k))
diff --git a/Physlib/QuantumMechanics/OneDimension/Operators/Position.lean b/Physlib/QuantumMechanics/OneDimension/Operators/Position.lean
index d0e8de9a2..5db2c171a 100644
--- a/Physlib/QuantumMechanics/OneDimension/Operators/Position.lean
+++ b/Physlib/QuantumMechanics/OneDimension/Operators/Position.lean
@@ -64,15 +64,11 @@ open SchwartzMap
 def positionOperatorSchwartz : 𝓢(ℝ, ℂ) →L[ℂ] 𝓢(ℝ, ℂ) := Distribution.powOneMul ℂ
 
 lemma positionOperatorSchwartz_apply_fun (ψ : 𝓢(ℝ, ℂ)) :
-    (positionOperatorSchwartz ψ) = fun x => x * ψ x := by
-  simp [positionOperatorSchwartz]
-  rfl
+    (positionOperatorSchwartz ψ) = fun x => x * ψ x := rfl
 
 @[simp]
 lemma positionOperatorSchwartz_apply (ψ : 𝓢(ℝ, ℂ)) (x : ℝ) :
-    (positionOperatorSchwartz ψ) x = x * ψ x := by
-  simp [positionOperatorSchwartz]
-  rfl
+    (positionOperatorSchwartz ψ) x = x * ψ x := rfl
 
 /-- The unbounded position operator, whose domain is Schwartz maps. -/
 def positionOperatorUnbounded : UnboundedOperator schwartzIncl schwartzIncl_injective :=
@@ -86,8 +82,7 @@ def positionOperatorUnbounded : UnboundedOperator schwartzIncl schwartzIncl_inje
 
 lemma positionStates_generalized_eigenvector_positionOperatorUnbounded (x : ℝ) :
     positionOperatorUnbounded.IsGeneralizedEigenvector (positionState x) x := by
-  dsimp [positionOperatorUnbounded]
-  rw [UnboundedOperator.isGeneralizedEigenvector_ofSelfCLM_iff]
+  rw [positionOperatorUnbounded, UnboundedOperator.isGeneralizedEigenvector_ofSelfCLM_iff]
   intro ψ
   simp [positionState_apply]
 
diff --git a/Physlib/Relativity/CliffordAlgebra.lean b/Physlib/Relativity/CliffordAlgebra.lean
index f1029532c..9627892c6 100644
--- a/Physlib/Relativity/CliffordAlgebra.lean
+++ b/Physlib/Relativity/CliffordAlgebra.lean
@@ -91,8 +91,7 @@ representation. -/
 @[simp]
 def γSet : Set (Matrix (Fin 4) (Fin 4) ℂ) := {γ i | i : Fin 4}
 
-lemma γ_in_γSet (μ : Fin 4) : γ μ ∈ γSet := by
-  simp [γSet]
+lemma γ_in_γSet (μ : Fin 4) : γ μ ∈ γSet := by simp [γSet]
 
 /-- The algebra generated by the gamma matrices in the Dirac representation. -/
 def diracAlgebra : Subalgebra ℝ (Matrix (Fin 4) (Fin 4) ℂ) :=
@@ -188,8 +187,7 @@ lemma ofCliffordAlgebra_range_eq_top : ofCliffordAlgebra.range = ⊤ := by
 
 /-- The homomorphism `ofCliffordAlgebra` is surjective. -/
 theorem ofCliffordAlgebra_surjective : Function.Surjective ofCliffordAlgebra := by
-  rw [← AlgHom.range_eq_top]
-  exact ofCliffordAlgebra_range_eq_top
+  simpa [← AlgHom.range_eq_top] using ofCliffordAlgebra_range_eq_top
 
 end γ
 
diff --git a/Physlib/Relativity/LorentzAlgebra/ExponentialMap.lean b/Physlib/Relativity/LorentzAlgebra/ExponentialMap.lean
index a906c0aab..9c10b89ad 100644
--- a/Physlib/Relativity/LorentzAlgebra/ExponentialMap.lean
+++ b/Physlib/Relativity/LorentzAlgebra/ExponentialMap.lean
@@ -61,16 +61,14 @@ lemma exp_transpose_of_mem_algebra (A : lorentzAlgebra) :
       val_inv := minkowskiMatrix.sq,
       inv_val := minkowskiMatrix.sq }
   rw [show -(η * A.1 * η) = η * (-A.1) * η by noncomm_ring]
-  erw [NormedSpace.exp_units_conj P_gl (-A.1)]
-  rfl
+  exact NormedSpace.exp_units_conj P_gl (-A.1)
 
 set_option backward.isDefEq.respectTransparency false in
 /--
 The exponential of an element of the Lorentz algebra is a member of the Lorentz group.
 -/
 theorem exp_mem_lorentzGroup (A : lorentzAlgebra) : NormedSpace.exp A.1 ∈ LorentzGroup 3 := by
-  rw [LorentzGroup.mem_iff_transpose_mul_minkowskiMatrix_mul_self]
-  rw [← Matrix.exp_transpose]
+  rw [LorentzGroup.mem_iff_transpose_mul_minkowskiMatrix_mul_self, ← Matrix.exp_transpose]
   rw [exp_transpose_of_mem_algebra A]
   calc
     (η * NormedSpace.exp (-A.1) * η) * η * NormedSpace.exp A.1
@@ -108,8 +106,7 @@ variable {n R ι : Type*} [Fintype n]-- [DecidableEq n]
 @[simp]
 lemma trace_reindex [Semiring R] [Fintype ι] (e : n ≃ ι) (A : Matrix n n R) :
     trace (A.submatrix e.symm e.symm) = trace A := by
-  simp only [trace, diag_apply, submatrix_apply]
-  exact e.symm.sum_comp (fun i : n => A i i)
+  simpa only [trace, diag_apply, submatrix_apply] using e.symm.sum_comp (fun i : n => A i i)
 
 variable {n R ι : Type*} [Fintype n] [DecidableEq n]
 
@@ -177,5 +174,4 @@ theorem exp_isOrthochronous (A : lorentzAlgebra) :
 restricted Lorentz group. -/
 theorem exp_mem_restricted_lorentzGroup (A : lorentzAlgebra) :
     (⟨NormedSpace.exp A.1, exp_mem_lorentzGroup A⟩ : LorentzGroup 3) ∈
-    LorentzGroup.restricted 3 := by
-  exact ⟨exp_isProper A, exp_isOrthochronous A⟩
+    LorentzGroup.restricted 3 := ⟨exp_isProper A, exp_isOrthochronous A⟩
diff --git a/Physlib/Relativity/LorentzGroup/Basic.lean b/Physlib/Relativity/LorentzGroup/Basic.lean
index 7ae8ca105..f82962cf4 100644
--- a/Physlib/Relativity/LorentzGroup/Basic.lean
+++ b/Physlib/Relativity/LorentzGroup/Basic.lean
@@ -60,12 +60,10 @@ variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
 
 -/
 
-lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := by
-  rfl
+lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := Iff.rfl
 
 lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
-  rw [mem_iff_self_mul_dual]
-  exact _root_.mul_eq_one_comm
+  simpa [mem_iff_self_mul_dual] using (_root_.mul_eq_one_comm (a := Λ) (b := dual Λ))
 
 lemma mem_iff_transpose : Λ ∈ LorentzGroup d ↔ Λᵀ ∈ LorentzGroup d := by
   refine Iff.intro (fun h ↦ ?_) (fun h ↦ ?_)
@@ -81,8 +79,7 @@ lemma mem_iff_transpose : Λ ∈ LorentzGroup d ↔ Λᵀ ∈ LorentzGroup d :=
     noncomm_ring
 
 lemma mem_iff_neg_mem : Λ ∈ LorentzGroup d ↔ -Λ ∈ LorentzGroup d := by
-  rw [mem_iff_self_mul_dual, mem_iff_self_mul_dual]
-  simp [dual]
+  simp [mem_iff_self_mul_dual, dual]
 
 lemma mem_mul (hΛ : Λ ∈ LorentzGroup d) (hΛ' : Λ' ∈ LorentzGroup d) : Λ * Λ' ∈ LorentzGroup d := by
   rw [mem_iff_dual_mul_self, dual_mul]
@@ -91,9 +88,7 @@ lemma mem_mul (hΛ : Λ ∈ LorentzGroup d) (hΛ' : Λ' ∈ LorentzGroup d) : Λ
   · rw [(mem_iff_dual_mul_self).mp hΛ]
     simp [(mem_iff_dual_mul_self).mp hΛ']
 
-lemma one_mem : 1 ∈ LorentzGroup d := by
-  rw [mem_iff_dual_mul_self]
-  simp
+lemma one_mem : 1 ∈ LorentzGroup d := by simp [mem_iff_dual_mul_self]
 
 lemma dual_mem (h : Λ ∈ LorentzGroup d) : dual Λ ∈ LorentzGroup d := by
   rw [mem_iff_dual_mul_self, dual_dual]
@@ -105,8 +100,7 @@ A matrix `Λ` is in the Lorentz group if and only if it satisfies `Λᵀ * η *
 lemma mem_iff_transpose_mul_minkowskiMatrix_mul_self
     (Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) :
     Λ ∈ LorentzGroup d ↔ Λᵀ * η * Λ = η := by
-  rw [mem_iff_dual_mul_self]
-  rw [dual]
+  rw [mem_iff_dual_mul_self, dual]
   constructor
   · intro h
     have h' : η * ((η * Λᵀ * η) * Λ) = η * 1 := congr_arg (η * ·) h
@@ -176,8 +170,7 @@ lemma isClosedEmbedding_val {d : ℕ} :
 
 -/
 lemma inv_eq_dual (Λ : LorentzGroup d) :
-    (Λ⁻¹ : LorentzGroup d) = ⟨minkowskiMatrix.dual Λ.1, LorentzGroup.dual_mem Λ.2⟩ := by
-  rfl
+    (Λ⁻¹ : LorentzGroup d) = ⟨minkowskiMatrix.dual Λ.1, LorentzGroup.dual_mem Λ.2⟩ := rfl
 
 lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp Λ.2)).symm
 
@@ -429,8 +422,7 @@ def toComplex : LorentzGroup d →* Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) 
   map_mul' Λ Λ' := by
     ext i j
     simp only [lorentzGroupIsGroup_mul_coe, map_apply, ofRealHom_eq_coe]
-    simp only [← Matrix.map_mul]
-    rfl
+    simp [← Matrix.map_mul]
 
 /-- The image of a Lorentz transformation under `toComplex` is invertible. -/
 instance (M : LorentzGroup d) : Invertible (toComplex M) where
@@ -474,8 +466,7 @@ lemma toComplex_mulVec_ofReal (v : Fin 1 ⊕ Fin d → ℝ) (Λ : LorentzGroup d
     toComplex Λ *ᵥ (ofRealHom ∘ v) = ofRealHom ∘ (Λ *ᵥ v) := by
   simp only [toComplex, MonoidHom.coe_mk, OneHom.coe_mk]
   funext i
-  rw [← RingHom.map_mulVec]
-  rfl
+  exact (RingHom.map_mulVec ofRealHom Λ.val v i).symm
 
 /-- The parity transformation. -/
 def parity : LorentzGroup d := ⟨minkowskiMatrix, by
@@ -489,8 +480,7 @@ def parity : LorentzGroup d := ⟨minkowskiMatrix, by
 -/
 
 lemma eq_of_mulVec_eq {Λ Λ' : LorentzGroup d}
-    (h : ∀ (x : Fin 1 ⊕ Fin d → ℝ), Λ.1 *ᵥ x = Λ'.1 *ᵥ x) : Λ = Λ' := by
-  apply Subtype.ext
-  exact ext_of_mulVec_single fun i => h (Pi.single i 1)
+    (h : ∀ (x : Fin 1 ⊕ Fin d → ℝ), Λ.1 *ᵥ x = Λ'.1 *ᵥ x) : Λ = Λ' :=
+  Subtype.ext (ext_of_mulVec_single fun i => h (Pi.single i 1))
 
 end LorentzGroup
diff --git a/Physlib/Relativity/LorentzGroup/Boosts/Basic.lean b/Physlib/Relativity/LorentzGroup/Boosts/Basic.lean
index a45850ad4..58aae9217 100644
--- a/Physlib/Relativity/LorentzGroup/Boosts/Basic.lean
+++ b/Physlib/Relativity/LorentzGroup/Boosts/Basic.lean
@@ -30,8 +30,7 @@ def γ (β : ℝ) : ℝ := 1 / Real.sqrt (1 - β^2)
 lemma γ_sq (β : ℝ) (hβ : |β| < 1) : (γ β)^2 = 1 / (1 - β^2) := by
   simp only [γ, one_div, inv_pow, _root_.inv_inj]
   refine Real.sq_sqrt ?_
-  simp only [sub_nonneg, sq_le_one_iff_abs_le_one]
-  exact le_of_lt hβ
+  simpa only [sub_nonneg, sq_le_one_iff_abs_le_one] using le_of_lt hβ
 
 @[simp]
 lemma γ_zero : γ 0 = 1 := by simp [γ]
@@ -209,50 +208,39 @@ lemma boost_inverse (i : Fin d) {β : ℝ} (hβ : |β| < 1) :
 
 @[simp]
 lemma boost_inl_0_inl_0 (i : Fin d) {β : ℝ} (hβ : |β| < 1) :
-    (boost i β hβ).1 (Sum.inl 0) (Sum.inl 0) = γ β := by
-  simp [boost]
+    (boost i β hβ).1 (Sum.inl 0) (Sum.inl 0) = γ β := by simp [boost]
 
 @[simp]
 lemma boost_inr_self_inr_self (i : Fin d) {β : ℝ} (hβ : |β| < 1) :
-    (boost i β hβ).1 (Sum.inr i) (Sum.inr i) = γ β := by
-  simp [boost]
+    (boost i β hβ).1 (Sum.inr i) (Sum.inr i) = γ β := by simp [boost]
 
 @[simp]
 lemma boost_inl_0_inr_self (i : Fin d) {β : ℝ} (hβ : |β| < 1) :
-    (boost i β hβ).1 (Sum.inl 0) (Sum.inr i) = - γ β * β := by
-  simp [boost]
+    (boost i β hβ).1 (Sum.inl 0) (Sum.inr i) = - γ β * β := by simp [boost]
 
 @[simp]
 lemma boost_inr_self_inl_0 (i : Fin d) {β : ℝ} (hβ : |β| < 1) :
-    (boost i β hβ).1 (Sum.inr i) (Sum.inl 0) = - γ β * β := by
-  simp [boost]
+    (boost i β hβ).1 (Sum.inr i) (Sum.inl 0) = - γ β * β := by simp [boost]
 
 lemma boost_inl_0_inr_other {i j : Fin d} {β : ℝ} (hβ : |β| < 1) (hij : j ≠ i) :
-    (boost i β hβ).1 (Sum.inl 0) (Sum.inr j) = 0 := by
-  simp [boost, hij]
+    (boost i β hβ).1 (Sum.inl 0) (Sum.inr j) = 0 := by simp [boost, hij]
 
 lemma boost_inr_other_inl_0 {i j : Fin d} {β : ℝ} (hβ : |β| < 1) (hij : j ≠ i) :
-    (boost i β hβ).1 (Sum.inr j) (Sum.inl 0) = 0 := by
-  simp [boost, hij]
+    (boost i β hβ).1 (Sum.inr j) (Sum.inl 0) = 0 := by simp [boost, hij]
 
 lemma boost_inr_self_inr_other {i j : Fin d} {β : ℝ} (hβ : |β| < 1) (hij : j ≠ i) :
-    (boost i β hβ).1 (Sum.inr i) (Sum.inr j) = 0 := by
-  simp [boost, hij, Ne.symm hij]
+    (boost i β hβ).1 (Sum.inr i) (Sum.inr j) = 0 := by simp [boost, hij, Ne.symm hij]
 
 lemma boost_inr_other_inr_self {i j : Fin d} {β : ℝ} (hβ : |β| < 1) (hij : j ≠ i) :
-    (boost i β hβ).1 (Sum.inr j) (Sum.inr i) = 0 := by
-  simp [boost, hij]
+    (boost i β hβ).1 (Sum.inr j) (Sum.inr i) = 0 := by simp [boost, hij]
 
 lemma boost_inr_other_inr {i j k : Fin d} {β : ℝ} (hβ : |β| < 1) (hij : j ≠ i) :
-    (boost i β hβ).1 (Sum.inr j) (Sum.inr k) = if j = k then 1 else 0:= by
-  simp [boost, hij]
+    (boost i β hβ).1 (Sum.inr j) (Sum.inr k) = if j = k then 1 else 0:= by simp [boost, hij]
 
 lemma boost_inr_inr_other {i j k : Fin d} {β : ℝ} (hβ : |β| < 1) (hij : j ≠ i) :
     (boost i β hβ).1 (Sum.inr k) (Sum.inr j) = if j = k then 1 else 0:= by
   rw [← boost_transpose_eq_self]
-  simp only [transpose, transpose_apply]
-  rw [boost_inr_other_inr]
-  exact hij
+  simpa only [transpose, transpose_apply] using boost_inr_other_inr hβ hij
 /-!
 
 ## Properties of boosts in the zero-direction
@@ -302,8 +290,7 @@ lemma boost_zero_inr_nat_succ_inr_0 {d : ℕ} {β : ℝ} (hβ : |β| < 1) (i : 
 lemma boost_zero_inr_succ_inr_succ {d : ℕ} {β : ℝ} (hβ : |β| < 1) (i1 i2 : Fin d) :
     (boost (0 : Fin d.succ) β hβ).1 (Sum.inr i1.succ) (Sum.inr i2.succ) =
     if i1 = i2 then 1 else 0 := by
-  rw [boost_inr_inr_other hβ (Fin.succ_ne_zero i2)]
-  simp [Fin.succ_inj, eq_comm]
+  simp [boost_inr_inr_other hβ (Fin.succ_ne_zero i2), Fin.succ_inj, eq_comm]
 
 end LorentzGroup
 
diff --git a/Physlib/Relativity/LorentzGroup/Boosts/Generalized.lean b/Physlib/Relativity/LorentzGroup/Boosts/Generalized.lean
index 55c9e9457..166e49a8f 100644
--- a/Physlib/Relativity/LorentzGroup/Boosts/Generalized.lean
+++ b/Physlib/Relativity/LorentzGroup/Boosts/Generalized.lean
@@ -45,8 +45,7 @@ variable {d : ℕ}
 /-- An auxiliary linear map used in the definition of a generalised boost. -/
 def genBoostAux₁ (u v : Velocity d) : Vector d →ₗ[ℝ] Vector d where
   toFun x := (2 * ⟪x, u⟫ₘ) • v
-  map_add' x y := by
-    simp [map_add, mul_add, _root_.add_smul]
+  map_add' x y := by simp [map_add, mul_add, _root_.add_smul]
   map_smul' c x := by
     simp only [map_smul, RingHom.id_apply, smul_smul]
     simp only [FunLike.coe_smul, Pi.smul_apply, smul_eq_mul]
@@ -105,15 +104,13 @@ lemma genBoostAux₂_basis_minkowskiProduct (u v : Velocity d) (μ ν : Fin 1 
     ⟪genBoostAux₂ u v (Vector.basis μ), genBoostAux₂ u v (Vector.basis ν)⟫ₘ =
     2 * η μ μ * η ν ν * (u.1 μ + v.1 μ) * (u.1 ν + v.1 ν)
     * (1 + ⟪u, v.1⟫ₘ)⁻¹ := by
-  rw [genBoostAux₂_apply_basis, genBoostAux₂_apply_basis]
-  rw [map_smul, map_smul]
+  rw [genBoostAux₂_apply_basis, genBoostAux₂_apply_basis, map_smul, map_smul]
   have h1 : ⟪u.1 + v.1, u.1 + v.1⟫ₘ = 2 * (1 + ⟪u.1, v.1⟫ₘ) := by
     simp only [map_add, add_apply, Velocity.minkowskiProduct_self_eq_one]
     rw [minkowskiProduct_symm]
     ring
   dsimp
-  have h2 : (1 + ⟪u.1, v.1⟫ₘ) ≠ 0 := by
-    exact Velocity.one_add_minkowskiProduct_ne_zero u v
+  have h2 : (1 + ⟪u.1, v.1⟫ₘ) ≠ 0 := Velocity.one_add_minkowskiProduct_ne_zero u v
   field_simp [h2]
   have h2 : (minkowskiProduct ↑v) u.1  = ⟪u.1, v.1⟫ₘ := by rw [minkowskiProduct_symm]
   simp only [map_add, _root_.smul_add, _root_.neg_smul, add_apply, _root_.neg_apply, smul_apply,
@@ -124,16 +121,14 @@ lemma genBoostAux₂_basis_minkowskiProduct (u v : Velocity d) (μ ν : Fin 1 
 lemma genBoostAux₁_basis_genBoostAux₂_minkowskiProduct (u v : Velocity d) (μ ν : Fin 1 ⊕ Fin d) :
     ⟪genBoostAux₁ u v (Vector.basis μ), genBoostAux₂ u v (Vector.basis ν)⟫ₘ =
     - 2 * η μ μ * η ν ν * u.1 μ * (u.1 ν + v.1 ν) := by
-  rw [genBoostAux₁_apply_basis, genBoostAux₂_apply_basis]
-  rw [map_smul, map_smul]
+  rw [genBoostAux₁_apply_basis, genBoostAux₂_apply_basis, map_smul, map_smul]
   have h1 : ⟪v.1, u.1 + v.1⟫ₘ = (1 + ⟪u.1, v.1⟫ₘ) := by
     simp only [map_add, Velocity.minkowskiProduct_self_eq_one]
     rw [minkowskiProduct_symm]
     ring
   simp only [smul_apply, map_add, Velocity.minkowskiProduct_self_eq_one, smul_eq_mul, neg_mul,
     neg_inj]
-  have h2 : (1 + ⟪u.1, v.1⟫ₘ) ≠ 0 := by
-    exact Velocity.one_add_minkowskiProduct_ne_zero u v
+  have h2 : (1 + ⟪u.1, v.1⟫ₘ) ≠ 0 := Velocity.one_add_minkowskiProduct_ne_zero u v
   field_simp [h2]
   rw [minkowskiProduct_symm]
   ring
@@ -174,8 +169,7 @@ lemma basis_minkowskiProduct_genBoostAux₁_add_genBoostAux₂ (u v : Velocity d
     rw [genBoostAux₁_apply_basis, genBoostAux₂_apply_basis]
     rw [map_smul, map_smul]
     simp
-  have h2 : (1 + ⟪u.1, v.1⟫ₘ) ≠ 0 := by
-    exact Velocity.one_add_minkowskiProduct_ne_zero u v
+  have h2 : (1 + ⟪u.1, v.1⟫ₘ) ≠ 0 := Velocity.one_add_minkowskiProduct_ne_zero u v
   field_simp
   ring
 
diff --git a/Physlib/Relativity/LorentzGroup/Orthochronous/Basic.lean b/Physlib/Relativity/LorentzGroup/Orthochronous/Basic.lean
index 1a9a3add9..3db8ce3fb 100644
--- a/Physlib/Relativity/LorentzGroup/Orthochronous/Basic.lean
+++ b/Physlib/Relativity/LorentzGroup/Orthochronous/Basic.lean
@@ -37,8 +37,7 @@ def IsOrthochronous : Prop := 0 ≤ Λ.1 (Sum.inl 0) (Sum.inl 0)
 /-- A Lorentz transformation is `orthochronous` if and only if its first column is
   future pointing. -/
 lemma isOrthochronous_iff_toVector_timeComponet_nonneg :
-    IsOrthochronous Λ ↔ 0 ≤ (toVector Λ).timeComponent := by
-  simp [IsOrthochronous, timeComponent]
+    IsOrthochronous Λ ↔ 0 ≤ (toVector Λ).timeComponent := by simp [IsOrthochronous, timeComponent]
 
 /-- A Lorentz transformation is orthochronous if and only if its transpose is orthochronous. -/
 lemma isOrthochronous_iff_transpose :
@@ -94,13 +93,10 @@ lemma not_isOrthochronous_iff_le_zero : ¬ IsOrthochronous Λ ↔ Λ.1 (Sum.inl
   constructor <;> intro h <;> rcases h1 with h1 | h1 <;> linarith
 
 lemma not_isOrthochronous_iff_toVector_timeComponet_nonpos :
-    ¬ IsOrthochronous Λ ↔ (toVector Λ).timeComponent ≤ 0:= by
-  rw [not_isOrthochronous_iff_le_zero]
-  simp
+    ¬ IsOrthochronous Λ ↔ (toVector Λ).timeComponent ≤ 0:= by simp [not_isOrthochronous_iff_le_zero]
 
 /-- The identity Lorentz transformation is orthochronous. -/
-lemma id_isOrthochronous : @IsOrthochronous d 1 := by
-  simp [IsOrthochronous]
+lemma id_isOrthochronous : @IsOrthochronous d 1 := by simp [IsOrthochronous]
 
 /-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
 def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => Λ.1 (Sum.inl 0) (Sum.inl 0),
@@ -159,8 +155,7 @@ def orthchroMap : C(LorentzGroup d, ℤ₂) :=
 /-- A Lorentz transformation which is orthochronous maps under `orthchroMap` to `1`
   in `ℤ₂` (the identity element). -/
 lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
-    orthchroMap Λ = 1 := by
-  simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
+    orthchroMap Λ = 1 := by simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
 
 /-- A Lorentz transformation which is not-orthochronous maps under `orthchroMap` to
   the non-identity element of `ℤ₂`. -/
@@ -168,8 +163,7 @@ lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochron
     orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
   simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
     orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]
-  rw [if_neg (by norm_num)]
-  rfl
+  exact if_neg (by norm_num)
 
 /-- The product of two orthochronous Lorentz transformations is orthochronous. -/
 lemma isOrthochronous_mul {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
@@ -231,7 +225,6 @@ to the same element via the homomorphism from `LorentzGroup` to `ℤ₂`. -/
 lemma isOrthochronous_iff_of_orthchroMap_eq {Λ Λ' : LorentzGroup d}
     (h : orthchroMap Λ = orthchroMap Λ') : IsOrthochronous Λ ↔ IsOrthochronous Λ' := by
   rw [IsOrthochronous.iff_in_orthchroRep_ker, IsOrthochronous.iff_in_orthchroRep_ker]
-  rw [MonoidHom.mem_ker, MonoidHom.mem_ker]
   change orthchroMap Λ = 1 ↔ orthchroMap Λ' = 1
   rw [h]
 
diff --git a/Physlib/Relativity/LorentzGroup/Proper.lean b/Physlib/Relativity/LorentzGroup/Proper.lean
index 98bbc1201..a2a051ead 100644
--- a/Physlib/Relativity/LorentzGroup/Proper.lean
+++ b/Physlib/Relativity/LorentzGroup/Proper.lean
@@ -43,8 +43,7 @@ local notation "ℤ₂" => Multiplicative (ZMod 2)
 instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
 
 /-- The topological space defined by `ℤ₂` is discrete. -/
-instance : DiscreteTopology ℤ₂ := by
-  exact discreteTopology_iff_forall_isOpen.mpr fun _ => trivial
+instance : DiscreteTopology ℤ₂ := discreteTopology_iff_forall_isOpen.mpr fun _ => trivial
 
 /-- The instance of a topological group on `ℤ₂` defined via the discrete topology. -/
 instance : IsTopologicalGroup ℤ₂ := IsTopologicalGroup.mk
@@ -74,8 +73,7 @@ lemma detContinuous_eq_one (Λ : LorentzGroup d) :
   refine Iff.intro (fun h => ?_) (fun h => ?_)
   · by_contra hn
     have h' := h hn
-    change (1 : Fin 2) = (0 : Fin 2) at h'
-    simp only [Fin.isValue, one_ne_zero] at h'
+    exact (show (1 : Fin 2) ≠ 0 by decide) h'
   · intro h'
     exact False.elim (h' h)
 
@@ -86,8 +84,7 @@ lemma detContinuous_eq_zero (Λ : LorentzGroup d) :
   refine Iff.intro (fun h => ?_) (fun h => ?_)
   · by_contra hn
     rw [if_pos] at h
-    · change (0 : Fin 2) = (1 : Fin 2) at h
-      simp only [Fin.isValue, zero_ne_one] at h
+    · exact (show (0 : Fin 2) ≠ 1 by decide) h
     · cases' det_eq_one_or_neg_one Λ with h2 h2
       · simp_all only [ite_true]
       · simp_all only [not_true_eq_false]
@@ -109,14 +106,10 @@ lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
     simp only [Nat.reduceAdd]
   | .inl h1, .inr h2 =>
     rw [h1, h2, (detContinuous_eq_one Λ).mpr h1, (detContinuous_eq_zero Λ').mpr h2]
-    change (0 : Fin 2) = (1 : Fin 2) ↔ _
-    simp only [zero_ne_one, false_iff]
-    linarith
+    exact iff_of_false (by decide) (by linarith)
   | .inr h1, .inl h2 =>
     rw [h1, h2, (detContinuous_eq_zero Λ).mpr h1, (detContinuous_eq_one Λ').mpr h2]
-    change (1 : Fin 2) = (0 : Fin 2) ↔ _
-    simp only [one_ne_zero, false_iff]
-    linarith
+    exact iff_of_false (show (1 : Fin 2) ≠ 0 by decide) (by linarith)
 
 /-- The representation taking a Lorentz matrix to its determinant. -/
 @[simps!]
@@ -193,8 +186,7 @@ lemma isProper_iff (Λ : LorentzGroup d) : IsProper Λ ↔ detRep Λ = 1 := by
   simp only [IsProper, lorentzGroupIsGroup_one_coe, det_one]
 
 /-- The identity Lorentz transformation is proper. -/
-lemma isProper_id : @IsProper d 1 := by
-  simp [IsProper]
+lemma isProper_id : @IsProper d 1 := by simp [IsProper]
 
 /-- If two Lorentz transformations are in the same connected component, and one is proper then
   the other is also proper. -/
diff --git a/Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean b/Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean
index 83ca99d28..46ba4a485 100644
--- a/Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean
+++ b/Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean
@@ -52,8 +52,7 @@ def toRotation {d} (Λ : LorentzGroup.restricted d) : Rotations d :=
     erw [generalizedBoost_apply_fst]
     simp
   · refine isProper_mul ?_ Λ.2.1
-    rw [generalizedBoost_inv]
-    exact generalizedBoost_isProper _ _⟩
+    simpa [generalizedBoost_inv] using generalizedBoost_isProper _ _⟩
 
 @[fun_prop]
 lemma toRotation_continuous {d} :
diff --git a/Physlib/Relativity/LorentzGroup/Rotations.lean b/Physlib/Relativity/LorentzGroup/Rotations.lean
index f476e2c64..4103c4fbe 100644
--- a/Physlib/Relativity/LorentzGroup/Rotations.lean
+++ b/Physlib/Relativity/LorentzGroup/Rotations.lean
@@ -46,8 +46,7 @@ def Rotations (d) : Subgroup (LorentzGroup d) where
     · simpa [inv_eq_dual, IsProper] using h.2
 
 lemma mem_rotations_iff {d} (Λ : LorentzGroup d) :
-    Λ ∈ Rotations d ↔ Λ.1 (Sum.inl 0) (Sum.inl 0) = 1 ∧ IsProper Λ := by
-  rfl
+    Λ ∈ Rotations d ↔ Λ.1 (Sum.inl 0) (Sum.inl 0) = 1 ∧ IsProper Λ := Iff.rfl
 
 @[simp]
 lemma transpose_mem_rotations {d} (Λ : LorentzGroup d) :
@@ -78,8 +77,7 @@ def ofSpecialOrthogonal {d} :
     let M : Matrix (Fin d) (Fin d) ℝ := fun i j => Λ.1 (Sum.inr i) (Sum.inr j)
     have h1 : Matrix.fromBlocks 1 0 0 M = Λ.1 := by
       have h1 : LorentzGroup.toVector Λ = Lorentz.Vector.basis (Sum.inl 0) := by
-        rw [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one]
-        exact h.1
+        simpa [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one] using h.1
       have h2 : LorentzGroup.toVector (LorentzGroup.transpose Λ) =
           Lorentz.Vector.basis (Sum.inl 0) := by
         rw [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one]
@@ -123,16 +121,14 @@ def ofSpecialOrthogonal {d} :
     simp only [Submonoid.coe_mul, MulMemClass.mk_mul_mk]
     apply Subtype.ext
     simp [Matrix.fromBlocks_multiply]
-  left_inv Λ := by
-    simp
+  left_inv Λ := by simp
   right_inv Λ := by
     match Λ with
     | ⟨Λ, h⟩ =>
     let M : Matrix (Fin d) (Fin d) ℝ := fun i j => Λ.1 (Sum.inr i) (Sum.inr j)
     have h1 : Matrix.fromBlocks 1 0 0 M = Λ.1 := by
       have h1 : LorentzGroup.toVector Λ = Lorentz.Vector.basis (Sum.inl 0) := by
-        rw [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one]
-        exact h.1
+        simpa [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one] using h.1
       have h2 : LorentzGroup.toVector (LorentzGroup.transpose Λ) =
           Lorentz.Vector.basis (Sum.inl 0) := by
         rw [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one]
@@ -181,8 +177,7 @@ lemma rotations_subset_restricted (d) : Rotations d ≤ LorentzGroup.restricted
 @[simp]
 lemma toVector_rotation {d} (Λ : Rotations d) :
     LorentzGroup.toVector Λ.1= Lorentz.Vector.basis (Sum.inl 0) := by
-  rw [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one]
-  exact Λ.2.1
+  simpa [LorentzGroup.toVector_eq_basis_iff_timeComponent_eq_one] using Λ.2.1
 
 end LorentzGroup
 
diff --git a/Physlib/Relativity/LorentzGroup/ToVector.lean b/Physlib/Relativity/LorentzGroup/ToVector.lean
index 08af8b687..42c01950b 100644
--- a/Physlib/Relativity/LorentzGroup/ToVector.lean
+++ b/Physlib/Relativity/LorentzGroup/ToVector.lean
@@ -25,17 +25,14 @@ open Vector
 def toVector {d : ℕ} (Λ : LorentzGroup d) : Lorentz.Vector d := Λ • basis (Sum.inl 0)
 
 lemma toVector_mul {d : ℕ} (Λ₁ Λ₂ : LorentzGroup d) :
-    toVector (Λ₁ * Λ₂) = Λ₁ • toVector Λ₂ := by
-  rw [toVector, toVector, smul_smul]
+    toVector (Λ₁ * Λ₂) = Λ₁ • toVector Λ₂ := by rw [toVector, toVector, smul_smul]
 
 lemma toVector_neg {d : ℕ} (Λ : LorentzGroup d) :
-    toVector (-Λ) = -toVector Λ := by
-  simp [toVector, neg_smul]
+    toVector (-Λ) = -toVector Λ := by simp [toVector, neg_smul]
 
 @[simp]
 lemma toVector_apply {d : ℕ} (Λ : LorentzGroup d)
-    (i : Fin 1 ⊕ Fin d) : (toVector Λ) i = Λ.1 i (Sum.inl 0) := by
-  simp [toVector, smul_eq_sum]
+    (i : Fin 1 ⊕ Fin d) : (toVector Λ) i = Λ.1 i (Sum.inl 0) := by simp [toVector, smul_eq_sum]
 
 lemma toVector_eq_fun {d : ℕ} (Λ : LorentzGroup d) :
     toVector Λ = (fun i => Λ.1 i (Sum.inl 0)) := by
@@ -54,13 +51,11 @@ lemma toVector_continuous {d : ℕ} : Continuous (toVector (d := d)) := by
   fun_prop
 
 lemma toVector_timeComponent {d : ℕ} (Λ : LorentzGroup d) :
-    (toVector Λ).timeComponent = Λ.1 (Sum.inl 0) (Sum.inl 0) := by
-  simp
+    (toVector Λ).timeComponent = Λ.1 (Sum.inl 0) (Sum.inl 0) := by simp
 
 @[simp]
 lemma toVector_minkowskiProduct_self {d : ℕ} (Λ : LorentzGroup d) :
-    ⟪toVector Λ, toVector Λ⟫ₘ = 1 := by
-  simp [toVector, minkowskiMatrix.inl_0_inl_0]
+    ⟪toVector Λ, toVector Λ⟫ₘ = 1 := by simp [toVector, minkowskiMatrix.inl_0_inl_0]
 
 lemma one_le_abs_timeComponent {d : ℕ} (Λ : LorentzGroup d) :
     1 ≤ |Λ.1 (Sum.inl 0) (Sum.inl 0)| := by
diff --git a/Physlib/Relativity/MinkowskiMatrix.lean b/Physlib/Relativity/MinkowskiMatrix.lean
index d06f5350e..a7c890f57 100644
--- a/Physlib/Relativity/MinkowskiMatrix.lean
+++ b/Physlib/Relativity/MinkowskiMatrix.lean
@@ -113,8 +113,7 @@ We prove some simple properties related to the components of the Minkowski matri
 
 /-- The `time-time` component of the Minkowski matrix is `1`. -/
 @[simp]
-lemma inl_0_inl_0 : @minkowskiMatrix d (Sum.inl 0) (Sum.inl 0) = 1 := by
-  rfl
+lemma inl_0_inl_0 : @minkowskiMatrix d (Sum.inl 0) (Sum.inl 0) = 1 := by rfl
 
 /-- The space diagonal components of the Minkowski matrix are `-1`. -/
 @[simp]
@@ -126,8 +125,7 @@ lemma inr_i_inr_i (i : Fin d) : @minkowskiMatrix d (Sum.inr i) (Sum.inr i) = -1
 lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 := by
   aesop (add safe forward as_diagonal)
 
-lemma η_diag_ne_zero {μ : Fin 1 ⊕ Fin d} : η μ μ ≠ 0 := by
-  aesop (add safe forward as_diagonal)
+lemma η_diag_ne_zero {μ : Fin 1 ⊕ Fin d} : η μ μ ≠ 0 := by aesop (add safe forward as_diagonal)
 
 /-!
 
@@ -140,8 +138,7 @@ as well as other properties related to squaring the Minkowski matrix.
 
 /-- The Minkowski matrix is self-inverting. -/
 @[simp]
-lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
-  simp [as_block, fromBlocks_multiply]
+lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by simp [as_block, fromBlocks_multiply]
 
 /-- Multiplying any element on the diagonal of the Minkowski matrix by itself gives `1`. -/
 @[simp]
@@ -162,8 +159,7 @@ The Minkowski matrix is symmetric, due to it being diagonal.
 
 /-- The Minkowski matrix is symmetric. -/
 @[simp]
-lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
-  simp [as_diagonal]
+lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by simp [as_diagonal]
 
 /-!
 
@@ -177,8 +173,7 @@ We show the determinant of the Minkowski matrix is equal to `(-1)^d` where
 /-- The determinant of the Minkowski matrix is equal to `-1` to the power
   of the number of spatial dimensions. -/
 @[simp]
-lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
-  simp [as_diagonal]
+lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by simp [as_diagonal]
 
 /-!
 
@@ -205,14 +200,12 @@ We show properties of the action of the Minkowski matrix on vectors.
 /-- The time components of a vector acted on by the Minkowski matrix remains unchanged. -/
 @[simp]
 lemma mulVec_inl_0 (v : (Fin 1 ⊕ Fin d) → ℝ) :
-    (η *ᵥ v) (Sum.inl 0) = v (Sum.inl 0) := by
-  simp [as_diagonal, mulVec_diagonal]
+    (η *ᵥ v) (Sum.inl 0) = v (Sum.inl 0) := by simp [as_diagonal, mulVec_diagonal]
 
 /-- The space components of a vector acted on by the Minkowski matrix swaps sign. -/
 @[simp]
 lemma mulVec_inr_i (v : (Fin 1 ⊕ Fin d) → ℝ) (i : Fin d) :
-    (η *ᵥ v) (Sum.inr i) = - v (Sum.inr i) := by
-  simp [as_diagonal, mulVec_diagonal]
+    (η *ᵥ v) (Sum.inr i) = - v (Sum.inr i) := by simp [as_diagonal, mulVec_diagonal]
 
 /-!
 
@@ -285,8 +278,7 @@ lemma dual_dual : Function.Involutive (@dual d) := by
 
 /-- The Minkowski dual commutes with the transpose. -/
 @[simp]
-lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
-  simp [dual, mul_assoc]
+lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by simp [dual, mul_assoc]
 
 /-!
 
@@ -296,8 +288,7 @@ lemma dual_transpose : dual Λᵀ = (dual Λ)ᵀ := by
 
 /-- The Minkowski dual preserves the Minkowski matrix. -/
 @[simp]
-lemma dual_eta : @dual d η = η := by
-  simp [dual]
+lemma dual_eta : @dual d η = η := by simp [dual]
 
 /-!
 
@@ -324,13 +315,11 @@ We show a number of properties related to the components of the duals.
 /-- Expansion of the components of the Minkowski dual in terms of the components
   of the original matrix. -/
 lemma dual_apply (μ ν : Fin 1 ⊕ Fin d) :
-    dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by
-  simp [dual, as_diagonal]
+    dual Λ μ ν = η μ μ * Λ ν μ * η ν ν := by simp [dual, as_diagonal]
 
 /-- The components of the Minkowski dual of a matrix multiplied by the Minkowski matrix
   in terms of the original matrix. -/
 lemma dual_apply_minkowskiMatrix (μ ν : Fin 1 ⊕ Fin d) :
-    dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by
-  simp [dual_apply, mul_assoc]
+    dual Λ μ ν * η ν ν = η μ μ * Λ ν μ := by simp [dual_apply, mul_assoc]
 
 end minkowskiMatrix
diff --git a/Physlib/Relativity/PauliMatrices/AsTensor.lean b/Physlib/Relativity/PauliMatrices/AsTensor.lean
index d1fcdbb6f..7bbfebbe8 100644
--- a/Physlib/Relativity/PauliMatrices/AsTensor.lean
+++ b/Physlib/Relativity/PauliMatrices/AsTensor.lean
@@ -42,8 +42,7 @@ lemma asTensor_expand_complexContrBasis : asTensor =
     complexContrBasis (Sum.inl 0) ⊗ₜ leftRightToMatrix.symm (pauliBasis (Sum.inl 0))
     + complexContrBasis (Sum.inr 0) ⊗ₜ leftRightToMatrix.symm (pauliBasis (Sum.inr 0))
     + complexContrBasis (Sum.inr 1) ⊗ₜ leftRightToMatrix.symm (pauliBasis (Sum.inr 1))
-    + complexContrBasis (Sum.inr 2) ⊗ₜ leftRightToMatrix.symm (pauliBasis (Sum.inr 2)) := by
-  rfl
+    + complexContrBasis (Sum.inr 2) ⊗ₜ leftRightToMatrix.symm (pauliBasis (Sum.inr 2)) := rfl
 
 /-- The expansion of the pauli matrix `σ₀` in terms of a basis of tensor product vectors. -/
 lemma leftRightToMatrix_σSA_inl_0_expand : leftRightToMatrix.symm (pauliBasis (Sum.inl 0)) =
@@ -100,8 +99,7 @@ def asConsTensor :
     map_add' := fun x y => by
       simp only [add_smul]
     map_smul' := fun m x => by
-      simp only [smul_smul]
-      rfl
+      simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
     isIntertwining' M := by
       refine LinearMap.ext fun x : ℂ => ?_
       change x • asTensor =
@@ -173,9 +171,7 @@ def asConsTensor :
 
 /-- The map `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ leftHanded ⊗ rightHanded` corresponding
   to Pauli matrices, when evaluated on `1` corresponds to the tensor `PauliMatrix.asTensor`. -/
-lemma asConsTensor_apply_one : asConsTensor (1 : ℂ) = asTensor := by
-  change (1 : ℂ) • asTensor = asTensor
-  simp only [one_smul]
+lemma asConsTensor_apply_one : asConsTensor (1 : ℂ) = asTensor := by simp [asConsTensor]
 
 end
 end PauliMatrix
diff --git a/Physlib/Relativity/PauliMatrices/Basic.lean b/Physlib/Relativity/PauliMatrices/Basic.lean
index 471937671..1f5c31824 100644
--- a/Physlib/Relativity/PauliMatrices/Basic.lean
+++ b/Physlib/Relativity/PauliMatrices/Basic.lean
@@ -60,8 +60,7 @@ scoped[PauliMatrix] notation "σ2" => σ (Sum.inr 1)
 /-- The Pauli matrix corresponding to the matrix `!![1, 0; 0, -1]`. -/
 scoped[PauliMatrix] notation "σ3" => σ (Sum.inr 2)
 
-lemma pauliMatrix_inl_zero_eq_one : pauliMatrix (Sum.inl 0) = 1 := by
-  dsimp [pauliMatrix]
+lemma pauliMatrix_inl_zero_eq_one : pauliMatrix (Sum.inl 0) = 1 := rfl
 
 /-!
 
@@ -73,8 +72,7 @@ lemma pauliMatrix_selfAdjoint (μ : Fin 1 ⊕ Fin 3) :
     (σ μ)ᴴ = σ μ := by
   fin_cases μ
   all_goals
-    dsimp [pauliMatrix]
-    rw [eta_fin_two _ᴴ]
+    rw [pauliMatrix.eq_def, eta_fin_two _ᴴ]
     simp
   ext i j
   fin_cases i <;> fin_cases j
@@ -92,8 +90,7 @@ lemma pauliMatrix_mul_self (μ : Fin 1 ⊕ Fin 3) :
     (σ μ) * (σ μ) = 1 := by
   fin_cases μ
   all_goals
-    dsimp [pauliMatrix]
-    simp [one_fin_two]
+    simp [pauliMatrix, one_fin_two]
 
 instance pauliMatrixInvertiable (μ : Fin 1 ⊕ Fin 3) : Invertible (σ μ) := by
   use σ μ
@@ -143,28 +140,24 @@ lemma σ1_σ1_trace : Matrix.trace (σ1 * σ1) = 2 := by simp
 
 /-- The trace of `σ1` multiplied by `σ2` is equal to `0`. -/
 @[simp]
-lemma σ1_σ2_trace : Matrix.trace (σ1 * σ2) = 0 := by
-  simp [pauliMatrix]
+lemma σ1_σ2_trace : Matrix.trace (σ1 * σ2) = 0 := by simp [pauliMatrix]
 
 /-- The trace of `σ1` multiplied by `σ3` is equal to `0`. -/
 @[simp]
-lemma σ1_σ3_trace : Matrix.trace (σ1 * σ3) = 0 := by
-  simp [pauliMatrix]
+lemma σ1_σ3_trace : Matrix.trace (σ1 * σ3) = 0 := by simp [pauliMatrix]
 
 /-- The trace of `σ2` multiplied by `σ0` is equal to `0`. -/
 lemma σ2_σ0_trace : Matrix.trace (σ2 * σ0) = 0 := by simp [pauliMatrix]
 
 /-- The trace of `σ2` multiplied by `σ1` is equal to `0`. -/
-lemma σ2_σ1_trace : Matrix.trace (σ2 * σ1) = 0 := by
-  simp [pauliMatrix]
+lemma σ2_σ1_trace : Matrix.trace (σ2 * σ1) = 0 := by simp [pauliMatrix]
 
 /-- The trace of `σ2` multiplied by `σ2` is equal to `2`. -/
 lemma σ2_σ2_trace : Matrix.trace (σ2 * σ2) = 2 := by simp
 
 /-- The trace of `σ2` multiplied by `σ3` is equal to `0`. -/
 @[simp]
-lemma σ2_σ3_trace : Matrix.trace (σ2 * σ3) = 0 := by
-  simp [pauliMatrix]
+lemma σ2_σ3_trace : Matrix.trace (σ2 * σ3) = 0 := by simp [pauliMatrix]
 
 /-- The trace of `σ3` multiplied by `σ0` is equal to `0`. -/
 lemma σ3_σ0_trace : Matrix.trace (σ3 * σ0) = 0 := by simp [pauliMatrix]
diff --git a/Physlib/Relativity/SL2C/Basic.lean b/Physlib/Relativity/SL2C/Basic.lean
index 3f7ba29f2..673763862 100644
--- a/Physlib/Relativity/SL2C/Basic.lean
+++ b/Physlib/Relativity/SL2C/Basic.lean
@@ -72,8 +72,7 @@ lemma toSelfAdjointMap_apply (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
         (star_eq_conjTranspose A.1).symm.trans $ selfAdjoint.mem_iff.mp A.2]⟩ := rfl
 
 lemma toSelfAdjointMap_apply_det (M : SL(2, ℂ)) (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
-    det ((toSelfAdjointMap M) A).1 = det A.1 := by
-  simp [toSelfAdjointMap]
+    det ((toSelfAdjointMap M) A).1 = det A.1 := by simp [toSelfAdjointMap]
 
 lemma toSelfAdjointMap_apply_pauliBasis'_inl (M : SL(2, ℂ)) :
     toSelfAdjointMap M (PauliMatrix.pauliBasis' (Sum.inl 0)) =
@@ -152,8 +151,7 @@ lemma toMatrix_apply_contrMod (M : SL(2, ℂ)) (v : ContrMod 3) :
     *ᵥ (((Finsupp.linearEquivFunOnFinite ℝ ℝ (Fin 1 ⊕ Fin 3))
     (PauliMatrix.pauliBasis'.repr a)))) = _
   apply congrArg
-  erw [LinearMap.toMatrix_mulVec_repr]
-  rfl
+  exact LinearMap.toMatrix_mulVec_repr PauliMatrix.pauliBasis' _ _ a
 
 lemma toMatrix_mem_lorentzGroup (M : SL(2, ℂ)) : toMatrix M ∈ LorentzGroup 3 := by
   rw [LorentzGroup.mem_iff_norm]
@@ -167,17 +165,14 @@ lemma toMatrix_mem_lorentzGroup (M : SL(2, ℂ)) : toMatrix M ∈ LorentzGroup 3
 @[simps!]
 def toLorentzGroup : SL(2, ℂ) →* LorentzGroup 3 where
   toFun M := ⟨toMatrix M, toMatrix_mem_lorentzGroup M⟩
-  map_one' := by
-    simp only [_root_.map_one]
-    rfl
+  map_one' := Subtype.ext (by simp)
   map_mul' M N := by
     ext1
     simp only [_root_.map_mul, lorentzGroupIsGroup_mul_coe]
 
 lemma toLorentzGroup_eq_pauliBasis' (M : SL(2, ℂ)) :
     toLorentzGroup M = LinearMap.toMatrix
-    PauliMatrix.pauliBasis' PauliMatrix.pauliBasis' (toSelfAdjointMap M) := by
-  rfl
+    PauliMatrix.pauliBasis' PauliMatrix.pauliBasis' (toSelfAdjointMap M) := rfl
 
 lemma toSelfAdjointMap_basis (i : Fin 1 ⊕ Fin 3) :
     toSelfAdjointMap M (PauliMatrix.pauliBasis' i) =
@@ -235,8 +230,7 @@ lemma toLorentzGroup_fst_col (M : SL(2, ℂ)) :
   refine sub_eq_zero.mp ?_
   refine Fintype.linearIndependent_iff.mp PauliMatrix.pauliBasis'.linearIndependent
     (fun x => ((toLorentzGroup M).1 x (Sum.inl 0) - k x)) ?_ μ
-  simp only [sub_smul]
-  exact h1x
+  simpa only [sub_smul] using h1x
 
 /-- The first element of the image of `SL(2, ℂ)` in the Lorentz group. -/
 lemma toLorentzGroup_inl_inl (M : SL(2, ℂ)) :
diff --git a/Physlib/Relativity/SL2C/SelfAdjoint.lean b/Physlib/Relativity/SL2C/SelfAdjoint.lean
index d8e873171..1290658d7 100644
--- a/Physlib/Relativity/SL2C/SelfAdjoint.lean
+++ b/Physlib/Relativity/SL2C/SelfAdjoint.lean
@@ -122,8 +122,7 @@ lemma toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (detM
   have he' : Mᴴ = !![conj x, 0; _, conj y] :=
     Matrix.ext fun | 0, 0 | 1, 0 | 1, 1 => rfl | 0, 1 => by simp [hM10]
   have hxy : x * y = 1 := by rw [show x * y = M.det by rw [he]; simp, detM]
-  have detA_one : normSq x * normSq y = 1 := by
-    rw [← Complex.normSq_mul, hxy, Complex.normSq_one]
+  have detA_one : normSq x * normSq y = 1 := by rw [← Complex.normSq_mul, hxy, Complex.normSq_one]
   have detD_one : D.det = 1 :=
     let z := x * conj y
     have k₀ : (M * E₂ * Mᴴ) 0 1 = z := by rw [he', he]; simp [E₂, z]
diff --git a/Physlib/Relativity/Special/ProperTime.lean b/Physlib/Relativity/Special/ProperTime.lean
index 4ca7759fa..0771bc087 100644
--- a/Physlib/Relativity/Special/ProperTime.lean
+++ b/Physlib/Relativity/Special/ProperTime.lean
@@ -36,9 +36,7 @@ def properTime {d : ℕ} (q p : SpaceTime d) : ℝ :=
 lemma properTime_pos_ofTimeLike {d : ℕ} (q p : SpaceTime d)
     (h : causalCharacter (p - q) = .timeLike) :
     0 < properTime q p := by
-  rw [properTime]
-  refine sqrt_pos_of_pos ?_
-  exact (timeLike_iff_norm_sq_pos (p - q)).mp h
+  simpa [properTime] using sqrt_pos_of_pos ((timeLike_iff_norm_sq_pos (p - q)).mp h)
 
 lemma properTime_zero_ofLightLike {d : ℕ} (q p : SpaceTime d)
     (h : causalCharacter (p - q) = .lightLike) :
diff --git a/Physlib/Relativity/Tensors/Basic.lean b/Physlib/Relativity/Tensors/Basic.lean
index efc05c52f..0c9805737 100644
--- a/Physlib/Relativity/Tensors/Basic.lean
+++ b/Physlib/Relativity/Tensors/Basic.lean
@@ -102,8 +102,7 @@ lemma congr_mid {n : ℕ} {c : Fin n → C} (c' : C) (p : Pure S c)
   subst hi
   simp only [LinearEquiv.cast_apply, cast_eq]
   symm
-  apply congr_right
-  exact h
+  exact congr_right p i j h
 
 lemma map_mid_move_left {n n1 : ℕ} {c : Fin n → C} {c1 : Fin n1 → C} (p : Pure S c)
     (p' : Pure S c1) {c' : C}
@@ -136,8 +135,7 @@ lemma update_eq_function_update {n : ℕ} {c : Fin n → C} [inst : DecidableEq
 
 @[simp]
 lemma update_same {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n)] (p : Pure S c) (i : Fin n)
-    (x : V (c i)) : (update p i x) i = x := by
-  simp [update]
+    (x : V (c i)) : (update p i x) i = x := by simp [update]
 
 lemma update_diff {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n)] (p : Pure S c) (i j : Fin n)
     (x : V (c i)) (hij : i ≠ j) : (update p i x) j = p j := by
@@ -149,8 +147,7 @@ lemma update_succAbove_apply {n : ℕ} {c : Fin (n + 1) → C} [inst : Decidable
     (p : Pure S c) (i : Fin (n + 1)) (j : Fin n) (x : V (c (i.succAbove j))) :
     update p (i.succAbove j) x i = p i := by
   simp only [update]
-  rw [Function.update_of_ne]
-  exact Fin.ne_succAbove i j
+  simp [Function.update_of_ne, Fin.ne_succAbove i j]
 
 @[simp]
 lemma toTensor_update_add {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n)] (p : Pure S c)
@@ -161,8 +158,7 @@ lemma toTensor_update_add {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n
 @[simp]
 lemma toTensor_update_smul {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n)] (p : Pure S c)
     (i : Fin n) (r : k) (y : V (c i)) :
-    (update p i (r • y)).toTensor = r • (update p i y).toTensor := by
-  simp [toTensor, update]
+    (update p i (r • y)).toTensor = r • (update p i y).toTensor := by simp [toTensor, update]
 
 /-- Given a list of indices `c` of length `n + 1`, a pure tensor `p` and an `i : Fin (n + 1)`, then
   `drop p i` is the tensor `p` with it's `i`th part dropped.
@@ -184,8 +180,7 @@ lemma update_succAbove_drop {n : ℕ} {c : Fin (n + 1) → C} [inst : DecidableE
   · rw [Function.update_of_ne h, Function.update_of_ne]
     · rfl
     · simp only [ne_eq]
-      rw [Function.Injective.eq_iff (Fin.succAbove_right_injective (p := i))]
-      exact h
+      simpa [Function.Injective.eq_iff (Fin.succAbove_right_injective (p := i))] using h
 
 @[simp]
 lemma update_drop_self {n : ℕ} {c : Fin (n + 1) → C} [inst : DecidableEq (Fin (n + 1))]
@@ -193,8 +188,7 @@ lemma update_drop_self {n : ℕ} {c : Fin (n + 1) → C} [inst : DecidableEq (Fi
     (update p i x).drop i = p.drop i := by
   ext k
   simp only [Function.comp_apply, drop, update]
-  rw [Function.update_of_ne]
-  exact Fin.succAbove_ne i k
+  simp [Function.update_of_ne, Fin.succAbove_ne i k]
 
 /-!
 
@@ -219,8 +213,7 @@ lemma component_eq_drop {n : ℕ} {c : Fin (n + 1) → C} (p : Pure S c) (i : Fi
     p.component φ = ((b (c i)).repr (p i) (φ i)) *
     ((drop p i).component (fun j => φ (i.succAbove j))) := by
   simp only [component, Function.comp_apply]
-  rw [Fin.prod_univ_succAbove _ i]
-  rfl
+  exact Fin.prod_univ_succAbove _ i
 
 @[simp]
 lemma component_update_add {n : ℕ} [inst : DecidableEq (Fin n)]
@@ -258,13 +251,11 @@ noncomputable def componentMap {n : ℕ} (c : Fin n → C) :
     exact component_update_add p i x y b
   map_update_smul' p i x y := by
     ext b
-    change component (update p i (x • y)) b = x * component (update p i y) b
     exact component_update_smul p i x y b
 
 @[simp]
 lemma componentMap_apply {n : ℕ} (c : Fin n → C)
-    (p : Pure S c) : componentMap c p = p.component := by
-  rfl
+    (p : Pure S c) : componentMap c p = p.component := rfl
 
 /-- Given an component idx `b` in `ComponentIdx c`, `basisVector c b` is the pure tensor
   formed by `S.basis (c i) (b i)`. -/
@@ -288,8 +279,7 @@ lemma component_basisVector {n : ℕ} (c : Fin n → C) (b1 b2 : ComponentIdx (S
       simp_all only [not_forall]
       obtain ⟨w, h⟩ := h
       refine Finset.prod_eq_zero (Finset.mem_univ i) ?_
-      rw [Finsupp.single_eq_of_ne]
-      exact fun a => hi (id (Eq.symm a))
+      exact Finsupp.single_eq_of_ne fun a => hi (id (Eq.symm a))
 
 end Pure
 
@@ -327,10 +317,8 @@ lemma componentMap_pure {n : ℕ} (c : Fin n → C)
 def ofComponents {n : ℕ} (c : Fin n → C) :
     (ComponentIdx (S := S) c → k) →ₗ[k] S.Tensor c where
   toFun f := ∑ b, f b • (Pure.basisVector c b).toTensor
-  map_add' fb gb := by
-    simp [add_smul, Finset.sum_add_distrib]
-  map_smul' fb r := by
-    simp [smul_smul, Finset.smul_sum]
+  map_add' fb gb := by simp [add_smul, Finset.sum_add_distrib]
+  map_smul' fb r := by simp [smul_smul, Finset.smul_sum]
 
 @[simp]
 lemma componentMap_ofComponents {n : ℕ} (c : Fin n → C) (f : ComponentIdx c → k) :
@@ -487,8 +475,6 @@ lemma actionP_cast {g : G} {p : Pure S c} (h : c = c1) :
 lemma drop_actionP {n : ℕ} {c : Fin (n + 1) → C} {i : Fin (n + 1)} {p : Pure S c} (g : G) :
     (g • p).drop i = g • (p.drop i) := by
   ext j
-  rw [drop, actionP_eq, actionP_eq]
-  simp only [Function.comp_apply]
   rfl
 
 end Pure
@@ -506,19 +492,16 @@ lemma actionT_eq {g : G} {t : S.Tensor c} : g • t =
     PiTensorProduct.map (fun i => rep (c i) g) t := rfl
 
 noncomputable instance actionT : MulAction G (S.Tensor c) where
-  one_smul t := by
-    simp [actionT_eq]
+  one_smul t := by simp [actionT_eq]
   mul_smul g g' t := by
     simp [actionT_eq]
     change _ = (PiTensorProduct.map (fun i => rep (c i) g) ∘ₗ
       (PiTensorProduct.map (fun i => rep (c i) g'))) t
-    rw [← PiTensorProduct.map_comp]
-    rfl
+    exact LinearMap.congr_fun (PiTensorProduct.map_comp _ _) t
 
 lemma actionT_pure {g : G} {p : Pure S c} :
     g • p.toTensor = Pure.toTensor (g • p) := by
-  rw [actionT_eq, Pure.toTensor, PiTensorProduct.map_tprod]
-  rfl
+  exact PiTensorProduct.map_tprod (R := k) (fun i => rep (c i) g) p
 
 lemma actionT_add {g : G} {t1 t2 : S.Tensor c} :
     g • (t1 + t2) = g • t1 + g • t2 := by
@@ -531,14 +514,10 @@ lemma actionT_smul {g : G} {r : k} {t : S.Tensor c} :
   rw [actionT_eq, actionT_eq (S := S)]
   simp
 
-lemma actionT_zero {g : G} : g • (0 : S.Tensor c) = 0 := by
-  simp [actionT_eq]
+lemma actionT_zero {g : G} : g • (0 : S.Tensor c) = 0 := by simp [actionT_eq]
 
 lemma actionT_neg {g : G} {t : S.Tensor c} :
-    g • (-t) = -(g • t) := by
-  rw [actionT_eq]
-  simp only [map_neg, neg_inj]
-  rfl
+    g • (-t) = -(g • t) := by simp [actionT_eq]
 
 noncomputable instance : DistribMulAction G (S.Tensor c) where
   smul_zero g := by simp [actionT_zero]
@@ -775,8 +754,7 @@ noncomputable def toField {c : Fin 0 → C} : S.Tensor c →ₗ[k] k :=
   (PiTensorProduct.isEmptyEquiv (Fin 0)).toLinearMap
 
 lemma toField_default {c : Fin 0 → C} :
-    toField (Pure.toTensor default : S.Tensor c) = 1 := by
-  simp [toField, Pure.toTensor]
+    toField (Pure.toTensor default : S.Tensor c) = 1 := by simp [toField, Pure.toTensor]
 
 lemma toField_injective {c : Fin 0 → C} :
     Function.Injective (toField : S.Tensor c → k) :=
@@ -805,8 +783,7 @@ lemma toField_basis_default {c : Fin 0 → C} :
   simp [basis_apply]
 
 lemma toField_basis {c : Fin 0 → C} (b : ComponentIdx (S := S) c) :
-    toField (basis c b) = 1 := by
-  simp [basis_apply]
+    toField (basis c b) = 1 := by simp [basis_apply]
 
 lemma toField_eq_repr {c : Fin 0 → C} (t : Tensor S c) :
     t.toField = (basis c).repr t (fun j => Fin.elim0 j) := by
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Matrix/Pre.lean b/Physlib/Relativity/Tensors/ComplexTensor/Matrix/Pre.lean
index d0175eb43..576b8bde7 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Matrix/Pre.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Matrix/Pre.lean
@@ -118,8 +118,7 @@ lemma contrContrToMatrix_ρ (v : (ContrℂModule ⊗[ℂ] ContrℂModule)) (M :
     have h1 := (LinearMap.toMatrix_mulVec_repr (complexContrBasis.tensorProduct complexContrBasis)
       (complexContrBasis.tensorProduct complexContrBasis)
       (TensorProduct.map (ContrℂModule.SL2CRep M) (ContrℂModule.SL2CRep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
@@ -162,8 +161,7 @@ lemma coCoToMatrix_ρ (v : (CoℂModule ⊗[ℂ] CoℂModule)) (M : SL(2,ℂ)) :
     have h1 := (LinearMap.toMatrix_mulVec_repr (complexCoBasis.tensorProduct complexCoBasis)
       (complexCoBasis.tensorProduct complexCoBasis)
       (TensorProduct.map (CoℂModule.SL2CRep M) (CoℂModule.SL2CRep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
@@ -205,8 +203,7 @@ lemma contrCoToMatrix_ρ (v : (ContrℂModule ⊗[ℂ] CoℂModule)) (M : SL(2,
     have h1 := (LinearMap.toMatrix_mulVec_repr (complexContrBasis.tensorProduct complexCoBasis)
       (complexContrBasis.tensorProduct complexCoBasis)
       (TensorProduct.map (ContrℂModule.SL2CRep M) (CoℂModule.SL2CRep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
@@ -248,8 +245,7 @@ lemma coContrToMatrix_ρ (v : (CoℂModule ⊗[ℂ] ContrℂModule)) (M : SL(2,
     have h1 := (LinearMap.toMatrix_mulVec_repr (complexCoBasis.tensorProduct complexContrBasis)
       (complexCoBasis.tensorProduct complexContrBasis)
       (TensorProduct.map (CoℂModule.SL2CRep M) (ContrℂModule.SL2CRep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean b/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean
index d9f832f19..13fc2fb0e 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean
@@ -85,14 +85,10 @@ open Tensor
 -/
 
 lemma coMetric_eq_fromConstPair : η' = fromConstPair (S := complexLorentzTensor)
-    (c1 := .down) (c2 := .down) Lorentz.coMetric := by
-  rw [Lorentz.coMetric]
-  rfl
+    (c1 := .down) (c2 := .down) Lorentz.coMetric := rfl
 
 lemma contrMetric_eq_fromConstPair : η = fromConstPair (S := complexLorentzTensor)
-    (c1 := .up) (c2 := .up) Lorentz.contrMetric := by
-  rw [Lorentz.contrMetric]
-  rfl
+    (c1 := .up) (c2 := .up) Lorentz.contrMetric := rfl
 
 lemma leftMetric_eq_fromConstPair : εL = fromConstPair Fermion.leftMetric := rfl
 
@@ -109,34 +105,28 @@ lemma dualRightMetric_eq_fromConstPair : εR' = fromConstPair Fermion.dualRightM
 -/
 
 lemma coMetric_eq_fromPairT : η' = fromPairT (Lorentz.coMetricVal) := by
-  rw [coMetric_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Lorentz.coMetric_apply_one
+  simpa only [coMetric_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Lorentz.coMetric_apply_one
 
 lemma contrMetric_eq_fromPairT : η = fromPairT (Lorentz.contrMetricVal) := by
-  rw [contrMetric_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Lorentz.contrMetric_apply_one
+  simpa only [contrMetric_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Lorentz.contrMetric_apply_one
 
 lemma leftMetric_eq_fromPairT : εL = fromPairT (Fermion.leftMetricVal) := by
-  rw [leftMetric_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.leftMetric_apply_one
+  simpa only [leftMetric_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.leftMetric_apply_one
 
 lemma rightMetric_eq_fromPairT : εR = fromPairT (Fermion.rightMetricVal) := by
-  rw [rightMetric_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.rightMetric_apply_one
+  simpa only [rightMetric_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.rightMetric_apply_one
 
 lemma dualLeftMetric_eq_fromPairT : εL' = fromPairT (Fermion.dualLeftMetricVal) := by
-  rw [dualLeftMetric_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.dualLeftMetric_apply_one
+  simpa only [dualLeftMetric_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.dualLeftMetric_apply_one
 
 lemma dualRightMetric_eq_fromPairT : εR' = fromPairT (Fermion.dualRightMetricVal) := by
-  rw [dualRightMetric_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.dualRightMetric_apply_one
+  simpa only [dualRightMetric_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.dualRightMetric_apply_one
 
 /-!
 
@@ -160,8 +150,7 @@ lemma coMetric_eq_complexCoBasisFin4 : η' =
     - fromPairT (complexCoBasisFin4 2 ⊗ₜ[ℂ] complexCoBasisFin4 2)
     - fromPairT (complexCoBasisFin4 3 ⊗ₜ[ℂ] complexCoBasisFin4 3) := by
   rw [coMetric_eq_complexCoBasis]
-  simp [complexCoBasisFin4]
-  rfl
+  simp [complexCoBasisFin4, finSumFinEquiv, Fin.addCases]
 
 open Lorentz in
 lemma contrMetric_eq_complexContrBasis : η =
@@ -179,8 +168,7 @@ lemma contrMetric_eq_complexContrBasisFin4 : η =
     - fromPairT (complexContrBasisFin4 2 ⊗ₜ[ℂ] complexContrBasisFin4 2)
     - fromPairT (complexContrBasisFin4 3 ⊗ₜ[ℂ] complexContrBasisFin4 3) := by
   rw [contrMetric_eq_complexContrBasis]
-  simp [complexContrBasisFin4]
-  rfl
+  simp [complexContrBasisFin4, finSumFinEquiv, Fin.addCases]
 
 open Fermion in
 lemma leftMetric_eq_leftBasis : εL =
@@ -403,34 +391,22 @@ lemma dualRightMetric_eq_ofRat : εR' = ofRat fun f =>
 
 open TensorSpecies
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `coMetric` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_coMetric (g : SL(2,ℂ)) : g • η' = η' := by
-  rw [metricTensor_invariant]
+lemma actionT_coMetric (g : SL(2,ℂ)) : g • η' = η' := metricTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `contrMetric` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_contrMetric (g : SL(2,ℂ)) : g • η = η := by
-  rw [metricTensor_invariant]
+lemma actionT_contrMetric (g : SL(2,ℂ)) : g • η = η := metricTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `leftMetric` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_leftMetric (g : SL(2,ℂ)) : g • εL = εL := by
-  rw [metricTensor_invariant]
+lemma actionT_leftMetric (g : SL(2,ℂ)) : g • εL = εL := metricTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `rightMetric` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_rightMetric (g : SL(2,ℂ)) : g • εR = εR := by
-  rw [metricTensor_invariant]
+lemma actionT_rightMetric (g : SL(2,ℂ)) : g • εR = εR := metricTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `dualLeftMetric` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_dualLeftMetric (g : SL(2,ℂ)) : g • εL' = εL' := by
-  rw [metricTensor_invariant]
+lemma actionT_dualLeftMetric (g : SL(2,ℂ)) : g • εL' = εL' := metricTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `dualRightMetric` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_dualRightMetric (g : SL(2,ℂ)) : g • εR' = εR' := by
-  rw [metricTensor_invariant]
+lemma actionT_dualRightMetric (g : SL(2,ℂ)) : g • εR' = εR' := metricTensor_invariant g
 
 end complexLorentzTensor
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Pre.lean b/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Pre.lean
index 0be7cec11..ce8b812b6 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Pre.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Metrics/Pre.lean
@@ -55,9 +55,7 @@ def contrMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
       a' • contrMetricVal
     map_add' := fun x y => by
       simp only [add_smul]
-    map_smul' := fun m x => by
-      simp only [smul_smul]
-      rfl
+    map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
     isIntertwining' M := by
       refine LinearMap.ext fun x : ℂ => ?_
       change x • contrMetricVal =
@@ -69,9 +67,7 @@ def contrMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
       apply congrArg
       simp only [LorentzGroup.toComplex_mul_minkowskiMatrix_mul_transpose]
 
-lemma contrMetric_apply_one : contrMetric (1 : ℂ) = contrMetricVal := by
-  change (1 : ℂ) • contrMetricVal = contrMetricVal
-  simp only [one_smul]
+lemma contrMetric_apply_one : contrMetric (1 : ℂ) = contrMetricVal := by simp [contrMetric]
 
 /-- The metric `ηᵢᵢ` as an element of `(complexCo ⊗ complexCo).V`. -/
 def coMetricVal : (CoℂModule ⊗[ℂ] CoℂModule) :=
@@ -105,9 +101,7 @@ def coMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     a' • coMetricVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • coMetricVal =
@@ -121,9 +115,7 @@ def coMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     simp only [LorentzGroup.inv_eq_dual, SL2C.toLorentzGroup_apply_coe,
       LorentzGroup.toComplex_transpose_mul_minkowskiMatrix_mul_self]
 
-lemma coMetric_apply_one : coMetric (1 : ℂ) = coMetricVal := by
-  change (1 : ℂ) • coMetricVal = coMetricVal
-  simp only [one_smul]
+lemma coMetric_apply_one : coMetric (1 : ℂ) = coMetricVal := by simp [coMetric]
 
 /-!
 
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/OfRat.lean b/Physlib/Relativity/Tensors/ComplexTensor/OfRat.lean
index 5490f8761..28942f482 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/OfRat.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/OfRat.lean
@@ -50,8 +50,7 @@ noncomputable def ofRat {n : ℕ} {c : Fin n → complexLorentzTensor.Color} :
 lemma ofRat_basis_repr_apply {n : ℕ} {c : Fin n → complexLorentzTensor.Color}
     (f : (ComponentIdx c) → RatComplexNum)
     (b :(ComponentIdx c)) :
-  (Tensor.basis c).repr (ofRat f) b = toComplexNum (f b) := by
-  simp [ofRat]
+  (Tensor.basis c).repr (ofRat f) b = toComplexNum (f b) := by simp [ofRat]
 
 lemma basis_eq_ofRat {n : ℕ} {c : Fin n → complexLorentzTensor.Color}
     (b : (ComponentIdx c)) :
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean b/Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean
index 947062b86..0ad7057c2 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean
@@ -84,29 +84,17 @@ scoped[complexLorentzTensor] notation "δR" => rightDualRightUnit
 
 -/
 
-lemma coContrUnit_eq_fromConstPair : δ' = fromConstPair Lorentz.coContrUnit := by
-  rw [Lorentz.coContrUnit]
-  rfl
+lemma coContrUnit_eq_fromConstPair : δ' = fromConstPair Lorentz.coContrUnit := rfl
 
-lemma contrCoUnit_eq_fromConstPair : δ = fromConstPair Lorentz.contrCoUnit := by
-  rw [Lorentz.contrCoUnit]
-  rfl
+lemma contrCoUnit_eq_fromConstPair : δ = fromConstPair Lorentz.contrCoUnit := rfl
 
-lemma dualLeftLeftUnit_eq_fromConstPair : δL' = fromConstPair Fermion.dualLeftLeftUnit := by
-  rw [Fermion.dualLeftLeftUnit]
-  rfl
+lemma dualLeftLeftUnit_eq_fromConstPair : δL' = fromConstPair Fermion.dualLeftLeftUnit := rfl
 
-lemma leftDualLeftUnit_eq_fromConstPair : δL = fromConstPair Fermion.leftDualLeftUnit := by
-  rw [Fermion.leftDualLeftUnit]
-  rfl
+lemma leftDualLeftUnit_eq_fromConstPair : δL = fromConstPair Fermion.leftDualLeftUnit := rfl
 
-lemma dualRightRightUnit_eq_fromConstPair : δR' = fromConstPair Fermion.dualRightRightUnit := by
-  rw [Fermion.dualRightRightUnit]
-  rfl
+lemma dualRightRightUnit_eq_fromConstPair : δR' = fromConstPair Fermion.dualRightRightUnit := rfl
 
-lemma rightDualRightUnit_eq_fromConstPair : δR = fromConstPair Fermion.rightDualRightUnit := by
-  rw [Fermion.rightDualRightUnit]
-  rfl
+lemma rightDualRightUnit_eq_fromConstPair : δR = fromConstPair Fermion.rightDualRightUnit := rfl
 
 /-!
 
@@ -115,34 +103,28 @@ lemma rightDualRightUnit_eq_fromConstPair : δR = fromConstPair Fermion.rightDua
 -/
 
 lemma coContrUnit_eq_fromPairT : δ' = fromPairT (Lorentz.coContrUnitVal) := by
-  rw [coContrUnit_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Lorentz.coContrUnit_apply_one
+  simpa only [coContrUnit_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Lorentz.coContrUnit_apply_one
 
 lemma contrCoUnit_eq_fromPairT : δ = fromPairT (Lorentz.contrCoUnitVal) := by
-  rw [contrCoUnit_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Lorentz.contrCoUnit_apply_one
+  simpa only [contrCoUnit_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Lorentz.contrCoUnit_apply_one
 
 lemma dualLeftLeftUnit_eq_fromPairT : δL' = fromPairT (Fermion.dualLeftLeftUnitVal) := by
-  rw [dualLeftLeftUnit_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.dualLeftLeftUnit_apply_one
+  simpa only [dualLeftLeftUnit_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.dualLeftLeftUnit_apply_one
 
 lemma leftDualLeftUnit_eq_fromPairT : δL = fromPairT (Fermion.leftDualLeftUnitVal) := by
-  rw [leftDualLeftUnit_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.leftDualLeftUnit_apply_one
+  simpa only [leftDualLeftUnit_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.leftDualLeftUnit_apply_one
 
 lemma dualRightRightUnit_eq_fromPairT : δR' = fromPairT (Fermion.dualRightRightUnitVal) := by
-  rw [dualRightRightUnit_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.dualRightRightUnit_apply_one
+  simpa only [dualRightRightUnit_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.dualRightRightUnit_apply_one
 
 lemma rightDualRightUnit_eq_fromPairT : δR = fromPairT (Fermion.rightDualRightUnitVal) := by
-  rw [rightDualRightUnit_eq_fromConstPair, fromConstPair]
-  congr 1
-  exact Fermion.rightDualRightUnit_apply_one
+  simpa only [rightDualRightUnit_eq_fromConstPair, fromConstPair] using
+    congrArg fromPairT Fermion.rightDualRightUnit_apply_one
 
 /-!
 
@@ -153,52 +135,44 @@ lemma rightDualRightUnit_eq_fromPairT : δR = fromPairT (Fermion.rightDualRightU
 open Lorentz in
 lemma coContrUnit_eq_complexCoBasis_complexContrBasis : δ' =
     ∑ i, fromPairT (complexCoBasis i ⊗ₜ[ℂ] complexContrBasis i) := by
-  rw [coContrUnit_eq_fromPairT, coContrUnitVal_expand_tmul]
-  rfl
+  exact (coContrUnit_eq_fromPairT.trans (congrArg fromPairT coContrUnitVal_expand_tmul)).trans rfl
 
 open Lorentz in
 lemma coContrUnit_eq_complexCoBasisFin4_complexContrBasisFin4 : δ' =
     ∑ i, fromPairT (complexCoBasisFin4 i ⊗ₜ[ℂ] complexContrBasisFin4 i) := by
-  rw [coContrUnit_eq_complexCoBasis_complexContrBasis]
-  rw [← finSumFinEquiv.symm.sum_comp]
+  rw [coContrUnit_eq_complexCoBasis_complexContrBasis, ← finSumFinEquiv.symm.sum_comp]
   simp [complexCoBasisFin4, complexContrBasisFin4]
 
 open Lorentz in
 lemma contrCoUnit_eq_complexContrBasis_complexCoBasis : δ =
     ∑ i, fromPairT (complexContrBasis i ⊗ₜ[ℂ] complexCoBasis i) := by
-  rw [contrCoUnit_eq_fromPairT, contrCoUnitVal_expand_tmul]
-  rfl
+  exact (contrCoUnit_eq_fromPairT.trans (congrArg fromPairT contrCoUnitVal_expand_tmul)).trans rfl
 
 open Lorentz in
 lemma contrCoUnit_eq_complexContrBasisFin4_complexCoBasisFin4 : δ =
     ∑ i, fromPairT (complexContrBasisFin4 i ⊗ₜ[ℂ] complexCoBasisFin4 i) := by
-  rw [contrCoUnit_eq_complexContrBasis_complexCoBasis]
-  rw [← finSumFinEquiv.symm.sum_comp]
+  rw [contrCoUnit_eq_complexContrBasis_complexCoBasis, ← finSumFinEquiv.symm.sum_comp]
   simp [complexContrBasisFin4, complexCoBasisFin4]
 
 open Fermion in
 lemma dualLeftLeftUnit_eq_dualLeftBasis_leftBasis : δL' =
     ∑ i, fromPairT (dualLeftBasis i ⊗ₜ[ℂ] leftBasis i) := by
-  rw [dualLeftLeftUnit_eq_fromPairT, dualLeftLeftUnitVal_expand_tmul]
-  rfl
+  simp [dualLeftLeftUnit_eq_fromPairT, dualLeftLeftUnitVal_expand_tmul]
 
 open Fermion in
 lemma leftDualLeftUnit_eq_leftBasis_dualLeftBasis : δL =
     ∑ i, fromPairT (leftBasis i ⊗ₜ[ℂ] dualLeftBasis i) := by
-  rw [leftDualLeftUnit_eq_fromPairT, leftDualLeftUnitVal_expand_tmul]
-  rfl
+  simp [leftDualLeftUnit_eq_fromPairT, leftDualLeftUnitVal_expand_tmul]
 
 open Fermion in
 lemma dualRightRightUnit_eq_dualRightBasis_rightBasis : δR' =
     ∑ i, fromPairT (dualRightBasis i ⊗ₜ[ℂ] rightBasis i) := by
-  rw [dualRightRightUnit_eq_fromPairT, dualRightRightUnitVal_expand_tmul]
-  rfl
+  simp [dualRightRightUnit_eq_fromPairT, dualRightRightUnitVal_expand_tmul]
 
 open Fermion in
 lemma rightDualRightUnit_eq_rightBasis_dualRightBasis : δR =
     ∑ i, fromPairT (rightBasis i ⊗ₜ[ℂ] dualRightBasis i) := by
-  rw [rightDualRightUnit_eq_fromPairT, rightDualRightUnitVal_expand_tmul]
-  rfl
+  simp [rightDualRightUnit_eq_fromPairT, rightDualRightUnitVal_expand_tmul]
 
 /-!
 
@@ -210,66 +184,42 @@ lemma coContrUnit_eq_basis : δ' =
     ∑ i, Tensor.basis (S := complexLorentzTensor)
       ![Color.down, Color.up] (fun | 0 => i | 1 => i) := by
   rw [coContrUnit_eq_complexCoBasisFin4_complexContrBasisFin4]
-  conv_lhs =>
-    enter [2, x]
-    change fromPairT ((complexLorentzTensor.basis .down x) ⊗ₜ[ℂ]
-      (complexLorentzTensor.basis .up _))
-    rw [fromPairT_apply_basis_repr]
+  simp [fromPairT_apply_basis_repr]
   rfl
 
 lemma contrCoUnit_eq_basis : δ =
     ∑ i, Tensor.basis (S := complexLorentzTensor)
       ![Color.up, Color.down] (fun | 0 => i | 1 => i) := by
   rw [contrCoUnit_eq_complexContrBasisFin4_complexCoBasisFin4]
-  conv_lhs =>
-    enter [2, x]
-    change fromPairT ((complexLorentzTensor.basis .up x) ⊗ₜ[ℂ]
-      (complexLorentzTensor.basis .down _))
-    rw [fromPairT_apply_basis_repr]
+  simp [fromPairT_apply_basis_repr]
   rfl
 
 lemma dualLeftLeftUnit_eq_basis : δL' =
     ∑ i, Tensor.basis (S := complexLorentzTensor)
       ![Color.downL, Color.upL] (fun | 0 => i | 1 => i) := by
   rw [dualLeftLeftUnit_eq_dualLeftBasis_leftBasis]
-  conv_lhs =>
-    enter [2, x]
-    change fromPairT ((complexLorentzTensor.basis .downL x) ⊗ₜ[ℂ]
-      (complexLorentzTensor.basis .upL _))
-    rw [fromPairT_apply_basis_repr]
+  simp [fromPairT_apply_basis_repr]
   rfl
 
 lemma leftDualLeftUnit_eq_basis : δL =
     ∑ i, Tensor.basis (S := complexLorentzTensor)
       ![Color.upL, Color.downL] (fun | 0 => i | 1 => i) := by
   rw [leftDualLeftUnit_eq_leftBasis_dualLeftBasis]
-  conv_lhs =>
-    enter [2, x]
-    change fromPairT ((complexLorentzTensor.basis .upL x) ⊗ₜ[ℂ]
-      (complexLorentzTensor.basis .downL _))
-    rw [fromPairT_apply_basis_repr]
+  simp [fromPairT_apply_basis_repr]
   rfl
 
 lemma dualRightRightUnit_eq_basis : δR' =
     ∑ i, Tensor.basis (S := complexLorentzTensor)
       ![Color.downR, Color.upR] (fun | 0 => i | 1 => i) := by
   rw [dualRightRightUnit_eq_dualRightBasis_rightBasis]
-  conv_lhs =>
-    enter [2, x]
-    change fromPairT ((complexLorentzTensor.basis .downR x) ⊗ₜ[ℂ]
-      (complexLorentzTensor.basis .upR _))
-    rw [fromPairT_apply_basis_repr]
+  simp [fromPairT_apply_basis_repr]
   rfl
 
 lemma rightDualRightUnit_eq_basis : δR =
     ∑ i, Tensor.basis (S := complexLorentzTensor)
       ![Color.upR, Color.downR] (fun | 0 => i | 1 => i) := by
   rw [rightDualRightUnit_eq_rightBasis_dualRightBasis]
-  conv_lhs =>
-    enter [2, x]
-    change fromPairT ((complexLorentzTensor.basis .upR x) ⊗ₜ[ℂ]
-      (complexLorentzTensor.basis .downR _))
-    rw [fromPairT_apply_basis_repr]
+  simp [fromPairT_apply_basis_repr]
   rfl
 
 /-!
@@ -344,34 +294,22 @@ lemma rightDualRightUnit_eq_ofRat : δR = ofRat fun f =>
 
 -/
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `coContrUnit` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_coContrUnit (g : SL(2,ℂ)) : g • δ' = δ' := by
-  rw [unitTensor_invariant]
+lemma actionT_coContrUnit (g : SL(2,ℂ)) : g • δ' = δ' := unitTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `contrCoUnit` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_contrCoUnit (g : SL(2,ℂ)) : g • δ = δ := by
-  rw [unitTensor_invariant]
+lemma actionT_contrCoUnit (g : SL(2,ℂ)) : g • δ = δ := unitTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `dualLeftLeftUnit` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_dualLeftLeftUnit (g : SL(2,ℂ)) : g • δL' = δL' := by
-  rw [unitTensor_invariant]
+lemma actionT_dualLeftLeftUnit (g : SL(2,ℂ)) : g • δL' = δL' := unitTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `leftDualLeftUnit` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_leftDualLeftUnit (g : SL(2,ℂ)) : g • δL = δL := by
-  rw [unitTensor_invariant]
+lemma actionT_leftDualLeftUnit (g : SL(2,ℂ)) : g • δL = δL := unitTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `dualRightRightUnit` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_dualRightRightUnit (g : SL(2,ℂ)) : g • δR' = δR' := by
-  rw [unitTensor_invariant]
+lemma actionT_dualRightRightUnit (g : SL(2,ℂ)) : g • δR' = δR' := unitTensor_invariant g
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The tensor `rightDualRightUnit` is invariant under the action of `SL(2,ℂ)`. -/
-lemma actionT_rightDualRightUnit (g : SL(2,ℂ)) : g • δR = δR := by
-  rw [unitTensor_invariant]
+lemma actionT_rightDualRightUnit (g : SL(2,ℂ)) : g • δR = δR := unitTensor_invariant g
 
 end complexLorentzTensor
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Units/Pre.lean b/Physlib/Relativity/Tensors/ComplexTensor/Units/Pre.lean
index b766a96ff..d0bc3ba8d 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Units/Pre.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Units/Pre.lean
@@ -57,9 +57,7 @@ def contrCoUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     a' • contrCoUnitVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • contrCoUnitVal =
@@ -71,9 +69,7 @@ def contrCoUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     apply congrArg
     simp
 
-lemma contrCoUnit_apply_one : contrCoUnit (1 : ℂ) = contrCoUnitVal := by
-  change (1 : ℂ) • contrCoUnitVal = contrCoUnitVal
-  rw [one_smul]
+lemma contrCoUnit_apply_one : contrCoUnit (1 : ℂ) = contrCoUnitVal := by simp [contrCoUnit]
 
 /-- The co-contra unit for complex lorentz vectors. Usually denoted `δᵢⁱ`. -/
 def coContrUnitVal : CoℂModule ⊗[ℂ] ContrℂModule :=
@@ -108,9 +104,7 @@ def coContrUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     a' • coContrUnitVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • coContrUnitVal =
@@ -124,9 +118,7 @@ def coContrUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     refine transpose_eq_one.mp ?h.h.h.a
     simp
 
-lemma coContrUnit_apply_one : coContrUnit (1 : ℂ) = coContrUnitVal := by
-  change (1 : ℂ) • coContrUnitVal = coContrUnitVal
-  rw [one_smul]
+lemma coContrUnit_apply_one : coContrUnit (1 : ℂ) = coContrUnitVal := by simp [coContrUnit]
 
 /-!
 
@@ -168,16 +160,14 @@ open CategoryTheory
 lemma contrCoUnit_symm :
     contrCoUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap
       (TensorProduct.comm ℂ _ _ (coContrUnit (1 : ℂ))) := by
-  rw [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
-  rw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]
-  rfl
+  simp [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul,
+    coContrUnit_apply_one, coContrUnitVal_expand_tmul]
 
 lemma coContrUnit_symm :
     (coContrUnit (1 : ℂ)) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap
       (TensorProduct.comm ℂ _ _ (contrCoUnit (1 : ℂ))) := by
-  rw [coContrUnit_apply_one, coContrUnitVal_expand_tmul]
-  rw [contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
-  rfl
+  simp [coContrUnit_apply_one, coContrUnitVal_expand_tmul,
+    contrCoUnit_apply_one, contrCoUnitVal_expand_tmul]
 
 end Lorentz
 end
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean b/Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean
index a7f32eae1..c8eca1e95 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean
@@ -27,39 +27,33 @@ open TensorSpecies
 open Tensor
 
 /-- Swapping indices of `coContrUnit` returns `contrCoUnit`: `{δ' | μ ν = δ | ν μ}ᵀ`. -/
-lemma coContrUnit_symm : {δ' | μ ν = δ | ν μ}ᵀ := by
-  rw [coContrUnit, unitTensor_eq_permT_dual]
-  rfl
+lemma coContrUnit_symm : {δ' | μ ν = δ | ν μ}ᵀ :=
+  unitTensor_eq_permT_dual (S := complexLorentzTensor) Color.up
 
 /-- Swapping indices of `contrCoUnit` returns `coContrUnit`: `{δ | μ ν = δ' | ν μ}ᵀ`. -/
-lemma contrCoUnit_symm : {δ | μ ν = δ' | ν μ}ᵀ := by
-  rw [contrCoUnit, unitTensor_eq_permT_dual]
-  rfl
+lemma contrCoUnit_symm : {δ | μ ν = δ' | ν μ}ᵀ :=
+  unitTensor_eq_permT_dual (S := complexLorentzTensor) Color.down
 
 /-- Swapping indices of `dualLeftLeftUnit` returns
   `leftDualLeftUnit`: `{δL' | α α' = δL | α' α}ᵀ`. -/
-lemma dualLeftLeftUnit_symm : {δL' | α α' = δL | α' α}ᵀ := by
-  rw [dualLeftLeftUnit, unitTensor_eq_permT_dual]
-  rfl
+lemma dualLeftLeftUnit_symm : {δL' | α α' = δL | α' α}ᵀ :=
+  unitTensor_eq_permT_dual (S := complexLorentzTensor) Color.upL
 
 /-- Swapping indices of `leftDualLeftUnit` returns
   `dualLeftLeftUnit`: `{δL | α α' = δL' | α' α}ᵀ`. -/
-lemma leftDualLeftUnit_symm : {δL | α α' = δL' | α' α}ᵀ := by
-  rw [leftDualLeftUnit, unitTensor_eq_permT_dual]
-  rfl
+lemma leftDualLeftUnit_symm : {δL | α α' = δL' | α' α}ᵀ :=
+  unitTensor_eq_permT_dual (S := complexLorentzTensor) Color.downL
 
 /-- Swapping indices of `dualRightRightUnit` returns `rightDualRightUnit`:
 `{δR' | β β' = δR | β' β}ᵀ`.
 -/
-lemma dualRightRightUnit_symm : {δR' | β β' = δR | β' β}ᵀ := by
-  rw [dualRightRightUnit, unitTensor_eq_permT_dual]
-  rfl
+lemma dualRightRightUnit_symm : {δR' | β β' = δR | β' β}ᵀ :=
+  unitTensor_eq_permT_dual (S := complexLorentzTensor) Color.upR
 
 /-- Swapping indices of `rightDualRightUnit` returns `dualRightRightUnit`:
 `{δR | β β' = δR' | β' β}ᵀ`.
 -/
-lemma rightDualRightUnit_symm : {δR | β β' = δR' | β' β}ᵀ := by
-  rw [rightDualRightUnit, unitTensor_eq_permT_dual]
-  rfl
+lemma rightDualRightUnit_symm : {δR | β β' = δR' | β' β}ᵀ :=
+  unitTensor_eq_permT_dual (S := complexLorentzTensor) Color.downR
 
 end complexLorentzTensor
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean b/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean
index 583458750..00d103ec7 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean
@@ -31,9 +31,7 @@ def complexContrBasis : Basis (Fin 1 ⊕ Fin 3) ℂ ContrℂModule :=
 
 @[simp]
 lemma complexContrBasis_toFin13ℂ (i :Fin 1 ⊕ Fin 3) :
-    (complexContrBasis i).toFin13ℂ = Pi.single i 1 := by
-  simp only [complexContrBasis, Basis.coe_ofEquivFun]
-  rfl
+    (complexContrBasis i).toFin13ℂ = Pi.single i 1 := by simp [complexContrBasis]
 
 @[simp]
 lemma complexContrBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
@@ -46,44 +44,37 @@ lemma complexContrBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
 
 lemma complexContrBasis_ρ_val (M : SL(2,ℂ)) (v : ContrℂModule) :
     ((ContrℂModule.SL2CRep M) v).val =
-    LorentzGroup.toComplex (SL2C.toLorentzGroup M) *ᵥ v.val := by
-  rfl
+    LorentzGroup.toComplex (SL2C.toLorentzGroup M) *ᵥ v.val := rfl
 
 /-- The standard basis of complex contravariant Lorentz vectors indexed by `Fin 4`. -/
 def complexContrBasisFin4 : Basis (Fin 4) ℂ ContrℂModule :=
   Basis.reindex complexContrBasis finSumFinEquiv
 
 lemma complexContrBasisFin4_eq_reindex :
-    complexContrBasisFin4 = complexContrBasis.reindex finSumFinEquiv :=
-  rfl
+    complexContrBasisFin4 = complexContrBasis.reindex finSumFinEquiv := rfl
 
 lemma complexContrBasis_reindex_apply_eq_fin4 (j : Fin 4) :
-    (complexContrBasis.reindex finSumFinEquiv) j = complexContrBasisFin4 j :=
-  rfl
+    (complexContrBasis.reindex finSumFinEquiv) j = complexContrBasisFin4 j := rfl
 
 @[simp]
 lemma complexContrBasisFin4_apply_zero :
     complexContrBasisFin4 0 = complexContrBasis (Sum.inl 0) := by
-  simp only [complexContrBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexContrBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexContrBasisFin4_apply_one :
     complexContrBasisFin4 1 = complexContrBasis (Sum.inr 0) := by
-  simp only [complexContrBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexContrBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexContrBasisFin4_apply_two :
     complexContrBasisFin4 2 = complexContrBasis (Sum.inr 1) := by
-  simp only [complexContrBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexContrBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexContrBasisFin4_apply_three :
     complexContrBasisFin4 3 = complexContrBasis (Sum.inr 2) := by
-  simp only [complexContrBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexContrBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexContrBasisFin4_apply_succ (i : Fin 3) :
@@ -98,8 +89,7 @@ def complexCoBasis : Basis (Fin 1 ⊕ Fin 3) ℂ CoℂModule :=
 
 @[simp]
 lemma complexCoBasis_toFin13ℂ (i :Fin 1 ⊕ Fin 3) : (complexCoBasis i).toFin13ℂ = Pi.single i 1 := by
-  simp only [complexCoBasis, Basis.coe_ofEquivFun]
-  rfl
+  simp [complexCoBasis]
 
 @[simp]
 lemma complexCoBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
@@ -112,44 +102,37 @@ lemma complexCoBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 1 ⊕ Fin 3) :
 
 lemma CoℂModule.SL2CRep_val (M : SL(2,ℂ)) (v : CoℂModule) :
     ((CoℂModule.SL2CRep M) v).val =
-    (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ *ᵥ v.val := by
-  rfl
+    (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ *ᵥ v.val := rfl
 
 /-- The standard basis of complex covariant Lorentz vectors indexed by `Fin 4`. -/
 def complexCoBasisFin4 : Basis (Fin 4) ℂ CoℂModule :=
   Basis.reindex complexCoBasis finSumFinEquiv
 
 lemma complexCoBasisFin4_eq_reindex :
-    complexCoBasisFin4 = complexCoBasis.reindex finSumFinEquiv :=
-  rfl
+    complexCoBasisFin4 = complexCoBasis.reindex finSumFinEquiv := rfl
 
 lemma complexCoBasis_reindex_apply_eq_fin4 (j : Fin 4) :
-    (complexCoBasis.reindex finSumFinEquiv) j = complexCoBasisFin4 j :=
-  rfl
+    (complexCoBasis.reindex finSumFinEquiv) j = complexCoBasisFin4 j := rfl
 
 @[simp]
 lemma complexCoBasisFin4_apply_zero :
     complexCoBasisFin4 0 = complexCoBasis (Sum.inl 0) := by
-  simp only [complexCoBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexCoBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexCoBasisFin4_apply_one :
     complexCoBasisFin4 1 = complexCoBasis (Sum.inr 0) := by
-  simp only [complexCoBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexCoBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexCoBasisFin4_apply_two :
     complexCoBasisFin4 2 = complexCoBasis (Sum.inr 1) := by
-  simp only [complexCoBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexCoBasisFin4, finSumFinEquiv, Fin.addCases]
 
 @[simp]
 lemma complexCoBasisFin4_apply_three :
     complexCoBasisFin4 3 = complexCoBasis (Sum.inr 2) := by
-  simp only [complexCoBasisFin4, Basis.reindex_apply]
-  rfl
+  simp [complexCoBasisFin4, finSumFinEquiv, Fin.addCases]
 
 /-!
 
@@ -197,15 +180,13 @@ lemma inclCongrRealLorentz_ρ (M : SL(2, ℂ)) (v : ContrMod 3) :
     inclCongrRealLorentz (ContrMod.rep (SL2C.toLorentzGroup M) v) := by
   apply Lorentz.ContrℂModule.ext
   rw [complexContrBasis_ρ_val, inclCongrRealLorentz_val, inclCongrRealLorentz_val]
-  rw [LorentzGroup.toComplex_mulVec_ofReal]
-  rfl
+  exact LorentzGroup.toComplex_mulVec_ofReal _ _
 
 lemma SL2CRep_ρ_basis (M : SL(2, ℂ)) (i : Fin 1 ⊕ Fin 3) :
     (ContrℂModule.SL2CRep M) (complexContrBasis i) =
     ∑ j, (SL2C.toLorentzGroup M).1 j i •
     complexContrBasis j := by
-  rw [complexContrBasis_of_real, inclCongrRealLorentz_ρ]
-  rw [Contr.ρ_stdBasis, map_sum]
+  rw [complexContrBasis_of_real, inclCongrRealLorentz_ρ, Contr.ρ_stdBasis, map_sum]
   apply congrArg
   funext j
   simp only [LinearMap.map_smulₛₗ, ofRealHom_eq_coe, coe_smul]
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Contraction.lean b/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Contraction.lean
index ed154264d..59d50aee4 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Contraction.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Contraction.lean
@@ -35,18 +35,15 @@ def contrCoContrBi : ContrℂModule →ₗ[ℂ] CoℂModule →ₗ[ℂ] ℂ wher
       rw [dotProduct_add]
     map_smul' := by
       intro r φ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' ψ ψ':= by
     refine LinearMap.ext (fun φ => ?_)
     simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
     rw [add_dotProduct]
   map_smul' r ψ := by
     refine LinearMap.ext (fun φ => ?_)
-    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
-    rw [smul_dotProduct]
-    rfl
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk, smul_dotProduct,
+      LinearMap.smul_apply, RingHom.id_apply]
 
 /-- The bi-linear map corresponding to contraction of a covariant Lorentz vector with a
   contravariant Lorentz vector. -/
@@ -59,18 +56,15 @@ def contrContrCoBi : CoℂModule →ₗ[ℂ] ContrℂModule →ₗ[ℂ] ℂ wher
       rw [dotProduct_add]
     map_smul' := by
       intro r ψ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' φ φ' := by
     refine LinearMap.ext (fun ψ => ?_)
     simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
     rw [add_dotProduct]
   map_smul' r φ := by
     refine LinearMap.ext (fun ψ => ?_)
-    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
-    rw [smul_dotProduct]
-    rfl
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk, smul_dotProduct,
+      LinearMap.smul_apply, RingHom.id_apply]
 
 /-- The linear map from complexContr ⊗ complexCo to ℂ given by
     summing over components of contravariant Lorentz vector and
@@ -89,8 +83,7 @@ def contrCoContraction : (ContrℂModule.SL2CRep.tprod CoℂModule.SL2CRep).Inte
     simp
 
 lemma contrCoContraction_hom_tmul (ψ : ContrℂModule) (φ : CoℂModule) :
-    contrCoContraction (ψ ⊗ₜ φ) = ψ.toFin13ℂ ⬝ᵥ φ.toFin13ℂ := by
-  rfl
+    contrCoContraction (ψ ⊗ₜ φ) = ψ.toFin13ℂ ⬝ᵥ φ.toFin13ℂ := by rfl
 
 lemma contrCoContraction_basis (i j : Fin 4) :
     contrCoContraction (complexContrBasisFin4 i ⊗ₜ complexCoBasisFin4 j) =
@@ -108,9 +101,7 @@ lemma contrCoContraction_basis' (i j : Fin 1 ⊕ Fin 3) :
     if i = j then (1 : ℂ) else 0 := by
   rw [contrCoContraction_hom_tmul]
   simp only [complexContrBasis_toFin13ℂ, complexCoBasis_toFin13ℂ, dotProduct_single, mul_one]
-  rw [Pi.single_apply]
-  refine ite_congr ?h₁ (congrFun rfl) (congrFun rfl)
-  exact Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a)
+  simp [Pi.single_apply, eq_comm]
 
 /-- The linear map from complexCo ⊗ complexContr to ℂ given by
     summing over components of covariant Lorentz vector and
@@ -128,8 +119,7 @@ def coContrContraction : (CoℂModule.SL2CRep.tprod ContrℂModule.SL2CRep).Inte
     simp
 
 lemma coContrContraction_hom_tmul (φ : CoℂModule) (ψ : ContrℂModule) :
-    coContrContraction (φ ⊗ₜ ψ) = φ.toFin13ℂ ⬝ᵥ ψ.toFin13ℂ := by
-  rfl
+    coContrContraction (φ ⊗ₜ ψ) = φ.toFin13ℂ ⬝ᵥ ψ.toFin13ℂ := by rfl
 
 lemma coContrContraction_basis (i j : Fin 4) :
     coContrContraction (complexCoBasisFin4 i ⊗ₜ complexContrBasisFin4 j) =
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean b/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean
index ff5bf5b72..9a1a24891 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean
@@ -164,8 +164,7 @@ def lorentzGroupRep : Representation ℂ (LorentzGroup 3) CoℂModule where
       LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
     refine (congrFun (congrArg _ ?_) _)
     simp only [_root_.map_mul]
-    rw [Matrix.mul_inv_rev]
-    exact transpose_mul _ _
+    simp [Matrix.mul_inv_rev]
 
 /-- The representation of the SL(2, ℂ) on `ContrℂModule` induced by the representation of the
   Lorentz group. -/
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean
index 0a3c356a8..12c5b0c2e 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean
@@ -92,8 +92,7 @@ def dualLeftHandedRep : Representation ℂ SL(2,ℂ) DualLeftHandedWeyl where
     simp only [SpecialLinearGroup.coe_mul, LinearMap.coe_mk, AddHom.coe_mk, Module.End.mul_apply,
       LinearEquiv.apply_symm_apply, mulVec_mulVec, EmbeddingLike.apply_eq_iff_eq]
     refine (congrFun (congrArg _ ?_) _)
-    rw [Matrix.mul_inv_rev]
-    exact transpose_mul _ _
+    simp [Matrix.mul_inv_rev]
 
 /-- The standard basis on dual-left-handed Weyl fermions. -/
 def dualLeftBasis : Basis (Fin 2) ℂ DualLeftHandedWeyl := Basis.ofEquivFun
@@ -207,8 +206,7 @@ def leftHandedToDual : leftHandedRep.IntertwiningMap dualLeftHandedRep where
     simp only [mulVec_add, LinearEquiv.map_add]
   map_smul' := by
     intro a ψ
-    simp only [mulVec_smul, LinearEquiv.map_smul]
-    rfl
+    simp only [mulVec_smul, LinearEquiv.map_smul, RingHom.id_apply]
   isIntertwining' := by
     intro M
     refine LinearMap.ext (fun ψ => ?_)
@@ -237,9 +235,7 @@ def leftHandedDualTo : dualLeftHandedRep.IntertwiningMap leftHandedRep where
     rw [mulVec_add, LinearEquiv.map_add]
   map_smul' := by
     intro a ψ
-    simp only [LinearEquiv.map_smul]
-    rw [mulVec_smul, LinearEquiv.map_smul]
-    rfl
+    simp only [mulVec_smul, LinearEquiv.map_smul, RingHom.id_apply]
   isIntertwining' := by
     intro M
     refine LinearMap.ext (fun ψ => ?_)
@@ -267,16 +263,14 @@ def leftHandedDualEquiv : leftHandedRep.Equiv dualLeftHandedRep := by
     rw [leftHandedDualTo_hom_apply, leftHandedToDual_hom_apply]
     rw [DualLeftHandedWeyl.toFin2ℂ, LinearEquiv.apply_symm_apply, mulVec_mulVec]
     rw [show (!![0, -1; (1 : ℂ), 0] * !![0, 1; -1, 0]) = 1 by simpa using Eq.symm one_fin_two]
-    rw [one_mulVec]
-    rfl
+    simp
   · intro ψ
     simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
       Representation.IntertwiningMap.coe_toLinearMap]
     rw [leftHandedDualTo_hom_apply, leftHandedToDual_hom_apply, LeftHandedWeyl.toFin2ℂ,
       LinearEquiv.apply_symm_apply, mulVec_mulVec]
     rw [show (!![0, (1 : ℂ); -1, 0] * !![0, -1; 1, 0]) = 1 by simpa using Eq.symm one_fin_two]
-    rw [one_mulVec]
-    rfl
+    simp
 
 /-- `leftHandedDualEquiv` acting on an element `ψ : leftHanded` corresponds
   to multiplying `ψ` by the matrix `!![0, 1; -1, 0]`. -/
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean
index 4748e2992..69a6f5724 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean
@@ -42,18 +42,15 @@ def leftDualBi : LeftHandedWeyl →ₗ[ℂ] DualLeftHandedWeyl →ₗ[ℂ] ℂ w
       rw [dotProduct_add]
     map_smul' := by
       intro r φ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' ψ ψ':= by
     refine LinearMap.ext (fun φ => ?_)
     simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
     rw [add_dotProduct]
   map_smul' r ψ := by
     refine LinearMap.ext (fun φ => ?_)
-    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
-    rw [smul_dotProduct]
-    rfl
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk, smul_dotProduct,
+      LinearMap.smul_apply, RingHom.id_apply]
 
 /-- The bi-linear map corresponding to contraction of a dual-left-handed Weyl fermion with a
   left-handed Weyl fermion. -/
@@ -66,9 +63,7 @@ def dualLeftBi : DualLeftHandedWeyl →ₗ[ℂ] LeftHandedWeyl →ₗ[ℂ] ℂ w
       rw [dotProduct_add]
     map_smul' := by
       intro r φ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' ψ ψ':= by
     refine LinearMap.ext (fun φ => ?_)
     simp only [map_add, add_dotProduct, vec2_dotProduct, Fin.isValue, LinearMap.coe_mk,
@@ -89,18 +84,15 @@ def rightDualBi : RightHandedWeyl →ₗ[ℂ] DualRightHandedWeyl →ₗ[ℂ] 
       rw [dotProduct_add]
     map_smul' := by
       intro r φ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' ψ ψ':= by
     refine LinearMap.ext (fun φ => ?_)
     simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
     rw [add_dotProduct]
   map_smul' r ψ := by
     refine LinearMap.ext (fun φ => ?_)
-    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
-    rw [smul_dotProduct]
-    rfl
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk, smul_dotProduct,
+      LinearMap.smul_apply, RingHom.id_apply]
 
 /-- The bi-linear map corresponding to contraction of a dual-right-handed Weyl fermion with a
   right-handed Weyl fermion. -/
@@ -113,9 +105,7 @@ def dualRightBi : DualRightHandedWeyl →ₗ[ℂ] RightHandedWeyl →ₗ[ℂ] 
       rw [dotProduct_add]
     map_smul' := by
       intro r φ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' ψ ψ':= by
     refine LinearMap.ext (fun φ => ?_)
     simp only [map_add, add_dotProduct, vec2_dotProduct, Fin.isValue, LinearMap.coe_mk,
@@ -140,8 +130,7 @@ def leftDualContraction : (leftHandedRep.tprod dualLeftHandedRep).IntertwiningMa
 
 lemma leftDualContraction_hom_tmul (ψ : LeftHandedWeyl)
     (φ : DualLeftHandedWeyl) :
-    leftDualContraction (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ := by
-  rfl
+    leftDualContraction (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ := by rfl
 
 lemma leftDualContraction_basis (i j : Fin 2) :
     leftDualContraction (leftBasis i ⊗ₜ dualLeftBasis j) = if i.1 = j.1 then (1 : ℂ) else 0 := by
@@ -166,8 +155,7 @@ def dualLeftContraction : (dualLeftHandedRep.tprod leftHandedRep).IntertwiningMa
     simp
 
 lemma dualLeftContraction_hom_tmul (φ : DualLeftHandedWeyl) (ψ : LeftHandedWeyl) :
-    dualLeftContraction (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ := by
-  rfl
+    dualLeftContraction (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ := by rfl
 
 lemma dualLeftContraction_basis (i j : Fin 2) :
     dualLeftContraction (dualLeftBasis i ⊗ₜ leftBasis j) = if i.1 = j.1 then (1 : ℂ) else 0 := by
@@ -206,8 +194,7 @@ def rightDualContraction : (rightHandedRep.tprod dualRightHandedRep).Intertwinin
 
 lemma rightDualContraction_hom_tmul (ψ : RightHandedWeyl)
     (φ : DualRightHandedWeyl) :
-    rightDualContraction (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ := by
-  rfl
+    rightDualContraction (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ := by rfl
 
 lemma rightDualContraction_basis (i j : Fin 2) :
     rightDualContraction (rightBasis i ⊗ₜ dualRightBasis j) =
@@ -247,8 +234,7 @@ def dualRightContraction : (dualRightHandedRep.tprod rightHandedRep).Intertwinin
 
 lemma dualRightContraction_hom_tmul (φ : DualRightHandedWeyl)
     (ψ : RightHandedWeyl) :
-    dualRightContraction (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ := by
-  rfl
+    dualRightContraction (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ := by rfl
 
 lemma dualRightContraction_basis (i j : Fin 2) :
     dualRightContraction (dualRightBasis i ⊗ₜ rightBasis j) =
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean
index 60f8174b6..216bb7fa2 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean
@@ -33,8 +33,7 @@ def metricRaw : Matrix (Fin 2) (Fin 2) ℂ := !![0, 1; -1, 0]
 /-- Multiplying an element of `SL(2, ℂ)` on the left with the metric `𝓔` is equivalent
   to multiplying the inverse-transpose of that element on the right with the metric. -/
 lemma comm_metricRaw (M : SL(2,ℂ)) : M.1 * metricRaw = metricRaw * (M.1⁻¹)ᵀ := by
-  rw [metricRaw]
-  rw [Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
+  rw [metricRaw, Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
   rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
       Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ]
   simp only [Fin.isValue, mul_zero, mul_neg, mul_one, zero_add, add_zero, transpose_apply, of_apply,
@@ -43,8 +42,7 @@ lemma comm_metricRaw (M : SL(2,ℂ)) : M.1 * metricRaw = metricRaw * (M.1⁻¹)
     empty_vecMul, neg_smul, neg_cons, neg_neg, neg_empty, empty_mul, Equiv.symm_apply_apply]
 
 lemma metricRaw_comm (M : SL(2,ℂ)) : metricRaw * M.1 = (M.1⁻¹)ᵀ * metricRaw := by
-  rw [metricRaw]
-  rw [Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
+  rw [metricRaw, Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
   rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
       Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ]
   simp only [Fin.isValue, zero_mul, one_mul, zero_add, neg_mul, add_zero, transpose_apply, of_apply,
@@ -54,16 +52,14 @@ lemma metricRaw_comm (M : SL(2,ℂ)) : metricRaw * M.1 = (M.1⁻¹)ᵀ * metricR
     empty_vecMul, add_cons, empty_add_empty, empty_mul, Equiv.symm_apply_apply]
 
 lemma star_comm_metricRaw (M : SL(2,ℂ)) : M.1.map star * metricRaw = metricRaw * ((M.1)⁻¹)ᴴ := by
-  rw [metricRaw]
-  rw [Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
+  rw [metricRaw, Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
   rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
       eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᴴ]
   rw [eta_fin_two (!![M.1 0 0, M.1 0 1; M.1 1 0, M.1 1 1].map star)]
   simp
 
 lemma metricRaw_comm_star (M : SL(2,ℂ)) : metricRaw * M.1.map star = ((M.1)⁻¹)ᴴ * metricRaw := by
-  rw [metricRaw]
-  rw [Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
+  rw [metricRaw, Lorentz.SL2C.inverse_coe, eta_fin_two M.1]
   rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two,
       eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᴴ]
   rw [eta_fin_two (!![M.1 0 0, M.1 0 1; M.1 1 0, M.1 1 1].map star)]
@@ -96,9 +92,7 @@ def leftMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     a' • leftMetricVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • leftMetricVal =
@@ -112,9 +106,7 @@ def leftMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     simp only [SpecialLinearGroup.det_coe, isUnit_iff_ne_zero, ne_eq, one_ne_zero,
       not_false_eq_true, mul_nonsing_inv, transpose_one, mul_one]
 
-lemma leftMetric_apply_one : leftMetric (1 : ℂ) = leftMetricVal := by
-  change (1 : ℂ) • leftMetricVal = leftMetricVal
-  simp only [one_smul]
+lemma leftMetric_apply_one : leftMetric (1 : ℂ) = leftMetricVal := by simp [leftMetric]
 
 /-- The metric `εₐₐ` as an element of `(dualLeftHanded ⊗ dualLeftHanded).V`. -/
 def dualLeftMetricVal : (DualLeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl) :=
@@ -139,9 +131,7 @@ def dualLeftMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
       a' • dualLeftMetricVal
     map_add' := fun x y => by
       simp only [add_smul]
-    map_smul' := fun m x => by
-      simp only [smul_smul]
-      rfl
+    map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
     isIntertwining' M := by
       refine LinearMap.ext fun x : ℂ => ?_
       change x • dualLeftMetricVal =
@@ -156,8 +146,7 @@ def dualLeftMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
         not_false_eq_true, mul_nonsing_inv, mul_one]
 
 lemma dualLeftMetric_apply_one : dualLeftMetric (1 : ℂ) = dualLeftMetricVal := by
-  change (1 : ℂ) • dualLeftMetricVal = dualLeftMetricVal
-  simp only [one_smul]
+  simp [dualLeftMetric]
 
 /-- The metric `ε^{dot a}^{dot a}` as an element of `(rightHanded ⊗ rightHanded).V`. -/
 def rightMetricVal : (RightHandedWeyl ⊗[ℂ] RightHandedWeyl) :=
@@ -186,9 +175,7 @@ def rightMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
     a' • rightMetricVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • rightMetricVal =
@@ -209,9 +196,7 @@ def rightMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
       simp
     · rw [← rightRightToMatrix_ρ_symm metricRaw M]
 
-lemma rightMetric_apply_one : rightMetric (1 : ℂ) = rightMetricVal := by
-  change (1 : ℂ) • rightMetricVal = rightMetricVal
-  simp only [one_smul]
+lemma rightMetric_apply_one : rightMetric (1 : ℂ) = rightMetricVal := by simp [rightMetric]
 
 /-- The metric `ε_{dot a}_{dot a}` as an element of `(dualRightHanded ⊗ dualRightHanded).V`. -/
 def dualRightMetricVal : DualRightHandedWeyl ⊗[ℂ] DualRightHandedWeyl :=
@@ -237,9 +222,7 @@ def dualRightMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
       a' • dualRightMetricVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • dualRightMetricVal =
@@ -263,8 +246,7 @@ def dualRightMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap
       rfl
 
 lemma dualRightMetric_apply_one : dualRightMetric (1 : ℂ) = dualRightMetricVal := by
-  change (1 : ℂ) • dualRightMetricVal = dualRightMetricVal
-  simp only [one_smul]
+  simp [dualRightMetric]
 
 /-!
 
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean
index 3f638348d..d6ef3db03 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean
@@ -742,9 +742,8 @@ lemma dualLeftDualRightToMatrix_ρ_symm_selfAdjoint (v : Matrix (Fin 2) (Fin 2)
 lemma leftRightToMatrix_ρ_symm_selfAdjoint (v : Matrix (Fin 2) (Fin 2) ℂ)
     (hv : IsSelfAdjoint v) (M : SL(2,ℂ)) :
     TensorProduct.map (leftHandedRep M) (rightHandedRep M) (leftRightToMatrix.symm v) =
-    leftRightToMatrix.symm (SL2C.toSelfAdjointMap M ⟨v, hv⟩) := by
-  rw [leftRightToMatrix_ρ_symm]
-  rfl
+    leftRightToMatrix.symm (SL2C.toSelfAdjointMap M ⟨v, hv⟩) :=
+  leftRightToMatrix_ρ_symm v M
 
 end
 end Fermion
diff --git a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean
index 244f4e7d2..7390a73c3 100644
--- a/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean
+++ b/Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean
@@ -47,9 +47,7 @@ def leftDualLeftUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMa
     a' • leftDualLeftUnitVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • leftDualLeftUnitVal =
@@ -62,8 +60,7 @@ def leftDualLeftUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMa
     simp
 
 lemma leftDualLeftUnit_apply_one : leftDualLeftUnit (1 : ℂ) = leftDualLeftUnitVal := by
-  change (1 : ℂ) • leftDualLeftUnitVal = leftDualLeftUnitVal
-  simp only [one_smul]
+  simp [leftDualLeftUnit]
 
 /-- The dual-left-left unit `δₐᵃ` as an element of `(dualLeftHanded ⊗ leftHanded).V`. -/
 def dualLeftLeftUnitVal : (DualLeftHandedWeyl ⊗[ℂ] LeftHandedWeyl) :=
@@ -87,9 +84,7 @@ def dualLeftLeftUnit :
       a' • dualLeftLeftUnitVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • dualLeftLeftUnitVal =
@@ -104,8 +99,7 @@ def dualLeftLeftUnit :
 
 /-- Applying the morphism `dualLeftLeftUnit` to `1` returns `dualLeftLeftUnitVal`. -/
 lemma dualLeftLeftUnit_apply_one : dualLeftLeftUnit (1 : ℂ) = dualLeftLeftUnitVal := by
-  change (1 : ℂ) • dualLeftLeftUnitVal = dualLeftLeftUnitVal
-  simp only [one_smul]
+  simp [dualLeftLeftUnit]
 
 /-- The right-dual-right unit `δ^{dot a}_{dot a}` as an element of
   `(rightHanded ⊗ dualRightHanded).V`. -/
@@ -130,9 +124,7 @@ def rightDualRightUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).Intertwining
     a' • rightDualRightUnitVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • rightDualRightUnitVal =
@@ -151,8 +143,7 @@ def rightDualRightUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).Intertwining
     simp
 
 lemma rightDualRightUnit_apply_one : rightDualRightUnit (1 : ℂ) = rightDualRightUnitVal := by
-  change (1 : ℂ) • rightDualRightUnitVal = rightDualRightUnitVal
-  simp only [one_smul]
+  simp [rightDualRightUnit]
 
 /-- The dual-right-right unit `δ_{dot a}^{dot a}` as an element of
   `(rightHanded ⊗ dualRightHanded).V`. -/
@@ -177,9 +168,7 @@ def dualRightRightUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).Intertwining
     a' • dualRightRightUnitVal
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℂ => ?_
     change x • dualRightRightUnitVal =
@@ -196,8 +185,7 @@ def dualRightRightUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).Intertwining
     simp
 
 lemma dualRightRightUnit_apply_one : dualRightRightUnit (1 : ℂ) = dualRightRightUnitVal := by
-  change (1 : ℂ) • dualRightRightUnitVal = dualRightRightUnitVal
-  simp only [one_smul]
+  simp [dualRightRightUnit]
 
 /-!
 
@@ -259,30 +247,26 @@ open CategoryTheory
 lemma dualLeftLeftUnit_symm :
     dualLeftLeftUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap
     (TensorProduct.comm ℂ _ _ (leftDualLeftUnit (1 : ℂ))) := by
-  rw [dualLeftLeftUnit_apply_one, dualLeftLeftUnitVal_expand_tmul]
-  rw [leftDualLeftUnit_apply_one, leftDualLeftUnitVal_expand_tmul]
-  rfl
+  simp [dualLeftLeftUnit_apply_one, dualLeftLeftUnitVal_expand_tmul,
+    leftDualLeftUnit_apply_one, leftDualLeftUnitVal_expand_tmul]
 
 lemma leftDualLeftUnit_symm :
     leftDualLeftUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap
       (TensorProduct.comm ℂ _ _ (dualLeftLeftUnit (1 : ℂ))) := by
-  rw [dualLeftLeftUnit_apply_one, dualLeftLeftUnitVal_expand_tmul]
-  rw [leftDualLeftUnit_apply_one, leftDualLeftUnitVal_expand_tmul]
-  rfl
+  simp [dualLeftLeftUnit_apply_one, dualLeftLeftUnitVal_expand_tmul,
+    leftDualLeftUnit_apply_one, leftDualLeftUnitVal_expand_tmul]
 
 lemma dualRightRightUnit_symm :
     dualRightRightUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap
       (TensorProduct.comm ℂ _ _ (rightDualRightUnit (1 : ℂ))) := by
-  rw [dualRightRightUnit_apply_one, dualRightRightUnitVal_expand_tmul]
-  rw [rightDualRightUnit_apply_one, rightDualRightUnitVal_expand_tmul]
-  rfl
+  simp [dualRightRightUnit_apply_one, dualRightRightUnitVal_expand_tmul,
+    rightDualRightUnit_apply_one, rightDualRightUnitVal_expand_tmul]
 
 lemma rightDualRightUnit_symm :
     rightDualRightUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap
       (TensorProduct.comm ℂ _ _ (dualRightRightUnit (1 : ℂ))) := by
-  rw [dualRightRightUnit_apply_one, dualRightRightUnitVal_expand_tmul]
-  rw [rightDualRightUnit_apply_one, rightDualRightUnitVal_expand_tmul]
-  rfl
+  simp [dualRightRightUnit_apply_one, dualRightRightUnitVal_expand_tmul,
+    rightDualRightUnit_apply_one, rightDualRightUnitVal_expand_tmul]
 
 end
 end Fermion
diff --git a/Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean b/Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean
index b09c2f539..0b4ebea8f 100644
--- a/Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean
+++ b/Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean
@@ -98,8 +98,7 @@ lemma mem_iff_apply_succSuccAbove_eq {n : ℕ} {c : Fin (n + 1 + 1) → C}
 lemma mem_self_of_dropPair {n : ℕ} {c : Fin (n + 1 + 1) → C}
     {i j : Fin (n + 1 + 1)}
     (b : ComponentIdx (c)) :
-    b ∈ DropPairSection (S := S) (b.dropPair i j) := by
-  simp [DropPairSection]
+    b ∈ DropPairSection (S := S) (b.dropPair i j) := by simp [DropPairSection]
 
 /-- Given a `b` in `ComponentIdx (c ∘ Fin.succSuccAbove i j))` and
   an `x` in `Fin (S.repDim (c i)) × Fin (S.repDim (c j))`, the corresponding
@@ -118,8 +117,7 @@ def ofFin {n : ℕ} {c : Fin (n + 1 + 1) → C}
 lemma ofFin_apply_fst {n : ℕ} {c : Fin (n + 1 + 1) → C}
     {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))
     (x : basisIdx (c i) × basisIdx (c j)) :
-    ofFin (S := S) hij b x i = x.1 := by
-  simp [ofFin]
+    ofFin (S := S) hij b x i = x.1 := by simp [ofFin]
 
 @[simp]
 lemma ofFin_apply_snd {n : ℕ} {c : Fin (n + 1 + 1) → C}
@@ -163,22 +161,19 @@ def ofFinEquiv {n : ℕ} {c : Fin n.succ.succ → C}
       symm
       apply ComponentIdx.congr_right
       simp
-  left_inv x := by
-    simp
+  left_inv x := by simp
 
 @[simp]
 lemma ofFinEquiv_apply_fst {n : ℕ} {c : Fin (n + 1 + 1) → C}
     {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))
     (x : basisIdx (c i) × basisIdx (c j)) :
-    (ofFinEquiv (S := S) hij b x).1 i = x.1 := by
-  simp [ofFinEquiv]
+    (ofFinEquiv (S := S) hij b x).1 i = x.1 := by simp [ofFinEquiv]
 
 @[simp]
 lemma ofFinEquiv_apply_snd {n : ℕ} {c : Fin (n + 1 + 1) → C}
     {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))
     (x : basisIdx (c i) × basisIdx (c j)) :
-    (ofFinEquiv (S := S) hij b x).1 j = x.2 := by
-  simp [ofFinEquiv]
+    (ofFinEquiv (S := S) hij b x).1 j = x.2 := by simp [ofFinEquiv]
 
 end DropPairSection
 
diff --git a/Physlib/Relativity/Tensors/ComponentIdx/Single.lean b/Physlib/Relativity/Tensors/ComponentIdx/Single.lean
index cb521a7ad..42773a904 100644
--- a/Physlib/Relativity/Tensors/ComponentIdx/Single.lean
+++ b/Physlib/Relativity/Tensors/ComponentIdx/Single.lean
@@ -61,10 +61,8 @@ def ComponentIdx.single {c : C} :
   left_inv b := by
     ext i
     cases Fin.fin_one_eq_zero i
-    simp [basisIdxCongr]
     rfl
   right_inv b := by
-    simp [basisIdxCongr]
     rfl
 
 @[simp]
diff --git a/Physlib/Relativity/Tensors/Constructors.lean b/Physlib/Relativity/Tensors/Constructors.lean
index b37d9ddf6..b57ae0e9c 100644
--- a/Physlib/Relativity/Tensors/Constructors.lean
+++ b/Physlib/Relativity/Tensors/Constructors.lean
@@ -126,8 +126,7 @@ noncomputable def fromPairT {c1 c2 : C} :
 lemma fromPairT_tmul {c1 c2 : C} (x : V c1)
     (y : V c2) : fromPairT (x ⊗ₜ[k] y) =
     permT id (And.intro Function.bijective_id (fun i => by fin_cases i <;> rfl))
-    (prodT (fromSingleT (S := S) x) (fromSingleT y)) := by
-  rfl
+    (prodT (fromSingleT (S := S) x) (fromSingleT y)) := rfl
 
 lemma fromPairT_eq_pure {c1 c2 : C} (x : V c1) (y : V c2) :
     fromPairT (S := S) (x ⊗ₜ[k] y) = Pure.toTensor (fun | 0 => x | 1 => y) := by
@@ -413,8 +412,7 @@ lemma fromPairT_basis_repr {c c1 : C}
     rw [prodT_pure, permT_pure]
     rw [basis_repr_pure]
     simp [Pure.component]
-    rw [mul_comm]
-    rfl
+    exact mul_comm _ _
   · intro x y hx hy
     simp_all [P]
 
@@ -476,17 +474,14 @@ noncomputable def fromTripleT {c1 c2 c3 : C} :
     let x2 :=
       TensorProduct.lift prodT (TensorProduct.map LinearMap.id (TensorProduct.lift prodT) x1)
     permT id (And.intro Function.bijective_id (fun i => by fin_cases i <;> rfl)) x2
-  map_add' x y := by
-    simp
-  map_smul' r x := by
-    simp
+  map_add' x y := by simp
+  map_smul' r x := by simp
 
 lemma fromTripleT_tmul {c1 c2 c3 : C} (x : V c1)
     (y : V c2) (z : V c3) :
     fromTripleT (x ⊗ₜ[k] (y ⊗ₜ[k] z)) =
     permT id (And.intro Function.bijective_id (fun i => by fin_cases i <;> rfl))
-      (prodT (fromSingleT (S := S) x) (prodT (fromSingleT y) (fromSingleT z))) := by
-  rfl
+      (prodT (fromSingleT (S := S) x) (prodT (fromSingleT y) (fromSingleT z))) := rfl
 
 lemma actionT_fromTripleT {c1 c2 c3 : C}
     (x : V c1 ⊗[k] (V c2 ⊗[k] V c3)) (g : G) :
@@ -539,11 +534,7 @@ lemma fromTripleT_basis_repr {c c1 c2 : C}
       rw [prodT_pure, prodT_pure, permT_pure]
       rw [basis_repr_pure]
       simp [Pure.component, Fin.prod_univ_three]
-      conv_rhs =>
-        rw [mul_assoc, mul_comm]
-        enter [1]
-        rw [mul_comm]
-      rfl
+      ac_rfl
     · intro y1 y2 hx hy
       simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue,
         Basis.tensorProduct_repr_tmul_apply, smul_eq_mul, P1, P] at hx hy
diff --git a/Physlib/Relativity/Tensors/Contraction/Products.lean b/Physlib/Relativity/Tensors/Contraction/Products.lean
index 0d95a6abb..2451026a3 100644
--- a/Physlib/Relativity/Tensors/Contraction/Products.lean
+++ b/Physlib/Relativity/Tensors/Contraction/Products.lean
@@ -133,8 +133,7 @@ lemma prodT_contrT_snd {n n1 : ℕ} {c : Fin (n + 1 + 1) → C}
     apply induction_on_pure
     · intro p1
       simp only [Nat.add_eq, finSumFinEquiv_apply_right, contrT_pure, P2, P]
-      rw [Pure.prodP_contrP_snd, prodT_pure, contrT_pure]
-      rfl
+      simp [Pure.prodP_contrP_snd, prodT_pure, contrT_pure]
     · intro r t h1
       simp_all only [map_smul, LinearMap.smul_apply, P2, P]
     · intro t1 t2 h1 h2
diff --git a/Physlib/Relativity/Tensors/Contraction/Pure.lean b/Physlib/Relativity/Tensors/Contraction/Pure.lean
index 87e256b57..1f39f39e0 100644
--- a/Physlib/Relativity/Tensors/Contraction/Pure.lean
+++ b/Physlib/Relativity/Tensors/Contraction/Pure.lean
@@ -52,7 +52,6 @@ lemma dropPair_equivariant {n : ℕ} {c : Fin (n + 1 + 1) → C}
     (i j : Fin (n + 1 + 1)) (hij : i ≠ j) (p : Pure S c) (g : G) :
     dropPair i j hij (g • p) = g • dropPair i j hij p := by
   ext m
-  simp only [dropPair, actionP_eq]
   rfl
 
 lemma dropPair_symm (i j : Fin (n + 1 + 1)) (hij : i ≠ j)
@@ -88,8 +87,7 @@ lemma dropPair_update_fst {n : ℕ} [inst : DecidableEq (Fin (n + 1 +1))] {c : F
     dropPair i j hij (p.update i x) = dropPair i j hij p := by
   ext m
   simp only [Function.comp_apply, dropPair, update]
-  rw [Function.update_of_ne]
-  exact Ne.symm (fst_ne_succSuccAbove_pre i j m)
+  simp [Function.update_of_ne, Ne.symm (fst_ne_succSuccAbove_pre i j m)]
 
 @[simp]
 lemma dropPair_update_snd {n : ℕ} [inst : DecidableEq (Fin (n + 1 +1))] {c : Fin (n + 1 + 1) → C}
diff --git a/Physlib/Relativity/Tensors/Contraction/SuccSuccAbove.lean b/Physlib/Relativity/Tensors/Contraction/SuccSuccAbove.lean
index 8b6eefa7a..eab20821b 100644
--- a/Physlib/Relativity/Tensors/Contraction/SuccSuccAbove.lean
+++ b/Physlib/Relativity/Tensors/Contraction/SuccSuccAbove.lean
@@ -169,8 +169,7 @@ lemma succSuccAbove_symm (i j : Fin (n + 1 + 1)) :
 set_option warning.simp.varHead false in
 @[simp, nolint simpVarHead]
 lemma apply_succSuccAbove_symm {c : Fin (n + 1 + 1) → C} (i j : Fin (n + 1 + 1))
-    (k : Fin n) : c (succSuccAbove i j k) = c (succSuccAbove j i k) := by
-  rw [succSuccAbove_symm]
+    (k : Fin n) : c (succSuccAbove i j k) = c (succSuccAbove j i k) := by rw [succSuccAbove_symm]
 
 lemma succSuccAbove_apply_eq_orderIsoOfFin {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (m : Fin n) :
     (succSuccAbove i j) m = (Finset.orderIsoOfFin {i, j}ᶜ
@@ -199,8 +198,7 @@ lemma succSuccAbove_ne_fst (i j : Fin (n + 1 + 1)) (m : Fin n) :
 @[simp]
 lemma snd_ne_succSuccAbove_pre (i j : Fin (n + 1 + 1)) (m : Fin n) :
     ¬ j = (succSuccAbove i j) m := by
-  rw [succSuccAbove_symm]
-  exact fst_ne_succSuccAbove_pre j i m
+  simpa [succSuccAbove_symm] using fst_ne_succSuccAbove_pre j i m
 
 @[simp]
 lemma succSuccAbove_ne_snd (i j : Fin (n + 1 + 1)) (m : Fin n) :
@@ -344,9 +342,7 @@ lemma succSuccAbove_comm_apply (i1 j1 : Fin (n + 1 + 1 + 1 + 1)) (i2 j2 : Fin (n
     succSuccAbove i2' j2' (succSuccAbove i1' j1' m) =
     succSuccAbove i1 j1 (succSuccAbove i2 j2 m) := by
   intro i2' j2' hi2j2' i1' j1'
-  change _ = (succSuccAbove i1 j1 ∘ succSuccAbove i2 j2) m
-  rw [succSuccAbove_comm i1 j1 i2 j2 hij1 hij2]
-  rfl
+  exact (congr_fun (succSuccAbove_comm i1 j1 i2 j2 hij1 hij2) m).symm
 
 /-!
 
diff --git a/Physlib/Relativity/Tensors/Dual.lean b/Physlib/Relativity/Tensors/Dual.lean
index eecf2559b..3cf8f940c 100644
--- a/Physlib/Relativity/Tensors/Dual.lean
+++ b/Physlib/Relativity/Tensors/Dual.lean
@@ -30,15 +30,12 @@ namespace Tensor
 noncomputable def fromDualMap {c : C} : S.Tensor ![S.τ c] →ₗ[k] S.Tensor ![c] where
   toFun t := permT id (by simp; rfl)
     (contrT 1 1 2 (by simp; rfl) (prodT (metricTensor c) t))
-  map_add' t1 t2 := by
-    simp
-  map_smul' r t := by
-    simp
+  map_add' t1 t2 := by simp
+  map_smul' r t := by simp
 
 lemma fromDualMap_apply {c : C} (t : S.Tensor ![S.τ c]) :
     fromDualMap t = permT id (by simp; rfl)
-      (contrT 1 1 2 (by simp; rfl) (prodT (metricTensor c) t)) := by
-  rfl
+      (contrT 1 1 2 (by simp; rfl) (prodT (metricTensor c) t)) := rfl
 
 /-- The linear map taking a tensor based on the color `c` to a tensor
   based on the color `S.τ c`, defined by contraction with the metric tensor. -/
@@ -47,17 +44,14 @@ noncomputable def toDualMap {c : C} : S.Tensor ![c] →ₗ[k] S.Tensor ![S.τ c]
     simp; rfl) (contrT 1 1 2 (by
     change _ ∧ S.τ (S.τ c) = c
     simp) (prodT (metricTensor (S.τ c)) t))
-  map_add' t1 t2 := by
-    simp
-  map_smul' r t := by
-    simp
+  map_add' t1 t2 := by simp
+  map_smul' r t := by simp
 
 lemma toDualMap_apply {c : C} (t : S.Tensor ![c]) :
     toDualMap t = permT id (by
       simp; rfl) (contrT 1 1 2 (by
       change _ ∧ S.τ (S.τ c) = c
-      simp) (prodT (metricTensor (S.τ c)) t)) := by
-  rfl
+      simp) (prodT (metricTensor (S.τ c)) t)) := rfl
 
 set_option backward.isDefEq.respectTransparency false in
 @[simp]
@@ -109,8 +103,7 @@ lemma fromDualMap_eq_permT_toDualMap {c : C} (t : S.Tensor ![S.τ c]) :
 
 lemma toDualMap_eq_permT_fromDualMap {c : C} (t : S.Tensor ![c]) :
     toDualMap t = (fromDualMap (permT id (by simp) t)) := by
-  rw [fromDualMap_eq_permT_toDualMap]
-  rw [toDualMap_apply, toDualMap_apply]
+  rw [fromDualMap_eq_permT_toDualMap, toDualMap_apply, toDualMap_apply]
   conv_rhs =>
     enter [2, 2]
     rw [prodT_permT_right]
diff --git a/Physlib/Relativity/Tensors/Evaluation.lean b/Physlib/Relativity/Tensors/Evaluation.lean
index f29006d19..d618a2f80 100644
--- a/Physlib/Relativity/Tensors/Evaluation.lean
+++ b/Physlib/Relativity/Tensors/Evaluation.lean
@@ -45,16 +45,14 @@ noncomputable def evalPCoeff (i : Fin (n + 1)) (φ : basisIdx (c i)) (p : Pure S
 lemma evalPCoeff_update_self (i : Fin (n + 1)) [inst : DecidableEq (Fin (n + 1))]
     (φ : basisIdx (c i)) (p : Pure S c)
     (x : V (c i)) :
-    evalPCoeff i φ (p.update i x) = (b (c i)).repr x φ := by
-  simp [evalPCoeff]
+    evalPCoeff i φ (p.update i x) = (b (c i)).repr x φ := by simp [evalPCoeff]
 
 @[simp]
 lemma evalPCoeff_update_succAbove (i : Fin (n + 1)) [inst : DecidableEq (Fin (n + 1))]
     (j : Fin n)
     (φ : basisIdx (c i)) (p : Pure S c)
     (x : V (c (i.succAbove j))) :
-    evalPCoeff i φ (p.update (i.succAbove j) x) = evalPCoeff i φ p := by
-  simp [evalPCoeff]
+    evalPCoeff i φ (p.update (i.succAbove j) x) = evalPCoeff i φ p := by simp [evalPCoeff]
 
 lemma evalPCoeff_basisVector (i : Fin (n + 1)) (φ : basisIdx (c i)) (b' : ComponentIdx (S := S) c) :
     evalPCoeff i φ (Pure.basisVector c b') = if b' i = φ then (1 : k) else 0 := by
diff --git a/Physlib/Relativity/Tensors/LeviCivita/Basic.lean b/Physlib/Relativity/Tensors/LeviCivita/Basic.lean
index f3e60002c..76361fe20 100644
--- a/Physlib/Relativity/Tensors/LeviCivita/Basic.lean
+++ b/Physlib/Relativity/Tensors/LeviCivita/Basic.lean
@@ -50,8 +50,7 @@ scoped[realLorentzTensor] notation "ε4" => leviCivita
 lemma leviCivita_eq_ofInt : ε4 =
     TensorInt.toTensor (S := realLorentzTensor 3)
     (c := ![Color.up, Color.up, Color.up, Color.up]) fun f =>
-    generalizedKroneckerDelta (fun i => finSumFinEquiv (f i)) (id : Fin 4 → Fin 4) :=
-  rfl
+    generalizedKroneckerDelta (fun i => finSumFinEquiv (f i)) (id : Fin 4 → Fin 4) := rfl
 
 /-- The components of the Levi-Civita tensor in the standard basis are the generalized
 Kronecker delta of the multi-index against the identity. -/
diff --git a/Physlib/Relativity/Tensors/MetricTensor.lean b/Physlib/Relativity/Tensors/MetricTensor.lean
index e9e1a9678..f8f6159f7 100644
--- a/Physlib/Relativity/Tensors/MetricTensor.lean
+++ b/Physlib/Relativity/Tensors/MetricTensor.lean
@@ -36,25 +36,21 @@ lemma metricTensor_congr {c c1 : C} (h : c = c1) :
 
 @[simp]
 lemma metricTensor_invariant {c : C} (g : G) :
-    g • S.metricTensor c = metricTensor c := by
-  rw [metricTensor, actionT_fromConstPair]
+    g • S.metricTensor c = metricTensor c := by rw [metricTensor, actionT_fromConstPair]
 
 lemma permT_fromPairTContr_metric_metric {c : C} :
     permT ![1, 0] (And.intro (by decide) (fun i => by fin_cases i <;> rfl))
     (fromPairTContr ((S.metric c) (1 : k))
     ((S.metric ((S.τ c))) (1 : k))) = (unitTensor c) := by
   rw [fromPairTContr, ← fromPairT_comm]
-  change _ = fromPairT ((S.unit c) (1 : k))
-  rw [← S.contr_metric]
-  rfl
+  exact congr_arg fromPairT (S.contr_metric c)
 
 lemma fromPairTContr_metric_metric_eq_permT_unit {c : C} :
     fromPairTContr ((S.metric c) (1 : k))
     ((S.metric (S.τ c)) (1 : k)) =
     permT ![1, 0] (And.intro (by decide) (fun i => by fin_cases i <;> rfl))
     (unitTensor c) := by
-  rw [← permT_fromPairTContr_metric_metric]
-  rw [permT_permT]
+  rw [← permT_fromPairTContr_metric_metric, permT_permT]
   symm
   apply permT_congr_eq_id
   decide
@@ -69,20 +65,16 @@ lemma contrT_metricTensor_metricTensor {c : C} :
   rw [metricTensor, metricTensor, fromConstPair, fromConstPair]
   rw [fromPairT_contr_fromPairT_eq_fromPairTContr]
   erw [fromPairTContr_metric_metric_eq_permT_unit]
-  rw [permT_permT]
-  rfl
+  simp [permT_permT]
 
 lemma contrT_metricTensor_metricTensor_eq_dual_unit {c : C} :
     contrT 2 1 2 (by simp; rfl) (prodT (metricTensor c) (metricTensor (S.τ c))) =
       permT ![0, 1] (And.intro (by decide) (fun i => by
         fin_cases i
-        · change S.τ (S.τ c) = c
-          simp
+        · exact S.τ_involution c
         · rfl))
       (unitTensor (S := S) (S.τ c)) := by
-  rw [contrT_metricTensor_metricTensor]
-  rw [unitTensor_eq_permT_dual]
-  rw [permT_permT]
+  rw [contrT_metricTensor_metricTensor, unitTensor_eq_permT_dual, permT_permT]
   apply permT_congr
   · decide
   · rfl
diff --git a/Physlib/Relativity/Tensors/OfInt.lean b/Physlib/Relativity/Tensors/OfInt.lean
index b50bf4139..c0a0b1e27 100644
--- a/Physlib/Relativity/Tensors/OfInt.lean
+++ b/Physlib/Relativity/Tensors/OfInt.lean
@@ -41,9 +41,7 @@ noncomputable def toTensor {n : ℕ} {c : Fin n → C} (f : TensorInt S c) :
 
 lemma basis_repr_apply {n : ℕ} {c : Fin n → C}
     (f : TensorInt S c) (b : ComponentIdx c) :
-    (Tensor.basis c).repr (toTensor f) b = Int.cast (f b) := by
-  simp only [toTensor, Basis.repr_symm_apply, Basis.repr_linearCombination]
-  rfl
+    (Tensor.basis c).repr (toTensor f) b = Int.cast (f b) := by simp [toTensor]
 
 end TensorInt
 
diff --git a/Physlib/Relativity/Tensors/Product.lean b/Physlib/Relativity/Tensors/Product.lean
index 16c405c4b..d6d14d281 100644
--- a/Physlib/Relativity/Tensors/Product.lean
+++ b/Physlib/Relativity/Tensors/Product.lean
@@ -121,15 +121,13 @@ def Pure.prodP {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}
 lemma Pure.prodP_apply_castAdd {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}
     (p1 : Pure S c) (p2 : Pure S c1) (i : Fin n1) :
     Pure.prodP p1 p2 (Fin.castAdd n2 i) =
-    LinearEquiv.cast (R := k) (by simp) (p1 i) := by
-  simp [Pure.prodP]
+    LinearEquiv.cast (R := k) (by simp) (p1 i) := by simp [Pure.prodP]
 
 @[simp]
 lemma Pure.prodP_apply_natAdd {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}
     (p1 : Pure S c) (p2 : Pure S c1) (i : Fin n2) :
     Pure.prodP p1 p2 (Fin.natAdd n1 i) =
-    LinearEquiv.cast (R := k) (by simp) (p2 i) := by
-  simp [Pure.prodP]
+    LinearEquiv.cast (R := k) (by simp) (p2 i) := by simp [Pure.prodP]
 
 lemma Pure.prodP_apply_finSumFinEquiv {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}
     (p1 : Pure S c) (p2 : Pure S c1) (i : Fin n1 ⊕ Fin n2) :
@@ -445,8 +443,7 @@ noncomputable def prodT {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C} :
 
 lemma prodT_pure {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}
     (t : Pure S c) (t1 : Pure S c1) :
-    (t.toTensor).prodT (t1.toTensor) = (Pure.prodP t t1).toTensor := by
-  simp [prodT, Pure.toTensor]
+    (t.toTensor).prodT (t1.toTensor) = (Pure.prodP t t1).toTensor := by simp [prodT, Pure.toTensor]
 
 /-!
 
diff --git a/Physlib/Relativity/Tensors/RealTensor/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/Basic.lean
index b95714532..152b69768 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Basic.lean
@@ -136,13 +136,11 @@ These re-express fields of `realLorentzTensor d` in terms of `Lorentz` data.
 
 @[simp]
 lemma basisIdxCongr_eq_refl {d : ℕ} {c1 c2 : realLorentzTensor.Color} (h : c1 = c2) :
-    TensorSpecies.basisIdxCongr (basisIdx := fun _ => Fin 1 ⊕ Fin d) h = Equiv.refl _ := by
-  rfl
+    TensorSpecies.basisIdxCongr (basisIdx := fun _ => Fin 1 ⊕ Fin d) h = Equiv.refl _ := rfl
 
 lemma basisIdxCongr_apply {d : ℕ} {c1 c2 : realLorentzTensor.Color} (h : c1 = c2)
     (i : Fin 1 ⊕ Fin d) :
-    TensorSpecies.basisIdxCongr (basisIdx := fun _ => Fin 1 ⊕ Fin d) h i = i := by
-  simp
+    TensorSpecies.basisIdxCongr (basisIdx := fun _ => Fin 1 ⊕ Fin d) h i = i := by simp
 
 /-!
 
@@ -224,8 +222,7 @@ lemma contrT_toField {d} (c : Fin 2 → Color)
   rw [contrT_eq_sum_evalT, map_sum]
   congr
   ext μ
-  simp only [toField_permT]
-  rfl
+  exact toField_permT _ _ _
 
 open ComponentIdx in
 lemma contrT_basis_repr_apply_eq_fin {n d: ℕ} {c : Fin (n + 1 + 1) → realLorentzTensor.Color}
diff --git a/Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean
index e9636635c..ffe5c83d2 100644
--- a/Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean
@@ -79,8 +79,7 @@ instance isNormedAddCommGroup (d : ℕ) : NormedAddCommGroup (CoVector d) where
 
 instance isNormedSpace (d : ℕ) : NormedSpace ℝ (CoVector d) where
   norm_smul_le c v := by
-    simp only [norm_eq_equivEuclid, map_smul]
-    exact norm_smul_le c (equivEuclid d v)
+    simpa only [norm_eq_equivEuclid, map_smul] using norm_smul_le c (equivEuclid d v)
 
 open InnerProductSpace
 
diff --git a/Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean b/Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean
index a79c5cbc6..094fa9566 100644
--- a/Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean
@@ -52,8 +52,7 @@ lemma rep_apply_eq_sum (d : ℕ) (Λ : LorentzGroup d) (v : CoVector d) (k : Fin
     rep Λ v k = ∑ j, (Λ⁻¹).1 j k • v j := rfl
 
 lemma rep_apply_eq_sum_coe (d : ℕ) (Λ : LorentzGroup d) (v : CoVector d) (k : Fin 1 ⊕ Fin d) :
-    rep Λ v k = ∑ j, Λ.1⁻¹ j k • v j := by
-  rw [rep_apply_eq_sum, LorentzGroup.coe_inv]
+    rep Λ v k = ∑ j, Λ.1⁻¹ j k • v j := by rw [rep_apply_eq_sum, LorentzGroup.coe_inv]
 
 lemma rep_apply_basis {d} (μ : Fin 1 ⊕ Fin d) (Λ : LorentzGroup d) :
     rep Λ (basis μ) = ∑ j, Λ.1⁻¹ μ j • basis j := by
diff --git a/Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean b/Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean
index 5a5c6537f..9d3241f95 100644
--- a/Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean
@@ -173,13 +173,11 @@ lemma smul_sub {d : ℕ} (Λ : LorentzGroup d) (p q : CoVector d) :
 
 set_option backward.isDefEq.respectTransparency false in
 lemma smul_zero {d : ℕ} (Λ : LorentzGroup d) :
-    Λ • (0 : CoVector d) = 0 := by
-  rw [smul_eq_mulVec, Matrix.mulVec_zero]
+    Λ • (0 : CoVector d) = 0 := by rw [smul_eq_mulVec, Matrix.mulVec_zero]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma smul_neg {d : ℕ} (Λ : LorentzGroup d) (p : CoVector d) :
-    Λ • (-p) = - (Λ • p) := by
-  rw [smul_eq_mulVec, smul_eq_mulVec, Matrix.mulVec_neg]
+    Λ • (-p) = - (Λ • p) := by rw [smul_eq_mulVec, smul_eq_mulVec, Matrix.mulVec_neg]
 
 /-- The Lorentz action on vectors as a continuous linear map. -/
 def actionCLM {d : ℕ} (Λ : LorentzGroup d) :
diff --git a/Physlib/Relativity/Tensors/RealTensor/Matrix/Pre.lean b/Physlib/Relativity/Tensors/RealTensor/Matrix/Pre.lean
index 7162b8956..bc0ecb84c 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Matrix/Pre.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Matrix/Pre.lean
@@ -35,8 +35,7 @@ lemma contrContrToMatrixRe_symm_expand_tmul (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1
   rw [Finsupp.linearCombination_apply_of_mem_supported ℝ (s := Finset.univ)]
   · rw [Fintype.sum_prod_type]
     refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
-    erw [Basis.tensorProduct_apply (contrBasis d) (contrBasis d) i j]
-    rfl
+    exact congrArg₂ (· • ·) rfl (Basis.tensorProduct_apply (contrBasis d) (contrBasis d) i j)
   · simp
 
 /-- Equivalence of `CoMod ⊗ CoMod` to `(1 + d) x (1 + d)` real matrices. -/
@@ -53,8 +52,7 @@ lemma coCoToMatrixRe_symm_expand_tmul (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ F
   rw [Finsupp.linearCombination_apply_of_mem_supported ℝ (s := Finset.univ)]
   · rw [Fintype.sum_prod_type]
     refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
-    erw [Basis.tensorProduct_apply (coBasis d) (coBasis d) i j]
-    rfl
+    exact congrArg₂ (· • ·) rfl (Basis.tensorProduct_apply (coBasis d) (coBasis d) i j)
   · simp
 
 /-- Equivalence of `ContrMod d ⊗ CoMod d` to `(1 + d) x (1 + d)` real matrices. -/
@@ -72,8 +70,7 @@ lemma contrCoToMatrixRe_symm_expand_tmul (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 
   rw [Finsupp.linearCombination_apply_of_mem_supported ℝ (s := Finset.univ)]
   · rw [Fintype.sum_prod_type]
     refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
-    erw [Basis.tensorProduct_apply _ _ i j]
-    rfl
+    exact congrArg₂ (· • ·) rfl (Basis.tensorProduct_apply _ _ i j)
   · simp
 
 /-- Equivalence of `CoMod d ⊗ ContrMod d` to `(1 + d) x (1 + d)` real matrices. -/
@@ -91,8 +88,7 @@ lemma coContrToMatrixRe_symm_expand_tmul (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 
   rw [Finsupp.linearCombination_apply_of_mem_supported ℝ (s := Finset.univ)]
   · rw [Fintype.sum_prod_type]
     refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_))
-    erw [Basis.tensorProduct_apply _ _ i j]
-    rfl
+    exact congrArg₂ (· • ·) rfl (Basis.tensorProduct_apply _ _ i j)
   · simp
 
 /-!
@@ -117,8 +113,7 @@ lemma contrContrToMatrixRe_ρ {d : ℕ} (v : (ContrMod d ⊗[ℝ] ContrMod d)) (
     have h1 := (LinearMap.toMatrix_mulVec_repr ((contrBasis d).tensorProduct (contrBasis d))
       ((contrBasis d).tensorProduct (contrBasis d))
       (TensorProduct.map (ContrMod.rep M) (ContrMod.rep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
@@ -154,8 +149,7 @@ lemma coCoToMatrixRe_ρ {d : ℕ} (v : (CoMod d ⊗[ℝ] CoMod d)) (M : LorentzG
     have h1 := (LinearMap.toMatrix_mulVec_repr ((coBasis d).tensorProduct (coBasis d))
       ((coBasis d).tensorProduct (coBasis d))
       (TensorProduct.map (CoMod.rep M) (CoMod.rep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
@@ -191,8 +185,7 @@ lemma contrCoToMatrixRe_ρ {d : ℕ} (v : (ContrMod d ⊗[ℝ] CoMod d)) (M : Lo
     have h1 := (LinearMap.toMatrix_mulVec_repr ((contrBasis d).tensorProduct (coBasis d))
       ((contrBasis d).tensorProduct (coBasis d))
       (TensorProduct.map (ContrMod.rep M) (CoMod.rep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
@@ -228,8 +221,7 @@ lemma coContrToMatrixRe_ρ {d : ℕ} (v : (CoMod d ⊗[ℝ] ContrMod d)) (M : Lo
     have h1 := (LinearMap.toMatrix_mulVec_repr ((coBasis d).tensorProduct (contrBasis d))
       ((coBasis d).tensorProduct (contrBasis d))
       (TensorProduct.map (CoMod.rep M) (ContrMod.rep M)) v)
-    erw [h1]
-    rfl
+    exact h1.symm
   rw [TensorProduct.toMatrix_map]
   funext i j
   change ∑ k, ((kroneckerMap (fun x1 x2 => x1 * x2)
diff --git a/Physlib/Relativity/Tensors/RealTensor/Metrics/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/Metrics/Basic.lean
index 61061f0d7..dddd3b1b1 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Metrics/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Metrics/Basic.lean
@@ -60,13 +60,11 @@ open Tensor
 
 lemma coMetric_eq_fromConstPair {d : ℕ} :
     η' d = fromConstPair (S := realLorentzTensor d) (c1 := .down) (c2 := .down)
-      (Lorentz.preCoMetric d) := by
-  rfl
+      (Lorentz.preCoMetric d) := rfl
 
 lemma contrMetric_eq_fromConstPair {d : ℕ} :
     η d = fromConstPair (S := realLorentzTensor d)
-      (c1 := .up) (c2 := .up) (Lorentz.preContrMetric d) := by
-  rfl
+      (c1 := .up) (c2 := .up) (Lorentz.preContrMetric d) := rfl
 
 lemma coMetric_eq_fromPairT {d : ℕ} :
     η' d = fromPairT (S := realLorentzTensor d) (c1 := .down) (c2 := .down)
@@ -87,8 +85,7 @@ lemma contrMetric_eq_fromPairT {d : ℕ} :
 /-- The tensor `coMetric` is invariant under the action of `LorentzGroup d`. -/
 @[simp]
 lemma actionT_coMetric {d : ℕ} (g : LorentzGroup d) :
-    g • η' d = η' d:= by
-  erw [TensorSpecies.metricTensor_invariant]
+    g • η' d = η' d := TensorSpecies.metricTensor_invariant g
 
 /-- The tensor `contrMetric` is invariant under the action of `LorentzGroup d`. -/
 @[simp]
diff --git a/Physlib/Relativity/Tensors/RealTensor/ToComplex.lean b/Physlib/Relativity/Tensors/RealTensor/ToComplex.lean
index 8ce430965..b73df9dd3 100644
--- a/Physlib/Relativity/Tensors/RealTensor/ToComplex.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/ToComplex.lean
@@ -93,16 +93,14 @@ lemma colorToComplex_match_up {n} {c : Fin n → realLorentzTensor.Color} {j}
     (match c j with
       | .up => complexLorentzTensor.Color.up
       | .down => complexLorentzTensor.Color.down)
-      = complexLorentzTensor.Color.up := by
-  rw [hc]
+      = complexLorentzTensor.Color.up := by rw [hc]
 
 lemma colorToComplex_match_down {n} {c : Fin n → realLorentzTensor.Color} {j}
     (hc : c j = realLorentzTensor.Color.down) :
     (match c j with
       | .up => complexLorentzTensor.Color.up
       | .down => complexLorentzTensor.Color.down)
-      = complexLorentzTensor.Color.down := by
-  rw [hc]
+      = complexLorentzTensor.Color.down := by rw [hc]
 
 lemma colorToComplex_comp_eq_match {n} (c : Fin n → realLorentzTensor.Color) (j : Fin n) :
     (colorToComplex ∘ c) j =
@@ -131,8 +129,7 @@ lemma ComponentIdx.complexify_apply {n} {c : Fin n → realLorentzTensor.Color}
 @[simp]
 lemma ComponentIdx.complexify_toFun_apply {n} {c : Fin n → realLorentzTensor.Color}
     (f : ComponentIdx (S := realLorentzTensor) c) (j : Fin n) :
-    (ComponentIdx.complexify.toFun f) j = (ComponentIdx.complexify f) j :=
-  rfl
+    (ComponentIdx.complexify.toFun f) j = (ComponentIdx.complexify f) j := rfl
 
 /-!
 
@@ -156,8 +153,7 @@ noncomputable def toComplex {n} {c : Fin n → realLorentzTensor.Color} :
     rw [Finset.smul_sum]
     congr
     funext i
-    rw [← smul_smul]
-    rfl
+    exact (smul_smul _ _ _).symm
   map_add' c v := by
     simp only [map_add, Finsupp.coe_add, Pi.add_apply]
     rw [← Finset.sum_add_distrib]
@@ -313,14 +309,12 @@ noncomputable def toComplexVector (c : realLorentzTensor.Color) :
       simp only [map_add, Finsupp.coe_add, Pi.add_apply, Nat.reduceAdd, ← Finset.sum_add_distrib]
       congr
       funext x
-      rw [add_smul]
-      rfl
+      exact add_smul _ _ _
     | Color.down =>
       simp only [map_add, Finsupp.coe_add, Pi.add_apply, Nat.reduceAdd, ← Finset.sum_add_distrib]
       congr
       funext x
-      rw [add_smul]
-      rfl
+      exact add_smul _ _ _
   map_smul' r v := by
     match c with
     | Color.up =>
@@ -329,16 +323,14 @@ noncomputable def toComplexVector (c : realLorentzTensor.Color) :
       rw [Finset.smul_sum]
       congr
       funext x
-      rw [← smul_smul]
-      rfl
+      exact (smul_smul _ _ _).symm
     | Color.down =>
       simp only [map_smul, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul, Nat.reduceAdd,
         Complex.ofRealHom_eq_coe, Complex.coe_smul]
       rw [Finset.smul_sum]
       congr
       funext x
-      rw [← smul_smul]
-      rfl
+      exact (smul_smul _ _ _).symm
 
 lemma toComplexVector_up_eq_inclCongrRealLorentz (v : Lorentz.ContrMod 3) :
     toComplexVector Color.up v = Lorentz.inclCongrRealLorentz v := by
@@ -349,14 +341,12 @@ lemma toComplexVector_up_eq_inclCongrRealLorentz (v : Lorentz.ContrMod 3) :
             Lorentz.complexContrBasisFin4, Lorentz.ContrMod.toFin1dℝ_eq_val]
           rfl
     _ = ∑ i, v.toFin1dℝ i • Lorentz.inclCongrRealLorentz
-        (Lorentz.ContrMod.stdBasis i) := by
-          simp only [Lorentz.complexContrBasis_of_real]
+        (Lorentz.ContrMod.stdBasis i) := by simp only [Lorentz.complexContrBasis_of_real]
     _ = Lorentz.inclCongrRealLorentz
         (∑ i, v.toFin1dℝ i • Lorentz.ContrMod.stdBasis i) := by
           rw [map_sum]
           simp only [LinearMap.map_smulₛₗ, Complex.ofRealHom_eq_coe, Complex.coe_smul]
-    _ = Lorentz.inclCongrRealLorentz v := by
-          rw [← Lorentz.ContrMod.stdBasis_decomp v]
+    _ = Lorentz.inclCongrRealLorentz v := by rw [← Lorentz.ContrMod.stdBasis_decomp v]
 
 lemma toComplexVector_down_eq_inclCoRealLorentz (v : Lorentz.CoMod 3) :
     toComplexVector Color.down v = Lorentz.inclCoRealLorentz v := by
@@ -366,14 +356,12 @@ lemma toComplexVector_down_eq_inclCoRealLorentz (v : Lorentz.CoMod 3) :
           simp [toComplexVector, Lorentz.coBasis_repr_apply, Lorentz.complexCoBasisFin4]
           rfl
     _ = ∑ i, v.toFin1dℝ i • Lorentz.inclCoRealLorentz
-        (Lorentz.CoMod.stdBasis i) := by
-          simp only [Lorentz.complexCoBasis_of_real]
+        (Lorentz.CoMod.stdBasis i) := by simp only [Lorentz.complexCoBasis_of_real]
     _ = Lorentz.inclCoRealLorentz
         (∑ i, v.toFin1dℝ i • Lorentz.CoMod.stdBasis i) := by
           rw [map_sum]
           simp only [LinearMap.map_smulₛₗ, Complex.ofRealHom_eq_coe, Complex.coe_smul]
-    _ = Lorentz.inclCoRealLorentz v := by
-          rw [← Lorentz.CoMod.stdBasis_decomp v]
+    _ = Lorentz.inclCoRealLorentz v := by rw [← Lorentz.CoMod.stdBasis_decomp v]
 
 /-- The function which turns a real pure tensor into a complex one. -/
 noncomputable def toComplexPure {c : Fin n → Color} (p : Pure realLorentzTensor c) :
@@ -515,9 +503,7 @@ lemma toComplex_equivariant {n} {c : Fin n → realLorentzTensor.Color}
   · rw [actionT_pure, toComplex_pure, actionT_pure, actionP_toComplexPure, toComplex_pure]
   · simp
     rw [← h]
-    change Λ • (r : ℂ) • toComplex t = _
-    rw [actionT_smul]
-    rfl
+    exact actionT_smul (g := Λ) (r := (r : ℂ)) (t := toComplex t)
   · simp_all
 
 /-!
@@ -842,8 +828,7 @@ lemma contrT_toComplex {n : ℕ}
             (p := Pure.basisVector (colorToComplex ∘ c) (ComponentIdx.complexify b))]
         exact toComplex_contrP_basisVector h b
       · -- zero tensor
-        dsimp [P]
-        simp
+        simp [P]
       · -- scalar multiplication
         intro r t ht
         dsimp [P] at ht ⊢
diff --git a/Physlib/Relativity/Tensors/RealTensor/Units/Pre.lean b/Physlib/Relativity/Tensors/RealTensor/Units/Pre.lean
index 605ae3f2c..04d3dcb45 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Units/Pre.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Units/Pre.lean
@@ -53,9 +53,7 @@ def preContrCoUnit (d : ℕ := 3) :
   toFun := fun a => a • preContrCoUnitVal d
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℝ => ?_
     simp only [LinearMap.coe_comp, Function.comp_apply]
@@ -69,8 +67,7 @@ def preContrCoUnit (d : ℕ := 3) :
     simp
 
 lemma preContrCoUnit_apply_one {d : ℕ} : (preContrCoUnit d) (1 : ℝ) = preContrCoUnitVal d := by
-  change (1 : ℝ) • preContrCoUnitVal d = preContrCoUnitVal d
-  rw [one_smul]
+  simp [preContrCoUnit]
 
 /-- The co-contra unit for complex lorentz vectors. Usually denoted `δᵢⁱ`. -/
 def preCoContrUnitVal (d : ℕ := 3) : CoMod d ⊗[ℝ] ContrMod d :=
@@ -104,9 +101,7 @@ def preCoContrUnit (d : ℕ) : (Representation.trivial ℝ (LorentzGroup d) ℝ)
   toFun := fun a => a • preCoContrUnitVal d
   map_add' := fun x y => by
     simp only [add_smul]
-  map_smul' := fun m x => by
-    simp only [smul_smul]
-    rfl
+  map_smul' := fun m x => by simp only [smul_eq_mul, smul_smul, RingHom.id_apply]
   isIntertwining' M := by
     refine LinearMap.ext fun x : ℝ => ?_
     simp only [LinearMap.coe_comp, Function.comp_apply]
@@ -122,8 +117,7 @@ def preCoContrUnit (d : ℕ) : (Representation.trivial ℝ (LorentzGroup d) ℝ)
     simp
 
 lemma preCoContrUnit_apply_one {d : ℕ} : (preCoContrUnit d) (1 : ℝ) = preCoContrUnitVal d := by
-  change (1 : ℝ) • preCoContrUnitVal d = preCoContrUnitVal d
-  rw [one_smul]
+  simp [preCoContrUnit]
 
 /-!
 
@@ -147,9 +141,7 @@ lemma contr_preContrCoUnit {d : ℕ} (x : CoMod d) :
     rfl
   rw [h1]
   have h2 : coContrContract.toLinearMap.rTensor _ (∑ i, (x ⊗ₜ[ℝ] contrBasis d i) ⊗ₜ[ℝ] coBasis d i)
-      = ∑ i, ((coContrContract) (x ⊗ₜ[ℝ] contrBasis d i)) ⊗ₜ[ℝ] coBasis d i := by
-    rw [map_sum]
-    rfl
+      = ∑ i, ((coContrContract) (x ⊗ₜ[ℝ] contrBasis d i)) ⊗ₜ[ℝ] coBasis d i := map_sum _ _ _
   erw [h2]
   obtain ⟨c, rfl⟩ := (Submodule.mem_span_range_iff_exists_fun ℝ).mp (Basis.mem_span (coBasis d) x)
   have h3 (i : Fin 1 ⊕ Fin d) : (coContrContract)
@@ -162,8 +154,7 @@ lemma contr_preContrCoUnit {d : ℕ} (x : CoMod d) :
   conv_lhs =>
     enter [2, 2, i]
     rw [h3 i]
-  rw [map_sum]
-  rfl
+  exact map_sum _ _ _
 
 /-- Contraction on the right with `coContrUnit`. -/
 lemma contr_preCoContrUnit {d : ℕ} (x : ContrMod d) :
@@ -181,9 +172,7 @@ lemma contr_preCoContrUnit {d : ℕ} (x : ContrMod d) :
     rfl
   rw [h1]
   have h2 :contrCoContract.toLinearMap.rTensor _ (∑ i, (x ⊗ₜ[ℝ] coBasis d i) ⊗ₜ[ℝ] contrBasis d i)
-      = ∑ i, ((contrCoContract) (x ⊗ₜ[ℝ] coBasis d i)) ⊗ₜ[ℝ] contrBasis d i := by
-    rw [map_sum]
-    rfl
+      = ∑ i, ((contrCoContract) (x ⊗ₜ[ℝ] coBasis d i)) ⊗ₜ[ℝ] contrBasis d i := map_sum _ _ _
   erw [h2]
   obtain ⟨c, rfl⟩ := (Submodule.mem_span_range_iff_exists_fun ℝ).mp
     (Basis.mem_span (contrBasis d) x)
@@ -197,8 +186,7 @@ lemma contr_preCoContrUnit {d : ℕ} (x : ContrMod d) :
   conv_lhs =>
     enter [2, 2, i]
     rw [h3 i]
-  rw [map_sum]
-  rfl
+  exact map_sum _ _ _
 
 /-!
 
@@ -209,16 +197,14 @@ lemma contr_preCoContrUnit {d : ℕ} (x : ContrMod d) :
 lemma preContrCoUnit_symm {d : ℕ} :
     (preContrCoUnit d) (1 : ℝ) = LinearMap.lTensor _ (LinearEquiv.cast (by rfl)).toLinearMap
       (TensorProduct.comm ℝ _ _ ((preCoContrUnit d) (1 : ℝ))) := by
-  rw [preContrCoUnit_apply_one, preContrCoUnitVal_expand_tmul]
-  rw [preCoContrUnit_apply_one, preCoContrUnitVal_expand_tmul]
-  simp
+  simp [preContrCoUnit_apply_one, preContrCoUnitVal_expand_tmul, preCoContrUnit_apply_one,
+    preCoContrUnitVal_expand_tmul]
 
 lemma preCoContrUnit_symm {d : ℕ} :
     (preCoContrUnit d) (1 : ℝ) = LinearMap.lTensor _ (LinearEquiv.cast (by simp)).toLinearMap
     (TensorProduct.comm ℝ _ _ ((preContrCoUnit d) (1 : ℝ))) := by
-  rw [preContrCoUnit_apply_one, preContrCoUnitVal_expand_tmul]
-  rw [preCoContrUnit_apply_one, preCoContrUnitVal_expand_tmul]
-  simp
+  simp [preContrCoUnit_apply_one, preContrCoUnitVal_expand_tmul, preCoContrUnit_apply_one,
+    preCoContrUnitVal_expand_tmul]
 
 end Lorentz
 end
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean
index 529398ff4..f4f59992b 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean
@@ -91,8 +91,7 @@ instance isNormedAddCommGroup (d : ℕ) : NormedAddCommGroup (Vector d) where
 
 instance isNormedSpace (d : ℕ) : NormedSpace ℝ (Vector d) where
   norm_smul_le c v := by
-    simp only [norm_eq_equivEuclid, map_smul]
-    exact norm_smul_le c (equivEuclid d v)
+    simpa only [norm_eq_equivEuclid, map_smul] using norm_smul_le c (equivEuclid d v)
 open InnerProductSpace
 
 instance (d : ℕ) : Inner ℝ (Vector d) where
@@ -153,7 +152,6 @@ lemma apply_sum {d : ℕ} {ι : Type} [Fintype ι] (f : ι → Vector d) (i : Fi
     simp at h'
     rw [← @e.sum_comp, ← @e.sum_comp, h']
   · intro f i
-    simp only [Finset.univ_eq_empty, Finset.sum_empty]
     rfl
   · intro ι _ h f i
     rw [Fintype.sum_option, apply_add, h, Fintype.sum_option]
@@ -409,10 +407,8 @@ def ofTemporalComponent {d : ℕ} : ℝ →L[ℝ] Vector d where
   Lorentz Vector with only non-zero spatial components. -/
 def ofSpatialComponent {d : ℕ} : EuclideanSpace ℝ (Fin d) →L[ℝ] Vector d where
   toFun xs := ∑ i, xs i • basis (Sum.inr i)
-  map_add' xs ys := by
-    simp [add_smul, Finset.sum_add_distrib]
-  map_smul' c xs := by
-    simp [smul_smul, Finset.smul_sum]
+  map_add' xs ys := by simp [add_smul, Finset.sum_add_distrib]
+  map_smul' c xs := by simp [smul_smul, Finset.smul_sum]
 
 /-!
 
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/Basic.lean
index 028fafe7d..d066f5eec 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/Basic.lean
@@ -56,8 +56,7 @@ lemma spaceLike_iff_norm_sq_neg {d : ℕ} (p : Vector d) :
     simp only [reduceCtorEq, h, lt_self_iff_false]
   · split
     · rename_i h
-      simp only [reduceCtorEq, false_iff, not_lt]
-      exact le_of_lt h
+      simpa only [reduceCtorEq, false_iff, not_lt] using le_of_lt h
     · rename_i h1 h2
       simp only [true_iff]
       rw [not_lt_iff_eq_or_lt] at h2
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/LightLike.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/LightLike.lean
index c919ddf4f..3b39e714d 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/LightLike.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/LightLike.lean
@@ -30,8 +30,7 @@ lemma lightLike_iff_norm_sq_zero {d : ℕ} (p : Vector d) :
   -- Zero vector has zero Minkowski norm squared
 @[simp]
 lemma causalCharacter_zero {d : ℕ} : causalCharacter (0 : Vector d) =
-    CausalCharacter.lightLike := by
-  simp [causalCharacter]
+    CausalCharacter.lightLike := by simp [causalCharacter]
 
 /-- Causally preceding is reflexive -/
 @[simp]
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean
index 1febf830f..f157866d1 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Causality/TimeLike.lean
@@ -27,8 +27,7 @@ namespace Vector
 @[simp]
 lemma timelike_neg_time_component_product {d : ℕ} (v w : Vector d)
     (hv_neg : v (Sum.inl 0) < 0) (hw_neg : w (Sum.inl 0) < 0) :
-    v (Sum.inl 0) * w (Sum.inl 0) > 0 := by
-  exact mul_pos_of_neg_of_neg hv_neg hw_neg
+    v (Sum.inl 0) * w (Sum.inl 0) > 0 := mul_pos_of_neg_of_neg hv_neg hw_neg
 
 /-- For timelike vectors, the Minkowski inner product is positive -/
 lemma timeLike_iff_norm_sq_pos {d : ℕ} (p : Vector d) :
@@ -52,16 +51,14 @@ lemma timelike_time_dominates_space {d : ℕ} {v : Vector d}
   rw [minkowskiProduct_toCoord] at hv
   simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial]
   have h_spatial_sum : ∑ x, spatialPart v x * spatialPart v x =
-                    ∑ i, v (Sum.inr i) * v (Sum.inr i) := by
-      simp only
+                    ∑ i, v (Sum.inr i) * v (Sum.inr i) := by simp only
   have h_time : timeComponent v = v (Sum.inl 0) := rfl
   rw [h_spatial_sum, h_time]
   have h_norm_pos : 0 < v (Sum.inl 0) * v (Sum.inl 0) -
                   ∑ i, v (Sum.inr i) * v (Sum.inr i) := hv
   -- Rearrange the inequality
   have h : ∑ i, v (Sum.inr i) * v (Sum.inr i) <
-          v (Sum.inl 0) * v (Sum.inl 0) := by
-    exact lt_of_sub_pos h_norm_pos
+          v (Sum.inl 0) * v (Sum.inl 0) := lt_of_sub_pos h_norm_pos
   exact h
 
 /-- For nonzero timelike vectors, the time component is nonzero -/
@@ -102,8 +99,7 @@ lemma timeLike_iff_time_lt_space {d : ℕ} {v : Vector d} :
 @[simp]
 lemma timeComponent_squared_pos_of_timelike {d : ℕ} {v : Vector d}
     (hv : causalCharacter v = .timeLike) :
-    0 < (timeComponent v)^2 := by
-  exact pow_two_pos_of_ne_zero (time_component_ne_zero_of_timelike hv)
+    0 < (timeComponent v)^2 := pow_two_pos_of_ne_zero (time_component_ne_zero_of_timelike hv)
 
 /-- For timelike vectors, the spatial norm squared is strictly less
     than the time component squared -/
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/MinkowskiProduct.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/MinkowskiProduct.lean
index 2dbe8d943..26261b0ce 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/MinkowskiProduct.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/MinkowskiProduct.lean
@@ -76,8 +76,7 @@ lemma minkowskiProductMap_add_fst {d : ℕ} (p q r : Vector d) :
 @[simp]
 lemma minkowskiProductMap_add_snd {d : ℕ} (p q r : Vector d) :
     minkowskiProductMap p (q + r) = minkowskiProductMap p q + minkowskiProductMap p r := by
-  rw [minkowskiProductMap_symm, minkowskiProductMap_add_fst]
-  rw [minkowskiProductMap_symm q p, minkowskiProductMap_symm r p]
+  simpa [minkowskiProductMap_symm] using minkowskiProductMap_add_fst q r p
 
 @[simp]
 lemma minkowskiProductMap_smul_fst {d : ℕ} (c : ℝ) (p q : Vector d) :
@@ -127,9 +126,8 @@ lemma minkowskiProduct_apply {d : ℕ} (p q : Vector d) :
     ⟪p, q⟫ₘ = minkowskiProductMap p q := rfl
 
 lemma minkowskiProduct_symm {d : ℕ} (p q : Vector d) :
-    ⟪p, q⟫ₘ = ⟪q, p⟫ₘ := by
-  rw [minkowskiProduct_apply, minkowskiProductMap_symm]
-  rfl
+    ⟪p, q⟫ₘ = ⟪q, p⟫ₘ :=
+  minkowskiProductMap_symm p q
 
 lemma minkowskiProduct_toCoord {d : ℕ} (p q : Vector d) :
     ⟪p, q⟫ₘ = p (Sum.inl 0) * q (Sum.inl 0) - ∑ i, p (Sum.inr i) * q (Sum.inr i) := by
@@ -255,8 +253,7 @@ def IsLorentz {d : ℕ} (f : Vector d →ₗ[ℝ] Vector d) :
     Prop := ∀ p q : Vector d, ⟪f p, f q⟫ₘ = ⟪p, q⟫ₘ
 
 lemma isLorentz_iff {d : ℕ} (f : Vector d →ₗ[ℝ] Vector d) :
-    IsLorentz f ↔ ∀ p q : Vector d, ⟪f p, f q⟫ₘ = ⟪p, q⟫ₘ := by
-  rfl
+    IsLorentz f ↔ ∀ p q : Vector d, ⟪f p, f q⟫ₘ = ⟪p, q⟫ₘ := Iff.rfl
 
 lemma isLorentz_iff_basis {d : ℕ} (f : Vector d →ₗ[ℝ] Vector d) :
     IsLorentz f ↔ ∀ μ ν : Fin 1 ⊕ Fin d, ⟪f (basis μ), f (basis ν)⟫ₘ = ⟪basis μ, basis ν⟫ₘ := by
@@ -281,8 +278,7 @@ lemma isLorentz_iff_comp_adjoint_eq_id {d : ℕ} (f : Vector d →ₗ[ℝ] Vecto
 
 lemma isLorentz_iff_toMatrix_mem_lorentzGroup {d : ℕ} (f : Vector d →ₗ[ℝ] Vector d) :
     IsLorentz f ↔ LinearMap.toMatrix Vector.basis Vector.basis f ∈ LorentzGroup d := by
-  rw [isLorentz_iff_comp_adjoint_eq_id]
-  rw [LorentzGroup.mem_iff_dual_mul_self]
+  rw [isLorentz_iff_comp_adjoint_eq_id, LorentzGroup.mem_iff_dual_mul_self]
   trans LinearMap.toMatrix Vector.basis Vector.basis (adjoint f ∘ₗ f) =
     LinearMap.toMatrix Vector.basis Vector.basis (LinearMap.id : Vector d →ₗ[ℝ] Vector d)
   · exact Iff.symm (EmbeddingLike.apply_eq_iff_eq (LinearMap.toMatrix basis basis))
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Basic.lean
index f2f6c988c..edd564fa4 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Basic.lean
@@ -39,14 +39,10 @@ lemma contrBasis_ρ_apply {d : ℕ} (M : LorentzGroup d) (i j : Fin 1 ⊕ Fin d)
 
 @[simp]
 lemma contrBasis_toFin1dℝ {d : ℕ} (i : Fin 1 ⊕ Fin d) :
-    (contrBasis d i).toFin1dℝ = Pi.single i 1 := by
-  simp only [ContrMod.toFin1dℝ, contrBasis, Basis.coe_ofEquivFun]
-  rfl
+    (contrBasis d i).toFin1dℝ = Pi.single i 1 := by simp [ContrMod.toFin1dℝ, contrBasis]
 
 lemma contrBasis_repr_apply {d : ℕ} (p : ContrMod d) (i : Fin 1 ⊕ Fin d) :
-    (contrBasis d).repr p i = p.val i := by
-  simp only [contrBasis, Basis.ofEquivFun_repr_apply]
-  rfl
+    (contrBasis d).repr p i = p.val i := rfl
 
 /-- The standard basis of contravariant Lorentz vectors indexed by `Fin (1 + d)`. -/
 def contrBasisFin (d : ℕ := 3) : Basis (Fin (1 + d)) ℝ (ContrMod d) :=
@@ -61,8 +57,7 @@ lemma contrBasisFin_repr_apply {d : ℕ} (p : ContrMod d) (i : Fin (1 + d)) :
     (contrBasisFin d).repr p i = p.val (finSumFinEquiv.symm i) := by rfl
 
 lemma continuous_contr {T : Type} [TopologicalSpace T] (f : T → ContrMod d)
-    (h : Continuous (fun i => (f i).toFin1dℝ)) : Continuous f := by
-  exact continuous_induced_rng.mpr h
+    (h : Continuous (fun i => (f i).toFin1dℝ)) : Continuous f := continuous_induced_rng.mpr h
 
 set_option backward.isDefEq.respectTransparency false in
 lemma contr_continuous {T : Type} [TopologicalSpace T] (f : ContrMod d → T)
@@ -86,15 +81,11 @@ lemma coBasis_ρ_apply {d : ℕ} (M : LorentzGroup d) (i j : Fin 1 ⊕ Fin d) :
   simp [LorentzGroup.transpose, ← LorentzGroup.coe_inv]
 
 lemma coBasis_repr_apply {d : ℕ} (p : CoMod d) (i : Fin 1 ⊕ Fin d) :
-    (coBasis d).repr p i = p.val i := by
-  simp only [coBasis, Basis.ofEquivFun_repr_apply]
-  rfl
+    (coBasis d).repr p i = p.val i := rfl
 
 @[simp]
 lemma coBasis_toFin1dℝ {d : ℕ} (i : Fin 1 ⊕ Fin d) :
-    (coBasis d i).toFin1dℝ = Pi.single i 1 := by
-  simp only [coBasis, Basis.coe_ofEquivFun]
-  rfl
+    (coBasis d i).toFin1dℝ = Pi.single i 1 := by simp [coBasis]
 
 /-- The standard basis of covariant Lorentz vectors indexed by `Fin (1 + d)`. -/
 def coBasisFin (d : ℕ := 3) : Basis (Fin (1 + d)) ℝ (CoMod d) :=
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Contraction.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Contraction.lean
index 1d82b73a5..2fa2efe5c 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Contraction.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Contraction.lean
@@ -38,18 +38,15 @@ def contrModCoModBi (d : ℕ) : ContrMod d →ₗ[ℝ] CoMod d →ₗ[ℝ] ℝ w
       rw [dotProduct_add]
     map_smul' := by
       intro r φ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' ψ ψ':= by
     refine LinearMap.ext (fun φ => ?_)
     simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
     rw [add_dotProduct]
   map_smul' r ψ := by
     refine LinearMap.ext (fun φ => ?_)
-    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
-    rw [smul_dotProduct]
-    rfl
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk, smul_dotProduct,
+      LinearMap.smul_apply, RingHom.id_apply]
 
 /-- The bi-linear map corresponding to contraction of a covariant Lorentz vector with a
   contravariant Lorentz vector. -/
@@ -62,18 +59,15 @@ def coModContrModBi (d : ℕ) : CoMod d →ₗ[ℝ] ContrMod d →ₗ[ℝ] ℝ w
       rw [dotProduct_add]
     map_smul' := by
       intro r ψ
-      simp only [LinearEquiv.map_smul]
-      rw [dotProduct_smul]
-      rfl}
+      simp only [LinearEquiv.map_smul, dotProduct_smul, RingHom.id_apply]}
   map_add' φ φ' := by
     refine LinearMap.ext (fun ψ => ?_)
     simp only [map_add, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply]
     rw [add_dotProduct]
   map_smul' r φ := by
     refine LinearMap.ext (fun ψ => ?_)
-    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk]
-    rw [smul_dotProduct]
-    rfl
+    simp only [LinearEquiv.map_smul, LinearMap.coe_mk, AddHom.coe_mk, smul_dotProduct,
+      LinearMap.smul_apply, RingHom.id_apply]
 
 /-- The linear map from ContrMod d ⊗ CoMod d to ℝ given by
     summing over components of contravariant Lorentz vector and
@@ -91,15 +85,13 @@ def contrCoContract : ((ContrMod.rep).tprod (CoMod.rep)).IntertwiningMap
     change (Λ.1 *ᵥ ψ.toFin1dℝ) ⬝ᵥ ((LorentzGroup.transpose Λ⁻¹).1 *ᵥ φ.toFin1dℝ) = _
     rw [dotProduct_mulVec, LorentzGroup.transpose_val,
       vecMul_transpose, mulVec_mulVec, LorentzGroup.coe_inv, inv_mul_of_invertible Λ.1]
-    simp only [one_mulVec]
-    rfl
+    simp [contrModCoModBi]
 
 /-- Notation for `contrCoContract` acting on a tmul. -/
 local notation "⟪" ψ "," φ "⟫ₘ" => contrCoContract (ψ ⊗ₜ φ)
 
 lemma contrCoContract_hom_tmul (ψ : ContrMod d) (φ : CoMod d) :
-    ⟪ψ, φ⟫ₘ = ψ.toFin1dℝ ⬝ᵥ φ.toFin1dℝ := by
-  rfl
+    ⟪ψ, φ⟫ₘ = ψ.toFin1dℝ ⬝ᵥ φ.toFin1dℝ := rfl
 
 /-- The linear map from CoMod d ⊗ ContrMod d to ℝ given by
     summing over components of contravariant Lorentz vector and
@@ -114,15 +106,13 @@ def coContrContract : ((CoMod.rep (d := d)).tprod (ContrMod.rep (d := d))).Inter
     change ((LorentzGroup.transpose Λ⁻¹).1 *ᵥ ψ.toFin1dℝ) ⬝ᵥ (Λ.1 *ᵥ φ.toFin1dℝ) = _
     rw [dotProduct_mulVec, LorentzGroup.transpose_val, mulVec_transpose, vecMul_vecMul,
       LorentzGroup.coe_inv, inv_mul_of_invertible Λ.1]
-    simp only [vecMul_one]
-    rfl
+    simp [coModContrModBi]
 
 /-- Notation for `coContrContract` acting on a tmul. -/
 local notation "⟪" φ "," ψ "⟫ₘ" => coContrContract (φ ⊗ₜ ψ)
 
 lemma coContrContract_hom_tmul (φ : CoMod d) (ψ : ContrMod d) :
-    ⟪φ, ψ⟫ₘ = φ.toFin1dℝ ⬝ᵥ ψ.toFin1dℝ := by
-  rfl
+    ⟪φ, ψ⟫ₘ = φ.toFin1dℝ ⬝ᵥ ψ.toFin1dℝ := rfl
 
 /-!
 
@@ -159,10 +149,7 @@ def contrContrContractField : ContrMod d ⊗[ℝ] ContrMod d →ₗ[ℝ] ℝ :=
 local notation "⟪" ψ "," φ "⟫ₘ" => contrContrContractField (ψ ⊗ₜ φ)
 
 lemma contrContrContract_hom_tmul (φ : ContrMod d) (ψ : ContrMod d) :
-    ⟪φ, ψ⟫ₘ = φ.toFin1dℝ ⬝ᵥ η *ᵥ ψ.toFin1dℝ:= by
-  simp only [contrContrContractField]
-  erw [contrCoContract_hom_tmul]
-  rfl
+    ⟪φ, ψ⟫ₘ = φ.toFin1dℝ ⬝ᵥ η *ᵥ ψ.toFin1dℝ := rfl
 
 /-- The linear map from CoMod d ⊗ CoMod d to ℝ induced by the homomorphism
   `Co.toContr` and the contraction `coContrContract`. -/
@@ -206,8 +193,7 @@ lemma as_sum_toSpace : ⟪x, y⟫ₘ = x.val (Sum.inl 0) * y.val (Sum.inl 0) -
   rw [as_sum]
   congr
   funext i
-  rw [mul_comm]
-  rfl
+  exact mul_comm _ _
 
 lemma stdBasis_inl {d : ℕ} :
     ⟪@ContrMod.stdBasis d (Sum.inl 0), ContrMod.stdBasis (Sum.inl 0)⟫ₘ = (1 : ℝ) := by
@@ -235,8 +221,7 @@ lemma dual_mulVec_right : ⟪x, dual Λ *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ :=
   simp only [dual, ← mul_assoc, minkowskiMatrix.sq, one_mul]
   rw [← mulVec_mulVec, dotProduct_mulVec, vecMul_transpose]
 
-lemma dual_mulVec_left : ⟪dual Λ *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by
-  rw [symm, dual_mulVec_right, symm]
+lemma dual_mulVec_left : ⟪dual Λ *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by rw [symm, dual_mulVec_right, symm]
 
 lemma right_parity : ⟪x, ContrMod.rep LorentzGroup.parity y⟫ₘ = ∑ i, x.val i * y.val i := by
   rw [as_sum]
@@ -358,9 +343,8 @@ lemma inl_sq_eq (v : ContrMod d) : v.val (Sum.inl 0) ^ 2 =
 
 lemma le_inl_sq (v : ContrMod d) : ⟪v, v⟫ₘ ≤ v.val (Sum.inl 0) ^ 2 := by
   rw [inl_sq_eq]
-  apply (le_add_iff_nonneg_right _).mpr
-  refine Fintype.sum_nonneg ?hf
-  exact fun i => pow_two_nonneg (v.val (Sum.inr i))
+  exact (le_add_iff_nonneg_right _).mpr
+    (Fintype.sum_nonneg fun i => pow_two_nonneg (v.val (Sum.inr i)))
 
 lemma ge_abs_inner_product (v w : ContrMod d) : v.val (Sum.inl 0) * w.val (Sum.inl 0) -
     ‖⟪v.toSpace, w.toSpace⟫_ℝ‖ ≤ ⟪v, w⟫ₘ := by
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Modules.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Modules.lean
index 72d7b7d89..c95ee408e 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Modules.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Modules.lean
@@ -87,8 +87,7 @@ lemma stdBasis_toFin1dℝEquiv_apply_same (μ : Fin 1 ⊕ Fin d) :
     toFin1dℝEquiv (stdBasis μ) μ = 1 := by
   simp only [stdBasis, Basis.ofEquivFun, Basis.coe_ofRepr, LinearEquiv.trans_symm,
     LinearEquiv.symm_symm, LinearEquiv.trans_apply, Finsupp.linearEquivFunOnFinite_single]
-  rw [@LinearEquiv.apply_symm_apply]
-  exact Pi.single_eq_same μ 1
+  simp
 
 @[simp]
 lemma stdBasis_apply_same (μ : Fin 1 ⊕ Fin d) : (stdBasis μ).val μ = 1 :=
@@ -98,8 +97,7 @@ lemma stdBasis_toFin1dℝEquiv_apply_ne {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν
     toFin1dℝEquiv (stdBasis μ) ν = 0 := by
   simp only [stdBasis, Basis.ofEquivFun, Basis.coe_ofRepr, LinearEquiv.trans_symm,
     LinearEquiv.symm_symm, LinearEquiv.trans_apply, Finsupp.linearEquivFunOnFinite_single]
-  rw [@LinearEquiv.apply_symm_apply]
-  exact Pi.single_eq_of_ne' h 1
+  simpa using Pi.single_eq_of_ne' h 1
 
 @[simp]
 lemma stdBasis_inl_apply_inr (i : Fin d) : (stdBasis (Sum.inl 0)).val (Sum.inr i) = 0 := by
@@ -141,33 +139,27 @@ scoped[Lorentz] infixr:73 " *ᵥ " => ContrMod.mulVec
 
 @[simp]
 lemma mulVec_toFin1dℝ (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v : ContrMod d) :
-    (M *ᵥ v).toFin1dℝ = M *ᵥ v.toFin1dℝ := by
-  rfl
+    (M *ᵥ v).toFin1dℝ = M *ᵥ v.toFin1dℝ := rfl
 
 @[simp]
 lemma mulVec_val (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v : ContrMod d) :
-    (M *ᵥ v).val = M *ᵥ v.val := by
-  rfl
+    (M *ᵥ v).val = M *ᵥ v.val := rfl
 
 lemma mulVec_sub (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v w : ContrMod d) :
-    M *ᵥ (v - w) = M *ᵥ v - M *ᵥ w := by
-  simp only [mulVec, LinearMap.map_sub]
+    M *ᵥ (v - w) = M *ᵥ v - M *ᵥ w := by simp only [mulVec, LinearMap.map_sub]
 
 lemma sub_mulVec (M N : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v : ContrMod d) :
-    (M - N) *ᵥ v = M *ᵥ v - N *ᵥ v := by
-  simp only [mulVec, map_sub, LinearMap.sub_apply]
+    (M - N) *ᵥ v = M *ᵥ v - N *ᵥ v := by simp only [mulVec, map_sub, LinearMap.sub_apply]
 
 lemma mulVec_add (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v w : ContrMod d) :
-    M *ᵥ (v + w) = M *ᵥ v + M *ᵥ w := by
-  simp only [mulVec, LinearMap.map_add]
+    M *ᵥ (v + w) = M *ᵥ v + M *ᵥ w := by simp only [mulVec, LinearMap.map_add]
 
 @[simp]
 lemma one_mulVec (v : ContrMod d) : (1 : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) *ᵥ v = v := by
   simp [mulVec, _root_.map_one]
 
 lemma mulVec_mulVec (M N : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v : ContrMod d) :
-    M *ᵥ (N *ᵥ v) = (M * N) *ᵥ v := by
-  simp [mulVec, _root_.map_mul]
+    M *ᵥ (N *ᵥ v) = (M * N) *ᵥ v := by simp [mulVec, _root_.map_mul]
 
 /-!
 
@@ -201,12 +193,10 @@ def toSpace (v : ContrMod d) : EuclideanSpace ℝ (Fin d) := WithLp.toLp 2 (v.va
 def rep : Representation ℝ (LorentzGroup d) (ContrMod d) where
   toFun g := Matrix.toLinAlgEquiv stdBasis g
   map_one' := EmbeddingLike.map_eq_one_iff.mpr rfl
-  map_mul' x y := by
-    simp only [lorentzGroupIsGroup_mul_coe, _root_.map_mul]
+  map_mul' x y := by simp only [lorentzGroupIsGroup_mul_coe, _root_.map_mul]
 
 lemma rep_apply_toFin1dℝ (g : LorentzGroup d) (ψ : ContrMod d) :
-    (rep g ψ).toFin1dℝ = g.1 *ᵥ ψ.toFin1dℝ := by
-  rfl
+    (rep g ψ).toFin1dℝ = g.1 *ᵥ ψ.toFin1dℝ := rfl
 
 /-!
 
@@ -241,9 +231,8 @@ lemma toSelfAdjoint_apply_coe (x : ContrMod 3) : (toSelfAdjoint x).1 =
     x.toFin1dℝ (Sum.inl 0) • PauliMatrix.pauliMatrix (Sum.inl 0)
     - x.toFin1dℝ (Sum.inr 0) • PauliMatrix.pauliMatrix (Sum.inr 0)
     - x.toFin1dℝ (Sum.inr 1) • PauliMatrix.pauliMatrix (Sum.inr 1)
-    - x.toFin1dℝ (Sum.inr 2) • PauliMatrix.pauliMatrix (Sum.inr 2) := by
-  rw [toSelfAdjoint_apply]
-  rfl
+    - x.toFin1dℝ (Sum.inr 2) • PauliMatrix.pauliMatrix (Sum.inr 2) :=
+  congrArg Subtype.val (toSelfAdjoint_apply x)
 
 lemma toSelfAdjoint_stdBasis (i : Fin 1 ⊕ Fin 3) :
     toSelfAdjoint (stdBasis i) = PauliMatrix.pauliBasis' i := by
@@ -290,8 +279,7 @@ instance : TopologicalSpace (ContrMod d) := TopologicalSpace.induced
 
 open Topology
 
-lemma toFin1dℝEquiv_isInducing : IsInducing (@ContrMod.toFin1dℝEquiv d) := by
-  exact { eq_induced := rfl }
+lemma toFin1dℝEquiv_isInducing : IsInducing (@ContrMod.toFin1dℝEquiv d) := { eq_induced := rfl }
 
 lemma toFin1dℝEquiv_symm_isInducing : IsInducing ((@ContrMod.toFin1dℝEquiv d).symm) := by
   let x := Equiv.toHomeomorphOfIsInducing (@ContrMod.toFin1dℝEquiv d).toEquiv
@@ -353,8 +341,7 @@ lemma stdBasis_toFin1dℝEquiv_apply_same (μ : Fin 1 ⊕ Fin d) :
     toFin1dℝEquiv (stdBasis μ) μ = 1 := by
   simp only [stdBasis, Basis.ofEquivFun, Basis.coe_ofRepr, LinearEquiv.trans_symm,
     LinearEquiv.symm_symm, LinearEquiv.trans_apply, Finsupp.linearEquivFunOnFinite_single]
-  rw [@LinearEquiv.apply_symm_apply]
-  exact Pi.single_eq_same μ 1
+  simp
 
 @[simp]
 lemma stdBasis_apply_same (μ : Fin 1 ⊕ Fin d) : (stdBasis μ).val μ = 1 :=
@@ -364,8 +351,7 @@ lemma stdBasis_toFin1dℝEquiv_apply_ne {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν
     toFin1dℝEquiv (stdBasis μ) ν = 0 := by
   simp only [stdBasis, Basis.ofEquivFun, Basis.coe_ofRepr, LinearEquiv.trans_symm,
     LinearEquiv.symm_symm, LinearEquiv.trans_apply, Finsupp.linearEquivFunOnFinite_single]
-  rw [@LinearEquiv.apply_symm_apply]
-  exact Pi.single_eq_of_ne' h 1
+  simpa using Pi.single_eq_of_ne' h 1
 
 lemma stdBasis_apply (μ ν : Fin 1 ⊕ Fin d) : (stdBasis μ).val ν = if μ = ν then 1 else 0 := by
   simp only [stdBasis, Basis.coe_ofEquivFun]
@@ -402,13 +388,11 @@ scoped[Lorentz] infixr:73 " *ᵥ " => CoMod.mulVec
 
 @[simp]
 lemma mulVec_toFin1dℝ (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v : CoMod d) :
-    (M *ᵥ v).toFin1dℝ = M *ᵥ v.toFin1dℝ := by
-  rfl
+    (M *ᵥ v).toFin1dℝ = M *ᵥ v.toFin1dℝ := rfl
 
 @[simp]
 lemma mulVec_val (M : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ) (v : CoMod d) :
-    (M *ᵥ v).val = M *ᵥ v.val := by
-  rfl
+    (M *ᵥ v).val = M *ᵥ v.val := rfl
 /-!
 
 ## The representation
diff --git a/Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean b/Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean
index 245f32c5f..bc57d3676 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean
@@ -138,8 +138,7 @@ lemma smul_eq_sum {d : ℕ} (i : Fin 1 ⊕ Fin d) (Λ : LorentzGroup d) (p : Vec
   · intro p
     have toTensor_symm_pure_val : ∀ (q : Pure (realLorentzTensor d) ![.up]) (j : Fin 1 ⊕ Fin d),
         (toTensor (self := tensorial)).symm q.toTensor j = (q 0).val j := fun q j => by
-      rw [toTensor_symm_pure, contrBasis_repr_apply]
-      rfl
+      exact toTensor_symm_pure q j
     rw [actionT_pure]
     simp only [toTensor_symm_pure_val]
     show (Λ.1 *ᵥ (p 0)).val i = ∑ j, Λ.1 i j * (p 0).val j
@@ -169,13 +168,11 @@ lemma smul_sub {d : ℕ} (Λ : LorentzGroup d) (p q : Vector d) :
 
 set_option backward.isDefEq.respectTransparency false in
 lemma smul_zero {d : ℕ} (Λ : LorentzGroup d) :
-    Λ • (0 : Vector d) = 0 := by
-  rw [smul_eq_mulVec, Matrix.mulVec_zero]
+    Λ • (0 : Vector d) = 0 := by rw [smul_eq_mulVec, Matrix.mulVec_zero]
 
 set_option backward.isDefEq.respectTransparency false in
 lemma smul_neg {d : ℕ} (Λ : LorentzGroup d) (p : Vector d) :
-    Λ • (-p) = - (Λ • p) := by
-  rw [smul_eq_mulVec, smul_eq_mulVec, Matrix.mulVec_neg]
+    Λ • (-p) = - (Λ • p) := by rw [smul_eq_mulVec, smul_eq_mulVec, Matrix.mulVec_neg]
 
 lemma neg_smul {d} (Λ : LorentzGroup d) (p : Vector d) :
     (-Λ) • p = - (Λ • p) := by
diff --git a/Physlib/Relativity/Tensors/RealTensor/Velocity/Basic.lean b/Physlib/Relativity/Tensors/RealTensor/Velocity/Basic.lean
index 0720cd529..661ca703b 100644
--- a/Physlib/Relativity/Tensors/RealTensor/Velocity/Basic.lean
+++ b/Physlib/Relativity/Tensors/RealTensor/Velocity/Basic.lean
@@ -35,11 +35,9 @@ variable {d : ℕ}
 noncomputable instance : TopologicalSpace (Velocity d) := instTopologicalSpaceSubtype
 
 @[ext]
-lemma ext {v w : Velocity d} (h : v.1 = w.1) : v = w := by
-  exact SetCoe.ext h
+lemma ext {v w : Velocity d} (h : v.1 = w.1) : v = w := SetCoe.ext h
 
-lemma mem_iff {v : Vector d} : v ∈ Velocity d ↔ ⟪v, v⟫ₘ = (1 : ℝ) ∧ 0 < v.timeComponent := by
-  rfl
+lemma mem_iff {v : Vector d} : v ∈ Velocity d ↔ ⟪v, v⟫ₘ = (1 : ℝ) ∧ 0 < v.timeComponent := Iff.rfl
 
 @[simp]
 lemma minkowskiProduct_self_eq_one (v : Velocity d) : ⟪v.1, v.1⟫ₘ = (1 : ℝ) := v.2.1
@@ -53,8 +51,7 @@ lemma timeComponent_nonneg (v : Velocity d) : 0 ≤ v.1.timeComponent := by
 
 @[simp]
 lemma timeComponent_abs (v : Velocity d) :
-    |v.1.timeComponent| = v.1.timeComponent := by
-  simp
+    |v.1.timeComponent| = v.1.timeComponent := by simp
 
 lemma norm_spatialPart_le_timeComponent (v : Velocity d) :
     ‖v.1.spatialPart‖ ≤ ‖v.1.timeComponent‖ := by
@@ -68,8 +65,7 @@ lemma norm_spatialPart_le_timeComponent (v : Velocity d) :
 
 lemma norm_spatialPart_sq_eq (v : Velocity d) :
     ‖v.1.spatialPart‖ ^ 2 = (v.1 (Sum.inl 0))^2 - 1 := by
-  rw [← minkowskiProduct_self_eq_one v]
-  rw [minkowskiProduct_self_eq_timeComponent_spatialPart]
+  rw [← minkowskiProduct_self_eq_one v, minkowskiProduct_self_eq_timeComponent_spatialPart]
   simp [timeComponent]
 
 lemma zero_le_minkowskiProduct (u v : Velocity d) :
@@ -87,12 +83,10 @@ lemma zero_le_minkowskiProduct (u v : Velocity d) :
   · exact real_inner_le_norm u.1.spatialPart v.1.spatialPart
 
 lemma one_add_minkowskiProduct_ne_zero (u v : Velocity d) :
-    1 + ⟪u.1, v.1⟫ₘ ≠ 0 := by
-  linarith [zero_le_minkowskiProduct u v]
+    1 + ⟪u.1, v.1⟫ₘ ≠ 0 := by linarith [zero_le_minkowskiProduct u v]
 
 lemma minkowskiProduct_continuous_snd (u : Vector d) :
-    Continuous fun (x : Velocity d) => ⟪u, x.1⟫ₘ := by
-  fun_prop
+    Continuous fun (x : Velocity d) => ⟪u, x.1⟫ₘ := by fun_prop
 
 @[fun_prop]
 lemma minkowskiProduct_continuous_fst (u : Vector d) :
@@ -101,8 +95,7 @@ lemma minkowskiProduct_continuous_fst (u : Vector d) :
     (fun (x : Velocity d) => ⟪u, x.1⟫ₘ) := by
     ext x
     rw [minkowskiProduct_symm]
-  rw [h1]
-  exact minkowskiProduct_continuous_snd u
+  simpa [h1] using minkowskiProduct_continuous_snd u
 
 /-!
 
@@ -155,10 +148,8 @@ noncomputable def pathFromZero (u : Velocity d) : Path zero u where
         · apply mul_nonneg
           · exact sq_nonneg _
           · exact sq_nonneg _⟩
-  continuous_toFun := by
-    fun_prop
-  source' := by
-    simp
+  continuous_toFun := by fun_prop
+  source' := by simp
   target' := by
     ext1
     simp only [Set.Icc.coe_one,
diff --git a/Physlib/Relativity/Tensors/Reindexing.lean b/Physlib/Relativity/Tensors/Reindexing.lean
index 0997b6ca5..913bef1c8 100644
--- a/Physlib/Relativity/Tensors/Reindexing.lean
+++ b/Physlib/Relativity/Tensors/Reindexing.lean
@@ -72,8 +72,7 @@ lemma auto {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}
 
 @[simp]
 lemma on_id {n : ℕ} {c c1 : Fin n → C} :
-    IsReindexing c c1 (id : Fin n → Fin n) ↔ ∀ i, c i = c1 i := by
-  simp [IsReindexing]
+    IsReindexing c c1 (id : Fin n → Fin n) ↔ ∀ i, c i = c1 i := by simp [IsReindexing]
 
 lemma on_id_symm {n : ℕ} {c c1 : Fin n → C} (h : IsReindexing c1 c id) :
     IsReindexing c c1 (id : Fin n → Fin n) := by
@@ -98,15 +97,11 @@ def toEquiv {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}
 
 lemma apply_inv_apply {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}
     (σ : Fin m → Fin n) (h : IsReindexing c c1 σ) (x : Fin m) :
-    h.inv σ (σ x) = x := by
-  change h.toEquiv (h.toEquiv.symm x) = x
-  simp
+    h.inv σ (σ x) = x := by simpa [inv, toEquiv] using h.toEquiv.apply_symm_apply x
 
 lemma inv_apply_apply {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}
     (σ : Fin m → Fin n) (h : IsReindexing c c1 σ) (x : Fin n) :
-    σ (h.inv σ x) = x := by
-  change h.toEquiv.symm (h.toEquiv x) = x
-  simp
+    σ (h.inv σ x) = x := by simpa [inv, toEquiv] using h.toEquiv.symm_apply_apply x
 
 lemma preserve_color {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}
     {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) :
@@ -124,8 +119,7 @@ lemma inv_perserve_color {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}
   obtain ⟨x, rfl⟩ := h.toEquiv.symm.surjective x
   change c1 (h.toEquiv _) = _
   simp only [Equiv.apply_symm_apply]
-  rw [h.preserve_color]
-  rfl
+  exact h.preserve_color _
 
 set_option warning.simp.varHead false in
 @[simp, nolint simpVarHead]
@@ -133,8 +127,7 @@ lemma toEquiv_symm_perserve_color {n m : ℕ} {c : Fin n → C} {c1 : Fin m →
     {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) (x : Fin m) :
     c (h.toEquiv.symm x) = c1 x := by
   obtain ⟨x, rfl⟩ := h.toEquiv.surjective x
-  rw [h.preserve_color]
-  rfl
+  exact (h.preserve_color (h.toEquiv x)).symm
 
 /-!
 
diff --git a/Physlib/Relativity/Tensors/Tensorial.lean b/Physlib/Relativity/Tensors/Tensorial.lean
index c301054f1..0ea6dc78a 100644
--- a/Physlib/Relativity/Tensors/Tensorial.lean
+++ b/Physlib/Relativity/Tensors/Tensorial.lean
@@ -108,9 +108,7 @@ noncomputable instance self {n : ℕ} (S : TensorSpecies k C G V basisIdx rep b)
 @[simp]
 lemma self_toTensor_apply {n : ℕ} (S : TensorSpecies k C G V basisIdx rep b)
     (c : Fin n → C) (t : S.Tensor c) :
-    Tensorial.toTensor t = t := by
-  rw [Tensorial.toTensor]
-  rfl
+    Tensorial.toTensor t = t := rfl
 
 /-!
 
@@ -141,8 +139,7 @@ noncomputable instance mulAction [Tensorial S c M] : MulAction G M where
   mul_smul g h m := by
     change _ = toTensor.symm (g • toTensor (toTensor.symm (h • toTensor m)))
     simp only [LinearEquiv.apply_symm_apply]
-    rw [← mul_smul]
-    rfl
+    exact congrArg toTensor.symm (mul_smul g h (toTensor m))
 
 /-!
 
@@ -151,9 +148,7 @@ noncomputable instance mulAction [Tensorial S c M] : MulAction G M where
 -/
 
 lemma smul_eq {g : G} {t : M} [Tensorial S c M] :
-    g • t = toTensor.symm (g • toTensor t) := by
-  rw [Tensorial.toTensor]
-  rfl
+    g • t = toTensor.symm (g • toTensor t) := rfl
 
 lemma toTensor_smul {g : G} {t : M} [Tensorial S c M] :
     toTensor (g • t) = g • toTensor t := by
@@ -393,7 +388,6 @@ def toTensorCLM [IsTopologicalAddGroup M]
   continuous_toFun := by
     let e : M →L[k] (S.Tensor c) := LinearMap.toContinuousLinearMap
       (Tensorial.toTensor (M := M))
-    change Continuous e
     exact ContinuousLinearMap.continuous e
   continuous_invFun := by apply IsModuleTopology.continuous_of_linearMap
 
diff --git a/Physlib/Relativity/Tensors/UnitTensor.lean b/Physlib/Relativity/Tensors/UnitTensor.lean
index 106246b89..7d7e10233 100644
--- a/Physlib/Relativity/Tensors/UnitTensor.lean
+++ b/Physlib/Relativity/Tensors/UnitTensor.lean
@@ -38,8 +38,7 @@ lemma unitTensor_congr {c c1 : C} (h : c = c1) :
 lemma unitTensor_eq_permT_dual (c : C) :
     S.unitTensor c = permT ![1, 0] (And.intro (by decide) (fun i => by fin_cases i <;> simp))
     (unitTensor (S.τ c)) := by
-  rw [unitTensor, fromConstPair, S.unit_symm]
-  rw [unitTensor, fromConstPair]
+  rw [unitTensor, fromConstPair, S.unit_symm, unitTensor, fromConstPair]
   simp [fromPairT]
   generalize (S.unit (S.τ c)) 1 = u at *
   induction' u using TensorProduct.induction_on with x y
@@ -65,15 +64,12 @@ lemma unitTensor_eq_permT_dual (c : C) :
 lemma dual_unitTensor_eq_permT_unitTensor (c : C) :
     S.unitTensor (S.τ c) = permT ![1, 0] (And.intro (by decide) (fun i => by fin_cases i <;> simp))
       (unitTensor c) := by
-  rw [unitTensor_eq_permT_dual]
-  rw [unitTensor_congr (by simp : c = S.τ (S.τ c))]
+  rw [unitTensor_eq_permT_dual, unitTensor_congr (by simp : c = S.τ (S.τ c))]
   simp
 
 lemma unit_fromSingleTContrFromPairT_eq_fromSingleT {c : C} (x : V c) :
     fromSingleTContrFromPairT x ((S.unit c) (1 : k)) =
-    fromSingleT x := by
-  conv_rhs => rw [← S.contr_unit c x]
-  rfl
+    fromSingleT x := congrArg fromSingleT (S.contr_unit c x)
 
 /-- This lemma represents the de-categorification of `S.contr_unit`. -/
 @[simp]
@@ -89,12 +85,8 @@ set_option backward.isDefEq.respectTransparency false in
 lemma contrT_unitTensor_dual_single {c : C} (x : Tensor S ![S.τ c]) :
     contrT 1 1 2 (by simp; rfl) (prodT (unitTensor c) x) =
     permT id (by simp; rfl) x := by
-  rw [unitTensor_eq_permT_dual]
-  rw [prodT_permT_left]
-  rw [contrT_permT]
-  rw [prodT_swap]
-  rw [contrT_permT]
-  rw [permT_permT]
+  rw [unitTensor_eq_permT_dual, prodT_permT_left, contrT_permT, prodT_swap, contrT_permT,
+    permT_permT]
   conv_lhs =>
     enter [2]
     change contrT 1 1 0 _ _
@@ -111,7 +103,6 @@ lemma contrT_unitTensor_dual_single {c : C} (x : Tensor S ![S.τ c]) :
 
 @[simp]
 lemma unitTensor_invariant {c : C} (g : G) :
-    g • S.unitTensor c = S.unitTensor c := by
-  rw [unitTensor, actionT_fromConstPair]
+    g • S.unitTensor c = S.unitTensor c := by rw [unitTensor, actionT_fromConstPair]
 
 end TensorSpecies
diff --git a/Physlib/SpaceAndTime/Space/Basic.lean b/Physlib/SpaceAndTime/Space/Basic.lean
index 794987c57..aad41c358 100644
--- a/Physlib/SpaceAndTime/Space/Basic.lean
+++ b/Physlib/SpaceAndTime/Space/Basic.lean
@@ -160,13 +160,8 @@ lemma vadd_transitive {d} (s1 s2 : Space d) :
   simp
 
 noncomputable instance : AddAction (EuclideanSpace ℝ (Fin d)) (Space d) where
-  zero_vadd s := by
-    ext i
-    simp
-  add_vadd v1 v2 s := by
-    ext i
-    simp only [vadd_apply, PiLp.add_apply]
-    ring
+  zero_vadd s := eq_of_apply fun i => by simp
+  add_vadd v1 v2 s := eq_of_apply fun i => by simp [add_assoc]
 
 /-!
 
@@ -281,7 +276,6 @@ instance (d : ℕ) : IsManifold (𝓡 d) ⊤ (Space d) :=
 
 lemma chartAt_eq (d : ℕ) (p : Space d) :
     chartAt (EuclideanSpace ℝ (Fin d)) p =
-    (homEuclideanSpaceSpace d).symm.toOpenPartialHomeomorph :=
-  rfl
+    (homEuclideanSpaceSpace d).symm.toOpenPartialHomeomorph := rfl
 
 end Space
diff --git a/Physlib/SpaceAndTime/Space/ConstantSliceDist.lean b/Physlib/SpaceAndTime/Space/ConstantSliceDist.lean
index e1f7f1918..b8ea3bd4e 100644
--- a/Physlib/SpaceAndTime/Space/ConstantSliceDist.lean
+++ b/Physlib/SpaceAndTime/Space/ConstantSliceDist.lean
@@ -157,10 +157,8 @@ lemma schwartzMap_mul_iteratedFDeriv_integrable_slice_symm {d : ℕ} (n m : ℕ)
     · fun_prop
     apply Continuous.norm
     apply Continuous.comp'
-    apply ContDiff.continuous_iteratedFDeriv (n := (n + 1 : ℕ))
-    exact Nat.cast_le.mpr (by omega)
-    have hη := η.smooth'
-    apply hη.of_le (ENat.LEInfty.out)
+    exact (η.smooth'.of_le (ENat.LEInfty.out)).continuous_iteratedFDeriv
+      (n := (n + 1 : ℕ)) (Nat.cast_le.mpr (by omega))
     fun_prop
   · filter_upwards with t
     apply le_trans _ (hbound x t)
@@ -210,8 +208,7 @@ lemma schwartzMap_fderiv_left_integrable_slice_symm {d : ℕ} (η : 𝓢(Space d
     simp only [Nat.succ_eq_add_one, one_mul]
     change fderiv ℝ (η ∘ fun r => ((slice i).symm (r, x))) r 1
     rw [fderiv_comp _ (by
-      apply Differentiable.differentiableAt
-      exact η.smooth'.differentiable (by simp))
+      exact (η.smooth'.differentiable (by simp)).differentiableAt)
       (by fun_prop)]
     simp only [Nat.succ_eq_add_one, ContinuousLinearMap.coe_comp, Function.comp_apply,
       fderiv_slice_symm_left_apply]
@@ -236,10 +233,8 @@ lemma schwartzMap_iteratedFDeriv_slice_symm_integrable {n} {d : ℕ} (η : 𝓢(
   · fun_prop
   · apply Continuous.aestronglyMeasurable
     apply Continuous.comp'
-    apply ContDiff.continuous_iteratedFDeriv (n := (n + 1 : ℕ))
-    exact Nat.cast_le.mpr (by omega)
-    have hη := η.smooth'
-    apply hη.of_le (ENat.LEInfty.out)
+    exact (η.smooth'.of_le (ENat.LEInfty.out)).continuous_iteratedFDeriv
+      (n := (n + 1 : ℕ)) (Nat.cast_le.mpr (by omega))
     fun_prop
 
 /-!
@@ -523,8 +518,7 @@ lemma schwartzMap_mul_pow_slice_integral_iteratedFDeriv_norm_le {d : ℕ} (n m :
         refine mul_le_mul_of_nonneg ?_ (by rfl) (by positivity) (by positivity)
         refine integral_mono ?_ ?_ ?_
         · fun_prop
-        · apply Integrable.const_mul
-          exact hrt
+        · exact hrt.const_mul k
         · refine Pi.le_def.mpr ?_
           intro t
           convert hbound x t using 1
@@ -576,8 +570,7 @@ def sliceSchwartz {d : ℕ} (i : Fin d.succ) :
     rfl
 
 lemma sliceSchwartz_apply {d : ℕ} (i : Fin d.succ) (η : 𝓢(Space d.succ, ℝ)) (x : Space d) :
-    sliceSchwartz i η x = ∫ (r : ℝ), η ((slice i).symm (r, x)) := by
-  rfl
+    sliceSchwartz i η x = ∫ (r : ℝ), η ((slice i).symm (r, x)) := rfl
 /-!
 
 ## B. Constant slice distribution
@@ -598,8 +591,7 @@ def constantSliceDist {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d :
 
 lemma constantSliceDist_apply {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M]
     {d : ℕ} (i : Fin d.succ) (f : (Space d) →d[ℝ] M) (η : 𝓢(Space d.succ, ℝ)) :
-    constantSliceDist i f η = f (sliceSchwartz i η) := by
-  rfl
+    constantSliceDist i f η = f (sliceSchwartz i η) := rfl
 
 /-!
 
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/Basic.lean b/Physlib/SpaceAndTime/Space/Derivatives/Basic.lean
index b0b16689f..f8d3087a5 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/Basic.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/Basic.lean
@@ -93,8 +93,7 @@ lemma deriv_eq_fderiv_basis [AddCommGroup M] [Module ℝ M] [TopologicalSpace M]
 lemma fderiv_eq_sum_deriv {M d} [AddCommGroup M] [Module ℝ M] [TopologicalSpace M]
     (f : Space d → M) (x y : Space d) :
     fderiv ℝ f x y = ∑ i : Fin d, y i • ∂[i] f x := by
-  have h1 : y = ∑ i, y i • basis i := by
-    exact Eq.symm (OrthonormalBasis.sum_repr basis y)
+  have h1 : y = ∑ i, y i • basis i := Eq.symm (OrthonormalBasis.sum_repr basis y)
   conv_lhs => rw [h1]
   simp [deriv_eq_fderiv_basis]
 
@@ -139,8 +138,7 @@ lemma deriv_eq_mfderiv_manifoldStructure {M d} [NormedAddCommGroup M] [NormedSpa
       (modelDiffeo.mdifferentiable WithTop.top_ne_zero).mdifferentiableAt]
     simp only [Function.comp_apply, modelDiffeo_apply, mfderiv_eq_fderiv,
       ContinuousLinearMap.coe_comp]
-    rw [basis_eq_mfderiv_modelDiffeo_single]
-    rfl
+    exact congrArg (fun v => (fderiv ℝ f x) v) (basis_eq_mfderiv_modelDiffeo_single (d := d) μ x)
   · rw [deriv_eq, fderiv_zero_of_not_differentiableAt hf,
       mfderiv_zero_of_not_mdifferentiableAt <|
       mdifferentiable_manifoldStructure_iff_differentiable.mp.mt hf]
@@ -171,9 +169,7 @@ lemma deriv_add [NormedAddCommGroup M] [NormedSpace ℝ M]
     ∂[u] (f1 + f2) = ∂[u] f1 + ∂[u] f2 := by
   rw [deriv_eq_fderiv_fun]
   ext x
-  rw [fderiv_add]
-  rfl
-  repeat fun_prop
+  exact congrArg (fun L => L (basis u)) (fderiv_add (hf1 x) (hf2 x))
 
 /-- Derivatives on space distribute coordinate-wise over addition. -/
 lemma deriv_coord_add (f1 f2 : Space d → EuclideanSpace ℝ (Fin d))
@@ -194,9 +190,7 @@ lemma deriv_sub [NormedAddCommGroup M] [NormedSpace ℝ M]
     ∂[u] (f1 - f2) = ∂[u] f1 - ∂[u] f2 := by
   rw [deriv_eq_fderiv_fun]
   ext x
-  rw [fderiv_sub]
-  rfl
-  repeat fun_prop
+  exact congrArg (fun L => L (basis u)) (fderiv_sub (hf1 x) (hf2 x))
 
 /-!
 
@@ -209,8 +203,7 @@ lemma deriv_smul [NormedAddCommGroup M] [NormedSpace ℝ M] [NontriviallyNormedF
     [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 M] {c : Space d → 𝕜} {f : Space d → M}
     (hc : DifferentiableAt ℝ c x) (hf : DifferentiableAt ℝ f x) :
     ∂[u] (c • f) x = c x • ∂[u] f x + ∂[u] c x • f x := by
-  rw [deriv_eq_fderiv_basis, deriv_eq_fderiv_basis, deriv_eq_fderiv_basis, fderiv_smul hc hf]
-  rfl
+  simp [deriv_eq_fderiv_basis, fderiv_smul hc hf]
 
 /-- Space derivatives on scalar times function. -/
 lemma deriv_const_smul [NormedAddCommGroup M] [NormedSpace ℝ M] [Semiring R]
@@ -244,9 +237,8 @@ lemma deriv_commute [NormedAddCommGroup M] [NormedSpace ℝ M]
   simp only [fderiv_fun_const, Pi.ofNat_apply, ContinuousLinearMap.comp_zero, zero_add,
     ContinuousLinearMap.flip_apply]
   rw [IsSymmSndFDerivAt.eq]
-  apply ContDiffAt.isSymmSndFDerivAt
-  exact ContDiff.contDiffAt hf
-  simp only [minSmoothness_of_isRCLikeNormedField, le_refl]
+  exact (ContDiff.contDiffAt hf).isSymmSndFDerivAt
+    (by simp only [minSmoothness_of_isRCLikeNormedField, le_refl])
   repeat fun_prop
 
 /-!
@@ -292,8 +284,7 @@ lemma deriv_component (μ ν : Fin d) (x : Space d) :
 
 lemma deriv_component_sq {d : ℕ} {ν μ : Fin d} (x : Space d) :
     (deriv ν (fun x => (x μ) ^ 2) x) = if ν = μ then 2 * x μ else 0:= by
-  rw [deriv_eq_fderiv_basis]
-  rw [fderiv_fun_pow]
+  rw [deriv_eq_fderiv_basis, fderiv_fun_pow]
   simp only [Nat.add_one_sub_one, pow_one, nsmul_eq_mul, Nat.cast_ofNat,
     FunLike.coe_smul, Pi.smul_apply, smul_eq_mul]
   rw [← deriv_eq_fderiv_basis, deriv_component]
@@ -324,7 +315,6 @@ lemma deriv_lorentz_vector {d ν μ} {f : Space d → Lorentz.Vector d}
   change fderiv ℝ (Lorentz.Vector.coordCLM μ ∘ fun x => f x) x (basis ν) = _
   rw [fderiv_comp]
   · simp
-    rw [← deriv_eq_fderiv_basis]
     rfl
   · fun_prop
   · fun_prop
@@ -438,8 +428,7 @@ lemma deriv_eq_inner_self (x : Space d) (i : Fin d) :
 @[simp]
 lemma deriv_inner_left {d} (x1 x2 : Space d) (i : Fin d) :
     deriv i (fun x => ⟪x, x2⟫_ℝ) x1 = x2 i := by
-  rw [deriv_eq_fderiv_basis]
-  rw [fderiv_inner_apply]
+  rw [deriv_eq_fderiv_basis, fderiv_inner_apply]
   simp only [fderiv_fun_const, Pi.zero_apply, _root_.zero_apply, inner_zero_right,
     fderiv_fun_id, ContinuousLinearMap.coe_id', id_eq, basis_inner, zero_add]
   · fun_prop
@@ -448,8 +437,7 @@ lemma deriv_inner_left {d} (x1 x2 : Space d) (i : Fin d) :
 @[simp]
 lemma deriv_inner_right {d} (x1 x2 : Space d) (i : Fin d) :
     deriv i (fun x => ⟪x1, x⟫_ℝ) x2 = x1 i := by
-  rw [deriv_eq_fderiv_basis]
-  rw [fderiv_inner_apply]
+  rw [deriv_eq_fderiv_basis, fderiv_inner_apply]
   simp only [fderiv_fun_id, ContinuousLinearMap.coe_id', id_eq, inner_basis, fderiv_fun_const,
     Pi.ofNat_apply, _root_.zero_apply, inner_zero_left, add_zero]
   · fun_prop
@@ -494,14 +482,11 @@ noncomputable def distDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
   toFun f :=
     let ev : (Space d →L[ℝ] M) →L[ℝ] M := {
       toFun v := v (basis μ)
-      map_add' v1 v2 := by
-        simp only [_root_.add_apply]
-      map_smul' a v := by
-        simp
+      map_add' v1 v2 := by simp only [_root_.add_apply]
+      map_smul' a v := by simp
     }
     ev.comp (Distribution.fderivD ℝ f)
-  map_add' f1 f2 := by
-    simp
+  map_add' f1 f2 := by simp
   map_smul' a f := by simp
 
 @[inherit_doc distDeriv]
@@ -515,8 +500,7 @@ macro "∂ᵈ[" i:term "]" : term => `(distDeriv $i)
 
 lemma distDeriv_apply {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (μ : Fin d) (f : (Space d) →d[ℝ] M) (ε : 𝓢(Space d, ℝ)) :
-    (∂ᵈ[μ] f) ε = fderivD ℝ f ε (basis μ) := by
-  simp [distDeriv, Distribution.fderivD]
+    (∂ᵈ[μ] f) ε = fderivD ℝ f ε (basis μ) := by simp [distDeriv, Distribution.fderivD]
 
 /-!
 
@@ -541,8 +525,7 @@ lemma schwartMap_fderiv_comm {d}
     ContinuousLinearMap.flip_apply]
   rw [IsSymmSndFDerivAt.eq]
   apply ContDiffAt.isSymmSndFDerivAt (n := 2)
-  · refine ContDiff.contDiffAt ?_
-    exact h2
+  · exact h2.contDiffAt
   · simp
   · fun_prop
   · exact differentiableAt_const (basis μ)
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/Div.lean b/Physlib/SpaceAndTime/Space/Derivatives/Div.lean
index 92f8daac2..765495193 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/Div.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/Div.lean
@@ -71,9 +71,7 @@ lemma div_eq_sum_fderiv {d} (f : Space d → EuclideanSpace ℝ (Fin d))
   congr
   funext i
   rw [← Space.deriv_eq]
-  rw [deriv_euclid]
-  rfl
-  fun_prop
+  exact deriv_euclid (ν := i) hf x
 
 /-!
 
@@ -83,10 +81,8 @@ lemma div_eq_sum_fderiv {d} (f : Space d → EuclideanSpace ℝ (Fin d))
 
 @[simp]
 lemma div_zero : ∇ ⬝ (0 : Space d → EuclideanSpace ℝ (Fin d)) = 0 := by
-  unfold div Space.deriv Finset.sum
-  simp only [Pi.ofNat_apply, fderiv_fun_const, _root_.zero_apply, Multiset.map_const',
-    Finset.card_val, Finset.card_univ, Fintype.card_fin, Multiset.sum_replicate, smul_zero]
-  rfl
+  ext x
+  simp [div, Space.deriv]
 
 /-!
 
@@ -96,10 +92,8 @@ lemma div_zero : ∇ ⬝ (0 : Space d → EuclideanSpace ℝ (Fin d)) = 0 := by
 
 @[simp]
 lemma div_const : ∇ ⬝ (fun _ : Space d => v) = 0 := by
-  unfold div Space.deriv Finset.sum
-  simp only [fderiv_fun_const, Pi.ofNat_apply, _root_.zero_apply, Multiset.map_const',
-    Finset.card_val, Finset.card_univ, Fintype.card_fin, Multiset.sum_replicate, smul_zero]
-  rfl
+  ext x
+  simp [div, Space.deriv]
 
 /-!
 
@@ -212,9 +206,7 @@ lemma distDiv_apply_eq_sum_fderivD {d}
 
 lemma distDiv_apply_eq_sum_distDeriv {d}
     (f : (Space d) →d[ℝ] EuclideanSpace ℝ (Fin d)) (η : 𝓢(Space d, ℝ)) :
-    (∇ᵈ ⬝ f) η = ∑ i, ∂ᵈ[i] f η i := by
-  rw [distDiv_apply_eq_sum_fderivD]
-  rfl
+    (∇ᵈ ⬝ f) η = ∑ i, ∂ᵈ[i] f η i := distDiv_apply_eq_sum_fderivD f η
 
 /-!
 
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/Grad.lean b/Physlib/SpaceAndTime/Space/Derivatives/Grad.lean
index 38eb0f25c..174330d76 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/Grad.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/Grad.lean
@@ -82,9 +82,8 @@ scoped[Space] notation "∇" => grad
 
 @[simp]
 lemma grad_zero : ∇ (0 : Space d → ℝ) = 0 := by
-  unfold grad Space.deriv
-  simp only [fderiv_zero, Pi.zero_apply, _root_.zero_apply]
-  rfl
+  ext x i
+  simp [grad, Space.deriv]
 
 /-!
 
@@ -96,13 +95,8 @@ lemma grad_zero : ∇ (0 : Space d → ℝ) = 0 := by
 lemma grad_add (f1 f2 : Space d → ℝ)
     (hf1 : Differentiable ℝ f1) (hf2 : Differentiable ℝ f2) :
     ∇ (f1 + f2) = ∇ f1 + ∇ f2 := by
-  unfold grad
   ext x i
-  simp only [Pi.add_apply]
-  rw [deriv_add]
-  rfl
-  exact hf1
-  exact hf2
+  simp [grad, deriv_add f1 f2 hf1 hf2]
 
 @[simp]
 lemma grad_fun_add_const (f : Space d → ℝ) (c : ℝ) :
@@ -118,9 +112,8 @@ lemma grad_fun_add_const (f : Space d → ℝ) (c : ℝ) :
 
 @[simp]
 lemma grad_const : ∇ (fun _ : Space d => c) = 0 := by
-  unfold grad Space.deriv
-  simp only [fderiv_fun_const, Pi.ofNat_apply, _root_.zero_apply]
-  rfl
+  ext x i
+  simp [grad, Space.deriv]
 
 /-!
 
@@ -132,12 +125,8 @@ lemma grad_const : ∇ (fun _ : Space d => c) = 0 := by
 lemma grad_smul (f : Space d → ℝ) (k : ℝ)
     (hf : Differentiable ℝ f) :
     ∇ (k • f) = k • ∇ f := by
-  unfold grad
   ext x i
-  simp only [Pi.smul_apply]
-  rw [deriv_const_smul]
-  rfl
-  exact hf
+  simp [grad, deriv_const_smul k hf]
 
 /-!
 
@@ -148,11 +137,8 @@ lemma grad_smul (f : Space d → ℝ) (k : ℝ)
 @[to_fun]
 lemma grad_neg (f : Space d → ℝ) :
     ∇ (- f) = - ∇ f := by
-  unfold grad
   ext x i
-  simp only [Pi.neg_apply]
-  rw [Space.deriv_eq, fderiv_neg]
-  rfl
+  simp [grad, Space.deriv, fderiv_neg]
 
 /-!
 
@@ -183,13 +169,7 @@ lemma grad_eq_sum {d} (f : Space d → ℝ) (x : Space d) :
 -/
 
 lemma grad_apply {d} (f : Space d → ℝ) (x : Space d) (i : Fin d) :
-    (∇ f x) i = ∂[i] f x := by
-  rw [grad_eq_sum]
-  change WithLp.linearEquiv 2 ℝ (Fin d → ℝ) (∑ x_1, (fderiv ℝ f x) (basis x_1) •
-    EuclideanSpace.single x_1 1) i = _
-  rw [map_sum, Finset.sum_apply]
-  simp [Pi.single_apply]
-  rfl
+    (∇ f x) i = ∂[i] f x := rfl
 
 /-!
 
@@ -201,9 +181,7 @@ open InnerProductSpace
 
 lemma grad_inner_single {d} (f : Space d → ℝ) (x : Space d) (i : Fin d) :
     ⟪∇ f x, EuclideanSpace.single i 1⟫_ℝ = ∂[i] f x := by
-  simp only [EuclideanSpace.inner_single_right, conj_trivial,
-    one_mul]
-  exact rfl
+  simpa only [EuclideanSpace.inner_single_right, conj_trivial, one_mul] using grad_apply f x i
 
 lemma grad_inner_eq {d} (f : Space d → ℝ) (x : Space d) (y : EuclideanSpace ℝ (Fin d)) :
     ⟪∇ f x, y⟫_ℝ = ∑ i, y i * ∂[i] f x:= by
@@ -214,9 +192,7 @@ lemma grad_inner_eq {d} (f : Space d → ℝ) (x : Space d) (y : EuclideanSpace
   simp [inner_smul_right, grad_inner_single]
 
 lemma inner_grad_eq {d} (f : Space d → ℝ) (x : EuclideanSpace ℝ (Fin d)) (y : Space d) :
-    ⟪x, ∇ f y⟫_ℝ = ∑ i, x i * ∂[i] f y := by
-  rw [← grad_inner_eq]
-  exact real_inner_comm (∇ f y) x
+    ⟪x, ∇ f y⟫_ℝ = ∑ i, x i * ∂[i] f y := by simpa [← grad_inner_eq] using real_inner_comm (∇ f y) x
 
 lemma grad_inner_repr_eq {d} (f : Space d → ℝ) (x y : Space d) :
     ⟪∇ f x, (Space.basis).repr y⟫_ℝ = fderiv ℝ f x y := by
@@ -261,8 +237,7 @@ lemma gradient_apply_eq_grad {d} (f : Space d → ℝ) (x : Space d) :
   simp
 
 lemma gradient_eq_sum {d} (f : Space d → ℝ) (x : Space d) :
-    gradient f x = ∑ i, ∂[i] f x • basis i := by
-  simp [gradient_eq_grad, grad_eq_sum f x]
+    gradient f x = ∑ i, ∂[i] f x • basis i := by simp [gradient_eq_grad, grad_eq_sum f x]
 
 lemma euclid_gradient_eq_sum {d} (f : EuclideanSpace ℝ (Fin d) → ℝ) (x : EuclideanSpace ℝ (Fin d)) :
     gradient f x = ∑ i, fderiv ℝ f x (EuclideanSpace.single i 1) • EuclideanSpace.single i 1 := by
@@ -467,8 +442,7 @@ lemma integrable_isDistBounded_inner_grad_schwartzMap_spherical {d : ℕ}
     change Integrable ((fun x => ⟪f x, Space.grad η x⟫_ℝ) ∘ Subtype.val)
       (.comap (Subtype.val (p := fun x => x ∈ ({0}ᶜ : Set _))) volume)
     rw [← MeasureTheory.integrableOn_iff_comap_subtypeVal]
-    apply Integrable.integrableOn
-    exact integrable_isDistBounded_inner_grad_schwartzMap hf η
+    exact Integrable.integrableOn (integrable_isDistBounded_inner_grad_schwartzMap hf η)
     simp
   have he := (MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd
     (volume (α := Space d)))
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/Iterated.lean b/Physlib/SpaceAndTime/Space/Derivatives/Iterated.lean
index af9506c20..ecda75e79 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/Iterated.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/Iterated.lean
@@ -91,8 +91,7 @@ private lemma iteratedDerivList_const_smul (L : List (Fin d)) (c : ℝ) {f : Spa
 
 @[simp]
 lemma iteratedDeriv_zero [AddCommGroup M] [Module ℝ M] [TopologicalSpace M]
-    (f : Space d → M) : ∂^[0] f = f := by
-  simp [iteratedDeriv, Physlib.MultiIndex.toList_zero]
+    (f : Space d → M) : ∂^[0] f = f := by simp [iteratedDeriv, Physlib.MultiIndex.toList_zero]
 
 @[simp]
 lemma iteratedDeriv_increment_zero [NeZero d] [AddCommGroup M] [Module ℝ M] [TopologicalSpace M]
@@ -120,14 +119,12 @@ lemma iteratedDeriv_const_smul (I : MultiIndex d) (c : ℝ) {f : Space d → ℝ
 /-- Iterated spatial derivatives preserve smoothness for scalar-valued functions. -/
 lemma iteratedDeriv_contDiff (I : MultiIndex d) {f : Space d → ℝ}
     (hf : ContDiff ℝ ∞ f) :
-    ContDiff ℝ ∞ (∂^[I] f) := by
-  simpa [iteratedDeriv] using iteratedDerivList_contDiff I.toList hf
+    ContDiff ℝ ∞ (∂^[I] f) := by simpa [iteratedDeriv] using iteratedDerivList_contDiff I.toList hf
 
 /-- The topological support of a spatial derivative is contained in that of the original
 function. -/
 lemma tsupport_deriv_subset (i : Fin d) {f : Space d → ℝ} :
-    tsupport (deriv i f) ⊆ tsupport f := by
-  simpa [deriv_eq_fderiv_fun] using
+    tsupport (deriv i f) ⊆ tsupport f := by simpa [deriv_eq_fderiv_fun] using
     (tsupport_fderiv_apply_subset (𝕜 := ℝ) (f := fun x => f x) (v := basis i))
 
 private lemma iteratedDerivList_commute_deriv (L : List (Fin d)) (i : Fin d)
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean b/Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean
index dcdb7a4b6..a2a7abcb2 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean
@@ -57,9 +57,7 @@ scoped[Space] notation "Δ" => laplacian
 -/
 
 lemma laplacian_eq_sum_snd_deriv {d} (f : Space d → ℝ) :
-    Δ f = fun x => ∑ i, ∂[i] (∂[i] f) x := by
-  unfold laplacian div grad
-  simp
+    Δ f = fun x => ∑ i, ∂[i] (∂[i] f) x := rfl
 
 /-!
 
@@ -100,7 +98,6 @@ scoped[Space] notation "Δᵈ" => distLaplacian
 
 @[simp]
 lemma distLaplacian_const {d : ℕ} (c : ℝ) :
-    Δᵈ (Distribution.const ℝ (Space d) c) = 0 := by
-  simp [distLaplacian, distGrad_const]
+    Δᵈ (Distribution.const ℝ (Space d) c) = 0 := by simp [distLaplacian, distGrad_const]
 
 end Space
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/MatrixDiv.lean b/Physlib/SpaceAndTime/Space/Derivatives/MatrixDiv.lean
index c7a61b815..3899baae6 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/MatrixDiv.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/MatrixDiv.lean
@@ -69,8 +69,7 @@ noncomputable def matrixDiv (d : ℕ) (T : Space d → Matrix (Fin d) (Fin d) 
 @[simp]
 lemma matrixDiv_apply (d : ℕ) (T : Space d → Matrix (Fin d) (Fin d) ℝ)
     (x : Space d) (i : Fin d) :
-    matrixDiv d T x i = ∑ j, ∂[j] (fun x => T x i j) x := by
-  simp [matrixDiv, div]
+    matrixDiv d T x i = ∑ j, ∂[j] (fun x => T x i j) x := rfl
 
 /-!
 
@@ -82,7 +81,6 @@ lemma matrixDiv_apply (d : ℕ) (T : Space d → Matrix (Fin d) (Fin d) ℝ)
 lemma matrixDiv_zero (d : ℕ) :
     matrixDiv d (0 : Space d → Matrix (Fin d) (Fin d) ℝ) = 0 := by
   ext x i
-  change (∑ j : Fin d, ∂[j] (fun _ : Space d => (0 : ℝ)) x) = 0
   simp
 
 /-!
@@ -95,7 +93,6 @@ lemma matrixDiv_zero (d : ℕ) :
 lemma matrixDiv_const (d : ℕ) (T : Matrix (Fin d) (Fin d) ℝ) :
     matrixDiv d (fun _ : Space d => T) = 0 := by
   ext x i
-  change (∑ j : Fin d, ∂[j] (fun _ : Space d => T i j) x) = 0
   simp
 
 /-!
diff --git a/Physlib/SpaceAndTime/Space/Derivatives/MultiIndex.lean b/Physlib/SpaceAndTime/Space/Derivatives/MultiIndex.lean
index 999ac2613..9b8d358d4 100644
--- a/Physlib/SpaceAndTime/Space/Derivatives/MultiIndex.lean
+++ b/Physlib/SpaceAndTime/Space/Derivatives/MultiIndex.lean
@@ -97,17 +97,14 @@ lemma add_apply (I J : MultiIndex d) (i : Fin d) : (I + J) i = I i + J i := rfl
 
 @[simp]
 lemma increment_apply_same (I : MultiIndex d) (i : Fin d) :
-    increment I i i = I i + 1 := by
-  simp [increment]
+    increment I i i = I i + 1 := by simp [increment]
 
 @[simp]
 lemma increment_apply_ne (I : MultiIndex d) {i j : Fin d} (h : j ≠ i) :
-    increment I i j = I j := by
-  simp [increment, Pi.single_eq_of_ne h]
+    increment I i j = I j := by simp [increment, Pi.single_eq_of_ne h]
 
 @[simp]
-lemma order_zero : order (0 : MultiIndex d) = 0 := by
-  simp [order]
+lemma order_zero : order (0 : MultiIndex d) = 0 := by simp [order]
 
 lemma order_add (I J : MultiIndex d) : order (I + J) = order I + order J := by
   simp [order, Finset.sum_add_distrib]
@@ -125,8 +122,7 @@ lemma order_single (i : Fin d) : order (⟨Pi.single i 1⟩ : MultiIndex d) = 1
 
 @[simp]
 lemma order_increment (I : MultiIndex d) (i : Fin d) :
-    order (increment I i) = order I + 1 := by
-  simp [increment, order, Finset.sum_add_distrib]
+    order (increment I i) = order I + 1 := by simp [increment, order, Finset.sum_add_distrib]
 
 /-!
 ### A.3. Canonical ordered lists of directions
@@ -142,9 +138,7 @@ def toList : {d : ℕ} → MultiIndex d → List (Fin d)
   | _ + 1, I => List.replicate (I 0) 0 ++ (toList (tail I)).map Fin.succ
 
 @[simp]
-lemma tail_zero : tail (0 : MultiIndex d.succ) = 0 := by
-  ext i
-  rfl
+lemma tail_zero : tail (0 : MultiIndex d.succ) = 0 := rfl
 
 @[simp]
 lemma tail_increment_zero (I : MultiIndex d.succ) : tail (increment I 0) = tail I := by
@@ -193,8 +187,7 @@ lemma toList_single (i : Fin d) : toList (increment 0 i : MultiIndex d) = [i] :=
         have htail :
             tail (increment (0 : MultiIndex d.succ) j.succ) = increment (0 : MultiIndex d) j := by
           rw [tail_increment_succ, tail_zero]
-        have hzero : increment (0 : MultiIndex d.succ) j.succ 0 = 0 := by
-          simp [increment]
+        have hzero : increment (0 : MultiIndex d.succ) j.succ 0 = 0 := by simp [increment]
         simp [toList, hzero, htail, ih j]
 
 end MultiIndex
diff --git a/Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean b/Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean
index 15950e40a..d1a5600c9 100644
--- a/Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean
+++ b/Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean
@@ -135,13 +135,10 @@ transport of `toAffineIsometryMulEquiv` acting on `EuclideanSpace`:
 `chart (g • p) = (toAffineIsometryMulEquiv g) (chart p)`. -/
 lemma chartEuclidean_smul (g : EuclideanGroup d) (p : Space d) :
     Space.chartEuclidean d (g • p) = toAffineIsometryMulEquiv g (Space.chartEuclidean d p) := by
-  rw [Space.chartEuclidean_apply]
-  rw [toAffineIsometryMulEquiv_apply, toAffineIsometryHom_apply]
-  have h_left : g • p -ᵥ (0 : Space d) = g.linear • (p -ᵥ (0 : Space d)) + g.translation := by
-    exact
-      (eq_vadd_iff_vsub_eq (g • p) (g.linear • (p -ᵥ (0 : Space d)) + g.translation)
-            ((0 : Space d))).mp
-        rfl
+  rw [Space.chartEuclidean_apply, toAffineIsometryMulEquiv_apply, toAffineIsometryHom_apply]
+  have h_left : g • p -ᵥ (0 : Space d) = g.linear • (p -ᵥ (0 : Space d)) + g.translation :=
+    (eq_vadd_iff_vsub_eq (g • p) (g.linear • (p -ᵥ (0 : Space d)) + g.translation)
+      ((0 : Space d))).mp rfl
   simp [h_left]
   exact add_comm' (g.linear • (p -ᵥ (0 : Space d))) g.translation
 end EuclideanGroup
diff --git a/Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean b/Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean
index 5e57a6106..bbffbfde9 100644
--- a/Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean
+++ b/Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean
@@ -76,12 +76,8 @@ noncomputable def toAffineIsometryHom :
       AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) where
   toFun A := AffineIsometryEquiv.constVAdd ℝ (EuclideanSpace ℝ (Fin n)) A.translation *
     (orthogonalToLinearIsometryEquiv A.linear).toAffineIsometryEquiv
-  map_one' := by
-    apply AffineIsometryEquiv.ext
-    intro x; simp
-  map_mul' A B := by
-    apply AffineIsometryEquiv.ext
-    intro x
+  map_one' := AffineIsometryEquiv.ext fun x => by simp
+  map_mul' A B := AffineIsometryEquiv.ext fun x => by
     simp [mul_smul, add_assoc]
 
 /-- Unfolds `toAffineIsometryHom` into its translation and linear factors. -/
@@ -96,10 +92,8 @@ noncomputable def _root_.AffineIsometryEquiv.toAffineEquivHom :
     AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) →*
       AffineEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) where
   toFun e := e.toAffineEquiv
-  map_one' := by
-    apply AffineEquiv.ext; intro x; trivial
-  map_mul' e e' := by
-    apply AffineEquiv.ext; intro x; simp
+  map_one' := AffineEquiv.ext fun x => by trivial
+  map_mul' e e' := AffineEquiv.ext fun x => by simp
 
 /-- The inclusion of the Euclidean group into Mathlib's affine automorphism group: the composite
 of the two legs `toAffineIsometryHom` and `AffineIsometryEquiv.toAffineEquivHom`. -/
@@ -176,8 +170,7 @@ this proves `right_inv` of `toAffineIsometryMulEquiv`. -/
     show Matrix.toLin _ _
         (LinearMap.toMatrix _ _
           (L.toLinearEquiv : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (Fin n))) x = L x
-    rw [Matrix.toLin_toMatrix]
-    rfl
+    exact LinearMap.congr_fun (Matrix.toLin_toMatrix _ _ _) x
 
 /-- `EuclideanGroup n ≃* AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) _`: the Euclidean group
 is the full group of affine isometries of Euclidean space. This upgrades `toAffineIsometryHom` to
diff --git a/Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean b/Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean
index cb378c177..30342bed7 100644
--- a/Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean
+++ b/Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean
@@ -142,8 +142,7 @@ lemma translationVector.incl_range :
   constructor
   · rintro ⟨v, hv⟩
     show g.linear.val = 1
-    rw [← hv]
-    rfl
+    exact congrArg (fun g : EuclideanGroup n => g.linear.val) hv.symm
   · intro h
     rw [Set.mem_range]
     refine ⟨g.translation, ?_⟩
@@ -154,8 +153,7 @@ lemma translationVector.incl_range :
 
 /-- The translation by the zero vector is the identity of the Euclidean group. -/
 lemma translation_zero : translationVector.incl n
-    (Multiplicative.ofAdd (0 : EuclideanSpace ℝ (Fin n))) = 1 := by
-  simp
+    (Multiplicative.ofAdd (0 : EuclideanSpace ℝ (Fin n))) = 1 := by simp
 
 /-- The subgroup of `EuclideanGroup n` whose elements fix the origin
 (translation = 0). This is the copy of `O(n)` sitting inside `E(n)`. -/
@@ -248,9 +246,7 @@ lemma RotationsAbout.fromOrigin_comp_toOrigin :
   intro x
   apply Subtype.ext
   simp only [MonoidHom.coe_comp, Function.comp_apply, MonoidHom.id_apply, SetLike.coe_eq_coe]
-  unfold RotationsAbout.toOrigin
-  unfold RotationsAbout.fromOrigin
-  simp [mul_assoc]
+  simp [RotationsAbout.toOrigin, RotationsAbout.fromOrigin, mul_assoc]
 
 /-- `RotationsAbout.fromOrigin p` followed by `RotationsAbout.toOrigin p` is the identity; the
 backward leg of the isomorphism `RotationsAboutEquiv`. -/
@@ -261,9 +257,7 @@ lemma RotationsAbout.toOrigin_comp_fromOrigin :
   intro x
   apply Subtype.ext
   simp only [MonoidHom.coe_comp, Function.comp_apply, MonoidHom.id_apply, SetLike.coe_eq_coe]
-  unfold RotationsAbout.toOrigin
-  unfold RotationsAbout.fromOrigin
-  simp [mul_assoc]
+  simp [RotationsAbout.toOrigin, RotationsAbout.fromOrigin, mul_assoc]
 
 /-- API feature: conjugation by the translation `T(p)` exhibits the rotations about `p` as
 isomorphic to the rotations about the origin `RotationGroup n`. -/
@@ -355,10 +349,8 @@ noncomputable def specialOrthogonal.fromRotation (n : ℕ) :
 the forward leg of the isomorphism `specialOrthogonalEquiv`. -/
 lemma specialOrthogonal.fromRotation_comp_toRotation :
     (specialOrthogonal.fromRotation n).comp (specialOrthogonal.toRotation n) =
-      MonoidHom.id (Matrix.specialOrthogonalGroup (Fin n) ℝ) := by
-  apply MonoidHom.ext
-  intro x
-  rfl
+      MonoidHom.id (Matrix.specialOrthogonalGroup (Fin n) ℝ) :=
+  MonoidHom.ext fun _ => rfl
 
 /-- `specialOrthogonal.fromRotation n` followed by `specialOrthogonal.toRotation n` is the identity;
 the backward leg of the isomorphism `specialOrthogonalEquiv`. -/
diff --git a/Physlib/SpaceAndTime/Space/Integrals/Basic.lean b/Physlib/SpaceAndTime/Space/Integrals/Basic.lean
index 1a957df94..411404d21 100644
--- a/Physlib/SpaceAndTime/Space/Integrals/Basic.lean
+++ b/Physlib/SpaceAndTime/Space/Integrals/Basic.lean
@@ -58,8 +58,7 @@ lemma volume_metricBall_three :
 lemma volume_metricBall_two :
     volume (Metric.ball (0 : Space 2) 1) = ENNReal.ofReal Real.pi := by
   rw [InnerProductSpace.volume_ball_of_dim_even (k := 1)]
-  simp [finrank_eq_dim]
-  simp [finrank_eq_dim]
+  all_goals simp [finrank_eq_dim]
 
 @[simp]
 lemma volume_metricBall_two_real :
@@ -110,13 +109,11 @@ lemma integral_volume_eq_spherical (d : ℕ) [NeZero d] (f : Space d → F)
   rw [MeasureTheory.integral_subtype_comap (by simp), ← setIntegral_univ]
   simp [f']
   refine integral_congr_ae ?_
-  have h1 : ∀ᵐ x ∂(volume (α := Space d)), x ≠ 0 := by
-    exact Measure.ae_ne volume 0
+  have h1 : ∀ᵐ x ∂(volume (α := Space d)), x ≠ 0 := Measure.ae_ne volume 0
   filter_upwards [Measure.ae_ne volume 0] with x hx
   congr
   simp [smul_smul]
-  have hx : ‖x‖ ≠ 0 := by
-    simpa using hx
+  have hx : ‖x‖ ≠ 0 := by simpa using hx
   field_simp
   simp
 
@@ -236,8 +233,7 @@ lemma lintegral_volume_eq_spherical (d : ℕ) [NeZero d]
   filter_upwards [Measure.ae_ne volume 0] with x hx
   congr
   simp [smul_smul]
-  have hx : ‖x‖ ≠ 0 := by
-    simpa using hx
+  have hx : ‖x‖ ≠ 0 := by simpa using hx
   field_simp
   rw [one_smul]
 
diff --git a/Physlib/SpaceAndTime/Space/Integrals/NormPow.lean b/Physlib/SpaceAndTime/Space/Integrals/NormPow.lean
index 20c0fb61e..7fa4215f5 100644
--- a/Physlib/SpaceAndTime/Space/Integrals/NormPow.lean
+++ b/Physlib/SpaceAndTime/Space/Integrals/NormPow.lean
@@ -60,12 +60,10 @@ lemma radial_jacobian_zpow_mul_self
     r ^ (d - 1) * (r ^ q * r)
         = r ^ (d - 1: ℤ) * (r ^ q * r ^ (1 : ℤ)) := by
             rw [← zpow_natCast, zpow_one, ← Nat.cast_pred (Nat.pos_of_neZero d)]
-    _ = r ^ ((d - 1: ℤ) + (q + 1)) := by
-            rw [← zpow_add₀ hz q 1, ← zpow_add₀ hz (d - 1: ℤ) (q + 1)]
+    _ = r ^ ((d - 1: ℤ) + (q + 1)) := by rw [← zpow_add₀ hz q 1, ← zpow_add₀ hz (d - 1: ℤ) (q + 1)]
     _ = r ^ (p : ℤ) := by
             congr 1; omega
-    _ = r ^ p := by
-            rw [zpow_natCast]
+    _ = r ^ p := by rw [zpow_natCast]
 
 private lemma radial_jacobian_zpow
     {d p : ℕ} [NeZero d] {q : ℤ} (hp_int : (p : ℤ) = q + (d : ℤ))
@@ -78,8 +76,7 @@ private lemma radial_jacobian_zpow
             rw [← Nat.cast_pred (Nat.pos_of_neZero d), ← zpow_natCast, ← zpow_add₀ hz]
     _ = r ^ ((p - 1 : ℕ) : ℤ) := by
             congr 1; omega
-    _ = r ^ (p - 1) := by
-            rw [zpow_natCast]
+    _ = r ^ (p - 1) := by rw [zpow_natCast]
 
 lemma radial_norm_power_spherical_integral_eq_space_integral
     {d p : ℕ} [NeZero d] {q : ℤ} (hp_int : (p : ℤ) = q + (d : ℤ))
diff --git a/Physlib/SpaceAndTime/Space/Integrals/RadialAngularMeasure.lean b/Physlib/SpaceAndTime/Space/Integrals/RadialAngularMeasure.lean
index bfc387228..741c371fc 100644
--- a/Physlib/SpaceAndTime/Space/Integrals/RadialAngularMeasure.lean
+++ b/Physlib/SpaceAndTime/Space/Integrals/RadialAngularMeasure.lean
@@ -69,13 +69,11 @@ def radialAngularMeasure {d : ℕ} : Measure (Space d) :=
 -/
 
 lemma radialAngularMeasure_eq_volume_withDensity {d : ℕ} : radialAngularMeasure =
-    volume.withDensity (fun x : Space d => ENNReal.ofReal (1 / ‖x‖ ^ (d - 1))) := by
-  rfl
+    volume.withDensity (fun x : Space d => ENNReal.ofReal (1 / ‖x‖ ^ (d - 1))) := rfl
 
 @[simp]
 lemma radialAngularMeasure_zero_eq_volume :
-    radialAngularMeasure (d := 0) = volume := by
-  simp [radialAngularMeasure]
+    radialAngularMeasure (d := 0) = volume := by simp [radialAngularMeasure]
 
 /-!
 
@@ -104,8 +102,7 @@ lemma integral_radialAngularMeasure {d : ℕ} (f : Space d → F) :
 
 lemma lintegral_radialMeasure {d : ℕ} (f : Space d → ENNReal) (hf : Measurable f) :
     ∫⁻ x, f x ∂radialAngularMeasure = ∫⁻ x, ENNReal.ofReal (1 / ‖x‖ ^ (d - 1)) * f x := by
-  dsimp [radialAngularMeasure]
-  rw [lintegral_withDensity_eq_lintegral_mul]
+  rw [radialAngularMeasure, lintegral_withDensity_eq_lintegral_mul]
   simp only [one_div, Pi.mul_apply]
   all_goals fun_prop
 
diff --git a/Physlib/SpaceAndTime/Space/IsDistBounded.lean b/Physlib/SpaceAndTime/Space/IsDistBounded.lean
index ac8f3095a..30963783f 100644
--- a/Physlib/SpaceAndTime/Space/IsDistBounded.lean
+++ b/Physlib/SpaceAndTime/Space/IsDistBounded.lean
@@ -115,14 +115,12 @@ lemma aestronglyMeasurable {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
 @[fun_prop]
 lemma aeStronglyMeasurable_schwartzMap_smul {d : ℕ} {f : Space d → F}
     (hf : IsDistBounded f) (η : 𝓢(Space d, ℝ)) :
-    AEStronglyMeasurable (fun x => η x • f x) := by
-  fun_prop
+    AEStronglyMeasurable (fun x => η x • f x) := by fun_prop
 
 @[fun_prop]
 lemma aeStronglyMeasurable_fderiv_schwartzMap_smul {d : ℕ} {f : Space d → F}
     (hf : IsDistBounded f) (η : 𝓢(Space d, ℝ)) (y : Space d) :
-    AEStronglyMeasurable (fun x => fderiv ℝ η x y • f x) := by
-  fun_prop
+    AEStronglyMeasurable (fun x => fderiv ℝ η x y • f x) := by fun_prop
 
 @[fun_prop]
 lemma aeStronglyMeasurable_inv_pow {d r : ℕ} {f : Space d → F}
@@ -204,8 +202,7 @@ lemma integrable_space {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
   by_cases hp' : p = 0
   · subst hp'
     simp only [zpow_zero, mul_one]
-    apply Integrable.norm
-    exact η.integrable
+    exact Integrable.norm η.integrable
   suffices h1 : Integrable (fun x => ‖η x‖ * ‖x‖ ^ (p + d)) (radialAngularMeasure (d := (d + 1))) by
     rw [integrable_radialAngularMeasure_iff] at h1
     convert h1 using 1
@@ -221,8 +218,7 @@ lemma integrable_space {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
       simp [zero_pow_eq, zero_zpow_eq, hp']
       omega
     field_simp
-    rw [zpow_add₀ hr]
-    rfl
+    exact (zpow_add₀ hr p d).symm
   convert integrable_pow_mul_iteratedFDeriv radialAngularMeasure η (p + d).toNat 0 using 1
   funext x
   simp only [Real.norm_eq_abs, norm_iteratedFDeriv_zero]
@@ -236,8 +232,7 @@ lemma integrable_space {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
 @[fun_prop]
 lemma integrable_space_mul {d : ℕ} {f : Space d → ℝ} (hf : IsDistBounded f)
     (η : 𝓢(Space d, ℝ)) :
-    Integrable (fun x : Space d => η x * f x) volume := by
-  exact hf.integrable_space η
+    Integrable (fun x : Space d => η x * f x) volume := hf.integrable_space η
 
 @[fun_prop]
 lemma integrable_space_fderiv {d : ℕ} {f : Space d → F} (hf : IsDistBounded f)
@@ -635,8 +630,7 @@ lemma add {d : ℕ} {f g : Space d → F}
 
 @[fun_prop]
 lemma fun_add {d : ℕ} {f g : Space d → F}
-    (hf : IsDistBounded f) (hg : IsDistBounded g) : IsDistBounded (fun x => f x + g x) := by
-  exact hf.add hg
+    (hf : IsDistBounded f) (hg : IsDistBounded g) : IsDistBounded (fun x => f x + g x) := hf.add hg
 
 /-!
 
@@ -905,8 +899,7 @@ lemma pow_shift {d : ℕ} (n : ℤ)
   · simp
     exact hn
   · intro x
-    simp
-    rfl
+    simp [sub_eq_add_neg]
 
 @[fun_prop]
 lemma inv_shift {d : ℕ} (g : Space d) (hd : 2 ≤ d := by omega) :
@@ -916,8 +909,7 @@ lemma inv_shift {d : ℕ} (g : Space d) (hd : 2 ≤ d := by omega) :
   simp
 @[fun_prop]
 lemma nat_pow {d : ℕ} (n : ℕ) :
-    IsDistBounded (d := d) (fun x => ‖x‖ ^ n) := by
-  exact IsDistBounded.pow (d := d) (n : ℤ) (by omega)
+    IsDistBounded (d := d) (fun x => ‖x‖ ^ n) := IsDistBounded.pow (d := d) (n : ℤ) (by omega)
 
 @[fun_prop]
 lemma norm_add_nat_pow {d : ℕ} (n : ℕ) (a : ℝ) :
@@ -1038,8 +1030,7 @@ lemma zpow_smul_self {d : ℕ} (n : ℤ) (hn : - (d - 1 : ℕ) - 1 ≤ n) :
 lemma zpow_smul_repr_self {d : ℕ} (n : ℤ) (hn : - (d - 1 : ℕ) - 1 ≤ n) :
     IsDistBounded (d := d) (fun x => ‖x‖ ^ n • basis.repr x) := by
   apply IsDistBounded.congr (f := fun x => ‖x‖ ^ n • x)
-  · apply zpow_smul_self
-    exact hn
+  · exact zpow_smul_self n hn
   · apply AEMeasurable.aestronglyMeasurable
     fun_prop
   · intro x
@@ -1158,8 +1149,7 @@ lemma norm_smul_isDistBounded {d : ℕ} [NormedSpace ℝ F] {f : Space d → F}
   · apply IsDistBounded.congr (f := fun x => ∑ i, (c i * (‖x‖ * ‖x + g i‖ ^ (p i))))
     · apply IsDistBounded.sum_fun
       intro i _
-      apply IsDistBounded.const_mul_fun
-      exact norm_smul_zpow (p i) (g i) (p_bound i)
+      exact IsDistBounded.const_mul_fun (norm_smul_zpow (p i) (g i) (p_bound i)) (c i)
     · fun_prop
     · intro x
       congr
@@ -1190,16 +1180,14 @@ lemma component_smul_isDistBounded {d : ℕ} [NormedSpace ℝ F] {f : Space d 
   apply IsDistBounded.mono (f := fun x => ‖x‖ • f x)
   · fun_prop
   · apply AEStronglyMeasurable.smul
-    · have h1 : AEStronglyMeasurable (fun x => Space.coordCLM i x) := by
-        fun_prop
+    · have h1 : AEStronglyMeasurable (fun x => Space.coordCLM i x) := by fun_prop
       convert h1 using 1
       funext i
       simp [coordCLM_apply, coord_apply]
     · fun_prop
   · intro x
     simp [norm_smul]
-    apply mul_le_mul ?_ (by rfl) (by positivity) (by positivity)
-    exact abs_eval_le_norm x i
+    exact mul_le_mul (abs_eval_le_norm x i) (by rfl) (by positivity) (by positivity)
 
 @[fun_prop]
 lemma component_mul_isDistBounded {d : ℕ} {f : Space d → ℝ}
@@ -1260,8 +1248,7 @@ lemma isDistBounded_smul_inner_of_smul_norm {d : ℕ} [NormedSpace ℝ F] {f : S
   · fun_prop
   · intro x
     simp [norm_smul]
-    refine mul_le_mul_of_nonneg_right ?_ (by positivity)
-    exact abs_eval_le_norm x i
+    exact mul_le_mul_of_nonneg_right (abs_eval_le_norm x i) (by positivity)
 
 @[fun_prop]
 lemma isDistBounded_mul_inner {d : ℕ} {f : Space d → ℝ}
@@ -1293,15 +1280,13 @@ lemma mul_inner_pow_neg_two {d : ℕ} (y : Space d) (hd : 2 ≤ d := by omega) :
       by_cases hx : x = 0
       · subst hx
         simp
-      have hx' : ‖x‖ ≠ 0 := by
-        simpa using hx
+      have hx' : ‖x‖ ≠ 0 := by simpa using hx
       field_simp
   · apply AEMeasurable.aestronglyMeasurable
     fun_prop
   · intro x
     simp
-    apply mul_le_mul_of_nonneg _ (by rfl) (by positivity) (by positivity)
-    exact abs_real_inner_le_norm y x
+    exact mul_le_mul_of_nonneg (abs_real_inner_le_norm y x) (by rfl) (by positivity) (by positivity)
 
 end constructors
 end IsDistBounded
diff --git a/Physlib/SpaceAndTime/Space/LengthUnit.lean b/Physlib/SpaceAndTime/Space/LengthUnit.lean
index e4d32fea5..12741e552 100644
--- a/Physlib/SpaceAndTime/Space/LengthUnit.lean
+++ b/Physlib/SpaceAndTime/Space/LengthUnit.lean
@@ -36,8 +36,7 @@ structure LengthUnit where
 namespace LengthUnit
 
 @[simp]
-lemma val_ne_zero (x : LengthUnit) : x.val ≠ 0 := by
-  exact Ne.symm (ne_of_lt x.property)
+lemma val_ne_zero (x : LengthUnit) : x.val ≠ 0 := Ne.symm (ne_of_lt x.property)
 
 lemma val_pos (x : LengthUnit) : 0 < x.val := x.property
 
@@ -72,9 +71,7 @@ lemma div_pos (x y : LengthUnit) : (0 : ℝ≥0) < x/ y := by
 
 @[simp]
 lemma div_self (x : LengthUnit) :
-    x / x = (1 : ℝ≥0) := by
-  simp [div_eq_val, x.val_ne_zero]
-  rfl
+    x / x = (1 : ℝ≥0) := NNReal.eq (_root_.div_self x.val_ne_zero)
 
 lemma div_symm (x y : LengthUnit) :
     x / y = (y / x)⁻¹ := NNReal.eq <| by
@@ -100,19 +97,16 @@ def scale (r : ℝ) (x : LengthUnit) (hr : 0 < r := by norm_num) : LengthUnit :=
 
 @[simp]
 lemma scale_div_self (x : LengthUnit) (r : ℝ) (hr : 0 < r) :
-    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by
-  simp [scale, div_eq_val]
+    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by simp [scale, div_eq_val]
 
 @[simp]
 lemma self_div_scale (x : LengthUnit) (r : ℝ) (hr : 0 < r) :
     x / scale r x hr = (⟨1/r, _root_.div_nonneg (by simp) (le_of_lt hr)⟩ : ℝ≥0) := by
   simp [scale, div_eq_val]
-
   field_simp
 
 @[simp]
-lemma scale_one (x : LengthUnit) : scale 1 x = x := by
-  simp [scale]
+lemma scale_one (x : LengthUnit) : scale 1 x = x := by simp [scale]
 
 @[simp]
 lemma scale_div_scale (x1 x2 : LengthUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0 < r2) :
@@ -125,8 +119,7 @@ lemma scale_div_scale (x1 x2 : LengthUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0
 @[simp]
 lemma scale_scale (x : LengthUnit) (r1 r2 : ℝ) (hr1 : 0 < r1) (hr2 : 0 < r2) :
     scale r1 (scale r2 x hr2) hr1 = scale (r1 * r2) x (mul_pos hr1 hr2) := by
-  simp [scale]
-  ring
+  simp [scale, mul_assoc]
 /-!
 
 ## Specific choices of Length units
diff --git a/Physlib/SpaceAndTime/Space/Module.lean b/Physlib/SpaceAndTime/Space/Module.lean
index 94afdb17f..ee705c263 100644
--- a/Physlib/SpaceAndTime/Space/Module.lean
+++ b/Physlib/SpaceAndTime/Space/Module.lean
@@ -64,26 +64,13 @@ lemma add_val {d: ℕ} (x y : Space d) :
 
 @[simp]
 lemma add_apply {d : ℕ} (x y : Space d) (i : Fin d) :
-    (x + y) i = x i + y i := by
-  simp [add_val]
+    (x + y) i = x i + y i := by simp [add_val]
 
 instance {d} : AddCommMonoid (Space d) where
-  add_assoc a b c:= by
-    apply eq_of_val
-    simp only [add_val]
-    ring
-  zero_add a := by
-    apply eq_of_val
-    simp only [zero_val, add_val, add_eq_right]
-    rfl
-  add_zero a := by
-    apply eq_of_val
-    simp only [zero_val, add_val, add_eq_left]
-    rfl
-  add_comm a b := by
-    apply eq_of_val
-    simp only [add_val]
-    ring
+  add_assoc a b c := eq_of_apply fun i => by simp [add_assoc]
+  zero_add a := eq_of_apply fun i => by simp
+  add_zero a := eq_of_apply fun i => by simp
+  add_comm a b := eq_of_apply fun i => by simp [add_comm]
   nsmul n a := ⟨fun i => n • a.val i⟩
 
 @[simp]
@@ -102,9 +89,8 @@ lemma eq_vadd_zero {d} (s : Space d) :
 
 @[simp]
 lemma add_vadd_zero {d} (v1 v2 : EuclideanSpace ℝ (Fin d)) :
-    (v1 +ᵥ (0 : Space d)) + (v2 +ᵥ (0 : Space d)) = (v1 + v2) +ᵥ (0 : Space d) := by
-  ext i
-  simp
+    (v1 +ᵥ (0 : Space d)) + (v2 +ᵥ (0 : Space d)) = (v1 + v2) +ᵥ (0 : Space d) :=
+  eq_of_apply fun i => by simp
 
 /-!
 
@@ -125,32 +111,16 @@ lemma smul_apply {d : ℕ} (c : ℝ) (p : Space d) (i : Fin d) :
 
 @[simp]
 lemma smul_vadd_zero {d} (k : ℝ) (v : EuclideanSpace ℝ (Fin d)) :
-    k • (v +ᵥ (0 : Space d)) = (k • v) +ᵥ (0 : Space d) := by
-  ext i
-  simp
+    k • (v +ᵥ (0 : Space d)) = (k • v) +ᵥ (0 : Space d) :=
+  eq_of_apply fun i => by simp
 
 instance {d} : Module ℝ (Space d) where
-  one_smul x := by
-    ext i
-    simp
-  mul_smul a b x := by
-    ext i
-    simp only [smul_apply]
-    ring
-  smul_add a x y := by
-    ext i
-    simp only [smul_apply, add_apply]
-    ring
-  smul_zero a := by
-    ext i
-    simp
-  add_smul a b x := by
-    ext i
-    simp only [smul_apply, add_apply]
-    ring
-  zero_smul x := by
-    ext i
-    simp
+  one_smul x := eq_of_apply fun i => by simp
+  mul_smul a b x := eq_of_apply fun i => by simp [mul_assoc]
+  smul_add a x y := eq_of_apply fun i => by simp [mul_add]
+  smul_zero a := eq_of_apply fun i => by simp
+  add_smul a b x := eq_of_apply fun i => by simp [add_mul]
+  zero_smul x := eq_of_apply fun i => by simp
 
 /-!
 
@@ -161,8 +131,7 @@ instance {d} : Module ℝ (Space d) where
 noncomputable instance {d} : Norm (Space d) where
   norm p := √ (∑ i, (p i)^2)
 
-lemma norm_eq {d} (p : Space d) : ‖p‖ = √ (∑ i, (p i) ^ 2) := by
-  rfl
+lemma norm_eq {d} (p : Space d) : ‖p‖ = √ (∑ i, (p i) ^ 2) := rfl
 
 @[simp]
 lemma abs_eval_le_norm {d} (p : Space d) (i : Fin d) :
@@ -179,14 +148,12 @@ lemma norm_sq_eq {d} (p : Space d) :
   refine Real.sq_sqrt ?_
   positivity
 
-lemma point_dim_zero_eq (p : Space 0) : p = 0 := by
-  ext i
-  fin_cases i
+lemma point_dim_zero_eq (p : Space 0) : p = 0 :=
+  eq_of_apply fun i => by fin_cases i
 
 @[simp]
 lemma norm_vadd_zero {d} (v : EuclideanSpace ℝ (Fin d)) :
-    ‖v +ᵥ (0 : Space d)‖ = ‖v‖ := by
-  simp [norm_eq, PiLp.norm_eq_of_L2]
+    ‖v +ᵥ (0 : Space d)‖ = ‖v‖ := by simp [norm_eq, PiLp.norm_eq_of_L2]
 
 instance : Neg (Space d) where
   neg p := ⟨fun i => - (p.val i)⟩
@@ -201,12 +168,8 @@ lemma neg_apply {d : ℕ} (p : Space d) (i : Fin d) :
 
 noncomputable instance {d} : AddCommGroup (Space d) where
   zsmul z p := ⟨fun i => z * p.val i⟩
-  neg_add_cancel p := by
-    ext i
-    simp
-  zsmul_zero' p := by
-    ext i
-    simp
+  neg_add_cancel p := eq_of_apply fun i => by simp
+  zsmul_zero' p := eq_of_apply fun i => by simp
   zsmul_succ' n p := by
     ext i
     simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, Int.cast_add, Int.cast_natCast,
@@ -220,8 +183,7 @@ noncomputable instance {d} : AddCommGroup (Space d) where
 
 @[simp]
 lemma sub_apply {d} (p q : Space d) (i : Fin d) :
-    (p - q) i = p i - q i := by
-  simp [sub_eq_add_neg, neg_apply, add_apply]
+    (p - q) i = p i - q i := rfl
 
 @[simp]
 lemma sub_val {d} (p q : Space d) :
@@ -229,9 +191,8 @@ lemma sub_val {d} (p q : Space d) :
 
 @[simp]
 lemma vadd_zero_sub_vadd_zero {d} (v1 v2 : EuclideanSpace ℝ (Fin d)) :
-    (v1 +ᵥ (0 : Space d)) - (v2 +ᵥ (0 : Space d)) = (v1 - v2) +ᵥ (0 : Space d) := by
-  ext i
-  simp [sub_apply, vadd_apply]
+    (v1 +ᵥ (0 : Space d)) - (v2 +ᵥ (0 : Space d)) = (v1 - v2) +ᵥ (0 : Space d) :=
+  eq_of_apply fun i => by simp [sub_apply, vadd_apply]
 
 @[simp]
 lemma dist_eq_norm {d} (p q : Space d) :
@@ -319,8 +280,7 @@ instance {d : ℕ} : BorelSpace (Space d) where
 -/
 
 lemma inner_eq_sum {d} (p q : Space d) :
-    inner ℝ p q = ∑ i, p i * q i := by
-  simp [inner]
+    inner ℝ p q = ∑ i, p i * q i := inner_apply p q
 
 @[simp]
 lemma sum_apply {ι : Type} [Fintype ι] (f : ι → Space d) (i : Fin d) :
@@ -362,30 +322,20 @@ noncomputable def basis {d} : OrthonormalBasis (Fin d) ℝ (Space d) where
   repr := {
     toFun p := WithLp.toLp 2 fun i => p i
     invFun := fun v => ⟨v⟩
-    left_inv := by
-      intro p
-      rfl
-    right_inv := by
-      intro v
-      rfl
-    map_add' := by
-      intro v1 v2
-      rfl
-    map_smul' := by
-      intro c v
-      rfl
+    left_inv := fun p => rfl
+    right_inv := fun v => rfl
+    map_add' := fun v1 v2 => rfl
+    map_smul' := fun c v => rfl
     norm_map' := by
       intro x
       simp [norm_eq, PiLp.norm_eq_of_L2]}
 
 lemma apply_eq_basis_repr_apply {d} (p : Space d) (i : Fin d) :
-    p i = basis.repr p i := by
-  simp [basis]
+    p i = basis.repr p i := rfl
 
 @[simp]
 lemma basis_repr_apply {d} (p : Space d) (i : Fin d) :
-    basis.repr p i = p i := by
-  simp [apply_eq_basis_repr_apply]
+    basis.repr p i = p i := rfl
 
 @[simp]
 lemma basis_repr_symm_apply {d} (v : EuclideanSpace ℝ (Fin d)) (i : Fin d) :
@@ -398,18 +348,15 @@ lemma basis_apply {d} (i j : Fin d) :
   exact Lean.Grind.eq_congr' rfl rfl
 
 @[simp]
-lemma basis_self {d} (i : Fin d) : basis i i = 1 := by
-  simp [basis_apply]
+lemma basis_self {d} (i : Fin d) : basis i i = 1 := by simp [basis_apply]
 
 @[simp high]
 lemma inner_basis {d} (p : Space d) (i : Fin d) :
-    inner ℝ p (basis i) = p i := by
-  simp [inner_eq_sum, basis_apply]
+    inner ℝ p (basis i) = p i := by simp [inner_eq_sum, basis_apply]
 
 @[simp high]
 lemma basis_inner {d} (i : Fin d) (p : Space d) :
-    inner ℝ (basis i) p = p i := by
-  simp [inner_eq_sum, basis_apply]
+    inner ℝ (basis i) p = p i := by simp [inner_eq_sum, basis_apply]
 
 open InnerProductSpace
 
@@ -475,8 +422,7 @@ noncomputable def coord (μ : Fin d) (p : Space d) : ℝ :=
   inner ℝ p (basis μ)
 
 lemma coord_apply (μ : Fin d) (p : Space d) :
-    coord μ p = p μ := by
-  simp [coord]
+    coord μ p = p μ := by simp [coord]
 
 /-- The standard coordinate functions of Space based on `Fin d`, as a continuous linear map. -/
 noncomputable def coordCLM {d} (μ : Fin d) : Space d →L[ℝ] ℝ where
@@ -497,8 +443,7 @@ lemma coord_contDiff {i} : ContDiff ℝ ∞ (fun x : Space d => x.coord i) := by
   fun_prop
 
 lemma coordCLM_apply (μ : Fin d) (p : Space d) :
-    coordCLM μ p = coord μ p := by
-  rfl
+    coordCLM μ p = coord μ p := rfl
 
 @[inherit_doc coord]
 scoped notation "𝔁" => coord
@@ -652,12 +597,10 @@ noncomputable def oneEquiv : Space 1 ≃ₗᵢ[ℝ] ℝ where
     exact Eq.symm (Real.sqrt_sq_eq_abs (x 0))
 
 lemma oneEquiv_coe :
-    (oneEquiv : Space 1 → ℝ) = fun x => x 0 := by
-  rfl
+    (oneEquiv : Space 1 → ℝ) = fun x => x 0 := rfl
 
 lemma oneEquiv_symm_coe :
-    (oneEquiv.symm : ℝ → Space 1) = (fun x => ⟨fun _ => x⟩) := by
-  rfl
+    (oneEquiv.symm : ℝ → Space 1) = (fun x => ⟨fun _ => x⟩) := rfl
 
 lemma oneEquiv_symm_apply (x : ℝ) (i : Fin 1) :
     oneEquiv.symm x i = x := by
diff --git a/Physlib/SpaceAndTime/Space/Norm/Basic.lean b/Physlib/SpaceAndTime/Space/Norm/Basic.lean
index 99f6cb005..db6c0bd0d 100644
--- a/Physlib/SpaceAndTime/Space/Norm/Basic.lean
+++ b/Physlib/SpaceAndTime/Space/Norm/Basic.lean
@@ -159,8 +159,7 @@ open Space
 
 lemma deriv_normPowerSeries {d} (n : ℕ) (x : Space d) (i : Fin d) :
     ∂[i] (normPowerSeries n) x = x i * (normPowerSeries n x)⁻¹ := by
-  rw [deriv_eq_fderiv_basis]
-  rw [normPowerSeries_eq]
+  rw [deriv_eq_fderiv_basis, normPowerSeries_eq]
   rw [fderiv_sqrt]
   simp only [one_div, mul_inv_rev, fderiv_add_const, FunLike.coe_smul, Pi.smul_apply,
     smul_eq_mul]
@@ -279,8 +278,7 @@ lemma normPowerSeries_zpow_le_norm_sq_add_one {d} (n : ℕ) (m : ℤ) (x : Space
   | .ofNat m =>
     trans (‖x‖ + 1) ^ m
     · simp
-      refine pow_le_pow_left₀ (by simp) ?_ m
-      exact normPowerSeries_le_norm_sq_add_one n x
+      exact pow_le_pow_left₀ (by simp) (normPowerSeries_le_norm_sq_add_one n x) m
     · simp
   | .negSucc m =>
     trans (‖x‖ ^ (m + 1))⁻¹; swap
@@ -289,8 +287,7 @@ lemma normPowerSeries_zpow_le_norm_sq_add_one {d} (n : ℕ) (m : ℤ) (x : Space
     simp only [zpow_negSucc]
     refine inv_anti₀ ?_ ?_
     · positivity
-    refine pow_le_pow_left₀ (by simp) ?_ (m + 1)
-    exact norm_le_normPowerSeries n x
+    exact pow_le_pow_left₀ (by simp) (norm_le_normPowerSeries n x) (m + 1)
 
 lemma normPowerSeries_inv_le {d} (n : ℕ) (x : Space d) (hx : x ≠ 0) :
     (normPowerSeries n x)⁻¹ ≤ ‖x‖⁻¹ := by
@@ -448,8 +445,7 @@ lemma deriv_normPowerSeries_zpow {d : ℕ} {n : ℕ} (m : ℤ) (x : Space d) (i
   simp only [basis_inner]
   field_simp
   ring_nf
-  have h1 : normPowerSeries n x ^ (-1 + m) = normPowerSeries n x ^ ((-2 + m) + 1) := by
-    ring_nf
+  have h1 : normPowerSeries n x ^ (-1 + m) = normPowerSeries n x ^ ((-2 + m) + 1) := by ring_nf
   rw [h1, zpow_add₀]
   simp only [Int.reduceNeg, zpow_one]
   ring
@@ -510,8 +506,7 @@ lemma gradient_dist_normPowerSeries_zpow {d : ℕ} {n : ℕ} (m : ℤ) :
   simp [distGrad_inner_eq]
   rw [Distribution.fderivD_apply, distOfFunction_apply, distOfFunction_inner]
   calc _
-    _ = - ∫ (x : Space d), fderiv ℝ η x (basis.repr.symm y) * normPowerSeries n x ^ m := by
-      rfl
+    _ = - ∫ (x : Space d), fderiv ℝ η x (basis.repr.symm y) * normPowerSeries n x ^ m := rfl
     _ = ∫ (x : Space d), η x * fderiv ℝ (normPowerSeries n · ^ m) x (basis.repr.symm y) := by
       rw [integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable]
       · fun_prop
@@ -736,8 +731,7 @@ lemma gradient_dist_normPowerSeries_log {d : ℕ} {n : ℕ} :
   simp [distGrad_inner_eq]
   rw [Distribution.fderivD_apply, distOfFunction_apply, distOfFunction_inner]
   calc _
-    _ = - ∫ (x : Space d), fderiv ℝ η x (basis.repr.symm y) * Real.log (normPowerSeries n x) := by
-      rfl
+    _ = - ∫ (x : Space d), fderiv ℝ η x (basis.repr.symm y) * Real.log (normPowerSeries n x) := rfl
     _ = ∫ (x : Space d), η x *
         fderiv ℝ (fun x => Real.log (normPowerSeries n x)) x (basis.repr.symm y) := by
       rw [integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable]
@@ -808,8 +802,7 @@ lemma gradient_dist_normPowerSeries_log_tendsTo_distGrad_norm {d : ℕ} (hd : 2
     intro x hx
     simp at hx
     simp
-    apply mul_le_mul (by rfl) _ (by positivity) (by positivity)
-    exact normPowerSeries_log_le n x hx
+    exact mul_le_mul (by rfl) (normPowerSeries_log_le n x hx) (by positivity) (by positivity)
   · rw [Filter.eventually_iff_exists_mem]
     use {0}ᶜ
     constructor
@@ -996,8 +989,7 @@ private lemma radial_power_deriv_integral_by_parts
           simp [fderiv_smul_const]
         rw [iteratedFDeriv_succ_const]
         rfl) (by use 1, 1; simp [norm_smul]) η
-  have hη'_apply (x : ℝ) : η' x = η (x • n.1) := by
-    simp [η']
+  have hη'_apply (x : ℝ) : η' x = η (x • n.1) := by simp [η']
   have hmul_iter_apply :
       ∀ k x, ((Physlib.Distribution.powOneMul ℝ)^[k] η') x = x ^ k * η' x := by
     intro k
@@ -1028,8 +1020,7 @@ private lemma radial_power_deriv_integral_by_parts
           ∂(.comap Subtype.val volume)
         =
         ∫ (x : ℝ) in Set.Ioi (0 : ℝ),
-          x ^ (p - 1) * η (x • n.1) := by
-    exact MeasureTheory.integral_subtype_comap (μ := volume)
+          x ^ (p - 1) * η (x • n.1) := MeasureTheory.integral_subtype_comap (μ := volume)
       (s := Set.Ioi (0 : ℝ)) measurableSet_Ioi
       (fun x : ℝ => x ^ (p - 1) * η (x • n.1))
   rw [hleft_subtype, hright_subtype]
@@ -1074,9 +1065,7 @@ private lemma radial_power_deriv_integral_by_parts
         ring_nf
         simp [hη'_apply, mul_assoc])
         measurableSet_Ioi
-    · have hcont :
-          ContinuousAt (fun x : ℝ => x ^ p * η (x • n.1)) (0 : ℝ) := by
-        fun_prop
+    · have hcont : ContinuousAt (fun x : ℝ => x ^ p * η (x • n.1)) (0 : ℝ) := by fun_prop
       have hlim := tendsto_nhdsWithin_of_tendsto_nhds
         (s := Set.Ioi (0 : ℝ)) hcont.tendsto
       simp only [ne_eq, hp.ne', not_false_eq_true, zero_pow, zero_smul, zero_mul] at hlim
@@ -1121,8 +1110,7 @@ private lemma distDiv_norm_zpow_smul_repr_self_apply_eq_radial_deriv
   calc
     (∇ᵈ ⬝ (distOfFunction (fun x : Space d => ‖x‖ ^ q • basis.repr x)
       (IsDistBounded.zpow_smul_repr_self q (by omega)))) η
-        = - ∫ x, F x := by
-            rw [distDiv_ofFunction]
+        = - ∫ x, F x := by rw [distDiv_ofFunction]
     _ = - ∫ r, F (r.2.1 • r.1.1)
         ∂(volume (α := Space d).toSphere.prod
           (Measure.volumeIoiPow (Module.finrank ℝ (Space d) - 1))) := by
@@ -1165,13 +1153,9 @@ lemma distDiv_norm_zpow_smul_repr_self_eq_smul
           (IsDistBounded.pow q (by omega))) := by
   ext η
   let p : ℕ := Int.toNat (q + (d : ℤ))
-  have hp_int : (p : ℤ) = q + (d : ℤ) := by
-    dsimp [p]
-    exact Int.toNat_of_nonneg (le_of_lt hq)
+  have hp_int : (p : ℤ) = q + (d : ℤ) := by simpa [p] using Int.toNat_of_nonneg (le_of_lt hq)
   have hp_pos : 0 < p := by
-    have : (0 : ℤ) < (p : ℤ) := by
-      rw [hp_int]
-      exact hq
+    have : (0 : ℤ) < (p : ℤ) := by simpa [hp_int] using hq
     exact_mod_cast this
   have hcoef : (((q + d : ℤ) : ℝ)) = (p : ℝ) := by
     exact_mod_cast hp_int.symm
@@ -1446,8 +1430,7 @@ lemma distLaplacian_fundamentalSolution_norm_zpow {d : ℕ} :
             ‖x‖ ^ (- (d : ℤ)) • basis.repr x)
           (IsDistBounded.zpow_smul_repr_self (- (d : ℤ)) (by omega))) =
           (d * (volume (α := Space d)).real
-            (Metric.ball 0 1)) • diracDelta ℝ 0 := by
-      exact distDiv_inv_pow_eq_dim (d := d)
+            (Metric.ball 0 1)) • diracDelta ℝ 0 := distDiv_inv_pow_eq_dim (d := d)
     rw [hdiv]
     rw [smul_smul]
     ring_nf
diff --git a/Physlib/SpaceAndTime/Space/Origin.lean b/Physlib/SpaceAndTime/Space/Origin.lean
index 360d10492..c50edaec0 100644
--- a/Physlib/SpaceAndTime/Space/Origin.lean
+++ b/Physlib/SpaceAndTime/Space/Origin.lean
@@ -33,8 +33,7 @@ lemma zero_val {d : ℕ} : (0 : Space d).val = fun _ => 0 := rfl
 
 @[simp]
 lemma zero_apply {d : ℕ} (i : Fin d) :
-    (0 : Space d) i = 0 := by
-  simp [zero_val]
+    (0 : Space d) i = 0 := rfl
 
 /-- A Euclidean vector, based at the chosen origin, viewed as a point of `Space d`. -/
 noncomputable def vectorToSpace {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : Space d :=
@@ -42,14 +41,11 @@ noncomputable def vectorToSpace {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : Spa
 
 @[simp]
 lemma vectorToSpace_apply {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) (i : Fin d) :
-    vectorToSpace v i = v i := by
-  simp [vectorToSpace]
+    vectorToSpace v i = v i := by simp [vectorToSpace]
 
 @[simp]
 lemma vectorToSpace_vsub_zero {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) :
-    vectorToSpace v -ᵥ (0 : Space d) = v := by
-  ext i
-  simp [vectorToSpace]
+    vectorToSpace v -ᵥ (0 : Space d) = v := by simp [vectorToSpace]
 
 /-- The standard chart `Space d ≃ᵃⁱ[ℝ] EuclideanSpace ℝ (Fin d)`, `p ↦ p -ᵥ 0`, identifying a point
 with its coordinate vector relative to the origin (the vector-space zero `(0 : Space d)`). -/
diff --git a/Physlib/SpaceAndTime/Space/Slice.lean b/Physlib/SpaceAndTime/Space/Slice.lean
index 4487e19d3..c0f1e181f 100644
--- a/Physlib/SpaceAndTime/Space/Slice.lean
+++ b/Physlib/SpaceAndTime/Space/Slice.lean
@@ -105,13 +105,11 @@ lemma slice_symm_apply {d : ℕ} (i : Fin d.succ) (r : ℝ) (x : Space d) :
 
 @[simp]
 lemma slice_symm_apply_self {d : ℕ} (i : Fin d.succ) (r : ℝ) (x : Space d) :
-    (slice i).symm (r, x) i = r := by
-  simp [slice_symm_apply]
+    (slice i).symm (r, x) i = r := by simp [slice_symm_apply]
 
 @[simp]
 lemma slice_symm_apply_succAbove {d : ℕ} (i : Fin d.succ) (r : ℝ) (x : Space d) (j : Fin d) :
-    (slice i).symm (r, x) (Fin.succAbove i j) = x j := by
-  simp [slice_symm_apply]
+    (slice i).symm (r, x) (Fin.succAbove i j) = x j := by simp [slice_symm_apply]
 
 /-!
 
@@ -172,8 +170,7 @@ lemma norm_left_le_norm_slice_symm {d : ℕ} (i : Fin d.succ) (r : ℝ) (x : Spa
 
 @[simp]
 lemma fderiv_slice_symm {d : ℕ} (i : Fin d.succ) (p1 : ℝ × Space d) :
-    fderiv ℝ (slice i).symm p1 = (slice i).symm := by
-  rw [ContinuousLinearEquiv.fderiv]
+    fderiv ℝ (slice i).symm p1 = (slice i).symm := by rw [ContinuousLinearEquiv.fderiv]
 
 lemma fderiv_slice_symm_left_apply {d : ℕ} (i : Fin d.succ) (x : Space d) (r1 r2 : ℝ) :
     (fderiv ℝ (fun r => (slice i).symm (r, x))) r1 r2 = (slice i).symm (r2, 0) := by
diff --git a/Physlib/SpaceAndTime/Space/Translations.lean b/Physlib/SpaceAndTime/Space/Translations.lean
index 7ce7c116d..f42cb3d85 100644
--- a/Physlib/SpaceAndTime/Space/Translations.lean
+++ b/Physlib/SpaceAndTime/Space/Translations.lean
@@ -68,8 +68,7 @@ noncomputable def translateSchwartz {d : ℕ} (a : EuclideanSpace ℝ (Fin d)) :
           trans (1 + (‖x‖ + ‖a‖)) + ‖x‖ * ‖a‖
           · simp
             positivity
-          ring_nf
-          rfl) (by
+          nlinarith) (by
           use 1, (1 + ‖a‖)
           intro x
           simp only [pow_one]
@@ -79,8 +78,7 @@ noncomputable def translateSchwartz {d : ℕ} (a : EuclideanSpace ℝ (Fin d)) :
           trans (1 + (‖a‖ + ‖x - basis.repr.symm a‖)) + ‖a‖ * ‖x - basis.repr.symm a‖
           · simp
             positivity
-          ring_nf
-          rfl)
+          nlinarith)
 
 @[simp]
 lemma translateSchwartz_apply {d : ℕ} (a : EuclideanSpace ℝ (Fin d))
@@ -89,9 +87,7 @@ lemma translateSchwartz_apply {d : ℕ} (a : EuclideanSpace ℝ (Fin d))
 
 lemma translateSchwartz_coe_eq {d : ℕ} (a : EuclideanSpace ℝ (Fin d))
     (η : 𝓢(Space d, X)) :
-    (translateSchwartz a η : Space d → X) = fun x => η (x - basis.repr.symm a) := by
-  ext
-  simp
+    (translateSchwartz a η : Space d → X) = fun x => η (x - basis.repr.symm a) := rfl
 
 /-- The continuous linear map translating distributions. -/
 noncomputable def distTranslate {d : ℕ} (a : EuclideanSpace ℝ (Fin d)) :
@@ -100,8 +96,7 @@ noncomputable def distTranslate {d : ℕ} (a : EuclideanSpace ℝ (Fin d)) :
   map_add' T1 T2 := by
     ext η
     simp
-  map_smul' c T := by
-    simp
+  map_smul' c T := by simp
 
 lemma distTranslate_apply {d : ℕ} (a : EuclideanSpace ℝ (Fin d))
     (T : (Space d) →d[ℝ] X) (η : 𝓢(Space d, ℝ)) :
diff --git a/Physlib/SpaceAndTime/SpaceTime/Basic.lean b/Physlib/SpaceAndTime/SpaceTime/Basic.lean
index 3852a03a7..58d321fae 100644
--- a/Physlib/SpaceAndTime/SpaceTime/Basic.lean
+++ b/Physlib/SpaceAndTime/SpaceTime/Basic.lean
@@ -101,27 +101,21 @@ open TensorSpecies
   This is denoted `𝔁 μ p`, where `𝔁` is typed with `\MCx`. -/
 def coord {d : ℕ} (μ : Fin (1 + d)) : SpaceTime d →ₗ[ℝ] ℝ where
   toFun x := x (finSumFinEquiv.symm μ)
-  map_add' x1 x2 := by
-    simp
-  map_smul' c x := by
-    simp
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
 
 @[inherit_doc coord]
 scoped notation "𝔁" => coord
 
 lemma coord_apply {d : ℕ} (μ : Fin (1 + d)) (y : SpaceTime d) :
-    𝔁 μ y = y (finSumFinEquiv.symm μ) := by
-  rfl
+    𝔁 μ y = y (finSumFinEquiv.symm μ) := rfl
 
 /-- The continuous linear map from a point in space time to one of its coordinates. -/
 def coordCLM (μ : Fin 1 ⊕ Fin d) : SpaceTime d →L[ℝ] ℝ where
   toFun x := x μ
-  map_add' x1 x2 := by
-    simp
-  map_smul' c x := by
-    simp
-  cont := by
-    fun_prop
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+  cont := by fun_prop
 
 /-!
 
@@ -204,8 +198,7 @@ def space {d : ℕ} : SpaceTime d →L[ℝ] Space d where
   map_smul' c x := by
     ext i
     simp
-  cont := by
-    fun_prop
+  cont := by fun_prop
 
 /-!
 
@@ -258,8 +251,7 @@ def time {d : ℕ} (c : SpeedOfLight := 1) : SpaceTime d →ₗ[ℝ] Time where
 
 @[simp]
 lemma time_val_toCoord_symm {d : ℕ} (c : SpeedOfLight) (f : Fin 1 ⊕ Fin d → ℝ) :
-    (time c f).val = f (Sum.inl 0) / c := by
-  simp [time, Lorentz.Vector.timeComponent]
+    (time c f).val = f (Sum.inl 0) / c := by simp [time, Lorentz.Vector.timeComponent]
 
 /-!
 
@@ -295,8 +287,7 @@ def toTimeAndSpace {d : ℕ} (c : SpeedOfLight := 1) : SpaceTime d ≃L[ℝ] Tim
         simp
       ext i
       simp
-    map_smul' := by
-      simp
+    map_smul' := by simp
   }
 
 @[simp]
@@ -448,15 +439,13 @@ def timeSpaceBasisEquiv {d : ℕ} (c : SpeedOfLight) :
     match μ with
     | Sum.inl 0 =>
       field_simp
-    | Sum.inr i =>
-      rfl
+    | Sum.inr i => rfl
   right_inv x := by
     funext μ
     match μ with
     | Sum.inl 0 =>
       field_simp
-    | Sum.inr i =>
-      rfl
+    | Sum.inr i => rfl
   map_add' x y := by
     funext μ
     match μ with
@@ -584,8 +573,7 @@ lemma toTimeAndSpace_symm_measurePreserving {d : ℕ} (c : SpeedOfLight) :
   · fun_prop
   rw [Space.volume_eq_addHaar, Time.volume_eq_basis_addHaar, ← Module.Basis.prod_addHaar,
     Module.Basis.map_addHaar]
-  rw [← timeSpaceBasis_addHaar c]
-  rfl
+  exact timeSpaceBasis_addHaar c
 
 /-!
 
@@ -604,10 +592,8 @@ lemma spaceTime_integral_eq_time_space_integral {M} [NormedAddCommGroup M]
     apply MeasureTheory.MeasurePreserving.integral_comp
     · refine { measurable := ?_, map_eq := ?_ }
       · fun_prop
-      have hs : volume (α := Space d) = Space.basis.toBasis.addHaar := by
-        exact Space.volume_eq_addHaar
-      have ht : volume (α := Time) = Time.basis.toBasis.addHaar := by
-        exact Time.volume_eq_basis_addHaar
+      have hs : volume (α := Space d) = Space.basis.toBasis.addHaar := Space.volume_eq_addHaar
+      have ht : volume (α := Time) = Time.basis.toBasis.addHaar := Time.volume_eq_basis_addHaar
       rw [hs, ht]
       rw [← Module.Basis.prod_addHaar]
       rw [Module.Basis.map_addHaar]
diff --git a/Physlib/SpaceAndTime/SpaceTime/Derivatives.lean b/Physlib/SpaceAndTime/SpaceTime/Derivatives.lean
index d7de1137f..3e88e2bd0 100644
--- a/Physlib/SpaceAndTime/SpaceTime/Derivatives.lean
+++ b/Physlib/SpaceAndTime/SpaceTime/Derivatives.lean
@@ -88,8 +88,7 @@ scoped notation "∂_" => deriv
 lemma deriv_eq {M : Type} [AddCommGroup M] [Module ℝ M] [TopologicalSpace M]
     {d : ℕ} (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → M) (y : SpaceTime d) :
     ∂_ μ f y =
-    fderiv ℝ f y (Lorentz.Vector.basis μ) := by
-  rfl
+    fderiv ℝ f y (Lorentz.Vector.basis μ) := rfl
 
 /-- The derivative of a function from `SpaceTime d` to a manifold along the `μ`
 coordinate, as a tangent vector at the value of the function. -/
@@ -244,9 +243,8 @@ lemma deriv_commute {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : 
   simp only [fderiv_fun_const, Pi.ofNat_apply, ContinuousLinearMap.comp_zero, zero_add,
     ContinuousLinearMap.flip_apply]
   rw [IsSymmSndFDerivAt.eq]
-  · apply ContDiffAt.isSymmSndFDerivAt
-    exact hf.contDiffAt
-    simp only [minSmoothness_of_isRCLikeNormedField, le_refl]
+  · exact hf.contDiffAt.isSymmSndFDerivAt
+      (by simp only [minSmoothness_of_isRCLikeNormedField, le_refl])
   · have h1 := hf.differentiable (by norm_cast); fun_prop
   · fun_prop
   · have h1 := hf.differentiable (by norm_cast); fun_prop
@@ -374,10 +372,8 @@ noncomputable def distDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
   toFun f :=
     let ev : (SpaceTime d →L[ℝ] M) →L[ℝ] M := {
       toFun v := v (Lorentz.Vector.basis μ)
-      map_add' v1 v2 := by
-        simp only [_root_.add_apply]
-      map_smul' a v := by
-        simp
+      map_add' _ _ := rfl
+      map_smul' _ _ := rfl
     }
     ev.comp (Distribution.fderivD ℝ f)
   map_add' f1 f2 := by
diff --git a/Physlib/SpaceAndTime/SpaceTime/LorentzAction.lean b/Physlib/SpaceAndTime/SpaceTime/LorentzAction.lean
index a84380c9a..9ba4c507f 100644
--- a/Physlib/SpaceAndTime/SpaceTime/LorentzAction.lean
+++ b/Physlib/SpaceAndTime/SpaceTime/LorentzAction.lean
@@ -81,8 +81,7 @@ def schwartzAction {d} : LorentzGroup d →* 𝓢(SpaceTime d, ℝ) →L[ℝ] 
     ext η x
     simp only [_root_.mul_inv_rev, compCLM_apply, Function.comp_apply,
       Lorentz.Vector.actionCLM_apply]
-    rw [SemigroupAction.mul_smul]
-    rfl
+    exact congr_arg η (SemigroupAction.mul_smul Λ₂⁻¹ Λ₁⁻¹ x)
 
 /-!
 
@@ -93,8 +92,7 @@ def schwartzAction {d} : LorentzGroup d →* 𝓢(SpaceTime d, ℝ) →L[ℝ] 
 lemma schwartzAction_mul_apply {d} (Λ₁ Λ₂ : LorentzGroup d)
     (η : 𝓢(SpaceTime d, ℝ)) :
     schwartzAction Λ₂ (schwartzAction (Λ₁) η) =
-    schwartzAction (Λ₂ * Λ₁) η := by
-  simp
+    schwartzAction (Λ₂ * Λ₁) η := by simp
 
 lemma schwartzAction_apply {d} (Λ : LorentzGroup d)
     (η : 𝓢(SpaceTime d, ℝ)) (x : SpaceTime d) :
diff --git a/Physlib/SpaceAndTime/SpaceTime/TimeSlice.lean b/Physlib/SpaceAndTime/SpaceTime/TimeSlice.lean
index 077a59bc5..e4efdaa9f 100644
--- a/Physlib/SpaceAndTime/SpaceTime/TimeSlice.lean
+++ b/Physlib/SpaceAndTime/SpaceTime/TimeSlice.lean
@@ -135,9 +135,8 @@ noncomputable def distTimeSlice {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     congr
     ext x
     simp
-  map_add' f1 f2 := by
-    simp
-  map_smul' a f := by simp
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
   continuous_toFun := ((compCLMOfContinuousLinearEquiv ℝ (toTimeAndSpace c)).precomp M).continuous
   continuous_invFun :=
     ((compCLMOfContinuousLinearEquiv ℝ (toTimeAndSpace c).symm).precomp M).continuous
@@ -179,8 +178,7 @@ lemma distTimeSlice_distDeriv_inl {M d} [NormedAddCommGroup M] [NormedSpace ℝ
   rw [toTimeAndSpace_basis_inl']
   rw [map_smul]
   simp only [one_div, smul_eq_mul]
-  · apply Differentiable.differentiableAt
-    exact SchwartzMap.differentiable κ
+  · exact (SchwartzMap.differentiable κ).differentiableAt
   · fun_prop
 
 lemma distDeriv_inl_distTimeSlice_symm {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
@@ -220,8 +218,7 @@ lemma distTimeSlice_distDeriv_inr {M d} [NormedAddCommGroup M] [NormedSpace ℝ
   simp only [toTimeAndSpace_fderiv, ContinuousLinearMap.coe_comp, ContinuousLinearEquiv.coe_coe,
     Function.comp_apply]
   rw [toTimeAndSpace_basis_inr]
-  · apply Differentiable.differentiableAt
-    exact SchwartzMap.differentiable κ
+  · exact (SchwartzMap.differentiable κ).differentiableAt
   · fun_prop
 
 lemma distDeriv_inr_distTimeSlice_symm {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
diff --git a/Physlib/SpaceAndTime/Time/Basic.lean b/Physlib/SpaceAndTime/Time/Basic.lean
index 20e3fbe16..d3fe73bfe 100644
--- a/Physlib/SpaceAndTime/Time/Basic.lean
+++ b/Physlib/SpaceAndTime/Time/Basic.lean
@@ -80,10 +80,8 @@ structure Time where
 
 namespace Time
 
-lemma val_injective : Function.Injective val := by
-  intro t1 t2 h
-  ext
-  exact h
+lemma val_injective : Function.Injective val :=
+  fun _ _ h => Time.ext h
 
 /-!
 
@@ -128,8 +126,7 @@ lemma zero_val : val 0 = 0 := by
   norm_cast
 
 @[simp]
-lemma eq_zero_iff (t : Time) : t = 0 ↔ t.val = 0 := by
-  aesop
+lemma eq_zero_iff (t : Time) : t = 0 ↔ t.val = 0 := by aesop
 
 @[simp]
 lemma one_val : val 1 = 1 := by
@@ -137,8 +134,7 @@ lemma one_val : val 1 = 1 := by
   norm_cast
 
 @[simp]
-lemma eq_one_iff (t : Time) : t = 1 ↔ t.val = 1 := by
-  aesop
+lemma eq_one_iff (t : Time) : t = 1 ↔ t.val = 1 := by aesop
 
 /-!
 
@@ -188,14 +184,7 @@ instance : PartialOrder Time where
   le_antisymm t1 t2 h1 h2 := by simp_all [le_def]; ext; exact le_antisymm h1 h2
 
 lemma lt_def (t1 t2 : Time) :
-    t1 < t2 ↔ t1.val < t2.val := by
-  constructor
-  · intro h
-    exact lt_iff_le_not_ge.mpr h
-  · intro h
-    apply lt_iff_le_not_ge.mpr
-    simp_all [le_def]
-    apply le_of_lt h
+    t1 < t2 ↔ t1.val < t2.val := Iff.symm lt_iff_le_not_ge
 
 /-!
 
@@ -371,9 +360,7 @@ noncomputable def basis : OrthonormalBasis (Fin 1) ℝ Time where
   repr := {
     toFun := fun x => WithLp.toLp 2 (fun _ => x)
     invFun := fun f => ⟨f 0⟩
-    left_inv := by
-      intro x
-      rfl
+    left_inv := fun _ => rfl
     right_inv := by
       intro f
       ext i
@@ -395,8 +382,7 @@ noncomputable def basis : OrthonormalBasis (Fin 1) ℝ Time where
       rw [@PiLp.norm_eq_of_L2]
       simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Real.norm_eq_abs, sq_abs,
         Finset.sum_const, Finset.card_singleton, one_smul]
-      rw [Real.sqrt_sq_eq_abs]
-      rfl
+      exact Real.sqrt_sq_eq_abs _
   }
 
 @[simp]
@@ -428,13 +414,11 @@ lemma finRank_eq_one : Module.finrank ℝ Time = 1 := by
     ext
     simp [one_val]
 
-instance : FiniteDimensional ℝ Time := by
-  refine Module.finite_of_rank_eq_one ?_
-  simp
+instance : FiniteDimensional ℝ Time :=
+  Module.finite_of_rank_eq_one rank_eq_one
 
 lemma volume_eq_basis_addHaar :
-    (volume (α := Time)) = basis.toBasis.addHaar := by
-  exact (OrthonormalBasis.addHaar_eq_volume _).symm
+    (volume (α := Time)) = basis.toBasis.addHaar := (OrthonormalBasis.addHaar_eq_volume _).symm
 
 /-!
 
@@ -468,14 +452,10 @@ noncomputable def toRealLIE : Time ≃ₗᵢ[ℝ] ℝ where
   right_inv x := by rfl
   map_add' := by simp
   map_smul' := by simp
-  norm_map' x := by
-    simp
-    rfl
+  norm_map' _ := rfl
 
 lemma eq_one_smul (t : Time) :
-    t = t.val • 1 := by
-  ext
-  simp [one_val]
+    t = t.val • 1 := Time.ext (by simp [one_val])
 
 @[fun_prop]
 lemma val_measurable : Measurable Time.val := by
diff --git a/Physlib/SpaceAndTime/Time/Derivatives.lean b/Physlib/SpaceAndTime/Time/Derivatives.lean
index 00c2dd385..5f3eb6b6e 100644
--- a/Physlib/SpaceAndTime/Time/Derivatives.lean
+++ b/Physlib/SpaceAndTime/Time/Derivatives.lean
@@ -100,16 +100,14 @@ lemma deriv_eq_mfderiv [NormedAddCommGroup M] [NormedSpace ℝ M]
 open Manifold in
 lemma deriv_eq_manifoldDeriv [NormedAddCommGroup M] [NormedSpace ℝ M]
     (f : Time → M) (t : Time) :
-    deriv f t = manifoldDeriv 𝓘(ℝ, M) f t := by
-  rw [deriv_eq_mfderiv, manifoldDeriv_eq]
+    deriv f t = manifoldDeriv 𝓘(ℝ, M) f t := by rw [deriv_eq_mfderiv, manifoldDeriv_eq]
 
 open Manifold in
 @[simp]
 lemma manifoldDeriv_const {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]
     [ChartedSpace H N] (n : N) :
-    manifoldDeriv I (fun _ : Time => n) t = 0 := by
-  simp [manifoldDeriv]
+    manifoldDeriv I (fun _ : Time => n) t = 0 := by simp [manifoldDeriv]
 
 /-!
 
@@ -120,14 +118,10 @@ lemma manifoldDeriv_const {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ
 lemma deriv_smul (f : Time → EuclideanSpace ℝ (Fin d)) (k : ℝ)
     (hf : Differentiable ℝ f) :
     ∂ₜ (fun t => k • f t) t = k • ∂ₜ (fun t => f t) t := by
-  rw [deriv, fderiv_fun_const_smul]
-  rfl
-  fun_prop
+  exact congrArg (fun g => g 1) (fderiv_fun_const_smul (c := k) hf.differentiableAt)
 
 lemma deriv_neg [NormedAddCommGroup M] [NormedSpace ℝ M] (f : Time → M) :
-    ∂ₜ (-f) t = -∂ₜ f t := by
-  rw [deriv, fderiv_neg]
-  rfl
+    ∂ₜ (-f) t = -∂ₜ f t := by simp [deriv, fderiv_neg]
 
 /-- Quotient rule for `Time.deriv` on real-valued functions: if `c` and `g` are
   differentiable at `t` and `g t ≠ 0`, then
@@ -229,7 +223,6 @@ lemma deriv_lorentzVector {d : ℕ} {f : Time → Lorentz.Vector d}
   change fderiv ℝ (Lorentz.Vector.coordCLM i ∘ fun x => f x) t 1 = _
   rw [fderiv_comp]
   · simp
-    rw [← deriv_eq]
     rfl
   · fun_prop
   · fun_prop
diff --git a/Physlib/SpaceAndTime/Time/TimeMan.lean b/Physlib/SpaceAndTime/Time/TimeMan.lean
index 772d26586..bf94d4044 100644
--- a/Physlib/SpaceAndTime/Time/TimeMan.lean
+++ b/Physlib/SpaceAndTime/Time/TimeMan.lean
@@ -71,8 +71,7 @@ lemma val_injective : Function.Injective TimeMan.val := by
 
 lemma val_isOpenEmbedding : Topology.IsOpenEmbedding TimeMan.val where
   eq_induced := rfl
-  isOpen_range := by
-    simp
+  isOpen_range := by simp
   injective := val_injective
 
 lemma isOpen_iff {s : Set TimeMan} :
@@ -83,13 +82,8 @@ lemma isOpen_iff {s : Set TimeMan} :
 def valHomeomorphism : TimeMan ≃ₜ ℝ where
   toFun := TimeMan.val
   invFun := fun t => { val := t }
-  left_inv := by
-    intro t
-    cases t
-    rfl
-  right_inv := by
-    intro t
-    rfl
+  left_inv := fun ⟨_⟩ => rfl
+  right_inv := fun _ => rfl
   continuous_toFun := by fun_prop
   continuous_invFun := by
     refine { isOpen_preimage := ?_ }
@@ -112,8 +106,7 @@ def valHomeomorphism : TimeMan ≃ₜ ℝ where
 instance : ChartedSpace ℝ TimeMan where
   atlas := { valHomeomorphism.toOpenPartialHomeomorph }
   chartAt _ := valHomeomorphism.toOpenPartialHomeomorph
-  mem_chart_source := by
-    simp
+  mem_chart_source := by simp
   chart_mem_atlas := by
     intro x
     simp
@@ -128,17 +121,14 @@ instance : IsManifold 𝓘(ℝ, ℝ) ω TimeMan where
     subst h1 h2
     exact symm_trans_mem_contDiffGroupoid valHomeomorphism.toOpenPartialHomeomorph
 
-lemma val_contDiff : ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) ω TimeMan.val := by
-  refine contMDiffOn_univ.mp ?_
-  exact contMDiffOn_chart (x := (⟨0⟩ : TimeMan))
+lemma val_contDiff : ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) ω TimeMan.val :=
+  contMDiffOn_univ.mp (contMDiffOn_chart (x := (⟨0⟩ : TimeMan)))
 
 /-- The choice of map `Time.val` from `TimeMan` to `ℝ` as a diffeomorphism. -/
 noncomputable def valDiffeomorphism : TimeMan ≃ₘ^ω⟮𝓘(ℝ, ℝ), 𝓘(ℝ, ℝ)⟯ ℝ where
   toEquiv := valHomeomorphism.toEquiv
   contMDiff_toFun := val_contDiff
-  contMDiff_invFun := by
-    refine contMDiffOn_univ.mp ?_
-    exact contMDiffOn_chart_symm (x := (⟨0⟩ : TimeMan))
+  contMDiff_invFun := contMDiffOn_univ.mp (contMDiffOn_chart_symm (x := (⟨0⟩ : TimeMan)))
 
 /-!
 
diff --git a/Physlib/SpaceAndTime/Time/TimeTransMan.lean b/Physlib/SpaceAndTime/Time/TimeTransMan.lean
index 9dd46be15..1d9d5a054 100644
--- a/Physlib/SpaceAndTime/Time/TimeTransMan.lean
+++ b/Physlib/SpaceAndTime/Time/TimeTransMan.lean
@@ -84,16 +84,12 @@ lemma val_range : Set.range val = Set.univ := by
 lemma val_inducing : Topology.IsInducing TimeTransMan.val where
   eq_induced := rfl
 
-lemma val_injective : Function.Injective TimeTransMan.val := by
-  intro t1 t2 h
-  cases t1
-  cases t2
-  simp_all
+lemma val_injective : Function.Injective TimeTransMan.val :=
+  fun _ _ h => ext_of h
 
 lemma val_isOpenEmbedding : Topology.IsOpenEmbedding TimeTransMan.val where
   eq_induced := rfl
-  isOpen_range := by
-    simp
+  isOpen_range := by simp
   injective := val_injective
 
 lemma isOpen_iff {s : Set TimeTransMan} :
@@ -104,13 +100,8 @@ lemma isOpen_iff {s : Set TimeTransMan} :
 def valHomeomorphism : TimeTransMan ≃ₜ ℝ where
   toFun := TimeTransMan.val
   invFun := fun t => { val := t }
-  left_inv := by
-    intro t
-    cases t
-    rfl
-  right_inv := by
-    intro t
-    rfl
+  left_inv := fun ⟨_⟩ => rfl
+  right_inv := fun _ => rfl
   continuous_toFun := by fun_prop
   continuous_invFun := by
     refine { isOpen_preimage := ?_ }
@@ -133,8 +124,7 @@ def valHomeomorphism : TimeTransMan ≃ₜ ℝ where
 instance : ChartedSpace ℝ TimeTransMan where
   atlas := { valHomeomorphism.toOpenPartialHomeomorph }
   chartAt _ := valHomeomorphism.toOpenPartialHomeomorph
-  mem_chart_source := by
-    simp
+  mem_chart_source := by simp
   chart_mem_atlas := by
     intro x
     simp
@@ -149,17 +139,14 @@ instance : IsManifold 𝓘(ℝ, ℝ) ω TimeTransMan where
     subst h1 h2
     exact symm_trans_mem_contDiffGroupoid valHomeomorphism.toOpenPartialHomeomorph
 
-lemma val_contDiff : ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) ω TimeTransMan.val := by
-  refine contMDiffOn_univ.mp ?_
-  exact contMDiffOn_chart (x := (⟨0⟩ : TimeTransMan))
+lemma val_contDiff : ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) ω TimeTransMan.val :=
+  contMDiffOn_univ.mp (contMDiffOn_chart (x := (⟨0⟩ : TimeTransMan)))
 
 /-- The choice of map `Time.val` from `TimeTransMan` to `ℝ` as a diffeomorphism. -/
 noncomputable def valDiffeomorphism : TimeTransMan ≃ₘ^ω⟮𝓘(ℝ, ℝ), 𝓘(ℝ, ℝ)⟯ ℝ where
   toEquiv := valHomeomorphism.toEquiv
   contMDiff_toFun := val_contDiff
-  contMDiff_invFun := by
-    refine contMDiffOn_univ.mp ?_
-    exact contMDiffOn_chart_symm (x := (⟨0⟩ : TimeTransMan))
+  contMDiff_invFun := contMDiffOn_univ.mp (contMDiffOn_chart_symm (x := (⟨0⟩ : TimeTransMan)))
 
 /-!
 
@@ -175,14 +162,8 @@ lemma vadd_val (p : ℝ) (t : TimeTransMan) :
     (p +ᵥ t).val = p + t.val := rfl
 
 instance : AddAction ℝ TimeTransMan where
-  zero_vadd t := by
-    cases t
-    ext
-    simp
-  add_vadd p1 p2 t := by
-    ext
-    simp only [vadd_val]
-    ring
+  zero_vadd t := ext_of (by simp)
+  add_vadd p1 p2 t := ext_of (by simp [add_assoc])
 
 /-!
 
@@ -235,8 +216,7 @@ lemma diff_eq_val (x : TimeUnit) (t1 t2 : TimeTransMan) :
 
 @[simp]
 lemma diff_self (x : TimeUnit) (t : TimeTransMan) :
-    diff x t t = 0 := by
-  simp [diff_eq_val]
+    diff x t t = 0 := by simp [diff_eq_val]
 
 lemma diff_fst_injective (x : TimeUnit) (t : TimeTransMan) : Function.Injective (diff x · t) := by
   intro t1 t2 h
@@ -307,8 +287,7 @@ lemma addTime_eq_val (x : TimeUnit) (r : ℝ) (t : TimeTransMan) :
   simp [diff_eq_val]
 
 lemma addTime_val (x : TimeUnit) (r : ℝ) (t : TimeTransMan) :
-    (addTime x r t).val = x.1 * r + t.val := by
-  rw [addTime_eq_val]
+    (addTime x r t).val = x.1 * r + t.val := by rw [addTime_eq_val]
 
 /-!
 
@@ -329,10 +308,8 @@ noncomputable def neg (zero : TimeTransMan) (t : TimeTransMan) : TimeTransMan :=
   negMetric zero default t
 
 lemma neg_eq_negMetric (zero : TimeTransMan) (x : TimeUnit) (t : TimeTransMan) :
-    neg zero t = negMetric zero x t := by
-  simp [neg, negMetric]
-  ext
-  simp [addTime_val, diff_eq_val]
+    neg zero t = negMetric zero x t :=
+  ext_of (by simp [neg, negMetric, addTime_val, diff_eq_val])
 
 /-!
 
@@ -375,23 +352,18 @@ noncomputable def toTime (zero : TimeTransMan) (x : TimeUnit) : TimeTransMan ≃
 
 @[simp]
 lemma toTime_zero (zero : TimeTransMan) (x : TimeUnit) :
-    toTime zero x zero = 0 := by
-  ext
-  simp [toTime, diff_eq_val]
+    toTime zero x zero = 0 := Time.ext (by simp [toTime, diff_eq_val])
 
 @[simp]
 lemma toTime_symm_zero_add (zero : TimeTransMan) (x : TimeUnit) :
-    (toTime zero x).symm 0 = zero := by
-  ext
-  simp [toTime, addTime_val, diff_eq_val]
+    (toTime zero x).symm 0 = zero :=
+  ext_of (by simp [toTime, addTime_val, diff_eq_val])
 
 lemma toTime_val (zero : TimeTransMan) (x : TimeUnit) (t : TimeTransMan) :
     (toTime zero x t).val = diff x t zero := by rfl
 
 lemma toTime_symm_val (zero : TimeTransMan) (x : TimeUnit) (r : Time) :
-    (toTime zero x).symm r = addTime x r zero := by
-  ext
-  simp [toTime, addTime_val, diff_eq_val]
+    (toTime zero x).symm r = addTime x r zero := rfl
 
 @[simp]
 lemma toTime_addTime (zero : TimeTransMan) (x : TimeUnit) (r : ℝ) (τ : TimeTransMan) :
@@ -430,9 +402,8 @@ lemma toTime_neg (zero : TimeTransMan) (x : TimeUnit) (t : TimeTransMan) :
   ring
 
 lemma toTime_symm_neg (zero : TimeTransMan) (x : TimeUnit) (t : Time) :
-    (toTime zero x).symm (- t) = neg zero ((toTime zero x).symm t) := by
-  ext
-  simp [toTime_symm_val, addTime_val, diff_eq_val, neg, negMetric]
+    (toTime zero x).symm (- t) = neg zero ((toTime zero x).symm t) :=
+  ext_of (by simp [toTime_symm_val, addTime_val, diff_eq_val, neg, negMetric])
 
 lemma toTime_symm_sub (zero : TimeTransMan) (x : TimeUnit)
     (t1 t2 : Time) : (toTime zero x).symm (t1 - t2) =
diff --git a/Physlib/SpaceAndTime/Time/TimeUnit.lean b/Physlib/SpaceAndTime/Time/TimeUnit.lean
index 2da8bd01b..e88c95027 100644
--- a/Physlib/SpaceAndTime/Time/TimeUnit.lean
+++ b/Physlib/SpaceAndTime/Time/TimeUnit.lean
@@ -41,8 +41,7 @@ structure TimeUnit : Type where
 namespace TimeUnit
 
 @[simp]
-lemma val_ne_zero (x : TimeUnit) : x.val ≠ 0 := by
-  exact Ne.symm (ne_of_lt x.property)
+lemma val_ne_zero (x : TimeUnit) : x.val ≠ 0 := Ne.symm (ne_of_lt x.property)
 
 lemma val_pos (x : TimeUnit) : 0 < x.val := x.property
 
@@ -75,9 +74,7 @@ lemma div_pos (x y : TimeUnit) : (0 : ℝ≥0) < x/ y := by
 
 @[simp]
 lemma div_self (x : TimeUnit) :
-    x / x = (1 : ℝ≥0) := by
-  simp [div_eq_val, x.val_ne_zero]
-  rfl
+    x / x = (1 : ℝ≥0) := NNReal.eq (_root_.div_self x.val_ne_zero)
 
 lemma div_symm (x y : TimeUnit) :
     x / y = (y / x)⁻¹ := NNReal.eq <| by
@@ -103,12 +100,10 @@ def scale (r : ℝ) (x : TimeUnit) (hr : 0 < r := by norm_num) : TimeUnit :=
 
 @[simp]
 lemma scale_div_self (x : TimeUnit) (r : ℝ) (hr : 0 < r) :
-    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by
-  simp [scale, div_eq_val]
+    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by simp [scale, div_eq_val]
 
 @[simp]
-lemma scale_one (x : TimeUnit) : scale 1 x = x := by
-  simp [scale]
+lemma scale_one (x : TimeUnit) : scale 1 x = x := by simp [scale]
 
 @[simp]
 lemma scale_div_scale (x1 x2 : TimeUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0 < r2) :
@@ -127,8 +122,7 @@ lemma self_div_scale (x : TimeUnit) (r : ℝ) (hr : 0 < r) :
 @[simp]
 lemma scale_scale (x : TimeUnit) (r1 r2 : ℝ) (hr1 : 0 < r1) (hr2 : 0 < r2) :
     scale r1 (scale r2 x hr2) hr1 = scale (r1 * r2) x (mul_pos hr1 hr2) := by
-  simp [scale]
-  ring
+  simp [scale, mul_assoc]
 /-!
 
 ## Specific choices of time units
diff --git a/Physlib/SpaceAndTime/TimeAndSpace/Basic.lean b/Physlib/SpaceAndTime/TimeAndSpace/Basic.lean
index 480e656a9..cae49e6df 100644
--- a/Physlib/SpaceAndTime/TimeAndSpace/Basic.lean
+++ b/Physlib/SpaceAndTime/TimeAndSpace/Basic.lean
@@ -97,14 +97,12 @@ lemma space_apply (tx : TimeAndSpace d) :
 /-- The time projection is nonexpanding for the product metric. -/
 lemma dist_time_le (tx ty : TimeAndSpace d) :
     dist (time tx) (time ty) ≤ dist tx ty := by
-  change dist tx.1 ty.1 ≤ dist tx ty
   rw [Prod.dist_eq]
   exact le_max_left (dist tx.1 ty.1) (dist tx.2 ty.2)
 
 /-- The spatial projection is nonexpanding for the product metric. -/
 lemma dist_space_le (tx ty : TimeAndSpace d) :
     dist (space tx) (space ty) ≤ dist tx ty := by
-  change dist tx.2 ty.2 ≤ dist tx ty
   rw [Prod.dist_eq]
   exact le_max_right (dist tx.1 ty.1) (dist tx.2 ty.2)
 
@@ -373,33 +371,27 @@ noncomputable def distTimeDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
   toFun f :=
     let ev : ((Time × Space d) →L[ℝ] M) →L[ℝ] M := {
       toFun v := v (1, 0)
-      map_add' v1 v2 := by
-        simp only [_root_.add_apply]
-      map_smul' a v := by
-        simp
+      map_add' _ _ := rfl
+      map_smul' _ _ := rfl
     }
     ev.comp (Distribution.fderivD ℝ f)
-  map_add' f1 f2 := by
-    simp
+  map_add' f1 f2 := by simp
   map_smul' a f := by simp
 
 lemma distTimeDeriv_apply {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (f : (Time × Space d) →d[ℝ] M) (ε : 𝓢(Time × Space d, ℝ)) :
-    (distTimeDeriv f) ε = fderivD ℝ f ε (1, 0) := by
-  simp [distTimeDeriv]
+    (distTimeDeriv f) ε = fderivD ℝ f ε (1, 0) := by simp [distTimeDeriv]
 
 lemma distTimeDeriv_apply' {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (f : (Time × Space d) →d[ℝ] M) (ε : 𝓢(Time × Space d, ℝ)) :
     (distTimeDeriv f) ε =
     -f (SchwartzMap.evalCLM ℝ (Time × Space d) ℝ (1, 0)
-    ((fderivCLM ℝ (Time × Space d) ℝ) ε)) := by
-  rw [distTimeDeriv_apply, fderivD_apply]
+    ((fderivCLM ℝ (Time × Space d) ℝ) ε)) := by rw [distTimeDeriv_apply, fderivD_apply]
 
 lemma apply_fderiv_eq_distTimeDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (f : (Time × Space d) →d[ℝ] M) (ε : 𝓢(Time × Space d, ℝ)) :
     f (SchwartzMap.evalCLM ℝ (Time × Space d) ℝ (1, 0) ((fderivCLM ℝ (Time × Space d) ℝ) ε)) =
-    - (distTimeDeriv f) ε := by
-  simp [distTimeDeriv_apply']
+    - (distTimeDeriv f) ε := by simp [distTimeDeriv_apply']
 
 /-!
 
@@ -425,34 +417,28 @@ noncomputable def distSpaceDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M
   toFun f :=
     let ev : (Time × Space d →L[ℝ] M) →L[ℝ] M := {
       toFun v := v (0, basis i)
-      map_add' v1 v2 := by
-        simp only [_root_.add_apply]
-      map_smul' a v := by
-        simp
+      map_add' _ _ := rfl
+      map_smul' _ _ := rfl
     }
     ev.comp (Distribution.fderivD ℝ f)
-  map_add' f1 f2 := by
-    simp
+  map_add' f1 f2 := by simp
   map_smul' a f := by simp
 
 lemma distSpaceDeriv_apply {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (i : Fin d) (f : (Time × Space d) →d[ℝ] M) (ε : 𝓢(Time × Space d, ℝ)) :
-    (distSpaceDeriv i f) ε = fderivD ℝ f ε (0, basis i) := by
-  simp [distSpaceDeriv]
+    (distSpaceDeriv i f) ε = fderivD ℝ f ε (0, basis i) := by simp [distSpaceDeriv]
 
 lemma distSpaceDeriv_apply' {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (i : Fin d) (f : (Time × Space d) →d[ℝ] M) (ε : 𝓢(Time × Space d, ℝ)) :
     (distSpaceDeriv i f) ε =
     - f ((SchwartzMap.evalCLM ℝ (Time × Space d) ℝ (0, basis i))
-    ((fderivCLM ℝ (Time × Space d) ℝ) ε)) := by
-  rw [distSpaceDeriv_apply, fderivD_apply]
+    ((fderivCLM ℝ (Time × Space d) ℝ) ε)) := by rw [distSpaceDeriv_apply, fderivD_apply]
 
 lemma apply_fderiv_eq_distSpaceDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
     (i : Fin d) (f : (Time × Space d) →d[ℝ] M) (ε : 𝓢(Time × Space d, ℝ)) :
     f ((SchwartzMap.evalCLM ℝ (Time × Space d) ℝ (0, basis i))
     ((fderivCLM ℝ (Time × Space d) ℝ) ε)) =
-    - (distSpaceDeriv i f) ε := by
-  simp [distSpaceDeriv_apply']
+    - (distSpaceDeriv i f) ε := by simp [distSpaceDeriv_apply']
 
 /-!
 
@@ -537,8 +523,7 @@ noncomputable def distSpaceGrad {d} :
     simp
 
 lemma distSpaceGrad_apply {d} (f : (Time × Space d) →d[ℝ] ℝ) (ε : 𝓢(Time × Space d, ℝ)) :
-    distSpaceGrad f ε = fun i => distSpaceDeriv i f ε := by
-  rfl
+    distSpaceGrad f ε = fun i => distSpaceDeriv i f ε := rfl
 
 /-!
 
diff --git a/Physlib/SpaceAndTime/TimeAndSpace/ConstantTimeDist.lean b/Physlib/SpaceAndTime/TimeAndSpace/ConstantTimeDist.lean
index 3cb0daf08..56358bdbe 100644
--- a/Physlib/SpaceAndTime/TimeAndSpace/ConstantTimeDist.lean
+++ b/Physlib/SpaceAndTime/TimeAndSpace/ConstantTimeDist.lean
@@ -152,7 +152,6 @@ lemma time_integral_hasFDerivAt {d : ℕ} (η : 𝓢(Time × Space d, ℝ)) (x
     fun x₀ t => fderiv ℝ (fun x : Space d => η (t, x)) x₀
   have hF : ∀ t, ∀ x, HasFDerivAt (F · t) (F' x t) x := by
     intro t x
-    dsimp [F, F']
     refine DifferentiableAt.hasFDerivAt ?_
     have hf := η.smooth'
     apply Differentiable.differentiableAt
@@ -210,8 +209,7 @@ lemma time_integral_hasFDerivAt {d : ℕ} (η : 𝓢(Time × Space d, ℝ)) (x
     · filter_upwards with x
       fun_prop
     · simp [F]
-      have hf : Integrable η (volume.prod volume) := by
-        exact η.integrable
+      have hf : Integrable η (volume.prod volume) := η.integrable
       apply MeasureTheory.Integrable.comp_measurable
       · haveI : (Measure.map (fun t => (t, x₀)) (volume (α := Time))).HasTemperateGrowth := by
           refine { exists_integrable := ?_ }
@@ -273,8 +271,7 @@ lemma time_integral_hasFDerivAt {d : ℕ} (η : 𝓢(Time × Space d, ℝ)) (x
       · rfl
       fun_prop
       fun_prop
-      · apply Differentiable.differentiableAt
-        exact η.smooth'.differentiable (by simp)
+      · exact (η.smooth'.differentiable (by simp)).differentiableAt
       fun_prop
     · apply Integrable.const_mul
       convert hrt using 1
@@ -380,8 +377,7 @@ lemma integrable_fderiv_space {d : ℕ} (η : 𝓢(Time × Space d, ℝ)) (x : S
     · exact hx x t
     · fun_prop
     · fun_prop
-    · apply Differentiable.differentiableAt
-      exact η.smooth'.differentiable (by simp)
+    · exact (η.smooth'.differentiable (by simp)).differentiableAt
     · fun_prop
   rw [← MeasureTheory.integrable_norm_iff]
   apply Integrable.mono' (g := fun t => k * ‖ContinuousLinearMap.prod (0 : Space d →L[ℝ] Time)
@@ -449,8 +445,7 @@ lemma time_integral_contDiff {d : ℕ} (n : ℕ) (η : 𝓢(Time × Space d, ℝ
           ContinuousLinearMap.coe_id', id_eq, SchwartzMap.lineDerivOp_apply_eq_fderiv]
         fun_prop
         fun_prop
-        · apply Differentiable.differentiableAt
-          exact η.smooth'.differentiable (by simp)
+        · exact (η.smooth'.differentiable (by simp)).differentiableAt
         fun_prop
       rw [hl2]
       apply ih
@@ -755,9 +750,7 @@ lemma time_integral_iteratedFDeriv_eq {d : ℕ} (n : ℕ) (η : 𝓢(Time × Spa
       (ContinuousLinearMap.id ℝ (Space d)))) := by
   ext y
   rw [time_integral_iteratedFDeriv_apply]
-  rw [← ContinuousMultilinearMap.integral_apply]
-  rfl
-  exact iteratedFDeriv_integrable η x
+  exact (ContinuousMultilinearMap.integral_apply (iteratedFDeriv_integrable η x) _).symm
 
 /-!
 
@@ -812,8 +805,7 @@ lemma time_integral_mul_pow_iteratedFDeriv_norm_le {d : ℕ} (n m : ℕ) :
           ‖((0 : Space d →L[ℝ] Time).prod (.id ℝ (Space d)))‖ ^ n := by
         refine mul_le_mul_of_nonneg ?_ (by rfl) (by positivity) (by positivity)
         refine integral_mono ?_ ?_ ?_
-        · apply Integrable.const_mul
-          exact iteratedFDeriv_norm_integrable η x
+        · exact (iteratedFDeriv_norm_integrable η x).const_mul (‖x‖ ^ m)
         · exact iteratedFDeriv_norm_mul_pow_integrable n m η x
         · refine Pi.le_def.mpr ?_
           intro t
@@ -825,8 +817,7 @@ lemma time_integral_mul_pow_iteratedFDeriv_norm_le {d : ℕ} (n m : ℕ) :
         refine mul_le_mul_of_nonneg ?_ (by rfl) (by positivity) (by positivity)
         refine integral_mono ?_ ?_ ?_
         · exact iteratedFDeriv_norm_mul_pow_integrable n m η x
-        · apply Integrable.const_mul
-          exact hrt
+        · exact hrt.const_mul k
         · refine Pi.le_def.mpr ?_
           intro t
           convert! hbound t using 1
@@ -934,8 +925,7 @@ lemma constantTime_distSpaceDeriv {M : Type} {d : ℕ} [NormedAddCommGroup M] [N
     ContinuousLinearMap.coe_id', id_eq]
   · fun_prop
   · fun_prop
-  · apply Differentiable.differentiableAt
-    exact η.smooth'.differentiable (by simp)
+  · exact (η.smooth'.differentiable (by simp)).differentiableAt
   · fun_prop
 
 /-!
@@ -965,8 +955,7 @@ lemma constantTime_distSpaceDiv {d : ℕ} (f : (Space d) →d[ℝ] EuclideanSpac
   rw [Space.distSpaceDiv_apply_eq_sum_distSpaceDeriv, Space.distDiv_apply_eq_sum_distDeriv]
   congr
   funext i
-  rw [constantTime_distSpaceDeriv]
-  rfl
+  exact congrArg (fun g => (g η).ofLp i) (constantTime_distSpaceDeriv i f)
 
 /-!
 
@@ -1013,8 +1002,7 @@ lemma constantTime_distTimeDeriv {M : Type} [NormedAddCommGroup M] [NormedSpace
         ContinuousLinearMap.coe_id', id_eq, _root_.zero_apply]
       · fun_prop
       · fun_prop
-      · apply Differentiable.differentiableAt
-        exact η.smooth'.differentiable (by simp)
+      · exact (η.smooth'.differentiable (by simp)).differentiableAt
       · fun_prop
     _ = ∫ (t : Time), (fun t => 1) t * fderiv ℝ (fun t => η (t, x)) t 1 := by simp
     _ = - ∫ (t : Time), fderiv ℝ (fun t => 1) t 1 * (fun t => η (t, x)) t := by
@@ -1025,8 +1013,7 @@ lemma constantTime_distTimeDeriv {M : Type} [NormedAddCommGroup M] [NormedSpace
           simp only [one_mul]
           change (fderiv ℝ (η ∘ fun t => (t, x)) t) 1
           rw [fderiv_comp _ (by
-            apply Differentiable.differentiableAt
-            exact η.smooth'.differentiable (by simp))
+            exact (η.smooth'.differentiable (by simp)).differentiableAt)
             (by fun_prop), DifferentiableAt.fderiv_prodMk (by fun_prop) (by fun_prop)]
           simp only [fderiv_fun_id, fderiv_fun_const, Pi.ofNat_apply,
             ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
diff --git a/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean b/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean
index a7e4575c9..e77aade77 100644
--- a/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean
+++ b/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean
@@ -82,13 +82,11 @@ lemma snd_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :
 
 /-- The Euclidean-group action fixes the time projection. -/
 lemma time_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :
-    time (g • tx) = time tx := by
-  simp [time]
+    time (g • tx) = time tx := by simp [time]
 
 /-- The Euclidean-group action applies to the space projection. -/
 lemma space_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :
-    space (g • tx) = g • space tx := by
-  simp [space]
+    space (g • tx) = g • space tx := by simp [space]
 
 /-- The Euclidean-group action on `TimeAndSpace d` preserves product distance. -/
 lemma dist_smul (g : EuclideanGroup d) (tx ty : TimeAndSpace d) :
diff --git a/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean b/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean
index e4e3b76ef..db3528f69 100644
--- a/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean
+++ b/Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean
@@ -70,8 +70,7 @@ noncomputable def schwartzEuclideanAction {d : ℕ} :
     ext η tx
     simp only [_root_.mul_inv_rev, SchwartzMap.compCLMOfAntilipschitz_apply,
       Function.comp_apply]
-    rw [mul_smul]
-    rfl
+    exact congrArg η (mul_smul _ _ _)
 
 /-- Pointwise formula for the monoid-homomorphism form of the Schwartz-map pullback action. -/
 @[simp]
diff --git a/Physlib/StatisticalMechanics/BoltzmannConstant.lean b/Physlib/StatisticalMechanics/BoltzmannConstant.lean
index 9836dfe8b..212eabbb6 100644
--- a/Physlib/StatisticalMechanics/BoltzmannConstant.lean
+++ b/Physlib/StatisticalMechanics/BoltzmannConstant.lean
@@ -39,7 +39,6 @@ lemma kB_pos : 0 < kB := kBAx.2
 lemma kB_nonneg : 0 ≤ kB := le_of_lt kBAx.2
 
 /-- The Boltzmann constant is not equal to zero. -/
-lemma kB_ne_zero : kB ≠ 0 := by
-  linarith [kB_pos]
+lemma kB_ne_zero : kB ≠ 0 := by linarith [kB_pos]
 
 end Constants
diff --git a/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean b/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean
index ff8e5c1aa..4a7f1aead 100644
--- a/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean
+++ b/Physlib/StatisticalMechanics/CanonicalEnsemble/Basic.lean
@@ -285,8 +285,7 @@ instance [NeZero 𝓒.μ] [NeZero 𝓒1.μ] : NeZero (𝓒 + 𝓒1).μ := by
       (𝓒.μ Set.univ) * (𝓒1.μ Set.univ) := by
     rw [← @Measure.prod_prod]
     simp
-  rw [h1]
-  exact NeZero.pos (𝓒.μ Set.univ * 𝓒1.μ Set.univ)
+  simpa [h1] using NeZero.pos (𝓒.μ Set.univ * 𝓒1.μ Set.univ)
 
 instance μ_neZero_congr [NeZero 𝓒.μ] (e : ι1 ≃ᵐ ι) :
     NeZero (𝓒.congr e).μ := by
@@ -371,26 +370,21 @@ lemma μBolt_ne_zero_of_μ_ne_zero (T : Temperature) (h : 𝓒.μ ≠ 0) :
   simp only [Measure.coe_zero, Pi.zero_apply]
   have hs : {x | ENNReal.ofReal (rexp (-(↑T.β * 𝓒.energy x))) ≠ 0} = Set.univ := by
     ext i
-    simp only [ne_eq, ENNReal.ofReal_eq_zero, not_le, Set.mem_setOf_eq, Set.mem_univ, iff_true]
-    exact exp_pos _
+    simp [ENNReal.ofReal_eq_zero, exp_pos]
   conv =>
     enter [1, s]
     rw [MeasureTheory.withDensity_apply_eq_zero' (by fun_prop), hs]
     simp
   simpa using h
 
-instance (T : Temperature) [NeZero 𝓒.μ] : NeZero (𝓒.μBolt T) := by
-  refine { out := ?_ }
-  apply μBolt_ne_zero_of_μ_ne_zero
-  exact Ne.symm (NeZero.ne' 𝓒.μ)
+instance (T : Temperature) [NeZero 𝓒.μ] : NeZero (𝓒.μBolt T) :=
+  ⟨μBolt_ne_zero_of_μ_ne_zero 𝓒 T (Ne.symm (NeZero.ne' 𝓒.μ))⟩
 
 instance (T : Temperature) [IsFiniteMeasure (𝓒.μBolt T)] [IsFiniteMeasure (𝓒1.μBolt T)] :
-    IsFiniteMeasure ((𝓒 + 𝓒1).μBolt T) := by
-  simp only [μBolt_add]; infer_instance
+    IsFiniteMeasure ((𝓒 + 𝓒1).μBolt T) := by simp only [μBolt_add]; infer_instance
 
 instance (T : Temperature) [IsFiniteMeasure (𝓒.μBolt T)] (n : ℕ) :
-    IsFiniteMeasure ((nsmul n 𝓒).μBolt T) := by
-  simp [μBolt_nsmul]; infer_instance
+    IsFiniteMeasure ((nsmul n 𝓒).μBolt T) := by simp [μBolt_nsmul]; infer_instance
 
 /-!
 
@@ -452,8 +446,7 @@ lemma mathematicalPartitionFunction_eq_zero_iff (T : Temperature) [IsFiniteMeasu
   let s : Set ι := {x | 0 < rexp (-(T.β * 𝓒.energy x))}
   have h : s = Set.univ := by
     ext i
-    simp [s]
-    exact exp_pos (-(T.β * 𝓒.energy i))
+    simpa [s] using exp_pos (-(T.β * 𝓒.energy i))
   change 𝓒.μ s = 0 ↔ 𝓒.μ = 0
   rw [h]
   simp only [Measure.measure_univ_eq_zero]
@@ -470,8 +463,7 @@ lemma mathematicalPartitionFunction_comp_ofβ_apply (β : ℝ≥0) :
 measure is non-zero and the Boltzmann measure is finite. -/
 lemma mathematicalPartitionFunction_pos (T : Temperature)
     [IsFiniteMeasure (𝓒.μBolt T)] [NeZero 𝓒.μ] :
-    0 < 𝓒.mathematicalPartitionFunction T := by
-  simp [mathematicalPartitionFunction]
+    0 < 𝓒.mathematicalPartitionFunction T := by simp [mathematicalPartitionFunction]
 
 open NNReal Constants
 
@@ -494,8 +486,7 @@ lemma probability_add {T : Temperature} (i : ι × ι1) :
 
 @[simp]
 lemma probability_congr (e : ι1 ≃ᵐ ι) (T : Temperature) (i : ι1) :
-    (𝓒.congr e).probability T i = 𝓒.probability T (e i) := by
-  simp [probability]
+    (𝓒.congr e).probability T i = 𝓒.probability T (e i) := by simp [probability]
 
 lemma probability_nsmul (n : ℕ) (T : Temperature) (f : Fin n → ι) :
     (nsmul n 𝓒).probability T f = ∏ i, 𝓒.probability T (f i) := by
@@ -603,8 +594,7 @@ lemma integrable_energy_congr (T : Temperature) (e : ι1 ≃ᵐ ι)
     Integrable (𝓒.congr e).energy ((𝓒.congr e).μProd T) := by
   simp [μProd_congr]
   refine (integrable_map_equiv e.symm (𝓒.congr e).energy).mpr ?_
-  simp only [congr_energy_comp_symmm]
-  exact h
+  simpa only [congr_energy_comp_symmm] using h
 
 @[fun_prop]
 lemma integrable_energy_nsmul (n : ℕ) (T : Temperature)
@@ -656,8 +646,7 @@ lemma meanEnergy_add {T : Temperature}
     simp
   rw [integral_add h1 (integrable_const _)]
   rw [integral_const]
-  simp
-  rfl
+  simp [meanEnergy]
   · have h1 := integrable_energy_add 𝓒 𝓒1 T h1 h2
     simp_all only [μProd_add]
     exact h1
@@ -725,9 +714,7 @@ lemma differentialEntropy_nonneg_of_prob_le_one
     intro i
     have hpos := probability_pos (𝓒:=𝓒) (T:=T) i
     have hle := hP_le_one i
-    have hle' : 𝓒.probability T i ≤ Real.exp 0 := by
-      simpa [Real.exp_zero] using hle
-    exact (log_le_iff_le_exp hpos).mpr hle'
+    exact (log_le_iff_le_exp hpos).mpr (by simpa [Real.exp_zero] using hle)
   have hInt0 : Integrable (fun _ : ι => (0 : ℝ)) (𝓒.μProd T) := integrable_const _
   have hIntLe : (∫ i, Real.log (𝓒.probability T i) ∂𝓒.μProd T)
       ≤ (∫ _i, (0 : ℝ) ∂𝓒.μProd T) :=
@@ -772,8 +759,7 @@ lemma partitionFunction_pos
 
 lemma partitionFunction_congr
     (𝓒 : CanonicalEnsemble ι) (e : ι1 ≃ᵐ ι) (T : Temperature) :
-    (𝓒.congr e).partitionFunction T = 𝓒.partitionFunction T := by
-  simp [partitionFunction]
+    (𝓒.congr e).partitionFunction T = 𝓒.partitionFunction T := by simp [partitionFunction]
 
 lemma partitionFunction_add
     (𝓒 : CanonicalEnsemble ι) (𝓒1 : CanonicalEnsemble ι1)
@@ -794,13 +780,11 @@ lemma partitionFunction_nsmul
 
 lemma partitionFunction_dof_zero
     (𝓒 : CanonicalEnsemble ι) (T : Temperature) (h : 𝓒.dof = 0) :
-    𝓒.partitionFunction T = 𝓒.mathematicalPartitionFunction T := by
-  simp [partitionFunction, h]
+    𝓒.partitionFunction T = 𝓒.mathematicalPartitionFunction T := by simp [partitionFunction, h]
 
 lemma partitionFunction_phase_space_unit_one
     (𝓒 : CanonicalEnsemble ι) (T : Temperature) (h : 𝓒.phaseSpaceunit = 1) :
-    𝓒.partitionFunction T = 𝓒.mathematicalPartitionFunction T := by
-  simp [partitionFunction, h]
+    𝓒.partitionFunction T = 𝓒.mathematicalPartitionFunction T := by simp [partitionFunction, h]
 
 lemma log_partitionFunction
     (𝓒 : CanonicalEnsemble ι) (T : Temperature)
@@ -812,8 +796,7 @@ lemma log_partitionFunction
   have hden : 0 < 𝓒.phaseSpaceunit ^ 𝓒.dof := pow_pos 𝓒.hPos _
   have hlogpow :
       Real.log (𝓒.phaseSpaceunit ^ 𝓒.dof)
-        = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
-    simp
+        = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by simp
   simp [partitionFunction, Real.log_div hZ.ne' hden.ne', hlogpow,
         sub_eq_add_neg]
 
@@ -846,8 +829,7 @@ lemma helmholtzFreeEnergy_def
 
 lemma helmholtzFreeEnergy_congr
     (𝓒 : CanonicalEnsemble ι) (e : ι1 ≃ᵐ ι) (T : Temperature) :
-    (𝓒.congr e).helmholtzFreeEnergy T = 𝓒.helmholtzFreeEnergy T := by
-  simp [helmholtzFreeEnergy]
+    (𝓒.congr e).helmholtzFreeEnergy T = 𝓒.helmholtzFreeEnergy T := by simp [helmholtzFreeEnergy]
 
 lemma helmholtzFreeEnergy_dof_zero
     (𝓒 : CanonicalEnsemble ι) (T : Temperature) (h : 𝓒.dof = 0) :
@@ -874,14 +856,12 @@ lemma helmholtzFreeEnergy_add
   calc
     (𝓒 + 𝓒1).helmholtzFreeEnergy T
         = -kB * T.val * Real.log ((𝓒 + 𝓒1).partitionFunction T) := rfl
-    _ = -kB * T.val * Real.log (𝓒.partitionFunction T * 𝓒1.partitionFunction T) := by
-          rw [hPF]
+    _ = -kB * T.val * Real.log (𝓒.partitionFunction T * 𝓒1.partitionFunction T) := by rw [hPF]
     _ = -kB * T.val *
           (Real.log (𝓒.partitionFunction T) + Real.log (𝓒1.partitionFunction T)) := by
           rw [Real.log_mul hpf₁.ne' hpf₂.ne']
     _ = (-kB * T.val) * Real.log (𝓒.partitionFunction T)
-        + (-kB * T.val) * Real.log (𝓒1.partitionFunction T) := by
-          ring
+        + (-kB * T.val) * Real.log (𝓒1.partitionFunction T) := by ring
     _ = 𝓒.helmholtzFreeEnergy T + 𝓒1.helmholtzFreeEnergy T := by
           simp [helmholtzFreeEnergy, mul_comm, mul_assoc]
 
@@ -898,10 +878,8 @@ lemma helmholtzFreeEnergy_nsmul
   calc
     (nsmul n 𝓒).helmholtzFreeEnergy T
         = -kB * T.val * Real.log ((nsmul n 𝓒).partitionFunction T) := rfl
-    _ = -kB * T.val * ((n : ℝ) * Real.log (𝓒.partitionFunction T)) := by
-          rw [hlog]
-    _ = (n : ℝ) * (-kB * T.val * Real.log (𝓒.partitionFunction T)) := by
-          ring
+    _ = -kB * T.val * ((n : ℝ) * Real.log (𝓒.partitionFunction T)) := by rw [hlog]
+    _ = (n : ℝ) * (-kB * T.val * Real.log (𝓒.partitionFunction T)) := by ring
     _ = n * 𝓒.helmholtzFreeEnergy T := by
           simp [helmholtzFreeEnergy, mul_comm, mul_left_comm, mul_assoc]
 
@@ -998,18 +976,15 @@ lemma integral_physicalProbability_base
 
 lemma physicalProbability_dof_zero
     (T : Temperature) (h : 𝓒.dof = 0) (i : ι) :
-    𝓒.physicalProbability T i = 𝓒.probability T i := by
-  simp [physicalProbability, h]
+    𝓒.physicalProbability T i = 𝓒.probability T i := by simp [physicalProbability, h]
 
 lemma physicalProbability_phase_space_unit_one
     (T : Temperature) (h : 𝓒.phaseSpaceunit = 1) (i : ι) :
-    𝓒.physicalProbability T i = 𝓒.probability T i := by
-  simp [physicalProbability, h]
+    𝓒.physicalProbability T i = 𝓒.probability T i := by simp [physicalProbability, h]
 
 lemma physicalProbability_congr (e : ι1 ≃ᵐ ι) (T : Temperature) (i : ι1) :
     (𝓒.congr e).physicalProbability T i
-      = 𝓒.physicalProbability T (e i) := by
-  simp [physicalProbability, probability]
+      = 𝓒.physicalProbability T (e i) := by simp [physicalProbability, probability]
 
 lemma physicalProbability_add
     {ι1} [MeasurableSpace ι1]
diff --git a/Physlib/StatisticalMechanics/CanonicalEnsemble/Finite.lean b/Physlib/StatisticalMechanics/CanonicalEnsemble/Finite.lean
index f1f80ed17..1b3b04e0d 100644
--- a/Physlib/StatisticalMechanics/CanonicalEnsemble/Finite.lean
+++ b/Physlib/StatisticalMechanics/CanonicalEnsemble/Finite.lean
@@ -81,10 +81,8 @@ instance [IsFinite 𝓒] [IsFinite 𝓒1] : IsFinite (𝓒 + 𝓒1) where
     · exact ht
     · exact hs
     · exact MeasurableSet.prod hs ht
-  dof_eq_zero := by
-    simp [IsFinite.dof_eq_zero (𝓒:=𝓒), IsFinite.dof_eq_zero (𝓒:=𝓒1)]
-  phase_space_unit_eq_one := by
-    simp [IsFinite.phase_space_unit_eq_one (𝓒:=𝓒)]
+  dof_eq_zero := by simp [IsFinite.dof_eq_zero (𝓒:=𝓒), IsFinite.dof_eq_zero (𝓒:=𝓒1)]
+  phase_space_unit_eq_one := by simp [IsFinite.phase_space_unit_eq_one (𝓒:=𝓒)]
 
 instance [IsFinite 𝓒] (e : ι1 ≃ᵐ ι) : IsFinite (congr 𝓒 e) where
   μ_eq_count := by
@@ -104,10 +102,8 @@ instance [IsFinite 𝓒] (e : ι1 ≃ᵐ ι) : IsFinite (congr 𝓒 e) where
     · exact Set.toFinite (⇑e '' s)
     · exact hs
     · exact (MeasurableEquiv.measurableSet_preimage e.symm).mpr hs
-  dof_eq_zero := by
-    simp [IsFinite.dof_eq_zero (𝓒:=𝓒)]
-  phase_space_unit_eq_one := by
-    simp [IsFinite.phase_space_unit_eq_one (𝓒:=𝓒)]
+  dof_eq_zero := by simp [IsFinite.dof_eq_zero (𝓒:=𝓒)]
+  phase_space_unit_eq_one := by simp [IsFinite.phase_space_unit_eq_one (𝓒:=𝓒)]
 
 instance [IsFinite 𝓒] (n : ℕ) : IsFinite (nsmul n 𝓒) where
   μ_eq_count := by
@@ -126,8 +122,7 @@ instance [IsFinite 𝓒] (n : ℕ) : IsFinite (nsmul n 𝓒) where
     | succ n ih =>
       haveI : IsFinite (nsmul n 𝓒) := {
         μ_eq_count := ih
-        dof_eq_zero := by
-          simp [CanonicalEnsemble.dof_nsmul, IsFinite.dof_eq_zero (𝓒:=𝓒)]
+        dof_eq_zero := by simp [CanonicalEnsemble.dof_nsmul, IsFinite.dof_eq_zero (𝓒:=𝓒)]
         phase_space_unit_eq_one := by
           simp [CanonicalEnsemble.phase_space_unit_nsmul,
             IsFinite.phase_space_unit_eq_one (𝓒:=𝓒)]
@@ -138,8 +133,7 @@ instance [IsFinite 𝓒] (n : ℕ) : IsFinite (nsmul n 𝓒) where
             (MeasurableEquiv.piFinSuccAbove (fun _ => ι) 0)).μ
           = Measure.count := by erw [IsFinite.μ_eq_count]; aesop
       rw [← h]; rw [← @nsmul_succ]
-  dof_eq_zero := by
-    simp [CanonicalEnsemble.dof_nsmul, IsFinite.dof_eq_zero (𝓒:=𝓒)]
+  dof_eq_zero := by simp [CanonicalEnsemble.dof_nsmul, IsFinite.dof_eq_zero (𝓒:=𝓒)]
   phase_space_unit_eq_one := by
     simp [CanonicalEnsemble.phase_space_unit_nsmul,
       IsFinite.phase_space_unit_eq_one (𝓒:=𝓒)]
@@ -153,8 +147,7 @@ instance [IsFinite 𝓒] [Nonempty ι] : NeZero 𝓒.μ := by
   refine ⟨?_⟩
   intro hμ
   obtain ⟨i₀⟩ := (inferInstance : Nonempty ι)
-  have hone : 𝓒.μ {i₀} = 1 := by
-    simp [IsFinite.μ_eq_count (𝓒:=𝓒)]
+  have hone : 𝓒.μ {i₀} = 1 := by simp [IsFinite.μ_eq_count (𝓒:=𝓒)]
   simp_all only [Measure.coe_zero, Pi.zero_apply, zero_ne_one]
 
 /-- The Shannon entropy of a finite canonical ensemble.
@@ -168,8 +161,7 @@ lemma mathematicalPartitionFunction_of_fintype [IsFinite 𝓒] (T : Temperature)
     𝓒.mathematicalPartitionFunction T = ∑ i, exp (- β T * 𝓒.energy i) := by
   rw [mathematicalPartitionFunction_eq_integral, MeasureTheory.integral_fintype]
   simp [IsFinite.μ_eq_count]
-  · rw [IsFinite.μ_eq_count]
-    exact Integrable.of_finite
+  · simp [IsFinite.μ_eq_count]
 
 lemma partitionFunction_of_fintype [IsFinite 𝓒] (T : Temperature) :
     𝓒.partitionFunction T = ∑ i, exp (- T.β * 𝓒.energy i) := by
@@ -182,21 +174,17 @@ lemma μBolt_of_fintype (T : Temperature) [IsFinite 𝓒] (i : ι) :
   rw [μBolt]
   simp only [neg_mul]
   rw [@measureReal_def]
-  simp [IsFinite.μ_eq_count]
-  exact Real.exp_nonneg _
+  simpa [IsFinite.μ_eq_count] using Real.exp_nonneg _
 
 instance {T} [IsFinite 𝓒] : IsFiniteMeasure (𝓒.μBolt T) := by
-  rw [μBolt]
-  refine isFiniteMeasure_withDensity_ofReal ?_
-  exact HasFiniteIntegral.of_finite
+  simpa [μBolt] using isFiniteMeasure_withDensity_ofReal HasFiniteIntegral.of_finite
 
 @[simp]
 lemma μProd_of_fintype (T : Temperature) [IsFinite 𝓒] (i : ι) :
     (𝓒.μProd T).real {i} = 𝓒.probability T i := by
   rw [μProd]
   simp [probability]
-  rw [inv_mul_eq_div]
-  rfl
+  rw [inv_mul_eq_div, mathematicalPartitionFunction, measureReal_def]
 
 lemma meanEnergy_of_fintype [IsFinite 𝓒] (T : Temperature) :
     𝓒.meanEnergy T = ∑ i, 𝓒.energy i * 𝓒.probability T i := by
@@ -270,8 +258,7 @@ lemma entropy_nonneg [MeasurableSingletonClass ι] [IsFinite 𝓒] [Nonempty ι]
     Fintype.sum_nonpos fun i =>
       mul_nonpos_iff.2 <| Or.inl ⟨probability_nonneg_finite 𝓒 T i,
         Real.log_nonpos (probability_nonneg_finite 𝓒 T i) (probability_le_one 𝓒 T i)⟩
-  rw [shannonEntropy, neg_mul, neg_nonneg]
-  exact mul_nonpos_iff.2 <| Or.inl ⟨kB_pos.le, hsum⟩
+  simpa [shannonEntropy, neg_mul, neg_nonneg] using mul_nonpos_iff.2 <| Or.inl ⟨kB_pos.le, hsum⟩
 
 lemma shannonEntropy_eq_differentialEntropy
     [MeasurableSingletonClass ι] [IsFinite 𝓒] (T : Temperature) :
@@ -353,8 +340,7 @@ noncomputable def meanEnergyBetaReal (b : ℝ) : ℝ :=
 lemma meanEnergy_Beta_eq_finite [MeasurableSingletonClass ι] [IsFinite 𝓒] (b : ℝ) (hb : 0 < b) :
     𝓒.meanEnergyBeta b = 𝓒.meanEnergyBetaReal b := by
   let T := Temperature.ofβ (Real.toNNReal b)
-  have hT_beta : (T.β : ℝ) = b := by
-    simp [T, Real.toNNReal_of_nonneg hb.le]
+  have hT_beta : (T.β : ℝ) = b := by simp [T, Real.toNNReal_of_nonneg hb.le]
   rw [meanEnergyBeta, meanEnergy_of_fintype 𝓒 T, meanEnergyBetaReal]
   refine Finset.sum_congr rfl fun i _ => ?_
   simp [CanonicalEnsemble.probability, probabilityBetaReal,
@@ -393,8 +379,7 @@ lemma deriv_mathematicalPartitionFunctionBetaReal (b : ℝ) :
   unfold mathematicalPartitionFunctionBetaReal meanEnergyNumerator
   have hd : ∀ c : ℝ, HasDerivAt (fun x => Real.exp (-x * c)) (-c * Real.exp (-b * c)) b := by
     intro c
-    have h : HasDerivAt (fun x => -x * c) (-c) b := by
-      simpa using (hasDerivAt_id b).neg.mul_const c
+    have h : HasDerivAt (fun x => -x * c) (-c) b := by simpa using (hasDerivAt_id b).neg.mul_const c
     simpa [mul_comm] using h.exp
   have h_sum : HasDerivAt (fun x => ∑ i, Real.exp (-x * 𝓒.energy i))
       (∑ i, -𝓒.energy i * Real.exp (-b * 𝓒.energy i)) b :=
@@ -407,8 +392,7 @@ lemma deriv_meanEnergyNumerator (b : ℝ) :
   unfold meanEnergyNumerator
   have hd : ∀ c : ℝ, HasDerivAt (fun x => Real.exp (-x * c)) (-c * Real.exp (-b * c)) b := by
     intro c
-    have h : HasDerivAt (fun x => -x * c) (-c) b := by
-      simpa using (hasDerivAt_id b).neg.mul_const c
+    have h : HasDerivAt (fun x => -x * c) (-c) b := by simpa using (hasDerivAt_id b).neg.mul_const c
     simpa [mul_comm] using h.exp
   have h_sum : HasDerivAt (fun x => ∑ i, 𝓒.energy i * Real.exp (-x * 𝓒.energy i))
       (∑ i, -(𝓒.energy i)^2 * Real.exp (-b * 𝓒.energy i)) b := by
diff --git a/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean b/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean
index f08cee5bf..aa8b9ceaf 100644
--- a/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean
+++ b/Physlib/StatisticalMechanics/CanonicalEnsemble/Lemmas.lean
@@ -80,8 +80,7 @@ lemma helmholtzFreeEnergy_eq_helmholtzMathematicalFreeEnergy_add_correction (T :
   simp_rw [helmholtzFreeEnergy, mathematicalHelmholtzFreeEnergy, partitionFunction,
     Real.log_div hZ_pos.ne' h_pow_pos.ne']
   have h_log_pow : Real.log (𝓒.phaseSpaceunit ^ 𝓒.dof)
-      = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
-    simp
+      = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by simp
   simp [sub_eq_add_neg, h_log_pow, mul_add, add_comm, mul_comm, mul_left_comm, mul_assoc]
 
 /-- General identity: S_diff = kB β ⟨E⟩ + kB log Z_math.
@@ -122,8 +121,7 @@ lemma log_probability
     Real.log (𝓒.probability T i)
       = - (β T) * 𝓒.energy i - Real.log (𝓒.mathematicalPartitionFunction T) := by
   have hZpos := mathematicalPartitionFunction_pos (𝓒:=𝓒) (T:=T)
-  unfold probability
-  simp [Real.log_div, hZpos.ne', Real.log_exp, sub_eq_add_neg]
+  simp [probability, Real.log_div, hZpos.ne', Real.log_exp, sub_eq_add_neg]
 
 /-- Auxiliary identity: `kB · β = 1 / T`.
 `β` is defined as `1 / (kB · T)` (see `Temperature.β`). -/
@@ -155,8 +153,7 @@ lemma thermodynamicEntropy_eq_differentialEntropy_sub_correction
             - Real.log (𝓒.mathematicalPartitionFunction T) := by
     intro i
     have : 0 < 𝓒.mathematicalPartitionFunction T := hZpos
-    unfold probability
-    simp [Real.log_div (Real.exp_pos _).ne' this.ne', Real.log_exp, sub_eq_add_neg]
+    simp [probability, Real.log_div (Real.exp_pos _).ne' this.ne', Real.log_exp, sub_eq_add_neg]
   have h_log_phys_pt :
       ∀ i, Real.log (𝓒.physicalProbability T i)
         = Real.log (𝓒.probability T i)
@@ -190,8 +187,7 @@ lemma thermodynamicEntropy_eq_differentialEntropy_sub_correction
     simp_all only [neg_mul, physicalProbability_def]
   have h_int_const_eval :
       ∫ _i, (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit ∂ 𝓒.μProd T
-        = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
-    simp [integral_const]
+        = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by simp [integral_const]
   unfold thermodynamicEntropy differentialEntropy
   rw [h_int_rewrite,
       integral_add h_int_log_prob h_int_const,
@@ -201,8 +197,7 @@ lemma thermodynamicEntropy_eq_differentialEntropy_sub_correction
             (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit)
         =
         (-kB * ∫ i, Real.log (𝓒.probability T i) ∂ 𝓒.μProd T)
-          - kB * (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
-    ring
+          - kB * (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by ring
   simp [this, sub_eq_add_neg, mul_assoc]
 
 /-- No semiclassical correction when `dof = 0`. -/
@@ -251,8 +246,7 @@ theorem helmholtzFreeEnergy_eq_meanEnergy_sub_temp_mul_thermodynamicEntropy
   have hZpos := 𝓒.mathematicalPartitionFunction_pos T
   have hhpos : 0 < 𝓒.phaseSpaceunit ^ 𝓒.dof := pow_pos 𝓒.hPos _
   have hTne : (T.val : ℝ) ≠ 0 := by exact_mod_cast hT.ne'
-  have hkβT : T.val * (kB * (T.β : ℝ)) = 1 := by
-    rw [kB_mul_beta T hT, mul_one_div, div_self hTne]
+  have hkβT : T.val * (kB * (T.β : ℝ)) = 1 := by rw [kB_mul_beta T hT, mul_one_div, div_self hTne]
   rw [helmholtzFreeEnergy_def, partitionFunction_def, Real.log_div hZpos.ne' hhpos.ne',
     Real.log_pow, 𝓒.thermodynamicEntropy_eq_differentialEntropy_sub_correction (T := T) hE,
     𝓒.differentialEntropy_eq_kB_beta_meanEnergy_add_kB_log_mathZ (T := T) hE]
@@ -312,8 +306,7 @@ theorem differentialEntropy_eq_meanEnergy_sub_helmholtz_div_temp_add_correction
     have hy : 0 < 𝓒.phaseSpaceunit ^ 𝓒.dof := hpow_pos
     have hlog_pow :
         Real.log (𝓒.phaseSpaceunit ^ 𝓒.dof)
-          = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
-      simp
+          = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by simp
     calc
       Real.log Zmath
           = Real.log (Zphys * 𝓒.phaseSpaceunit ^ 𝓒.dof) := by simp [hZdef]
@@ -339,8 +332,7 @@ theorem differentialEntropy_eq_meanEnergy_sub_helmholtz_div_temp_add_correction
             + kB * (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
               simp [hS_phys, add_comm, add_left_comm]
     _ = (E - 𝓒.helmholtzFreeEnergy T) / T.val
-            + kB * 𝓒.dof * Real.log 𝓒.phaseSpaceunit := by
-              rw [hEF]
+            + kB * 𝓒.dof * Real.log 𝓒.phaseSpaceunit := by rw [hEF]
 
 /-- Discrete / normalized specialization of the previous theorem.
 If either `dof = 0` (no semiclassical correction) or `phaseSpaceUnit = 1`
@@ -384,10 +376,8 @@ lemma integral_bolt_eq_integral_mul_exp
       = ∫ x, φ x * Real.exp (-T.β * 𝓒.energy x) ∂ 𝓒.μ := by
   unfold μBolt
   set f : ι → ℝ≥0∞ := fun x => ENNReal.ofReal (Real.exp (-T.β * 𝓒.energy x))
-  have hf_meas : Measurable f := by
-    fun_prop
-  have hf_lt_top : ∀ᵐ x ∂ 𝓒.μ, f x < ∞ := by
-    simp [f]
+  have hf_meas : Measurable f := by fun_prop
+  have hf_lt_top : ∀ᵐ x ∂ 𝓒.μ, f x < ∞ := by simp [f]
   have h :=
     integral_withDensity_eq_integral_toReal_smul
       (μ := 𝓒.μ) hf_meas hf_lt_top φ
@@ -420,8 +410,7 @@ lemma meanEnergy_eq_ratio_of_integrals
   unfold meanEnergy μProd
   have h_scale :
       ∫ x, 𝓒.energy x ∂ ((𝓒.μBolt T Set.univ)⁻¹ • 𝓒.μBolt T)
-        = ((𝓒.μBolt T Set.univ)⁻¹).toReal * ∫ x, 𝓒.energy x ∂ 𝓒.μBolt T := by
-    simp
+        = ((𝓒.μBolt T Set.univ)⁻¹).toReal * ∫ x, 𝓒.energy x ∂ 𝓒.μBolt T := by simp
   have h_energy_bolt_raw :=
     integral_energy_bolt (𝓒:=𝓒) (T:=T)
   have h_den :
@@ -430,20 +419,16 @@ lemma meanEnergy_eq_ratio_of_integrals
     exact (mathematicalPartitionFunction_eq_integral (𝓒:=𝓒) (T:=T))
   have h_inv_toReal :
       ((𝓒.μBolt T Set.univ)⁻¹).toReal
-        = 1 / (𝓒.μBolt T Set.univ).toReal := by
-    simp [one_div, ENNReal.toReal_inv]
+        = 1 / (𝓒.μBolt T Set.univ).toReal := by simp [one_div, ENNReal.toReal_inv]
   calc
     ∫ x, 𝓒.energy x ∂ ((𝓒.μBolt T Set.univ)⁻¹ • 𝓒.μBolt T)
         = ((𝓒.μBolt T Set.univ)⁻¹).toReal * ∫ x, 𝓒.energy x ∂ 𝓒.μBolt T := h_scale
     _ = ((𝓒.μBolt T Set.univ)⁻¹).toReal *
-          (∫ x, 𝓒.energy x * Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) := by
-          simp [h_energy_bolt_raw]
+          (∫ x, 𝓒.energy x * Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) := by simp [h_energy_bolt_raw]
     _ = (1 / (𝓒.μBolt T Set.univ).toReal) *
-          (∫ x, 𝓒.energy x * Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) := by
-          simp [h_inv_toReal]
+          (∫ x, 𝓒.energy x * Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) := by simp [h_inv_toReal]
     _ = (∫ x, 𝓒.energy x * Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) /
-          (∫ x, Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) := by
-          simp [h_den, div_eq_mul_inv, mul_comm]
+          (∫ x, Real.exp (- T.β * 𝓒.energy x) ∂ 𝓒.μ) := by simp [h_den, div_eq_mul_inv, mul_comm]
 
 /-- The mean energy is the negative derivative of the logarithm of the
 (mathematical) partition function with respect to β = 1/(kB T).
@@ -509,8 +494,7 @@ lemma meanEnergy_eq_neg_deriv_log_mathZ_of_beta
             (- ∫ i, 𝓒.energy i * Real.exp (-(T.β : ℝ) * 𝓒.energy i) ∂𝓒.μ)) := by ring
     _ = - (derivWithin
             (fun β : ℝ => Real.log (∫ i, Real.exp (-β * 𝓒.energy i) ∂𝓒.μ))
-            (Set.Ioi 0) (T.β : ℝ)) := by
-          rw [h_deriv_log]
+            (Set.Ioi 0) (T.β : ℝ)) := by rw [h_deriv_log]
 
 section Ratios
 
@@ -538,8 +522,7 @@ lemma log_phys_eq_log_math_sub_const_on_Ioi
       (Set.Ioi (0 : ℝ)) := by
   intro β hβ
   have hβpos : 0 < β := hβ
-  have hβnn : Real.toNNReal β = β := by
-    simp [Real.toNNReal, hβpos.le]
+  have hβnn : Real.toNNReal β = β := by simp [Real.toNNReal, hβpos.le]
   have _inst : IsFiniteMeasure (𝓒.μBolt (Temperature.ofβ (Real.toNNReal β))) :=
     h_fin β hβpos
   have hZpos :
@@ -550,16 +533,14 @@ lemma log_phys_eq_log_math_sub_const_on_Ioi
   have h_pow_pos : 0 < 𝓒.phaseSpaceunit ^ 𝓒.dof := pow_pos 𝓒.hPos _
   have h_log_pow :
       Real.log (𝓒.phaseSpaceunit ^ 𝓒.dof)
-        = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by
-    simp
+        = (𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit := by simp
   have :
       Real.log (𝓒.partitionFunction (Temperature.ofβ (Real.toNNReal β))) =
         -((𝓒.dof : ℝ) * Real.log 𝓒.phaseSpaceunit)
           + Real.log (∫ i, Real.exp (-β * 𝓒.energy i) ∂ 𝓒.μ) := by
     have h_integral_pos : 0 < ∫ i, Real.exp (-β * 𝓒.energy i) ∂ 𝓒.μ := by
       have h_eq : ∫ i, Real.exp (-β * 𝓒.energy i) ∂ 𝓒.μ =
-        ∫ i, Real.exp (-(Real.toNNReal β).val * 𝓒.energy i) ∂ 𝓒.μ := by
-        simp [hβnn]
+        ∫ i, Real.exp (-(Real.toNNReal β).val * 𝓒.energy i) ∂ 𝓒.μ := by simp [hβnn]
       rw [h_eq]
       simp [mathematicalPartitionFunction_eq_integral
         (𝓒:=𝓒) (T:=Temperature.ofβ (Real.toNNReal β))] at hZpos
@@ -674,8 +655,7 @@ theorem energyVariance_eq_meanSquareEnergy_sub_meanEnergy_sq
   rw [integral_const_mul]
   rw [integral_const]
   have hμProb : (𝓒.μProd T) Set.univ = 1 := by simp
-  have hμReal : (𝓒.μProd T).real Set.univ = 1 := by
-    simp [measureReal_def, hμProb]
+  have hμReal : (𝓒.μProd T).real Set.univ = 1 := by simp [measureReal_def, hμProb]
   calc
     ∫ i, (𝓒.energy i)^2 ∂𝓒.μProd T
         - 2 * U * ∫ i, 𝓒.energy i ∂𝓒.μProd T
@@ -716,8 +696,7 @@ lemma heatCapacity_eq_deriv_meanEnergyBeta
   unfold heatCapacity meanEnergy_T
   have h_U_eq_comp : (𝓒.meanEnergy_T) = fun t : ℝ => (𝓒.meanEnergyBeta) (betaFromReal t) := by
     funext t
-    dsimp [meanEnergy_T, meanEnergyBeta, betaFromReal]
-    simp
+    simp [meanEnergy_T, meanEnergyBeta, betaFromReal]
   let dUdβ := derivWithin (𝓒.meanEnergyBeta) (Set.Ioi 0) (T.β : ℝ)
   have h_chain := chain_rule_T_beta (F:=𝓒.meanEnergyBeta) (F':=dUdβ) T hT_pos hU_deriv
   have h_UD :
diff --git a/Physlib/StatisticalMechanics/CanonicalEnsemble/TwoState.lean b/Physlib/StatisticalMechanics/CanonicalEnsemble/TwoState.lean
index b67dc0c5b..e249ba004 100644
--- a/Physlib/StatisticalMechanics/CanonicalEnsemble/TwoState.lean
+++ b/Physlib/StatisticalMechanics/CanonicalEnsemble/TwoState.lean
@@ -50,12 +50,10 @@ lemma twoState_partitionFunction_apply_eq_cosh (E₀ E₁ : ℝ) (T : Temperatur
   ring_nf
 
 @[simp]
-lemma twoState_energy_fst (E₀ E₁ : ℝ) : (twoState E₀ E₁).energy 0 = E₀ := by
-  rfl
+lemma twoState_energy_fst (E₀ E₁ : ℝ) : (twoState E₀ E₁).energy 0 = E₀ := rfl
 
 @[simp]
-lemma twoState_energy_snd (E₀ E₁ : ℝ) : (twoState E₀ E₁).energy 1 = E₁ := by
-  rfl
+lemma twoState_energy_snd (E₀ E₁ : ℝ) : (twoState E₀ E₁).energy 1 = E₁ := rfl
 
 /-- Probability of the first state (energy `E₀`) in closed form. -/
 lemma twoState_probability_fst (E₀ E₁ : ℝ) (T : Temperature) :
diff --git a/Physlib/StatisticalMechanics/MicroCanonicalEnsemble/IdealGas.lean b/Physlib/StatisticalMechanics/MicroCanonicalEnsemble/IdealGas.lean
index ddef990f3..4713b123c 100644
--- a/Physlib/StatisticalMechanics/MicroCanonicalEnsemble/IdealGas.lean
+++ b/Physlib/StatisticalMechanics/MicroCanonicalEnsemble/IdealGas.lean
@@ -90,9 +90,8 @@ lemma partitionZ_eq (hV : 0 < V) (hβ : 0 < β) :
   simp_rw [show ∀ (x y i p_i), eq_pm (x, y) (i, Sum.inl p_i) = x (i, p_i) from fun _ _ _ _ => rfl,
     show ∀ (x y i m_i), eq_pm (x, y) (i, Sum.inr m_i) = y (i, m_i) from fun _ _ _ _ => rfl]
   have h_measurable_box : Measurable fun (a : (Fin n × Fin 3 → ℝ))
-      => ∃ x_1 x_2, V ^ (3⁻¹:ℝ) / 2 < |a (x_1, x_2)| := by
-    refine Measurable.exists fun i => Measurable.exists fun j => ?_
-    exact measurable_const.lt (by fun_prop)
+      => ∃ x_1 x_2, V ^ (3⁻¹:ℝ) / 2 < |a (x_1, x_2)| :=
+    Measurable.exists fun i => Measurable.exists fun j => measurable_const.lt (by fun_prop)
   have h_measurability : Measurable fun x : (Fin n × Fin 3 → ℝ) × (Fin n × Fin 3 → ℝ) =>
       if ∃ x_1 x_2, V ^ (3⁻¹:ℝ) / 2 < |x.1 (x_1, x_2)| then 0
       else Real.exp (-(β * ∑ x_1 : Fin n × Fin 3, x.2 (x_1.1, x_1.2) ^ 2 / 2)) := by
@@ -196,8 +195,7 @@ theorem ideal_gas_law (hV : 0 < V) (hT : 0 < T) :
     let P := IdealGas.pressure (n,V) T;
     let R := 1;
     P * V = n * R * T := by
-  dsimp [pressure]
-  rw [← derivWithin_of_isOpen (s := Set.Ioi 0) isOpen_Ioi hV]
+  rw [pressure, ← derivWithin_of_isOpen (s := Set.Ioi 0) isOpen_Ioi hV]
   rw [derivWithin_congr (f := fun V' ↦ -n * T *
     (Real.log V' + (3/2) * Real.log (2 * Real.pi * T))) ?_ ?_]
   rw [derivWithin_of_isOpen (s := Set.Ioi 0) isOpen_Ioi hV]
diff --git a/Physlib/StringTheory/FTheory/SU5/Fluxes/NoExotics/Completeness.lean b/Physlib/StringTheory/FTheory/SU5/Fluxes/NoExotics/Completeness.lean
index 986e2a3b2..d27a05f0f 100644
--- a/Physlib/StringTheory/FTheory/SU5/Fluxes/NoExotics/Completeness.lean
+++ b/Physlib/StringTheory/FTheory/SU5/Fluxes/NoExotics/Completeness.lean
@@ -272,10 +272,9 @@ lemma mem_elemsNoExotics_of_noExotics (F : FluxesFive) (hNE : F.NoExotics) (hnZ
   which obeys `NoExotics` is an element of `elemsNoExotics`, and every
   element of `elemsNoExotics` obeys `NoExotics`. -/
 lemma noExotics_iff_mem_elemsNoExotics (F : FluxesFive) :
-    F.NoExotics ∧ F.HasNoZero ↔ F ∈ elemsNoExotics := by
-  constructor
-  · exact fun ⟨h1, h2⟩ => mem_elemsNoExotics_of_noExotics F h1 h2
-  · exact fun h => ⟨noExotics_of_mem_elemsNoExotics F h, hasNoZero_of_mem_elemsNoExotics F h⟩
+    F.NoExotics ∧ F.HasNoZero ↔ F ∈ elemsNoExotics :=
+  ⟨fun ⟨h1, h2⟩ => mem_elemsNoExotics_of_noExotics F h1 h2,
+    fun h => ⟨noExotics_of_mem_elemsNoExotics F h, hasNoZero_of_mem_elemsNoExotics F h⟩⟩
 
 end FluxesFive
 
@@ -444,10 +443,9 @@ lemma mem_elemsNoExotics_of_noExotics (F : FluxesTen) (hNE : F.NoExotics) (hnZ :
   which obeys `NoExotics` is an element of `elemsNoExotics`, and every
   element of `elemsNoExotics` obeys `NoExotics`. -/
 lemma noExotics_iff_mem_elemsNoExotics (F : FluxesTen) :
-    F.NoExotics ∧ F.HasNoZero ↔ F ∈ elemsNoExotics := by
-  constructor
-  · exact fun h => mem_elemsNoExotics_of_noExotics F h.1 h.2
-  · exact fun h => ⟨noExotics_of_mem_elemsNoExotics F h, hasNoZero_of_mem_elemsNoExotics F h⟩
+    F.NoExotics ∧ F.HasNoZero ↔ F ∈ elemsNoExotics :=
+  ⟨fun h => mem_elemsNoExotics_of_noExotics F h.1 h.2,
+    fun h => ⟨noExotics_of_mem_elemsNoExotics F h, hasNoZero_of_mem_elemsNoExotics F h⟩⟩
 
 end FluxesTen
 end SU5
diff --git a/Physlib/StringTheory/FTheory/SU5/Quanta/FiveQuanta.lean b/Physlib/StringTheory/FTheory/SU5/Quanta/FiveQuanta.lean
index 69f125ad4..6ea02abd0 100644
--- a/Physlib/StringTheory/FTheory/SU5/Quanta/FiveQuanta.lean
+++ b/Physlib/StringTheory/FTheory/SU5/Quanta/FiveQuanta.lean
@@ -252,8 +252,7 @@ lemma reduce_sum_eq_sum_toCharges {M} [AddCommMonoid M] (x : FiveQuanta 𝓩) (f
           (((x.filter (fun f => f.1 = q5)).map (fun y => f q5 y.2))).sum := by
         congr
         funext q5
-        rw [AddMonoidHom.map_multiset_sum, Multiset.map_map]
-        rfl
+        simp [AddMonoidHom.map_multiset_sum, Multiset.map_map]
       _ = (x.toCharges.dedup.bind fun q5 =>
           ((x.filter (fun f => f.1 = q5)).map (fun y => f q5 y.2))).sum := by
         rw [Multiset.sum_bind]
@@ -290,8 +289,7 @@ lemma reduce_sum_eq_sum_toCharges {M} [AddCommMonoid M] (x : FiveQuanta 𝓩) (f
         · rw [Multiset.sum_eq_zero]
           refine Eq.symm (Multiset.count_eq_zero_of_notMem ?_)
           intro h
-          have h_mem : p.1 ∈ Multiset.map Prod.fst x := by
-            simp_all
+          have h_mem : p.1 ∈ Multiset.map Prod.fst x := by simp_all
           (expose_names; exact h_mem_1 h_mem)
           intro p' hp
           simp at hp
@@ -370,8 +368,7 @@ lemma mem_powerset_sum_of_mem_reduce_toFluxesFive {F : FiveQuanta 𝓩}
   use (Multiset.map (fun x => x.2) (Multiset.filter (fun x => x.1 = q) F))
   simp only [and_true]
   rw [toFluxesFive]
-  refine Multiset.map_le_map ?_
-  exact Multiset.filter_le (fun x => x.1 = q) F
+  exact Multiset.map_le_map (Multiset.filter_le (fun x => x.1 = q) F)
 
 lemma mem_powerset_sum_of_mem_reduce_toFluxesFive_filter {F : FiveQuanta 𝓩}
     {f : Fluxes} (hf : f ∈ F.reduce.toFluxesFive) :
@@ -388,8 +385,7 @@ lemma mem_powerset_sum_of_mem_reduce_toFluxesFive_filter {F : FiveQuanta 𝓩}
   apply And.intro
   simp only [Multiset.mem_powerset]
   rw [toFluxesFive]
-  refine Multiset.map_le_map ?_
-  exact Multiset.filter_le (fun x => x.1 = q) F
+  exact Multiset.map_le_map (Multiset.filter_le (fun x => x.1 = q) F)
   simp [Multiset.empty_eq_zero, ne_eq, Multiset.map_eq_zero, Multiset.filter_eq_nil,
     Prod.forall, not_forall, Decidable.not_not]
   rw [toCharges, Multiset.mem_map] at hq
@@ -566,8 +562,7 @@ def decompose (x : FiveQuanta 𝓩) : FiveQuanta 𝓩 :=
 -/
 
 lemma decompose_add (x y : FiveQuanta 𝓩) :
-    (x + y).decompose = x.decompose + y.decompose := by
-  simp [decompose]
+    (x + y).decompose = x.decompose + y.decompose := by simp [decompose]
 
 /-!
 
diff --git a/Physlib/StringTheory/FTheory/SU5/Quanta/TenQuanta.lean b/Physlib/StringTheory/FTheory/SU5/Quanta/TenQuanta.lean
index c0b4841c6..544a5085c 100644
--- a/Physlib/StringTheory/FTheory/SU5/Quanta/TenQuanta.lean
+++ b/Physlib/StringTheory/FTheory/SU5/Quanta/TenQuanta.lean
@@ -215,8 +215,7 @@ lemma reduce_filter (x : TenQuanta 𝓩) (q : 𝓩) (h : q ∈ x.toCharges) :
   simp only [Function.comp_apply]
   have hx : (Multiset.filter (fun x => x = q) x.toCharges.dedup) = {q} := by
     refine (Multiset.Nodup.ext ?_ ?_).mpr ?_
-    · refine Multiset.Nodup.filter (fun x => x = q) ?_
-      exact Multiset.nodup_dedup x.toCharges
+    · exact Multiset.Nodup.filter (fun x => x = q) (Multiset.nodup_dedup x.toCharges)
     · exact Multiset.nodup_singleton q
     intro a
     simp only [Multiset.mem_filter, Multiset.mem_dedup, Multiset.mem_singleton,
@@ -267,8 +266,7 @@ lemma reduce_sum_eq_sum_toCharges {M} [AddCommMonoid M] (x : TenQuanta 𝓩) (f
           (((x.filter (fun f => f.1 = q)).map (fun y => f q y.2))).sum := by
         congr
         funext q5
-        rw [AddMonoidHom.map_multiset_sum, Multiset.map_map]
-        rfl
+        simp [AddMonoidHom.map_multiset_sum, Multiset.map_map]
       _ = (x.toCharges.dedup.bind fun q =>
           ((x.filter (fun f => f.1 = q)).map (fun y => f q y.2))).sum := by
         rw [Multiset.sum_bind]
@@ -305,8 +303,7 @@ lemma reduce_sum_eq_sum_toCharges {M} [AddCommMonoid M] (x : TenQuanta 𝓩) (f
         · rw [Multiset.sum_eq_zero]
           refine Eq.symm (Multiset.count_eq_zero_of_notMem ?_)
           intro h
-          have h_mem : p.1 ∈ Multiset.map Prod.fst x := by
-            simp_all
+          have h_mem : p.1 ∈ Multiset.map Prod.fst x := by simp_all
           (expose_names; exact h_mem_1 h_mem)
           intro p' hp
           simp at hp
@@ -386,8 +383,7 @@ lemma mem_powerset_sum_of_mem_reduce_toFluxesTen {F : TenQuanta 𝓩}
   use (Multiset.map (fun x => x.2) (Multiset.filter (fun x => x.1 = q) F))
   simp only [and_true]
   rw [toFluxesTen]
-  refine Multiset.map_le_map ?_
-  exact Multiset.filter_le (fun x => x.1 = q) F
+  exact Multiset.map_le_map (Multiset.filter_le (fun x => x.1 = q) F)
 
 lemma mem_powerset_sum_of_mem_reduce_toFluxesTen_filter {F : TenQuanta 𝓩}
     {f : Fluxes} (hf : f ∈ F.reduce.toFluxesTen) :
@@ -403,8 +399,7 @@ lemma mem_powerset_sum_of_mem_reduce_toFluxesTen_filter {F : TenQuanta 𝓩}
     Multiset.filter_eq_nil, Prod.forall, not_forall, Decidable.not_not, and_true]
   apply And.intro
   rw [toFluxesTen]
-  refine Multiset.map_le_map ?_
-  exact Multiset.filter_le (fun x => x.1 = q) F
+  exact Multiset.map_le_map (Multiset.filter_le (fun x => x.1 = q) F)
   simpa [toCharges] using hq
 
 /-!
@@ -642,8 +637,7 @@ def decompose (x : TenQuanta 𝓩) : TenQuanta 𝓩 :=
 -/
 
 lemma decompose_add (x y : TenQuanta 𝓩) :
-    (x + y).decompose = x.decompose + y.decompose := by
-  simp [decompose]
+    (x + y).decompose = x.decompose + y.decompose := by simp [decompose]
 
 /-!
 
@@ -953,8 +947,7 @@ lemma mem_liftCharge_of_exists_toCharges_toFluxesTen (c : Finset 𝓩) {x : TenQ
           refine (Multiset.cons_le_of_notMem ?_).mpr ⟨?_, ?_⟩
           · simp
           · rw [← hp2_2]
-            apply Multiset.erase_subset p1 x
-            exact hp2
+            exact Multiset.erase_subset p1 x hp2
           simp only [Multiset.singleton_le]
           rw [← hp3_2]
           apply Multiset.erase_subset p1 x
@@ -973,10 +966,9 @@ lemma mem_liftCharge_iff_exists (c : Finset 𝓩) {x : TenQuanta 𝓩} :
     x ∈ liftCharge c ↔
     ∃ a : TenQuanta 𝓩, a.reduce = x ∧ a.toCharges.toFinset = c ∧
     (a.toFluxesTen = {⟨1, 0⟩, ⟨1, 0⟩, ⟨1, 0⟩}
-    ∨ a.toFluxesTen = {⟨1, 1⟩, ⟨1, -1⟩, ⟨1, 0⟩}) := by
-  constructor
-  · exact exists_toCharges_toFluxesTen_of_mem_liftCharge c
-  · exact mem_liftCharge_of_exists_toCharges_toFluxesTen c
+    ∨ a.toFluxesTen = {⟨1, 1⟩, ⟨1, -1⟩, ⟨1, 0⟩}) :=
+  ⟨exists_toCharges_toFluxesTen_of_mem_liftCharge c,
+    mem_liftCharge_of_exists_toCharges_toFluxesTen c⟩
 
 /-!
 
diff --git a/Physlib/Thermodynamics/Temperature/Basic.lean b/Physlib/Thermodynamics/Temperature/Basic.lean
index 4cf0db442..d28ece09f 100644
--- a/Physlib/Thermodynamics/Temperature/Basic.lean
+++ b/Physlib/Thermodynamics/Temperature/Basic.lean
@@ -69,8 +69,7 @@ lemma ofβ_eq : ofβ = fun β => ⟨⟨1 / (kB * β), by
     · exact zero_le_one
     · apply mul_nonneg
       · exact kB_nonneg
-      · exact β.2⟩⟩ := by
-  rfl
+      · exact β.2⟩⟩ := rfl
 
 @[simp]
 lemma β_ofβ (β' : ℝ≥0) : β (ofβ β') = β' := by
@@ -192,9 +191,8 @@ lemma beta_fun_T_formula (t : ℝ) (ht : 0 < t) :
 
 /-- On `Ioi 0`, `Beta_fun_T t` equals `1 / (kB * t)`. -/
 lemma beta_fun_T_eq_on_Ioi :
-    EqOn betaFromReal (fun t : ℝ => 1 / (kB * t)) (Set.Ioi 0) := by
-  intro t ht
-  exact beta_fun_T_formula t ht
+    EqOn betaFromReal (fun t : ℝ => 1 / (kB * t)) (Set.Ioi 0) :=
+  fun t ht => beta_fun_T_formula t ht
 
 lemma deriv_beta_wrt_T (T : Temperature) (hT_pos : 0 < T.val) :
     HasDerivWithinAt betaFromReal (-1 / (kB * (T.val : ℝ)^2)) (Set.Ioi 0) (T.val : ℝ) := by
@@ -225,11 +223,9 @@ lemma chain_rule_T_beta {F : ℝ → ℝ} {F' : ℝ}
     rw [beta_fun_T_eq_on_Ioi ht]
     exact one_div_pos.mpr (mul_pos kB_pos ht)
   have h_beta_at_T : betaFromReal (T.val : ℝ) = (T.β : ℝ) := by
-    rw [beta_fun_T_eq_on_Ioi (show (T.val : ℝ) ∈ Set.Ioi 0 from hT_pos), β_toReal]
-    rfl
+    exact (beta_fun_T_eq_on_Ioi (show (T.val : ℝ) ∈ Set.Ioi 0 from hT_pos)).trans (β_toReal T).symm
   have hF_deriv' : HasDerivWithinAt F F' (Set.Ioi 0) (betaFromReal (T.val : ℝ)) := by
-    rw [h_beta_at_T]
-    exact hF_deriv
+    simpa [h_beta_at_T] using hF_deriv
   exact hF_deriv'.comp (T.val : ℝ) (deriv_beta_wrt_T (T := T) hT_pos) h_map
 
 end Temperature
diff --git a/Physlib/Thermodynamics/Temperature/TemperatureUnits.lean b/Physlib/Thermodynamics/Temperature/TemperatureUnits.lean
index 0b61fad11..fc27fab5d 100644
--- a/Physlib/Thermodynamics/Temperature/TemperatureUnits.lean
+++ b/Physlib/Thermodynamics/Temperature/TemperatureUnits.lean
@@ -40,8 +40,7 @@ structure TemperatureUnit where
 namespace TemperatureUnit
 
 @[simp]
-lemma val_ne_zero (x : TemperatureUnit) : x.val ≠ 0 := by
-  exact Ne.symm (ne_of_lt x.property)
+lemma val_ne_zero (x : TemperatureUnit) : x.val ≠ 0 := Ne.symm (ne_of_lt x.property)
 
 lemma val_pos (x : TemperatureUnit) : 0 < x.val := x.property
 
@@ -74,9 +73,7 @@ lemma div_pos (x y : TemperatureUnit) : (0 : ℝ≥0) < x/ y := by
 
 @[simp]
 lemma div_self (x : TemperatureUnit) :
-    x / x = (1 : ℝ≥0) := by
-  simp [div_eq_val, x.val_ne_zero]
-  rfl
+    x / x = (1 : ℝ≥0) := NNReal.eq (_root_.div_self x.val_ne_zero)
 
 lemma div_symm (x y : TemperatureUnit) :
     x / y = (y / x)⁻¹ := NNReal.eq <| by
@@ -102,8 +99,7 @@ def scale (r : ℝ) (x : TemperatureUnit) (hr : 0 < r := by norm_num) : Temperat
 
 @[simp]
 lemma scale_div_self (x : TemperatureUnit) (r : ℝ) (hr : 0 < r) :
-    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by
-  simp [scale, div_eq_val]
+    scale r x hr / x = (⟨r, le_of_lt hr⟩ : ℝ≥0) := by simp [scale, div_eq_val]
 
 @[simp]
 lemma self_div_scale (x : TemperatureUnit) (r : ℝ) (hr : 0 < r) :
@@ -113,8 +109,7 @@ lemma self_div_scale (x : TemperatureUnit) (r : ℝ) (hr : 0 < r) :
   field_simp
 
 @[simp]
-lemma scale_one (x : TemperatureUnit) : scale 1 x = x := by
-  simp [scale]
+lemma scale_one (x : TemperatureUnit) : scale 1 x = x := by simp [scale]
 
 @[simp]
 lemma scale_div_scale (x1 x2 : TemperatureUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr2 : 0 < r2) :
@@ -127,8 +122,7 @@ lemma scale_div_scale (x1 x2 : TemperatureUnit) {r1 r2 : ℝ} (hr1 : 0 < r1) (hr
 @[simp]
 lemma scale_scale (x : TemperatureUnit) (r1 r2 : ℝ) (hr1 : 0 < r1) (hr2 : 0 < r2) :
     scale r1 (scale r2 x hr2) hr1 = scale (r1 * r2) x (mul_pos hr1 hr2) := by
-  simp [scale]
-  ring
+  simp [scale, mul_assoc]
 
 /-!
 
diff --git a/Physlib/Units/Basic.lean b/Physlib/Units/Basic.lean
index b604f276d..aeb71166e 100644
--- a/Physlib/Units/Basic.lean
+++ b/Physlib/Units/Basic.lean
@@ -105,8 +105,7 @@ noncomputable def dimScale (u1 u2 : UnitChoices) :Dimension →* ℝ≥0 where
     (u1.mass / u2.mass) ^ (d.mass : ℝ) *
     (u1.charge / u2.charge) ^ (d.charge : ℝ) *
     (u1.temperature / u2.temperature) ^ (d.temperature : ℝ)
-  map_one' := by
-    simp
+  map_one' := by simp
   map_mul' d1 d2 := by
     simp only [Dimension.length_mul, Rat.cast_add, Dimension.time_mul, Dimension.mass_mul,
       Dimension.charge_mul, Dimension.temperature_mul]
@@ -125,13 +124,11 @@ lemma dimScale_apply (u1 u2 : UnitChoices) (d : Dimension) :
 
 @[simp]
 lemma dimScale_self (u : UnitChoices) (d : Dimension) :
-    dimScale u u d = 1 := by
-  simp [dimScale]
+    dimScale u u d = 1 := by simp [dimScale]
 
 @[simp]
 lemma dimScale_one (u1 u2 : UnitChoices) :
-    dimScale u1 u2 1 = 1 := by
-  simp [dimScale]
+    dimScale u1 u2 1 = 1 := by simp [dimScale]
 
 lemma dimScale_transitive (u1 u2 u3 : UnitChoices) (d : Dimension) :
     dimScale u1 u2 d * dimScale u2 u3 d = dimScale u1 u3 d := by
@@ -152,8 +149,7 @@ lemma dimScale_transitive (u1 u2 u3 : UnitChoices) (d : Dimension) :
 
 @[simp]
 lemma dimScale_mul_symm (u1 u2 : UnitChoices) (d : Dimension) :
-    dimScale u1 u2 d * dimScale u2 u1 d = 1 := by
-  rw [dimScale_transitive, dimScale_self]
+    dimScale u1 u2 d * dimScale u2 u1 d = 1 := by rw [dimScale_transitive, dimScale_self]
 
 @[simp]
 lemma dimScale_coe_mul_symm (u1 u2 : UnitChoices) (d : Dimension) :
@@ -164,8 +160,7 @@ lemma dimScale_coe_mul_symm (u1 u2 : UnitChoices) (d : Dimension) :
 
 @[simp]
 lemma dimScale_ne_zero (u1 u2 : UnitChoices) (d : Dimension) :
-    dimScale u1 u2 d ≠ 0 := by
-  simp [dimScale]
+    dimScale u1 u2 d ≠ 0 := by simp [dimScale]
 
 lemma dimScale_symm (u1 u2 : UnitChoices) (d : Dimension) :
     dimScale u1 u2 d = (dimScale u2 u1 d)⁻¹ := by
@@ -179,8 +174,7 @@ lemma dimScale_symm (u1 u2 : UnitChoices) (d : Dimension) :
 
 lemma dimScale_of_inv_eq_swap (u1 u2 : UnitChoices) (d : Dimension) :
     dimScale u1 u2 d⁻¹ = dimScale u2 u1 d := by
-  simp only [map_inv]
-  conv_rhs => rw[dimScale_symm]
+  simpa only [map_inv] using (dimScale_symm u2 u1 d).symm
 
 @[simp]
 lemma smul_dimScale_injective {M : Type} [MulAction ℝ≥0 M] (u1 u2 : UnitChoices) (d : Dimension)
@@ -309,8 +303,7 @@ def HasDimension {M : Type} [CarriesDimension M] (f : UnitChoices → M) : Prop
 
 lemma hasDimension_iff {M : Type} [CarriesDimension M] (f : UnitChoices → M) :
     HasDimension f ↔ ∀ u1 u2 : UnitChoices, f u2 =
-    UnitChoices.dimScale u1 u2 (dim M) • f u1 := by
-  rfl
+    UnitChoices.dimScale u1 u2 (dim M) • f u1 := Iff.rfl
 
 /-- The subtype of functions `UnitChoices → M`, for which `M` carries a dimension,
   which `HasDimension`. -/
@@ -333,12 +326,10 @@ instance {M : Type} [CarriesDimension M] : MulAction ℝ≥0 (Dimensionful M) wh
     rw [smul_comm]⟩
   one_smul f := by
     ext u
-    change (1 : ℝ≥0) • f.1 u = f.1 u
-    simp
+    exact one_smul ℝ≥0 (f.1 u)
   mul_smul a b f := by
     ext u
-    change (a * b) • f.1 u = a • (b • f.1 u)
-    rw [smul_smul]
+    exact (smul_smul a b (f.1 u)).symm
 
 @[simp]
 lemma Dimensionful.smul_apply {M : Type} [CarriesDimension M]
@@ -356,8 +347,7 @@ noncomputable def CarriesDimension.toDimensionful {M : Type} [CarriesDimension M
       simp [smul_smul]
       rw [mul_comm, UnitChoices.dimScale_transitive]}
   invFun f := f.1 u
-  left_inv m := by
-    simp
+  left_inv m := by simp
   right_inv f := by
     simp only
     ext u1
diff --git a/Physlib/Units/Dimension.lean b/Physlib/Units/Dimension.lean
index 3b59e8a3b..4e5b4da29 100644
--- a/Physlib/Units/Dimension.lean
+++ b/Physlib/Units/Dimension.lean
@@ -100,9 +100,7 @@ lemma one_temperature : (1 : Dimension).temperature = 0 := rfl
 instance : CommGroup Dimension where
   mul_assoc a b c := by
     ext
-    all_goals
-      simp only [length_mul, time_mul, mass_mul, charge_mul, temperature_mul]
-      ring
+    all_goals simp [add_assoc]
   one_mul a := by
     ext
     all_goals
@@ -117,9 +115,7 @@ instance : CommGroup Dimension where
     all_goals simp
   mul_comm a b := by
     ext
-    all_goals
-      simp only [length_mul, time_mul, mass_mul, charge_mul, temperature_mul]
-      ring
+    all_goals simp [add_comm]
 
 @[simp]
 lemma inv_length (d : Dimension) : d⁻¹.length = -d.length := rfl
diff --git a/Physlib/Units/FDeriv.lean b/Physlib/Units/FDeriv.lean
index f015b8ebe..eeb028d08 100644
--- a/Physlib/Units/FDeriv.lean
+++ b/Physlib/Units/FDeriv.lean
@@ -42,8 +42,7 @@ lemma fderiv_apply_scaleUnit (u1 u2 : UnitChoices) (x dm : M1)
   change (fderiv ℝ ((u2.dimScale u1 (dim M2)).1 • fun mx => f
       ((u1.dimScale u2 (dim M1)).1 • mx)) ((u2.dimScale u1 (dim M1)).1 • x)) dm = _
   rw [fderiv_const_smul (by fun_prop), fderiv_comp_smul]
-  simp [smul_smul]
-  rfl
+  simp [smul_smul, NNReal.smul_def]
 
 open ContinuousLinearUnitDependent in
 /-- If a function is dimensionally valid then so is it's derivative. -/
diff --git a/Physlib/Units/Integral.lean b/Physlib/Units/Integral.lean
index ae28cffb6..083ce84ce 100644
--- a/Physlib/Units/Integral.lean
+++ b/Physlib/Units/Integral.lean
@@ -44,10 +44,8 @@ noncomputable instance (M : Type)
     simp [HasDim.scaleUnit_apply]
     · exact measurable_const_smul (α := M) ↑(u1.dimScale u2 (dim M)).1
     · exact measurable_const_smul (α := M) ↑(u2.dimScale u3 (dim M)).1
-  scaleUnit_id u μ := by
-    simp [scaleUnit_id]
-  scaleUnit_mul u1 u2 r μ := by
-    simp
+  scaleUnit_id u μ := by simp [scaleUnit_id]
+  scaleUnit_mul u1 u2 r μ := by simp
 
 variable {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] [HasDim M]
     [MeasurableSpace M] [MeasurableConstSMul ℝ M]
@@ -83,8 +81,7 @@ lemma integral_isDimensionallyCorrect (d : Dimension) :
     then do the integral, and then we take the result back to `u2`.
     If the integral is dimensionally correct, this should be the same as just doing the
     original integral in `u2` units i.e. `∫ x, f x ∂μ`. -/
-    _ = scaleUnit u1 u2 (∫ x, (scaleUnit u2 u1 f) x ∂(scaleUnit u2 u1 μ)) := by
-      simp
+    _ = scaleUnit u1 u2 (∫ x, (scaleUnit u2 u1 f) x ∂(scaleUnit u2 u1 μ)) := by simp
     /- Since we have assumed `μ` has dimension `d`, `(scaleUnit u2 μ u1)`
       is equal to `(u2.dimScale u1 d) • μ` -/
     _ = scaleUnit u1 u2 (u2.dimScale u1 d • ∫ (x : M), scaleUnit u2 u1 f x ∂ μ) := by
@@ -95,8 +92,7 @@ lemma integral_isDimensionallyCorrect (d : Dimension) :
       u2.dimScale u1 (dim G * d⁻¹) • ∫ (x : M), f x ∂ μ) := by
       rw [hf]
       congr
-      erw [MeasureTheory.integral_smul]
-      rfl
+      exact MeasureTheory.integral_smul _ _
     /- What remains is a simple cancellation of the dimensional scales. -/
     _ = (u1.dimScale u2 (dim G)) • ((u2.dimScale u1 d) •
         u2.dimScale u1 (dim G * d⁻¹) • ∫ (x : M), f x ∂ μ) := by
diff --git a/Physlib/Units/UnitDependent.lean b/Physlib/Units/UnitDependent.lean
index 735268ecf..a90342013 100644
--- a/Physlib/Units/UnitDependent.lean
+++ b/Physlib/Units/UnitDependent.lean
@@ -84,8 +84,7 @@ class ContinuousLinearUnitDependent (M : Type) [AddCommMonoid M] [Module ℝ M]
 @[simp]
 lemma UnitDependent.scaleUnit_symm_apply {M : Type} [UnitDependent M]
     (u1 u2 : UnitChoices) (m : M) :
-    scaleUnit u2 u1 (scaleUnit u1 u2 m) = m := by
-  rw [scaleUnit_trans, scaleUnit_id]
+    scaleUnit u2 u1 (scaleUnit u1 u2 m) = m := by rw [scaleUnit_trans, scaleUnit_id]
 
 @[simp]
 lemma UnitDependent.scaleUnit_injective {M : Type} [UnitDependent M]
@@ -246,8 +245,7 @@ noncomputable instance {M1 : Type} [CarriesDimension M1] : MulUnitDependent M1 w
     rw [smul_smul, mul_comm, UnitChoices.dimScale_transitive]
   scaleUnit_trans' u1 u2 u3 m := by
     simp [toDimensionful, smul_smul, UnitChoices.dimScale_transitive]
-  scaleUnit_id u m := by
-    simp [toDimensionful, UnitChoices.dimScale_self]
+  scaleUnit_id u m := by simp [toDimensionful, UnitChoices.dimScale_self]
   scaleUnit_mul u1 u2 r m := by
     simp [toDimensionful]
     exact smul_comm (u1.dimScale u2 (dim M1)) r m
@@ -262,13 +260,11 @@ noncomputable instance {M : Type} [AddCommMonoid M] [Module ℝ M] [HasDim M] :
   scaleUnit_add u1 u2 m1 m2 := by
     change (toDimensionful u1 (m1 + m2)).1 u2 = _
     rw [toDimensionful_apply_apply]
-    simp
-    rfl
+    exact smul_add _ _ _
   scaleUnit_smul u1 u2 r m := by
     change (toDimensionful u1 (r • m)).1 u2 = _
     rw [toDimensionful_apply_apply]
-    rw [smul_comm]
-    rfl
+    exact smul_comm _ _ _
 
 noncomputable instance {M : Type} [AddCommMonoid M] [Module ℝ M]
     [HasDim M] [TopologicalSpace M]
@@ -278,9 +274,7 @@ noncomputable instance {M : Type} [AddCommMonoid M] [Module ℝ M]
     conv =>
       enter [1, m]
       rw [toDimensionful_apply_apply]
-    change Continuous fun m => (u1.dimScale u2 (dim M)).1 • m
-    apply Continuous.const_smul
-    exact continuous_id'
+    exact Continuous.const_smul continuous_id' (u1.dimScale u2 (dim M)).1
 
 /-!
 
@@ -291,15 +285,11 @@ noncomputable instance {M : Type} [AddCommMonoid M] [Module ℝ M]
 noncomputable instance {M1 M2 : Type} [UnitDependent M2] :
     UnitDependent (M1 → M2) where
   scaleUnit u1 u2 f := fun m1 => scaleUnit u1 u2 (f m1)
-  scaleUnit_trans u1 u2 u3 f := by
-    funext m1
-    exact scaleUnit_trans u1 u2 u3 (f m1)
-  scaleUnit_trans' u1 u2 u3 f := by
-    funext m1
-    exact scaleUnit_trans' u1 u2 u3 (f m1)
-  scaleUnit_id u f := by
-    funext m1
-    exact scaleUnit_id u (f m1)
+  scaleUnit_trans u1 u2 u3 f := funext fun m1 =>
+    scaleUnit_trans u1 u2 u3 (f m1)
+  scaleUnit_trans' u1 u2 u3 f := funext fun m1 =>
+    scaleUnit_trans' u1 u2 u3 (f m1)
+  scaleUnit_id u f := funext fun m1 => scaleUnit_id u (f m1)
 
 @[simp]
 lemma UnitDependent.scaleUnit_apply_fun_right {M1 M2 : Type} [UnitDependent M2]
@@ -314,21 +304,15 @@ noncomputable instance {M1 M2 : Type} [AddCommMonoid M1] [Module ℝ M1]
       toFun m1 := scaleUnit u1 u2 (f m1)
       map_add' m1 m2 := by simp [scaleUnit_add]
       map_smul' := by simp [scaleUnit_smul]}
-  scaleUnit_trans u1 u2 u3 f := by
-    ext m1
-    exact scaleUnit_trans u1 u2 u3 (f m1)
-  scaleUnit_trans' u1 u2 u3 f := by
-    ext m1
-    exact scaleUnit_trans' u1 u2 u3 (f m1)
-  scaleUnit_id u f := by
-    ext m1
-    exact scaleUnit_id u (f m1)
-  scaleUnit_add u1 u2 f1 f2 := by
-    ext m
-    simp [scaleUnit_add]
-  scaleUnit_smul u1 u2 r f := by
-    ext m
-    simp [scaleUnit_smul]
+  scaleUnit_trans u1 u2 u3 f := LinearMap.ext fun m1 =>
+    scaleUnit_trans u1 u2 u3 (f m1)
+  scaleUnit_trans' u1 u2 u3 f := LinearMap.ext fun m1 =>
+    scaleUnit_trans' u1 u2 u3 (f m1)
+  scaleUnit_id u f := LinearMap.ext fun m1 => scaleUnit_id u (f m1)
+  scaleUnit_add u1 u2 f1 f2 := LinearMap.ext fun m =>
+    scaleUnit_add u1 u2 (f1 m) (f2 m)
+  scaleUnit_smul u1 u2 r f := LinearMap.ext fun m =>
+    scaleUnit_smul u1 u2 r (f m)
 
 open LinearUnitDependent ContinuousLinearUnitDependent in
 noncomputable instance {M1 M2 : Type} [AddCommGroup M1] [Module ℝ M1]
@@ -340,15 +324,11 @@ noncomputable instance {M1 M2 : Type} [AddCommGroup M1] [Module ℝ M1]
   scaleUnit u1 u2 f :=
     ContinuousLinearEquiv.arrowCongr (ContinuousLinearEquiv.refl ℝ _)
       (scaleUnitContLinearEquiv u1 u2) f
-  scaleUnit_trans u1 u2 u3 f := by
-    ext m1
-    exact scaleUnit_trans u1 u2 u3 (f m1)
-  scaleUnit_trans' u1 u2 u3 f := by
-    ext m1
-    exact scaleUnit_trans' u1 u2 u3 (f m1)
-  scaleUnit_id u f := by
-    ext m1
-    exact scaleUnit_id u (f m1)
+  scaleUnit_trans u1 u2 u3 f := ContinuousLinearMap.ext fun m1 =>
+    scaleUnit_trans u1 u2 u3 (f m1)
+  scaleUnit_trans' u1 u2 u3 f := ContinuousLinearMap.ext fun m1 =>
+    scaleUnit_trans' u1 u2 u3 (f m1)
+  scaleUnit_id u f := ContinuousLinearMap.ext fun m1 => scaleUnit_id u (f m1)
   scaleUnit_add u1 u2 f1 f2 := by simp
   scaleUnit_smul u1 u2 r f := by simp
   scaleUnit_cont u1 u2 := ContinuousLinearEquiv.continuous
@@ -357,15 +337,11 @@ noncomputable instance {M1 M2 : Type} [AddCommGroup M1] [Module ℝ M1]
 noncomputable instance {M1 M2 : Type} [UnitDependent M1] :
     UnitDependent (M1 → M2) where
   scaleUnit u1 u2 f := fun m1 => f (scaleUnit u2 u1 m1)
-  scaleUnit_trans u1 u2 u3 f := by
-    funext m1
-    simp [scaleUnit_trans]
-  scaleUnit_trans' u1 u2 u3 f := by
-    funext m1
-    simp [scaleUnit_trans']
-  scaleUnit_id u f := by
-    funext m1
-    simp [scaleUnit_id]
+  scaleUnit_trans u1 u2 u3 f := funext fun m1 =>
+    congrArg f (scaleUnit_trans u3 u2 u1 m1)
+  scaleUnit_trans' u1 u2 u3 f := funext fun m1 =>
+    congrArg f (scaleUnit_trans' u3 u2 u1 m1)
+  scaleUnit_id u f := funext fun m1 => congrArg f (scaleUnit_id u m1)
 
 @[simp]
 lemma UnitDependent.scaleUnit_apply_fun_left {M1 M2 : Type} [UnitDependent M1]
@@ -404,9 +380,8 @@ noncomputable instance instUnitDependentTwoSidedMul
   scaleUnit_id u f := by
     funext m1
     simp [scaleUnit_id]
-  scaleUnit_mul u1 u2 r f := by
-    funext m1
-    simp [MulUnitDependent.scaleUnit_mul]
+  scaleUnit_mul u1 u2 r f := funext fun m1 =>
+    MulUnitDependent.scaleUnit_mul u1 u2 r (f (scaleUnit u2 u1 m1))
 
 open LinearUnitDependent ContinuousLinearUnitDependent in
 noncomputable instance instContinuousLinearUnitDependentMap
@@ -472,24 +447,21 @@ lemma isDimensionallyCorrect_fun_iff {M1 M2 : Type} [UnitDependent M1] [UnitDepe
     {f : M1 → M2} :
     IsDimensionallyCorrect f ↔
     ∀ u1 u2 : UnitChoices, ∀ m, scaleUnit u1 u2 (f (scaleUnit u2 u1 m)) = f m := by
-  simp only [IsDimensionallyCorrect, funext_iff]
-  rfl
+  simp [IsDimensionallyCorrect, funext_iff]
 
 @[simp]
 lemma isDimensionallyCorrect_fun_left {M1 M2 : Type} [UnitDependent M1]
     {f : M1 → M2} :
     IsDimensionallyCorrect f ↔
     ∀ u1 u2 : UnitChoices, ∀ m, (f (scaleUnit u2 u1 m)) = f m := by
-  simp only [IsDimensionallyCorrect, funext_iff]
-  rfl
+  simp [IsDimensionallyCorrect, funext_iff]
 
 @[simp]
 lemma isDimensionallyCorrect_fun_right {M1 M2 : Type} [UnitDependent M2]
     {f : M1 → M2} :
     IsDimensionallyCorrect f ↔
     ∀ u1 u2 : UnitChoices, ∀ m, scaleUnit u1 u2 (f m) = f m := by
-  simp only [IsDimensionallyCorrect, funext_iff]
-  rfl
+  simp [IsDimensionallyCorrect, funext_iff]
 /-!
 
 ## Some type classes to help track dimensions
@@ -532,14 +504,8 @@ instance (M : Type) [MulAction ℝ≥0 M] [MulUnitDependent M] (d : Dimension) :
   smul a f := ⟨a • f.1, fun u1 u2 => by
     rw [smul_comm, ← f.2]
     rw [MulUnitDependent.scaleUnit_mul]⟩
-  one_smul f := by
-    ext
-    change (1 : ℝ≥0) • f.1 = f.1
-    simp
-  mul_smul a b f := by
-    ext
-    change (a * b) • f.1 = a • (b • f.1)
-    rw [smul_smul]
+  one_smul f := Subtype.ext (one_smul ℝ≥0 f.1)
+  mul_smul a b f := Subtype.ext (mul_smul a b f.1)
 
 instance (M : Type) [MulAction ℝ≥0 M] [MulUnitDependent M] (d : Dimension) :
     CarriesDimension (DimSet M d) where
@@ -548,9 +514,7 @@ instance (M : Type) [MulAction ℝ≥0 M] [MulUnitDependent M] (d : Dimension) :
 @[simp]
 lemma scaleUnit_dimSet_val {M : Type} [MulAction ℝ≥0 M] [MulUnitDependent M] (d : Dimension)
     (m : DimSet M d) (u1 u2 : UnitChoices) :
-    (scaleUnit u1 u2 m).1 = scaleUnit u1 u2 m.1 := by
-  rw [HasDim.scaleUnit_apply, m.2]
-  rfl
+    (scaleUnit u1 u2 m).1 = scaleUnit u1 u2 m.1 := (m.2 u1 u2).symm
 
 lemma DimSet.mem_iff {M : Type} [MulAction ℝ≥0 M] [MulUnitDependent M] (d : Dimension) (m : M) :
     m ∈ DimSet M d ↔ ∀ u1 u2, scaleUnit u1 u2 m = (UnitChoices.dimScale u1 u2 d) • m := by rfl
diff --git a/Physlib/Units/WithDim/Basic.lean b/Physlib/Units/WithDim/Basic.lean
index 5f9687c67..8ab69e863 100644
--- a/Physlib/Units/WithDim/Basic.lean
+++ b/Physlib/Units/WithDim/Basic.lean
@@ -75,47 +75,31 @@ lemma val_sub {d : Dimension} {M : Type} [Sub M] (m1 m2 : WithDim d M) :
 
 instance (d : Dimension) (M : Type) [AddSemigroup M] :
     AddSemigroup (WithDim d M) where
-  add_assoc m1 m2 m3 := by
-    ext
-    simp [add_assoc]
+  add_assoc m1 m2 m3 := WithDim.ext _ _ (add_assoc m1.val m2.val m3.val)
 
 instance (d : Dimension) (M : Type) [AddCommSemigroup M] :
     AddCommSemigroup (WithDim d M) where
-  add_comm m1 m2 := by
-    ext
-    simp [add_comm]
+  add_comm m1 m2 := WithDim.ext _ _ (add_comm m1.val m2.val)
 
 instance (d : Dimension) (M : Type) [AddMonoid M] :
     AddMonoid (WithDim d M) where
-  zero_add m := by
-    ext
-    simp [zero_add]
-  add_zero m := by
-    ext
-    simp [add_zero]
+  zero_add m := WithDim.ext _ _ (zero_add m.val)
+  add_zero m := WithDim.ext _ _ (add_zero m.val)
   nsmul := nsmulRec
 
 instance (d : Dimension) (M : Type) [AddCommMonoid M] :
     AddCommMonoid (WithDim d M) where
-  add_comm m1 m2 := by
-    ext
-    simp [add_comm]
+  add_comm m1 m2 := WithDim.ext _ _ (add_comm m1.val m2.val)
 
 instance (d : Dimension) (M : Type) [AddGroup M] :
     AddGroup (WithDim d M) where
-  sub_eq_add_neg m1 m2 := by
-    ext
-    simp [sub_eq_add_neg]
-  neg_add_cancel m := by
-    ext
-    simp [neg_add_cancel]
+  sub_eq_add_neg m1 m2 := WithDim.ext _ _ (sub_eq_add_neg m1.val m2.val)
+  neg_add_cancel m := WithDim.ext _ _ (neg_add_cancel m.val)
   zsmul := zsmulRec
 
 instance (d : Dimension) (M : Type) [AddCommGroup M] :
     AddCommGroup (WithDim d M) where
-  add_comm m1 m2 := by
-    ext
-    simp [add_comm]
+  add_comm m1 m2 := WithDim.ext _ _ (add_comm m1.val m2.val)
 
 instance (d : Dimension) (M : Type) [LE M] : LE (WithDim d M) where
   le m1 m2 := m1.val ≤ m2.val
@@ -133,27 +117,18 @@ lemma lt_def {d : Dimension} {M : Type} [LT M] (m1 m2 : WithDim d M) :
 
 instance (d : Dimension) (M : Type) [Preorder M] :
     Preorder (WithDim d M) where
-  le_refl m := by
-    exact le_refl m.val
-  le_trans m1 m2 m3 h12 h23 := by
-    change m1.val ≤ m3.val
-    exact le_trans h12 h23
-  lt_iff_le_not_ge m1 m2 := by
-    change m1.val < m2.val ↔ m1.val ≤ m2.val ∧ ¬ m2.val ≤ m1.val
-    exact lt_iff_le_not_ge
+  le_refl m := le_refl m.val
+  le_trans _ _ _ h12 h23 := le_trans h12 h23
+  lt_iff_le_not_ge _ _ := lt_iff_le_not_ge
 
 instance (d : Dimension) (M : Type) [PartialOrder M] :
     PartialOrder (WithDim d M) where
-  le_antisymm m1 m2 h12 h21 := by
-    ext
-    exact le_antisymm h12 h21
+  le_antisymm _ _ h12 h21 := WithDim.ext _ _ (le_antisymm h12 h21)
 
 instance (d : Dimension) (M : Type) [MulAction ℝ≥0 M] : MulAction ℝ≥0 (WithDim d M) where
   smul a m := ⟨a • m.val⟩
   one_smul m := ext _ _ (one_smul ℝ≥0 m.val)
-  mul_smul a b m := by
-    ext
-    exact mul_smul a b m.val
+  mul_smul a b m := WithDim.ext _ _ (mul_smul a b m.val)
 
 @[simp]
 lemma smul_val {d : Dimension} {M : Type} [MulAction ℝ≥0 M] (a : ℝ≥0) (m : WithDim d M) :
@@ -182,21 +157,17 @@ open UnitDependent
 
 @[simp]
 lemma val_mul_eq_mul {d1 d2 : Dimension} (m1 : WithDim d1 ℝ) (m2 : WithDim d2 ℝ) :
-    m1.val * m2.val = (m1 * m2).val := by
-  simp only [withDim_hMul_val]
+    m1.val * m2.val = (m1 * m2).val := rfl
 
 @[simp]
 lemma val_pow_two_eq_mul {d1 : Dimension} (m1 : WithDim d1 ℝ) :
-    m1.val ^ 2 = (m1 * m1).val := by
-  rw [sq]
-  rfl
+    m1.val ^ 2 = (m1 * m1).val := by simp [sq]
 
 @[simp]
 lemma scaleUnit_val_eq_scaleUnit_val {d : Dimension} (M : Type) [MulAction ℝ≥0 M]
     (u1 u2 : UnitChoices) (m1 m2 : WithDim d M) :
     (scaleUnit u1 u2 m1).val = (scaleUnit u1 u2 m2).val ↔ m1.val = m2.val := by
-  rw [← WithDim.ext_iff]
-  simp only [scaleUnit_injective]
+  rw [← WithDim.ext_iff, scaleUnit_injective]
   exact WithDim.ext_iff
 
 lemma scaleUnit_val_eq_scaleUnit_val_of_dim_eq {d1 d2 : Dimension} {M : Type} [MulAction ℝ≥0 M]
diff --git a/QuantumInfo/Capacity/Capacity.lean b/QuantumInfo/Capacity/Capacity.lean
index 0cf72a267..5a9a9f509 100644
--- a/QuantumInfo/Capacity/Capacity.lean
+++ b/QuantumInfo/Capacity/Capacity.lean
@@ -166,8 +166,7 @@ end AristotleLemmas
 theorem not_achievesRate_gt_log_dim_out (Λ : CPTPMap d₁ d₂) {R : ℝ} (hR : Real.logb 2 (Fintype.card d₂) < R): ¬Λ.AchievesRate R := by
   intro h;
   -- We show that the identity channel on the output space `d₂` emulates `Λ`. Since capacity is monotonic under emulation, `Q(Λ) ≤ Q(id_{d₂})`.
-  have h_emulate : (CPTPMap.id (dIn := d₂)).Emulates Λ := by
-    exact ⟨Λ, CPTPMap.id, by simp⟩
+  have h_emulate : (CPTPMap.id (dIn := d₂)).Emulates Λ := ⟨Λ, CPTPMap.id, by simp⟩
   -- If `Λ` achieves rate `R`, then `id_{d₂}` achieves rate `R`. This follows because if `Λ^{\otimes n}` emulates `B`, and `id^{\otimes n}` emulates `Λ^{\otimes n}` (by functoriality of tensor product), then `id^{\otimes n}` emulates `B`.
   have h_id_achieves : (CPTPMap.id (dIn := d₂)).AchievesRate R := by
     intro ε hε_pos
diff --git a/QuantumInfo/Channels/Bundled.lean b/QuantumInfo/Channels/Bundled.lean
index 63676a449..170a87735 100644
--- a/QuantumInfo/Channels/Bundled.lean
+++ b/QuantumInfo/Channels/Bundled.lean
@@ -176,9 +176,8 @@ variable [Fintype dIn] [Fintype dOut]
 namespace PMap
 
 @[ext]
-theorem ext {Λ₁ Λ₂ : PMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ := by
-  rw [PMap.mk.injEq]
-  exact HPMap.ext h
+theorem ext {Λ₁ Λ₂ : PMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ :=
+  (mk.injEq _ _ _ _).mpr (HPMap.ext h)
 
 theorem injective_toHPMap : (PMap.toHPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective :=
   fun _ _ ↦ (mk.injEq _ _ _ _).mpr
@@ -230,9 +229,8 @@ end CPMap
 namespace PTPMap
 
 @[ext]
-theorem ext {Λ₁ Λ₂ : PTPMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ := by
-  rw [PTPMap.mk.injEq]
-  exact PMap.ext h
+theorem ext {Λ₁ Λ₂ : PTPMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ :=
+  (mk.injEq _ _ _ _).mpr (PMap.ext h)
 
 theorem injective_toPMap : (PTPMap.toPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective :=
   fun _ _ ↦ (mk.injEq _ _ _ _).mpr
@@ -288,8 +286,7 @@ instance instMContinuousMapClass [DecidableEq dIn] [DecidableEq dOut] :
 
 -- @[norm_cast]
 theorem val_apply_MState [DecidableEq dIn] (M : PTPMap dIn dOut) (ρ : MState dIn) :
-    (M ρ : HermitianMat dOut ℂ) = (instFunLike.coe M) ρ := by
-  rfl
+    (M ρ : HermitianMat dOut ℂ) = (instFunLike.coe M) ρ := rfl
 
 --If we have a PTPMap, the input and output dimensions are always both nonempty (otherwise
 --we can't preserve trace) - or they're both empty. So `[Nonempty dIn]` will always suffice.
@@ -317,9 +314,8 @@ variable [DecidableEq dIn]
 
 /-- Two `CPTPMap`s are equal if their `MatrixMap`s are equal. -/
 @[ext]
-theorem ext {Λ₁ Λ₂ : CPTPMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ := by
-  rw [CPTPMap.mk.injEq]
-  exact PTPMap.ext h
+theorem ext {Λ₁ Λ₂ : CPTPMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ :=
+  (mk.injEq _ _ _ _).mpr (PTPMap.ext h)
 
 theorem injective_toPTPMap : (CPTPMap.toPTPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective :=
   fun _ _ ↦ (mk.injEq _ _ _ _).mpr
@@ -375,13 +371,11 @@ namespace PUMap
 variable [DecidableEq dIn] [DecidableEq dOut]
 
 @[ext]
-theorem ext {Λ₁ Λ₂ : PUMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ := by
-  rw [PUMap.mk.injEq]
-  exact PMap.ext h
+theorem ext {Λ₁ Λ₂ : PUMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ :=
+  (mk.injEq _ _ _ _).mpr (PMap.ext h)
 
-theorem injective_toPMap : (PUMap.toPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective := by
-  intro _ _ _
-  rwa [PUMap.mk.injEq]
+theorem injective_toPMap : (PUMap.toPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective :=
+  fun _ _ ↦ (mk.injEq _ _ _ _).mpr
 
 /-- `PUMap`s are functions from `HermitianMat`s to `HermitianMat`s. -/
 noncomputable instance instFunLike : FunLike (PUMap dIn dOut ℂ) (HermitianMat dIn ℂ) (HermitianMat dOut ℂ) where
@@ -418,13 +412,11 @@ namespace CPUMap
 variable [DecidableEq dIn] [DecidableEq dOut]
 
 @[ext]
-theorem ext {Λ₁ Λ₂ : CPUMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ := by
-  rw [CPUMap.mk.injEq, CPMap.mk.injEq]
-  exact PMap.ext h
+theorem ext {Λ₁ Λ₂ : CPUMap dIn dOut 𝕜} (h : Λ₁.map = Λ₂.map) : Λ₁ = Λ₂ :=
+  (mk.injEq _ _ _ _).mpr ((CPMap.mk.injEq _ _ _ _).mpr (PMap.ext h))
 
-theorem injective_toPMap : (CPMap.toPMap ∘ CPUMap.toCPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective := by
-  intro _ _ _
-  rwa [CPUMap.mk.injEq, CPMap.mk.injEq]
+theorem injective_toPMap : (CPMap.toPMap ∘ CPUMap.toCPMap (dIn := dIn) (dOut := dOut) (𝕜 := 𝕜)).Injective :=
+  fun _ _ h => (mk.injEq _ _ _ _).mpr ((CPMap.mk.injEq _ _ _ _).mpr h)
 
 /-- `CPUMap`s are functions from `HermitianMat`s to `HermitianMat`s. -/
 noncomputable instance instFunLike : FunLike (CPUMap dIn dOut ℂ) (HermitianMat dIn ℂ) (HermitianMat dOut ℂ) where
diff --git a/QuantumInfo/Channels/CPTP.lean b/QuantumInfo/Channels/CPTP.lean
index 6e89dff55..e24ce945f 100644
--- a/QuantumInfo/Channels/CPTP.lean
+++ b/QuantumInfo/Channels/CPTP.lean
@@ -56,9 +56,8 @@ variable (Λ : CPTPMap dIn dOut)
 def choi := Λ.map.choi_matrix
 
 /-- Two CPTPMaps are equal if their Choi matrices are equal. -/
-theorem choi_ext {Λ₁ Λ₂ : CPTPMap dIn dOut} (h : Λ₁.choi = Λ₂.choi) : Λ₁ = Λ₂ := by
-  ext1
-  exact MatrixMap.choi_equiv.injective h
+theorem choi_ext {Λ₁ Λ₂ : CPTPMap dIn dOut} (h : Λ₁.choi = Λ₂.choi) : Λ₁ = Λ₂ :=
+  CPTPMap.ext (MatrixMap.choi_equiv.injective h)
 
 /-- The Choi matrix of a channel is PSD. -/
 theorem choi_PSD_of_CPTP : Λ.map.choi_matrix.PosSemidef :=
@@ -81,12 +80,10 @@ def CPTP_of_choi_PSD_Tr {M : Matrix (dOut × dIn) (dOut × dIn) ℂ} (h₁ : M.P
 
 @[simp]
 theorem choi_of_CPTP_of_choi (M : Matrix (dOut × dIn) (dOut × dIn) ℂ) {h₁} {h₂} :
-    (CPTP_of_choi_PSD_Tr (M := M) h₁ h₂).choi = M := by
-  simp only [choi, CPTP_of_choi_PSD_Tr]
-  rw [MatrixMap.map_choi_inv]
+    (CPTP_of_choi_PSD_Tr (M := M) h₁ h₂).choi = M :=
+  MatrixMap.map_choi_inv M
 
-theorem mat_coe_eq_apply_mat [DecidableEq dOut] (ρ : MState dIn) : (Λ ρ).m = Λ.map ρ.m :=
-  rfl
+theorem mat_coe_eq_apply_mat [DecidableEq dOut] (ρ : MState dIn) : (Λ ρ).m = Λ.map ρ.m := rfl
 
 @[ext]
 theorem funext [DecidableEq dOut] {Λ₁ Λ₂ : CPTPMap dIn dOut} (h : ∀ ρ, Λ₁ ρ = Λ₂ ρ) : Λ₁ = Λ₂ :=
@@ -154,15 +151,13 @@ def id : CPTPMap dIn dIn where
 
 /-- The map `CPTPMap.id` leaves any matrix unchanged. -/
 @[simp]
-theorem id_map : (id (dIn := dIn)).map = LinearMap.id := by
-  rfl
+theorem id_map : (id (dIn := dIn)).map = LinearMap.id := rfl
 
 /-- The map `CPTPMap.id` leaves the input state unchanged. -/
 @[simp]
 theorem id_MState (ρ : MState dIn) : CPTPMap.id (dIn := dIn) ρ = ρ := by
-  apply MState.ext_m
-  rw [mat_coe_eq_apply_mat]
-  simp
+  ext
+  rfl
 
 /-- The map `CPTPMap.id` composed with any map is the same map. -/
 @[simp]
@@ -190,8 +185,7 @@ def ofEquiv (σ : dIn ≃ dOut) : CPTPMap dIn dOut where
 
 @[simp]
 theorem ofEquiv_apply (σ : dIn ≃ dOut) (ρ : MState dIn) :
-    ofEquiv σ ρ = ρ.relabel σ.symm := by
-  rfl
+    ofEquiv σ ρ = ρ.relabel σ.symm := rfl
 
 @[simp]
 theorem equiv_inverse (σ : dIn ≃ dOut)  : (ofEquiv σ) ∘ (ofEquiv σ.symm) = id (dIn := dOut) := by
@@ -216,8 +210,7 @@ def assoc' : CPTPMap (d₁ × d₂ × d₃) ((d₁ × d₂) × d₃) :=
   ofEquiv (Equiv.prodAssoc d₁ d₂ d₃).symm
 
 @[simp]
-theorem SWAP_eq_MState_SWAP (ρ : MState (d₁ × d₂)) : SWAP (d₁ := d₁) (d₂ := d₂) ρ = ρ.SWAP :=
-  rfl
+theorem SWAP_eq_MState_SWAP (ρ : MState (d₁ × d₂)) : SWAP (d₁ := d₁) (d₂ := d₂) ρ = ρ.SWAP := rfl
 
 @[simp]
 theorem assoc_eq_MState_assoc (ρ : MState ((d₁ × d₂) × d₃)) : assoc (d₁ := d₁) (d₂ := d₂) (d₃ := d₃) ρ = ρ.assoc :=
@@ -243,12 +236,8 @@ def traceLeft : CPTPMap (d₁ × d₂) d₂ :=
     --TODO: make Matrix.traceLeft a linear map, a `MatrixMap`.
   letI f (d) [Fintype d] [DecidableEq d]: Matrix (d₁ × d) (d₁ × d) ℂ →ₗ[ℂ] Matrix d d ℂ := {
     toFun x := Matrix.traceLeft x
-    map_add' := by
-      intros; ext
-      simp [Matrix.traceLeft, Finset.sum_add_distrib]
-    map_smul' := by
-      intros; ext
-      simp [Matrix.traceLeft, Finset.mul_sum]
+    map_add' x y := Matrix.traceLeft_add
+    map_smul' r x := Matrix.traceLeft_smul r
   }
   {
     toLinearMap := f d₂
@@ -277,13 +266,11 @@ def traceRight : CPTPMap (d₁ × d₂) d₁ :=
 
 @[simp]
 theorem traceLeft_eq_MState_traceLeft (ρ : MState (d₁ × d₂)) :
-    traceLeft (d₁ := d₁) (d₂ := d₂) ρ = ρ.traceLeft := by
-  rfl
+    traceLeft (d₁ := d₁) (d₂ := d₂) ρ = ρ.traceLeft := rfl
 
 @[simp]
 theorem traceRight_eq_MState_traceRight (ρ : MState (d₁ × d₂)) :
-    traceRight (d₁ := d₁) (d₂ := d₂) ρ = ρ.traceRight := by
-  rfl --It's actually pretty crazy that this is a definitional equality, cool
+    traceRight (d₁ := d₁) (d₂ := d₂) ρ = ρ.traceRight := rfl
 
 end trace
 
@@ -361,10 +348,8 @@ infixl:70 "⊗ᶜᵖ" => CPTPMap.prod
 @[simp]
 theorem prod_apply_prod (Λ₁ : CPTPMap dI₁ dO₁) (Λ₂ : CPTPMap dI₂ dO₂)
     (ρ₁ : MState dI₁) (ρ₂ : MState dI₂) :
-    (Λ₁ ⊗ᶜᵖ Λ₂) (ρ₁ ⊗ᴹ ρ₂) = (Λ₁ ρ₁) ⊗ᴹ (Λ₂ ρ₂) := by
-  apply MState.ext_m
-  exact
-    MatrixMap.kron_map_of_kron_state Λ₁.map Λ₂.map ρ₁.m ρ₂.m
+    (Λ₁ ⊗ᶜᵖ Λ₂) (ρ₁ ⊗ᴹ ρ₂) = (Λ₁ ρ₁) ⊗ᴹ (Λ₂ ρ₂) :=
+  MState.ext_m (MatrixMap.kron_map_of_kron_state Λ₁.map Λ₂.map ρ₁.m ρ₂.m)
 
 end prod
 
@@ -436,16 +421,14 @@ def ofUnitary (U : 𝐔[dIn]) : CPTPMap dIn dIn where
 
 /-- The unitary channel U conjugated by U. -/
 theorem ofUnitary_eq_conj (U : 𝐔[dIn]) (ρ : MState dIn) :
-    (ofUnitary U) ρ = ρ.U_conj U :=
-  rfl
+    (ofUnitary U) ρ = ρ.U_conj U := rfl
 
 /-- A channel is unitary iff it is `ofUnitary U`. -/
 def IsUnitary (Λ : CPTPMap dIn dIn) : Prop :=
   ∃ U, Λ = ofUnitary U
 
 /-- A channel is unitary iff it can be written as conjugation by a unitary. -/
-theorem IsUnitary_iff_U_conj (Λ : CPTPMap dIn dIn) : IsUnitary Λ ↔ ∃ U, ∀ ρ, Λ ρ = ρ.U_conj U := by
-  simp_rw [IsUnitary, ← ofUnitary_eq_conj, CPTPMap.funext_iff]
+theorem IsUnitary_iff_U_conj (Λ : CPTPMap dIn dIn) : IsUnitary Λ ↔ ∃ U, ∀ ρ, Λ ρ = ρ.U_conj U := by simp_rw [IsUnitary, ← ofUnitary_eq_conj, CPTPMap.funext_iff]
 
 theorem IsUnitary_equiv (σ : dIn ≃ dIn) : IsUnitary (ofEquiv σ) := by
   have h_unitary : ∃ U : Matrix dIn dIn ℂ, U * U.conjTranspose = 1 ∧ U.conjTranspose * U = 1 ∧ ∀ x : dIn, (∀ y : dIn, (U y x = 1) ↔ (y = σ x)) ∧ ∀ y : dIn, (U y x = 0) ↔ (y ≠ σ x) := by
@@ -571,9 +554,8 @@ private lemma exists_unitary_extending_isometry
   refine ⟨⟨Matrix.of (fun i j ↦ b j i), ?_⟩, ?_⟩
   · simp only [Matrix.mem_unitaryGroup_iff]
     ext1 i j
-    have := b.sum_inner_mul_inner (EuclideanSpace.single i 1) (EuclideanSpace.single j 1)
-    simp_all [inner]
-    exact this
+    simpa [Matrix.mul_apply, Matrix.one_apply, Matrix.star_apply, inner] using
+      b.sum_inner_mul_inner (EuclideanSpace.single i 1) (EuclideanSpace.single j 1)
   · simp [hb, u]
 
 omit [DecidableEq dOut] [Inhabited dOut] in
@@ -787,8 +769,7 @@ theorem prep_append_map_entry (X : Matrix dIn dIn ℂ)
     let prep := (id ⊗ᶜᵖ zero_prep)
     let append : CPTPMap dIn (dIn × Unit) := CPTPMap.ofEquiv (Equiv.prodPUnit dIn).symm
     (prep ∘ₘ append).map X (a₁, b₁c₁) (a₂, b₂c₂) =
-    X a₁ a₂ * τ.m b₁c₁ b₂c₂ := by
-  simp [purify_prep_append_entry]
+    X a₁ a₂ * τ.m b₁c₁ b₂c₂ := by simp [purify_prep_append_entry]
 
 /-- The complementary channel comes from tracing out the other half (the right half) of the purified channel `purify`. -/
 def complementary (Λ : CPTPMap dIn dOut) : CPTPMap dIn (dIn × dOut) :=
diff --git a/QuantumInfo/Channels/Dual.lean b/QuantumInfo/Channels/Dual.lean
index b9e38a470..acb18f0ba 100644
--- a/QuantumInfo/Channels/Dual.lean
+++ b/QuantumInfo/Channels/Dual.lean
@@ -137,11 +137,9 @@ theorem IsPositive.dual {M : MatrixMap dIn dOut ℂ} (h : M.IsPositive) : M.dual
   intro v
   have h_dual_pos : 0 ≤ (M (Matrix.vecMulVec v (star v)) * x).trace := by
     --TODO Cleanup. Should be all in terms of HermitianMat
-    apply Matrix.PosSemidef.trace_mul_nonneg;
-    · apply h;
-      exact Matrix.posSemidef_vecMulVec_self_star v;
-    · rw [← Matrix.posSemidef_iff_dotProduct_mulVec] at hx
-      exact hx;
+    exact Matrix.PosSemidef.trace_mul_nonneg
+      (h (Matrix.posSemidef_vecMulVec_self_star v))
+      (by rwa [← Matrix.posSemidef_iff_dotProduct_mulVec] at hx)
   convert h_dual_pos using 1;
   rw [ MatrixMap.Dual.trace_eq ];
   simp [ Matrix.vecMulVec, Matrix.mul_apply, Matrix.trace ];
@@ -170,13 +168,9 @@ lemma dual_unique
     (h : ∀ A B, (M A * B).trace = (A * M' B).trace) : M.dual = M' := by
   -- By definition of dual, we know that for any A and B, the trace of (M A) * B equals the trace of
   -- A * (M.dual B).
-  have h_dual : ∀ A : Matrix dIn dIn 𝕜, ∀ B : Matrix dOut dOut 𝕜, (M A * B).trace = (A * M.dual B).trace := by
-    exact fun A B => Dual.trace_eq M A B;
-  -- Since these two linear maps agree on all bases, they must be equal.
-  have h_eq : ∀ A : Matrix dIn dIn 𝕜, ∀ B : Matrix dOut dOut 𝕜, (A * M.dual B).trace = (A * M' B).trace := by
-    exact fun A B => h_dual A B ▸ h A B;
-  refine' LinearMap.ext fun B => _;
-  exact Matrix.ext_iff_trace_mul_left.mpr fun x => h_eq x B
+  refine LinearMap.ext fun B => ?_
+  exact Matrix.ext_iff_trace_mul_left.mpr fun A =>
+    (Dual.trace_eq M A B).symm.trans (h A B)
 
 /--
 The Choi matrix of the dual map is the transpose of the reindexed Choi matrix of the original map.
@@ -206,10 +200,8 @@ positive semidefinite.
 -/
 lemma dual_choi_matrix_posSemidef_of_posSemidef (M : MatrixMap dIn dOut 𝕜) (h : M.choi_matrix.PosSemidef) :
     M.dual.choi_matrix.PosSemidef := by
-  rw [ dual_choi_matrix ];
-  simp +zetaDelta at *;
-  apply_rules [ Matrix.PosSemidef.submatrix ];
-  convert h.transpose using 1
+  rw [dual_choi_matrix]
+  simpa +zetaDelta using Matrix.PosSemidef.submatrix h.transpose (Equiv.prodComm dIn dOut)
 
 /--
 The dual of the identity map is the identity map.
@@ -327,8 +319,7 @@ def dual (M : CPTPMap dIn dOut) : CPUMap dOut dIn where
   cp := .dual M.cp
 
 theorem dual_pos (M : CPTPMap dIn dOut) {T : HermitianMat dOut ℂ} (hT : 0 ≤ T) :
-    0 ≤ M.dual T := by
-  exact M.dual.pos_Hermitian hT
+    0 ≤ M.dual T := M.dual.pos_Hermitian hT
 
 /-- The dual of a CPTP map preserves POVMs. Stated here just for two-element POVMs, that is, an
 operator `T` between 0 and 1. -/
@@ -428,8 +419,7 @@ def HPMap.hermDual : HPMap dOut dIn :=
   HPMap.ofHermitianMat (LinearMapClass.linearMap f).adjoint
 
 @[simp]
-theorem HPMap.hermDual_hermDual : f.hermDual.hermDual = f := by
-  simp [hermDual]
+theorem HPMap.hermDual_hermDual : f.hermDual.hermDual = f := by simp [hermDual]
 
 open RealInnerProductSpace
 
@@ -466,8 +456,7 @@ theorem MatrixMap.IsPositive.hermDual (h : MatrixMap.IsPositive f.map) : f.hermD
 theorem HPMap.hermDual_Unital [DecidableEq dIn] [DecidableEq dOut] (h : MatrixMap.IsTracePreserving f.map) :
     f.hermDual.map.Unital := by
   suffices f.hermDual 1 = 1 by --todo: make this is an accessible 'constructor' for Unital
-    rw [HermitianMat.ext_iff] at this
-    exact this
+    rwa [HermitianMat.ext_iff] at this
   open RealInnerProductSpace in
   apply ext_inner_left ℝ
   intro v
@@ -487,8 +476,7 @@ def hermDual (M : PTPMap dIn dOut) : PUMap dOut dIn where
   unital := M.TP.hermDual
 
 theorem hermDual_pos (M : PTPMap dIn dOut) {T : HermitianMat dOut ℂ} (hT : 0 ≤ T) :
-    0 ≤ M.hermDual T := by
-  exact M.hermDual.pos_Hermitian hT
+    0 ≤ M.hermDual T := M.hermDual.pos_Hermitian hT
 
 /-- The dual of a PTP map preserves POVMs. Stated here just for two-element POVMs, that is, an
 operator `T` between 0 and 1. -/
diff --git a/QuantumInfo/Channels/MatrixMap.lean b/QuantumInfo/Channels/MatrixMap.lean
index c5eea82a5..ba94eb425 100644
--- a/QuantumInfo/Channels/MatrixMap.lean
+++ b/QuantumInfo/Channels/MatrixMap.lean
@@ -163,8 +163,7 @@ theorem exists_kraus (Φ : MatrixMap A B R) :
           (M₀ ∘ e) x * Matrix.single i₁ i₂ (1 : R) * ((N₀ ∘ e) x).conjTranspose) j₁ j₂
         =
       ∑ y : K, (M₀ y * Matrix.single i₁ i₂ (1 : R) * (N₀ y).conjTranspose) j₁ j₂ := by
-    rw [Matrix.sum_apply]
-    exact e.sum_comp
+    simpa [Matrix.sum_apply] using e.sum_comp
       (fun y : K => (M₀ y * Matrix.single i₁ i₂ (1 : R) * (N₀ y).conjTranspose) j₁ j₂)
   rw [hsum_reindex, Fintype.sum_prod_type, Fintype.sum_prod_type, Fintype.sum_prod_type]
   simp [M₀, N₀, Matrix.mul_apply, Matrix.single, ite_and]
@@ -192,13 +191,11 @@ def submatrix (f : B → A) : MatrixMap A B R where
   map_smul' := by simp [Matrix.submatrix_smul]
 
 @[simp]
-theorem submatrix_id : submatrix R _root_.id = id A R := by
-  ext1; simp
+theorem submatrix_id : submatrix R _root_.id = id A R := rfl
 
 @[simp]
 theorem submatrix_comp (f : C → B) (g : B → A) :
-    submatrix R f ∘ₗ submatrix R g = submatrix R (g ∘ f) := by
-  ext1; simp
+    submatrix R f ∘ₗ submatrix R g = submatrix R (g ∘ f) := rfl
 
 end submatrix
 
@@ -285,37 +282,29 @@ theorem kron_def [CommSemiring R] (M₁ : MatrixMap A B R) (M₂ : MatrixMap C D
 section kron_lemmas
 variable [CommSemiring R]
 
-theorem add_kron (ML₁ ML₂ : MatrixMap A B R) (MR : MatrixMap C D R) : (ML₁ + ML₂) ⊗ₖₘ MR = ML₁ ⊗ₖₘ MR + ML₂ ⊗ₖₘ MR := by
-  simp [kron, TensorProduct.map_add_left, Matrix.submatrix_add]
+theorem add_kron (ML₁ ML₂ : MatrixMap A B R) (MR : MatrixMap C D R) : (ML₁ + ML₂) ⊗ₖₘ MR = ML₁ ⊗ₖₘ MR + ML₂ ⊗ₖₘ MR := by simp [kron, TensorProduct.map_add_left, Matrix.submatrix_add]
 
-theorem kron_add (ML : MatrixMap A B R) (MR₁ MR₂ : MatrixMap C D R) : ML ⊗ₖₘ (MR₁ + MR₂) = ML ⊗ₖₘ MR₁ + ML ⊗ₖₘ  MR₂ := by
-  simp [kron, TensorProduct.map_add_right, Matrix.submatrix_add]
+theorem kron_add (ML : MatrixMap A B R) (MR₁ MR₂ : MatrixMap C D R) : ML ⊗ₖₘ (MR₁ + MR₂) = ML ⊗ₖₘ MR₁ + ML ⊗ₖₘ  MR₂ := by simp [kron, TensorProduct.map_add_right, Matrix.submatrix_add]
 
-theorem smul_kron (r : R) (ML : MatrixMap A B R) (MR : MatrixMap C D R) : (r • ML) ⊗ₖₘ MR = r • (ML ⊗ₖₘ MR) := by
-  simp [kron, TensorProduct.map_smul_left, Matrix.submatrix_smul]
+theorem smul_kron (r : R) (ML : MatrixMap A B R) (MR : MatrixMap C D R) : (r • ML) ⊗ₖₘ MR = r • (ML ⊗ₖₘ MR) := by simp [kron, TensorProduct.map_smul_left, Matrix.submatrix_smul]
 
-theorem kron_smul (r : R) (ML : MatrixMap A B R) (MR : MatrixMap C D R) : ML ⊗ₖₘ (r • MR) = r • (ML ⊗ₖₘ MR) := by
-  simp [kron, TensorProduct.map_smul_right, Matrix.submatrix_smul]
+theorem kron_smul (r : R) (ML : MatrixMap A B R) (MR : MatrixMap C D R) : ML ⊗ₖₘ (r • MR) = r • (ML ⊗ₖₘ MR) := by simp [kron, TensorProduct.map_smul_right, Matrix.submatrix_smul]
 
 @[simp]
-theorem zero_kron (MR : MatrixMap C D R) : (0 : MatrixMap A B R) ⊗ₖₘ MR = 0 := by
-  simp [kron]
+theorem zero_kron (MR : MatrixMap C D R) : (0 : MatrixMap A B R) ⊗ₖₘ MR = 0 := by simp [kron]
 
 @[simp]
-theorem kron_zero (ML : MatrixMap A B R) : ML ⊗ₖₘ (0 : MatrixMap C D R) = 0 := by
-  simp [kron]
+theorem kron_zero (ML : MatrixMap A B R) : ML ⊗ₖₘ (0 : MatrixMap C D R) = 0 := by simp [kron]
 
 variable [DecidableEq B] in
-theorem kron_id_id : (id A R ⊗ₖₘ id B R) = id (A × B) R := by
-  simp [kron]
+theorem kron_id_id : (id A R ⊗ₖₘ id B R) = id (A × B) R := by simp [kron]
 
 variable {Dl₁ Dl₂ Dl₃ Dr₁ Dr₂ Dr₃ : Type*}
   [Fintype Dl₁] [Fintype Dl₂] [Fintype Dl₃] [Fintype Dr₁] [Fintype Dr₂] [Fintype Dr₃]
   [DecidableEq Dl₁] [DecidableEq Dl₂] [DecidableEq Dr₁] [DecidableEq Dr₂] in
 /-- For maps L₁, L₂, R₁, and R₂, the product (L₂ ∘ₗ L₁) ⊗ₖₘ (R₂ ∘ₗ R₁) = (L₂ ⊗ₖₘ R₂) ∘ₗ (L₁ ⊗ₖₘ R₁) -/
 theorem kron_comp_distrib (L₁ : MatrixMap Dl₁ Dl₂ R) (L₂ : MatrixMap Dl₂ Dl₃ R) (R₁ : MatrixMap Dr₁ Dr₂ R)
-    (R₂ : MatrixMap Dr₂ Dr₃ R) : (L₂ ∘ₗ L₁) ⊗ₖₘ (R₂ ∘ₗ R₁) = (L₂ ⊗ₖₘ R₂) ∘ₗ (L₁ ⊗ₖₘ R₁) := by
-  simp [kron, TensorProduct.map_comp, ← Matrix.toLin_mul, Matrix.submatrix_mul_equiv, ← LinearMap.toMatrix_comp]
+    (R₂ : MatrixMap Dr₂ Dr₃ R) : (L₂ ∘ₗ L₁) ⊗ₖₘ (R₂ ∘ₗ R₁) = (L₂ ⊗ₖₘ R₂) ∘ₗ (L₁ ⊗ₖₘ R₁) := by simp [kron, TensorProduct.map_comp, ← Matrix.toLin_mul, Matrix.submatrix_mul_equiv, ← LinearMap.toMatrix_comp]
 
 end kron_lemmas
 
@@ -340,9 +329,8 @@ theorem kron_map_of_kron_state [CommRing R] (M₁ : MatrixMap A B R) (M₂ : Mat
   congr
   --TODO: Cleanup, these two branches are nearly identical (separate lemma?)
   · have h_linear : M₁ MA = ∑ i : A, ∑ i_1 : A, MA i i_1 • M₁ (Matrix.single i i_1 1) := by
-      have h_linear : M₁ MA = M₁ (∑ i : A, ∑ i_1 : A, Matrix.single i i_1 (MA i i_1)) := by
-        congr;
-        exact Matrix.matrix_eq_sum_single MA
+      have h_linear : M₁ MA = M₁ (∑ i : A, ∑ i_1 : A, Matrix.single i i_1 (MA i i_1)) :=
+        congrArg M₁ (Matrix.matrix_eq_sum_single MA)
       simp [ h_linear, Matrix.single]
       congr! 2 with i _ j _
       convert M₁.map_smul (MA i j) (Matrix.of fun i' j' ↦ if i = i' ∧ j = j' then 1 else 0) using 2
@@ -350,10 +338,8 @@ theorem kron_map_of_kron_state [CommRing R] (M₁ : MatrixMap A B R) (M₂ : Mat
       simp
     simp [h_linear, mul_comm, Matrix.sum_apply]
   · have h_expand : M₂ MC = ∑ i : C, ∑ j : C, MC i j • M₂ (Matrix.single i j 1) := by
-      have h_expand : MC = ∑ i : C, ∑ j : C, MC i j • Matrix.single i j 1 := by
-        ext i j
-        simp [Matrix.sum_apply, Matrix.single]
-        rw [ Finset.sum_eq_single i ] <;> aesop
+      have h_expand : MC = ∑ i : C, ∑ j : C, Matrix.single i j (MC i j) :=
+        Matrix.matrix_eq_sum_single MC
       conv_lhs => rw [ h_expand ];
       simp [map_sum]
       congr! 2 with i _ j _
@@ -394,12 +380,10 @@ theorem submatrix_kron_submatrix [CommSemiring R] (f : B → A) (g : D → C) :
   simp [Prod.map, Matrix.single, ite_and]
 
 theorem submatrix_kron_id [CommSemiring R] (f : B → A) :
-    submatrix R f ⊗ₖₘ id C R = submatrix R (Prod.map f _root_.id) := by
-  simp [← submatrix_kron_submatrix]
+    submatrix R f ⊗ₖₘ id C R = submatrix R (Prod.map f _root_.id) := submatrix_kron_submatrix f _root_.id
 
 theorem id_kron_submatrix [CommSemiring R] (f : B → A) :
-    id C R ⊗ₖₘ submatrix R f = submatrix R (Prod.map _root_.id f) := by
-  simp [← submatrix_kron_submatrix]
+    id C R ⊗ₖₘ submatrix R f = submatrix R (Prod.map _root_.id f) := submatrix_kron_submatrix _root_.id f
 
 end kron
 
@@ -453,8 +437,7 @@ theorem piProd_comp
   [∀ i, Fintype (d₂ i)] [∀ i, DecidableEq (d₂ i)]
   [∀ i, Fintype (d₃ i)] [∀ i, DecidableEq (d₃ i)]
   (Λ₁ : ∀ i, MatrixMap (d₁ i) (d₂ i) R) (Λ₂ : ∀ i, MatrixMap (d₂ i) (d₃ i) R) :
-    piProd (fun i ↦ (Λ₂ i) ∘ₗ (Λ₁ i)) = (piProd Λ₂) ∘ₗ (piProd Λ₁) := by
-  simp [piProd, PiTensorProduct.map_comp, ← Matrix.toLin_mul, ← LinearMap.toMatrix_comp]
+    piProd (fun i ↦ (Λ₂ i) ∘ₗ (Λ₁ i)) = (piProd Λ₂) ∘ₗ (piProd Λ₁) := by simp [piProd, PiTensorProduct.map_comp, ← Matrix.toLin_mul, ← LinearMap.toMatrix_comp]
 
 @[simp]
 theorem piProd_id :
diff --git a/QuantumInfo/Channels/Pinching.lean b/QuantumInfo/Channels/Pinching.lean
index d734407a7..07b5d2dc1 100644
--- a/QuantumInfo/Channels/Pinching.lean
+++ b/QuantumInfo/Channels/Pinching.lean
@@ -103,8 +103,7 @@ def pinching_map (ρ : MState d) : CPTPMap d d ℂ :=
 
 theorem pinchingMap_apply_M (σ ρ : MState d) : (pinching_map σ ρ).M =
   ⟨_, (MatrixMap.of_kraus_isCompletelyPositive
-    (HermitianMat.mat ∘ pinching_kraus σ)).IsPositive.IsHermitianPreserving ρ.M.H⟩ := by
-  rfl
+    (HermitianMat.mat ∘ pinching_kraus σ)).IsPositive.IsHermitianPreserving ρ.M.H⟩ := rfl
 
 theorem pinching_eq_sum_conj (σ ρ : MState d) : (pinching_map σ ρ).M =
     ∑ k, (pinching_kraus σ k).mat * ρ.M * (pinching_kraus σ k).mat := by
@@ -191,7 +190,7 @@ theorem pinching_bound (ρ σ : MState d) : ρ.M ≤ (↑(Fintype.card (spectrum
   simp only [HermitianMat.mat_finset_sum]
   simp only [Matrix.mul_sum, Matrix.sum_mul, Matrix.sum_mulVec, dotProduct_sum]
   simp only [MState.pure]
-  dsimp [MState.m]
+  dsimp only [MState.m]
   --This out to be Cauchy-Schwarz.
   have hschwarz := inner_mul_inner_self_le (𝕜 := ℂ) (E := EuclideanSpace ℂ (↑(spectrum ℝ σ.m)))
     (x := .toLp 2 fun i ↦ 1) (y := .toLp 2 fun k ↦ (
@@ -240,8 +239,7 @@ theorem ker_le_ker_pinching_of_PosDef (ρ σ : MState d) (hpos : σ.m.PosDef) :
     simp [LinearMap.ker_eq_bot', HermitianMat.ker] at this ⊢
     intro m hm
     simpa only [WithLp.ofLp_eq_zero] using this m congr($hm)
-  rw [h_ker]
-  exact bot_le
+  simp [h_ker]
 
 theorem pinching_idempotent (ρ σ : MState d) :
     (pinching_map σ) (pinching_map σ ρ) = (pinching_map σ ρ) := by
@@ -325,10 +323,9 @@ theorem pinching_map_ker_le (ρ σ : MState d) : (pinching_map σ ρ).M.ker ≤
   simp only [WithLp.toLp_sum, WithLp.toLp_ofLp] at hv_sum
   rw [← hv_sum]
   exact Submodule.sum_mem _ fun k _ ↦ by
-    have h1 := hv k;
-    simp only [HermitianMat.ker_conj ρ.nonneg, HermitianMat.conjTranspose_mat,
-      Submodule.mem_comap] at h1 ⊢
-    exact h1
+    change Matrix.toEuclideanLin (pinching_kraus σ k).mat v ∈ ρ.M.ker
+    simpa only [HermitianMat.ker_conj ρ.nonneg, HermitianMat.conjTranspose_mat,
+      Submodule.mem_comap] using hv k
 
 
 noncomputable section AristotleLemmas
diff --git a/QuantumInfo/Channels/Unbundled.lean b/QuantumInfo/Channels/Unbundled.lean
index 8cf0b84ef..df0315d23 100644
--- a/QuantumInfo/Channels/Unbundled.lean
+++ b/QuantumInfo/Channels/Unbundled.lean
@@ -86,8 +86,7 @@ theorem comp (h₁ : M.IsTracePreserving) (h₂ : M₂.IsTracePreserving) :
 
 /-- The identity MatrixMap IsTracePreserving. -/
 @[simp]
-theorem id : (id A R).IsTracePreserving := by
-  simp [IsTracePreserving, MatrixMap.id]
+theorem id : (id A R).IsTracePreserving := fun _ => rfl
 
 variable {R : Type*} [CommSemiring R] in
 /-- Unit linear combinations of IsTracePreserving maps are IsTracePreserving. -/
@@ -202,8 +201,7 @@ theorem map_1 (h : M.Unital) : M 1 = 1 :=
 
 /-- The identity `MatrixMap` is `Unital`. -/
 @[simp]
-theorem id : (id A R).Unital := by
-  simp [Unital, MatrixMap.id]
+theorem id : (id A R).Unital := rfl
 
 --TODO: Closed under composition, kronecker products, it's iff M.choi_matrix.traceLeft = 1...
 
@@ -308,13 +306,11 @@ theorem of_Fintype  {M : MatrixMap A B R} (h : IsCompletelyPositive M)
     simp only [EmbeddingLike.apply_eq_iff_eq, Equiv.symm_apply_apply]
   constructor
   · intro h₂
-    simp [MatrixMap.IsPositive]
-    exact h n
+    simpa [MatrixMap.IsPositive] using h n
   · intro h x hx
     specialize h (hx.submatrix (fun p : A × Fin n => (p.1, e.symm p.2)))
     rw [h_submatrix]
-    simp only [LinearMap.coe_comp, Function.comp_apply, submatrix_apply]
-    exact h.submatrix _
+    simpa only [LinearMap.coe_comp, Function.comp_apply, submatrix_apply] using h.submatrix _
 
 /- Every `MatrixMap` that `IsCompletelyPositive` also `IsPositiveMap`. -/
 theorem IsPositive {M : MatrixMap A B R}
@@ -365,15 +361,13 @@ theorem id : (id A R).IsCompletelyPositive := by
 theorem add {M₁ M₂ : MatrixMap A B R} (h₁ : M₁.IsCompletelyPositive) (h₂ : M₂.IsCompletelyPositive) :
     (M₁ + M₂).IsCompletelyPositive :=
   fun n _ h ↦ by
-  simp only [add_kron, LinearMap.add_apply]
-  exact Matrix.PosSemidef.add (h₁ n h) (h₂ n h)
+  simpa only [add_kron, LinearMap.add_apply] using Matrix.PosSemidef.add (h₁ n h) (h₂ n h)
 
 /-- Nonnegative scalings of `IsCompletelyPositive` maps are `IsCompletelyPositive`. -/
 theorem smul {M : MatrixMap A B R} (hM : M.IsCompletelyPositive) {x : R} (hx : 0 ≤ x) :
     (x • M).IsCompletelyPositive :=
-  fun n ρ h ↦ by
-    rw [MatrixMap.smul_kron]
-    exact (hM n h).smul hx
+  fun n _ h ↦ by
+    simpa only [smul_kron, LinearMap.smul_apply] using (hM n h).smul hx
 
 variable (A B) in
 /-- The zero map `IsCompletelyPositive`. -/
@@ -482,9 +476,7 @@ omit [DecidableEq A] in
 theorem of_kraus_isPositive (K : κ → Matrix B A ℂ) :
     (of_kraus K K).IsPositive := by
   rw [of_kraus]
-  apply IsPositive_sum
-  intro k
-  exact conj_isPositive (K k)
+  exact IsPositive_sum _ fun k => conj_isPositive (K k)
 
 theorem conj_kron (M : Matrix B A 𝕜) (N : Matrix D C 𝕜) [DecidableEq C] :
     conj M ⊗ₖₘ conj N = conj (M ⊗ₖ N) := by
@@ -552,14 +544,11 @@ theorem of_kraus_eq_sum_conj (K : κ → Matrix B A 𝕜) :
 
 theorem of_kraus_CP (K : κ → Matrix B A 𝕜) : (of_kraus K K).IsCompletelyPositive := by
   -- By definition of `MatrixMap.of_kraus`, we know that it is a sum of congruence maps.
-  have h_sum_congruence : MatrixMap.of_kraus K K = ∑ k, conj (K k) := by
-    -- By definition of `MatrixMap.of_kraus`, we know that it is equal to the sum of the congruence maps of each Kraus operator.
-    apply of_kraus_eq_sum_conj
-  have h_congruence_CP : ∀ k, (conj (K k)).IsCompletelyPositive := by
-    intro k; exact (by
-    convert congruence_CP ( K k ) using 1;
-    -- Since B is a finite type, we can use the fact that finite types have decidable equality.
-    apply Classical.decEq);
+  have h_sum_congruence : MatrixMap.of_kraus K K = ∑ k, conj (K k) :=
+    of_kraus_eq_sum_conj K
+  have h_congruence_CP : ∀ k, (conj (K k)).IsCompletelyPositive := fun k => by
+    convert congruence_CP (K k) using 1
+    exact Classical.decEq _
   exact h_sum_congruence.symm ▸ IsCompletelyPositive.finset_sum h_congruence_CP
 
 theorem exists_kraus_of_choi_PSD
@@ -598,8 +587,7 @@ theorem choi_PSD_iff_CP_map (M : MatrixMap A B R) :
     exact this ( Matrix.PosSemidef.outer_self_conj _ )
   · intro h_psd
     obtain ⟨K, hK⟩ := exists_kraus_of_choi_PSD M.choi_matrix h_psd
-    rw [choi_matrix_inj hK]
-    exact of_kraus_CP K
+    exact (choi_matrix_inj hK) ▸ of_kraus_CP K
 
 omit [Fintype B] [DecidableEq A] in
 theorem conj_eq_mulRightLinearMap_comp_mulRightLinearMap (y : Matrix B A R) :
diff --git a/QuantumInfo/ClassicalInfo/Distribution.lean b/QuantumInfo/ClassicalInfo/Distribution.lean
index 7ec3fd2f0..ac1ab7088 100644
--- a/QuantumInfo/ClassicalInfo/Distribution.lean
+++ b/QuantumInfo/ClassicalInfo/Distribution.lean
@@ -46,8 +46,7 @@ sum is 1. The fact that the values are at most 1 is derived as a consequence. -/
 def mk' (f : α → ℝ) (h₁ : ∀i, 0 ≤ f i) (hN : ∑ i, f i = 1) : ProbDistribution α :=
   have h₃ : ∀x, f x ≤ 1 := by
     intro x
-    simp [← hN, Fintype.sum_eq_sum_compl_add x]
-    exact Finset.sum_nonneg' h₁
+    simpa [← hN, Fintype.sum_eq_sum_compl_add x] using Finset.sum_nonneg' h₁
   ⟨ fun i ↦ ⟨f i, ⟨h₁ i, h₃ i⟩⟩, hN⟩
 
 instance instFunLikeProb : FunLike (ProbDistribution α) α Prob where
@@ -63,13 +62,11 @@ theorem normalized (d : ProbDistribution α) : Finset.sum Finset.univ (fun i ↦
 abbrev prob (d : ProbDistribution α) := (d : α → Prob)
 
 @[simp]
-theorem fun_eq_val (d : ProbDistribution α) : d.val = d :=
-  rfl
+theorem fun_eq_val (d : ProbDistribution α) : d.val = d := rfl
 
 @[simp]
 theorem funlike_apply (d : α → Prob) (h : _) (x : α) :
-    DFunLike.coe (self := instFunLikeProb) ⟨d, h⟩ x = d x :=
-  rfl
+    DFunLike.coe (self := instFunLikeProb) ⟨d, h⟩ x = d x := rfl
 
 @[ext]
 theorem ext {p q : ProbDistribution α} (h : ∀ x, p x = q x) : p = q :=
@@ -87,12 +84,10 @@ def constant (x : α) : ProbDistribution α :=
   ⟨fun y ↦ if x = y then 1 else 0,
     by simp [apply_ite]⟩
 
-theorem constant_def (x : α) : (constant x : α → Prob) = fun y ↦ if x = y then 1 else 0 := by
-  rfl
+theorem constant_def (x : α) : (constant x : α → Prob) = fun y ↦ if x = y then 1 else 0 := rfl
 
 @[simp]
-theorem constant_eq (x : α) : constant x y = if x = y then 1 else 0 := by
-  rfl
+theorem constant_eq (x : α) : constant x y = if x = y then 1 else 0 := rfl
 
 @[simp]
 theorem constant_def' (x y : α) : (constant x : α → Prob) y = if x = y then 1 else 0 := by
@@ -136,8 +131,7 @@ def prod (d1 : ProbDistribution α) (d2 : ProbDistribution β) : ProbDistributio
     simp [← Finset.mul_sum, Fintype.sum_prod_type]⟩
 
 @[simp]
-theorem prod_def (x : α) (y : β) : prod d1 d2 ⟨x, y⟩ = (d1 x) * (d2 y) :=
-  rfl
+theorem prod_def (x : α) (y : β) : prod d1 d2 ⟨x, y⟩ = (d1 x) * (d2 y) := rfl
 
 /-- Given a distribution on α, extend it to a distribution on `Sum α β` by
   giving it no support on `β`. -/
@@ -172,23 +166,19 @@ def congr (σ : α ≃ β) : ProbDistribution α ≃ ProbDistribution β := by
   case left_inv =>
     intro d
     ext i
-    unfold relabel
-    simp only [← fun_eq_val, Equiv.symm_apply_apply, Subtype.coe_eta]
+    simp only [relabel, ← fun_eq_val, Equiv.symm_apply_apply, Subtype.coe_eta]
   case right_inv =>
     intro d
     ext i
-    unfold relabel
-    simp only [← fun_eq_val, Equiv.apply_symm_apply, Subtype.coe_eta]
+    simp only [relabel, ← fun_eq_val, Equiv.apply_symm_apply, Subtype.coe_eta]
 
 @[simp]
-theorem congr_apply (σ : α ≃ β) (d : ProbDistribution α) (j : β): (congr σ d) j = d (σ.symm j) := by
-  rfl
+theorem congr_apply (σ : α ≃ β) (d : ProbDistribution α) (j : β): (congr σ d) j = d (σ.symm j) := rfl
 
 /-- The inverse and congruence operations for distributions commute -/
 @[simp]
 theorem congr_symm_apply (σ : α ≃ β) :
-    (ProbDistribution.congr σ).symm = ProbDistribution.congr σ.symm := by
-  rfl
+    (ProbDistribution.congr σ).symm = ProbDistribution.congr σ.symm := rfl
 
 /-- The distribution on Fin 2 corresponding to a coin with probability p.
   Chance p of 1, 1-p of 0. -/
@@ -196,12 +186,10 @@ def coin (p : Prob) : ProbDistribution (Fin 2) :=
   ⟨(if · = 0 then p else 1 - p), by simp⟩
 
 @[simp]
-theorem coin_val_zero (p : Prob) : coin p 0 = p := by
-  simp [coin]
+theorem coin_val_zero (p : Prob) : coin p 0 = p := rfl
 
 @[simp]
-theorem coin_val_one (p : Prob) : coin p 1 = 1 - p := by
-  simp [coin]
+theorem coin_val_one (p : Prob) : coin p 1 = 1 - p := rfl
 
 /-- Every distribution on two variable is some coin. -/
 theorem fin_two_eq_coin (d : ProbDistribution (Fin 2)) : d = coin (d 0) := by
@@ -260,10 +248,8 @@ theorem expect_val_eq_mixable_mix (d : ProbDistribution (Fin 2)) (x₁ x₂ : T)
   simp only [Mixable.mix, expect_val, DFunLike.coe, Mixable.to_U_of_mkT]
   calc
     ∑ i : Fin (Nat.succ 0).succ, (d i : ℝ) • Mixable.to_U (![x₁, x₂] i) =
-        ∑ i, (d i : ℝ) • Mixable.to_U (![x₁, x₂] i) := by
-      simp
-    _ = (d 0 : ℝ) • Mixable.to_U (![x₁, x₂] 0) + (d 1 : ℝ) • Mixable.to_U (![x₁, x₂] 1) := by
-      simp
+        ∑ i, (d i : ℝ) • Mixable.to_U (![x₁, x₂] i) := by simp
+    _ = (d 0 : ℝ) • Mixable.to_U (![x₁, x₂] 0) + (d 1 : ℝ) • Mixable.to_U (![x₁, x₂] 1) := by simp
     _ = (d 0 : ℝ) • Mixable.to_U x₁ + (1 - d 0).val • Mixable.to_U x₂ := by
       congr
       simpa only [Subtype.ext_iff, Prob.coe_one_minus, eq_sub_iff_add_eq, add_comm,
diff --git a/QuantumInfo/ClassicalInfo/Entropy.lean b/QuantumInfo/ClassicalInfo/Entropy.lean
index c7f579ad7..3c9e381a1 100644
--- a/QuantumInfo/ClassicalInfo/Entropy.lean
+++ b/QuantumInfo/ClassicalInfo/Entropy.lean
@@ -31,13 +31,11 @@ def H₁ : Prob → ℝ :=
 
 /-- H₁ of 0 is zero.-/
 @[simp]
-def H₁_zero_eq_zero : H₁ 0 = 0 := by
-  simp [H₁]
+def H₁_zero_eq_zero : H₁ 0 = 0 := Real.negMulLog_zero
 
 /-- H₁ of 1 is zero.-/
 @[simp]
-def H₁_one_eq_zero : H₁ 1 = 0 := by
-  simp [H₁]
+def H₁_one_eq_zero : H₁ 1 = 0 := Real.negMulLog_one
 
 /-- Entropy is nonnegative. -/
 theorem H₁_nonneg (p : Prob) : 0 ≤ H₁ p := by
@@ -112,17 +110,14 @@ theorem Hₛ_le_log_d (d : ProbDistribution α) : Hₛ d ≤ Real.log (Fintype.c
 
 /-- The shannon entropy of a constant variable is zero. -/
 @[simp]
-theorem Hₛ_constant_eq_zero {i : α} : Hₛ (ProbDistribution.constant i) = 0 := by
-  simp [Hₛ, apply_ite]
+theorem Hₛ_constant_eq_zero {i : α} : Hₛ (ProbDistribution.constant i) = 0 := by simp [Hₛ, apply_ite]
 
 /-- Shannon entropy of a uniform distribution is ln d. -/
 theorem Hₛ_uniform [Nonempty α] :
-    Hₛ (ProbDistribution.uniform (α := α)) = Real.log (Finset.univ.card (α := α)) := by
-  simp [Hₛ, ProbDistribution.prob, H₁, Real.negMulLog]
+    Hₛ (ProbDistribution.uniform (α := α)) = Real.log (Finset.univ.card (α := α)) := by simp [Hₛ, ProbDistribution.prob, H₁, Real.negMulLog]
 
 /-- Shannon entropy of two-event distribution. -/
-theorem Hₛ_coin (p : Prob) : Hₛ (ProbDistribution.coin p) = Real.binEntropy p := by
-  simp [Hₛ, H₁, ProbDistribution.coin, Real.binEntropy_eq_negMulLog_add_negMulLog_one_sub]
+theorem Hₛ_coin (p : Prob) : Hₛ (ProbDistribution.coin p) = Real.binEntropy p := by simp [Hₛ, H₁, ProbDistribution.coin, Real.binEntropy_eq_negMulLog_add_negMulLog_one_sub]
 
 lemma Hₛ_eq_of_multiset_map_eq (d₁ : ProbDistribution α) (d₂ : ProbDistribution β)
     (h : Multiset.map d₁.prob Finset.univ.val = Multiset.map d₂.prob Finset.univ.val) :
diff --git a/QuantumInfo/ClassicalInfo/Prob.lean b/QuantumInfo/ClassicalInfo/Prob.lean
index ab934e428..b401b43fe 100644
--- a/QuantumInfo/ClassicalInfo/Prob.lean
+++ b/QuantumInfo/ClassicalInfo/Prob.lean
@@ -53,24 +53,19 @@ instance instMul : Mul Prob :=
     ⟨mul_nonneg x.2.1 y.2.1, mul_le_one₀ x.2.2 y.2.1 y.2.2⟩⟩⟩
 
 @[simp, norm_cast]
-theorem coe_zero : (0 : Prob).val = 0 :=
-  rfl
+theorem coe_zero : (0 : Prob).val = 0 := rfl
 
 @[simp, norm_cast]
-theorem coe_one : (1 : Prob).val = 1 :=
-  rfl
+theorem coe_one : (1 : Prob).val = 1 := rfl
 
 @[simp, norm_cast]
-theorem coe_mul (x y : Prob) : (x * y).val = x.val * y.val :=
-  rfl
+theorem coe_mul (x y : Prob) : (x * y).val = x.val * y.val := rfl
 
 @[simp, norm_cast]
-theorem coe_inf (x y : Prob) : (x ⊓ y).val = x.val ⊓ y.val :=
-  rfl
+theorem coe_inf (x y : Prob) : (x ⊓ y).val = x.val ⊓ y.val := rfl
 
 @[simp, norm_cast]
-theorem coe_sup (x y : Prob) : (x ⊔ y).val = x.val ⊔ y.val :=
-  rfl
+theorem coe_sup (x y : Prob) : (x ⊔ y).val = x.val ⊔ y.val := rfl
 
 instance instCommMonoidWithZero : CommMonoidWithZero Prob where
   mul_assoc := by intros; ext; simp [mul_assoc]
@@ -120,16 +115,14 @@ theorem ne_iff {x y : Prob} : (x : ℝ) ≠ (y : ℝ) ↔ x ≠ y :=
   not_congr <| Prob.ext_iff.symm
 
 @[simp, norm_cast]
-theorem toReal_mul (x y : Prob) : (x * y : Prob) = (x : ℝ) * (y : ℝ) := by
-  simp only [coe_mul]
+theorem toReal_mul (x y : Prob) : (x * y : Prob) = (x : ℝ) * (y : ℝ) := rfl
 
 /-- Coercion `Prob → ℝ≥0`. -/
 @[coe] def toNNReal : Prob → ℝ≥0 :=
   fun p ↦ ⟨p.val, zero_le_coe⟩
 
 @[simp]
-theorem toNNReal_mk : toNNReal { val := x, property := hx} = { val := x, property := hx.1 } :=
-  rfl
+theorem toNNReal_mk : toNNReal { val := x, property := hx} = { val := x, property := hx.1 } := rfl
 
 instance : Coe Prob ℝ≥0 := ⟨toNNReal⟩
 
@@ -142,12 +135,10 @@ protected theorem eq_iff_nnreal (n m : Prob) : (n : ℝ≥0) = (m : ℝ≥0) ↔
   simp only [toNNReal_mk, Subtype.mk.injEq, NNReal]
 
 @[simp, norm_cast]
-theorem toNNReal_zero : (0 : Prob) = (0 : ℝ≥0) :=
-  rfl
+theorem toNNReal_zero : (0 : Prob) = (0 : ℝ≥0) := rfl
 
 @[simp, norm_cast]
-theorem toNNReal_one : (1 : Prob) = (1 : ℝ≥0) :=
-  rfl
+theorem toNNReal_one : (1 : Prob) = (1 : ℝ≥0) := rfl
 
 theorem ofNNReal_toNNReal : ENNReal.ofNNReal (toNNReal p) = ENNReal.ofReal (p : ℝ) := by
   simp [toNNReal, ENNReal.ofReal_eq_coe_nnreal]
@@ -170,15 +161,12 @@ instance instSub : Sub Prob where
     simpa using le_add_of_le_of_nonneg p.2.2 q.2.1
   ⟩
 
-theorem coe_sub (p q : Prob) : (p - q : Prob)  = (p.val - q.val) ⊔ (0 : ℝ) := by
-  rfl
+theorem coe_sub (p q : Prob) : (p - q : Prob)  = (p.val - q.val) ⊔ (0 : ℝ) := rfl
 
 @[simp, norm_cast]
-theorem coe_one_minus (p : Prob) : (1 - p : Prob) = 1 - (p : ℝ) := by
-  simp [coe_sub]
+theorem coe_one_minus (p : Prob) : (1 - p : Prob) = 1 - (p : ℝ) := by simp [coe_sub]
 
-theorem add_one_minus (p : Prob) : p.val + (1 - p).val = 1 := by
-  simp
+theorem add_one_minus (p : Prob) : p.val + (1 - p).val = 1 := by simp
 
 @[simp]
 theorem one_minus_inv (p : Prob) : 1 - (1 - p) = p := by
@@ -200,8 +188,7 @@ instance : Nontrivial Prob where
   exists_pair_ne := ⟨0, 1, by simp [← Prob.ne_iff]⟩
 
 @[simp]
-theorem top_eq_one : (⊤ : Prob) = 1 := by
-  rfl
+theorem top_eq_one : (⊤ : Prob) = 1 := rfl
 
 @[simp]
 theorem sub_zero (p : Prob) : p - 0 = p := by
@@ -291,12 +278,10 @@ instance instUniv [AddCommMonoid T] [Module ℝ T] : Mixable T T where
 
 @[simp]
 theorem mkT_instUniv [AddCommMonoid T] [Module ℝ T] {t : T} (h : ∃ t', to_U t' = t) :
-    instUniv.mkT h = ⟨t, rfl⟩ :=
-  rfl
+    instUniv.mkT h = ⟨t, rfl⟩ := rfl
 
 @[simp]
-theorem to_U_instUniv [AddCommMonoid T] [Module ℝ T] {t : T} : instUniv.to_U t = t :=
-  rfl
+theorem to_U_instUniv [AddCommMonoid T] [Module ℝ T] {t : T} : instUniv.to_U t = t := rfl
 
 section pi
 
@@ -325,13 +310,11 @@ instance instPi : Mixable ((i:D) → U i) ((i:D) → T i) where
 
 @[simp]
 theorem val_mkT_instPi (D : Type*) [inst : Mixable U T] {u : D → U} (h : ∃ t, to_U t = u) :
-    (instPi.mkT h).val = fun d ↦ (inst.mkT (instPi.lem_1 h d)).val :=
-  rfl
+    (instPi.mkT h).val = fun d ↦ (inst.mkT (instPi.lem_1 h d)).val := rfl
 
 @[simp]
 theorem to_U_instPi (D : Type*) [inst : Mixable U T] {t : D → T} :
-    (instPi).to_U t = fun d ↦ inst.to_U (t d) :=
-  rfl
+    (instPi).to_U t = fun d ↦ inst.to_U (t d) := rfl
 
 end pi
 
@@ -379,13 +362,11 @@ instance instMixable : Mixable ℝ Prob where
     · nlinarith
 
 @[simp]
-theorem to_U_mixable [AddCommMonoid T] [SMul ℝ T] (t : Prob) : instMixable.to_U t = t.val :=
-  rfl
+theorem to_U_mixable [AddCommMonoid T] [SMul ℝ T] (t : Prob) : instMixable.to_U t = t.val := rfl
 
 @[simp]
 theorem mkT_mixable (u : ℝ) (h : ∃ t : Prob, Mixable.to_U t = u) : Mixable.mkT h =
-    ⟨⟨u,Exists.casesOn h fun t ht ↦ ht ▸ t.2⟩, rfl⟩ :=
-  rfl
+    ⟨⟨u,Exists.casesOn h fun t ht ↦ ht ▸ t.2⟩, rfl⟩ := rfl
 
 /-- `Prob.mix` is an alias of `Mixable.mix` so it can be accessed from a probability with
 dot notation, e.g. `p.mix x y`. -/
@@ -423,20 +404,16 @@ theorem negLog_Antitone : Antitone negLog := by
     <;> exact lt_of_le_of_ne zero_le (unitInterval.coe_ne_zero.mpr (by assumption)).symm
 
 @[simp]
-theorem negLog_zero : —log (0 : Prob) = ⊤ := by
-  simp [negLog]
+theorem negLog_zero : —log (0 : Prob) = ⊤ := by simp [negLog]
 
 @[simp]
-theorem negLog_one : —log 1 = 0 := by
-  simp [negLog]; rfl
+theorem negLog_one : —log 1 = 0 := by simp [negLog]; rfl
 
 @[simp]
-theorem negLog_eq_top_iff {p : Prob} : —log p = ⊤ ↔ p = 0 := by
-  simp [negLog]
+theorem negLog_eq_top_iff {p : Prob} : —log p = ⊤ ↔ p = 0 := by simp [negLog]
 
 theorem negLog_pos_ENNReal {p : Prob} (hp : p ≠ 0) : —log p = .ofNNReal ⟨-Real.log p,
-    Left.nonneg_neg_iff.mpr (Real.log_nonpos p.2.1 p.2.2)⟩ := by
-  simp [negLog, hp]
+    Left.nonneg_neg_iff.mpr (Real.log_nonpos p.2.1 p.2.2)⟩ := by simp [negLog, hp]
 
 @[simp]
 theorem negLog_pos_Real {p : Prob} : (—log p).toReal = -Real.log p := by
diff --git a/QuantumInfo/Entropy/DPI.lean b/QuantumInfo/Entropy/DPI.lean
index a85a22508..de8febf8a 100644
--- a/QuantumInfo/Entropy/DPI.lean
+++ b/QuantumInfo/Entropy/DPI.lean
@@ -267,8 +267,7 @@ Proof: f_α(H_hat, ρ, σ) = α · Tr[(σ^γ ρ σ^γ)^α] - (α-1) · Tr[(σ^γ
 -/
 theorem f_alpha_at_optimizer (hα : 1 < α) (ρ σ : MState d) :
     f_alpha α (H_hat α ρ σ) ρ σ = Q̃_ α(ρ‖σ) := by
-  have h_inner : ⟪ρ.M, H_hat α ρ σ⟫_ℝ = ((ρ.M.conj (σ.M ^ ((1 - α) / (2 * α))).mat) ^ α).trace := by
-    exact inner_rho_H_hat hα ρ σ
+  have h_inner : ⟪ρ.M, H_hat α ρ σ⟫_ℝ = ((ρ.M.conj (σ.M ^ ((1 - α) / (2 * α))).mat) ^ α).trace := inner_rho_H_hat hα ρ σ
   have h_conj : (H_hat α ρ σ).conj (σ.M ^ ((α - 1) / (2 * α))).mat =
       (ρ.M.conj (σ.M ^ ((1 - α) / (2 * α))).mat) ^ (α - 1) := by
     convert H_hat_conj_sigma (hα := hα) (ρ := ρ) (σ := σ) using 1
@@ -317,8 +316,7 @@ lemma supportProj_mul_of_ker_le {A B : HermitianMat d ℂ}
           simp [add_comm]
           simp [← Matrix.ext_iff]
           intro i j; exact (by
-          have h_support : A.kerProj + A.supportProj = 1 := by
-            exact kerProj_add_supportProj A
+          have h_support : A.kerProj + A.supportProj = 1 := kerProj_add_supportProj A
           convert! congr_arg (fun f => f i j) h_support using 1)
         rw [← Matrix.add_mulVec, h_support, Matrix.one_mulVec]
       have hsup : B.mat *ᵥ (A.kerProj.mat *ᵥ x.ofLp) = 0 := by
@@ -326,8 +324,7 @@ lemma supportProj_mul_of_ker_le {A B : HermitianMat d ℂ}
         have h_support : A.mat * A.kerProj.mat = 0 := by
           have h_support : A.mat * A.kerProj.mat = A.mat * (1 - A.supportProj.mat) := by
             congr
-            have h_support : A.kerProj + A.supportProj = 1 := by
-              exact kerProj_add_supportProj A
+            have h_support : A.kerProj + A.supportProj = 1 := kerProj_add_supportProj A
             exact eq_sub_of_add_eq <| congr_arg Subtype.val h_support
           rw [h_support, mul_sub, mul_one, sub_eq_zero]
           exact Eq.symm (mul_supportProj_of_ker_le fun ⦃x⦄ a => a)
@@ -342,8 +339,7 @@ lemma supportProj_mul_of_ker_le {A B : HermitianMat d ℂ}
       · simp [Matrix.mulVec, dotProduct]
       · simp [Matrix.mulVec, dotProduct]
         rfl
-    have h_support : B.mat = B.mat.conjTranspose := by
-      exact B.2.symm
+    have h_support : B.mat = B.mat.conjTranspose := B.2.symm
     have h_support : (B.mat * A.supportProj.mat).conjTranspose = A.supportProj.mat * B.mat := by
       simp [Matrix.conjTranspose_mul]
     lia
@@ -435,8 +431,7 @@ theorem traceFunctional_eq_iSup_f_alpha (hα : 1 < α) (ρ σ : MState d) (hker
     Q̃_ α(ρ‖σ) = ⨆ (H : {H : HermitianMat d ℂ // 0 ≤ H}), f_alpha α H.1 ρ σ := by
   rw [@ciSup_eq_of_forall_le_of_forall_lt_exists_gt]
   · intro i
-    rw [← f_alpha_at_optimizer hα ρ σ]
-    exact f_alpha_le_at_optimizer hα ρ σ i i.2 hker
+    simpa [← f_alpha_at_optimizer hα ρ σ] using f_alpha_le_at_optimizer hα ρ σ i i.2 hker
   · intro w hw
     exact ⟨⟨H_hat α ρ σ, H_hat_nonneg ρ σ⟩, hw.trans_le (f_alpha_at_optimizer hα ρ σ ▸ le_rfl)⟩
 
@@ -535,9 +530,8 @@ theorem iSup_f_alpha_jointly_convex (hα : 1 < α)
       ∑ i, w i * (⨆ (H : {H : HermitianMat d ℂ // 0 ≤ H}), f_alpha α H.1 (ρs i) (σs i)) := by
   apply ciSup_le
   intro H
-  have h_sum : f_alpha α H.1 ρ_mix σ_mix ≤ ∑ i, w i * (f_alpha α H.1 (ρs i) (σs i)) := by
-    apply f_alpha_jointly_convex hα H.1 H.2 w hw_nonneg hw_sum ρs σs ρ_mix σ_mix hρ_mix hσ_mix
-  exact h_sum.trans (Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left
+  exact (f_alpha_jointly_convex hα H.1 H.2 w hw_nonneg hw_sum ρs σs ρ_mix σ_mix
+    hρ_mix hσ_mix).trans (Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left
     (le_ciSup (f_alpha_bddAbove hα (ρs i) (σs i) (hker i)) H) (hw_nonneg i))
 
 /-- If for all i, ker(σs i) ≤ ker(ρs i), then ker(∑ w i • σs i) ≤ ker(∑ w i • ρs i),
@@ -849,9 +843,7 @@ theorem sandwichedTraceFunctional_self (hα : 0 < α) (ρ : MState d) :
       · exact div_ne_zero (sub_ne_zero_of_ne (Ne.symm h)) (mul_ne_zero two_ne_zero hα.ne')
       · nlinarith [mul_div_cancel₀ (1 - α) (by positivity : (2 * α) ≠ 0)]
     have h_simp : (ρ.M ^ (1 + 2 * ((1 - α) / (2 * α)))) ^ α =
-        ρ.M ^ ((1 + 2 * ((1 - α) / (2 * α))) * α) := by
-      rw [← rpow_mul]
-      exact le_of_lt this
+        ρ.M ^ ((1 + 2 * ((1 - α) / (2 * α))) * α) := (HermitianMat.rpow_mul (le_of_lt this)).symm
     field_simp at *
     simp_all only [add_sub_cancel, one_div, rpow_one, MState.tr]
 
@@ -877,8 +869,7 @@ def MState.conjTensorUnitary (ρ : MState (dA × dB)) (V : Matrix.unitaryGroup d
 the twirling set gives `ρ_A ⊗ uniform_B`. We state the HermitianMat-level
 equality needed for the joint convexity argument. -/
 theorem MState.conjTensorUnitary_M (ρ : MState (dA × dB)) (V : Matrix.unitaryGroup dB ℂ) :
-    (ρ.conjTensorUnitary V).M = ρ.M.conj ((1 : Matrix.unitaryGroup dA ℂ) ⊗ᵤ V).val := by
-  rfl
+    (ρ.conjTensorUnitary V).M = ρ.M.conj ((1 : Matrix.unitaryGroup dA ℂ) ⊗ᵤ V).val := rfl
 
 /-- The trace functional is invariant under `1_A ⊗ V` conjugation. -/
 theorem sandwichedTraceFunctional_conj_tensorUnitary
@@ -1065,8 +1056,7 @@ This follows from `ker_conj` (which expresses `(A.conj B).ker` as a `comap`) and
 lemma ker_conj_le_of_ker_le {n : Type*} [Fintype n] [DecidableEq n]
     {A B : HermitianMat n ℂ} (hA : 0 ≤ A) (hB : 0 ≤ B) (h : A.ker ≤ B.ker)
     (C : Matrix n n ℂ) : (A.conj C).ker ≤ (B.conj C).ker := by
-  rw [ker_conj hA, ker_conj hB]
-  exact Submodule.comap_mono h
+  simpa [ker_conj hA, ker_conj hB] using Submodule.comap_mono h
 
 /-- Unitary conjugation preserves the kernel ordering between MStates.
 If `σ.M.ker ≤ ρ.M.ker`, then `(σ.conjTensorUnitary V).M.ker ≤ (ρ.conjTensorUnitary V).M.ker`. -/
@@ -1099,8 +1089,7 @@ theorem sandwichedTraceFunctional_mono_traceRight [Nonempty dB]
     field_simp
   -- Step 3: By joint convexity (α > 1)
   have hw_sum : ∑ i : κ, (Fintype.card κ : ℝ)⁻¹ = 1 := by
-    rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
-    exact mul_inv_cancel₀ hcard_ne
+    simp [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, hcard_ne]
   set ρ_mix := ρ.traceRight ⊗ᴹ MState.uniform (d := dB)
   set σ_mix := σ.traceRight ⊗ᴹ MState.uniform (d := dB)
   have hρ_mix : ρ_mix.M = ∑ i : κ, (Fintype.card κ : ℝ)⁻¹ • (ρ.conjTensorUnitary (V i)).M :=
@@ -1241,8 +1230,7 @@ theorem sandwichedRenyiEntropy_conj_unitary (hα : 0 < α) (ρ σ : MState d)
       · use (U.val.conjTranspose.toEuclideanLin x)
         simp_all [ker, Matrix.toEuclideanLin]
         simp_all [lin, Matrix.toLpLin]
-        have h_unitary : (U.val * U.val.conjTranspose) = 1 := by
-          exact U.2.2
+        have h_unitary : (U.val * U.val.conjTranspose) = 1 := U.2.2
         generalize_proofs at *; (
         apply_fun (U.val.conjTranspose *ᵥ ·) at hx
         simp_all [Matrix.mul_assoc, Matrix.mulVec_mulVec]
@@ -1284,14 +1272,12 @@ The sandwiched Rényi divergence is invariant under tensoring with a fixed pure
 `D̃_α(ρ ⊗ |ψ⟩⟨ψ| ‖ σ ⊗ |ψ⟩⟨ψ|) = D̃_α(ρ ‖ σ)`.
 -/
 theorem sandwichedRenyiEntropy_tensor_pure (hα : 0 < α) (ρ σ : MState d₁) (ψ : Ket d₂) :
-    D̃_ α(ρ ⊗ᴹ MState.pure ψ‖σ ⊗ᴹ MState.pure ψ) = D̃_ α(ρ‖σ) := by
-  simp [hα]
+    D̃_ α(ρ ⊗ᴹ MState.pure ψ‖σ ⊗ᴹ MState.pure ψ) = D̃_ α(ρ‖σ) := by simp [hα]
 
 /-- The sandwiched Rényi divergence is invariant under SWAP. -/
 @[simp]
 theorem sandwichedRenyiEntropy_SWAP (ρ σ : MState (dA × dB)) :
-    D̃_ α(ρ.SWAP‖σ.SWAP) = D̃_ α(ρ‖σ) := by
-  exact sandwichedRelRentropy_relabel ρ σ _
+    D̃_ α(ρ.SWAP‖σ.SWAP) = D̃_ α(ρ‖σ) := sandwichedRelRentropy_relabel ρ σ _
 
 /-
 Monotonicity of the sandwiched Rényi divergence under traceRight for `α > 1`,
diff --git a/QuantumInfo/Entropy/Relative.lean b/QuantumInfo/Entropy/Relative.lean
index 4775e59d6..49db5cee4 100644
--- a/QuantumInfo/Entropy/Relative.lean
+++ b/QuantumInfo/Entropy/Relative.lean
@@ -265,9 +265,8 @@ lemma HermitianMat.rpow_neg_mul_rpow_eq_supportProj
   rw [ h_cfc, ← mat_cfc_mul ];
   have h_cfc : ∀ x : ℝ, 0 ≤ x → x ^ (-p) * x ^ p = if x = 0 then 0 else 1 := by
     intro x hx; by_cases hx' : x = 0 <;> simp [ hx', Real.rpow_neg, hx, hp ] ;
-  have h_cfc_eq : (A.cfc (fun x => x ^ (-p) * x ^ p)) = (A.cfc (fun x => if x = 0 then 0 else 1)) := by
-    apply cfc_congr_of_nonneg hA;
-    exact fun x hx => h_cfc x hx;
+  have h_cfc_eq : (A.cfc (fun x => x ^ (-p) * x ^ p)) = (A.cfc (fun x => if x = 0 then 0 else 1)) :=
+    cfc_congr_of_nonneg hA (fun x hx => h_cfc x hx)
   convert! congr_arg ( fun x : HermitianMat d ℂ => x.val ) h_cfc_eq using 1;
   exact congr_arg ( fun x : HermitianMat d ℂ => x.val ) ( supportProj_eq_cfc A )
 
@@ -280,10 +279,8 @@ lemma HermitianMat.supportProj_mul_self (A : HermitianMat d ℂ) :
     have h_supportProj_mul_A : ∀ (v : EuclideanSpace ℂ d), v ∈ LinearMap.range A.val.toEuclideanLin → (A.supportProj.val.toEuclideanLin v) = v := by
       intro v hv
       have h_supportProj_mul_A : (A.supportProj.val.toEuclideanLin v) = (Submodule.orthogonalProjectionOnto (LinearMap.range A.val.toEuclideanLin) v) := by
-        simp only [val_eq_coe, Submodule.coe_orthogonalProjectionOnto_apply]
+        simp only [val_eq_coe]
         simp [supportProj, projector]
-        simp only [Submodule.starProjection]
-        simp
         have key : ∀ (f : EuclideanSpace ℂ d →ₗ[ℂ] EuclideanSpace ℂ d),
             Matrix.toEuclideanLin
               ((LinearMap.toMatrix (EuclideanSpace.basisFun d ℂ).toBasis (EuclideanSpace.basisFun d ℂ).toBasis) f) = f := by
@@ -314,8 +311,7 @@ lemma HermitianMat.mul_supportProj_of_ker_le {A B : HermitianMat d ℂ}
   (h : LinearMap.ker B.lin.toLinearMap ≤ LinearMap.ker A.lin.toLinearMap) :
     A.mat * B.supportProj.mat = A.mat := by
   -- Since $B.supportProj$ is the projection onto $range B$, we have $B.supportProj * B.mat = B.mat$.
-  have h_supportProj_mul_B : B.supportProj.mat * B.mat = B.mat := by
-    exact supportProj_mul_self B
+  have h_supportProj_mul_B : B.supportProj.mat * B.mat = B.mat := supportProj_mul_self B
   have h_range_A_subset_range_B : LinearMap.range A.lin.toLinearMap ≤ LinearMap.range B.lin.toLinearMap := by
     have h_orthogonal_complement : LinearMap.range (B.lin : EuclideanSpace ℂ d →ₗ[ℂ] EuclideanSpace ℂ d) = (LinearMap.ker (B.lin : EuclideanSpace ℂ d →ₗ[ℂ] EuclideanSpace ℂ d))ᗮ := by
       have h_orthogonal_complement : ∀ (T : EuclideanSpace ℂ d →ₗ[ℂ] EuclideanSpace ℂ d), T = T.adjoint → LinearMap.range T = (LinearMap.ker T)ᗮ := by
@@ -361,8 +357,7 @@ lemma HermitianMat.mul_supportProj_of_ker_le {A B : HermitianMat d ℂ}
 
 lemma HermitianMat.inner_supportProj_of_ker_le {A B : HermitianMat d ℂ}
   (h : LinearMap.ker B.lin.toLinearMap ≤ LinearMap.ker A.lin.toLinearMap) :
-    ⟪A, B.supportProj⟫ = A.trace := by
-  rw [inner_def, mul_supportProj_of_ker_le h, trace]
+    ⟪A, B.supportProj⟫ = A.trace := by rw [inner_def, mul_supportProj_of_ker_le h, trace]
 
 lemma supportProj_inner_density (h : σ.M.ker ≤ ρ.M.ker) :
     ⟪σ.M.supportProj, ρ.M⟫_ℝ = 1 := by
@@ -439,8 +434,7 @@ private theorem sandwichedRelRentropy_nonneg_α_gt_1 (h : σ.M.ker ≤ ρ.M.ker)
   positivity
 
 private lemma sandwichedRelRentropy.trace_at_one (ρ σ : MState d) :
-    ((ρ.M.conj (σ.M ^ ((1 - (1:ℝ)) / (2 * (1:ℝ)))).mat) ^ (1:ℝ)).trace = 1 := by
-  simp
+    ((ρ.M.conj (σ.M ^ ((1 - (1:ℝ)) / (2 * (1:ℝ)))).mat) ^ (1:ℝ)).trace = 1 := by simp
 
 /-
 For fixed PSD B, the derivative of α ↦ Tr[B^α] at α = 1 is ⟪B, B.log⟫ = Tr[B log B].
@@ -508,14 +502,11 @@ private lemma trace_conj_eq_inner_rpow {ρ σ : MState d} {t : ℝ} (ht : t ≠
     rw [ h_cyclic, two_mul ]
     ring_nf
     have h_exp : (σ.M ^ (t + t)).mat = (σ.M ^ t).mat * (σ.M ^ t).mat := by
-      have h_nonneg : 0 ≤ σ.M := by
-        exact σ.nonneg
-      have h_ne_zero : t + t ≠ 0 := by
-        exact fun h => ht ( by linarith )
+      have h_nonneg : 0 ≤ σ.M := σ.nonneg
+      have h_ne_zero : t + t ≠ 0 := fun h => ht ( by linarith )
       exact HermitianMat.mat_rpow_add h_nonneg h_ne_zero
     rw [ mul_two, h_exp ]
-  have h_inner : ⟪ρ.M, σ.M ^ (2 * t)⟫ = ((ρ.M.mat * (σ.M ^ (2 * t)).mat).trace).re := by
-    exact rfl
+  have h_inner : ⟪ρ.M, σ.M ^ (2 * t)⟫ = ((ρ.M.mat * (σ.M ^ (2 * t)).mat).trace).re := rfl
   simp_all
   convert congr_arg Complex.re h_cyclic using 1 ; simp [ HermitianMat.conj ] ; ring!;
   rw [ Matrix.trace_mul_comm ]
@@ -527,8 +518,7 @@ private def eigenWeight (ρ σ : MState d) (i : d) : ℝ :=
 private lemma inner_cfc_eq_sum_eigenWeight (ρ σ : MState d) (f : ℝ → ℝ) :
     ⟪ρ.M, σ.M.cfc f⟫ = ∑ i, f (σ.M.H.eigenvalues i) * eigenWeight ρ σ i := by
   -- By definition of the inner product in the context of Hermitian matrices, we can expand it using the trace.
-  have h_inner : ⟪ρ.M, σ.M.cfc f⟫ = RCLike.re (Matrix.trace (ρ.M.mat * (σ.M.cfc f).mat)) := by
-    exact rfl;
+  have h_inner : ⟪ρ.M, σ.M.cfc f⟫ = RCLike.re (Matrix.trace (ρ.M.mat * (σ.M.cfc f).mat)) := rfl;
   have h_trace : Matrix.trace (ρ.M.mat * (σ.M.cfc f).mat) = ∑ i, f (σ.M.H.eigenvalues i) * (star (σ.M.H.eigenvectorBasis i) ⬝ᵥ ρ.M.mat.mulVec (σ.M.H.eigenvectorBasis i)) := by
     rw [ Matrix.trace ];
     have h_cfc_def : (σ.M.cfc f).mat = ∑ i, (f (Matrix.IsHermitian.eigenvalues σ.M.H i)) • Matrix.of (fun x y => (σ.M.H.eigenvectorBasis i x) * (star (σ.M.H.eigenvectorBasis i y))) := by
@@ -663,14 +653,12 @@ private lemma B_of_nonneg (ρ σ : MState d) (α : ℝ) : 0 ≤ B_of ρ σ α :=
 
 -- The function g(M) = Tr[M^s] - Tr[M] satisfies g(M) = 0 when s = 1.
 private lemma trace_rpow_sub_trace_at_one (M : HermitianMat d ℂ) :
-    (M ^ (1 : ℝ)).trace - M.trace = 0 := by
-  simp [HermitianMat.rpow_one]
+    (M ^ (1 : ℝ)).trace - M.trace = 0 := by simp [HermitianMat.rpow_one]
 
 -- The cross term function value at α = 1 is zero.
 private lemma cross_term_at_one (ρ σ : MState d) :
     ((B_of ρ σ 1) ^ (1 : ℝ)).trace - (B_of ρ σ 1).trace
-    - (ρ.M ^ (1 : ℝ)).trace + 1 = 0 := by
-  simp [B_of_one, HermitianMat.rpow_one, ρ.tr]
+    - (ρ.M ^ (1 : ℝ)).trace + 1 = 0 := by simp [B_of_one, HermitianMat.rpow_one, ρ.tr]
 
 /-
 Scalar rpow cross term with just continuity: for a continuous function b with
@@ -837,11 +825,10 @@ private lemma B_of_continuousAt (ρ σ : MState d) (h : σ.M.ker ≤ ρ.M.ker) :
     ContinuousAt (B_of ρ σ) 1 := by
   -- Use the fact that the composition of continuous functions is continuous.
   have h_cont : ContinuousAt (fun α : ℝ => ρ.M.conj (σ.M ^ ((1 - α) / (2 * α))).mat) 1 := by
-    have h_inner : ContinuousAt (fun r : ℝ => ρ.M.conj (σ.M ^ r).mat) 0 := by
-      -- Apply the hypothesis `h_cont` directly to conclude the proof.
-      exact conj_rpow_continuousAt_zero σ.M ρ.M h
-    have h_exp : ContinuousAt (fun α : ℝ => (1 - α) / (2 * α)) 1 := by
-      exact ContinuousAt.div ( continuousAt_const.sub continuousAt_id ) ( continuousAt_const.mul continuousAt_id ) ( by norm_num )
+    have h_inner : ContinuousAt (fun r : ℝ => ρ.M.conj (σ.M ^ r).mat) 0 :=
+      conj_rpow_continuousAt_zero σ.M ρ.M h
+    have h_exp : ContinuousAt (fun α : ℝ => (1 - α) / (2 * α)) 1 :=
+      ContinuousAt.div ( continuousAt_const.sub continuousAt_id ) ( continuousAt_const.mul continuousAt_id ) ( by norm_num )
     exact ContinuousAt.comp ( by simpa using h_inner ) h_exp |> ContinuousAt.congr <| Filter.eventuallyEq_of_mem ( Ioi_mem_nhds zero_lt_one ) fun x hx ↦ by aesop;
   exact h_cont
 
@@ -985,12 +972,10 @@ private lemma trace_cfc_tendsto_of_tendsto (f : ℝ → ℝ)
   have h_cfc_cont : ContinuousWithinAt (fun A : HermitianMat d ℂ => A.cfc f) {A : HermitianMat d ℂ | 0 ≤ A} ρ := by
     have h_cont : ContinuousOn (fun A : HermitianMat d ℂ => A.cfc f) {A : HermitianMat d ℂ | 0 ≤ A} := by
       have h_cont_trace : ContinuousOn (fun A : HermitianMat d ℂ => (A.cfc f)) {A : HermitianMat d ℂ | 0 ≤ A} := by
-        have h_cont_cfc : ContinuousOn (fun A : HermitianMat d ℂ => A.cfc f) {A : HermitianMat d ℂ | spectrum ℝ A.mat ⊆ Set.Ici 0} := by
-          intro A hA
-          exact HermitianMat.continuousWithinAt_cfc_of_continuousOn hf hA
-        have h_spectrum_subset : ∀ A : HermitianMat d ℂ, 0 ≤ A → spectrum ℝ A.mat ⊆ Set.Ici 0 := by
-          intro A hA
-          exact (HermitianMat.posSemidef_iff_spectrum_Ici A).mp hA
+        have h_cont_cfc : ContinuousOn (fun A : HermitianMat d ℂ => A.cfc f) {A : HermitianMat d ℂ | spectrum ℝ A.mat ⊆ Set.Ici 0} :=
+          fun A hA => HermitianMat.continuousWithinAt_cfc_of_continuousOn hf hA
+        have h_spectrum_subset : ∀ A : HermitianMat d ℂ, 0 ≤ A → spectrum ℝ A.mat ⊆ Set.Ici 0 :=
+          fun A hA => (HermitianMat.posSemidef_iff_spectrum_Ici A).mp hA
         exact h_cont_cfc.mono fun A hA => h_spectrum_subset A hA
       exact h_cont_trace
     exact h_cont _ ( by simp [ ρ.2 ] ) |> ContinuousWithinAt.mono <| Set.Subset.refl _;
@@ -1551,8 +1536,7 @@ theorem sandwichedRelRentropy_additive_alpha_ne_one {α : ℝ} (hα : α ≠ 1)
     · norm_num [ add_div ];
       exact rfl;
     · exact False.elim ( ‹¬ ( σ₁ ⊗ᴹ σ₂ |> MState.M |> HermitianMat.ker ) ≤ ( ρ₁ ⊗ᴹ ρ₂ |> MState.M |> HermitianMat.ker ) › ( by simpa [ HermitianMat.ker ] using ker_prod_le_iff _ _ _ _ |>.2 h_ker ) );
-  · have h_ker_prod : ¬((σ₁ ⊗ᴹ σ₂).M.ker ≤ (ρ₁ ⊗ᴹ ρ₂).M.ker) := by
-      simp_all  [ ker_prod_le_iff ]
+  · have h_ker_prod : ¬((σ₁ ⊗ᴹ σ₂).M.ker ≤ (ρ₁ ⊗ᴹ ρ₂).M.ker) := by simp_all  [ ker_prod_le_iff ]
     rw [not_and_or] at h_ker
     rcases h_ker with h_ker | h_ker
     · simp [SandwichedRelRentropy, h_ker_prod, h_ker, hα0]
@@ -1640,8 +1624,7 @@ lemma maps_to_Iio_of_Ioi_1 : Set.MapsTo (fun α : ℝ => (1 - α) / (2 * α)) (S
 --PR'ed: #35494
 @[simp]
 theorem frontier_singleton {X : Type*} [TopologicalSpace X] [T1Space X] [PerfectSpace X]
-    (p : X) : frontier {p} = {p} := by
-  simp [frontier]
+    (p : X) : frontier {p} = {p} := by simp [frontier]
 
 private theorem sandwichedRelRentropy.continuousOn_Ioi_1_aux (ρ σ : MState d) :
     ContinuousOn (fun (α : ℝ) ↦ ((HermitianMat.conj (σ.M ^ ((1 - α) / (2 * α))).mat) ρ.M ^ α)) (Set.Ioi 1) := by
@@ -1796,8 +1779,7 @@ theorem qRelativeEnt_eq_neg_Sᵥₙ_add (ρ σ : MState d) :
 /-- The quantum relative entropy is unchanged by `MState.relabel` -/
 @[simp]
 theorem qRelativeEnt_relabel (ρ σ : MState d) (e : d₂ ≃ d) :
-    𝐃(ρ.relabel e‖σ.relabel e) = 𝐃(ρ‖σ) := by
-  simp [qRelativeEnt]
+    𝐃(ρ.relabel e‖σ.relabel e) = 𝐃(ρ‖σ) := by simp [qRelativeEnt]
 
 @[simp]
 theorem sandwichedRelRentropy_of_unique [Unique d] (ρ σ : MState d) :
@@ -1810,8 +1792,7 @@ theorem sandwichedRelRentropy_of_unique [Unique d] (ρ σ : MState d) :
 
 @[simp]
 theorem qRelEntropy_of_unique [Unique d] (ρ σ : MState d) :
-    𝐃(ρ‖σ) = 0 := by
-  exact sandwichedRelRentropy_of_unique ρ σ
+    𝐃(ρ‖σ) = 0 := sandwichedRelRentropy_of_unique ρ σ
 
 theorem sandwichedRelRentropy_heq_congr
       {d₁ d₂ : Type u} [Fintype d₁] [DecidableEq d₁] [Fintype d₂] [DecidableEq d₂]
@@ -1835,9 +1816,8 @@ theorem sandwichedRelRentropy_congr {α : ℝ}
   simp
 
 theorem qRelEntropy_heq_congr {d₁ d₂ : Type u} [Fintype d₁] [DecidableEq d₁] [Fintype d₂] [DecidableEq d₂]
-      {ρ₁ σ₁ : MState d₁} {ρ₂ σ₂ : MState d₂} (hd : d₁ = d₂) (hρ : ρ₁ ≍ ρ₂) (hσ : σ₁ ≍ σ₂) :
-    𝐃(ρ₁‖σ₁) = 𝐃(ρ₂‖σ₂) := by
-  exact sandwichedRelRentropy_heq_congr hd hρ hσ
+    {ρ₁ σ₁ : MState d₁} {ρ₂ σ₂ : MState d₂} (hd : d₁ = d₂) (hρ : ρ₁ ≍ ρ₂) (hσ : σ₁ ≍ σ₂) :
+    𝐃(ρ₁‖σ₁) = 𝐃(ρ₂‖σ₂) := sandwichedRelRentropy_heq_congr hd hρ hσ
 
 /-- Quantum relative entropy when σ has full rank -/
 theorem qRelativeEnt_rank {ρ σ : MState d} [σ.M.NonSingular] :
@@ -1943,8 +1923,7 @@ private lemma eigenWeight_eq_zero_iff (ρ x : MState d) (i : d) :
             intro v w
             have h_inner : star v ⬝ᵥ (ρ.M.mat.mulVec w) = star (star w ⬝ᵥ (ρ.M.mat.mulVec v)) := by
               have h_inner : star v ⬝ᵥ (ρ.M.mat.mulVec w) = star (star w ⬝ᵥ (ρ.M.mat.mulVec v)) := by
-                have h_inner : ρ.M.mat = star ρ.M.mat := by
-                  exact ρ.M.2.symm ▸ rfl
+                have h_inner : ρ.M.mat = star ρ.M.mat := ρ.M.2.symm ▸ rfl
                 conv_rhs => rw [ h_inner ]
                 simp [ Matrix.mulVec, dotProduct ]
                 ring_nf
@@ -1962,8 +1941,7 @@ private lemma eigenWeight_eq_zero_iff (ρ x : MState d) (i : d) :
   refine ⟨h_forward, fun h ↦ ?_⟩
   -- Since ρ e_i = 0, we have e_i^* ρ e_i = 0.
   have h_zero : (Matrix.vecMul (star (x.M.H.eigenvectorBasis i : d → ℂ)) ρ.M.mat) ⬝ᵥ (x.M.H.eigenvectorBasis i : d → ℂ) = 0 := by
-    have h_zero : ρ.M.mat.mulVec (x.M.H.eigenvectorBasis i : d → ℂ) = 0 := by
-      exact congr(WithLp.ofLp $h)
+    have h_zero : ρ.M.mat.mulVec (x.M.H.eigenvectorBasis i : d → ℂ) = 0 := congr(WithLp.ofLp $h)
     convert congr_arg ( fun v => star ( x.M.H.eigenvectorBasis i : d → ℂ ) ⬝ᵥ v ) h_zero using 1
     simp [ Matrix.dotProduct_mulVec]
     ring_nf
@@ -2122,8 +2100,7 @@ theorem qRelativeEnt_joint_convexity :
   sorry
 
 @[simp]
-theorem qRelEntropy_self (ρ : MState d) : 𝐃(ρ‖ρ) = 0 := by
-  simp [qRelativeEnt]
+theorem qRelEntropy_self (ρ : MState d) : 𝐃(ρ‖ρ) = 0 := by simp [qRelativeEnt]
 
 @[aesop (rule_sets := [finiteness]) unsafe apply]
 theorem qRelativeEnt_ne_top {ρ σ : MState d} [σ.M.NonSingular] : 𝐃(ρ‖σ) ≠ ⊤ := by
@@ -2213,9 +2190,8 @@ theorem qMutualInfo_as_qRelativeEnt (ρ : MState (dA × dB)) :
     simpa [HermitianMat.lin, Matrix.toEuclideanLin, A, B, ← Matrix.mul_kronecker_mul] using hmul
   have prod_marginals_ker_le : (ρ.traceRight ⊗ᴹ ρ.traceLeft).M.ker ≤ ρ.M.ker := by
     let P : HermitianMat (dA × dB) ℂ := ρ.traceRight.M.supportProj ⊗ₖ ρ.traceLeft.M.supportProj
-    have hP : ρ.M.support ≤ P.support := by
-      intro x hx
-      exact ⟨x, by simpa [P] using fixed_support_kron_prod hx⟩
+    have hP : ρ.M.support ≤ P.support :=
+      fun x hx => ⟨x, by simpa [P] using fixed_support_kron_prod hx⟩
     have hkerP : P.ker ≤ ρ.M.ker := by
       simpa [HermitianMat.support_orthogonal_eq_range] using Submodule.orthogonal_le hP
     exact (show (ρ.traceRight ⊗ᴹ ρ.traceLeft).M.ker ≤ P.ker by
@@ -2314,8 +2290,7 @@ private lemma pos_of_MState_le_smul {σ₁ σ₂ : MState d} (hσ : σ₂.M ≤
 open ComplexOrder in
 private lemma HermitianMat.inner_log_mono_of_posDef_of_le {A B C : HermitianMat d 𝕜}
     (hC : 0 ≤ C) (hA : A.mat.PosDef) (hAB : A ≤ B) :
-    ⟪C, A.log⟫ ≤ ⟪C, B.log⟫ := by
-  exact inner_mono hC (log_mono hA hAB)
+    ⟪C, A.log⟫ ≤ ⟪C, B.log⟫ := inner_mono hC (log_mono hA hAB)
 
 open ComplexOrder in
 private lemma posDef_add_eps {A : HermitianMat d ℂ} (hA : 0 ≤ A) {ε : ℝ} (hε : 0 < ε) :
@@ -2325,12 +2300,10 @@ private lemma posDef_add_eps {A : HermitianMat d ℂ} (hA : 0 ≤ A) {ε : ℝ}
   constructor
   · exact HermitianMat.H _
   · intro x hx_ne_zero
-    have h_inner : star x ⬝ᵥ (A.val.mulVec x) ≥ 0 := by
-      simp [hA]
+    have h_inner : star x ⬝ᵥ (A.val.mulVec x) ≥ 0 := by simp [hA]
     have h_eps : star x ⬝ᵥ (ε • 1 : Matrix d d ℂ).mulVec x = ε * star x ⬝ᵥ x := by
       simp [Matrix.mulVec, dotProduct, Finset.mul_sum, mul_left_comm, Matrix.one_apply]
-    have h_pos : 0 < ε * star x ⬝ᵥ x := by
-      simp [*]
+    have h_pos : 0 < ε * star x ⬝ᵥ x := by simp [*]
     simpa [Matrix.add_mulVec, h_eps] using add_pos_of_nonneg_of_pos h_inner h_pos
 
 private lemma log_add_eps_eq_cfc (A : HermitianMat d 𝕜) (ε : ℝ) :
@@ -2438,9 +2411,7 @@ theorem qRelEntropy_le_add_of_le_smul (ρ : MState d) {σ₁ σ₂ : MState d} (
         <;> simp only [sup_of_le_left, *]
         <;> norm_cast at *
         <;> linarith [Real.log_nonneg one_le_two]
-      have h_final' : qRelativeEnt ρ σ₁ ≤ qRelativeEnt ρ σ₂ + ENNReal.ofReal (Real.log α) := by
-        exact_mod_cast h_final
-      exact h_final'
+      exact_mod_cast h_final
     · by_contra h_contra;
       have hker_le : σ₁.M.ker ≤ σ₂.M.ker := by
         apply_rules [ HermitianMat.ker_le_of_le_smul, hσ ];
@@ -2452,5 +2423,4 @@ theorem qRelEntropy_le_add_of_le_smul (ρ : MState d) {σ₁ σ₂ : MState d} (
   · simp [hker, SandwichedRelRentropy, qRelativeEnt]
 
 theorem qRelativeEnt_op_le {ρ σ : MState d} (h : ρ.M ≤ α • σ.M) :
-    𝐃(ρ‖σ) ≤ .ofReal (Real.log α) := by
-  simpa using qRelEntropy_le_add_of_le_smul ρ h
+    𝐃(ρ‖σ) ≤ .ofReal (Real.log α) := by simpa using qRelEntropy_le_add_of_le_smul ρ h
diff --git a/QuantumInfo/Entropy/SSA.lean b/QuantumInfo/Entropy/SSA.lean
index 7b169c0c8..e9011a35d 100644
--- a/QuantumInfo/Entropy/SSA.lean
+++ b/QuantumInfo/Entropy/SSA.lean
@@ -188,8 +188,7 @@ lemma PosDef_traceRight [Nonempty dB] (A : HermitianMat (dA × dB) ℂ) (hA : A.
     rw [Finset.sum_comm_cycle]
 
 lemma PosDef_traceLeft [Nonempty dA] (A : HermitianMat (dA × dB) ℂ) (hA : A.mat.PosDef) :
-    A.traceLeft.mat.PosDef := by
-  exact PosDef_traceRight (A.reindex (Equiv.prodComm _ _)) (hA.reindex _)
+    A.traceLeft.mat.PosDef := PosDef_traceRight (A.reindex (Equiv.prodComm _ _)) (hA.reindex _)
 
 /--
 V_rho is an isometry.
@@ -200,8 +199,7 @@ theorem V_rho_isometry [Nonempty dB] (ρAB : HermitianMat (dA × dB) ℂ) (hρ :
   have h_pos_def : (ρAB.traceRight⁻¹.sqrt : Matrix dA dA ℂ) * (ρAB.traceRight : Matrix dA dA ℂ) * (ρAB.traceRight⁻¹.sqrt : Matrix dA dA ℂ) = 1 := by
     convert HermitianMat.sqrt_inv_mul_self_mul_sqrt_inv_eq_one _;
     exact PosDef_traceRight _ hρ
-  rw [← h_pos_def]
-  exact V_rho_conj_mul_self_eq ρAB hρ
+  simpa [← h_pos_def] using V_rho_conj_mul_self_eq ρAB hρ
 
 /--
 V_sigma is an isometry.
@@ -364,8 +362,7 @@ theorem HermitianMat.PosDef_kronecker {m n : Type*} [Fintype m] [DecidableEq m]
     [Fintype n] [DecidableEq n]
     (A : HermitianMat m ℂ) (B : HermitianMat n ℂ)
     (hA : A.mat.PosDef) (hB : B.mat.PosDef) :
-    (A ⊗ₖ B).mat.PosDef := by
-  exact Matrix.PosDef.kron hA hB
+    (A ⊗ₖ B).mat.PosDef := Matrix.PosDef.kron hA hB
 
 /- Reindex of PosDef is PosDef. -/
 theorem HermitianMat.PosDef_reindex {d d₂ : Type*} [Fintype d] [DecidableEq d]
@@ -572,8 +569,7 @@ private lemma T₂_sq_le_one [Nonempty dB]
       · aesop
     · aesop
 
-private lemma PERM_isometry : (PERM_mat dA dB dC)ᴴ * PERM_mat dA dB dC = 1 := by
-  simp [PERM_mat]
+private lemma PERM_isometry : (PERM_mat dA dB dC)ᴴ * PERM_mat dA dB dC = 1 := by simp [PERM_mat]
 
 /-- Element-wise identity: W_mat = ∑_{b*} V_rho * V_sigma†.
     This is the key computation from Eq. (6) of Lin-Kim-Hsieh. -/
@@ -804,12 +800,10 @@ theorem operator_ineq_SSA [Nonempty dA] [Nonempty dB] [Nonempty dC]
       convert HermitianMat.inv_kronecker _ _ using 1;
       · infer_instance;
       · exact ⟨ ⟨ Classical.arbitrary dB, Classical.arbitrary dC ⟩ ⟩;
-      · have h_trace_right_pos_def : (ρAB.traceRight).mat.PosDef := by
-          exact PosDef_traceRight ρAB hρ
+      · have h_trace_right_pos_def : (ρAB.traceRight).mat.PosDef := PosDef_traceRight ρAB hρ
         exact ⟨by exact PosDef_traceRight ρAB hρ |>.isUnit⟩
       · have h_inv_symm : σBC⁻¹.NonSingular := by
-          have h_inv_symm : σBC.NonSingular := by
-            exact nonSingular_of_posDef hσ
+          have h_inv_symm : σBC.NonSingular := nonSingular_of_posDef hσ
           exact nonSingular_iff_inv.mpr h_inv_symm;
         exact h_inv_symm;
     convert congr_arg ( fun x : HermitianMat _ _ => x.reindex ( Equiv.prodAssoc dA dB dC ).symm ) h_inv_symm using 1;
@@ -832,8 +826,7 @@ theorem operator_ineq_SSA [Nonempty dA] [Nonempty dB] [Nonempty dC]
     have h_inv_symm : (σBC.traceLeft⁻¹)⁻¹ = σBC.traceLeft := by
       have h_inv_symm : (σBC.traceLeft⁻¹).mat * σBC.traceLeft.mat = 1 := by
         have h_inv_symm : (σBC.traceLeft⁻¹).mat * σBC.traceLeft.mat = 1 := by
-          have h_inv_symm : σBC.traceLeft.mat.PosDef := by
-            exact PosDef_traceLeft σBC hσ
+          have h_inv_symm : σBC.traceLeft.mat.PosDef := PosDef_traceLeft σBC hσ
           convert! Matrix.nonsing_inv_mul _ _;
           exact isUnit_iff_ne_zero.mpr h_inv_symm.det_pos.ne';
         exact h_inv_symm
diff --git a/QuantumInfo/Entropy/VonNeumann.lean b/QuantumInfo/Entropy/VonNeumann.lean
index 59e5e7717..d415d33b9 100644
--- a/QuantumInfo/Entropy/VonNeumann.lean
+++ b/QuantumInfo/Entropy/VonNeumann.lean
@@ -141,14 +141,12 @@ theorem Sᵥₙ_unit_zero [Unique d] (ρ : MState d) : Sᵥₙ ρ = 0 := by
 /-- Von Neumann entropy is invariant under relabeling of the basis. -/
 @[simp]
 theorem Sᵥₙ_relabel (ρ : MState d₁) (e : d₂ ≃ d₁) :
-    Sᵥₙ (ρ.relabel e) = Sᵥₙ ρ := by
-  simp [Sᵥₙ_eq_neg_trace_log]
+    Sᵥₙ (ρ.relabel e) = Sᵥₙ ρ := by simp [Sᵥₙ_eq_neg_trace_log]
 
 /-- Von Neumann entropy is unchanged under SWAP. TODO: All unitaries-/
 @[simp]
-theorem Sᵥₙ_of_SWAP_eq (ρ : MState (d₁ × d₂)) : Sᵥₙ ρ.SWAP = Sᵥₙ ρ := by
-  apply Hₛ_eq_of_multiset_map_eq
-  exact ρ.multiset_spectrum_relabel_eq (Equiv.prodComm d₁ d₂).symm
+theorem Sᵥₙ_of_SWAP_eq (ρ : MState (d₁ × d₂)) : Sᵥₙ ρ.SWAP = Sᵥₙ ρ :=
+  Hₛ_eq_of_multiset_map_eq _ _ (ρ.multiset_spectrum_relabel_eq (Equiv.prodComm d₁ d₂).symm)
 
 /-- Von Neumann entropy is unchanged under assoc. -/
 @[simp]
@@ -257,8 +255,7 @@ the same von Neumann entropy.
 private lemma Sᵥₙ_eq_of_nonzero_eigenvalues_eq (ρ₁ : MState d₁) (ρ₂ : MState d₂)
     (h : (Finset.univ.val.map ρ₁.spectrum.prob).filter (· ≠ 0) =
       (Finset.univ.val.map ρ₂.spectrum.prob).filter (· ≠ 0)) :
-    Sᵥₙ ρ₁ = Sᵥₙ ρ₂ := by
-  exact Hₛ_eq_of_nonzero_multiset_eq _ _ h
+    Sᵥₙ ρ₁ = Sᵥₙ ρ₂ := Hₛ_eq_of_nonzero_multiset_eq _ _ h
 
 /--
 Filtering non-zero elements commutes with mapping `RCLike.ofReal` for multisets.
@@ -331,8 +328,7 @@ theorem qConditionalEnt_of_pure_symm (ψ : Ket (d₁ × d₂)) :
 /-- Quantum mutual information is symmetric. -/
 @[simp]
 theorem qMutualInfo_symm (ρ : MState (d₁ × d₂)) :
-    qMutualInfo ρ.SWAP = qMutualInfo ρ := by
-  simp [qMutualInfo, add_comm]
+    qMutualInfo ρ.SWAP = qMutualInfo ρ := by simp [qMutualInfo, add_comm]
 
 /-- For a pure state, the entropy of one subsystem equals the entropy of its complement,
 even after relabeling. -/
diff --git a/QuantumInfo/ForMathlib/ComplexLaplaceTransform.lean b/QuantumInfo/ForMathlib/ComplexLaplaceTransform.lean
index 1b239ca22..6cc9c6a49 100644
--- a/QuantumInfo/ForMathlib/ComplexLaplaceTransform.lean
+++ b/QuantumInfo/ForMathlib/ComplexLaplaceTransform.lean
@@ -139,9 +139,7 @@ theorem measurable_complexLaplaceIntegrand
       have hUntop : y.untop hy = e := by
         apply WithTop.coe_inj.mp
         rw [WithTop.coe_untop, ← he]
-      have hUntopD : y.untopD 0 = e := by
-        rw [← he]
-        rfl
+      have hUntopD : y.untopD 0 = e := he ▸ rfl
       rw [hUntop, hUntopD]]
   exact Measurable.ite (hE (measurableSet_singleton (⊤ : WithTop ℝ))) measurable_const
     (Complex.continuous_exp.measurable.comp (by fun_prop :
@@ -187,9 +185,8 @@ theorem continuousAt_complexLaplaceTransform_of_mem_interior_convergenceDomain
 
 theorem continuousOn_complexLaplaceTransform_interior_convergenceDomain
     {α : Type*} [MeasureTheory.MeasureSpace α] {E : α → WithTop ℝ} :
-    ContinuousOn (ComplexLaplaceTransform E) (interior (ComplexLaplaceConvergenceDomain E)) := by
-  intro z hz
-  exact (continuousAt_complexLaplaceTransform_of_mem_interior_convergenceDomain hz).continuousWithinAt
+    ContinuousOn (ComplexLaplaceTransform E) (interior (ComplexLaplaceConvergenceDomain E)) :=
+  fun _ hz => (continuousAt_complexLaplaceTransform_of_mem_interior_convergenceDomain hz).continuousWithinAt
 
 private theorem integrable_uncurry_complexLaplaceIntegrand_horizontal
     {α : Type*} [MeasureTheory.MeasureSpace α]
diff --git a/QuantumInfo/ForMathlib/ContinuousLinearMap.lean b/QuantumInfo/ForMathlib/ContinuousLinearMap.lean
index e1e96b3d0..2353e2b24 100644
--- a/QuantumInfo/ForMathlib/ContinuousLinearMap.lean
+++ b/QuantumInfo/ForMathlib/ContinuousLinearMap.lean
@@ -32,8 +32,7 @@ variable [TopologicalSpace M] [AddCommMonoid M] [TopologicalSpace M₂] [AddComm
 variable [Module R M] [Module S M₂]
 
 theorem ker_mk (f : M →ₛₗ[σ] M₂) (hf : Continuous f.toFun) :
-    (ContinuousLinearMap.mk f hf).ker = LinearMap.ker f := by
-  rfl
+    (ContinuousLinearMap.mk f hf).ker = LinearMap.ker f := rfl
 
 end ContinuousLinearMap
 
diff --git a/QuantumInfo/ForMathlib/ContinuousSup.lean b/QuantumInfo/ForMathlib/ContinuousSup.lean
index a8ea0c1d0..04456d914 100644
--- a/QuantumInfo/ForMathlib/ContinuousSup.lean
+++ b/QuantumInfo/ForMathlib/ContinuousSup.lean
@@ -182,8 +182,7 @@ end ContinuousLinearMap
 theorem LinearMap.BilinForm.continuous_iSup_fst
   {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
   (f : LinearMap.BilinForm ℝ E) {S : Set E} (hS : Bornology.IsBounded S) :
-    Continuous fun x ↦ ⨆ y : S, f y x := by
-  exact LinearMap.BilinForm.continuous_iSup f.flip hS
+    Continuous fun x ↦ ⨆ y : S, f y x := LinearMap.BilinForm.continuous_iSup f.flip hS
   --Old "direct" proof:
   -- -- Since $f$ is continuous, there exists $C > 0$ such that for all $y \in S$ and $x \in E$, $|f y x| \leq C \|y\| \|x\|$.
   -- obtain ⟨C, hC1, hC2⟩ : ∃ C > 0, ∀ y ∈ S, ∀ x : E, |f y x| ≤ C * ‖y‖ * ‖x‖ := by
diff --git a/QuantumInfo/ForMathlib/Filter.lean b/QuantumInfo/ForMathlib/Filter.lean
index 503835a25..3691d1c66 100644
--- a/QuantumInfo/ForMathlib/Filter.lean
+++ b/QuantumInfo/ForMathlib/Filter.lean
@@ -53,7 +53,6 @@ theorem ENNReal.tendsto_toReal_iff_of_eventually_ne_top
     rw [EventuallyEq]
     peel hf with h
     simp [h]
-  have he₂ : (fun n ↦ (f n).toReal) = (fun n ↦ ((f n).toNNReal : ℝ≥0∞).toReal) :=
-    rfl
+  have he₂ : (fun n ↦ (f n).toReal) = (fun n ↦ ((f n).toNNReal : ℝ≥0∞).toReal) := rfl
   rw [Filter.tendsto_congr' he₁, he₂]
   exact tendsto_toReal_iff (by finiteness) hx
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/BlockDiagonal.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/BlockDiagonal.lean
index 6f0af1e74..073f0bdf4 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/BlockDiagonal.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/BlockDiagonal.lean
@@ -165,18 +165,15 @@ noncomputable def blockDiagonalHom : (L ℋ × L ℋ) →⋆ₐ[ℝ] L (HSum ℋ
 
 omit [Nontrivial ℋ] in
 @[simp] theorem blockDiagonalHom_apply (p : L ℋ × L ℋ) :
-    blockDiagonalHom (ℋ := ℋ) p = blockDiagonal (ℋ := ℋ) p.1 p.2 :=
-  rfl
+    blockDiagonalHom (ℋ := ℋ) p = blockDiagonal (ℋ := ℋ) p.1 p.2 := rfl
 
 omit [CompleteSpace ℋ] [Nontrivial ℋ] in
 @[simp] theorem hsumProj_blockDiagonal_zero (A B : L ℋ) (z : HSum ℋ) :
-    hsumProj ℋ 0 (blockDiagonal A B z) = A (hsumProj ℋ 0 z) := by
-  simp [blockDiagonal]
+    hsumProj ℋ 0 (blockDiagonal A B z) = A (hsumProj ℋ 0 z) := by simp [blockDiagonal]
 
 omit [CompleteSpace ℋ] [Nontrivial ℋ] in
 @[simp] theorem hsumProj_blockDiagonal_one (A B : L ℋ) (z : HSum ℋ) :
-    hsumProj ℋ 1 (blockDiagonal A B z) = B (hsumProj ℋ 1 z) := by
-  simp [blockDiagonal]
+    hsumProj ℋ 1 (blockDiagonal A B z) = B (hsumProj ℋ 1 z) := by simp [blockDiagonal]
 
 omit [CompleteSpace ℋ] [Nontrivial ℋ] in
 @[simp] theorem blockDiagonal_one :
@@ -210,16 +207,14 @@ theorem blockDiagonal_nonneg {A B : L ℋ} (hA : 0 ≤ A) (hB : 0 ≤ B) :
           (inner ℂ (A (hsumProj ℋ 0 z)) (hsumProj ℋ 0 z) +
             inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z))) =
           ↑(RCLike.re (inner ℂ (A (hsumProj ℋ 0 z)) (hsumProj ℋ 0 z)) +
-            RCLike.re (inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z))) := by
-            simp
+            RCLike.re (inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z))) := by simp
       _ = inner ℂ (A (hsumProj ℋ 0 z)) (hsumProj ℋ 0 z) +
           inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z) := by
             have hsumre :
                 (↑(RCLike.re (inner ℂ (A (hsumProj ℋ 0 z)) (hsumProj ℋ 0 z)) +
                     RCLike.re (inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z))) : ℂ) =
                   ((RCLike.re (inner ℂ (A (hsumProj ℋ 0 z)) (hsumProj ℋ 0 z)) : ℂ) +
-                    (RCLike.re (inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z)) : ℂ)) := by
-                  simp
+                    (RCLike.re (inner ℂ (B (hsumProj ℋ 1 z)) (hsumProj ℋ 1 z)) : ℂ)) := by simp
             rw [hsumre, hAz.1, hBz.1]
   · dsimp [blockDiagonal]
     erw [inner_add_left, hz0, hz1]
@@ -228,14 +223,12 @@ theorem blockDiagonal_nonneg {A B : L ℋ} (hA : 0 ≤ A) (hB : 0 ≤ B) :
 omit [CompleteSpace ℋ] [Nontrivial ℋ] in
 @[simp] theorem hsumProj_blockOp_zero (A00 A01 A10 A11 : L ℋ) (z : HSum ℋ) :
     hsumProj ℋ 0 (blockOp (ℋ := ℋ) A00 A01 A10 A11 z) =
-      A00 (hsumProj ℋ 0 z) + A01 (hsumProj ℋ 1 z) := by
-  simp [blockOp]
+      A00 (hsumProj ℋ 0 z) + A01 (hsumProj ℋ 1 z) := by simp [blockOp]
 
 omit [CompleteSpace ℋ] [Nontrivial ℋ] in
 @[simp] theorem hsumProj_blockOp_one (A00 A01 A10 A11 : L ℋ) (z : HSum ℋ) :
     hsumProj ℋ 1 (blockOp (ℋ := ℋ) A00 A01 A10 A11 z) =
-      A10 (hsumProj ℋ 0 z) + A11 (hsumProj ℋ 1 z) := by
-  simp [blockOp]
+      A10 (hsumProj ℋ 0 z) + A11 (hsumProj ℋ 1 z) := by simp [blockOp]
 
 omit [Nontrivial ℋ] in
 @[simp] theorem blockOp_star (A00 A01 A10 A11 : L ℋ) :
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/GeneralizedPerspectiveFunction.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/GeneralizedPerspectiveFunction.lean
index d92d8e73c..cca85cffe 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/GeneralizedPerspectiveFunction.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/GeneralizedPerspectiveFunction.lean
@@ -126,8 +126,7 @@ private lemma spectrum_convexCombo_Ioi {A B : L ℋ} {t : ℝ}
           (smul_le_smul_of_nonneg_left hrB_le ht0)
     have hLHS :
         (1 - t) • algebraMap ℝ (L ℋ) rA + t • algebraMap ℝ (L ℋ) rB =
-          algebraMap ℝ (L ℋ) rC := by
-      simp [rC, Algebra.smul_def]
+          algebraMap ℝ (L ℋ) rC := by simp [rC, Algebra.smul_def]
     simpa [hLHS] using hsum
   intro x hx
   simpa [C] using
@@ -173,14 +172,12 @@ private lemma hpow_continuousOn
 omit [Nontrivial ℋ] in
 private lemma hSqrt_selfAdjoint (h : ℝ → ℝ) (B : L ℋ) :
     IsSelfAdjoint (hSqrt (ℋ := ℋ) h B) := by
-  dsimp [hSqrt, cfcR]
-  exact cfc_predicate _ _
+  simp [hSqrt, cfcR]
 
 omit [Nontrivial ℋ] in
 private lemma hInvSqrt_selfAdjoint (h : ℝ → ℝ) (B : L ℋ) :
     IsSelfAdjoint (hInvSqrt (ℋ := ℋ) h B) := by
-  dsimp [hInvSqrt, cfcR]
-  exact cfc_predicate _ _
+  simp [hInvSqrt, cfcR]
 
 omit [Nontrivial ℋ] in
 private lemma hSqrt_mul_hInvSqrt_eq_one
@@ -249,8 +246,7 @@ omit [Nontrivial ℋ] in
 private lemma conj_le_conj {X Y T : L ℋ} (hXY : X ≤ Y) (hT : IsSelfAdjoint T) :
     T * X * T ≤ T * Y * T := by
   have hnonneg : 0 ≤ Y - X := sub_nonneg.mpr hXY
-  have hconj : 0 ≤ T * (Y - X) * T := by
-    simpa using hT.conjugate_nonneg hnonneg
+  have hconj : 0 ≤ T * (Y - X) * T := by simpa using hT.conjugate_nonneg hnonneg
   have hsub : T * (Y - X) * T = T * Y * T - T * X * T := by
     simp [sub_eq_add_neg, mul_add, add_mul, mul_assoc]
   exact sub_nonneg.mp (by simpa [hsub] using hconj)
@@ -325,11 +321,9 @@ private theorem theorem_2_5_forward_jointlyConvexOn_psd_pd_of_condV
   have hIR₂S₂ : IR₂ * S₂ = (1 : L ℋ) :=
     hInvSqrt_mul_hSqrt_eq_one (ℋ := ℋ) hB₂_sa hB₂_spec hcont hpos
   have hM₁_nonneg : (0 : L ℋ) ≤ M₁ := by
-    dsimp [M₁]
-    simpa [mul_assoc] using hIR₁_sa.conjugate_nonneg hA₁_nonneg
+    simpa [M₁, mul_assoc] using hIR₁_sa.conjugate_nonneg hA₁_nonneg
   have hM₂_nonneg : (0 : L ℋ) ≤ M₂ := by
-    dsimp [M₂]
-    simpa [mul_assoc] using hIR₂_sa.conjugate_nonneg hA₂_nonneg
+    simpa [M₂, mul_assoc] using hIR₂_sa.conjugate_nonneg hA₂_nonneg
   have hM₁_sa : IsSelfAdjoint M₁ := IsSelfAdjoint.of_nonneg hM₁_nonneg
   have hM₂_sa : IsSelfAdjoint M₂ := IsSelfAdjoint.of_nonneg hM₂_nonneg
   have hM₁_spec : spectrum ℝ M₁ ⊆ Set.Ici (0 : ℝ) := by
@@ -421,10 +415,8 @@ private theorem theorem_2_5_forward_jointlyConvexOn_psd_pd_of_condV
                 simp [T₁, hS₁_sa.star_eq, hIR_sa.star_eq, mul_assoc, smul_smul]
       _ = (1 - θ) • (S * IR * (S₁ * cfcR (ℋ := ℋ) f M₁ * S₁) * IR * S) := by
             rw [Real.mul_self_sqrt (sub_nonneg.mpr hθ1)]
-      _ = (1 - θ) • (S₁ * cfcR (ℋ := ℋ) f M₁ * S₁) := by
-            simp [mul_assoc, hSIR, hIRS]
-      _ = (1 - θ) • ((f Δ h) A₁ B₁) := by
-            rfl
+      _ = (1 - θ) • (S₁ * cfcR (ℋ := ℋ) f M₁ * S₁) := by simp [mul_assoc, hSIR, hIRS]
+      _ = (1 - θ) • ((f Δ h) A₁ B₁) := rfl
   have hright₂ :
       S * (star T₂ * cfcR (ℋ := ℋ) f M₂ * T₂) * S =
         θ • ((f Δ h) A₂ B₂) := by
@@ -435,17 +427,14 @@ private theorem theorem_2_5_forward_jointlyConvexOn_psd_pd_of_condV
                 simp [T₂, hS₂_sa.star_eq, hIR_sa.star_eq, mul_assoc, smul_smul]
       _ = θ • (S * IR * (S₂ * cfcR (ℋ := ℋ) f M₂ * S₂) * IR * S) := by
             rw [Real.mul_self_sqrt hθ0]
-      _ = θ • (S₂ * cfcR (ℋ := ℋ) f M₂ * S₂) := by
-            simp [mul_assoc, hSIR, hIRS]
-      _ = θ • ((f Δ h) A₂ B₂) := by
-            rfl
+      _ = θ • (S₂ * cfcR (ℋ := ℋ) f M₂ * S₂) := by simp [mul_assoc, hSIR, hIRS]
+      _ = θ • ((f Δ h) A₂ B₂) := rfl
   have hright :
       S * (star T₁ * cfcR (ℋ := ℋ) f M₁ * T₁ + star T₂ * cfcR (ℋ := ℋ) f M₂ * T₂) * S =
         (1 - θ) • ((f Δ h) A₁ B₁) + θ • ((f Δ h) A₂ B₂) := by
     rw [mul_add, add_mul, hright₁, hright₂]
   have hleft :
-      S * cfcR (ℋ := ℋ) f (IR * A * IR) * S = (f Δ h) A B := by
-    rfl
+      S * cfcR (ℋ := ℋ) f (IR * A * IR) * S = (f Δ h) A B := rfl
   simpa [hleft, hright] using houter
 
 -- Restricted forward form of Theorem 2.5 on the positive cone.
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/HilbertSchmidtOperatorSpace.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/HilbertSchmidtOperatorSpace.lean
index 212343dbc..1758b69bd 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/HilbertSchmidtOperatorSpace.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/HilbertSchmidtOperatorSpace.lean
@@ -515,27 +515,20 @@ noncomputable def rightMulHSStarAlgHom : (L ℋ)ᵐᵒᵖ →⋆ₐ[ℝ] L (HSOp
     simp [rightMulHS_star (ℋ := ℋ) (MulOpposite.unop A)]
 
 @[simp] theorem rightMulHSStarAlgHom_apply (A : (L ℋ)ᵐᵒᵖ) :
-    rightMulHSStarAlgHom (ℋ := ℋ) A = rightMulHS (ℋ := ℋ) (MulOpposite.unop A) :=
-  rfl
+    rightMulHSStarAlgHom (ℋ := ℋ) A = rightMulHS (ℋ := ℋ) (MulOpposite.unop A) := rfl
 
 /-- The `⋆`-algebra hom sending `A` to `op (star A)`. On selfadjoint operators this is just `op`. -/
 noncomputable def opStarHSStarAlgHom : L ℋ →⋆ₐ[ℝ] (L ℋ)ᵐᵒᵖ where
   toFun := fun A => MulOpposite.op (star A)
-  map_one' := by simp
-  map_mul' := by
-    intro A B
-    simp [star_mul]
-  map_zero' := by simp
-  map_add' := by
-    intro A B
-    simp
+  map_one' := congrArg MulOpposite.op (star_one (R := L ℋ))
+  map_mul' A B := congrArg MulOpposite.op (star_mul A B)
+  map_zero' := congrArg MulOpposite.op (star_zero (R := L ℋ))
+  map_add' A B := congrArg MulOpposite.op (star_add A B)
   commutes' := by
     intro r
     apply congrArg MulOpposite.op
     simp [Algebra.algebraMap_eq_smul_one]
-  map_star' := by
-    intro A
-    simp
+  map_star' A := rfl
 
 private noncomputable def opStarHSAlgEquiv : L ℋ ≃ₐ[ℝ] (L ℋ)ᵐᵒᵖ where
   toFun A := MulOpposite.op (star A)
@@ -550,8 +543,7 @@ private noncomputable def opStarHSAlgEquiv : L ℋ ≃ₐ[ℝ] (L ℋ)ᵐᵒᵖ
 
 omit [FiniteDimensional ℂ ℋ] in
 lemma op_isSelfAdjoint (A : L ℋ) (hA : IsSelfAdjoint A) :
-    IsSelfAdjoint (MulOpposite.op A : (L ℋ)ᵐᵒᵖ) := by
-  exact congrArg MulOpposite.op hA.star_eq
+    IsSelfAdjoint (MulOpposite.op A : (L ℋ)ᵐᵒᵖ) := congrArg MulOpposite.op hA.star_eq
 
 private noncomputable def opStarHSLinearMap : L ℋ →ₗ[ℝ] (L ℋ)ᵐᵒᵖ where
   toFun := fun A => MulOpposite.op (star A)
@@ -604,9 +596,8 @@ lemma cfc_op_eq_op_cfc [ContinuousFunctionalCalculus ℝ (L ℋ) IsSelfAdjoint]
     cfc (R := ℝ) (A := (L ℋ)ᵐᵒᵖ) (p := IsSelfAdjoint) f (MulOpposite.op A) =
       MulOpposite.op (cfcR f A) := by
   let φ : L ℋ →⋆ₐ[ℝ] (L ℋ)ᵐᵒᵖ := opStarHSStarAlgHom (ℋ := ℋ)
-  have hφ : Continuous φ := by
-    exact
-      (LinearMap.continuous_of_finiteDimensional (opStarHSLinearMap (ℋ := ℋ)))
+  have hφ : Continuous φ :=
+    LinearMap.continuous_of_finiteDimensional (opStarHSLinearMap (ℋ := ℋ))
   have hopA : IsSelfAdjoint (MulOpposite.op A : (L ℋ)ᵐᵒᵖ) := op_isSelfAdjoint (A := A) hA
   have hφA : IsSelfAdjoint (φ A) := hA.map φ
   have hcfcA :
@@ -627,9 +618,8 @@ lemma leftMulHS_cfcR [ContinuousFunctionalCalculus ℝ (L ℋ) IsSelfAdjoint] [N
     leftMulHS (ℋ := ℋ) (cfcR f A) =
       cfcR (ℋ := HSOp ℋ) f (leftMulHS (ℋ := ℋ) A) := by
   let φ : L ℋ →⋆ₐ[ℝ] L (HSOp ℋ) := leftMulHSStarAlgHom (ℋ := ℋ)
-  have hφ : Continuous φ := by
-    exact
-      (LinearMap.continuous_of_finiteDimensional (leftMulHSLinearMap (ℋ := ℋ)))
+  have hφ : Continuous φ :=
+    LinearMap.continuous_of_finiteDimensional (leftMulHSLinearMap (ℋ := ℋ))
   have hφA : IsSelfAdjoint (φ A) := hA.map φ
   have hmap := StarAlgHom.map_cfc (φ := φ) (f := f) (a := A)
     (hf := hf) (hφ := hφ) (ha := hA) (hφa := hφA)
@@ -642,9 +632,8 @@ lemma rightMulHS_cfcR [ContinuousFunctionalCalculus ℝ (L ℋ) IsSelfAdjoint]
     rightMulHS (ℋ := ℋ) (cfcR f A) =
       cfcR (ℋ := HSOp ℋ) f (rightMulHS (ℋ := ℋ) A) := by
   let φ : (L ℋ)ᵐᵒᵖ →⋆ₐ[ℝ] L (HSOp ℋ) := rightMulHSStarAlgHom (ℋ := ℋ)
-  have hφ : Continuous φ := by
-    exact
-      (LinearMap.continuous_of_finiteDimensional (rightMulHSLinearMap (ℋ := ℋ)))
+  have hφ : Continuous φ :=
+    LinearMap.continuous_of_finiteDimensional (rightMulHSLinearMap (ℋ := ℋ))
   have hopA : IsSelfAdjoint (MulOpposite.op A : (L ℋ)ᵐᵒᵖ) := op_isSelfAdjoint (A := A) hA
   have hφA : IsSelfAdjoint (φ (MulOpposite.op A)) := hopA.map φ
   have hmap := StarAlgHom.map_cfc (φ := φ) (f := f) (a := MulOpposite.op A)
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequality.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequality.lean
index edd08ffd7..1a843e17b 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequality.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequality.lean
@@ -63,8 +63,7 @@ private theorem nontrivial_hsumL_wrap [Nontrivial ℋ] : Nontrivial (L (HSum ℋ
   have hdiag_ne_zero : (blockDiagonal (ℋ := ℋ) (1 : L ℋ) 0 : L (HSum ℋ)) ≠ 0 := by
     intro h0
     have hz :
-        blockDiagonal (ℋ := ℋ) (1 : L ℋ) 0 (hsumIncl ℋ 0 w) = 0 := by
-      simp [h0]
+        blockDiagonal (ℋ := ℋ) (1 : L ℋ) 0 (hsumIncl ℋ 0 w) = 0 := by simp [h0]
     have hw0 : w = 0 := by
       have hz0 := congrArg (fun z : HSum ℋ => hsumProj ℋ 0 z) hz
       simpa [blockDiagonal] using hz0
@@ -130,10 +129,8 @@ private lemma cfcR_blockDiagonal_wrap (f : ℝ → ℝ)
         (hab := hpair) (ha := hA) (hb := hB))
   calc
     cfcR (ℋ := HSum ℋ) f (blockDiagonal (ℋ := ℋ) A B)
-        = cfc (R := ℝ) (A := L (HSum ℋ)) (p := IsSelfAdjoint) f (φ (A, B)) := by
-            simp [cfcR, φ]
-    _ = φ (cfc (R := ℝ) (A := L ℋ × L ℋ) (p := IsSelfAdjoint) f (A, B)) := by
-            simpa using hmap.symm
+        = cfc (R := ℝ) (A := L (HSum ℋ)) (p := IsSelfAdjoint) f (φ (A, B)) := by simp [cfcR, φ]
+    _ = φ (cfc (R := ℝ) (A := L ℋ × L ℋ) (p := IsSelfAdjoint) f (A, B)) := by simpa using hmap.symm
     _ = φ (cfcR (ℋ := ℋ) f A, cfcR (ℋ := ℋ) f B) := by
             rw [hprod]
     _ = blockDiagonal (ℋ := ℋ) (cfcR (ℋ := ℋ) f A) (cfcR (ℋ := ℋ) f B) := by
@@ -150,10 +147,9 @@ private lemma continuousOn_union_of_subset_Ici_wrap {f : ℝ → ℝ}
   · exact ht hx
 
 private lemma spectrum_Ici_of_nonneg_wrap {A : L ℋ} (hA0 : (0 : L ℋ) ≤ A) :
-    spectrum ℝ A ⊆ Set.Ici (0 : ℝ) := by
-  exact
-    (StarOrderedRing.nonneg_iff_spectrum_nonneg (R := ℝ) A
-      (ha := IsSelfAdjoint.of_nonneg hA0)).1 hA0
+    spectrum ℝ A ⊆ Set.Ici (0 : ℝ) :=
+  (StarOrderedRing.nonneg_iff_spectrum_nonneg (R := ℝ) A
+    (ha := IsSelfAdjoint.of_nonneg hA0)).1 hA0
 
 variable [Nontrivial ℋ]
 
@@ -161,8 +157,7 @@ omit [CompleteSpace ℋ] in
 private lemma spectrum_zero_subset_Ici_wrap :
     spectrum ℝ (0 : L ℋ) ⊆ Set.Ici (0 : ℝ) := by
   intro x hx
-  have hx0 : x = 0 := by
-    simpa using hx
+  have hx0 : x = 0 := by simpa using hx
   simp [Set.Ici, hx0]
 
 omit [Nontrivial ℋ] in
@@ -242,15 +237,12 @@ theorem theorem_2_5_2_iv_imp_v {f : ℝ → ℝ} (hiv : CondIVAll.{u} f)
         blockDiagonal (ℋ := ℋ) (star X * X + star Y * Y) 0 ≤ (1 : L (HSum ℋ)) := by
       exact sub_nonneg.mp (by simpa [hsub] using hblock_nonneg)
     simpa [hXtilde_star_mul] using hblock
-  have hXtilde_star_mul_nonneg : (0 : L (HSum ℋ)) ≤ star Xtilde * Xtilde := by
-    simp
+  have hXtilde_star_mul_nonneg : (0 : L (HSum ℋ)) ≤ star Xtilde * Xtilde := by simp
   have hXtilde_norm : ‖Xtilde‖ ≤ 1 := by
     have hnormSq : ‖star Xtilde * Xtilde‖ ≤ 1 :=
       (CStarAlgebra.norm_le_one_iff_of_nonneg _ hXtilde_star_mul_nonneg).2 hXtilde_star_mul_le
-    have hnormSq' : ‖Xtilde‖ * ‖Xtilde‖ ≤ 1 := by
-      simpa [CStarRing.norm_star_mul_self] using hnormSq
-    have hsq : ‖Xtilde‖ ^ 2 ≤ 1 := by
-      simpa [pow_two] using hnormSq'
+    have hnormSq' : ‖Xtilde‖ * ‖Xtilde‖ ≤ 1 := by simpa [CStarRing.norm_star_mul_self] using hnormSq
+    have hsq : ‖Xtilde‖ ^ 2 ≤ 1 := by simpa [pow_two] using hnormSq'
     nlinarith [norm_nonneg Xtilde]
   have hiv_hsum : CondIV (ℋ := HSum ℋ) f := @hiv (HSum ℋ) _ _ _ _
   have hcore := hiv_hsum (A := Atilde) (X := Xtilde) hAtilde_sa hAtilde_spec hXtilde_norm
@@ -295,8 +287,7 @@ theorem theorem_2_5_2_iv_imp_v {f : ℝ → ℝ} (hiv : CondIVAll.{u} f)
 /-- Uniform consequence of Theorem 2.5.2: `(i) → (v)` via `(iv)`. -/
 theorem theorem_2_5_2_i_all_imp_v {f : ℝ → ℝ} (hf : CondIAll.{u} f) :
     CondV (ℋ := ℋ) f := by
-  have hconv : OperatorConvex (ℋ := ℋ) f := by
-    exact hf.1
+  have hconv : OperatorConvex (ℋ := ℋ) f := hf.1
   have hcont : ContinuousOn f Set.univ :=
     operatorConvex_continuousOn_univ (ℋ := ℋ) hconv
   refine theorem_2_5_2_iv_imp_v (ℋ := ℋ) ?_ hcont
@@ -357,25 +348,20 @@ theorem theorem_2_5_2_i_ici_all_imp_v {f : ℝ → ℝ}
         blockDiagonal (ℋ := ℋ) (star X * X + star Y * Y) 0 ≤ (1 : L (HSum ℋ)) := by
       exact sub_nonneg.mp (by simpa [hsub] using hblock_nonneg)
     simpa [hXtilde_star_mul] using hblock
-  have hXtilde_star_mul_nonneg : (0 : L (HSum ℋ)) ≤ star Xtilde * Xtilde := by
-    simp
+  have hXtilde_star_mul_nonneg : (0 : L (HSum ℋ)) ≤ star Xtilde * Xtilde := by simp
   have hXtilde_norm : ‖Xtilde‖ ≤ 1 := by
     have hnormSq : ‖star Xtilde * Xtilde‖ ≤ 1 :=
       (CStarAlgebra.norm_le_one_iff_of_nonneg _ hXtilde_star_mul_nonneg).2 hXtilde_star_mul_le
-    have hnormSq' : ‖Xtilde‖ * ‖Xtilde‖ ≤ 1 := by
-      simpa [CStarRing.norm_star_mul_self] using hnormSq
-    have hsq : ‖Xtilde‖ ^ 2 ≤ 1 := by
-      simpa [pow_two] using hnormSq'
+    have hnormSq' : ‖Xtilde‖ * ‖Xtilde‖ ≤ 1 := by simpa [CStarRing.norm_star_mul_self] using hnormSq
+    have hsq : ‖Xtilde‖ ^ 2 ≤ 1 := by simpa [pow_two] using hnormSq'
     nlinarith [norm_nonneg Xtilde]
   have hiv_hsum : CondIV (ℋ := HSum ℋ) f :=
     theorem_2_5_2_i_ici_all_imp_iv (ℋ := HSum ℋ) (f := f) hfIci
   have hcore := hiv_hsum (A := Atilde) (X := Xtilde) hAtilde_sa hAtilde_spec hXtilde_norm
-  have hsum_nonneg : (0 : L ℋ) ≤ star X * A * X + star Y * B * Y := by
-    have hXA : (0 : L ℋ) ≤ star X * A * X := by
-      simpa [mul_assoc] using star_left_conjugate_nonneg hA0 X
-    have hYB : (0 : L ℋ) ≤ star Y * B * Y := by
-      simpa [mul_assoc] using star_left_conjugate_nonneg hB0 Y
-    exact add_nonneg hXA hYB
+  have hsum_nonneg : (0 : L ℋ) ≤ star X * A * X + star Y * B * Y :=
+    add_nonneg
+      (by simpa [mul_assoc] using star_left_conjugate_nonneg hA0 X)
+      (by simpa [mul_assoc] using star_left_conjugate_nonneg hB0 Y)
   have hsum_sa : IsSelfAdjoint (star X * A * X + star Y * B * Y) :=
     IsSelfAdjoint.of_nonneg hsum_nonneg
   have hmul_block :
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIImpIV.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIImpIV.lean
index c495d7723..8b5da62e6 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIImpIV.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIImpIV.lean
@@ -76,8 +76,7 @@ private noncomputable def blockSwap (X : L ℋ) : L (HSum ℋ) :=
   blockOp (ℋ := ℋ) 0 (star X) X 0
 
 private lemma blockSwap_star (X : L ℋ) :
-    star (blockSwap (ℋ := ℋ) X) = blockSwap (ℋ := ℋ) X := by
-  simp [blockSwap, blockOp_star]
+    star (blockSwap (ℋ := ℋ) X) = blockSwap (ℋ := ℋ) X := by simp [blockSwap, blockOp_star]
 
 private lemma blockSwap_sq (X : L ℋ) :
     blockSwap (ℋ := ℋ) X * blockSwap (ℋ := ℋ) X =
@@ -186,10 +185,8 @@ private lemma cfcR_blockDiagonal (f : ℝ → ℝ)
         (hab := hpair) (ha := hA) (hb := hB))
   calc
     cfcR (ℋ := HSum ℋ) f (blockDiagonal (ℋ := ℋ) A B)
-        = cfc (R := ℝ) (A := L (HSum ℋ)) (p := IsSelfAdjoint) f (φ (A, B)) := by
-          simp [cfcR, φ]
-    _ = φ (cfc (R := ℝ) (A := L ℋ × L ℋ) (p := IsSelfAdjoint) f (A, B)) := by
-          simpa using hmap.symm
+        = cfc (R := ℝ) (A := L (HSum ℋ)) (p := IsSelfAdjoint) f (φ (A, B)) := by simp [cfcR, φ]
+    _ = φ (cfc (R := ℝ) (A := L ℋ × L ℋ) (p := IsSelfAdjoint) f (A, B)) := by simpa using hmap.symm
     _ = φ (cfcR (ℋ := ℋ) f A, cfcR (ℋ := ℋ) f B) := by
           rw [hprod]
     _ = blockDiagonal (ℋ := ℋ) (cfcR (ℋ := ℋ) f A) (cfcR (ℋ := ℋ) f B) := by
@@ -329,18 +326,15 @@ private lemma complex_I_smul_real_I_smul_invTwo (r : ℝ) (T : L ℋ) :
     simpa using (smul_comm r (Complex.I : ℂ) ((2⁻¹ : ℝ) • T x))
   calc
     Complex.I • r • Complex.I • (2⁻¹ : ℝ) • T x
-        = Complex.I • (r • (Complex.I • ((2⁻¹ : ℝ) • T x))) := by
-            rfl
+        = Complex.I • (r • (Complex.I • ((2⁻¹ : ℝ) • T x))) := rfl
     _ = Complex.I • (Complex.I • (r • ((2⁻¹ : ℝ) • T x))) := by
             rw [hcomm]
     _ = ((Complex.I : ℂ) * Complex.I) • (r • ((2⁻¹ : ℝ) • T x)) := by
             rw [smul_smul]
-    _ = (-1 : ℂ) • (r • ((2⁻¹ : ℝ) • T x)) := by
-            norm_num
+    _ = (-1 : ℂ) • (r • ((2⁻¹ : ℝ) • T x)) := by norm_num
     _ = (-1 : ℂ) • (((r * 2⁻¹ : ℝ)) • T x) := by
             rw [smul_smul]
-    _ = -((2⁻¹ : ℝ) * r) • T x := by
-            simp [neg_smul, mul_comm]
+    _ = -((2⁻¹ : ℝ) * r) • T x := by simp [neg_smul, mul_comm]
 
 omit [CompleteSpace ℋ] in
 private lemma real_smul_complex_I_real_smul_complex_I_comm (s r : ℝ) (T : L ℋ) :
@@ -354,15 +348,13 @@ private lemma real_smul_complex_I_real_smul_complex_I_comm (s r : ℝ) (T : L 
             rw [smul_comm (s : ℝ) r (Complex.I • T)]
     _ = Complex.I • (r • (Complex.I • ((s : ℝ) • T))) := by
             rw [smul_comm (s : ℝ) (Complex.I : ℂ) T]
-    _ = Complex.I • r • Complex.I • (s : ℝ) • T := by
-            rfl
+    _ = Complex.I • r • Complex.I • (s : ℝ) • T := rfl
 
 omit [CompleteSpace ℋ] in
 private lemma half_add_half_eq (T : L ℋ) :
     (2⁻¹ : ℝ) • T + (2⁻¹ : ℝ) • T = T := by
   calc
-    (2⁻¹ : ℝ) • T + (2⁻¹ : ℝ) • T = (2⁻¹ + 2⁻¹ : ℝ) • T := by
-      simp [add_smul]
+    (2⁻¹ : ℝ) • T + (2⁻¹ : ℝ) • T = (2⁻¹ + 2⁻¹ : ℝ) • T := by simp [add_smul]
     _ = (1 : ℝ) • T := by norm_num
     _ = T := by simp
 
@@ -371,8 +363,7 @@ private lemma half_mul_real_add_half_mul_real_eq (r : ℝ) (T : L ℋ) :
     ((2⁻¹ : ℝ) * r) • T + ((2⁻¹ : ℝ) * r) • T = r • T := by
   calc
     ((2⁻¹ : ℝ) * r) • T + ((2⁻¹ : ℝ) * r) • T =
-        (((2⁻¹ : ℝ) * r) + ((2⁻¹ : ℝ) * r)) • T := by
-          simp [add_smul]
+        (((2⁻¹ : ℝ) * r) + ((2⁻¹ : ℝ) * r)) • T := by simp [add_smul]
     _ = r • T := by ring_nf
 
 private lemma rightEval_topLeft_scalar
@@ -385,8 +376,7 @@ private lemma rightEval_topLeft_scalar
   have hP :
       (2⁻¹ : ℝ) • (star X * (T * X)) +
           (2⁻¹ : ℝ) • (star X * (T * X)) =
-        star X * (T * X) := by
-    simpa using half_add_half_eq (ℋ := ℋ) (star X * (T * X))
+        star X * (T * X) := by simpa using half_add_half_eq (ℋ := ℋ) (star X * (T * X))
   have hQhalf :
       -((2⁻¹ : ℝ) • Complex.I • r • Complex.I • (R0 * R0)) +
           -((2⁻¹ : ℝ) • Complex.I • r • Complex.I • (R0 * R0)) =
@@ -405,8 +395,7 @@ private lemma rightEval_topLeft_scalar
     calc
       -((2⁻¹ : ℝ) • Complex.I • r • Complex.I • (R0 * R0)) +
           -((2⁻¹ : ℝ) • Complex.I • r • Complex.I • (R0 * R0)) =
-        ((2⁻¹ : ℝ) * r) • (R0 * R0) + ((2⁻¹ : ℝ) * r) • (R0 * R0) := by
-          simp [hterm]
+        ((2⁻¹ : ℝ) * r) • (R0 * R0) + ((2⁻¹ : ℝ) * r) • (R0 * R0) := by simp [hterm]
       _ = r • (R0 * R0) := half_mul_real_add_half_mul_real_eq (ℋ := ℋ) r (R0 * R0)
   calc
     (2⁻¹ : ℝ) • (star X * (T * X)) +
@@ -433,8 +422,7 @@ private lemma rightEval_bottomRight_scalar
     simpa using half_mul_real_add_half_mul_real_eq (ℋ := ℋ) r (X * star X)
   have hT :
       (2⁻¹ : ℝ) • (R1 * (T * R1)) + (2⁻¹ : ℝ) • (R1 * (T * R1)) =
-        R1 * (T * R1) := by
-    simpa using half_add_half_eq (ℋ := ℋ) (R1 * (T * R1))
+        R1 * (T * R1) := by simpa using half_add_half_eq (ℋ := ℋ) (R1 * (T * R1))
   calc
     (2⁻¹ * r) • (X * star X) +
         ((2⁻¹ * r) • (X * star X) +
@@ -494,8 +482,7 @@ private lemma blockSwap_norm_le_one [Nontrivial ℋ] (X : L ℋ) (hX : ‖X‖ 
         simp [sub_eq_add_neg, add_left_comm, add_comm]
     have hle' :
         blockDiagonal (ℋ := ℋ) (star X * X) (X * star X) ≤
-          blockDiagonal (ℋ := ℋ) (1 : L ℋ) (1 : L ℋ) := by
-      simpa [hsum] using hle
+          blockDiagonal (ℋ := ℋ) (1 : L ℋ) (1 : L ℋ) := by simpa [hsum] using hle
     simpa [blockDiagonal_one] using hle'
   have hSstarSle :
       star (blockSwap (ℋ := ℋ) X) * blockSwap (ℋ := ℋ) X ≤ (1 : L (HSum ℋ)) := by
@@ -506,8 +493,7 @@ private lemma blockSwap_norm_le_one [Nontrivial ℋ] (X : L ℋ) (hX : ‖X‖ 
     (CStarAlgebra.norm_le_one_iff_of_nonneg _ hSstarSnonneg).2 hSstarSle
   have hnormSq' : ‖blockSwap (ℋ := ℋ) X‖ * ‖blockSwap (ℋ := ℋ) X‖ ≤ 1 := by
     simpa [CStarRing.norm_star_mul_self] using hnormSq
-  have hsq : ‖blockSwap (ℋ := ℋ) X‖ ^ 2 ≤ 1 := by
-    simpa [pow_two] using hnormSq'
+  have hsq : ‖blockSwap (ℋ := ℋ) X‖ ^ 2 ≤ 1 := by simpa [pow_two] using hnormSq'
   have hnonneg : 0 ≤ ‖blockSwap (ℋ := ℋ) X‖ := norm_nonneg _
   nlinarith
 
@@ -522,10 +508,9 @@ private lemma continuousOn_union_of_subset_Ici {f : ℝ → ℝ}
   · exact ht hx
 
 private lemma spectrum_Ici_of_nonneg {A : L ℋ} (hA0 : (0 : L ℋ) ≤ A) :
-    spectrum ℝ A ⊆ Set.Ici (0 : ℝ) := by
-  exact
-    (StarOrderedRing.nonneg_iff_spectrum_nonneg (R := ℝ) A
-      (ha := IsSelfAdjoint.of_nonneg hA0)).1 hA0
+    spectrum ℝ A ⊆ Set.Ici (0 : ℝ) :=
+  (StarOrderedRing.nonneg_iff_spectrum_nonneg (R := ℝ) A
+    (ha := IsSelfAdjoint.of_nonneg hA0)).1 hA0
 
 variable [Nontrivial ℋ]
 
@@ -533,8 +518,7 @@ omit [CompleteSpace ℋ] in
 private lemma spectrum_zero_subset_Ici :
     spectrum ℝ (0 : L ℋ) ⊆ Set.Ici (0 : ℝ) := by
   intro x hx
-  have hx0 : x = 0 := by
-    simpa using hx
+  have hx0 : x = 0 := by simpa using hx
   simp [Set.Ici, hx0]
 
 
@@ -555,8 +539,7 @@ theorem theorem_2_5_2_i_ici_all_imp_iv {f : ℝ → ℝ} (hf : CondIciAll.{u} f)
   have hSsa : IsSelfAdjoint S := by
     change star S = S
     simpa [S] using blockSwap_star (ℋ := ℋ) X
-  have hSnorm : ‖S‖ ≤ 1 := by
-    simpa [S] using blockSwap_norm_le_one (ℋ := ℋ) X hX
+  have hSnorm : ‖S‖ ≤ 1 := by simpa [S] using blockSwap_norm_le_one (ℋ := ℋ) X hX
   letI : Algebra ℝ (L (HSum ℋ)) := by
     infer_instance
   have hU_mem : S + Complex.I • CFC.sqrt (1 - S ^ 2) ∈ unitary (L (HSum ℋ)) := by
@@ -653,8 +636,7 @@ theorem theorem_2_5_2_i_ici_all_imp_iv {f : ℝ → ℝ} (hf : CondIciAll.{u} f)
     · have hhalf : (2⁻¹ + 2⁻¹ : ℝ) = (1 : ℝ) := by norm_num
       calc
         (1 / 2 : ℝ) • (star X * A * X) + (1 / 2 : ℝ) • (star X * A * X)
-            = (2⁻¹ + 2⁻¹ : ℝ) • (star X * (A * X)) := by
-                simp [add_smul, mul_assoc]
+            = (2⁻¹ + 2⁻¹ : ℝ) • (star X * (A * X)) := by simp [add_smul, mul_assoc]
         _ = (1 : ℝ) • (star X * (A * X)) := by rw [hhalf]
         _ = star X * (A * X) := by simp
         _ = star X * A * X := by simp [mul_assoc]
@@ -809,8 +791,7 @@ theorem theorem_2_5_2_i_all_imp_iv {f : ℝ → ℝ} (hf : CondIAll.{u} f) :
   have hSsa : IsSelfAdjoint S := by
     change star S = S
     simpa [S] using blockSwap_star (ℋ := ℋ) X
-  have hSnorm : ‖S‖ ≤ 1 := by
-    simpa [S] using blockSwap_norm_le_one (ℋ := ℋ) X hX
+  have hSnorm : ‖S‖ ≤ 1 := by simpa [S] using blockSwap_norm_le_one (ℋ := ℋ) X hX
   letI : Algebra ℝ (L (HSum ℋ)) := by
     infer_instance
   have hU_mem : S + Complex.I • CFC.sqrt (1 - S ^ 2) ∈ unitary (L (HSum ℋ)) := by
@@ -908,8 +889,7 @@ theorem theorem_2_5_2_i_all_imp_iv {f : ℝ → ℝ} (hf : CondIAll.{u} f) :
     · have hhalf : (2⁻¹ + 2⁻¹ : ℝ) = (1 : ℝ) := by norm_num
       calc
         (1 / 2 : ℝ) • (star X * A * X) + (1 / 2 : ℝ) • (star X * A * X)
-            = (2⁻¹ + 2⁻¹ : ℝ) • (star X * (A * X)) := by
-                simp [add_smul, mul_assoc]
+            = (2⁻¹ + 2⁻¹ : ℝ) • (star X * (A * X)) := by simp [add_smul, mul_assoc]
         _ = (1 : ℝ) • (star X * (A * X)) := by rw [hhalf]
         _ = star X * (A * X) := by simp
         _ = star X * A * X := by simp [mul_assoc]
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIVtoV.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIVtoV.lean
index 02e3168cf..efad0c931 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIVtoV.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/JensenOperatorInequalityIVtoV.lean
@@ -75,8 +75,7 @@ private theorem nontrivial_hsumL : Nontrivial (L (HSum ℋ)) := by
   have hdiag_ne_zero : (blockDiagonal (ℋ := ℋ) (1 : L ℋ) 0 : L (HSum ℋ)) ≠ 0 := by
     intro h0
     have hz :
-        blockDiagonal (ℋ := ℋ) (1 : L ℋ) 0 (hsumIncl ℋ 0 w) = 0 := by
-      simp [h0]
+        blockDiagonal (ℋ := ℋ) (1 : L ℋ) 0 (hsumIncl ℋ 0 w) = 0 := by simp [h0]
     have hw0 : w = 0 := by
       have hz0 := congrArg (fun z : HSum ℋ => hsumProj ℋ 0 z) hz
       simpa [blockDiagonal] using hz0
@@ -146,10 +145,8 @@ private lemma cfcR_blockDiagonal (f : ℝ → ℝ)
         (hab := hpair) (ha := hA) (hb := hB))
   calc
     cfcR (ℋ := HSum ℋ) f (blockDiagonal (ℋ := ℋ) A B)
-        = cfc (R := ℝ) (A := L (HSum ℋ)) (p := IsSelfAdjoint) f (φ (A, B)) := by
-            simp [cfcR, φ]
-    _ = φ (cfc (R := ℝ) (A := L ℋ × L ℋ) (p := IsSelfAdjoint) f (A, B)) := by
-            simpa using hmap.symm
+        = cfc (R := ℝ) (A := L (HSum ℋ)) (p := IsSelfAdjoint) f (φ (A, B)) := by simp [cfcR, φ]
+    _ = φ (cfc (R := ℝ) (A := L ℋ × L ℋ) (p := IsSelfAdjoint) f (A, B)) := by simpa using hmap.symm
     _ = φ (cfcR (ℋ := ℋ) f A, cfcR (ℋ := ℋ) f B) := by
             rw [hprod]
     _ = blockDiagonal (ℋ := ℋ) (cfcR (ℋ := ℋ) f A) (cfcR (ℋ := ℋ) f B) := by
@@ -234,15 +231,12 @@ theorem theorem_2_5_2_iv_imp_v {f : ℝ → ℝ} (hiv : CondIVAll.{u} f)
         blockDiagonal (ℋ := ℋ) (star X * X + star Y * Y) 0 ≤ (1 : L (HSum ℋ)) := by
       exact sub_nonneg.mp (by simpa [hsub] using hblock_nonneg)
     simpa [hXtilde_star_mul] using hblock
-  have hXtilde_star_mul_nonneg : (0 : L (HSum ℋ)) ≤ star Xtilde * Xtilde := by
-    simp
+  have hXtilde_star_mul_nonneg : (0 : L (HSum ℋ)) ≤ star Xtilde * Xtilde := by simp
   have hXtilde_norm : ‖Xtilde‖ ≤ 1 := by
     have hnormSq : ‖star Xtilde * Xtilde‖ ≤ 1 :=
       (CStarAlgebra.norm_le_one_iff_of_nonneg _ hXtilde_star_mul_nonneg).2 hXtilde_star_mul_le
-    have hnormSq' : ‖Xtilde‖ * ‖Xtilde‖ ≤ 1 := by
-      simpa [CStarRing.norm_star_mul_self] using hnormSq
-    have hsq : ‖Xtilde‖ ^ 2 ≤ 1 := by
-      simpa [pow_two] using hnormSq'
+    have hnormSq' : ‖Xtilde‖ * ‖Xtilde‖ ≤ 1 := by simpa [CStarRing.norm_star_mul_self] using hnormSq
+    have hsq : ‖Xtilde‖ ^ 2 ≤ 1 := by simpa [pow_two] using hnormSq'
     nlinarith [norm_nonneg Xtilde]
   have hiv_hsum : CondIV (ℋ := HSum ℋ) f := @hiv (HSum ℋ) _ _ _ _
   have hcore := hiv_hsum (A := Atilde) (X := Xtilde) hAtilde_sa hAtilde_spec hXtilde_norm
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LiebAndoTrace.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LiebAndoTrace.lean
index d50805849..5c1d5d07c 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LiebAndoTrace.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LiebAndoTrace.lean
@@ -417,13 +417,11 @@ private lemma hmiddle_leftMul_rightMul
     exact (rightMulHS_cfcR (ℋ := ℋ) (fun x : ℝ ↦ x ^ s) (B ^ (-1 : ℝ)) hBinv_sa
       (by
         intro x hx
-        have hx0 : x ≠ 0 := by
-          intro hx0
-          exact spectrum.zero_notMem (R := ℝ) hBinv_unit (by simpa [hx0] using hx)
+        have hx0 : x ≠ 0 :=
+          fun hx0 => spectrum.zero_notMem (R := ℝ) hBinv_unit (by simpa [hx0] using hx)
         exact (Real.continuousAt_rpow_const x s (Or.inl hx0)).continuousWithinAt)).symm
   have hprod0 :
-      0 ≤ leftMulHS (ℋ := ℋ) A * rightMulHS (ℋ := ℋ) (B ^ (-1 : ℝ)) := by
-    exact (leftMulHS_rightMulHS_commute (ℋ := ℋ) A (B ^ (-1 : ℝ))).mul_nonneg
+      0 ≤ leftMulHS (ℋ := ℋ) A * rightMulHS (ℋ := ℋ) (B ^ (-1 : ℝ)) := (leftMulHS_rightMulHS_commute (ℋ := ℋ) A (B ^ (-1 : ℝ))).mul_nonneg
       (leftMulHS_nonneg (ℋ := ℋ) hA0)
       (rightMulHS_nonneg (ℋ := ℋ) hBinv0)
   have hprod_sa :
@@ -431,8 +429,7 @@ private lemma hmiddle_leftMul_rightMul
         (leftMulHS (ℋ := ℋ) A * rightMulHS (ℋ := ℋ) (B ^ (-1 : ℝ))) :=
     IsSelfAdjoint.of_nonneg hprod0
   have htop :
-      (⨆ α, ⨆ β, eigenspace T0 α ⊓ eigenspace T1 β) = ⊤ := by
-    exact LinearMap.IsSymmetric.iSup_iSup_eigenspace_inf_eigenspace_eq_top_of_commute
+      (⨆ α, ⨆ β, eigenspace T0 α ⊓ eigenspace T1 β) = ⊤ := LinearMap.IsSymmetric.iSup_iSup_eigenspace_inf_eigenspace_eq_top_of_commute
       hT0_symm hT1_symm hcomm
   have hjoint_ker :
       ∀ α β,
@@ -530,8 +527,7 @@ private lemma hmiddle_leftMul_rightMul
       simp [mul_comm]
     have hscal :
         (((α.re * β.re : ℝ) ^ s : ℝ) : ℂ) =
-          ((((α.re ^ s) * (β.re ^ s) : ℝ)) : ℂ) := by
-      exact congrArg (fun t : ℝ => (t : ℂ)) (Real.mul_rpow hαnonneg hβnonneg)
+          ((((α.re ^ s) * (β.re ^ s) : ℝ)) : ℂ) := congrArg (fun t : ℝ => (t : ℂ)) (Real.mul_rpow hαnonneg hβnonneg)
     simpa [D] using
       sub_eq_zero.mpr (hlhsx.trans (hscal ▸ hrhsx.symm))
   have hker_top : LinearMap.ker D.toLinearMap = ⊤ := by
@@ -617,8 +613,7 @@ private lemma phiK_operatorPowerMean_eq_liebTraceMap
     simp
   have hBinv_sa : IsSelfAdjoint (B ^ (-1 : ℝ)) := IsSelfAdjoint.of_nonneg hBinv0
   have hright_invHalf0 :
-      0 ≤ rightMulHS (ℋ := ℋ) (B ^ ((-1 : ℝ) / 2)) := by
-    exact rightMulHS_nonneg (ℋ := ℋ) hBinvHalf0
+      0 ≤ rightMulHS (ℋ := ℋ) (B ^ ((-1 : ℝ) / 2)) := rightMulHS_nonneg (ℋ := ℋ) hBinvHalf0
   have hmid_nonneg :
       0 ≤ cfcR (ℋ := HSOp ℋ) (fun x : ℝ ↦ x ^ ((-1 : ℝ) / 2)) (rightMulHS (ℋ := ℋ) B) *
           leftMulHS (ℋ := ℋ) A *
@@ -650,8 +645,7 @@ private lemma phiK_operatorPowerMean_eq_liebTraceMap
           (cfcR (ℋ := HSOp ℋ) (fun x : ℝ ↦ x ^ ((-1 : ℝ) / 2)) (rightMulHS (ℋ := ℋ) B) *
             leftMulHS (ℋ := ℋ) A *
             cfcR (ℋ := HSOp ℋ) (fun x : ℝ ↦ x ^ ((-1 : ℝ) / 2)) (rightMulHS (ℋ := ℋ) B)) =
-        leftMulHS (ℋ := ℋ) (A ^ s) * rightMulHS (ℋ := ℋ) (B ^ (-s)) := by
-    exact hmiddle_leftMul_rightMul (ℋ := ℋ) (s := s) ⟨hA_sa, hA_spec⟩ ⟨hB_sa, hB_spec⟩
+        leftMulHS (ℋ := ℋ) (A ^ s) * rightMulHS (ℋ := ℋ) (B ^ (-s)) := hmiddle_leftMul_rightMul (ℋ := ℋ) (s := s) ⟨hA_sa, hA_spec⟩ ⟨hB_sa, hB_spec⟩
   have happly :
       operatorPowerMean (ℋ := HSOp ℋ) s 1
           (leftMulHS (ℋ := ℋ) A) (rightMulHS (ℋ := ℋ) B) (ofOp (star K)) =
@@ -750,8 +744,7 @@ private lemma phiK_operatorPowerMean_eq_liebTraceMap
               congrArg Complex.re
                 (LinearMap.trace_mul_comm (R := ℂ) (M := ℋ) K.toLinearMap X.toLinearMap)
           simpa [X, mul_assoc] using htrace'.trans hcycle
-    _ = liebTraceMap (ℋ := ℋ) s K A B := by
-          rfl
+    _ = liebTraceMap (ℋ := ℋ) s K A B := rfl
 
 omit [FiniteDimensional ℂ ℋ] in
 /-- Convex combinations preserve `pdSet` (strict positivity). -/
@@ -915,8 +908,7 @@ private lemma liebTraceMap_mono_right
   have hdiff0 : 0 ≤ B₂ ^ (1 - s) - B₁ ^ (1 - s) := sub_nonneg.mpr hpow
   have hprod0 :
       0 ≤ leftMulHS (ℋ := ℋ) (A ^ s) *
-          rightMulHS (ℋ := ℋ) (B₂ ^ (1 - s) - B₁ ^ (1 - s)) := by
-    exact (leftMulHS_rightMulHS_commute (ℋ := ℋ) (A ^ s) (B₂ ^ (1 - s) - B₁ ^ (1 - s))).mul_nonneg
+          rightMulHS (ℋ := ℋ) (B₂ ^ (1 - s) - B₁ ^ (1 - s)) := (leftMulHS_rightMulHS_commute (ℋ := ℋ) (A ^ s) (B₂ ^ (1 - s) - B₁ ^ (1 - s))).mul_nonneg
       (leftMulHS_nonneg (ℋ := ℋ) hApow0)
       (rightMulHS_nonneg (ℋ := ℋ) hdiff0)
   have hphi : 0 ≤
@@ -993,8 +985,7 @@ private lemma liebTraceMap_antitone_right
   have hdiff0 : 0 ≤ B₁ ^ (1 - s) - B₂ ^ (1 - s) := sub_nonneg.mpr hpow
   have hprod0 :
       0 ≤ leftMulHS (ℋ := ℋ) (A ^ s) *
-          rightMulHS (ℋ := ℋ) (B₁ ^ (1 - s) - B₂ ^ (1 - s)) := by
-    exact (leftMulHS_rightMulHS_commute (ℋ := ℋ) (A ^ s) (B₁ ^ (1 - s) - B₂ ^ (1 - s))).mul_nonneg
+          rightMulHS (ℋ := ℋ) (B₁ ^ (1 - s) - B₂ ^ (1 - s)) := (leftMulHS_rightMulHS_commute (ℋ := ℋ) (A ^ s) (B₁ ^ (1 - s) - B₂ ^ (1 - s))).mul_nonneg
       (leftMulHS_nonneg (ℋ := ℋ) hApow0)
       (rightMulHS_nonneg (ℋ := ℋ) hdiff0)
   have hphi : 0 ≤
@@ -1068,11 +1059,9 @@ theorem liebTrace_jointlyConcaveOn_pdSet
       toOp T * ((1 - θ) • B₁ + θ • B₂)
     rw [mul_add, mul_smul_comm, mul_smul_comm]
   have hA_combo :
-      ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
+      ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
   have hB_combo :
-      ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
+      ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
   letI : Nontrivial (HSOp ℋ) := by
     delta HSOp
     infer_instance
@@ -1105,8 +1094,7 @@ theorem liebTrace_jointlyConcaveOn_pdSet
         phiK (ℋ := ℋ) K
           (operatorPowerMean (ℋ := HSOp ℋ) s 1
             (leftMulHS (ℋ := ℋ) ((1 - θ) • A₁ + θ • A₂))
-            (rightMulHS (ℋ := ℋ) ((1 - θ) • B₁ + θ • B₂))) := by
-    exact phiK_mono (ℋ := ℋ) K hconc
+            (rightMulHS (ℋ := ℋ) ((1 - θ) • B₁ + θ • B₂))) := phiK_mono (ℋ := ℋ) K hconc
   rw [phiK_weightedSum_operatorPowerMean_eq (ℋ := ℋ) (s := s) (θ := θ) K A₁ A₂ B₁ B₂
       hA₁ hA₂ hB₁ hB₂] at hphi_mono
   rw [phiK_operatorPowerMean_eq_liebTraceMap (ℋ := ℋ) (s := s) K
@@ -1133,11 +1121,9 @@ theorem liebTrace_jointlyConvexOn_pdSet
       toOp T * ((1 - θ) • B₁ + θ • B₂)
     rw [mul_add, mul_smul_comm, mul_smul_comm]
   have hA_combo :
-      ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
+      ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
   have hB_combo :
-      ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
+      ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
   letI : Nontrivial (HSOp ℋ) := by
     delta HSOp
     infer_instance
@@ -1170,8 +1156,7 @@ theorem liebTrace_jointlyConvexOn_pdSet
           ((1 - θ) • operatorPowerMean (ℋ := HSOp ℋ) s 1
               (leftMulHS (ℋ := ℋ) A₁) (rightMulHS (ℋ := ℋ) B₁) +
             θ • operatorPowerMean (ℋ := HSOp ℋ) s 1
-              (leftMulHS (ℋ := ℋ) A₂) (rightMulHS (ℋ := ℋ) B₂)) := by
-    exact phiK_mono (ℋ := ℋ) K hconv
+              (leftMulHS (ℋ := ℋ) A₂) (rightMulHS (ℋ := ℋ) B₂)) := phiK_mono (ℋ := ℋ) K hconv
   rw [phiK_operatorPowerMean_eq_liebTraceMap (ℋ := ℋ) (s := s) K
       ((1 - θ) • A₁ + θ • A₂) ((1 - θ) • B₁ + θ • B₂) hA_combo hB_combo] at hphi_mono
   rw [phiK_weightedSum_operatorPowerMean_eq (ℋ := ℋ) (s := s) (θ := θ) K A₁ A₂ B₁ B₂
@@ -1199,11 +1184,9 @@ theorem liebExtensionTrace_jointlyConcaveOn_pdSet
     field_simp [h1q.ne']
   intro A₁ A₂ B₁ B₂ θ hA₁ hA₂ hB₁ hB₂ hθ0 hθ1
   have hA_combo :
-      ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
+      ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
   have hB_combo :
-      ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
+      ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
   have hB₁β : B₁ ^ β ∈ pdSet (ℋ := ℋ) :=
     pdSet_rpow_of_mem_Icc_zero_one (ℋ := ℋ) hβ hB₁
   have hB₂β : B₂ ^ β ∈ pdSet (ℋ := ℋ) :=
@@ -1212,8 +1195,7 @@ theorem liebExtensionTrace_jointlyConcaveOn_pdSet
       (((1 - θ) • B₁ + θ • B₂) ^ β) ∈ pdSet (ℋ := ℋ) :=
     pdSet_rpow_of_mem_Icc_zero_one (ℋ := ℋ) hβ hB_combo
   have hBpow_combo :
-      ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) ∈ pdSet (ℋ := ℋ) := by
-    exact pdSet_convexCombo (ℋ := ℋ) hB₁β hB₂β hθ0 hθ1
+      ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hB₁β hB₂β hθ0 hθ1
   have hB₁_mem := hB₁
   have hB₂_mem := hB₂
   have hB_combo_mem := hB_combo
@@ -1265,8 +1247,7 @@ theorem liebExtensionTrace_jointlyConcaveOn_pdSet
           ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) ≤
         liebTraceMap (ℋ := ℋ) q K
           ((1 - θ) • A₁ + θ • A₂)
-          (((1 - θ) • B₁ + θ • B₂) ^ β) := by
-    exact liebTraceMap_mono_right (ℋ := ℋ) (s := q) hsmono K
+          (((1 - θ) • B₁ + θ • B₂) ^ β) := liebTraceMap_mono_right (ℋ := ℋ) (s := q) hsmono K
       ((1 - θ) • A₁ + θ • A₂)
       ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β))
       (((1 - θ) • B₁ + θ • B₂) ^ β)
@@ -1325,11 +1306,9 @@ theorem andoTrace_jointlyConvexOn_pdSet
       ring
     intro A₁ A₂ B₁ B₂ θ hA₁ hA₂ hB₁ hB₂ hθ0 hθ1
     have hA_combo :
-        ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := by
-      exact pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
+        ((1 - θ) • A₁ + θ • A₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hA₁ hA₂ hθ0 hθ1
     have hB_combo :
-        ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := by
-      exact pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
+        ((1 - θ) • B₁ + θ • B₂) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hB₁ hB₂ hθ0 hθ1
     have hB₁β : B₁ ^ β ∈ pdSet (ℋ := ℋ) :=
       pdSet_rpow_of_mem_Icc_zero_one (ℋ := ℋ) hβ hB₁
     have hB₂β : B₂ ^ β ∈ pdSet (ℋ := ℋ) :=
@@ -1338,8 +1317,7 @@ theorem andoTrace_jointlyConvexOn_pdSet
         (((1 - θ) • B₁ + θ • B₂) ^ β) ∈ pdSet (ℋ := ℋ) :=
       pdSet_rpow_of_mem_Icc_zero_one (ℋ := ℋ) hβ hB_combo
     have hBpow_combo :
-        ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) ∈ pdSet (ℋ := ℋ) := by
-      exact pdSet_convexCombo (ℋ := ℋ) hB₁β hB₂β hθ0 hθ1
+        ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) ∈ pdSet (ℋ := ℋ) := pdSet_convexCombo (ℋ := ℋ) hB₁β hB₂β hθ0 hθ1
     have hB₁_mem := hB₁
     have hB₂_mem := hB₂
     have hB_combo_mem := hB_combo
@@ -1391,8 +1369,7 @@ theorem andoTrace_jointlyConvexOn_pdSet
             (((1 - θ) • B₁ + θ • B₂) ^ β) ≤
           liebTraceMap (ℋ := ℋ) q K
             ((1 - θ) • A₁ + θ • A₂)
-            ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) := by
-      exact liebTraceMap_antitone_right (ℋ := ℋ) (s := q) hsanti K
+            ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β)) := liebTraceMap_antitone_right (ℋ := ℋ) (s := q) hsanti K
         ((1 - θ) • A₁ + θ • A₂)
         ((1 - θ) • (B₁ ^ β) + θ • (B₂ ^ β))
         (((1 - θ) • B₁ + θ • B₂) ^ β)
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzCore.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzCore.lean
index 93674e3e5..cab27837e 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzCore.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzCore.lean
@@ -156,8 +156,7 @@ variable [NonnegSpectrumClass ℝ 𝓐]
 
 omit [Nontrivial (𝓐)] [NonnegSpectrumClass ℝ 𝓐] in
 lemma conjugate_isPositive {X T : 𝓐} (hX : 0 ≤ X) (hT : IsSelfAdjoint T) :
-    0 ≤ T * X * T := by
-  simpa using hT.conjugate_nonneg hX
+    0 ≤ T * X * T := by simpa using hT.conjugate_nonneg hX
 
 omit [Nontrivial 𝓐] [NonnegSpectrumClass ℝ 𝓐] in
 theorem one_div_operatorAntitoneOn_Ioi :
@@ -167,24 +166,18 @@ theorem one_div_operatorAntitoneOn_Ioi :
   let f : ℝ → ℝ := fun x ↦ x
   have hA_sa : IsSelfAdjoint A := IsSelfAdjoint.of_nonneg A_nonneg
   have hB_sa : IsSelfAdjoint B := IsSelfAdjoint.of_nonneg B_nonneg
-  have hA_ne0 : ∀ x ∈ spectrum ℝ A, f x ≠ 0 := by
-    intro x hx
-    exact ne_of_gt (As hx)
-  have hB_ne0 : ∀ x ∈ spectrum ℝ B, f x ≠ 0 := by
-    intro x hx
-    exact ne_of_gt (Bs hx)
+  have hA_ne0 : ∀ x ∈ spectrum ℝ A, f x ≠ 0 :=
+    fun _ hx => ne_of_gt (As hx)
+  have hB_ne0 : ∀ x ∈ spectrum ℝ B, f x ≠ 0 :=
+    fun _ hx => ne_of_gt (Bs hx)
   let uA : (𝓐)ˣ :=
     cfcUnits (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) f A hA_ne0 (ha := hA_sa)
   let uB : (𝓐)ˣ :=
     cfcUnits (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) f B hB_ne0 (ha := hB_sa)
-  have huA_val : (uA : 𝓐) = A := by
-    simp [uA, cfcUnits, f, cfc_id' (R := ℝ) (a := A) (ha := hA_sa)]
-  have huB_val : (uB : 𝓐) = B := by
-    simp [uB, cfcUnits, f, cfc_id' (R := ℝ) (a := B) (ha := hB_sa)]
-  have huB_nonneg : 0 ≤ (uB : 𝓐) := by
-    simpa [huB_val] using B_nonneg
-  have hub_le_hua : (uB : 𝓐) ≤ (uA : 𝓐) := by
-    simpa [huA_val, huB_val] using BA
+  have huA_val : (uA : 𝓐) = A := by simp [uA, cfcUnits, f, cfc_id' (R := ℝ) (a := A) (ha := hA_sa)]
+  have huB_val : (uB : 𝓐) = B := by simp [uB, cfcUnits, f, cfc_id' (R := ℝ) (a := B) (ha := hB_sa)]
+  have huB_nonneg : 0 ≤ (uB : 𝓐) := by simpa [huB_val] using B_nonneg
+  have hub_le_hua : (uB : 𝓐) ≤ (uA : 𝓐) := by simpa [huA_val, huB_val] using BA
   have hinv : (↑uA⁻¹ : 𝓐) ≤ (↑uB⁻¹ : 𝓐) := by
     simpa using
       (CStarAlgebra.inv_le_inv (A := 𝓐) (a := uB) (b := uA) huB_nonneg hub_le_hua)
@@ -220,8 +213,7 @@ private lemma spectrum_convexCombo_Ioi {A B : 𝓐} {t : ℝ}
       simpa [C] using add_le_add (smul_le_smul_of_nonneg_left hrA_le (sub_nonneg.mpr ht1)) (smul_le_smul_of_nonneg_left hrB_le ht0)
     have hLHS :
         (1 - t) • algebraMap ℝ (𝓐) rA + t • algebraMap ℝ (𝓐) rB =
-          algebraMap ℝ (𝓐) rC := by
-      simp [rC, Algebra.smul_def]
+          algebraMap ℝ (𝓐) rC := by simp [rC, Algebra.smul_def]
     simpa [hLHS] using hsum
   intro x hx
   simpa [C] using (CFC.exists_pos_algebraMap_le_iff (A := 𝓐) (a := C) (ha := hC)).1 ⟨rC, hrC, hrC_le⟩ x hx
@@ -239,11 +231,9 @@ private lemma posSemidef_block_one_inv {A : 𝓐} (hA : IsSelfAdjoint A)
   have posV : Matrix.PosSemidef (Matrix.vecMulVec v (star v)) := by
     simpa using (Matrix.posSemidef_vecMulVec_self_star (R := 𝓐) v)
   have hsqrtA : IsSelfAdjoint sqrtA := by
-    dsimp [sqrtA, cfcR]
-    exact cfc_predicate _ _
+    simp [sqrtA, cfcR]
   have hinvSqrtA : IsSelfAdjoint invSqrtA := by
-    dsimp [invSqrtA, cfcR]
-    exact cfc_predicate _ _
+    simp [invSqrtA, cfcR]
   have hcont_sqrt : ContinuousOn (fun x : ℝ ↦ x ^ ((1 : ℝ) / 2)) (spectrum ℝ A) :=
     fun x hx => (Real.continuousAt_rpow_const x _ (Or.inl (ne_of_gt (As hx)))).continuousWithinAt
   have hcont_invSqrt : ContinuousOn (fun x : ℝ ↦ x ^ ((-1 : ℝ) / 2)) (spectrum ℝ A) :=
@@ -281,7 +271,6 @@ private lemma posSemidef_block_one_inv {A : 𝓐} (hA : IsSelfAdjoint A)
             rw [← Real.rpow_add (As hx), show ((1 : ℝ) / 2 + (1 : ℝ) / 2 : ℝ) = 1 from by ring, Real.rpow_one]
       _ = A := cfc_id' (R := ℝ) (a := A) (ha := hA)
   have invA_eq : cfcR (fun x : ℝ ↦ x ^ (-1 : ℝ)) A = cfcR (fun x : ℝ ↦ x⁻¹) A := by
-    dsimp [cfcR]
     apply cfc_congr
     intro x hx
     have hxne : x ≠ 0 := ne_of_gt (As hx)
@@ -309,8 +298,7 @@ private lemma schur_conj_eq_diagonal {C D invC : 𝓐} (hInvC_sa : IsSelfAdjoint
     fin_cases i <;> fin_cases j <;> simp [hInvC_sa.star_eq]
   have hP :
       star U * (!![C, 1; 1, D] : Matrix (Fin 2) (Fin 2) (𝓐)) =
-        !![C, 1; -invC * C + 1, -invC + D] := by
-    simp [hstarU, U]
+        !![C, 1; -invC * C + 1, -invC + D] := by simp [hstarU, U]
   have hQ :
       star U * (!![C, 1; 1, D] : Matrix (Fin 2) (Fin 2) (𝓐)) * U =
         !![C, 0; 0, D - invC] := by
@@ -375,16 +363,13 @@ theorem one_div_operatorConvexOn_Ioi :
   have hU : IsUnit U := by
     let V : Matrix (Fin 2) (Fin 2) (𝓐) := !![(1 : 𝓐), invC; 0, 1]
     refine ⟨⟨U, V, ?_, ?_⟩, rfl⟩
-    · dsimp [U, V]
-      simp [Matrix.one_fin_two]
-    · dsimp [U, V]
-      simp [Matrix.one_fin_two]
+    · simp [U, V, Matrix.one_fin_two]
+    · simp [U, V, Matrix.one_fin_two]
   have hconj :
       star U * (!![C, 1; 1, D] : Matrix (Fin 2) (Fin 2) (𝓐)) * U
         = Matrix.diagonal (fun i : Fin 2 => if i = 0 then C else D - invC) := by
     have hInvC_sa : IsSelfAdjoint invC := by
-      dsimp [invC, cfcR]
-      exact cfc_predicate _ _
+      simp [invC, cfcR]
     have hcont_inv : ContinuousOn (fun x : ℝ ↦ x⁻¹) (spectrum ℝ C) :=
       fun x hx => (continuousAt_inv₀ (ne_of_gt (specC hx))).continuousWithinAt
     have invC_mul_C : invC * C = (1 : 𝓐) := by
@@ -422,11 +407,9 @@ theorem one_div_operatorConvexOn_Ioi :
         Matrix.PosSemidef
           (star U * (!![C, 1; 1, D] : Matrix (Fin 2) (Fin 2) 𝓐) * U) := by
       -- rewrite the middle block matrix as `M`
-      rw [← hM]
-      exact posConj
+      simpa [← hM] using posConj
     -- rewrite the goal using the computed conjugation
-    rw [← hconj]
-    exact posConj'
+    simpa [← hconj] using posConj'
   have hinvC : invC ≤ D := by
     have hDinvC : 0 ≤ D - invC := by
       simpa using
@@ -444,16 +427,10 @@ theorem one_div_add_t_operatorAntitoneOn_Ici : ∀ (t : ℝ), 0 < t →
   let f : ℝ → ℝ := fun x => x + t
   have hA_sa : IsSelfAdjoint A := IsSelfAdjoint.of_nonneg A_nonneg
   have hB_sa : IsSelfAdjoint B := IsSelfAdjoint.of_nonneg B_nonneg
-  have hA_ne0 : ∀ x ∈ spectrum ℝ A, f x ≠ 0 := by
-    intro x hx
-    have hx0 : (0 : ℝ) ≤ x := by
-      simpa [Set.Ici] using (As hx)
-    exact ne_of_gt (add_pos_of_nonneg_of_pos hx0 ht)
-  have hB_ne0 : ∀ x ∈ spectrum ℝ B, f x ≠ 0 := by
-    intro x hx
-    have hx0 : (0 : ℝ) ≤ x := by
-      simpa [Set.Ici] using (Bs hx)
-    exact ne_of_gt (add_pos_of_nonneg_of_pos hx0 ht)
+  have hA_ne0 : ∀ x ∈ spectrum ℝ A, f x ≠ 0 :=
+    fun _ hx => ne_of_gt (add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using As hx) ht)
+  have hB_ne0 : ∀ x ∈ spectrum ℝ B, f x ≠ 0 :=
+    fun _ hx => ne_of_gt (add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using Bs hx) ht)
   let uA : (𝓐)ˣ :=
     cfcUnits (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) f A hA_ne0 (ha := hA_sa)
   let uB : (𝓐)ˣ :=
@@ -475,9 +452,7 @@ theorem one_div_add_t_operatorAntitoneOn_Ici : ∀ (t : ℝ), 0 < t →
     simp only [uB, cfcUnits, f]
     refine cfc_nonneg (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (a := B) ?_
     intro x hx
-    have hx0 : (0 : ℝ) ≤ x := by
-      simpa [Set.Ici] using (Bs hx)
-    exact le_of_lt (add_pos_of_nonneg_of_pos hx0 ht)
+    exact le_of_lt (add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using Bs hx) ht)
   have hub_le_hua : (uB : 𝓐) ≤ (uA : 𝓐) := by
     simpa [huA_val, huB_val, add_assoc, add_left_comm, add_comm] using add_le_add_right BA (algebraMap ℝ (𝓐) t)
   have hinv : (↑uA⁻¹ : 𝓐) ≤ (↑uB⁻¹ : 𝓐) := by
@@ -508,14 +483,12 @@ theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
       (hr := IsSelfAdjoint.all (t : ℝ)))
   have A_nonneg : 0 ≤ A := by
     have h0 : 0 ≤ cfcR (fun x : ℝ ↦ x) A := by
-      dsimp [cfcR]
       apply cfc_nonneg
       intro x hx
       simpa [Set.Ici] using (As hx)
     simpa [cfcR, cfc_id' (R := ℝ) (a := A) (ha := hA)] using h0
   have B_nonneg : 0 ≤ B := by
     have h0 : 0 ≤ cfcR (fun x : ℝ ↦ x) B := by
-      dsimp [cfcR]
       apply cfc_nonneg
       intro x hx
       simpa [Set.Ici] using (Bs hx)
@@ -523,18 +496,15 @@ theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
   have C_nonneg : 0 ≤ C := by
     simpa [C] using add_nonneg (smul_nonneg (sub_nonneg.mpr hθ1) A_nonneg) (smul_nonneg hθ0 B_nonneg)
   have hA_shift : cfcR shift A = A + T := by
-    dsimp [cfcR, shift, T]
-    simpa [cfc_id' (R := ℝ) (a := A) (ha := hA)] using
+    simpa [cfcR, shift, T, cfc_id' (R := ℝ) (a := A) (ha := hA)] using
       (cfc_add_const (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (r := t)
         (f := fun x : ℝ ↦ x) (a := A) (ha := hA))
   have hB_shift : cfcR shift B = B + T := by
-    dsimp [cfcR, shift, T]
-    simpa [cfc_id' (R := ℝ) (a := B) (ha := hB)] using
+    simpa [cfcR, shift, T, cfc_id' (R := ℝ) (a := B) (ha := hB)] using
       (cfc_add_const (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (r := t)
         (f := fun x : ℝ ↦ x) (a := B) (ha := hB))
   have hC_shift : cfcR shift C = C + T := by
-    dsimp [cfcR, shift, T]
-    simpa [cfc_id' (R := ℝ) (a := C) (ha := hC)] using
+    simpa [cfcR, shift, T, cfc_id' (R := ℝ) (a := C) (ha := hC)] using
       (cfc_add_const (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (r := t)
         (f := fun x : ℝ ↦ x) (a := C) (ha := hC))
   set A1 : 𝓐 := A + T
@@ -554,13 +524,11 @@ theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
           (f := shift) (ha := hA))
     have hx' : x ∈ shift '' spectrum ℝ A := by
       have hx0 : x ∈ spectrum ℝ (cfc shift A) := by
-        have hval : cfc shift A = A1 := by
-          simpa [cfcR, shift, A1] using hA_shift
+        have hval : cfc shift A = A1 := by simpa [cfcR, shift, A1] using hA_shift
         simpa [hval] using hx
       simpa [hs] using hx0
     rcases hx' with ⟨y, hy, rfl⟩
-    have hy0 : 0 ≤ y := by
-      simpa [Set.Ici] using (As hy)
+    have hy0 : 0 ≤ y := by simpa [Set.Ici] using (As hy)
     simpa [Set.Ioi] using (add_pos_of_nonneg_of_pos hy0 ht)
   have specB1 : spectrum ℝ B1 ⊆ Set.Ioi (0 : ℝ) := by
     intro x hx
@@ -570,58 +538,47 @@ theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
           (f := shift) (ha := hB))
     have hx' : x ∈ shift '' spectrum ℝ B := by
       have hx0 : x ∈ spectrum ℝ (cfc shift B) := by
-        have hval : cfc shift B = B1 := by
-          simpa [cfcR, shift, B1] using hB_shift
+        have hval : cfc shift B = B1 := by simpa [cfcR, shift, B1] using hB_shift
         simpa [hval] using hx
       simpa [hs] using hx0
     rcases hx' with ⟨y, hy, rfl⟩
-    have hy0 : 0 ≤ y := by
-      simpa [Set.Ici] using (Bs hy)
+    have hy0 : 0 ≤ y := by simpa [Set.Ici] using (Bs hy)
     simpa [Set.Ioi] using (add_pos_of_nonneg_of_pos hy0 ht)
   have hC1 : C1 = C + T := by
     subst A1 B1 C1
     simp [C, add_assoc, add_left_comm, add_comm, smul_add]
-  have hshift_ne0_A : ∀ x ∈ spectrum ℝ A, shift x ≠ 0 := by
-    intro x hx
-    have hx0 : 0 ≤ x := by
-      simpa [Set.Ici] using (As hx)
-    exact ne_of_gt (by simpa [shift] using (add_pos_of_nonneg_of_pos hx0 ht))
-  have hshift_ne0_B : ∀ x ∈ spectrum ℝ B, shift x ≠ 0 := by
-    intro x hx
-    have hx0 : 0 ≤ x := by
-      simpa [Set.Ici] using (Bs hx)
-    exact ne_of_gt (by simpa [shift] using (add_pos_of_nonneg_of_pos hx0 ht))
+  have hshift_ne0_A : ∀ x ∈ spectrum ℝ A, shift x ≠ 0 :=
+    fun _ hx => ne_of_gt (by
+      simpa [shift] using add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using As hx) ht)
+  have hshift_ne0_B : ∀ x ∈ spectrum ℝ B, shift x ≠ 0 :=
+    fun _ hx => ne_of_gt (by
+      simpa [shift] using add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using Bs hx) ht)
   have hshift_ne0_C : ∀ x ∈ spectrum ℝ C, shift x ≠ 0 :=
     fun x hx ↦ ne_of_gt (by simpa [shift] using (add_pos_of_nonneg_of_pos (spectrum_nonneg_of_nonneg C_nonneg hx) ht))
   have hA_inv : cfcR (fun x : ℝ ↦ (x + t)⁻¹) A = Ring.inverse A1 := by
     have h' : cfc (fun x : ℝ ↦ (shift x)⁻¹) A = Ring.inverse (cfc shift A) := by
       simpa [shift] using (cfc_inv (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint)
         (f := shift) (a := A) hshift_ne0_A (ha := hA))
-    have hval : cfc shift A = A1 := by
-      simpa [cfcR, shift, A1] using hA_shift
+    have hval : cfc shift A = A1 := by simpa [cfcR, shift, A1] using hA_shift
     simpa [cfcR, shift, hval] using h'
   have hB_inv : cfcR (fun x : ℝ ↦ (x + t)⁻¹) B = Ring.inverse B1 := by
     have h' : cfc (fun x : ℝ ↦ (shift x)⁻¹) B = Ring.inverse (cfc shift B) := by
       simpa [shift] using (cfc_inv (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint)
         (f := shift) (a := B) hshift_ne0_B (ha := hB))
-    have hval : cfc shift B = B1 := by
-      simpa [cfcR, shift, B1] using hB_shift
+    have hval : cfc shift B = B1 := by simpa [cfcR, shift, B1] using hB_shift
     simpa [cfcR, shift, hval] using h'
   have hC_inv : cfcR (fun x : ℝ ↦ (x + t)⁻¹) C = Ring.inverse (C + T) := by
     have h' : cfc (fun x : ℝ ↦ (shift x)⁻¹) C = Ring.inverse (cfc shift C) := by
       simpa [shift] using (cfc_inv (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint)
         (f := shift) (a := C) hshift_ne0_C (ha := hC))
-    have hval : cfc shift C = C + T := by
-      simpa [cfcR, shift] using hC_shift
+    have hval : cfc shift C = C + T := by simpa [cfcR, shift] using hC_shift
     simpa [cfcR, shift, hval] using h'
   have hA1_inv : cfcR (fun x : ℝ ↦ x⁻¹) A1 = Ring.inverse A1 := by
-    dsimp [cfcR]
-    simpa [cfc_id' (R := ℝ) (a := A1) (ha := hA1_sa)] using
+    simpa [cfcR, cfc_id' (R := ℝ) (a := A1) (ha := hA1_sa)] using
       (cfc_inv (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x)
         (a := A1) (fun x hx ↦ ne_of_gt (specA1 hx)) (ha := hA1_sa))
   have hB1_inv : cfcR (fun x : ℝ ↦ x⁻¹) B1 = Ring.inverse B1 := by
-    dsimp [cfcR]
-    simpa [cfc_id' (R := ℝ) (a := B1) (ha := hB1_sa)] using
+    simpa [cfcR, cfc_id' (R := ℝ) (a := B1) (ha := hB1_sa)] using
       (cfc_inv (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x)
         (a := B1) (fun x hx ↦ ne_of_gt (specB1 hx)) (ha := hB1_sa))
   have hA_eq : cfcR (fun x : ℝ ↦ (x + t)⁻¹) A = cfcR (fun x : ℝ ↦ x⁻¹) A1 := by
@@ -636,10 +593,8 @@ theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
           (f := shift) (ha := hC))
     have hx' : x ∈ shift '' spectrum ℝ C := by
       have hx0 : x ∈ spectrum ℝ (cfc shift C) := by
-        have hval : cfc shift C = C + T := by
-          simpa [cfcR, shift] using hC_shift
-        have hval' : cfc shift C = C1 := by
-          simpa [hC1] using hval
+        have hval : cfc shift C = C + T := by simpa [cfcR, shift] using hC_shift
+        have hval' : cfc shift C = C1 := by simpa [hC1] using hval
         simpa [hval'] using hx
       simpa [hs] using hx0
     rcases hx' with ⟨y, hy, rfl⟩
@@ -647,11 +602,9 @@ theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
     have : 0 < y + t := add_pos_of_nonneg_of_pos hy0 ht
     simpa [Set.Ioi] using this
   have hC1_ne0 : ∀ x ∈ spectrum ℝ C1, (x : ℝ) ≠ 0 := fun x hx ↦ ne_of_gt (specC1 hx)
-  have hC1_sa : IsSelfAdjoint C1 := by
-    simpa [hC1] using (hC.add hT)
+  have hC1_sa : IsSelfAdjoint C1 := by simpa [hC1] using (hC.add hT)
   have hC1_inv : cfcR (fun x : ℝ ↦ x⁻¹) C1 = Ring.inverse C1 := by
-    dsimp [cfcR]
-    simpa [cfc_id' (R := ℝ) (a := C1) (ha := hC1_sa)] using
+    simpa [cfcR, cfc_id' (R := ℝ) (a := C1) (ha := hC1_sa)] using
       (cfc_inv (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x)
         (a := C1) hC1_ne0 (ha := hC1_sa))
   have hC_eq : cfcR (fun x : ℝ ↦ (x + t)⁻¹) C = cfcR (fun x : ℝ ↦ x⁻¹) C1 := by
@@ -689,16 +642,12 @@ theorem ratio_add_t_operatorMonotoneOn_Ici : ∀ (t : ℝ), 0 < t →
     have hEqT :
         (spectrum ℝ T).EqOn (fun x : ℝ ↦ x / (x + t)) (fun x : ℝ ↦ 1 - t * invfun x) := by
       intro x hx
-      have hx0 : (0 : ℝ) ≤ x := by
-        simpa [Set.Ici] using (hspT hx)
+      have hx0 : (0 : ℝ) ≤ x := by simpa [Set.Ici] using (hspT hx)
       simp [invfun]
       field_simp [ne_of_gt (add_pos_of_nonneg_of_pos hx0 ht)]
       ring
-    have hT_ne0 : ∀ x ∈ spectrum ℝ T, x + t ≠ 0 := by
-      intro x hx
-      have hx0 : (0 : ℝ) ≤ x := by
-        simpa [Set.Ici] using (hspT hx)
-      exact ne_of_gt (add_pos_of_nonneg_of_pos hx0 ht)
+    have hT_ne0 : ∀ x ∈ spectrum ℝ T, x + t ≠ 0 :=
+      fun _ hx => ne_of_gt (add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using hspT hx) ht)
     have hT_cont : ContinuousOn invfun (spectrum ℝ T) := by
       simpa [invfun, one_div] using (continuousOn_id.add continuousOn_const).inv₀ hT_ne0
     dsimp [cfcR]
@@ -734,11 +683,9 @@ theorem ratio_add_t_operatorMonotoneOn_Ici : ∀ (t : ℝ), 0 < t →
         rw [hone, hmul]
   calc
     cfcR (fun x : ℝ ↦ x / (x + t)) B
-        = (1 : 𝓐) - t • cfcR invfun B := by
-          simpa using hrepr B hB0 hspB
+        = (1 : 𝓐) - t • cfcR invfun B := by simpa using hrepr B hB0 hspB
     _ ≤ (1 : 𝓐) - t • cfcR invfun A := hmono_core
-    _ = cfcR (fun x : ℝ ↦ x / (x + t)) A := by
-      simpa using (hrepr A hA0 hspA).symm
+    _ = cfcR (fun x : ℝ ↦ x / (x + t)) A := by simpa using (hrepr A hA0 hspA).symm
 
 theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
   OperatorConcaveOn (𝓐 := 𝓐) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ x / (x + t)) := by
@@ -759,11 +706,8 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
           = algebraMap ℝ (𝓐) (-1 : ℝ)
             + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) T := by
       let invfun : ℝ → ℝ := fun x ↦ 1 / (x + t)
-      have hne0 : ∀ x ∈ spectrum ℝ T, x + t ≠ 0 := by
-        intro x hx
-        have hx0 : (0 : ℝ) ≤ x := by
-          simpa [Set.Ici] using (Ts hx)
-        exact ne_of_gt (add_pos_of_nonneg_of_pos hx0 ht)
+      have hne0 : ∀ x ∈ spectrum ℝ T, x + t ≠ 0 :=
+        fun _ hx => ne_of_gt (add_pos_of_nonneg_of_pos (by simpa [Set.Ici] using Ts hx) ht)
       have hcont : ContinuousOn invfun (spectrum ℝ T) := by
         simpa [invfun, one_div] using (continuousOn_id.add continuousOn_const).inv₀ hne0
       dsimp [cfcR]
@@ -773,8 +717,7 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
                 (fun x : ℝ ↦ (-1 : ℝ) + t * invfun x) T := by
         apply cfc_congr
         intro x hx
-        have hx0 : (0 : ℝ) ≤ x := by
-          simpa [Set.Ici] using (Ts hx)
+        have hx0 : (0 : ℝ) ≤ x := by simpa [Set.Ici] using (Ts hx)
         have :
             - (x / (x + t)) = (-1 : ℝ) + t * (1 / (x + t)) := by
           field_simp [ne_of_gt (add_pos_of_nonneg_of_pos hx0 ht)]
@@ -795,12 +738,10 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
             simp [cfc_const_mul (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) t invfun T
               (hf := hcont)]
         _ = algebraMap ℝ (𝓐) (-1 : ℝ)
-              + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) T := by
-            simp [invfun]
+              + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) T := by simp [invfun]
     set AB : 𝓐 := (1 - u) • A + u • B
     have hAB : IsSelfAdjoint AB := by
-      dsimp [AB]
-      simpa using
+      simpa [AB] using
         (IsSelfAdjoint.smul (by simp [IsSelfAdjoint]) hA).add
           (IsSelfAdjoint.smul (by simp [IsSelfAdjoint]) hB)
     -- apply operator convexity of `-(x/(x+t))`
@@ -824,8 +765,7 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
       have hB0 : 0 ≤ B :=
         nonneg_of_spectrum_subset_Ici0 (T := B) hB Bs
       have hAB0 : 0 ≤ AB := by
-        dsimp [AB]
-        exact add_nonneg (smul_nonneg hu0' hA0) (smul_nonneg hu0 hB0)
+        simpa [AB] using add_nonneg (smul_nonneg hu0' hA0) (smul_nonneg hu0 hB0)
       have ABs : spectrum ℝ AB ⊆ Set.Ici (0 : ℝ) := by
         intro x hx
         have hx0 : (0 : ℝ) ≤ x :=
@@ -838,8 +778,7 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
         have hconv_inv_AB :
             cfcR (fun x : ℝ ↦ 1 / (x + t)) AB
               ≤ (1 - u) • cfcR (fun x : ℝ ↦ 1 / (x + t)) A
-                + u • cfcR (fun x : ℝ ↦ 1 / (x + t)) B := by
-          simpa [AB] using hconv_inv
+                + u • cfcR (fun x : ℝ ↦ 1 / (x + t)) B := by simpa [AB] using hconv_inv
         have hscale0 :
             t • cfcR (fun x : ℝ ↦ 1 / (x + t)) AB
               ≤ t •
@@ -853,8 +792,7 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
                     + u • cfcR (fun x : ℝ ↦ 1 / (x + t)) B) := hscale0
           _ =
               (t * (1 - u)) • cfcR (fun x : ℝ ↦ 1 / (x + t)) A
-                + (t * u) • cfcR (fun x : ℝ ↦ 1 / (x + t)) B := by
-            simp [smul_add, smul_smul]
+                + (t * u) • cfcR (fun x : ℝ ↦ 1 / (x + t)) B := by simp [smul_add, smul_smul]
       have hconst : (1 - u) • C + u • C = C := by
         simpa [add_smul, sub_add_cancel] using (add_smul (1 - u) u C).symm
       have hmain :
@@ -906,12 +844,10 @@ theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
                 =
                 ((1 - u) • C + u • C)
                   + ((t * (1 - u)) • cfcR (fun x : ℝ ↦ 1 / (x + t)) A
-                    + (t * u) • cfcR (fun x : ℝ ↦ 1 / (x + t)) B) := by
-                  simp [hconst, add_comm]
+                    + (t * u) • cfcR (fun x : ℝ ↦ 1 / (x + t)) B) := by simp [hconst, add_comm]
             _ =
               (1 - u) • (C + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) A)
-                + u • (C + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) B) := by
-                  simpa using hR'.symm
+                + u • (C + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) B) := by simpa using hR'.symm
         calc
           C + t • cfcR (fun x : ℝ ↦ 1 / (x + t)) AB
               ≤
@@ -981,8 +917,7 @@ private lemma cfcₙ_rpowIntegrand₀₁_eq_smul_cfcR_ratio {q : NNReal} (hq : q
       cfcₙ (Real.rpowIntegrand₀₁ (q : ℝ) t) X = r • cfcR ratio X := by
     calc
       cfcₙ (Real.rpowIntegrand₀₁ (q : ℝ) t) X =
-          cfcR (Real.rpowIntegrand₀₁ (q : ℝ) t) X := by
-            simpa using hcfcₙ
+          cfcR (Real.rpowIntegrand₀₁ (q : ℝ) t) X := by simpa using hcfcₙ
       _ = cfcR (fun x : ℝ ↦ r * ratio x) X := hcfc_congr
       _ = r • cfcR ratio X := hcfc_mul
       _ = r • cfcR ratio X := by simp [cfcR]
@@ -1080,8 +1015,7 @@ private lemma concaveOn_nnrpow_Ioo {q : NNReal} (hq : q ∈ Set.Ioo (0 : NNReal)
   refine hconc_int.congr ?_
   intro A hA
   -- `A ^ q` is the set integral of the integrand on `Ioi 0`
-  have hEq : A ^ q = ∫ t, F t A ∂ν := by
-    simpa [F, ν] using (hμ A hA).2
+  have hEq : A ^ q = ∫ t, F t A ∂ν := by simpa [F, ν] using (hμ A hA).2
   simp [hEq]
 
 private lemma concaveOn_rpow_Ioo {p : ℝ} (hp : p ∈ Set.Ioo (0 : ℝ) 1) :
@@ -1089,12 +1023,10 @@ private lemma concaveOn_rpow_Ioo {p : ℝ} (hp : p ∈ Set.Ioo (0 : ℝ) 1) :
   -- reduce to the `ℝ≥0` exponent case
   let q : NNReal := ⟨p, le_of_lt hp.1⟩
   have hq0 : (0 : NNReal) < q := by
-    have : (0 : ℝ) < (q : ℝ) := by
-      exact hp.1
+    have : (0 : ℝ) < (q : ℝ) := hp.1
     exact (NNReal.coe_pos).1 this
   have hq1 : q < (1 : NNReal) := by
-    have : (q : ℝ) < (1 : ℝ) := by
-      exact hp.2
+    have : (q : ℝ) < (1 : ℝ) := hp.2
     exact (NNReal.coe_lt_coe).1 (by simpa using this)
   have hq : q ∈ Set.Ioo (0 : NNReal) 1 := ⟨hq0, hq1⟩
   -- main lemma: concavity for `a ↦ a ^ q`
@@ -1169,8 +1101,7 @@ theorem power_Icc_zero_one_operatorConcaveOn_Ici : ∀ p ∈ Set.Icc (0 : ℝ) 1
   have hC0 : 0 ≤ (1 - u) • A + u • B :=
     add_nonneg (smul_nonneg hu0' hA0) (smul_nonneg hu0 hB0)
   set C : 𝓐 := (1 - u) • A + u • B
-  have hC_mem : C ∈ Set.Ici (0 : 𝓐) := by
-    simpa [C, Set.Ici] using hC0
+  have hC_mem : C ∈ Set.Ici (0 : 𝓐) := by simpa [C, Set.Ici] using hC0
   have hA_mem : A ∈ Set.Ici (0 : 𝓐) := by simpa [Set.Ici] using hA0
   have hB_mem : B ∈ Set.Ici (0 : 𝓐) := by simpa [Set.Ici] using hB0
   have hconcC : (1 - u) • (A ^ p) + u • (B ^ p) ≤ C ^ p := by
@@ -1470,8 +1401,7 @@ private lemma convexOn_nnrpow_Ioo_one_add {q : NNReal} (hq : q ∈ Set.Ioo (0 :
       simpa [add_comm, add_left_comm, add_assoc] using
         (CFC.nnrpow_add (A := 𝓐) (a := A) (x := (1 : NNReal)) (y := q) zero_lt_one hq0)
     simpa [CFC.nnrpow_one (A := 𝓐) A hA0] using h1
-  have hEq_q : A ^ q = ∫ t, F0 t A ∂ν := by
-    simpa [F0, ν] using (hμ A hA).2
+  have hEq_q : A ^ q = ∫ t, F0 t A ∂ν := by simpa [F0, ν] using (hμ A hA).2
   have hEq_mul :
       A * (∫ t, F0 t A ∂ν) = ∫ t, A * F0 t A ∂ν := by
     have h :
@@ -1499,10 +1429,9 @@ private lemma convexOn_rpow_Ioo_one_two {p : ℝ} (hp : p ∈ Set.Ioo (1 : ℝ)
     ConvexOn ℝ (Set.Ici (0 : 𝓐)) (fun A : 𝓐 ↦ A ^ p) := by
   -- reduce to the `ℝ≥0` exponent case with `p = 1 + q`, `q ∈ (0,1)`
   let q : NNReal := ⟨p - 1, sub_nonneg.mpr (le_of_lt hp.1)⟩
-  have hq0 : (0 : NNReal) < q := by
-    have : (0 : ℝ) < (q : ℝ) := by
-      exact_mod_cast sub_pos.mpr hp.1
-    exact (NNReal.coe_pos).1 this
+  have hq0 : (0 : NNReal) < q :=
+    (NNReal.coe_pos).1 (show (0 : ℝ) < (q : ℝ) by
+      exact_mod_cast sub_pos.mpr hp.1)
   have hq1 : q < (1 : NNReal) := by
     have : (q : ℝ) < (1 : ℝ) := by
       have : p - 1 < (1 : ℝ) := by linarith [hp.2]
@@ -1538,16 +1467,14 @@ private lemma cfcR_mul_self (T : 𝓐) (hT : IsSelfAdjoint T) :
       simpa using
         (cfc_mul (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint)
           (f := fun x : ℝ ↦ x) (g := fun x : ℝ ↦ x) (a := T))
-    _ = T * T := by
-      simp [cfc_id' (R := ℝ) (a := T) (ha := hT)]
+    _ = T * T := by simp [cfc_id' (R := ℝ) (a := T) (ha := hT)]
 
 omit [Nontrivial (𝓐)] in
 omit [PartialOrder 𝓐] [StarOrderedRing 𝓐] [NonnegSpectrumClass ℝ 𝓐] in
 private lemma sub_mul_sub (A B : 𝓐) :
     (A - B) * (A - B) = A * A - A * B - B * A + B * B := by
   calc
-    (A - B) * (A - B) = (A * A - B * A) - (A * B - B * B) := by
-      simp [mul_sub, sub_mul]
+    (A - B) * (A - B) = (A * A - B * A) - (A * B - B * B) := by simp [mul_sub, sub_mul]
     _ = A * A - A * B - B * A + B * B := by
       abel
 
@@ -1555,8 +1482,7 @@ omit [Nontrivial (𝓐)] in
 omit [PartialOrder 𝓐] [StarOrderedRing 𝓐] [NonnegSpectrumClass ℝ 𝓐] in
 private lemma smul_sub_mul_sub (α : ℝ) (A B : 𝓐) :
     α • (A * A - A * B - B * A + B * B) =
-      α • (A * A) - α • (A * B) - α • (B * A) + α • (B * B) := by
-  rw [smul_add, smul_sub, smul_sub]
+      α • (A * A) - α • (A * B) - α • (B * A) + α • (B * B) := by rw [smul_add, smul_sub, smul_sub]
 
 omit [Nontrivial (𝓐)] in
 omit [PartialOrder 𝓐] [StarOrderedRing 𝓐] [NonnegSpectrumClass ℝ 𝓐] in
@@ -1571,16 +1497,14 @@ private lemma square_convexity_diff_rhs (A B : 𝓐) (u : ℝ) :
       α • (A * A) - α • (A * B) - α • (B * A) + α • (B * B)
         =
         (u * (1 - u)) • (A * A) - (u * (1 - u)) • (A * B) - (u * (1 - u)) • (B * A)
-          + (u * (1 - u)) • (B * B) := by
-    simp [α]
+          + (u * (1 - u)) • (B * B) := by simp [α]
   calc
     (u * (1 - u)) • ((A - B) * (A - B)) = α • ((A - B) * (A - B)) := by simp [α]
     _ = α • (A * A - A * B - B * A + B * B) := hsmul
     _ = α • (A * A) - α • (A * B) - α • (B * A) + α • (B * B) :=
       smul_sub_mul_sub  (α := α) A B
     _ = (u * (1 - u)) • (A * A) - (u * (1 - u)) • (A * B) - (u * (1 - u)) • (B * A)
-          + (u * (1 - u)) • (B * B) := by
-      simp [hα]
+          + (u * (1 - u)) • (B * B) := by simp [hα]
 
 omit [Nontrivial (𝓐)] in
 omit [PartialOrder 𝓐] [StarOrderedRing 𝓐] [NonnegSpectrumClass ℝ 𝓐] in
@@ -1651,8 +1575,7 @@ private lemma square_convexity_diff_hCC_sum (A B : 𝓐) (u : ℝ) :
         exact Algebra.smul_mul_assoc (R := ℝ) (A := 𝓐) (1 - u) A ((1 - u) • A)
       _ = (1 - u) • ((1 - u) • (A * A)) := by
         rw [Algebra.mul_smul_comm]
-      _ = ((1 - u) * (1 - u)) • (A * A) := by
-        simp [smul_smul]
+      _ = ((1 - u) * (1 - u)) • (A * A) := by simp [smul_smul]
   have hAB' :
       ((1 - u) • A) * (u • B) = ((1 - u) * u) • (A * B) := by
     calc
@@ -1660,26 +1583,21 @@ private lemma square_convexity_diff_hCC_sum (A B : 𝓐) (u : ℝ) :
         exact Algebra.smul_mul_assoc (R := ℝ) (A := 𝓐) (1 - u) A (u • B)
       _ = (1 - u) • (u • (A * B)) := by
         rw [Algebra.mul_smul_comm]
-      _ = ((1 - u) * u) • (A * B) := by
-        simp [smul_smul]
+      _ = ((1 - u) * u) • (A * B) := by simp [smul_smul]
   have hBA' :
       (u • B) * ((1 - u) • A) = (u * (1 - u)) • (B * A) := by
     calc
       (u • B) * ((1 - u) • A) = u • (B * ((1 - u) • A)) := by
         exact Algebra.smul_mul_assoc (R := ℝ) (A := 𝓐) u B ((1 - u) • A)
-      _ = u • ((1 - u) • (B * A)) := by
-        simp [Algebra.mul_smul_comm]
-      _ = (u * (1 - u)) • (B * A) := by
-        simpa using (smul_smul u (1 - u) (B * A))
+      _ = u • ((1 - u) • (B * A)) := by simp [Algebra.mul_smul_comm]
+      _ = (u * (1 - u)) • (B * A) := by simpa using (smul_smul u (1 - u) (B * A))
   have hBB' :
       (u • B) * (u • B) = (u * u) • (B * B) := by
     calc
       (u • B) * (u • B) = u • (B * (u • B)) := by
         exact Algebra.smul_mul_assoc (R := ℝ) (A := 𝓐) u B (u • B)
-      _ = u • (u • (B * B)) := by
-        simp [Algebra.mul_smul_comm]
-      _ = (u * u) • (B * B) := by
-        simp [smul_smul]
+      _ = u • (u • (B * B)) := by simp [Algebra.mul_smul_comm]
+      _ = (u * u) • (B * B) := by simp [smul_smul]
   rw [hAA', hAB', hBA', hBB']
 
 omit [Nontrivial (𝓐)] in
@@ -1698,19 +1616,15 @@ private lemma square_convexity_diff_hCC (A B : 𝓐) (u : ℝ) :
     set Y : 𝓐 := u • B
     have hXY : (1 - u) • A + u • B = X + Y := by simp [X, Y]
     calc
-      ((1 - u) • A + u • B) * ((1 - u) • A + u • B) = (X + Y) * (X + Y) := by
-        simp [hXY]
-      _ = X * (X + Y) + Y * (X + Y) := by
-        simp [add_mul]
-      _ = (X * X + X * Y) + (Y * X + Y * Y) := by
-        simp [mul_add, add_assoc]
+      ((1 - u) • A + u • B) * ((1 - u) • A + u • B) = (X + Y) * (X + Y) := by simp [hXY]
+      _ = X * (X + Y) + Y * (X + Y) := by simp [add_mul]
+      _ = (X * X + X * Y) + (Y * X + Y * Y) := by simp [mul_add, add_assoc]
       _ = X * X + X * Y + Y * X + Y * Y := by
         abel
       _ = ((1 - u) • A) * ((1 - u) • A)
             + ((1 - u) • A) * (u • B)
             + (u • B) * ((1 - u) • A)
-            + (u • B) * (u • B) := by
-        simp [X, Y]
+            + (u • B) * (u • B) := by simp [X, Y]
   exact hexpand.trans (square_convexity_diff_hCC_sum  A B u)
 
 omit [PartialOrder 𝓐] [StarOrderedRing 𝓐] [NonnegSpectrumClass ℝ 𝓐] [Nontrivial (𝓐)] in
@@ -1760,8 +1674,7 @@ private lemma operatorConvexOn_pow_two_Ici :
   have hub : 0 ≤ u * (1 - u) := mul_nonneg hu0 hu0'
   have hdiff :
       (1 - u) • (A * A) + u • (B * B) - C * C
-        = (u * (1 - u)) • ((A - B) * (A - B)) := by
-    simpa [C] using (square_convexity_diff  A B u)
+        = (u * (1 - u)) • ((A - B) * (A - B)) := by simpa [C] using (square_convexity_diff  A B u)
   have hnonneg : 0 ≤ (1 - u) • (A * A) + u • (B * B) - C * C := by
     have hscale : 0 ≤ (u * (1 - u)) • ((A - B) * (A - B)) := smul_nonneg hub hsq
     simpa [hdiff] using hscale
@@ -1816,8 +1729,7 @@ theorem power_Icc_one_two_operatorConvexOn_Ici : ∀ p ∈ Set.Icc (1 : ℝ) 2,
   set C : 𝓐 := (1 - u) • A + u • B
   have hC0 : 0 ≤ C :=
     add_nonneg (smul_nonneg hu0' hA0) (smul_nonneg hu0 hB0)
-  have hC_mem : C ∈ Set.Ici (0 : 𝓐) := by
-    simpa [C, Set.Ici] using hC0
+  have hC_mem : C ∈ Set.Ici (0 : 𝓐) := by simpa [C, Set.Ici] using hC0
   have hA_mem : A ∈ Set.Ici (0 : 𝓐) := by simpa [Set.Ici] using hA0
   have hB_mem : B ∈ Set.Ici (0 : 𝓐) := by simpa [Set.Ici] using hB0
   have hab : (1 - u) + u = (1 : ℝ) := by ring
@@ -1842,10 +1754,8 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
   let q : ℝ := -p
   have hq : q ∈ Set.Icc (0 : ℝ) 1 := by
     constructor
-    · dsimp [q]
-      exact neg_nonneg.mpr hp.2
-    · dsimp [q]
-      simpa using (neg_le_neg hp.1)
+    · simpa [q] using neg_nonneg.mpr hp.2
+    · simpa [q] using (neg_le_neg hp.1)
   have hBAq : cfcR  (fun x : ℝ ↦ x ^ q) B ≤ cfcR  (fun x : ℝ ↦ x ^ q) A := by
     have hspA' : spectrum ℝ A ⊆ Set.Ici (0 : ℝ) := by
       intro x hx
@@ -1859,13 +1769,11 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
   let Aq : 𝓐 := cfcR  (fun x : ℝ ↦ x ^ q) A
   let Bq : 𝓐 := cfcR  (fun x : ℝ ↦ x ^ q) B
   have hAq0 : 0 ≤ Aq := by
-    dsimp [Aq, cfcR]
     refine cfc_nonneg ?_
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using hspA hx
     exact le_of_lt (Real.rpow_pos_of_pos hx0 _)
   have hBq0 : 0 ≤ Bq := by
-    dsimp [Bq, cfcR]
     refine cfc_nonneg ?_
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using hspB hx
@@ -1878,8 +1786,7 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
       exact (Real.continuousAt_rpow_const x q (Or.inl (ne_of_gt hx0))).continuousWithinAt
     have hspec :
         spectrum ℝ Aq = (fun x : ℝ ↦ x ^ q) '' spectrum ℝ A := by
-      dsimp [Aq, cfcR]
-      simpa using
+      simpa [Aq, cfcR] using
         (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ q)
           (a := A) (ha := hA_sa) (hf := hcontA))
     intro y hy
@@ -1895,8 +1802,7 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
       exact (Real.continuousAt_rpow_const x q (Or.inl (ne_of_gt hx0))).continuousWithinAt
     have hspec :
         spectrum ℝ Bq = (fun x : ℝ ↦ x ^ q) '' spectrum ℝ B := by
-      dsimp [Bq, cfcR]
-      simpa using
+      simpa [Bq, cfcR] using
         (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ q)
           (a := B) (ha := hB_sa) (hf := hcontB))
     intro y hy
@@ -1927,21 +1833,17 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
     have hcomp :
         cfcR  (fun y : ℝ ↦ 1 / y) (cfcR  (fun x : ℝ ↦ x ^ q) A) =
           cfcR  (fun x : ℝ ↦ 1 / (x ^ q)) A := by
-      dsimp [cfcR]
-      simpa [Function.comp] using
+      simpa [cfcR, Function.comp] using
         (cfc_comp' (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (g := fun y : ℝ ↦ 1 / y)
           (f := fun x : ℝ ↦ x ^ q) (a := A) (hg := hg) (hf := hcontA) (ha := hA_sa)).symm
     have hL :
         cfcR  (fun x : ℝ ↦ 1 / x) Aq =
-          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ q) A) := by
-      simp [Aq, one_div]
+          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ q) A) := by simp [Aq, one_div]
     rw [hL]
     have hcomp' :
         cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ q) A) =
-          cfcR  (fun x : ℝ ↦ (x ^ q)⁻¹) A := by
-      simpa [one_div] using hcomp
+          cfcR  (fun x : ℝ ↦ (x ^ q)⁻¹) A := by simpa [one_div] using hcomp
     rw [hcomp']
-    dsimp [cfcR]
     apply cfc_congr
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using hspA hx
@@ -1965,21 +1867,17 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
     have hcomp :
         cfcR  (fun y : ℝ ↦ 1 / y) (cfcR  (fun x : ℝ ↦ x ^ q) B) =
           cfcR  (fun x : ℝ ↦ 1 / (x ^ q)) B := by
-      dsimp [cfcR]
-      simpa [Function.comp] using
+      simpa [cfcR, Function.comp] using
         (cfc_comp' (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (g := fun y : ℝ ↦ 1 / y)
           (f := fun x : ℝ ↦ x ^ q) (a := B) (hg := hg) (hf := hcontB) (ha := hB_sa)).symm
     have hL :
         cfcR  (fun x : ℝ ↦ 1 / x) Bq =
-          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ q) B) := by
-      simp [Bq, one_div]
+          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ q) B) := by simp [Bq, one_div]
     rw [hL]
     have hcomp' :
         cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ q) B) =
-          cfcR  (fun x : ℝ ↦ (x ^ q)⁻¹) B := by
-      simpa [one_div] using hcomp
+          cfcR  (fun x : ℝ ↦ (x ^ q)⁻¹) B := by simpa [one_div] using hcomp
     rw [hcomp']
-    dsimp [cfcR]
     apply cfc_congr
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using hspB hx
@@ -1989,19 +1887,15 @@ theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-
   have hanti : cfcR  (fun x : ℝ ↦ x ^ p) A ≤ cfcR  (fun x : ℝ ↦ x ^ p) B := by
     calc
       cfcR  (fun x : ℝ ↦ x ^ p) A =
-          cfcR  (fun x : ℝ ↦ 1 / x) Aq := by
-            simpa using hcompA.symm
+          cfcR  (fun x : ℝ ↦ 1 / x) Aq := by simpa using hcompA.symm
       _ ≤ cfcR  (fun x : ℝ ↦ 1 / x) Bq := h_inv
-      _ = cfcR  (fun x : ℝ ↦ x ^ p) B := by
-            simpa using hcompB
+      _ = cfcR  (fun x : ℝ ↦ x ^ p) B := by simpa using hcompB
   have hneg : -cfcR  (fun x : ℝ ↦ x ^ p) B ≤ -cfcR  (fun x : ℝ ↦ x ^ p) A :=
     neg_le_neg hanti
   have hnegA :
-      cfcR  (fun x : ℝ ↦ -(x ^ p)) A = -cfcR  (fun x : ℝ ↦ x ^ p) A := by
-    simp [cfcR, cfc_neg]
+      cfcR  (fun x : ℝ ↦ -(x ^ p)) A = -cfcR  (fun x : ℝ ↦ x ^ p) A := by simp [cfcR, cfc_neg]
   have hnegB :
-      cfcR  (fun x : ℝ ↦ -(x ^ p)) B = -cfcR  (fun x : ℝ ↦ x ^ p) B := by
-    simp [cfcR, cfc_neg]
+      cfcR  (fun x : ℝ ↦ -(x ^ p)) B = -cfcR  (fun x : ℝ ↦ x ^ p) B := by simp [cfcR, cfc_neg]
   simpa [hnegA, hnegB] using hneg
 
 theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1 : ℝ) 0,
@@ -2014,10 +1908,8 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
   let r : ℝ := -p
   have hr : r ∈ Set.Icc (0 : ℝ) 1 := by
     constructor
-    · dsimp [r]
-      exact neg_nonneg.mpr hp.2
-    · dsimp [r]
-      simpa using (neg_le_neg hp.1)
+    · simpa [r] using neg_nonneg.mpr hp.2
+    · simpa [r] using (neg_le_neg hp.1)
   -- convex combination
   set C : 𝓐 := (1 - t) • A + t • B
   have hC : IsSelfAdjoint C := by
@@ -2045,8 +1937,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         (-Cr) ≤ (1 - t) • (-Ar) + t • (-Br) := by
       simpa [C, Ar, Br, Cr, cfcR, cfc_neg] using
         hconc (A := A) (B := B) (t := t) hA hB ht0 ht1 As0 Bs0
-    have h2 : (1 - t) • (-Ar) + t • (-Br) = -Dr := by
-      simp [Dr, smul_neg, add_comm]
+    have h2 : (1 - t) • (-Ar) + t • (-Br) = -Dr := by simp [Dr, smul_neg, add_comm]
     have h3 : (-Cr) ≤ (-Dr) := by
       calc
         (-Cr) ≤ (1 - t) • (-Ar) + t • (-Br) := h1
@@ -2056,35 +1947,28 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
   have h_inv1 :
       cfcR  (fun x : ℝ ↦ 1 / x) Cr ≤ cfcR  (fun x : ℝ ↦ 1 / x) Dr := by
     have hCr0 : 0 ≤ Cr := by
-      dsimp [Cr, cfcR]
       refine cfc_nonneg ?_
       intro x hx
       have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using Cs hx
       exact le_of_lt (Real.rpow_pos_of_pos hx0 r)
     have hAr0 : 0 ≤ Ar := by
-      dsimp [Ar, cfcR]
       refine cfc_nonneg ?_
       intro x hx
       have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using As hx
       exact le_of_lt (Real.rpow_pos_of_pos hx0 r)
     have hBr0 : 0 ≤ Br := by
-      dsimp [Br, cfcR]
       refine cfc_nonneg ?_
       intro x hx
       have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using Bs hx
       exact le_of_lt (Real.rpow_pos_of_pos hx0 r)
     have hDr0 : 0 ≤ Dr := by
-      dsimp [Dr]
-      exact add_nonneg (smul_nonneg (sub_nonneg.mpr ht1) hAr0) (smul_nonneg ht0 hBr0)
+      simpa [Dr] using add_nonneg (smul_nonneg (sub_nonneg.mpr ht1) hAr0) (smul_nonneg ht0 hBr0)
     have hCr_sa : IsSelfAdjoint Cr := by
-      dsimp [Cr, cfcR]
-      exact cfc_predicate _ _
+      simp [Cr, cfcR]
     have hAr_sa : IsSelfAdjoint Ar := by
-      dsimp [Ar, cfcR]
-      exact cfc_predicate _ _
+      simp [Ar, cfcR]
     have hBr_sa : IsSelfAdjoint Br := by
-      dsimp [Br, cfcR]
-      exact cfc_predicate _ _
+      simp [Br, cfcR]
     have hspCr : spectrum ℝ Cr ⊆ Set.Ioi (0 : ℝ) := by
       have hcontC : ContinuousOn (fun x : ℝ ↦ x ^ r) (spectrum ℝ C) := by
         intro x hx
@@ -2092,8 +1976,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         exact (Real.continuousAt_rpow_const x r (Or.inl (ne_of_gt hx0))).continuousWithinAt
       have hspec :
           spectrum ℝ Cr = (fun x : ℝ ↦ x ^ r) '' spectrum ℝ C := by
-        dsimp [Cr, cfcR]
-        simpa using
+        simpa [Cr, cfcR] using
           (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ r)
             (a := C) (ha := hC) (hf := hcontC))
       intro y hy
@@ -2108,8 +1991,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         exact (Real.continuousAt_rpow_const x r (Or.inl (ne_of_gt hx0))).continuousWithinAt
       have hspec :
           spectrum ℝ Ar = (fun x : ℝ ↦ x ^ r) '' spectrum ℝ A := by
-        dsimp [Ar, cfcR]
-        simpa using
+        simpa [Ar, cfcR] using
           (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ r)
             (a := A) (ha := hA) (hf := hcontA))
       intro y hy
@@ -2124,8 +2006,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         exact (Real.continuousAt_rpow_const x r (Or.inl (ne_of_gt hx0))).continuousWithinAt
       have hspec :
           spectrum ℝ Br = (fun x : ℝ ↦ x ^ r) '' spectrum ℝ B := by
-        dsimp [Br, cfcR]
-        simpa using
+        simpa [Br, cfcR] using
           (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ r)
             (a := B) (ha := hB) (hf := hcontB))
       intro y hy
@@ -2144,11 +2025,9 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         ≤ (1 - t) • cfcR  (fun x : ℝ ↦ 1 / x) Ar
           + t • cfcR  (fun x : ℝ ↦ 1 / x) Br := by
     have hAr_sa : IsSelfAdjoint Ar := by
-      dsimp [Ar, cfcR]
-      exact cfc_predicate _ _
+      simp [Ar, cfcR]
     have hBr_sa : IsSelfAdjoint Br := by
-      dsimp [Br, cfcR]
-      exact cfc_predicate _ _
+      simp [Br, cfcR]
     have hspAr : spectrum ℝ Ar ⊆ Set.Ioi (0 : ℝ) := by
       have hcontA : ContinuousOn (fun x : ℝ ↦ x ^ r) (spectrum ℝ A) := by
         intro x hx
@@ -2156,8 +2035,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         exact (Real.continuousAt_rpow_const x r (Or.inl (ne_of_gt hx0))).continuousWithinAt
       have hspec :
           spectrum ℝ Ar = (fun x : ℝ ↦ x ^ r) '' spectrum ℝ A := by
-        dsimp [Ar, cfcR]
-        simpa using
+        simpa [Ar, cfcR] using
           (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ r)
             (a := A) (ha := hA) (hf := hcontA))
       intro y hy
@@ -2172,8 +2050,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
         exact (Real.continuousAt_rpow_const x r (Or.inl (ne_of_gt hx0))).continuousWithinAt
       have hspec :
           spectrum ℝ Br = (fun x : ℝ ↦ x ^ r) '' spectrum ℝ B := by
-        dsimp [Br, cfcR]
-        simpa using
+        simpa [Br, cfcR] using
           (cfc_map_spectrum (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (f := fun x : ℝ ↦ x ^ r)
             (a := B) (ha := hB) (hf := hcontB))
       intro y hy
@@ -2202,27 +2079,22 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
     have hcomp :
         cfcR  (fun y : ℝ ↦ 1 / y) (cfcR  (fun x : ℝ ↦ x ^ r) A) =
           cfcR  (fun x : ℝ ↦ 1 / (x ^ r)) A := by
-      dsimp [cfcR]
-      simpa [Function.comp] using
+      simpa [cfcR, Function.comp] using
         (cfc_comp' (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (g := fun y : ℝ ↦ 1 / y)
           (f := fun x : ℝ ↦ x ^ r) (a := A) (hg := hg) (hf := hcontA) (ha := hA)).symm
     have hL :
         cfcR  (fun x : ℝ ↦ 1 / x) Ar =
-          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) A) := by
-      simp [Ar, one_div]
+          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) A) := by simp [Ar, one_div]
     rw [hL]
     have hcomp' :
         cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) A) =
-          cfcR  (fun x : ℝ ↦ (x ^ r)⁻¹) A := by
-      simpa [one_div] using hcomp
+          cfcR  (fun x : ℝ ↦ (x ^ r)⁻¹) A := by simpa [one_div] using hcomp
     rw [hcomp']
-    dsimp [cfcR]
     apply cfc_congr
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using As hx
     calc
-      (x ^ r)⁻¹ = x ^ (-r) := by
-        simpa using (Real.rpow_neg (le_of_lt hx0) r).symm
+      (x ^ r)⁻¹ = x ^ (-r) := by simpa using (Real.rpow_neg (le_of_lt hx0) r).symm
       _ = x ^ p := by simp [r]
   have hcompB :
       cfcR  (fun x : ℝ ↦ 1 / x) Br = cfcR  (fun x : ℝ ↦ x ^ p) B := by
@@ -2240,27 +2112,22 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
     have hcomp :
         cfcR  (fun y : ℝ ↦ 1 / y) (cfcR  (fun x : ℝ ↦ x ^ r) B) =
           cfcR  (fun x : ℝ ↦ 1 / (x ^ r)) B := by
-      dsimp [cfcR]
-      simpa [Function.comp] using
+      simpa [cfcR, Function.comp] using
         (cfc_comp' (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (g := fun y : ℝ ↦ 1 / y)
           (f := fun x : ℝ ↦ x ^ r) (a := B) (hg := hg) (hf := hcontB) (ha := hB)).symm
     have hL :
         cfcR  (fun x : ℝ ↦ 1 / x) Br =
-          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) B) := by
-      simp [Br, one_div]
+          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) B) := by simp [Br, one_div]
     rw [hL]
     have hcomp' :
         cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) B) =
-          cfcR  (fun x : ℝ ↦ (x ^ r)⁻¹) B := by
-      simpa [one_div] using hcomp
+          cfcR  (fun x : ℝ ↦ (x ^ r)⁻¹) B := by simpa [one_div] using hcomp
     rw [hcomp']
-    dsimp [cfcR]
     apply cfc_congr
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using Bs hx
     calc
-      (x ^ r)⁻¹ = x ^ (-r) := by
-        simpa using (Real.rpow_neg (le_of_lt hx0) r).symm
+      (x ^ r)⁻¹ = x ^ (-r) := by simpa using (Real.rpow_neg (le_of_lt hx0) r).symm
       _ = x ^ p := by simp [r]
   have hcompC :
       cfcR  (fun x : ℝ ↦ 1 / x) Cr = cfcR  (fun x : ℝ ↦ x ^ p) C := by
@@ -2278,27 +2145,22 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
     have hcomp :
         cfcR  (fun y : ℝ ↦ 1 / y) (cfcR  (fun x : ℝ ↦ x ^ r) C) =
           cfcR  (fun x : ℝ ↦ 1 / (x ^ r)) C := by
-      dsimp [cfcR]
-      simpa [Function.comp] using
+      simpa [cfcR, Function.comp] using
         (cfc_comp' (R := ℝ) (A := 𝓐) (p := IsSelfAdjoint) (g := fun y : ℝ ↦ 1 / y)
           (f := fun x : ℝ ↦ x ^ r) (a := C) (hg := hg) (hf := hcontC) (ha := hC)).symm
     have hL :
         cfcR  (fun x : ℝ ↦ 1 / x) Cr =
-          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) C) := by
-      simp [Cr, one_div]
+          cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) C) := by simp [Cr, one_div]
     rw [hL]
     have hcomp' :
         cfcR  (fun y : ℝ ↦ y⁻¹) (cfcR  (fun x : ℝ ↦ x ^ r) C) =
-          cfcR  (fun x : ℝ ↦ (x ^ r)⁻¹) C := by
-      simpa [one_div] using hcomp
+          cfcR  (fun x : ℝ ↦ (x ^ r)⁻¹) C := by simpa [one_div] using hcomp
     rw [hcomp']
-    dsimp [cfcR]
     apply cfc_congr
     intro x hx
     have hx0 : (0 : ℝ) < x := by simpa [Set.Ioi] using Cs hx
     calc
-      (x ^ r)⁻¹ = x ^ (-r) := by
-        simpa using (Real.rpow_neg (le_of_lt hx0) r).symm
+      (x ^ r)⁻¹ = x ^ (-r) := by simpa using (Real.rpow_neg (le_of_lt hx0) r).symm
       _ = x ^ p := by simp [r]
   -- finish
   have hmain :
@@ -2320,8 +2182,7 @@ theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1
     have hchain' :
         cfcR  (fun x : ℝ ↦ x⁻¹) Cr
           ≤ (1 - t) • cfcR  (fun x : ℝ ↦ x⁻¹) Ar
-            + t • cfcR  (fun x : ℝ ↦ x⁻¹) Br := by
-      simpa [one_div] using hchain
+            + t • cfcR  (fun x : ℝ ↦ x⁻¹) Br := by simpa [one_div] using hchain
     simpa [hcompA', hcompB', hcompC'] using hchain'
   simpa [C] using hmain
 
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzTheorem.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzTheorem.lean
index 7f89cad56..f2f0c8e49 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzTheorem.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/LownerHeinzTheorem.lean
@@ -158,10 +158,8 @@ theorem operatorConvex_convexOn_univ {f : ℝ → ℝ} (hf : OperatorConvex (ℋ
           (algebraMap ℝ (L ℋ) u) * (algebraMap ℝ (L ℋ) r) +
             (algebraMap ℝ (L ℋ) v) * (algebraMap ℝ (L ℋ) s) := by
               rw [Algebra.smul_def, Algebra.smul_def]
-      _ = algebraMap ℝ (L ℋ) (u * r) + algebraMap ℝ (L ℋ) (v * s) := by
-            simp
-      _ = algebraMap ℝ (L ℋ) (u * r + v * s) := by
-            simp
+      _ = algebraMap ℝ (L ℋ) (u * r) + algebraMap ℝ (L ℋ) (v * s) := by simp
+      _ = algebraMap ℝ (L ℋ) (u * r + v * s) := by simp
   have hop :
       cfcR (ℋ := ℋ) f
           ((1 - b) • (algebraMap ℝ (L ℋ) x) + b • (algebraMap ℝ (L ℋ) y)) ≤
@@ -183,8 +181,7 @@ theorem operatorConvex_convexOn_univ {f : ℝ → ℝ} (hf : OperatorConvex (ℋ
     halg_combo (u := 1 - b) (v := b) (r := f x) (s := f y)] at hop
   have hscalar :
       algebraMap ℝ (L ℋ) (f ((1 - b) * x + b * y)) ≤
-        algebraMap ℝ (L ℋ) ((1 - b) * f x + b * f y) := by
-    simpa [Algebra.smul_def] using hop
+        algebraMap ℝ (L ℋ) ((1 - b) * f x + b * f y) := by simpa [Algebra.smul_def] using hop
   have hspec_le :
       ∀ z ∈ spectrum ℝ (algebraMap ℝ (L ℋ) (f ((1 - b) * x + b * y))),
         z ≤ (1 - b) * f x + b * f y := by
@@ -229,51 +226,42 @@ theorem one_div_operatorConvexOn_Ioi :
 
 omit [Nontrivial ℋ] in
 theorem one_div_add_t_operatorAntitoneOn_Ici : ∀ (t : ℝ), 0 < t →
-    OperatorAntitoneOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ 1 / (x + t)) := by
-  intro t ht
-  exact (LownerHeinzCore.one_div_add_t_operatorAntitoneOn_Ici (𝓐 := L ℋ) t ht)
+    OperatorAntitoneOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ 1 / (x + t)) :=
+  fun t ht => LownerHeinzCore.one_div_add_t_operatorAntitoneOn_Ici (𝓐 := L ℋ) t ht
 
 theorem one_div_add_t_operatorConvexOn_Ici : ∀ (t : ℝ), 0 < t →
-    OperatorConvexOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ 1 / (x + t)) := by
-  intro t ht
-  exact (LownerHeinzCore.one_div_add_t_operatorConvexOn_Ici (𝓐 := L ℋ) t ht)
+    OperatorConvexOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ 1 / (x + t)) :=
+  fun t ht => LownerHeinzCore.one_div_add_t_operatorConvexOn_Ici (𝓐 := L ℋ) t ht
 
 omit [Nontrivial ℋ] in
 theorem ratio_add_t_operatorMonotoneOn_Ici : ∀ (t : ℝ), 0 < t →
-    OperatorMonotoneOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ x / (x + t)) := by
-  intro t ht
-  exact (LownerHeinzCore.ratio_add_t_operatorMonotoneOn_Ici (𝓐 := L ℋ) t ht)
+    OperatorMonotoneOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ x / (x + t)) :=
+  fun t ht => LownerHeinzCore.ratio_add_t_operatorMonotoneOn_Ici (𝓐 := L ℋ) t ht
 
 theorem ratio_add_t_operatorConcaveOn_Ici : ∀ (t : ℝ), 0 < t →
-    OperatorConcaveOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ x / (x + t)) := by
-  intro t ht
-  exact (LownerHeinzCore.ratio_add_t_operatorConcaveOn_Ici (𝓐 := L ℋ) t ht)
+    OperatorConcaveOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x : ℝ ↦ x / (x + t)) :=
+  fun t ht => LownerHeinzCore.ratio_add_t_operatorConcaveOn_Ici (𝓐 := L ℋ) t ht
 
 omit [Nontrivial ℋ] in
 theorem power_Icc_zero_one_operatorMonotoneOn_Ici : ∀ p ∈ Set.Icc (0 : ℝ) 1,
-    OperatorMonotoneOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x ↦ x ^ p) := by
-  intro p hp
-  exact (LownerHeinzCore.power_Icc_zero_one_operatorMonotoneOn_Ici (𝓐 := L ℋ) p hp)
+    OperatorMonotoneOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x ↦ x ^ p) :=
+  fun p hp => LownerHeinzCore.power_Icc_zero_one_operatorMonotoneOn_Ici (𝓐 := L ℋ) p hp
 
 theorem power_Icc_zero_one_operatorConcaveOn_Ici : ∀ p ∈ Set.Icc (0 : ℝ) 1,
-    OperatorConcaveOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x ↦ x ^ p) := by
-  intro p hp
-  exact (LownerHeinzCore.power_Icc_zero_one_operatorConcaveOn_Ici (𝓐 := L ℋ) p hp)
+    OperatorConcaveOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x ↦ x ^ p) :=
+  fun p hp => LownerHeinzCore.power_Icc_zero_one_operatorConcaveOn_Ici (𝓐 := L ℋ) p hp
 
 theorem power_Icc_one_two_operatorConvexOn_Ici : ∀ p ∈ Set.Icc (1 : ℝ) 2,
-    OperatorConvexOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x ↦ x ^ p) := by
-  intro p hp
-  exact (LownerHeinzCore.power_Icc_one_two_operatorConvexOn_Ici (𝓐 := L ℋ) p hp)
+    OperatorConvexOn (ℋ := ℋ) (Set.Ici (0 : ℝ)) (fun x ↦ x ^ p) :=
+  fun p hp => LownerHeinzCore.power_Icc_one_two_operatorConvexOn_Ici (𝓐 := L ℋ) p hp
 
 omit [Nontrivial ℋ] in
 theorem power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi : ∀ p ∈ Set.Icc (-1 : ℝ) 0,
-    OperatorMonotoneOn (ℋ := ℋ) (Set.Ioi (0 : ℝ)) (fun x ↦ -(x ^ p)) := by
-  intro p hp
-  exact (LownerHeinzCore.power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi (𝓐 := L ℋ) p hp)
+    OperatorMonotoneOn (ℋ := ℋ) (Set.Ioi (0 : ℝ)) (fun x ↦ -(x ^ p)) :=
+  fun p hp => LownerHeinzCore.power_Icc_neg_one_zero_neg_operatorMonotoneOn_Ioi (𝓐 := L ℋ) p hp
 
 theorem power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi : ∀ p ∈ Set.Icc (-1 : ℝ) 0,
-    OperatorConcaveOn (ℋ := ℋ) (Set.Ioi (0 : ℝ)) (fun x ↦ -(x ^ p)) := by
-  intro p hp
-  exact (LownerHeinzCore.power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi (𝓐 := L ℋ) p hp)
+    OperatorConcaveOn (ℋ := ℋ) (Set.Ioi (0 : ℝ)) (fun x ↦ -(x ^ p)) :=
+  fun p hp => LownerHeinzCore.power_Icc_neg_one_zero_neg_operatorConcaveOn_Ioi (𝓐 := L ℋ) p hp
 
 end LownerHeinzTheorem
diff --git a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/OperatorGeometricMean.lean b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/OperatorGeometricMean.lean
index cf15c94a6..9fbfc2a0f 100644
--- a/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/OperatorGeometricMean.lean
+++ b/QuantumInfo/ForMathlib/HayataGroup/TraceInequality/OperatorGeometricMean.lean
@@ -25,9 +25,8 @@ noncomputable def operatorPowerMean (α β : ℝ) (A B : L ℋ) : L ℋ :=
   GeneralizedPerspective (fun x : ℝ ↦ x ^ α) (fun x : ℝ ↦ x ^ β) A B
 
 private lemma rpow_continuousOn_Ici (p : ℝ) (hp : 0 ≤ p) :
-    ContinuousOn (fun x : ℝ ↦ x ^ p) (Set.Ici (0 : ℝ)) := by
-  intro x hx
-  exact (Real.continuousAt_rpow_const x p (Or.inr hp)).continuousWithinAt
+    ContinuousOn (fun x : ℝ ↦ x ^ p) (Set.Ici (0 : ℝ)) :=
+  fun x _ => (Real.continuousAt_rpow_const x p (Or.inr hp)).continuousWithinAt
 
 omit [Nontrivial ℋ] in
 private lemma operatorConcaveOn_Ioi_of_Ici {f : ℝ → ℝ}
@@ -44,9 +43,8 @@ private lemma pdSet_subset_psdSet : pdSet (ℋ := ℋ) ⊆ psdSet (ℋ := ℋ) :
   exact ⟨hA_sa, Set.Subset.trans hA_spec Set.Ioi_subset_Ici_self⟩
 
 private lemma rpow_pos_on_Ioi (p : ℝ) :
-    ∀ x ∈ Set.Ioi (0 : ℝ), 0 < x ^ p := by
-  intro x hx
-  exact Real.rpow_pos_of_pos hx p
+    ∀ x ∈ Set.Ioi (0 : ℝ), 0 < x ^ p :=
+  fun _ hx => Real.rpow_pos_of_pos hx p
 
 /--
 Theorem 1.1, concave range: the operator `(α, β)`-power mean is jointly concave on
@@ -68,14 +66,11 @@ theorem operatorPowerMean_jointlyConcaveOn_pdSet
         intro K _ _ _ _
         exact power_Icc_zero_one_operatorConcaveOn_Ici (ℋ := K) α hα)
       (hfcont := rpow_continuousOn_Ici α hα.1)
-      (hf0 := by
-        exact Real.rpow_nonneg (show (0 : ℝ) ≤ 0 by simp) α)
-      (hconc := by
-        exact operatorConcaveOn_Ioi_of_Ici (ℋ := ℋ)
-          (power_Icc_zero_one_operatorConcaveOn_Ici (ℋ := ℋ) β hβ))
-      (hcont := by
-        intro x hx
-        exact (Real.continuousAt_rpow_const x β (Or.inl (ne_of_gt hx))).continuousWithinAt)
+      (hf0 := Real.rpow_nonneg (show (0 : ℝ) ≤ 0 by simp) α)
+      (hconc := operatorConcaveOn_Ioi_of_Ici (ℋ := ℋ)
+        (power_Icc_zero_one_operatorConcaveOn_Ici (ℋ := ℋ) β hβ))
+      (hcont := fun x hx =>
+        (Real.continuousAt_rpow_const x β (Or.inl (ne_of_gt hx))).continuousWithinAt)
       (hpos := rpow_pos_on_Ioi β)
   simpa [operatorPowerMean] using
     hconc (A₁ := A₁) (A₂ := A₂) (B₁ := B₁) (B₂ := B₂) (θ := θ)
@@ -106,12 +101,10 @@ theorem operatorPowerMean_jointlyConvexOn_pdSet
         · have hα0 : α ≠ 0 := by linarith [hα.1]
           simp [Real.zero_rpow hα0]
       )
-      (hconc := by
-        exact operatorConcaveOn_Ioi_of_Ici (ℋ := ℋ)
-          (power_Icc_zero_one_operatorConcaveOn_Ici (ℋ := ℋ) β hβ))
-      (hcont := by
-        intro x hx
-        exact (Real.continuousAt_rpow_const x β (Or.inl (ne_of_gt hx))).continuousWithinAt)
+      (hconc := operatorConcaveOn_Ioi_of_Ici (ℋ := ℋ)
+        (power_Icc_zero_one_operatorConcaveOn_Ici (ℋ := ℋ) β hβ))
+      (hcont := fun x hx =>
+        (Real.continuousAt_rpow_const x β (Or.inl (ne_of_gt hx))).continuousWithinAt)
       (hpos := rpow_pos_on_Ioi β)
   simpa [operatorPowerMean] using
     hconv (A₁ := A₁) (A₂ := A₂) (B₁ := B₁) (B₂ := B₂) (θ := θ)
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Basic.lean b/QuantumInfo/ForMathlib/HermitianMat/Basic.lean
index 31eaa84ad..b8f8e3047 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Basic.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Basic.lean
@@ -28,8 +28,7 @@ section addgroup
 
 variable [AddGroup α] [StarAddMonoid α]
 
-theorem eq_IsHermitian : HermitianMat n α  = { m : Matrix n n α // m.IsHermitian} := by
-  rfl
+theorem eq_IsHermitian : HermitianMat n α  = { m : Matrix n n α // m.IsHermitian} := rfl
 
 @[coe] def mat : HermitianMat n α → Matrix n n α :=
   Subtype.val
@@ -41,12 +40,10 @@ theorem val_eq_coe (A : HermitianMat n α) : A.val = A := by
   rfl
 
 @[simp]
-theorem mat_mk (x : Matrix n n α) (h) : mat ⟨x, h⟩ = x := by
-  rfl
+theorem mat_mk (x : Matrix n n α) (h) : mat ⟨x, h⟩ = x := rfl
 
 @[simp]
-theorem mk_mat {A : HermitianMat n α} (h : A.mat.IsHermitian) : ⟨A.mat, h⟩ = A := by
-  rfl
+theorem mk_mat {A : HermitianMat n α} (h : A.mat.IsHermitian) : ⟨A.mat, h⟩ = A := rfl
 
 /-- Alias for HermitianMat.property or HermitianMat.2, this gets the fact that the value
   is actually `IsHermitian`.-/
@@ -61,8 +58,7 @@ instance instFun : FunLike (HermitianMat n α) n (n → α) where
   coe_injective _ _ h := HermitianMat.ext h
 
 @[simp]
-theorem mat_apply {A : HermitianMat n α} {i j : n} : A.mat i j = A i j := by
-  rfl
+theorem mat_apply {A : HermitianMat n α} {i j : n} : A.mat i j = A i j := rfl
 
 @[simp]
 theorem conjTranspose_mat (A : HermitianMat n α) :
@@ -77,31 +73,25 @@ instance [IsEmpty n] : Unique (HermitianMat n α) where
   uniq a := by ext; exact (IsEmpty.false ‹_›).elim
 
 @[simp, norm_cast]
-theorem mat_zero : (0 : HermitianMat n α).mat = 0 := by
-  rfl
+theorem mat_zero : (0 : HermitianMat n α).mat = 0 := rfl
 
 @[simp]
-theorem mk_zero (h : (0 : Matrix n n α).IsHermitian) : ⟨0, h⟩ = (0 : HermitianMat n α) := by
-  rfl
+theorem mk_zero (h : (0 : Matrix n n α).IsHermitian) : ⟨0, h⟩ = (0 : HermitianMat n α) := rfl
 
 @[simp]
-theorem zero_apply (i j : n) : (0 : HermitianMat n 𝕜) i j = 0 := by
-  rfl
+theorem zero_apply (i j : n) : (0 : HermitianMat n 𝕜) i j = 0 := rfl
 
 @[simp, norm_cast]
 theorem mat_add (A B : HermitianMat n α) :
-    (A + B).mat = A.mat + B.mat := by
-  rfl
+    (A + B).mat = A.mat + B.mat := rfl
 
 @[simp, norm_cast]
 theorem mat_sub (A B : HermitianMat n α) :
-    (A - B).mat = A.mat - B.mat := by
-  rfl
+    (A - B).mat = A.mat - B.mat := rfl
 
 @[simp, norm_cast]
 theorem mat_neg (A : HermitianMat n α) :
-    (-A).mat = -A.mat := by
-  rfl
+    (-A).mat = -A.mat := rfl
 
 section smul
 variable [SMul R α] [StarModule R α]
@@ -116,8 +106,7 @@ theorem mat_smul (c : R) (A : HermitianMat n α) :
 
 @[simp]
 theorem smul_apply (c : R) (A : HermitianMat n α) (i j : n) :
-    (c • A) i j = c • A i j := by
-  rfl
+    (c • A) i j = c • A i j := rfl
 end smul
 section topology
 
@@ -223,16 +212,13 @@ instance : One (HermitianMat n α) :=
       Matrix.one_apply, apply_ite (β := α), eq_comm]⟩
 
 @[simp, norm_cast]
-theorem mat_one : (1 : HermitianMat n α).mat = 1 := by
-  rfl
+theorem mat_one : (1 : HermitianMat n α).mat = 1 := rfl
 
 @[simp]
-theorem mk_one (h : (1 : Matrix n n α).IsHermitian) : ⟨1, h⟩ = (1 : HermitianMat n α) := by
-  rfl
+theorem mk_one (h : (1 : Matrix n n α).IsHermitian) : ⟨1, h⟩ = (1 : HermitianMat n α) := rfl
 
 @[simp]
-theorem one_apply (i j : n) : (1 : HermitianMat n α) i j = (1 : Matrix n n α) i j := by
-  rfl
+theorem one_apply (i j : n) : (1 : HermitianMat n α) i j = (1 : Matrix n n α) i j := rfl
 
 noncomputable instance : AddCommMonoidWithOne (HermitianMat n 𝕜) where
 
@@ -251,8 +237,7 @@ noncomputable instance instInv : Inv (HermitianMat m α) :=
   ⟨fun x ↦ ⟨x⁻¹, x.H.inv⟩⟩
 
 @[simp, norm_cast]
-theorem mat_inv : (A⁻¹).mat = A.mat⁻¹ := by
-  rfl
+theorem mat_inv : (A⁻¹).mat = A.mat⁻¹ := rfl
 
 @[simp]
 theorem zero_inv : ((0 : HermitianMat m α)⁻¹) = 0 := by
@@ -266,8 +251,7 @@ noncomputable instance instPow : Pow (HermitianMat m α) ℕ :=
   ⟨fun x n ↦ ⟨x ^ n, x.H.pow n⟩⟩
 
 @[simp, norm_cast]
-theorem mat_pow (n : ℕ) : (A ^ n).mat = A.mat ^ n := by
-  rfl
+theorem mat_pow (n : ℕ) : (A ^ n).mat = A.mat ^ n := rfl
 
 @[simp]
 theorem pow_zero : A ^ 0 = 1 := by
@@ -285,12 +269,10 @@ noncomputable instance instZPow : Pow (HermitianMat m α) ℤ :=
   ⟨fun x z ↦ ⟨x ^ z, x.H.zpow z⟩⟩
 
 @[simp]
-theorem mat_zpow (z : ℤ) : (A ^ z).mat = A.mat ^ z := by
-  rfl
+theorem mat_zpow (z : ℤ) : (A ^ z).mat = A.mat ^ z := rfl
 
 @[simp, norm_cast]
-theorem zpow_natCast : A ^ (n : ℤ) = A ^ n := by
-  rfl
+theorem zpow_natCast : A ^ (n : ℤ) = A ^ n := rfl
 
 @[simp]
 theorem zpow_zero : A ^ (0 : ℤ) = 1 := by
@@ -371,13 +353,11 @@ def conj {m} (B : Matrix m n α) : HermitianMat n α →+ HermitianMat m α wher
   map_zero' := by simp
 
 theorem conj_apply (B : Matrix m n α) (A : HermitianMat n α) :
-    conj B A = ⟨B * A.mat * B.conjTranspose, (conj B A).2⟩ := by
-  rfl
+    conj B A = ⟨B * A.mat * B.conjTranspose, (conj B A).2⟩ := rfl
 
 @[simp]
 theorem conj_apply_mat (B : Matrix m n α) (A : HermitianMat n α) :
-    (A.conj B).mat = B * A.mat * B.conjTranspose := by
-  rfl
+    (A.conj B).mat = B * A.mat * B.conjTranspose := rfl
 
 theorem conj_conj {m l} [Fintype m] (B : Matrix m n α) (C : Matrix l m α) :
     (A.conj B).conj C = A.conj (C * B) := by
@@ -387,12 +367,10 @@ theorem conj_conj {m l} [Fintype m] (B : Matrix m n α) (C : Matrix l m α) :
 variable (B : HermitianMat n α)
 
 @[simp]
-theorem conj_zero [DecidableEq n] : A.conj (0 : Matrix m n α) = 0 := by
-  simp [conj_apply]
+theorem conj_zero [DecidableEq n] : A.conj (0 : Matrix m n α) = 0 := by simp [conj_apply]
 
 @[simp]
-theorem conj_one [DecidableEq n] : A.conj 1 = A := by
-  simp [conj_apply]
+theorem conj_one [DecidableEq n] : A.conj 1 = A := by simp [conj_apply]
 
 @[simp]
 lemma conj_one_unitary [DecidableEq n] (U : Matrix.unitaryGroup n α) :
@@ -411,8 +389,7 @@ def conjLinear {m} (B : Matrix m n α) : HermitianMat n α →ₗ[R] HermitianMa
     simp
 
 @[simp]
-theorem conjLinear_apply (B : Matrix m n α) : conjLinear R B A = conj B A  := by
-  rfl
+theorem conjLinear_apply (B : Matrix m n α) : conjLinear R B A = conj B A  := rfl
 
 @[fun_prop]
 lemma continuous_conj (ρ : HermitianMat n 𝕜) : Continuous (ρ.conj (m := m) ·) := by
@@ -439,12 +416,10 @@ theorem isSymmetric : A.lin.IsSymmetric :=
   Matrix.isSymmetric_toEuclideanLin_iff.symm.mp A.H
 
 @[simp]
-theorem lin_zero : (0 : HermitianMat n 𝕜).lin = 0 := by
-  simp [lin]; rfl
+theorem lin_zero : (0 : HermitianMat n 𝕜).lin = 0 := by simp [lin]; rfl
 
 @[simp]
-theorem lin_one : (1 : HermitianMat n 𝕜).lin = 1 := by
-  simp [lin]; rfl
+theorem lin_one : (1 : HermitianMat n 𝕜).lin = 1 := by simp [lin]; rfl
 
 noncomputable def eigenspace (μ : 𝕜) : Submodule 𝕜 (EuclideanSpace 𝕜 n) :=
   Module.End.eigenspace A.lin μ
@@ -463,12 +438,10 @@ theorem ker_eq_eigenspace_zero : A.ker = A.eigenspace 0 := by
   simp [ker, eigenspace]
 
 @[simp]
-theorem ker_zero : (0 : HermitianMat n 𝕜).ker = ⊤ := by
-  simp [ker]
+theorem ker_zero : (0 : HermitianMat n 𝕜).ker = ⊤ := by simp [ker]
 
 @[simp]
-theorem ker_one : (1 : HermitianMat n 𝕜).ker = ⊥ := by
-  simp [ker]; rfl
+theorem ker_one : (1 : HermitianMat n 𝕜).ker = ⊥ := by simp [ker]; rfl
 
 theorem ker_pos_smul {c : ℝ} (hc : c ≠ 0) : (c • A).ker = A.ker := by
   ext x
@@ -484,8 +457,7 @@ theorem support_eq_sup_eigenspace_nonzero : A.support = ⨆ μ ≠ 0, A.eigenspa
   exact A.lin.support_eq_sup_eigenspace_nonzero A.isSymmetric
 
 @[simp]
-theorem support_zero : (0 : HermitianMat n 𝕜).support = ⊥ := by
-  simp [support]
+theorem support_zero : (0 : HermitianMat n 𝕜).support = ⊥ := by simp [support]
 
 @[simp]
 theorem support_one : (1 : HermitianMat n 𝕜).support = ⊤ := by
@@ -517,8 +489,7 @@ def diagonal (f : n → ℝ) : HermitianMat n 𝕜 :=
 variable (f g : n → ℝ)
 
 @[simp]
-theorem diagonal_mat : (diagonal 𝕜 f).mat = Matrix.diagonal (f · : n → 𝕜) := by
-  rfl
+theorem diagonal_mat : (diagonal 𝕜 f).mat = Matrix.diagonal (f · : n → 𝕜) := rfl
 
 @[simp]
 theorem diagonal_zero : (diagonal 𝕜 0) = (0 : HermitianMat n 𝕜) := by
@@ -573,8 +544,7 @@ scoped[HermitianMat] infixl:100 " ⊗ₖ " => HermitianMat.kronecker
 
 @[simp, norm_cast]
 theorem kronecker_mat (A : HermitianMat m α) (B : HermitianMat n α) :
-    (A ⊗ₖ B).mat = A.mat ⊗ₖ B.mat := by
-  rfl
+    (A ⊗ₖ B).mat = A.mat ⊗ₖ B.mat := rfl
 
 @[simp]
 theorem zero_kronecker (A : HermitianMat m α) : (0 : HermitianMat n α) ⊗ₖ A = 0 := by
@@ -691,8 +661,7 @@ theorem _root_.Matrix.range_mul_conjTranspose_of_ker_le_ker {A : Matrix d d 𝕜
             rw [← inner_self_eq_zero (𝕜 := 𝕜)]
             have : ∀ y, @inner 𝕜 _ _ y (Matrix.toEuclideanLin M x) = 0 := by
               intro y
-              rw [← LinearMap.adjoint_inner_left]
-              exact h _ ⟨y, rfl⟩
+              simpa [← LinearMap.adjoint_inner_left] using h _ ⟨y, rfl⟩
             exact this _
           · intro h y ⟨z, hz⟩
             rw [← hz, LinearMap.adjoint_inner_left, h, inner_zero_right]
@@ -741,8 +710,7 @@ theorem _root_.Matrix.IsHermitian.spectrum_rcLike {A : Matrix n n 𝕜} (hA : A.
 of the real spectrum. -/
 @[simp]
 theorem spectrum_rcLike (A : HermitianMat n 𝕜) :
-    spectrum 𝕜 A.mat = RCLike.ofReal '' spectrum ℝ A.mat := by
-  exact A.H.spectrum_rcLike.symm
+    spectrum 𝕜 A.mat = RCLike.ofReal '' spectrum ℝ A.mat := A.H.spectrum_rcLike.symm
 
 theorem ne_zero_iff_ne_zero_spectrum (A : HermitianMat n 𝕜) :
     A ≠ 0 ↔ ∃ x ∈ spectrum ℝ A.mat, x ≠ 0 := by
diff --git a/QuantumInfo/ForMathlib/HermitianMat/CFC.lean b/QuantumInfo/ForMathlib/HermitianMat/CFC.lean
index b342732de..e1187b559 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/CFC.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/CFC.lean
@@ -33,25 +33,21 @@ variable (A : HermitianMat d 𝕜) (f : ℝ → ℝ) (g : ℝ → ℝ) (q r : 
 /- Adding this to the `CStarAlgebra` aesop set allows `cfc_tac` to use it. -/
 omit [Fintype d] [DecidableEq d] in
 @[aesop safe apply (rule_sets := [CStarAlgebra])]
-theorem isSelfAdjoint : IsSelfAdjoint A.mat := by
-  exact A.H
+theorem isSelfAdjoint : IsSelfAdjoint A.mat := A.H
 
 /- Adding this to `fun_prop` allows `cfc_cont_tac` to use it. -/
 @[fun_prop]
 theorem continuousOn_finite {α β : Type*} (f : α → β) (S : Set α)
     [TopologicalSpace α] [TopologicalSpace β] [T1Space α] [Finite S] : ContinuousOn f S := by
-  rw [continuousOn_iff_continuous_restrict]
-  exact continuous_of_discreteTopology
+  simpa [continuousOn_iff_continuous_restrict] using continuous_of_discreteTopology
 
 @[simp]
-theorem conjTranspose_cfc : (cfc f A.mat).conjTranspose = cfc f A.mat := by
-  exact cfc_predicate f A.mat
+theorem conjTranspose_cfc : (cfc f A.mat).conjTranspose = cfc f A.mat := cfc_predicate f A.mat
 
 protected def cfc : HermitianMat d 𝕜 :=
   ⟨cfc f A.mat, cfc_predicate _ _⟩
 
-theorem cfc_eq : A.cfc f = ⟨cfc f A.mat, cfc_predicate f A.mat⟩ := by
-  rfl
+theorem cfc_eq : A.cfc f = ⟨cfc f A.mat, cfc_predicate f A.mat⟩ := rfl
 
 @[simp]
 theorem mat_cfc : (A.cfc f).mat = _root_.cfc f A.mat := by
@@ -93,8 +89,7 @@ variable {A B : HermitianMat d 𝕜}
 
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem _root_.Commute.cfc_left (hAB : Commute A.mat B.mat) :
-    Commute (A.cfc f).mat B.mat := by
-  exact hAB.cfc_real f
+    Commute (A.cfc f).mat B.mat := hAB.cfc_real f
 
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem _root_.Commute.cfc_right (hAB : Commute A.mat B.mat) :
@@ -102,8 +97,7 @@ theorem _root_.Commute.cfc_right (hAB : Commute A.mat B.mat) :
   (hAB.symm.cfc_left f).symm
 
 theorem cfc_commute (f g : ℝ → ℝ) (hAB : Commute A.mat B.mat) :
-    Commute (A.cfc f).mat (B.cfc g).mat := by
-  exact (hAB.cfc_right g).cfc_left f
+    Commute (A.cfc f).mat (B.cfc g).mat := (hAB.cfc_right g).cfc_left f
 
 @[aesop safe apply (rule_sets := [Commutes])]
 theorem cfc_self_commute (A : HermitianMat d 𝕜) (f g : ℝ → ℝ) :
@@ -120,8 +114,7 @@ theorem cfc_reindex (e : d ≃ d₂) : (A.reindex e).cfc f = (A.cfc f).reindex e
   exact Matrix.cfc_reindex f e
 
 theorem spectrum_cfc_eq_image (A : HermitianMat d 𝕜) (f : ℝ → ℝ) :
-    spectrum ℝ (A.cfc f).mat = f '' (spectrum ℝ A.mat) := by
-  exact cfc_map_spectrum f A.mat
+    spectrum ℝ (A.cfc f).mat = f '' (spectrum ℝ A.mat) := cfc_map_spectrum f A.mat
 
 set_option backward.isDefEq.respectTransparency false in
 /--
@@ -156,8 +149,7 @@ theorem cfc_eigenvalues (A : HermitianMat d 𝕜) :
 etc. We need these because (as above) `HermitianMat.cfc` is different from `_root_.cfc`. -/
 
 @[simp]
-nonrec theorem cfc_id : A.cfc id = A := by
-  simpa [HermitianMat.ext_iff] using cfc_id ℝ A.mat
+nonrec theorem cfc_id : A.cfc id = A := by simpa [HermitianMat.ext_iff] using cfc_id ℝ A.mat
 
 @[simp]
 nonrec theorem cfc_id' : A.cfc (·) = A :=
@@ -189,8 +181,7 @@ theorem mat_cfc_mul : (A.cfc (f * g)).mat = A.cfc f * A.cfc g := by
   simp only [mat_cfc]
   exact cfc_mul ..
 
-theorem mat_cfc_mul_apply : (A.cfc (fun x ↦ f x * g x)).mat = A.cfc f * A.cfc g := by
-  exact mat_cfc_mul ..
+theorem mat_cfc_mul_apply : (A.cfc (fun x ↦ f x * g x)).mat = A.cfc f * A.cfc g := mat_cfc_mul ..
 
 nonrec theorem cfc_comp : A.cfc (g ∘ f) = (A.cfc f).cfc g := by
   ext1; exact cfc_comp ..
@@ -386,8 +377,7 @@ theorem Matrix.PosDef.spectrum_subset_Ioi {d 𝕜 : Type*} [Fintype d] [Decidabl
     {A : Matrix d d 𝕜} (hA : A.PosDef) : spectrum ℝ A ⊆ Set.Ioi 0 := by
   intro x hx;
   -- Since $A$ is positive definite, all its eigenvalues are positive.
-  have h_eigenvalues_pos : ∀ i : d, 0 < hA.1.eigenvalues i := by
-    exact hA.eigenvalues_pos;
+  have h_eigenvalues_pos : ∀ i : d, 0 < hA.1.eigenvalues i := hA.eigenvalues_pos
   have h_spectrum_eq_range : spectrum ℝ A = Set.range (hA.1.eigenvalues) := by
     exact Matrix.IsHermitian.spectrum_real_eq_range_eigenvalues hA.left;
   aesop
@@ -458,8 +448,7 @@ lemma norm_cfc_sub_cfc_le_sqrt_card {A : HermitianMat d ℂ} {f g : ℝ → ℝ}
   · intro x hx
     apply le_csSup;
     · -- The supremum of a finite set of real numbers is finite.
-      have h_finite : Set.Finite (spectrum ℝ A.mat) := by
-        exact Set.toFinite _;
+      have h_finite : Set.Finite (spectrum ℝ A.mat) := Set.toFinite _
       obtain ⟨ M, hM ⟩ := h_finite.exists_finset_coe;
       refine' ⟨ ∑ x ∈ M, ‖f x - g x‖, Set.forall_mem_range.2 fun x => _ ⟩;
       rw [ ← hM ];
@@ -740,11 +729,9 @@ lemma continuousWithinAt_cfc_of_continuousOn {T : Set ℝ} {g : ℝ → ℝ}
     (hg : ContinuousOn g T) (hA₀ : spectrum ℝ A₀.mat ⊆ T) :
     ContinuousWithinAt (fun B ↦ B.cfc g) {B | spectrum ℝ B.mat ⊆ T} A₀ := by
   have h_ext : ∃ h : ℝ → ℝ, Continuous h ∧ ∀ x ∈ spectrum ℝ A₀.mat, h x = g x := by
-    have h_finite : Set.Finite (spectrum ℝ A₀.mat) := by
-      exact Set.toFinite _
+    have h_finite : Set.Finite (spectrum ℝ A₀.mat) := Set.toFinite _
     generalize_proofs at *; (
-    have h_cont : ContinuousOn g (spectrum ℝ A₀.val) := by
-      exact hg.mono hA₀
+    have h_cont : ContinuousOn g (spectrum ℝ A₀.val) := hg.mono hA₀
     generalize_proofs at *; (
     have := @ContinuousMap.exists_restrict_eq ℝ;
     specialize this ( show IsClosed ( spectrum ℝ A₀.val ) from h_finite.isClosed ) ( ContinuousMap.mk ( fun x => g x ) <| by exact continuousOn_iff_continuous_restrict.mp h_cont ) ; rcases this with ⟨ h, hh ⟩ ; exact ⟨ h, h.continuous, fun x hx => by simpa using congr_arg ( fun f => f ⟨ x, hx ⟩ ) hh ⟩ ;));
@@ -762,8 +749,7 @@ lemma continuousWithinAt_cfc_of_continuousOn {T : Set ℝ} {g : ℝ → ℝ}
           simpa [ hh_eq y hy ] using h_diff_small.abs;
         have := Metric.tendsto_nhdsWithin_nhds.mp h_diff_small ( ε / ( Real.sqrt ( Fintype.card d ) + 1 ) ) ( div_pos ε_pos ( add_pos_of_nonneg_of_pos ( Real.sqrt_nonneg _ ) zero_lt_one ) ) ; aesop;
       choose! δ hδ_pos hδ using h_diff_small;
-      have h_finite : Set.Finite (spectrum ℝ A₀.mat) := by
-        exact Set.toFinite _;
+      have h_finite : Set.Finite (spectrum ℝ A₀.mat) := Set.toFinite _;
       obtain ⟨δ_min, hδ_min_pos, hδ_min⟩ : ∃ δ_min > 0, ∀ y ∈ spectrum ℝ A₀.mat, δ_min ≤ δ y := by
         by_cases h_empty : spectrum ℝ A₀.mat = ∅;
         · exact ⟨ 1, zero_lt_one, by simp [ h_empty ] ⟩;
@@ -773,8 +759,7 @@ lemma continuousWithinAt_cfc_of_continuousOn {T : Set ℝ} {g : ℝ → ℝ}
     -- By the spectrum_subset_of_isOpen lemma, there exists a neighborhood U of A₀ such that the spectrum of B is within δ of the spectrum of A₀ for all B in U.
     obtain ⟨U, hU⟩ : ∃ U ∈ nhds A₀, ∀ B ∈ U, spectrum ℝ B.mat ⊆ {x | ∃ y ∈ spectrum ℝ A₀.mat, |x - y| < δ} := by
       have h_spectrum_subset : ∀ᶠ B in nhds A₀, spectrum ℝ B.mat ⊆ Metric.thickening δ (spectrum ℝ A₀.mat) := by
-        have h_open : IsOpen (Metric.thickening δ (spectrum ℝ A₀.mat)) := by
-          exact Metric.isOpen_thickening
+        have h_open : IsOpen (Metric.thickening δ (spectrum ℝ A₀.mat)) := Metric.isOpen_thickening
         have := spectrum_subset_of_isOpen A₀ ( Metric.thickening δ ( spectrum ℝ A₀.mat ) ) h_open ( Metric.self_subset_thickening δ_pos _ ) ; aesop;
       generalize_proofs at *; (
       exact ⟨ _, h_spectrum_subset, fun B hB => fun x hx => by simpa [ dist_eq_norm ] using Metric.mem_thickening_iff.mp ( hB hx ) ⟩)
@@ -876,8 +861,7 @@ lemma dist_lt_of_continuous_spectrum {X : Type*} [TopologicalSpace X]
         choose U_i V_i hU_i hV_i hx₀_i hV_i_i h_cont_i using h_cont; exact ⟨ U_i, V_i, hU_i, hV_i, hx₀_i, hV_i_i, h_cont_i ⟩ ;);
       -- The open set W := ⋃ᵢ V_i contains spectrum(A x₀) (since each λᵢ ∈ V_i and spectrum = range of eigenvalues). W is open as a union of open sets.
       set W := ⋃ i, V_i i with hW_def
-      have hW_open : IsOpen W := by
-        exact isOpen_iUnion hV_i
+      have hW_open : IsOpen W := isOpen_iUnion hV_i
       have hW_spectrum : spectrum ℝ (A x₀).mat ⊆ W := by
         intro t ht
         obtain ⟨i, hi⟩ : ∃ i, t = (A x₀).H.eigenvalues i := by
@@ -886,8 +870,7 @@ lemma dist_lt_of_continuous_spectrum {X : Type*} [TopologicalSpace X]
           generalize_proofs at *; (
           exact h_eigenvalues.subset ht |> Exists.imp fun i hi => hi.symm)
         aesop
-      have hW_subset : ∀ᶠ B in nhds (A x₀), spectrum ℝ B.mat ⊆ W := by
-        exact spectrum_subset_of_isOpen (A x₀) W hW_open hW_spectrum
+      have hW_subset : ∀ᶠ B in nhds (A x₀), spectrum ℝ B.mat ⊆ W := spectrum_subset_of_isOpen (A x₀) W hW_open hW_spectrum
       have hW_subset_S : ∀ᶠ y in nhdsWithin x₀ S, spectrum ℝ (A y).mat ⊆ W := by
         exact Filter.mem_of_superset ( hA₂.continuousWithinAt hx₀ |> fun h => h.eventually ( hW_subset ) ) fun y hy => hy
       obtain ⟨U', hU'⟩ : ∃ U' ∈ nhds x₀, ∀ y ∈ U' ∩ S, spectrum ℝ (A y).mat ⊆ W := by
@@ -895,8 +878,7 @@ lemma dist_lt_of_continuous_spectrum {X : Type*} [TopologicalSpace X]
       obtain ⟨U'', hU''⟩ : ∃ U'' ∈ nhds x₀, ∀ i, U'' ⊆ U_i i := by
         exact ⟨ ⋂ i, U_i i, Filter.mem_of_superset ( Filter.iInter_mem.mpr fun i => IsOpen.mem_nhds ( hU_i i ) ( h_cont.1 i ) ) fun x hx => by aesop, fun i => Set.iInter_subset _ i ⟩
       set U := U' ∩ U'' with hU_def
-      have hU_mem : U ∈ nhds x₀ := by
-        exact Filter.inter_mem hU'.1 hU''.1
+      have hU_mem : U ∈ nhds x₀ := Filter.inter_mem hU'.1 hU''.1
       have hU_subset : ∀ y ∈ U ∩ S, spectrum ℝ (A y).mat ⊆ W := by
         exact fun y hy => hU'.2 y ⟨ hy.1.1, hy.2 ⟩ |> Set.Subset.trans <| by simp [ hW_def ] ;
       have hU_cont : ∀ y ∈ U ∩ S, ∀ t ∈ spectrum ℝ (A y).mat, ‖f y t - f x₀ t‖ < ε := by
@@ -1240,14 +1222,12 @@ theorem inv_ge_one_of_le_one (hA : A.mat.PosDef) (h : A ≤ 1) : 1 ≤ A⁻¹ :=
     have h_cfc_nonneg : ∀ i, 0 ≤ (A.H.eigenvalues i)⁻¹ - 1 := by
       have h_pos : ∀ i, 0 < A.H.eigenvalues i ∧ A.H.eigenvalues i ≤ 1 := by
         -- Since $A$ is positive definite, all its eigenvalues are positive.
-        have h_pos : ∀ i, 0 < A.H.eigenvalues i := by
-          exact fun i => Matrix.PosDef.eigenvalues_pos hA i;
+        have h_pos : ∀ i, 0 < A.H.eigenvalues i := fun i => Matrix.PosDef.eigenvalues_pos hA i
         -- Since $A \leq 1$, for any eigenvalue $\lambda_i$ of $A$, we have $\lambda_i \leq 1$.
         have h_le_one : ∀ i, A.H.eigenvalues i ≤ 1 := by
           have h_le_one : ∀ i, A.H.eigenvalues i ≤ 1 := by
             intro i
-            have h_eigenvalue : A.mat.PosSemidef := by
-              exact hA.posSemidef
+            have h_eigenvalue : A.mat.PosSemidef := hA.posSemidef
             have h_eigenvalue_le_one : ∀ x : d → 𝕜, x ≠ 0 → (star x ⬝ᵥ A.mat.mulVec x) / (star x ⬝ᵥ x) ≤ 1 := by
               intro x hx_ne_zero
               have h_eigenvalue_le_one : (star x ⬝ᵥ (1 - A.mat).mulVec x) ≥ 0 := by
@@ -1267,8 +1247,7 @@ theorem inv_ge_one_of_le_one (hA : A.mat.PosDef) (h : A ≤ 1) : 1 ≤ A⁻¹ :=
     exact (cfc_nonneg_iff A fun x => x⁻¹ - 1).mpr h_cfc_nonneg;
   -- Since $A.cfc (fun x => x⁻¹ - 1) \geq 0$, we have $A.cfc (fun x => x⁻¹) \geq 1$.
   have h_cfc_ge_one : A.cfc (fun x => x⁻¹) ≥ 1 := by
-    have h_cfc_sub : A.cfc (fun x => x⁻¹ - 1) = A.cfc (fun x => x⁻¹) - A.cfc (fun _ => 1) := by
-      exact cfc_sub_apply A Inv.inv fun x => 1;
+    have h_cfc_sub : A.cfc (fun x => x⁻¹ - 1) = A.cfc (fun x => x⁻¹) - A.cfc (fun _ => 1) := cfc_sub_apply A Inv.inv fun x => 1;
     aesop;
   convert h_cfc_ge_one.le using 1;
   convert cfc_inv.symm;
@@ -1366,8 +1345,7 @@ lemma ker_cfc_le_ker_on_set
     (A.cfc f).ker ≤ A.ker := by
   intro x hx
   have h_inner : ∀ i, A.H.eigenvalues i ≠ 0 → inner ℂ (A.H.eigenvectorBasis i) x = 0 := by
-    have h_inner_zero : (A.cfc f).mat.mulVec x = 0 := by
-      exact (mem_ker_iff_mulVec_zero (A.cfc f) x).mp hx
+    have h_inner_zero : (A.cfc f).mat.mulVec x = 0 := (mem_ker_iff_mulVec_zero (A.cfc f) x).mp hx
     have h_inner_zero_expansion : ∑ i, (f (A.H.eigenvalues i) : ℂ) • inner ℂ (A.H.eigenvectorBasis i) x • A.H.eigenvectorBasis i = 0 := by
       convert h_inner_zero using 1;
       rw [ cfc_mulVec_expansion ];
@@ -1390,8 +1368,7 @@ lemma ker_cfc_le_ker_on_set
   exact mem_ker_iff_mulVec_zero A x
 
 lemma ker_cfc_le_ker (h : ∀ i, f i = 0 → i = 0) :
-    (A.cfc f).ker ≤ A.ker := by
-  exact ker_cfc_le_ker_on_set (Set.subset_univ _) (by simpa using h)
+    (A.cfc f).ker ≤ A.ker := ker_cfc_le_ker_on_set (Set.subset_univ _) (by simpa using h)
 
 lemma ker_cfc_le_ker_nonneg (hA : 0 ≤ A) (h : ∀ i ≥ 0, f i = 0 → i = 0) :
     (A.cfc f).ker ≤ A.ker := by
@@ -1403,8 +1380,7 @@ lemma ker_le_ker_cfc_on_set (hs : spectrum ℝ A.mat ⊆ s) (h : ∀ i ∈ s, i
   intro x hx
   have h_inner_zero : ∀ i, A.H.eigenvalues i ≠ 0 → inner ℂ (A.H.eigenvectorBasis i) x = 0 := by
     intro i hi
-    have h_inner_zero : A.mat.mulVec x = 0 := by
-      exact (mem_ker_iff_mulVec_zero A x).mp hx;
+    have h_inner_zero : A.mat.mulVec x = 0 := (mem_ker_iff_mulVec_zero A x).mp hx
     have := mulVec_eq_zero_iff_inner_eigenvector_zero A x; aesop;
   have h_mulVec_zero : (A.cfc f).mat.mulVec x = ∑ i, (f (A.H.eigenvalues i) : ℂ) • inner ℂ (A.H.eigenvectorBasis i) x • A.H.eigenvectorBasis i := by
     convert cfc_mulVec_expansion A f x using 1;
@@ -1416,8 +1392,7 @@ lemma ker_le_ker_cfc_on_set (hs : spectrum ℝ A.mat ⊆ s) (h : ∀ i ∈ s, i
   exact Or.inl ( h _ ( hs (Matrix.IsHermitian.eigenvalues_mem_spectrum_real (H A) j ) ) hj )
 
 lemma ker_le_ker_cfc (h : ∀ i, i = 0 → f i = 0) :
-    A.ker ≤ (A.cfc f).ker := by
-  exact ker_le_ker_cfc_on_set (Set.subset_univ _) (by simpa using h)
+    A.ker ≤ (A.cfc f).ker := ker_le_ker_cfc_on_set (Set.subset_univ _) (by simpa using h)
 
 lemma ker_le_ker_cfc_nonneg (hA : 0 ≤ A) (h : ∀ i ≥ 0, i = 0 → f i = 0) :
     A.ker ≤ (A.cfc f).ker := by
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Inner.lean b/QuantumInfo/ForMathlib/HermitianMat/Inner.lean
index a7278fad0..d30f0d640 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Inner.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Inner.lean
@@ -45,8 +45,7 @@ instance : Inner R (HermitianMat n α) where
   inner A B := selfadjMap (A.mat * B.mat).trace
 
 theorem inner_def (A B : HermitianMat n α) :
-    ⟪A, B⟫_R = selfadjMap (A.mat * B.mat).trace := by
-  rfl
+    ⟪A, B⟫_R = selfadjMap (A.mat * B.mat).trace := rfl
 
 end defs
 section semiring
@@ -64,12 +63,10 @@ protected theorem inner_add_left : ⟪A + B, C⟫_R = ⟪A, C⟫_R + ⟪B, C⟫_
   simp [inner_def, right_distrib]
 
 @[simp]
-protected theorem inner_zero_right : ⟪A, 0⟫_R = 0 := by
-  simp [inner_def]
+protected theorem inner_zero_right : ⟪A, 0⟫_R = 0 := by simp [inner_def]
 
 @[simp]
-protected theorem inner_zero_left : ⟪0, A⟫_R = 0 := by
-  simp [inner_def]
+protected theorem inner_zero_left : ⟪0, A⟫_R = 0 := by simp [inner_def]
 
 end semiring
 section ring
@@ -78,12 +75,10 @@ variable [CommRing R] [Ring α] [StarAddMonoid α] [Algebra R α] [IsMaximalSelf
 variable (A B C : HermitianMat n α)
 
 @[simp]
-protected theorem inner_neg_left : ⟪-A, B⟫_R = -⟪A, B⟫_R := by
-  simp [inner_def]
+protected theorem inner_neg_left : ⟪-A, B⟫_R = -⟪A, B⟫_R := by simp [inner_def]
 
 @[simp]
-protected theorem inner_neg_right : ⟪A, -B⟫_R = -⟪A, B⟫_R := by
-  simp [inner_def]
+protected theorem inner_neg_right : ⟪A, -B⟫_R = -⟪A, B⟫_R := by simp [inner_def]
 
 protected theorem inner_sub_left : ⟪A, B - C⟫_R = ⟪A, B⟫_R - ⟪A, C⟫_R := by
   simp [inner_def, mul_sub]
@@ -119,12 +114,10 @@ variable [CommSemiring R] [Ring α] [StarRing α] [Algebra R α] [IsMaximalSelfA
 variable (A B : HermitianMat n α)
 
 @[simp]
-theorem inner_one : ⟪A, 1⟫_R = A.trace := by
-  simp only [inner_def, mat_one,  mul_one, trace]
+theorem inner_one : ⟪A, 1⟫_R = A.trace := by simp only [inner_def, mat_one,  mul_one, trace]
 
 @[simp]
-theorem one_inner : ⟪1, A⟫_R = A.trace := by
-  simp only [inner_def, one_mul, mat_one, trace]
+theorem one_inner : ⟪1, A⟫_R = A.trace := by simp only [inner_def, one_mul, mat_one, trace]
 
 end starring
 section commring
@@ -140,8 +133,7 @@ theorem inner_eq_trace_mul : algebraMap R α ⟪A, B⟫_R = (A.mat * B.mat).trac
   rw [Finset.sum_comm]
   congr! <;> apply congrFun₂ (H _)
 
-theorem inner_comm : ⟪A, B⟫_R = ⟪B, A⟫_R := by
-  rw [inner_def, inner_def, Matrix.trace_mul_comm]
+theorem inner_comm : ⟪A, B⟫_R = ⟪B, A⟫_R := by rw [inner_def, inner_def, Matrix.trace_mul_comm]
 
 end commring
 
@@ -151,8 +143,7 @@ variable (A B : HermitianMat n α)
 
 /-- `HermitianMat.inner` reduces to `Matrix.trace (A * B)` when the elements are a `TrivialStar`. -/
 theorem inner_eq_trace_trivial : ⟪A, B⟫_α = (A.mat * B.mat).trace := by
-  rw [← inner_eq_trace_mul]
-  rfl
+  simpa using inner_eq_trace_mul A B
 
 end trivialstar
 
@@ -162,8 +153,7 @@ open ComplexOrder
 
 variable {n 𝕜 : Type*} [Fintype n] [RCLike 𝕜] (A B C : HermitianMat n 𝕜)
 
-theorem inner_eq_re_trace : ⟪A, B⟫ = RCLike.re (A.mat * B.mat).trace := by
-  rfl
+theorem inner_eq_re_trace : ⟪A, B⟫ = RCLike.re (A.mat * B.mat).trace := rfl
 
 theorem inner_eq_trace_rc : ⟪A, B⟫ = (A.mat * B.mat).trace := by
   rw [inner_eq_re_trace, ← RCLike.conj_eq_iff_re]
@@ -201,8 +191,7 @@ theorem inner_mono (hA : 0 ≤ A) : B ≤ C → ⟪A, B⟫ ≤ ⟪A, C⟫ := by
 
 theorem inner_mono' (hA : 0 ≤ A) : B ≤ C → ⟪B, A⟫ ≤ ⟪C, A⟫ := by
   intro hBC
-  rw [inner_comm B A, inner_comm C A]
-  exact inner_mono hA hBC
+  simpa [inner_comm B A, inner_comm C A] using inner_mono hA hBC
 
 /-- The inner product for PSD matrices is at most the product of their traces. -/
 theorem inner_le_mul_trace (hA : 0 ≤ A) (hB : 0 ≤ B) : ⟪A, B⟫ ≤ A.trace * B.trace := by
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Jordan.lean b/QuantumInfo/ForMathlib/HermitianMat/Jordan.lean
index cf43f296d..1f6dbf8ab 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Jordan.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Jordan.lean
@@ -37,16 +37,13 @@ theorem symmMul_comm : A.symmMul B = B.symmMul A := by
   rw [symmMul, symmMul, Subtype.mk.injEq, add_comm]
 
 @[simp]
-theorem symmMul_zero : A.symmMul 0 = 0:= by
-  simp [symmMul]
+theorem symmMul_zero : A.symmMul 0 = 0:= by simp [symmMul]
 
 @[simp]
-theorem zero_symmMul : symmMul 0 A = 0 := by
-  simp [symmMul]
+theorem zero_symmMul : symmMul 0 A = 0 := by simp [symmMul]
 
 theorem symmMul_toMat : (A.symmMul B).mat =
-    (2 : 𝕜)⁻¹ • (A.mat * B.mat + B.mat * A.mat) := by
-  rfl
+    (2 : 𝕜)⁻¹ • (A.mat * B.mat + B.mat * A.mat) := rfl
 
 variable [Invertible (2 : 𝕜)]
 
@@ -58,8 +55,7 @@ theorem symmMul_of_commute (hAB : Commute A.mat B.mat) :
   rw [smul_add, ← add_smul, inv_eq_one_div, ← add_div]
   rw [add_self_div_two, one_smul]
 
-theorem symmMul_self : (symmMul A A).mat = A.mat * A.mat := by
-  simp
+theorem symmMul_self : (symmMul A A).mat = A.mat * A.mat := by simp
 
 variable [DecidableEq d]
 
@@ -97,8 +93,7 @@ scoped instance : CommMagma (HermitianMat d 𝕜) where
 -- scoped instance : Mul (HermitianMat d 𝕜) :=
   -- CommMagma.toMul
 
-theorem mul_eq_symmMul : A * B = A.symmMul B := by
-  rfl
+theorem mul_eq_symmMul : A * B = A.symmMul B := rfl
 
 scoped instance : IsCommJordan (HermitianMat d 𝕜) where
   lmul_comm_rmul_rmul a b := by
diff --git a/QuantumInfo/ForMathlib/HermitianMat/LiebConcavity.lean b/QuantumInfo/ForMathlib/HermitianMat/LiebConcavity.lean
index db26cb40e..e649e19ff 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/LiebConcavity.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/LiebConcavity.lean
@@ -98,8 +98,7 @@ lemma Φ_mem_pdSet [Nonempty d] (A : HermitianMat d ℂ) (hA : A.mat.PosDef) :
 set_option synthInstance.maxHeartbeats 80000 in
 /-- `Φ` commutes with CFC for Hermitian matrices. -/
 lemma Φ_cfc (A : HermitianMat d ℂ) (f : ℝ → ℝ) :
-    Φ (cfc f A.mat) = cfc f (Φ A.mat) := by
-  exact StarAlgHomClass.map_cfc Φ f A.mat (hφ := Φ_continuous)
+    Φ (cfc f A.mat) = cfc f (Φ A.mat) := StarAlgHomClass.map_cfc Φ f A.mat (hφ := Φ_continuous)
     (ha := A.H.isSelfAdjoint)
 
 set_option synthInstance.maxHeartbeats 80000 in
@@ -120,8 +119,7 @@ lemma trace_Φ_eq (M : Matrix d d ℂ) :
 
 /-- `traceRe(Φ(M)) = re(Tr[M])` for any matrix M. -/
 lemma traceRe_Φ_general (M : Matrix d d ℂ) :
-    traceRe (Φ M) = Complex.re M.trace := by
-  simp [traceRe, trace_Φ_eq]
+    traceRe (Φ M) = Complex.re M.trace := by simp [traceRe, trace_Φ_eq]
 
 end HermitianMatBridge
 
@@ -183,8 +181,7 @@ omit [Fintype d] in
 /-- σ + εI → σ as ε → 0+. -/
 private lemma tendsto_add_eps (σ : HermitianMat d ℂ) :
     Filter.Tendsto (fun ε : ℝ ↦ σ + ε • (1 : HermitianMat d ℂ))
-      (nhdsWithin 0 (Set.Ioi 0)) (nhds σ) := by
-  exact tendsto_nhdsWithin_of_tendsto_nhds
+      (nhdsWithin 0 (Set.Ioi 0)) (nhds σ) := tendsto_nhdsWithin_of_tendsto_nhds
     (Continuous.tendsto' (by continuity) _ _ (by simp))
 
 /-! ### Helper lemmas for the core concavity proof -/
@@ -255,8 +252,7 @@ private lemma variational_eq_optimizer
   have h_exp : (X ^ p) ^ ((p - 1) / p) = X ^ (p - 1) := by
     rw [← rpow_mul hX, mul_div_cancel₀ _ (by positivity)]
   have h_inner : ⟪X, X ^ (p - 1)⟫_ℝ = (X ^ p).trace := by
-    have h_inner : ⟪X, X ^ (p - 1)⟫_ℝ = (X * (X ^ (p - 1)).mat).trace.re := by
-      exact Real.ext_cauchy rfl
+    have h_inner : ⟪X, X ^ (p - 1)⟫_ℝ = (X * (X ^ (p - 1)).mat).trace.re := Real.ext_cauchy rfl
     convert h_inner using 1
     have h_exp : (X ^ p).mat = X.mat * (X ^ (p - 1)).mat := by
       convert mat_rpow_add hX _
@@ -294,8 +290,7 @@ private lemma liebExtension_bridge [Nonempty d]
     rw [← traceRe_Φ_general]
     simp [liebExtensionTraceMap, Φ_rpow, hσ, hZ]
     rw [show star (Φ K.mat) = Φ K.mat from ?_]
-    have h_rewrite : IsSelfAdjoint (Φ K.mat) := by
-      exact Φ_isSelfAdjoint K
+    have h_rewrite : IsSelfAdjoint (Φ K.mat) := Φ_isSelfAdjoint K
     exact h_rewrite
   convert! h_joint_concave (Φ_mem_pdSet σ₁ hσ₁) (Φ_mem_pdSet σ₂ hσ₂)
     (Φ_mem_pdSet Z₁ hZ₁) (Φ_mem_pdSet Z₂ hZ₂) hθ₀ hθ₁ using 1
@@ -363,9 +358,8 @@ private lemma liebExtension_bridge_psd [Nonempty d]
   open scoped Topology in
   have h_cont : ∀ (ε : ℝ), 0 < ε → (1 - θ) * ⟪(σ₁ ^ q).conj K, (Z₁ + ε • 1) ^ r⟫_ℝ +
       θ * ⟪(σ₂ ^ q).conj K, (Z₂ + ε • 1) ^ r⟫_ℝ ≤ ⟪(((1 - θ) • σ₁ + θ • σ₂) ^ q).conj K,
-      ((1 - θ) • (Z₁ + ε • 1) + θ • (Z₂ + ε • 1)) ^ r⟫_ℝ := by
-    intro ε hε_pos
-    exact liebExtension_bridge hq hr hqr K σ₁ σ₂ (Z₁ + ε • 1) (Z₂ + ε • 1) hσ₁ hσ₂
+      ((1 - θ) • (Z₁ + ε • 1) + θ • (Z₂ + ε • 1)) ^ r⟫_ℝ := fun ε hε_pos =>
+    liebExtension_bridge hq hr hqr K σ₁ σ₂ (Z₁ + ε • 1) (Z₂ + ε • 1) hσ₁ hσ₂
       (psd_add_eps_posdef Z₁ hZ₁ ε hε_pos) (psd_add_eps_posdef Z₂ hZ₂ ε hε_pos) θ hθ₀ hθ₁
   -- Apply the continuity results to take the limit as ε approaches 0.
   have h_lim :
@@ -481,8 +475,7 @@ private lemma trace_conj_rpow_concave_pd [Nonempty d] {α : ℝ} (hα : 1 < α)
         rw [trace_lin]
         linarith [mul_le_mul_of_nonneg_left bridge hp_pos.le]
     _ ≤ (X_mix ^ p).trace := by
-        rw [hr_eq]
-        exact variational_lower_bound X_mix (a • Z₁ + b • Z₂) hX_mix hZ_mix hp_gt1
+        simpa [hr_eq] using variational_lower_bound X_mix (a • Z₁ + b • Z₂) hX_mix hZ_mix hp_gt1
 
 end VariationalAndBridge
 
diff --git a/QuantumInfo/ForMathlib/HermitianMat/LogExp.lean b/QuantumInfo/ForMathlib/HermitianMat/LogExp.lean
index 36063f7c9..6d5e0e6a0 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/LogExp.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/LogExp.lean
@@ -48,27 +48,23 @@ def exp (A : HermitianMat d 𝕜) : HermitianMat d 𝕜 :=
 /-- Primed because `Commute.exp_left` refers to `NormedSpace.exp` instead of `HermitianMat.exp`. -/
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem _root_.Commute.exp_left' (hAB : Commute A.mat B.mat) :
-    Commute (A.exp).mat B.mat := by
-  rw [exp]; commutes
+    Commute (A.exp).mat B.mat := by rw [exp]; commutes
 
 /-- Primed because `Commute.exp_right` refers to `NormedSpace.exp` instead of `HermitianMat.exp`. -/
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem _root_.Commute.exp_right' (hAB : Commute A.mat B.mat) :
-    Commute A.mat (B.exp).mat := by
-  rw [exp]; commutes
+    Commute A.mat (B.exp).mat := by rw [exp]; commutes
 
 @[simp]
 theorem reindex_exp (e : d ≃ d₂) : (A.reindex e).exp = A.exp.reindex e :=
   cfc_reindex A Real.exp e
 
 variable (A) in
-instance nonSingular_exp : NonSingular A.exp := by
-  exact cfc_nonSingular A Real.exp (fun i ↦ by positivity)
+instance nonSingular_exp : NonSingular A.exp := cfc_nonSingular A Real.exp (fun i ↦ by positivity)
 
 /-- The matrix exponential of a Hermitian matrix is nonnegative. -/
 theorem exp_nonneg (A : HermitianMat d 𝕜) : 0 ≤ A.exp := by
-  rw [exp, HermitianMat.cfc_nonneg_iff]
-  exact fun i ↦ le_of_lt (Real.exp_pos _)
+  simpa [exp, HermitianMat.cfc_nonneg_iff] using fun i ↦ le_of_lt (Real.exp_pos _)
 
 /-- The matrix exponential of a Hermitian matrix is strictly positive (Loewner order).
 Requires `Nonempty` since over an empty index type every matrix equals zero and `0 < 0`
@@ -104,25 +100,21 @@ def log (A : HermitianMat d 𝕜) : HermitianMat d 𝕜 :=
 
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem _root_.Commute.log_left (hAB : Commute A.mat B.mat) :
-    Commute (A.log).mat B.mat := by
-  rw [log]; commutes
+    Commute (A.log).mat B.mat := by rw [log]; commutes
 
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem _root_.Commute.log_right (hAB : Commute A.mat B.mat) :
-    Commute A.mat (B.log).mat := by
-  rw [log]; commutes
+    Commute A.mat (B.log).mat := by rw [log]; commutes
 
 @[simp]
 theorem reindex_log (e : d ≃ d₂) : (A.reindex e).log = A.log.reindex e :=
   cfc_reindex A Real.log e
 
 @[simp]
-theorem log_zero : (0 : HermitianMat d 𝕜).log = 0 := by
-  simp [log]
+theorem log_zero : (0 : HermitianMat d 𝕜).log = 0 := by simp [log]
 
 @[simp]
-theorem log_one : (1 : HermitianMat d 𝕜).log = 0 := by
-  simp [log]
+theorem log_one : (1 : HermitianMat d 𝕜).log = 0 := by simp [log]
 
 theorem log_smul_of_pos (A : HermitianMat d 𝕜) (hx : x ≠ 0) :
     (x • A).log = Real.log x • A.supportProj + A.log := by
@@ -131,8 +123,7 @@ theorem log_smul_of_pos (A : HermitianMat d 𝕜) (hx : x ≠ 0) :
   simp [cfc, log, supportProj_eq_cfc]
 
 theorem log_smul {A : HermitianMat d 𝕜} {x : ℝ} (hx : x ≠ 0) [NonSingular A] :
-    (x • A).log = Real.log x • 1 + A.log := by
-  simp [log_smul_of_pos A hx]
+    (x • A).log = Real.log x • 1 + A.log := by simp [log_smul_of_pos A hx]
 
 /-
 The inverse function is operator antitone for positive definite matrices.
@@ -365,9 +356,8 @@ theorem logApprox_eq_log_add_error
     convert! logApprox_eq_cfc_scalar x hx T hT using 1;
     apply cfc_congr_of_posDef hx;
     exact fun u hu => Eq.symm ( scalarLogApprox_eq u T hu.out hT );
-  have h_cfc_add : x.cfc (fun u => Real.log u + Real.log ((1 + T) / (u + T))) = x.cfc Real.log + x.cfc (fun u => Real.log ((1 + T) / (u + T))) := by
-    apply cfc_add;
-  exact h_logApprox.trans h_cfc_add
+  exact h_logApprox.trans (cfc_add (A := x) (f := Real.log)
+    (g := fun u => Real.log ((1 + T) / (u + T))))
 
 open ComplexOrder Filter Topology in
 open scoped Matrix.Norms.Frobenius in
@@ -448,8 +438,7 @@ lemma inv_convex {x y : HermitianMat d 𝕜} (hx : x.mat.PosDef) (hy : y.mat.Pos
       have h_block_pos : ∀ A : Matrix d d 𝕜, A.PosDef → (Matrix.fromBlocks A 1 1 A⁻¹).PosSemidef := by
         intro A hA
         have h_block_pos : (Matrix.fromBlocks A (1 : Matrix d d 𝕜) (1 : Matrix d d 𝕜) (A⁻¹)).PosSemidef := by
-          have h_inv_pos : A⁻¹.PosSemidef := by
-            exact hA.inv.posSemidef
+          have h_inv_pos : A⁻¹.PosSemidef := hA.inv.posSemidef
           have h_block_pos : (Matrix.fromBlocks A (1 : Matrix d d 𝕜) (1 : Matrix d d 𝕜) (A⁻¹)) = (Matrix.fromBlocks 1 0 A⁻¹ 1) * (Matrix.fromBlocks A 0 0 (A⁻¹ - A⁻¹ * A * A⁻¹)) * (Matrix.fromBlocks 1 A⁻¹ 0 1) := by
             simp [ Matrix.fromBlocks_multiply ];
             have := hA.det_pos;
@@ -629,8 +618,7 @@ lemma log_kron_diagonal {m n 𝕜 : Type*} [Fintype m] [DecidableEq m] [Fintype
     {d₁ : m → ℝ} {d₂ : n → ℝ} (h₁ : ∀ i, 0 < d₁ i) (h₂ : ∀ j, 0 < d₂ j) :
     (diagonal 𝕜 d₁ ⊗ₖ diagonal 𝕜 d₂).log =
     (diagonal 𝕜 d₁).log ⊗ₖ 1 + 1 ⊗ₖ (diagonal 𝕜 d₂).log := by
-  have h_eq : (diagonal 𝕜 d₁ ⊗ₖ diagonal 𝕜 d₂) = (diagonal 𝕜 (fun (i : m × n) => d₁ i.1 * d₂ i.2)) := by
-    exact kronecker_diagonal d₁ d₂
+  have h_eq : (diagonal 𝕜 d₁ ⊗ₖ diagonal 𝕜 d₂) = (diagonal 𝕜 (fun (i : m × n) => d₁ i.1 * d₂ i.2)) := kronecker_diagonal d₁ d₂
   convert congr_arg _ h_eq using 1;
   -- By definition of logarithm, we can rewrite the right-hand side.
   have h_rhs : (diagonal 𝕜 (fun (i : m × n) => d₁ i.1 * d₂ i.2)).log =
@@ -665,8 +653,7 @@ lemma log_conj_unitary (A : HermitianMat d 𝕜) (U : Matrix.unitaryGroup d 𝕜
 
 open RealInnerProductSpace in
 theorem inner_log_smul_of [NonSingular A] {x : ℝ} (hx : x ≠ 0) :
-    ⟪(x • A).log, B⟫ = Real.log x * B.trace + ⟪A.log, B⟫ := by
-  simp [log_smul hx, inner_add_left]
+    ⟪(x • A).log, B⟫ = Real.log x * B.trace + ⟪A.log, B⟫ := by simp [log_smul hx, inner_add_left]
 
 section kron
 
diff --git a/QuantumInfo/ForMathlib/HermitianMat/NonSingular.lean b/QuantumInfo/ForMathlib/HermitianMat/NonSingular.lean
index 94b66f777..175ed0757 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/NonSingular.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/NonSingular.lean
@@ -135,9 +135,7 @@ theorem nonSingular_iff_neg : NonSingular (-A) ↔ NonSingular A := by
   simp [nonSingular_iff_det_ne_zero, Matrix.det_neg]
 
 @[simp]
-theorem nonSingular_iff_inv : NonSingular (A⁻¹) ↔ NonSingular A := by
-  rw [nonsingular_iff_isUnit, nonsingular_iff_isUnit]
-  exact Matrix.isUnit_nonsing_inv_iff
+theorem nonSingular_iff_inv : NonSingular (A⁻¹) ↔ NonSingular A := by simp [nonsingular_iff_isUnit]
 
 @[simp]
 theorem nonSingular_iff_kronecker [Nonempty n] [Nonempty m] :
@@ -163,22 +161,17 @@ theorem nonSingular_empty [IsEmpty n] : NonSingular A := by
 
 variable [NonSingular A] [NonSingular B]
 
-theorem nonSingular_det_ne_zero : A.mat.det ≠ 0 := by
-  rwa [← nonSingular_iff_det_ne_zero]
+theorem nonSingular_det_ne_zero : A.mat.det ≠ 0 := by rwa [← nonSingular_iff_det_ne_zero]
 
 @[simp]
-theorem nonSingular_ker_bot : A.ker = ⊥ := by
-  rwa [← nonSingular_iff_ker_bot]
+theorem nonSingular_ker_bot : A.ker = ⊥ := by rwa [← nonSingular_iff_ker_bot]
 
 @[simp]
-theorem nonSingular_support_top : A.support = ⊤ := by
-  rwa [← nonSingular_iff_support_top]
+theorem nonSingular_support_top : A.support = ⊤ := by rwa [← nonSingular_iff_support_top]
 
-instance nonSingular_neg : NonSingular (-A) := by
-  rwa [nonSingular_iff_neg]
+instance nonSingular_neg : NonSingular (-A) := by rwa [nonSingular_iff_neg]
 
-instance nonSingular_inv : NonSingular (A⁻¹) := by
-  rwa [nonSingular_iff_inv]
+instance nonSingular_inv : NonSingular (A⁻¹) := by rwa [nonSingular_iff_inv]
 
 instance nonSingular_kron [Nonempty n] [Nonempty m] : NonSingular (A ⊗ₖ B) :=
   nonSingular_iff_kronecker.mpr ⟨inferInstance, inferInstance⟩
@@ -187,8 +180,7 @@ theorem nonSingular_conj (hC : IsUnit C) : NonSingular (A.conj C) := by
   rwa [nonSingular_iff_conj hC]
 
 instance nonSingular_conj_isometry {B : HermitianMat n 𝕜} [NonSingular B] :
-    NonSingular (A.conj B.mat) := by
-  simpa [nonSingular_iff_conj]
+    NonSingular (A.conj B.mat) := by simpa [nonSingular_iff_conj]
 
 theorem nonSingular_zero_notMem_spectrum : 0 ∉ spectrum ℝ A.mat := by
   rwa [← nonSingular_iff_zero_notMem_spectrum]
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Order.lean b/QuantumInfo/ForMathlib/HermitianMat/Order.lean
index 5f31efe28..12e1d4a03 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Order.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Order.lean
@@ -31,21 +31,17 @@ instance : IsOrderedAddMonoid (HermitianMat n 𝕜) :=
   inferInstanceAs (IsOrderedAddMonoid (selfAdjoint _))
 
 omit [Fintype n] in
-theorem le_iff : A ≤ B ↔ (B - A).mat.PosSemidef := by
-  rfl
+theorem le_iff : A ≤ B ↔ (B - A).mat.PosSemidef := Iff.rfl
 
 omit [Fintype n] in
-theorem zero_le_iff : 0 ≤ A ↔ A.mat.PosSemidef := by
-  rw [le_iff, sub_zero]
+theorem zero_le_iff : 0 ≤ A ↔ A.mat.PosSemidef := by rw [le_iff, sub_zero]
 
 theorem le_iff_mulVec_le : A ≤ B ↔
     ∀ x, star x ⬝ᵥ A.mat *ᵥ x ≤ star x ⬝ᵥ B.mat *ᵥ x := by
   simp [le_iff, Matrix.posSemidef_iff_dotProduct_mulVec, B.H.sub A.H, Matrix.sub_mulVec]
 
 instance [DecidableEq n] : ZeroLEOneClass (HermitianMat n 𝕜) where
-  zero_le_one := by
-    rw [zero_le_iff]
-    exact Matrix.PosSemidef.one
+  zero_le_one := by simpa [zero_le_iff] using Matrix.PosSemidef.one
 
 omit [Fintype n] in
 theorem lt_iff_posdef : A < B ↔ (B - A).mat.PosSemidef ∧ A ≠ B :=
@@ -54,8 +50,7 @@ theorem lt_iff_posdef : A < B ↔ (B - A).mat.PosSemidef ∧ A ≠ B :=
 instance : IsStrictOrderedModule ℝ (HermitianMat n 𝕜) where
   smul_lt_smul_of_pos_left a ha b b₂ hb := by
     rw [HermitianMat.lt_iff_posdef] at hb ⊢
-    simp only [← smul_sub, ne_eq, smul_right_inj ha.ne']
-    exact ⟨hb.left.smul ha.le, hb.right⟩
+    simpa only [← smul_sub, ne_eq, smul_right_inj ha.ne'] using ⟨hb.left.smul ha.le, hb.right⟩
   smul_lt_smul_of_pos_right a ha b b2 hb := by
     rw [HermitianMat.lt_iff_posdef] at ha ⊢
     rw [sub_zero] at ha
@@ -75,8 +70,7 @@ theorem posSemidef_iff_spectrum_Ici [DecidableEq n] (A : HermitianMat n 𝕜) :
   simp [A.H, Set.Ici.eq_1]
 
 theorem posSemidef_iff_spectrum_nonneg [DecidableEq n] (A : HermitianMat n 𝕜) :
-    0 ≤ A ↔ ∀ x ∈ spectrum ℝ A.mat, 0 ≤ x := by
-  exact A.posSemidef_iff_spectrum_Ici
+    0 ≤ A ↔ ∀ x ∈ spectrum ℝ A.mat, 0 ≤ x := A.posSemidef_iff_spectrum_Ici
 
 theorem trace_nonneg (hA : 0 ≤ A) : 0 ≤ A.trace := by
   exact (RCLike.nonneg_iff.mp (zero_le_iff.mp hA).trace_nonneg).1
@@ -136,8 +130,7 @@ theorem kronecker_self_mono (hA : 0 ≤ A) (hB : 0 ≤ B) (hAB : A ≤ B) :
       (HermitianMat.kronecker_add (A := A) (B := B) (C := -A)).symm
   have hEq : B ⊗ₖ B - A ⊗ₖ A = A ⊗ₖ (B - A) + (B - A) ⊗ₖ B := by
     calc
-      B ⊗ₖ B - A ⊗ₖ A = (A + (B - A)) ⊗ₖ B - A ⊗ₖ A := by
-        rw [show A + (B - A) = B by abel]
+      B ⊗ₖ B - A ⊗ₖ A = (A + (B - A)) ⊗ₖ B - A ⊗ₖ A := by rw [show A + (B - A) = B by abel]
       _ = (A ⊗ₖ B + (B - A) ⊗ₖ B) - A ⊗ₖ A := by rw [HermitianMat.add_kronecker]
       _ = (A ⊗ₖ B + -(A ⊗ₖ A)) + (B - A) ⊗ₖ B := by abel
       _ = A ⊗ₖ (B - A) + (B - A) ⊗ₖ B := by rw [hAC]
@@ -157,8 +150,7 @@ theorem kronecker_pos {A : HermitianMat m 𝕜} (hA : 0 < A) (hB : 0 < B) : 0 <
 
 omit [Fintype n] in
 open MatrixOrder in
-theorem posSemidef_to_nonneg {A : Matrix n n 𝕜} (hA : A.PosSemidef) : 0 ≤ A := by
-  exact hA.nonneg
+theorem posSemidef_to_nonneg {A : Matrix n n 𝕜} (hA : A.PosSemidef) : 0 ≤ A := hA.nonneg
 
 open MatrixOrder in
 theorem posDef_to_pos {A : Matrix n n 𝕜} (hA : A.PosDef) [Nonempty n] : 0 < A := by
@@ -237,8 +229,7 @@ omit [Fintype n] in
 theorem mat_posSemidef_to_nonneg (hA : A.mat.PosSemidef) : 0 ≤ A :=
   zero_le_iff.mpr hA
 
-theorem mat_posDef_to_pos [Nonempty n] (hA : A.mat.PosDef) : 0 < A := by
-  exact posDef_to_pos hA
+theorem mat_posDef_to_pos [Nonempty n] (hA : A.mat.PosDef) : 0 < A := posDef_to_pos hA
 
 open Lean Meta in
 /-- Given an expression `e` (a `HermitianMat`) and a proof expression `p` whose type may be
@@ -347,27 +338,21 @@ meta def evalHermitianMatConj : PositivityExt where eval {_u _α} _zα _pα e :=
   pure (.nonnegative (← mkAppM ``HermitianMat.conj_nonneg #[M, pfNonneg]))
 
 open MatrixOrder in
-example {A : Matrix n n ℂ} (hA : A.PosSemidef) : 0 ≤ A := by
-  positivity
+example {A : Matrix n n ℂ} (hA : A.PosSemidef) : 0 ≤ A := by positivity
 
 open MatrixOrder in
-example {A : Matrix n n ℂ} [Nonempty n] (hA : A.PosDef) : 0 < A := by
-  positivity
+example {A : Matrix n n ℂ} [Nonempty n] (hA : A.PosDef) : 0 < A := by positivity
 
-example (hA : A.mat.PosSemidef) : 0 ≤ A := by
-  positivity
+example (hA : A.mat.PosSemidef) : 0 ≤ A := by positivity
 
-example [Nonempty n] (hA : A.mat.PosDef) : 0 < A := by
-  positivity
+example [Nonempty n] (hA : A.mat.PosDef) : 0 < A := by positivity
 
 example [DecidableEq n] [DecidableEq m] [Nonempty n] [Nonempty m]
   (A B : HermitianMat n ℂ) (hA : 0 ≤ A) (hB : 0 ≤ B) (M : Matrix m n ℂ) :
-    0 < (2 : HermitianMat (n × m) ℂ) + (3 • A) ⊗ₖ (Real.pi • B).conj M := by
-  positivity
+    0 < (2 : HermitianMat (n × m) ℂ) + (3 • A) ⊗ₖ (Real.pi • B).conj M := by positivity
 
 example (A B : HermitianMat n ℂ) (hA : 0 < A) (hB : 0 < B) :
-    0 < ((37 • A) ⊗ₖ ((38 : ℝ) • B)).trace := by
-  positivity
+    0 < ((37 • A) ⊗ₖ ((38 : ℝ) • B)).trace := by positivity
 
 omit [Fintype n] in
 theorem convex_cone (hA : 0 ≤ A) (hB : 0 ≤ B) {c₁ c₂ : ℝ} (hc₁ : 0 ≤ c₁) (hc₂ : 0 ≤ c₂) :
@@ -497,8 +482,7 @@ theorem ker_conj [DecidableEq n] (hA : 0 ≤ A) (B : Matrix n n 𝕜) :
     · exact Eq.symm (LinearMap.map_zero (Matrix.toEuclideanLin B))
 
 theorem ker_le_of_le_smul {α : ℝ} [DecidableEq n] (hα : α ≠ 0) (hA : 0 ≤ A) (hAB : A ≤ α • B) : B.ker ≤ A.ker := by
-  rw [← ker_pos_smul B hα]
-  exact ker_antitone hA hAB
+  simpa [← ker_pos_smul B hα] using ker_antitone hA hAB
 
 /-- If a Hermitian matrix is bounded by `M * I`, then all its eigenvalues are at most `M`. -/
 theorem le_smul_one_imp_eigenvalues_le [DecidableEq n] (A : HermitianMat n ℂ) (M : ℝ)
@@ -525,9 +509,8 @@ open MatrixOrder in
 /-- If all eigenvalues of a Hermitian matrix are at most `M`, then it is bounded by `M * I`. -/
 theorem eigenvalues_le_imp_le_smul_one [DecidableEq n] (A : HermitianMat n ℂ) (M : ℝ)
     (h : ∀ i, A.H.eigenvalues i ≤ M) :
-    A ≤ M • (1 : HermitianMat n ℂ) := by
-  exact
-    (Matrix.PosSemidef.le_smul_one_of_eigenvalues_iff A.H M).mp h
+    A ≤ M • (1 : HermitianMat n ℂ) :=
+  (Matrix.PosSemidef.le_smul_one_of_eigenvalues_iff A.H M).mp h
 
 --TODO: Positivity extensions for traceLeft, traceRight, rpow, nat powers, inverse function,
 -- the various `proj` function (in Proj.lean), and the inner product.
@@ -651,8 +634,7 @@ variable [DecidableEq n] [DecidableEq m]
 open MatrixOrder
 
 -- Test: eigenvalues nonneg from PSD HermitianMat
-example (A : HermitianMat n ℂ) (hA : 0 < A) (i : n) : 0 ≤ A.H.eigenvalues i := by
-  positivity
+example (A : HermitianMat n ℂ) (hA : 0 < A) (i : n) : 0 ≤ A.H.eigenvalues i := by positivity
 
 -- Test: A.mat nonneg from A nonneg
 example (A : HermitianMat n ℂ) (hA : 0 ≤ A) : 0 ≤ A.mat := by positivity
@@ -677,8 +659,7 @@ example (M : Matrix n m ℝ) :
   positivity
 
 example (M : Matrix n n ℂ) (i : n) (A : HermitianMat n ℂ) (hA : 0 ≤ A) :
-    0 ≤ (A + ⟨_, M.isHermitian_mul_conjTranspose_self⟩ + 0).H.eigenvalues i := by
-  positivity
+    0 ≤ (A + ⟨_, M.isHermitian_mul_conjTranspose_self⟩ + 0).H.eigenvalues i := by positivity
 
 end tests
 end MatrixPositivity
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Peierls.lean b/QuantumInfo/ForMathlib/HermitianMat/Peierls.lean
index 5db4fab2e..6ad270653 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Peierls.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Peierls.lean
@@ -116,8 +116,7 @@ theorem trace_function_convex_univ (g : ℝ → ℝ) (hg : ConvexOn ℝ Set.univ
   -- Let $C = aA + bB$.
   set C : HermitianMat d ℂ := a • A + b • B
   -- By the properties of the trace and the convexity of $g$, we have:
-  have h_trace : (C.cfc g).trace = ∑ i, g (C.H.eigenvalues i) := by
-    exact trace_cfc_eq C g
+  have h_trace : (C.cfc g).trace = ∑ i, g (C.H.eigenvalues i) := trace_cfc_eq C g
   have h_sum : ∑ i, g (C.H.eigenvalues i) ≤
       a * ∑ i, g ((A.conj (star C.H.eigenvectorUnitary.val)).mat i i).re +
       b * ∑ i, g ((B.conj (star C.H.eigenvectorUnitary.val)).mat i i |> Complex.re) := by
@@ -174,8 +173,7 @@ theorem trace_function_convex_ici {g : ℝ → ℝ} (hg : ConvexOn ℝ (Set.Ici
   -- Let $C = aA + bB$.
   set C : HermitianMat d ℂ := a • A + b • B
   -- By the properties of the trace and the convexity of $g$, we have:
-  have h_trace : (C.cfc g).trace = ∑ i, g (C.H.eigenvalues i) := by
-    exact trace_cfc_eq C g
+  have h_trace : (C.cfc g).trace = ∑ i, g (C.H.eigenvalues i) := trace_cfc_eq C g
   have h_sum : ∑ i, g (C.H.eigenvalues i) ≤
       a * ∑ i, g ((A.conj (star C.H.eigenvectorUnitary.val)).mat i i).re +
       b * ∑ i, g ((B.conj (star C.H.eigenvectorUnitary.val)).mat i i).re := by
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Proj.lean b/QuantumInfo/ForMathlib/HermitianMat/Proj.lean
index 486d825c3..08a367255 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Proj.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Proj.lean
@@ -230,11 +230,9 @@ scoped notation "{" A " ≤ₚ " B "}" => projLE A B
 scoped notation "{" A " >ₚ " B "}" => projLT B A
 scoped notation "{" A " <ₚ " B "}" => projLT A B
 
-theorem projLE_def : {A ≤ₚ B} = (B - A).cfc (fun x ↦ if 0 ≤ x then 1 else 0) := by
-  rfl
+theorem projLE_def : {A ≤ₚ B} = (B - A).cfc (fun x ↦ if 0 ≤ x then 1 else 0) := rfl
 
-theorem projLT_def : {A <ₚ B} = (B - A).cfc (fun x ↦ if 0 < x then 1 else 0) := by
-  rfl
+theorem projLT_def : {A <ₚ B} = (B - A).cfc (fun x ↦ if 0 < x then 1 else 0) := rfl
 
 theorem projLE_sq : {A ≤ₚ B}^2 = {A ≤ₚ B} := by
   rw [projLE_def, ← cfc_pow, ← cfc_comp]
@@ -297,8 +295,7 @@ theorem projLE_mul_nonneg : 0 ≤ {A ≤ₚ B}.mat * (B - A).mat := by
 
 open MatrixOrder in
 theorem projLE_mul_le : {A ≤ₚ B}.mat * A.mat ≤ {A ≤ₚ B}.mat * B.mat := by
-  rw [← sub_nonneg, ← mul_sub_left_distrib]
-  exact projLE_mul_nonneg A B
+  simpa [← sub_nonneg, ← mul_sub_left_distrib] using projLE_mul_nonneg A B
 
 @[simp]
 theorem proj_le_add_lt : {A <ₚ B} + {B ≤ₚ A} = 1 := by
@@ -334,11 +331,9 @@ instance : PosPart (HermitianMat n 𝕜) where
 instance : NegPart (HermitianMat n 𝕜) where
   negPart A := A.cfc (fun x ↦ -x ⊔ 0)
 
-theorem posPart_eq_cfc_max : A⁺ = A.cfc (fun x ↦ x ⊔ 0) := by
-  rfl
+theorem posPart_eq_cfc_max : A⁺ = A.cfc (fun x ↦ x ⊔ 0) := rfl
 
-theorem negPart_eq_cfc_min : A⁻ = A.cfc (fun x ↦ -x ⊔ 0) := by
-  rfl
+theorem negPart_eq_cfc_min : A⁻ = A.cfc (fun x ↦ -x ⊔ 0) := rfl
 
 theorem posPart_eq_cfc_ite : A⁺ = A.cfc (fun x ↦ if 0 ≤ x then x else 0) := by
   simp only [← max_def', posPart_eq_cfc_max]
@@ -409,8 +404,7 @@ theorem projLE_inner_nonneg  : 0 ≤ ⟪{A ≤ₚ B}, (B - A)⟫ :=
   inner_mul_nonneg (projLE_mul_nonneg A B)
 
 theorem projLE_inner_le : ⟪{A ≤ₚ B}, A⟫ ≤ ⟪{A ≤ₚ B}, B⟫ := by
-  rw [← sub_nonneg, ← inner_sub_right]
-  exact projLE_inner_nonneg A B
+  simpa [← sub_nonneg, ← inner_sub_right] using projLE_inner_nonneg A B
 
 open RealInnerProductSpace in
 theorem inner_projLE_nonneg : 0 ≤ ⟪{A ≤ₚ B}, (B - A)⟫ :=
@@ -447,8 +441,7 @@ theorem negPart_Continuous : Continuous (·⁻ : HermitianMat n ℂ → _) := by
 
 -- variable {d : Type*} [Fintype d] [DecidableEq d] (A B : HermitianMat d ℂ)
 
-theorem one_sub_projLT : 1 - {B ≤ₚ A} = {A <ₚ B} := by
-  rw [sub_eq_iff_eq_add, proj_le_add_lt]
+theorem one_sub_projLT : 1 - {B ≤ₚ A} = {A <ₚ B} := by rw [sub_eq_iff_eq_add, proj_le_add_lt]
 
 open MatrixOrder ComplexOrder in
 theorem projLT_mul_nonneg : 0 ≤ {A <ₚ B}.mat * (B - A).mat := by
@@ -462,8 +455,7 @@ theorem projLT_mul_nonneg : 0 ≤ {A <ₚ B}.mat * (B - A).mat := by
 
 open MatrixOrder ComplexOrder in
 theorem proj_lt_mul_lt : {A <ₚ B}.mat * A.mat ≤ {A <ₚ B}.mat * B.mat := by
-  rw [← sub_nonneg, ← mul_sub_left_distrib]
-  exact A.projLT_mul_nonneg B
+  simpa [← sub_nonneg, ← mul_sub_left_distrib] using A.projLT_mul_nonneg B
 
 theorem inner_negPart_nonpos : ⟪A, A⁻⟫ ≤ 0 := by
   rw [← neg_le_neg_iff, neg_zero, ← inner_neg_right]
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Reindex.lean b/QuantumInfo/ForMathlib/HermitianMat/Reindex.lean
index 9cc20aea9..4e816a204 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Reindex.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Reindex.lean
@@ -28,8 +28,7 @@ def reindex (e : d ≃ d₂) : HermitianMat d₂ 𝕜 :=
   ⟨A.mat.reindex e e, A.H.submatrix e.symm⟩
 
 @[simp]
-theorem mat_reindex : (A.reindex e).mat = A.mat.reindex e e := by
-  rfl
+theorem mat_reindex : (A.reindex e).mat = A.mat.reindex e e := rfl
 
 /-! Our simp-normal form for expressions involving `HermitianMat.reindex` is that we try to push
 the reindexing as far out as possible, so that it can be absorbed by `HermitianMat.trace`, or
@@ -38,8 +37,7 @@ we push it to the right side. One downside is that we're not as likely to hit `r
 
 @[simp]
 theorem reindex_refl (A : HermitianMat d 𝕜) :
-    A.reindex (.refl _) = A := by
-  rfl
+    A.reindex (.refl _) = A := rfl
 
 @[simp]
 theorem reindex_reindex (A : HermitianMat d 𝕜) (e : d ≃ d₂) (f : d₂ ≃ d₃) :
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Rpow.lean b/QuantumInfo/ForMathlib/HermitianMat/Rpow.lean
index 8cea291bd..69b3c1179 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Rpow.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Rpow.lean
@@ -52,11 +52,9 @@ theorem rpow_conj_unitary (A : HermitianMat d 𝕜) (U : Matrix.unitaryGroup d 
     (HermitianMat.conj U.val A) ^ r = HermitianMat.conj U.val (A ^ r) := by
   exact A.cfc_conj_unitary (· ^ r) U
 
-theorem pow_eq_rpow : A ^ r = A.rpow r :=
-  rfl
+theorem pow_eq_rpow : A ^ r = A.rpow r := rfl
 
-theorem rpow_eq_cfc : A ^ r = A.cfc (· ^ r) :=
-  rfl
+theorem rpow_eq_cfc : A ^ r = A.cfc (· ^ r) := rfl
 
 theorem diagonal_pow (f : d → ℝ) :
     (diagonal 𝕜 f) ^ r = diagonal 𝕜 (fun i ↦ (f i) ^ r) := by
@@ -93,8 +91,7 @@ theorem continuousOn_rpow_neg (A : HermitianMat d ℂ) : ContinuousOn (fun x : 
   exact (Real.continuousAt_const_rpow' hx.ne).continuousWithinAt
 
 @[simp]
-theorem rpow_one : A ^ (1 : ℝ) = A := by
-  simp [rpow_eq_cfc]
+theorem rpow_one : A ^ (1 : ℝ) = A := by simp [rpow_eq_cfc]
 
 /--
 Functional calculus of Real.sqrt is equal to functional calculus of x^(1/2).
@@ -114,12 +111,10 @@ theorem one_rpow : (1 : HermitianMat d 𝕜) ^ r = 1 := by
     simp
 
 @[simp]
-lemma rpow_zero (A : HermitianMat d 𝕜) : A ^ (0 : ℝ) = 1 := by
-  simp [rpow_eq_cfc]
+lemma rpow_zero (A : HermitianMat d 𝕜) : A ^ (0 : ℝ) = 1 := by simp [rpow_eq_cfc]
 
 lemma rpow_diagonal (a : d → ℝ) (r : ℝ) :
-  (diagonal ℂ a) ^ r = diagonal ℂ (fun i => a i ^ r) := by
-    exact cfc_diagonal _ _
+  (diagonal ℂ a) ^ r = diagonal ℂ (fun i => a i ^ r) := cfc_diagonal _ _
 
 /-- Keeps in line with our simp-normal form for moving reindex outwards. -/
 @[simp]
@@ -154,9 +149,8 @@ theorem rpow_pos {A : HermitianMat d 𝕜} (hA : 0 < A) {p : ℝ} : 0 < A ^ p :=
   · exact fun i hi => Real.rpow_pos_of_pos hi _
   · rcases eq_or_ne p 0 with h | h <;> simp [h]
 
-theorem rpow_nonneg (hA : 0 ≤ A) {p : ℝ} : 0 ≤ A ^ p := by
-  apply cfc_nonneg_of_nonneg hA
-  exact fun i hi => Real.rpow_nonneg hi p
+theorem rpow_nonneg (hA : 0 ≤ A) {p : ℝ} : 0 ≤ A ^ p :=
+  cfc_nonneg_of_nonneg hA fun _ hi => Real.rpow_nonneg hi p
 
 open ComplexOrder in
 theorem inv_eq_rpow_neg_one (hA : A.mat.PosDef) : A⁻¹ = A ^ (-1 : ℝ) := by
@@ -169,10 +163,8 @@ theorem sandwich_self (hB : B.mat.PosDef) :
     (B.conj (B ^ (-1/2 : ℝ)).mat) = 1 := by
   have hB_inv_sqrt : (B ^ (-1 / 2 : ℝ)).mat * (B ^ (-1 / 2 : ℝ)).mat = (B ^ (-1 : ℝ)).mat := by
     rw [ ← mat_rpow_add ] <;> norm_num
-    rw [zero_le_iff]
-    exact hB.posSemidef
-  have hB_inv : (B ^ (-1 : ℝ)).mat = B.mat⁻¹ := by
-    rw [← inv_eq_rpow_neg_one hB, mat_inv]
+    simpa [zero_le_iff] using hB.posSemidef
+  have hB_inv : (B ^ (-1 : ℝ)).mat = B.mat⁻¹ := by rw [← inv_eq_rpow_neg_one hB, mat_inv]
   rw [ hB_inv ] at hB_inv_sqrt;
   ext1
   simp [mul_assoc];
@@ -186,10 +178,8 @@ lemma rpow_inv_eq_neg_rpow (hA : A.mat.PosDef) (p : ℝ) : (A ^ p)⁻¹ = A ^ (-
   ext i j;
   have h_inv : (A ^ p).mat * (A ^ (-p)).mat = 1 := by
     have h_inv : (A ^ p).mat * (A ^ (-p)).mat = 1 := by
-      have h_pow : (A ^ p).mat = A.cfc (fun x => x ^ p) := by
-        exact rfl
-      have h_pow_neg : (A ^ (-p)).mat = A.cfc (fun x => x ^ (-p)) := by
-        exact rfl
+      have h_pow : (A ^ p).mat = A.cfc (fun x => x ^ p) := rfl
+      have h_pow_neg : (A ^ (-p)).mat = A.cfc (fun x => x ^ (-p)) := rfl
       have h_inv : (A ^ p).mat * (A ^ (-p)).mat = A.cfc (fun x => x ^ p * x ^ (-p)) := by
         rw [ h_pow, h_pow_neg, ← mat_cfc_mul ];
         rfl;
@@ -200,14 +190,12 @@ lemma rpow_inv_eq_neg_rpow (hA : A.mat.PosDef) (p : ℝ) : (A ^ p)⁻¹ = A ^ (-
       rw [ h_inv, cfc_const ] ; norm_num;
     exact h_inv;
   -- By definition of matrix inverse, if $(A^p) * (A^{-p}) = 1$, then $(A^{-p})$ is the inverse of $(A^p)$.
-  have h_inv_def : (A ^ p).mat⁻¹ = (A ^ (-p)).mat := by
-    exact Matrix.inv_eq_right_inv h_inv;
+  have h_inv_def : (A ^ p).mat⁻¹ = (A ^ (-p)).mat := Matrix.inv_eq_right_inv h_inv
   convert! congr_fun ( congr_fun h_inv_def i ) j using 1
 
 open ComplexOrder in
 lemma sandwich_le_one (hB : B.mat.PosDef) (h : A ≤ B) :
-    (A.conj (B ^ (-1/2 : ℝ)).mat) ≤ 1 := by
-  exact le_trans (conj_mono h) (sandwich_self hB).le
+    (A.conj (B ^ (-1/2 : ℝ)).mat) ≤ 1 := le_trans (conj_mono h) (sandwich_self hB).le
 
 open ComplexOrder in
 lemma rpow_neg_mul_rpow_self (hA : A.mat.PosDef) (p : ℝ) :
@@ -232,13 +220,11 @@ lemma rpow_neg_mul_rpow_self (hA : A.mat.PosDef) (p : ℝ) :
 open ComplexOrder in
 lemma isUnit_rpow_toMat (hA : A.mat.PosDef) (p : ℝ) : IsUnit (A ^ p).mat := by
   have hA_inv : IsUnit (A ^ (-p)).mat := by
-    have hA_inv : (A ^ (-p)).mat * (A ^ p).mat = 1 := by
-      exact rpow_neg_mul_rpow_self hA p
+    have hA_inv : (A ^ (-p)).mat * (A ^ p).mat = 1 := rpow_neg_mul_rpow_self hA p
     exact IsUnit.of_mul_eq_one _ hA_inv
   -- Since $(A^{-p}) (A^p) = 1$, we have that $(A^p)$ is the inverse of $(A^{-p})$.
   have hA_inv : (A ^ p).mat = (A ^ (-p)).mat⁻¹ := by
-    have hA_inv : (A ^ (-p)).mat * (A ^ p).mat = 1 := by
-      exact rpow_neg_mul_rpow_self hA p;
+    have hA_inv : (A ^ (-p)).mat * (A ^ p).mat = 1 := rpow_neg_mul_rpow_self hA p
     exact Eq.symm (Matrix.inv_eq_right_inv hA_inv);
   aesop
 
@@ -364,8 +350,7 @@ theorem cfc_sq_rpow_eq_cfc_rpow
 
 /-- Tr[A^p] = ∑ᵢ λᵢ^p for a Hermitian matrix A. -/
 lemma trace_rpow_eq_sum (A : HermitianMat d ℂ) (p : ℝ) :
-    (A ^ p).trace = ∑ i, (A.H.eigenvalues i) ^ p := by
-  exact A.trace_cfc_eq (· ^ p)
+    (A ^ p).trace = ∑ i, (A.H.eigenvalues i) ^ p := A.trace_cfc_eq (· ^ p)
 
 /-! ## Loewner-Heinz Theorem
 The operator monotonicity of `x ↦ x ^ q` for `0 < q ≤ 1`:
@@ -447,8 +432,7 @@ theorem rpowApprox_eq_cfc_scalar (A : HermitianMat d ℂ) (hA : A.mat.PosDef) (q
               exact fun {f} hf => inv_cfc_eq_cfc_inv f hf
             apply h_inv_smul
             intro i
-            have h_eigenvalue_pos : 0 < A.H.eigenvalues i := by
-              exact Matrix.PosDef.eigenvalues_pos hA i
+            have h_eigenvalue_pos : 0 < A.H.eigenvalues i := Matrix.PosDef.eigenvalues_pos hA i
             exact ne_of_gt (add_pos h_eigenvalue_pos ht.left);
           rw [h_inv_def, h_inv_comp];
         exact h_inv
@@ -461,8 +445,7 @@ theorem rpowApprox_eq_cfc_scalar (A : HermitianMat d ℂ) (hA : A.mat.PosDef) (q
     have h_integrable : ∀ u : d, IntervalIntegrable (fun t : ℝ => t ^ q * (1 / (1 + t) - 1 / (A.H.eigenvalues u + t))) volume 0 T := by
       intro u
       have h_integrable : IntervalIntegrable (fun t : ℝ => t ^ q * (1 / (1 + t) - 1 / (A.H.eigenvalues u + t))) volume 0 T := by
-        have h_pos : 0 < A.H.eigenvalues u := by
-          exact Matrix.PosDef.eigenvalues_pos hA u
+        have h_pos : 0 < A.H.eigenvalues u := Matrix.PosDef.eigenvalues_pos hA u
         exact ContinuousOn.intervalIntegrable ( by exact ContinuousOn.mul ( continuousOn_id.rpow_const fun x hx => Or.inr <| by linarith ) <| ContinuousOn.sub ( continuousOn_const.div ( continuousOn_const.add continuousOn_id ) fun x hx => by linarith [ Set.mem_Icc.mp <| by simpa [ hT.le ] using hx ] ) ( continuousOn_const.div ( continuousOn_const.add continuousOn_id ) fun x hx => by linarith [ Set.mem_Icc.mp <| by simpa [ hT.le ] using hx ] ) ) ..;
       exact h_integrable
     exact integral_cfc_eq_cfc_integral _ _ _ h_integrable
@@ -614,8 +597,7 @@ theorem rpow_le_rpow_of_posDef (hA : A.mat.PosDef) (hAB : A ≤ B)
       convert le_of_tendsto_of_tendsto ( tendsto_rpowApprox hA hq ( lt_of_le_of_ne hq1 hq_eq_one ) ) ( tendsto_rpowApprox ( posDef_of_posDef_le hA hAB ) hq ( lt_of_le_of_ne hq1 hq_eq_one ) ) _ using 1
       generalize_proofs at *; (
       filter_upwards [ Filter.eventually_gt_atTop 0 ] with T hT using rpowApprox_mono hA ( posDef_of_posDef_le hA hAB ) hAB hq.le T hT |> le_trans <| by aesop;);
-    have h_rpow_pos : 0 < rpowConst q := by
-      exact rpowConst_pos hq ( lt_of_le_of_ne hq1 hq_eq_one );
+    have h_rpow_pos : 0 < rpowConst q := rpowConst_pos hq ( lt_of_le_of_ne hq1 hq_eq_one )
     simp_all
 
 open ComplexOrder Filter in
@@ -639,16 +621,13 @@ theorem rpow_le_rpow_of_le (hA : 0 ≤ A) (hAB : A ≤ B)
           ring_nf
           simp [ Matrix.mulVec, dotProduct, Finset.mul_sum _ _ _, mul_assoc, mul_left_comm];
           simp [ Matrix.one_apply]
-        have h_inner_nonneg : 0 ≤ star x ⬝ᵥ A.mat.mulVec x := by
-          exact inner_mulVec_nonneg hA x
-        have h_inner_pos : 0 < star x ⬝ᵥ x := by
-          simp_all
+        have h_inner_nonneg : 0 ≤ star x ⬝ᵥ A.mat.mulVec x := inner_mulVec_nonneg hA x
+        have h_inner_pos : 0 < star x ⬝ᵥ x := by simp_all
         exact h_inner.symm ▸ add_pos_of_nonneg_of_pos h_inner_nonneg ( mul_pos ( mod_cast hε_pos ) ( mod_cast h_inner_pos ) ) |> lt_of_lt_of_le <| le_rfl;
     have h_pos_def_Bε : (Bε ε).mat.PosDef := by
       convert posDef_of_posDef_le h_pos_def_Aε _ using 1
       exact add_le_add_left hAB _ |> le_trans ( by simp [ Aε ] ) ;
-    have h_le_Aε_Bε : Aε ε ≤ Bε ε := by
-      exact add_le_add_left hAB _ |> le_trans <| by simp [ Bε ] ;
+    have h_le_Aε_Bε : Aε ε ≤ Bε ε := add_le_add_left hAB _ |> le_trans <| by simp [ Bε ]
     exact ⟨h_pos_def_Aε, h_pos_def_Bε, h_le_Aε_Bε⟩
   -- By the continuity of the function $M \mapsto M^q$, we have $(Aε ε)^q \to A^q$ and $(Bε ε)^q \to B^q$ as $\epsilon \to 0^+$.
   have h_cont : Filter.Tendsto (fun ε => (Aε ε) ^ q) (nhdsWithin 0 (Set.Ioi 0)) (nhds (A ^ q)) ∧ Filter.Tendsto (fun ε => (Bε ε) ^ q) (nhdsWithin 0 (Set.Ioi 0)) (nhds (B ^ q)) := by
@@ -713,8 +692,7 @@ private lemma conj_rpow_le_one_of_conj_le_one_posDef
     have hconj := HermitianMat.conj_mono (M := (B ^ (-1 : ℝ)).mat) hAB
     have hleft : (A.conj B.mat).conj (B ^ (-1 : ℝ)).mat = A := by
       rw [HermitianMat.conj_conj]
-      have hmul : (B ^ (-1 : ℝ)).mat * B.mat = 1 := by
-        simpa using rpow_neg_mul_rpow_self hB 1
+      have hmul : (B ^ (-1 : ℝ)).mat * B.mat = 1 := by simpa using rpow_neg_mul_rpow_self hB 1
       exact (HermitianMat.conj_one (A := A)).symm ▸ by simp [hmul]
     have hright : (1 : HermitianMat d ℂ).conj (B ^ (-1 : ℝ)).mat = B ^ (-2 : ℝ) := by
       ext1
@@ -766,8 +744,7 @@ private lemma top_singular_le_of_self_mul_le_smul_one
     (hcard : 0 < Fintype.card e) :
     singularValuesSorted X ⟨0, hcard⟩ ≤ Real.sqrt α := by
   letI : Nonempty e := Fintype.card_pos_iff.mp hcard
-  let hne : (Finset.univ : Finset e).Nonempty := by
-    simp
+  let hne : (Finset.univ : Finset e).Nonempty := by simp
   rw [singularValuesSorted_zero_eq_sup X hcard]
   refine Finset.sup'_le _ _ ?_
   intro i hi
@@ -847,8 +824,7 @@ private lemma compound_top_singular_le_posDef
       have hconj := congrArg (fun H : HermitianMat _ ℂ => H.conj (Bk ^ (-1 : ℝ)).mat) hMk_eq_zero
       have hleft : (Ak.conj Bk.mat).conj (Bk ^ (-1 : ℝ)).mat = Ak := by
         rw [HermitianMat.conj_conj]
-        have : (Bk ^ (-1 : ℝ)).mat * Bk.mat = 1 := by
-          simpa using rpow_neg_mul_rpow_self hBk 1
+        have : (Bk ^ (-1 : ℝ)).mat * Bk.mat = 1 := by simpa using rpow_neg_mul_rpow_self hBk 1
         simp [this, HermitianMat.conj_one (A := Ak)]
       simpa [hleft] using hconj
     have hNk_zero : Nk.conjTranspose * Nk = 0 := by
@@ -886,8 +862,7 @@ private lemma compound_top_singular_le_posDef
                 apply HermitianMat.cfc_congr_of_nonneg (zero_le_iff.mpr hBk.posSemidef)
                 intro x hx
                 rw [Function.comp_apply, Real.mul_rpow (inv_nonneg.mpr hc_nonneg) hx]
-          _ = (c⁻¹) ^ r • (Bk ^ r) := by
-                rw [HermitianMat.cfc_const_mul, HermitianMat.rpow_eq_cfc]
+          _ = (c⁻¹) ^ r • (Bk ^ r) := by rw [HermitianMat.cfc_const_mul, HermitianMat.rpow_eq_cfc]
       rwa [hBk_scale, conj_smul_right] at hpow
     have hNk_le : ((Ak ^ r).conj (Bk ^ r).mat) ≤ c ^ (2 * r) • 1 := by
       have hmul := smul_le_smul_of_nonneg_left hpow' (show 0 ≤ c ^ (2 * r) by positivity)
@@ -898,13 +873,10 @@ private lemma compound_top_singular_le_posDef
             c ^ (2 * r) = c ^ (r + r) := by ring_nf
             _ = c ^ r * c ^ r := by rw [Real.rpow_add hc_pos]
             _ = (c ^ r) ^ 2 := by ring
-        have hinv : (c⁻¹) ^ r = (c ^ r)⁻¹ := by
-          rw [Real.inv_rpow hc_nonneg]
+        have hinv : (c⁻¹) ^ r = (c ^ r)⁻¹ := by rw [Real.inv_rpow hc_nonneg]
         calc
-          c ^ (2 * r) * ((c⁻¹) ^ r) ^ 2 = (c ^ r) ^ 2 * ((c ^ r)⁻¹) ^ 2 := by
-            rw [hpow, hinv]
-          _ = 1 := by
-            field_simp [pow_two, hcr_pos.ne']
+          c ^ (2 * r) * ((c⁻¹) ^ r) ^ 2 = (c ^ r) ^ 2 * ((c ^ r)⁻¹) ^ 2 := by rw [hpow, hinv]
+          _ = 1 := by field_simp [pow_two, hcr_pos.ne']
       rwa [smul_smul, hs, one_smul] at hmul
     have htop : singularValuesSorted Nk (compoundZero k hk) ≤ Real.sqrt (c ^ (2 * r)) :=
       top_singular_le_of_self_mul_le_smul_one Nk
@@ -1248,8 +1220,7 @@ private lemma intervalIntegrable_weighted_trace_mul_inv_shift {A : HermitianMat
             hp hT).add hs
   refine hg.congr ?_
   intro t ht
-  have htIoc : t ∈ Set.Ioc (0 : ℝ) T := by
-    simpa [Set.uIoc_of_le hT.le] using ht
+  have htIoc : t ∈ Set.Ioc (0 : ℝ) T := by simpa [Set.uIoc_of_le hT.le] using ht
   unfold g
   change ∑ i, t ^ (p - 1) * (A.H.eigenvalues i / (A.H.eigenvalues i + t)) =
     t ^ (p - 1) * (((A.mat * (A + t • 1).mat⁻¹).trace : ℂ)).re
@@ -1273,9 +1244,7 @@ private lemma tendsto_intervalIntegral_weighted_trace_mul_inv_shift {A : Hermiti
               apply intervalIntegral.integral_congr_ae
               filter_upwards with t ht
               rw [trace_mul_inv_shift_eq_sum_div hA (by
-                have htIoc : t ∈ Set.Ioc (0 : ℝ) T := by
-                  simpa [Set.uIoc_of_le hT.le] using ht
-                exact htIoc.1)]
+                exact (show t ∈ Set.Ioc (0 : ℝ) T by simpa [Set.uIoc_of_le hT.le] using ht).1)]
               simp [Finset.mul_sum]
       _ = ∑ i, ∫ t in (0)..T, t ^ (p - 1) * (A.H.eigenvalues i / (A.H.eigenvalues i + t)) := by
             rw [intervalIntegral.integral_finsetSum]
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Schatten.lean b/QuantumInfo/ForMathlib/HermitianMat/Schatten.lean
index 5a7a5236f..18be6542d 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Schatten.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Schatten.lean
@@ -39,8 +39,7 @@ theorem schattenNorm_hermitian_pow {A : HermitianMat d ℂ} (hA : 0 ≤ A) {p :
   ext
   simp only [HermitianMat.conjTranspose_mat, HermitianMat.mat_pow]
   convert rfl using 2
-  rw [sq]
-  exact Matrix.IsHermitian.cfc_eq _ _
+  simpa [sq] using Matrix.IsHermitian.cfc_eq _ _
 
 lemma schattenNorm_nonneg (A : Matrix d d ℂ) (p : ℝ) :
     0 ≤ schattenNorm A p := by
@@ -62,8 +61,7 @@ lemma schattenNorm_pow_eq
     · rw [ mul_comm, ← HermitianMat.rpow_mul ];
       exact hA;
     · -- Since $A$ is positive, $A^{k*p}$ is also positive, and the trace of a positive matrix is non-negative.
-      have h_pos : 0 ≤ A ^ (k * p) := by
-        exact HermitianMat.rpow_nonneg hA;
+      have h_pos : 0 ≤ A ^ (k * p) := HermitianMat.rpow_nonneg hA
       exact HermitianMat.trace_nonneg h_pos;
   · exact HermitianMat.rpow_nonneg hA
 
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Sqrt.lean b/QuantumInfo/ForMathlib/HermitianMat/Sqrt.lean
index 858d572df..776d0d4ea 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Sqrt.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Sqrt.lean
@@ -29,8 +29,7 @@ theorem sqrt_sq_eq_proj (A : HermitianMat d 𝕜) :
   grind [Pi.mul_apply, Real.mul_self_sqrt, Real.sqrt_eq_zero']
 
 theorem sqrt_sq (hA : 0 ≤ A) :
-    A.sqrt.mat * A.sqrt.mat = A := by
-  rw [sqrt_sq_eq_proj, posPart_eq_self hA]
+    A.sqrt.mat * A.sqrt.mat = A := by rw [sqrt_sq_eq_proj, posPart_eq_self hA]
 
 @[aesop unsafe apply 50% (rule_sets := [Commutes])]
 theorem commute_sqrt_left (hAB : Commute A.mat B.mat) :
@@ -50,8 +49,7 @@ lemma sqrt_inv_mul_self_mul_sqrt_inv_eq_one {A : HermitianMat d 𝕜} (hA : A.ma
     A⁻¹.sqrt.mat * A.mat * A⁻¹.sqrt.mat = 1 := by
   have h_inv_def : A⁻¹.sqrt.mat * A⁻¹.sqrt.mat = A⁻¹ := by
     apply HermitianMat.sqrt_sq
-    rw [zero_le_iff]
-    exact hA.inv.posSemidef
+    simpa [zero_le_iff] using hA.inv.posSemidef
   have h_inv_comm : Commute A⁻¹.sqrt.mat A.mat := by
     commutes
   rw [h_inv_comm, mul_assoc, h_inv_def]
@@ -85,8 +83,6 @@ meta def evalHermitianMatSqrt : PositivityExt where eval {_u _α} _zα _pα e :=
   catch _ =>
     pure (.nonnegative (← mkAppM ``HermitianMat.sqrt_nonneg #[A]))
 
-example {A : HermitianMat d ℂ} : 0 ≤ A.sqrt := by
-  positivity
+example {A : HermitianMat d ℂ} : 0 ≤ A.sqrt := by positivity
 
-example [Nonempty d] {A : HermitianMat d ℂ} : 0 < (1 + A.sqrt).sqrt  := by
-  positivity
+example [Nonempty d] {A : HermitianMat d ℂ} : 0 < (1 + A.sqrt).sqrt  := by positivity
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Trace.lean b/QuantumInfo/ForMathlib/HermitianMat/Trace.lean
index 3df4fc059..49043a307 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Trace.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Trace.lean
@@ -57,24 +57,20 @@ section semiring
 variable [CommSemiring R] [Ring α] [StarAddMonoid α] [Algebra R α] [IsMaximalSelfAdjoint R α]
 
 @[simp]
-theorem trace_zero : (0 : HermitianMat n α).trace = 0 := by
-  simp [trace]
+theorem trace_zero : (0 : HermitianMat n α).trace = 0 := by simp [trace]
 
 @[simp]
-theorem trace_add (A B : HermitianMat n α) : (A + B).trace = A.trace + B.trace := by
-  simp [trace]
+theorem trace_add (A B : HermitianMat n α) : (A + B).trace = A.trace + B.trace := by simp [trace]
 
 end semiring
 section ring
 
 variable [CommRing R] [Ring α] [StarAddMonoid α] [Algebra R α] [IsMaximalSelfAdjoint R α]
 @[simp]
-theorem trace_neg (A : HermitianMat n α) : (-A).trace = -A.trace := by
-  simp [trace]
+theorem trace_neg (A : HermitianMat n α) : (-A).trace = -A.trace := by simp [trace]
 
 @[simp]
-theorem trace_sub (A B : HermitianMat n α) : (A - B).trace = A.trace - B.trace := by
-  simp [trace]
+theorem trace_sub (A B : HermitianMat n α) : (A - B).trace = A.trace - B.trace := by simp [trace]
 
 end ring
 section starring
@@ -108,8 +104,7 @@ variable [Star α] [TrivialStar α] [CommSemiring α]
 /-- `HermitianMat.trace` reduces to `Matrix.trace` when the elements are a `TrivialStar`. -/
 @[simp]
 theorem trace_eq_trace_trivial (A : HermitianMat n ℝ) : A.trace = A.mat.trace := by
-  rw [← trace_eq_trace]
-  rfl
+  simpa using trace_eq_trace A
 
 end trivialstar
 
@@ -117,8 +112,7 @@ section RCLike
 
 variable {n m 𝕜 : Type*} [Fintype n] [Fintype m] [RCLike 𝕜]
 
-theorem trace_eq_re_trace (A : HermitianMat n 𝕜) : A.trace = RCLike.re A.mat.trace := by
-  rfl
+theorem trace_eq_re_trace (A : HermitianMat n 𝕜) : A.trace = RCLike.re A.mat.trace := rfl
 
 @[simp]
 theorem trace_one [DecidableEq n] : (1 : HermitianMat n 𝕜).trace = Fintype.card n := by
@@ -144,17 +138,14 @@ theorem sum_eigenvalues_eq_trace [DecidableEq n] (A : HermitianMat n 𝕜) :
 --Proving that traces are 0 or 1 is common enough that we have a convenience lemma here for turning
 --statements about HermitianMat traces into Matrix traces.
 theorem trace_eq_zero_iff (A : HermitianMat n 𝕜) : A.trace = 0 ↔ A.mat.trace = 0 := by
-  rw [← trace_eq_trace_rc]
-  exact ⟨mod_cast id, mod_cast id⟩
+  simp [← trace_eq_trace_rc]
 
 theorem trace_eq_one_iff (A : HermitianMat n 𝕜) : A.trace = 1 ↔ A.mat.trace = 1 := by
-  rw [← trace_eq_trace_rc]
-  exact ⟨mod_cast id, mod_cast id⟩
+  simp [← trace_eq_trace_rc]
 
 @[simp]
 theorem trace_reindex (A : HermitianMat n ℂ) (e : n ≃ m) :
-    (A.reindex e).trace = A.trace := by
-  simp [reindex, trace_eq_re_trace]
+    (A.reindex e).trace = A.trace := by simp [reindex, trace_eq_re_trace]
 
 end RCLike
 section partialTrace
@@ -172,8 +163,7 @@ def traceRight (A : HermitianMat (m × n) α) : HermitianMat m α :=
 variable (A B : HermitianMat (m × n) α)
 
 @[simp]
-theorem traceLeft_mat : A.traceLeft.mat = A.mat.traceLeft := by
-  rfl
+theorem traceLeft_mat : A.traceLeft.mat = A.mat.traceLeft := rfl
 
 @[simp]
 theorem traceLeft_add : (A + B).traceLeft = A.traceLeft + B.traceLeft := by
@@ -191,8 +181,7 @@ variable (A B : HermitianMat (n × m) α)
 
 @[simp]
 theorem traceRight_mat :
-    (traceRight A).mat = A.mat.traceRight := by
-  rfl
+    (traceRight A).mat = A.mat.traceRight := rfl
 
 @[simp]
 theorem traceRight_add : (A + B).traceRight = A.traceRight + B.traceRight := by
@@ -223,12 +212,10 @@ theorem traceRight_smul (r : ℝ) : (r • A).traceRight = r • A.traceRight :=
   ext1; simp
 
 @[simp]
-theorem traceLeft_trace : A.traceLeft.trace = A.trace := by
-  simp [trace_eq_re_trace]
+theorem traceLeft_trace : A.traceLeft.trace = A.trace := by simp [trace_eq_re_trace]
 
 @[simp]
-theorem traceRight_trace : A.traceRight.trace = A.trace := by
-  simp [trace_eq_re_trace]
+theorem traceRight_trace : A.traceRight.trace = A.trace := by simp [trace_eq_re_trace]
 
 end rcLike
 section kron
diff --git a/QuantumInfo/ForMathlib/HermitianMat/Unitary.lean b/QuantumInfo/ForMathlib/HermitianMat/Unitary.lean
index feab49853..fb8e708c4 100644
--- a/QuantumInfo/ForMathlib/HermitianMat/Unitary.lean
+++ b/QuantumInfo/ForMathlib/HermitianMat/Unitary.lean
@@ -20,8 +20,7 @@ variable {α : Type*} [NonUnitalNonAssocSemiring α] [StarRing α]
 variable {α β : Type*} [DecidableEq α] [Fintype α] [DecidableEq β] [Fintype β]
 
 @[simp]
-theorem neg_unitary_val (u : 𝐔[α]) : (-u).val = -u := by
-  rfl
+theorem neg_unitary_val (u : 𝐔[α]) : (-u).val = -u := rfl
 
 omit [DecidableEq α] [Fintype α] [DecidableEq β] [Fintype β] in
 open Kronecker in
@@ -42,8 +41,7 @@ scoped infixl:60 " ⊗ᵤ " => unitary_kron
 
 @[simp]
 theorem unitary_kron_apply (a : 𝐔[α]) (b : 𝐔[β]) (i₁ i₂ : α) (j₁ j₂ : β) :
-    (a ⊗ᵤ b) (i₁, j₁) (i₂, j₂) = (a i₁ i₂) * (b j₁ j₂) := by
-  rfl
+    (a ⊗ᵤ b) (i₁, j₁) (i₂, j₂) = (a i₁ i₂) * (b j₁ j₂) := rfl
 
 @[simp]
 theorem unitary_kron_one_one : (1 : 𝐔[α]) ⊗ᵤ (1 : 𝐔[β]) = (1 : 𝐔[α × β]) := by
diff --git a/QuantumInfo/ForMathlib/IsMaximalSelfAdjoint.lean b/QuantumInfo/ForMathlib/IsMaximalSelfAdjoint.lean
index 113394d4a..c66c117e0 100644
--- a/QuantumInfo/ForMathlib/IsMaximalSelfAdjoint.lean
+++ b/QuantumInfo/ForMathlib/IsMaximalSelfAdjoint.lean
@@ -54,11 +54,9 @@ namespace IsMaximalSelfAdjoint
 
 @[simp]
 theorem trivial_selfadjMap {R} [Star R] [TrivialStar R] [CommSemiring R] :
-    (selfadjMap : R →+ R) = .id R := by
-  rfl
+    (selfadjMap : R →+ R) = .id R := rfl
 
 @[simp]
-theorem RCLike_selfadjMap {α} [RCLike α] : (selfadjMap : α →+ ℝ) = RCLike.re := by
-  rfl
+theorem RCLike_selfadjMap {α} [RCLike α] : (selfadjMap : α →+ ℝ) = RCLike.re := rfl
 
 end IsMaximalSelfAdjoint
diff --git a/QuantumInfo/ForMathlib/Isometry.lean b/QuantumInfo/ForMathlib/Isometry.lean
index 133acee53..162928573 100644
--- a/QuantumInfo/ForMathlib/Isometry.lean
+++ b/QuantumInfo/ForMathlib/Isometry.lean
@@ -370,8 +370,7 @@ theorem LinearMap.IsSymmetric.directSum_isInternal_of_commute' {𝕜 E : Type*}
     ext μ₁₂
     simp only [DirectSum.zero_apply, ZeroMemClass.coe_zero]
     rw [← inner_self_eq_zero (𝕜 := 𝕜)]
-    have h_inner_zero : inner 𝕜 (x μ₁₂ : E) (x.coeAddMonoidHom _) = 0 := by
-      simp [hx]
+    have h_inner_zero : inner 𝕜 (x μ₁₂ : E) (x.coeAddMonoidHom _) = 0 := by simp [hx]
     rw [← h_inner_zero]
     simp only [DirectSum.coeAddMonoidHom_eq_dfinsuppSum, ZeroMemClass.coe_zero, implies_true,
       DFinsupp.sum_eq_sum_fintype, DFinsupp.equivFunOnFintype_apply]
diff --git a/QuantumInfo/ForMathlib/LimSupInf.lean b/QuantumInfo/ForMathlib/LimSupInf.lean
index fdc408a49..b0efa1f9d 100644
--- a/QuantumInfo/ForMathlib/LimSupInf.lean
+++ b/QuantumInfo/ForMathlib/LimSupInf.lean
@@ -265,16 +265,14 @@ lemma tendsto_of_block_sequence {α : Type*} [TopologicalSpace α] {x : ℕ →
   obtain ⟨k, hk⟩ : ∃ k, T k ≤ n ∧ n < T (k + 1) := by
     -- Since $T$ is strictly increasing, the set $\{k \mid T k \leq n\}$ is finite and non-empty.
     have h_finite : Set.Finite {k | T k ≤ n} := by
-      rw [Set.finite_iff_bddAbove]
-      exact ⟨_, (hT.id_le · |>.trans)⟩
+      exact Set.finite_iff_bddAbove.mpr ⟨_, (hT.id_le · |>.trans)⟩
     use h_finite.toFinset.max' ⟨a, h_finite.mem_toFinset.mpr hn⟩
     constructor
     · exact h_finite.mem_toFinset.mp (Finset.max'_mem _ _)
     · rw [← not_le]
       intro h
       exact not_lt_of_ge (Finset.le_max' _ _ (h_finite.mem_toFinset.mpr h)) (Nat.lt_succ_self _)
-  rw [hg k n hk]
-  exact ha k (le_of_not_gt fun hk' ↦ by linarith [hT.monotone hk'])
+  simpa [hg k n hk] using ha k (le_of_not_gt fun hk' ↦ by linarith [hT.monotone hk'])
 
 /-
 Given a lower bound sequence M and a property P that can always be satisfied eventually, there exists a strictly increasing sequence T bounded by M such that each interval [T_k, T_{k+1}) contains a witness for P.
@@ -399,8 +397,7 @@ lemma exists_liminf_zero_of_forall_liminf_limsup_le_with_UB (y₁ y₂ : ℝ≥0
       · aesop
       · aesop
   obtain ⟨N0, hN0⟩ : ∃ N0 : ℕ → ℕ, ∀ k, ∀ n ≥ N0 k, f₂ (x k) n ≤ y₂ + (k + 1 : ℝ≥0)⁻¹ := by
-    have h_limsup : ∀ k, Filter.limsup (f₂ (x k)) Filter.atTop ≤ y₂ := by
-      exact fun k => hf₂ _ ( hx.1 k );
+    have h_limsup : ∀ k, Filter.limsup (f₂ (x k)) Filter.atTop ≤ y₂ := fun k => hf₂ _ ( hx.1 k )
     have h_limsup_le : ∀ k, ∀ ε > 0, ∃ N, ∀ n ≥ N, f₂ (x k) n ≤ y₂ + ε := by
       intro k ε hε_pos
       have h_limsup_le : Filter.limsup (f₂ (x k)) Filter.atTop ≤ y₂ := h_limsup k;
diff --git a/QuantumInfo/ForMathlib/LinearEquiv.lean b/QuantumInfo/ForMathlib/LinearEquiv.lean
index 144eb99e1..d3e4d6826 100644
--- a/QuantumInfo/ForMathlib/LinearEquiv.lean
+++ b/QuantumInfo/ForMathlib/LinearEquiv.lean
@@ -61,13 +61,11 @@ def euclidean_of_relabel (e : d ≃ d₂) : EuclideanSpace 𝕜 d₂ ≃ₗ[𝕜
   (WithLp.linearEquiv 2 𝕜 _).trans ((of_relabel _ e).trans (WithLp.linearEquiv 2 𝕜 _).symm)
 
 @[simp]
-theorem of_relabel_refl : of_relabel R (.refl d) = LinearEquiv.refl R (d → R) := by
-  rfl
+theorem of_relabel_refl : of_relabel R (.refl d) = LinearEquiv.refl R (d → R) := rfl
 
 @[simp]
 theorem euclidean_of_relabel_refl : euclidean_of_relabel 𝕜 (.refl d) =
-    LinearEquiv.refl 𝕜 (EuclideanSpace 𝕜 d) := by
-  rfl
+    LinearEquiv.refl 𝕜 (EuclideanSpace 𝕜 d) := rfl
 
 end LinearEquiv
 
diff --git a/QuantumInfo/ForMathlib/Majorization.lean b/QuantumInfo/ForMathlib/Majorization.lean
index 5c8324e32..d0adaa010 100644
--- a/QuantumInfo/ForMathlib/Majorization.lean
+++ b/QuantumInfo/ForMathlib/Majorization.lean
@@ -59,10 +59,8 @@ noncomputable def singularValuesSorted (A : Matrix d d ℂ) :
 /-- Sorted singular values are nonneg. -/
 lemma singularValuesSorted_nonneg (A : Matrix d d ℂ) (i : Fin (Fintype.card d)) :
     0 ≤ singularValuesSorted A i := by
-  have h_nonneg : ∀ i, 0 ≤ (singularValues A i) := by
-    exact singularValues_nonneg A
-  have h_sorted_nonneg : ∀ {l : List ℝ}, (∀ x ∈ l, 0 ≤ x) → ∀ i < l.length, 0 ≤ l[i]! := by
-    aesop
+  have h_nonneg : ∀ i, 0 ≤ (singularValues A i) := singularValues_nonneg A
+  have h_sorted_nonneg : ∀ {l : List ℝ}, (∀ x ∈ l, 0 ≤ x) → ∀ i < l.length, 0 ≤ l[i]! := by aesop
   contrapose! h_sorted_nonneg
   use Multiset.sort (Finset.univ.val.map (singularValues A)) (· ≥ ·)
   refine' ⟨_, i, _, _⟩
@@ -102,8 +100,7 @@ lemma singularValuesSorted_antitone (A : Matrix d d ℂ) :
 lemma antitone_mul_of_antitone_nonneg {n : ℕ}
     {f g : Fin n → ℝ} (hf : Antitone f) (hg : Antitone g)
     (hf_nn : ∀ i, 0 ≤ f i) (hg_nn : ∀ i, 0 ≤ g i) :
-    Antitone (fun i => f i * g i) := by
-  exact fun i j hij => mul_le_mul (hf hij) (hg hij) (hg_nn _) (hf_nn _)
+    Antitone (fun i => f i * g i) := fun _ _ hij => mul_le_mul (hf hij) (hg hij) (hg_nn _) (hf_nn _)
 
 /-! ### Compound matrices and auxiliary lemmas for Horn's inequality
 
@@ -201,15 +198,13 @@ lemma cauchyBinet {m : ℕ} {n : Type*} [Fintype n] [DecidableEq n] [LinearOrder
           obtain ⟨a, ha⟩ : ∃ a : Fin m → Fin m, ∀ i, σ i = S.val.orderEmbOfFin S.property (a i) := by
             have h_exists_a : ∀ i, ∃ a : Fin m, σ i = S.val.orderEmbOfFin S.property a := by
               intro i
-              have h_exists_a : σ i ∈ S.val := by
-                exact hσ.2 ▸ Finset.mem_image_of_mem _ (Finset.mem_univ _)
+              have h_exists_a : σ i ∈ S.val := hσ.2 ▸ Finset.mem_image_of_mem _ (Finset.mem_univ _)
               have h_exists_a : Finset.image (fun a : Fin m => S.val.orderEmbOfFin S.property a) Finset.univ = S.val := by
                 refine' Finset.eq_of_subset_of_card_le (Finset.image_subset_iff.mpr fun a _ => Finset.orderEmbOfFin_mem _ _ _) _
                 rw [Finset.card_image_of_injective _ fun a b h => by simpa [Fin.ext_iff] using h]; simp [S.2]
               grind
             exact ⟨fun i => Classical.choose (h_exists_a i), fun i => Classical.choose_spec (h_exists_a i)⟩
-          have ha_inj : Function.Injective a := by
-            exact fun i j hij => hσ.1 <| by simp [ha, hij]
+          have ha_inj : Function.Injective a := fun i j hij => hσ.1 <| by simp [ha, hij]
           exact ⟨Equiv.ofBijective a ⟨ha_inj, Finite.injective_iff_surjective.mp ha_inj⟩, funext fun i => ha i ▸ rfl⟩
         · rintro ⟨a, rfl⟩
           constructor
@@ -639,8 +634,7 @@ lemma prod_singularValuesSorted_eq_compoundSV (M : Matrix d d ℂ) (k : ℕ)
     exact singularValuesSorted_antitone (compoundMatrix M k) (Fin.zero_le idx)
   · obtain ⟨j, hj⟩ := singularValuesSorted_mem_values (compoundMatrix M k) ⟨0, hcard⟩
     obtain ⟨S, hS⟩ := singularValues_compoundMatrix_rev M k j
-    rw [hj, hS]
-    exact prod_singularValues_subset_le_sorted_prod M k hk S
+    simpa [hj, hS] using prod_singularValues_subset_le_sorted_prod M k hk S
 
 /--
 The **Rayleigh quotient bound**:
@@ -799,8 +793,7 @@ lemma horn_weak_log_majorization (A B : Matrix d d ℂ) (k : ℕ)
 lemma rpow_antitone_of_nonneg_antitone {n : ℕ}
     {f : Fin n → ℝ} (hf : Antitone f) (hf_nn : ∀ i, 0 ≤ f i)
     {r : ℝ} (hr : 0 < r) :
-    Antitone (fun i => f i ^ r) := by
-  exact fun i j hij => Real.rpow_le_rpow (hf_nn _) (hf hij) hr.le
+    Antitone (fun i => f i ^ r) := fun _ _ hij => Real.rpow_le_rpow (hf_nn _) (hf hij) hr.le
 
 /-- Weak log-majorization is preserved under positive powers. -/
 lemma rpow_preserves_weak_log_maj {n : ℕ}
@@ -921,8 +914,7 @@ lemma weak_log_maj_sum_le {n : ℕ}
     · simp [Fin.sum_univ_castSucc, h_last]
       refine le_add_of_le_of_nonneg (ih (fun i => hx_nn _) (fun i => hy_nn _) (fun i j hij => hx_anti hij) (fun i j hij => hy_anti hij) ?_) (hy_nn _)
       intro k hk
-      simp
-      exact h_log_maj k (by linarith)
+      simpa using h_log_maj k (by linarith)
     · -- Since $x_{\text{last}} > 0$, we have $x_i > 0$ for all $i$.
       have hx_pos : ∀ i, 0 < x i := by
         exact fun i => lt_of_lt_of_le (lt_of_le_of_ne (hx_nn _) (Ne.symm h_last)) (hx_anti (Fin.le_last _))
diff --git a/QuantumInfo/ForMathlib/Matrix.lean b/QuantumInfo/ForMathlib/Matrix.lean
index a2aee46ca..5f8cdcfa2 100644
--- a/QuantumInfo/ForMathlib/Matrix.lean
+++ b/QuantumInfo/ForMathlib/Matrix.lean
@@ -421,8 +421,7 @@ theorem le_smul_one_of_eigenvalues_iff (hA : A.IsHermitian) (c : ℝ) :
   (∀ i, hA.eigenvalues i ≤ c) ↔ A ≤ c • (1 : Matrix n n 𝕜) := by
   let U : Matrix n n 𝕜 := ↑hA.eigenvectorUnitary
   have hU : U.conjTranspose = star U := by simp only [star]
-  have hU' : U * star U = 1 := by
-    simp only [SetLike.coe_mem, Unitary.mul_star_self_of_mem, U]
+  have hU' : U * star U = 1 := by simp only [SetLike.coe_mem, Unitary.mul_star_self_of_mem, U]
   have hc : c • (1 : Matrix n n 𝕜) = U * (c • 1) * U.conjTranspose := by
     simp only [Algebra.mul_smul_comm, mul_one, hU, Algebra.smul_mul_assoc, hU']
   have hc' : c • (1 : Matrix n n 𝕜) = diagonal (RCLike.ofReal ∘ fun _ : n ↦ c) := by
@@ -430,8 +429,7 @@ theorem le_smul_one_of_eigenvalues_iff (hA : A.IsHermitian) (c : ℝ) :
     simp only [smul_apply, one_apply, smul_ite, RCLike.real_smul_eq_coe_mul, mul_one, smul_zero,
       diagonal, Function.comp_apply, of_apply]
   have hAST : A = U * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * U.conjTranspose := by
-    rw [hU]
-    exact IsHermitian.spectral_theorem hA
+    simpa [hU] using IsHermitian.spectral_theorem hA
   constructor
   · intro h
     rw [hc, hc', hAST]
@@ -441,8 +439,7 @@ theorem le_smul_one_of_eigenvalues_iff (hA : A.IsHermitian) (c : ℝ) :
     simp only [Function.comp_apply, algebraMap_le_algebraMap, h i]
   intro hAc i
   replace hAc := conjTranspose_mul_mul_mono U hAc
-  have hU'CT : star U * U = 1 := by
-    simp only [SetLike.coe_mem, Unitary.star_mul_self_of_mem, U]
+  have hU'CT : star U * U = 1 := by simp only [SetLike.coe_mem, Unitary.star_mul_self_of_mem, U]
   have hcCT : U.conjTranspose * (c • 1) * U = c • (1 : Matrix n n 𝕜) := by
     simp only [Algebra.mul_smul_comm, mul_one, hU, Algebra.smul_mul_assoc, hU'CT]
   have hASTCT : U.conjTranspose * A * U = diagonal (RCLike.ofReal ∘ hA.eigenvalues) := by
@@ -459,8 +456,7 @@ theorem smul_one_le_of_eigenvalues_iff (hA : A.IsHermitian) (c : ℝ) :
   -- I did the lazy thing and just copied the previous proof
   let U : Matrix n n 𝕜 := ↑hA.eigenvectorUnitary
   have hU : U.conjTranspose = star U := by simp only [star]
-  have hU' : U * star U = 1 := by
-    simp only [SetLike.coe_mem, Unitary.mul_star_self_of_mem, U]
+  have hU' : U * star U = 1 := by simp only [SetLike.coe_mem, Unitary.mul_star_self_of_mem, U]
   have hc : c • (1 : Matrix n n 𝕜) = U * (c • 1) * U.conjTranspose := by
     simp only [Algebra.mul_smul_comm, mul_one, hU, Algebra.smul_mul_assoc, hU']
   have hc' : c • (1 : Matrix n n 𝕜) = diagonal (RCLike.ofReal ∘ fun _ : n ↦ c) := by
@@ -468,8 +464,7 @@ theorem smul_one_le_of_eigenvalues_iff (hA : A.IsHermitian) (c : ℝ) :
     simp only [smul_apply, one_apply, smul_ite, RCLike.real_smul_eq_coe_mul, mul_one, smul_zero,
       diagonal, Function.comp_apply, of_apply]
   have hAST : A = U * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * U.conjTranspose := by
-    rw [hU]
-    exact IsHermitian.spectral_theorem hA
+    simpa [hU] using IsHermitian.spectral_theorem hA
   constructor
   · intro h
     rw [hc, hc', hAST]
@@ -479,8 +474,7 @@ theorem smul_one_le_of_eigenvalues_iff (hA : A.IsHermitian) (c : ℝ) :
     simp only [Function.comp_apply, algebraMap_le_algebraMap, h i]
   intro hAc i
   replace hAc := conjTranspose_mul_mul_mono U hAc
-  have hU'CT : star U * U = 1 := by
-    simp only [SetLike.coe_mem, Unitary.star_mul_self_of_mem, U]
+  have hU'CT : star U * U = 1 := by simp only [SetLike.coe_mem, Unitary.star_mul_self_of_mem, U]
   have hcCT : U.conjTranspose * (c • 1) * U = c • (1 : Matrix n n 𝕜) := by
     simp only [Algebra.mul_smul_comm, mul_one, hU, Algebra.smul_mul_assoc, hU'CT]
   have hASTCT : U.conjTranspose * A * U = diagonal (RCLike.ofReal ∘ hA.eigenvalues) := by
@@ -748,8 +742,7 @@ theorem PosDef_iff_eigenvalues' (M : Matrix d d 𝕜) :
 --These is disgusting atm. There's cleaner versions of them headed to Mathlib. See #29526 and follow-ups
 theorem IsHermitian.cfc_eigenvalues {M : Matrix d d 𝕜} (hM : M.IsHermitian) (f : ℝ → ℝ) :
     ∃ (e : d ≃ d), Matrix.IsHermitian.eigenvalues (cfc_predicate f M) = f ∘ hM.eigenvalues ∘ e := by
-  have h_eigenvalues : Multiset.map hM.eigenvalues Finset.univ.val = Multiset.map (fun i => hM.eigenvalues i) Finset.univ.val := by
-    rfl
+  have h_eigenvalues : Multiset.map hM.eigenvalues Finset.univ.val = Multiset.map (fun i => hM.eigenvalues i) Finset.univ.val := rfl
   generalize_proofs at *;
   have h_eigenvalues_cfc : (IsHermitian.cfc hM f).charpoly.roots = Multiset.map (fun i => (f (hM.eigenvalues i) : 𝕜)) Finset.univ.val := by
     rw [ Matrix.IsHermitian.cfc, Matrix.charpoly ];
@@ -900,14 +893,12 @@ theorem PosSemidef.pos_of_mem_spectrum {A : Matrix d d 𝕜} (hA : A.PosSemidef)
   exact hA.eigenvalues_nonneg i
 
 theorem PosSemidef.pow_add {A : Matrix d d 𝕜} (hA : A.PosSemidef) {x y : ℝ} (hxy : x + y ≠ 0) :
-    cfc (· ^ (x + y) : ℝ → ℝ) A = cfc (fun r ↦ r ^ x * r ^ y : ℝ → ℝ) A := by
-  refine cfc_congr fun r hr ↦ ?_
-  exact Real.rpow_add' (hA.pos_of_mem_spectrum r hr) hxy
+    cfc (· ^ (x + y) : ℝ → ℝ) A = cfc (fun r ↦ r ^ x * r ^ y : ℝ → ℝ) A :=
+  cfc_congr fun r hr => Real.rpow_add' (hA.pos_of_mem_spectrum r hr) hxy
 
 theorem PosSemidef.pow_mul {A : Matrix d d 𝕜} {x y : ℝ} (hA : A.PosSemidef) :
-    cfc (· ^ (x * y) : ℝ → ℝ) A = cfc (fun r ↦ (r ^ x) ^ y : ℝ → ℝ) A := by
-  refine cfc_congr fun r hr ↦ ?_
-  exact Real.rpow_mul (hA.pos_of_mem_spectrum r hr) x y
+    cfc (· ^ (x * y) : ℝ → ℝ) A = cfc (fun r ↦ (r ^ x) ^ y : ℝ → ℝ) A :=
+  cfc_congr fun r hr => Real.rpow_mul (hA.pos_of_mem_spectrum r hr) x y
 
 end more_cfc
 
@@ -919,8 +910,7 @@ variable {d₁ d₂ : Type*} [Fintype d₁] [Fintype d₂]
 @[simp]
 theorem trace_submatrix
   (A : Matrix d₁ d₁ α) (e : d₂ ≃ d₁) :
-    (A.submatrix e e).trace = A.trace := by
-  simpa [Matrix.trace] using e.sum_comp (fun x ↦ A x x)
+    (A.submatrix e e).trace = A.trace := by simpa [Matrix.trace] using e.sum_comp (fun x ↦ A x x)
 
 end subm
 
@@ -1446,8 +1436,7 @@ The right partial trace of a matrix is equal to the left partial trace of the ma
 -/
 theorem traceRight_eq_traceLeft_reindex {n m R : Type*} [Fintype m] [AddCommMonoid R]
   (M : Matrix (n × m) (n × m) R) :
-    M.traceRight = (M.reindex (.prodComm ..) (.prodComm ..)).traceLeft := by
-  rfl
+    M.traceRight = (M.reindex (.prodComm ..) (.prodComm ..)).traceLeft := rfl
 
 open ComplexOrder in
 theorem PosSemidef.trace_pos {n 𝕜 : Type*} [Fintype n] [RCLike 𝕜]
@@ -1475,8 +1464,7 @@ theorem traceLeft_neg : (-A).traceLeft = -A.traceLeft := by
   ext : 2; simp [Matrix.traceLeft]
 
 @[simp]
-theorem traceLeft_sub : (A - B).traceLeft = A.traceLeft - B.traceLeft := by
-  simp [sub_eq_add_neg]
+theorem traceLeft_sub : (A - B).traceLeft = A.traceLeft - B.traceLeft := by simp [sub_eq_add_neg]
 
 variable {A B : Matrix (n × m) (n × m) α}
 
diff --git a/QuantumInfo/ForMathlib/MatrixNorm/TraceNorm.lean b/QuantumInfo/ForMathlib/MatrixNorm/TraceNorm.lean
index 7bffe61ca..14d942279 100644
--- a/QuantumInfo/ForMathlib/MatrixNorm/TraceNorm.lean
+++ b/QuantumInfo/ForMathlib/MatrixNorm/TraceNorm.lean
@@ -30,16 +30,14 @@ def traceNorm (A : Matrix m n R) : ℝ :=
   RCLike.re (CFC.sqrt (Aᴴ * A)).trace
 
 @[simp]
-theorem traceNorm_zero : traceNorm (0 : Matrix m n R) = 0 := by
-  simp [traceNorm]
+theorem traceNorm_zero : traceNorm (0 : Matrix m n R) = 0 := by simp [traceNorm]
 
 /-- The trace norm of the negative is equal to the trace norm. -/
 @[simp]
 theorem traceNorm_neg (A : Matrix m n R) : traceNorm (-A) = traceNorm A := by
   unfold traceNorm
   congr! 3
-  rw [Matrix.conjTranspose_neg, Matrix.neg_mul, Matrix.mul_neg]
-  exact neg_neg _
+  simp [Matrix.conjTranspose_neg, Matrix.neg_mul, Matrix.mul_neg]
 
 
 open MatrixOrder Isometry
@@ -228,9 +226,8 @@ theorem exists_svd_sqrt_eigenvalues (A : Matrix n n ℂ) :
     have hleft : (A * W.val) i j = A.mulVec (hH.eigenvectorBasis j).ofLp i := by
       simp [Matrix.mul_apply, Matrix.mulVec, dotProduct, W, Matrix.IsHermitian.eigenvectorUnitary_apply]
     by_cases hj : hH.eigenvalues j = 0
-    · have hzero : A.mulVec (hH.eigenvectorBasis j).ofLp = 0 := by
-        apply (WithLp.toLp_injective (p := 2))
-        exact inner_self_eq_zero.mp (by
+    · have hzero : A.mulVec (hH.eigenvectorBasis j).ofLp = 0 :=
+        WithLp.toLp_injective (p := 2) <| inner_self_eq_zero.mp (by
           rw [inner_A_mulVec_eq]
           rw [hH.mulVec_eigenvectorBasis j, hj]
           simp)
@@ -286,8 +283,7 @@ theorem singularValues_le_opNorm [DecidableEq n] (A : Matrix n n ℂ) (i : n) :
   let hH : (Aᴴ * A).IsHermitian := by
     simpa using (Matrix.isHermitian_mul_conjTranspose_self A.conjTranspose)
   have hmem : hH.eigenvalues i ∈ spectrum ℝ (Aᴴ * A) := by
-    rw [hH.spectrum_real_eq_range_eigenvalues]
-    exact ⟨i, rfl⟩
+    simp [hH.spectrum_real_eq_range_eigenvalues]
   have hsq : singularValues A i * singularValues A i ≤ ‖A‖ * ‖A‖ := by
     have hsv_sq : singularValues A i * singularValues A i = hH.eigenvalues i := by
       dsimp [singularValues]
@@ -315,9 +311,8 @@ theorem traceNorm_mul_le_opNorm_traceNorm [DecidableEq n] (A B : Matrix n n ℂ)
       rw [singularValuesSorted_zero_eq_sup A hcard]
       rw [Finset.sup'_le_iff]
       exact fun i _ => singularValues_le_opNorm A i
-    have hA_bound : ∀ i : Fin (Fintype.card n), singularValuesSorted A i ≤ ‖A‖ := by
-      intro i
-      exact ((singularValuesSorted_antitone A) (Fin.zero_le i)).trans htop
+    have hA_bound : ∀ i : Fin (Fintype.card n), singularValuesSorted A i ≤ ‖A‖ :=
+      fun i => ((singularValuesSorted_antitone A) (Fin.zero_le i)).trans htop
     calc
       (A * B).traceNorm = ∑ i : Fin (Fintype.card n), singularValuesSorted (A * B) i := by
         rw [traceNorm_eq_sum_singularValuesSorted]
@@ -327,10 +322,8 @@ theorem traceNorm_mul_le_opNorm_traceNorm [DecidableEq n] (A B : Matrix n n ℂ)
         refine Finset.sum_le_sum ?_
         intro i hi
         exact mul_le_mul_of_nonneg_right (hA_bound i) (singularValuesSorted_nonneg B i)
-      _ = ‖A‖ * ∑ i : Fin (Fintype.card n), singularValuesSorted B i := by
-        rw [Finset.mul_sum]
-      _ = ‖A‖ * B.traceNorm := by
-        rw [traceNorm_eq_sum_singularValuesSorted]
+      _ = ‖A‖ * ∑ i : Fin (Fintype.card n), singularValuesSorted B i := by rw [Finset.mul_sum]
+      _ = ‖A‖ * B.traceNorm := by rw [traceNorm_eq_sum_singularValuesSorted]
 
 omit [DecidableEq n] in
 /-- The trace norm is invariant under conjugate transpose. -/
@@ -371,12 +364,10 @@ omit [Fintype m] [RCLike R] [DecidableEq n] in
 theorem traceNorm_sandwich_le [DecidableEq n] {S M T : Matrix n n ℂ} (hS : ‖S‖ ≤ 1)
     (hT : ‖T‖ ≤ 1) : (S * M * T).traceNorm ≤ M.traceNorm :=
   calc (S * M * T).traceNorm
-      ≤ (M * T).traceNorm := by
-        exact le_trans
+      ≤ (M * T).traceNorm := le_trans
           (by simpa [Matrix.mul_assoc] using Matrix.traceNorm_mul_le_opNorm_traceNorm S (M * T))
           (by simpa using mul_le_mul_of_nonneg_right hS (Matrix.traceNorm_nonneg (M * T)))
-    _ ≤ M.traceNorm := by
-        exact le_trans (traceNorm_mul_le_traceNorm_opNorm M T)
+    _ ≤ M.traceNorm := le_trans (traceNorm_mul_le_traceNorm_opNorm M T)
           (by simpa using mul_le_mul_of_nonneg_left hT (Matrix.traceNorm_nonneg M))
 
 end
@@ -399,13 +390,11 @@ theorem abs_trace_le_traceNorm (A : Matrix n n ℂ) :
       simpa [Matrix.trace] using norm_sum_le (s := Finset.univ) (f := fun i => (C.val * D) i i)
     _ = ∑ i, ‖C.val i i‖ * Real.sqrt (hH.eigenvalues i) := by
       simp [D, Matrix.mul_apply, Matrix.diagonal, Real.norm_eq_abs, abs_of_nonneg]
-    _ ≤ ∑ i, Real.sqrt (hH.eigenvalues i) := by
-      exact Finset.sum_le_sum (fun i _ => by
+    _ ≤ ∑ i, Real.sqrt (hH.eigenvalues i) := Finset.sum_le_sum (fun i _ => by
         simpa using mul_le_mul_of_nonneg_right
           (entry_norm_bound_of_unitary C.property i i)
           (Real.sqrt_nonneg _))
-    _ = A.traceNorm := by
-      simpa [hH] using (traceNorm_eq_sum_sqrt_eigenvalues A).symm
+    _ = A.traceNorm := by simpa [hH] using (traceNorm_eq_sum_sqrt_eigenvalues A).symm
 
 end complexTraceNorm
 
diff --git a/QuantumInfo/ForMathlib/Misc.lean b/QuantumInfo/ForMathlib/Misc.lean
index 0e6dd5b76..4faae5a2c 100644
--- a/QuantumInfo/ForMathlib/Misc.lean
+++ b/QuantumInfo/ForMathlib/Misc.lean
@@ -169,8 +169,7 @@ lemma exists_equiv_of_multiset_map_eq {α β γ : Type*} [Fintype α] [Fintype 
     ∃ e : α ≃ β, f = g ∘ e := by
   -- Since the multisets of values are equal, the cardinalities of the domains must be equal (as the multiset size is the cardinality of the domain). Thus there exists a bijection `σ : α ≃ β`.
   obtain ⟨σ, hσ⟩ : ∃ σ : α ≃ β, Multiset.map f Finset.univ.val = Multiset.map (g ∘ σ) Finset.univ.val := by
-    have h_card : Fintype.card α = Fintype.card β := by
-      simpa using congr_arg Multiset.card h;
+    have h_card : Fintype.card α = Fintype.card β := by simpa using congr_arg Multiset.card h
     obtain σ := Fintype.equivOfCardEq h_card
     use σ
     have h_multiset_eq : Multiset.map g Finset.univ.val = Multiset.map (g ∘ σ) Finset.univ.val := by
@@ -178,6 +177,5 @@ lemma exists_equiv_of_multiset_map_eq {α β γ : Type*} [Fintype α] [Fintype 
       rw [ Multiset.map_map ]
     exact h.trans h_multiset_eq;
   -- By `Multiset.map_univ_eq_iff`, there exists `e' : α ≃ α` such that `f = (g ∘ σ) ∘ e'`.
-  obtain ⟨e', he'⟩ : ∃ e' : α ≃ α, f = (g ∘ σ) ∘ e' := by
-    exact (Multiset.map_univ_eq_iff f (g ∘ ⇑σ)).mp hσ;
+  obtain ⟨e', he'⟩ : ∃ e' : α ≃ α, f = (g ∘ σ) ∘ e' := (Multiset.map_univ_eq_iff f (g ∘ ⇑σ)).mp hσ
   exact ⟨ e'.trans σ, by simp_all [ Function.comp ]; grind⟩
diff --git a/QuantumInfo/ForMathlib/SionMinimax.lean b/QuantumInfo/ForMathlib/SionMinimax.lean
index c6e379814..f1c801510 100644
--- a/QuantumInfo/ForMathlib/SionMinimax.lean
+++ b/QuantumInfo/ForMathlib/SionMinimax.lean
@@ -121,8 +121,7 @@ theorem LowerSemicontinuousOn.bddBelow {α : Type*} [TopologicalSpace α] {S : S
     intros x hx
     specialize hg x hx (g x - 1) (sub_one_lt (g x))
     rw [eventually_nhdsWithin_iff] at hg
-    simp only [Set.mem_inter_iff, gt_iff_lt, and_imp]
-    exact ⟨_, hg, fun y hy hyS ↦ hy hyS⟩
+    simpa only [Set.mem_inter_iff, gt_iff_lt, and_imp] using ⟨_, hg, fun y hy hyS ↦ hy hyS⟩
   choose! U hU using h_neighborhood
   -- By the finite subcover property, there exists a finite subset t ⊆ S$ such that S ⊆ ⋃_{x ∈ t} U_x$.
   obtain ⟨t, ht⟩ : ∃ t, (∀ x ∈ t, x ∈ S) ∧ S ⊆ ⋃ x ∈ t, U x :=
@@ -164,16 +163,14 @@ theorem lowerSemicontinuousOn_iff_isClosed_preimage {f : α → γ} [IsClosed s]
       exact ⟨ o, ho, h.1, Set.eq_empty_iff_forall_notMem.2 fun z hz => h.2 ⟨ hz.1, hz.2.1 ⟩ |> not_le_of_gt <| Set.mem_Iic.1 hz.2.2 ⟩;
     · exact ⟨ sᶜ, IsClosed.isOpen_compl, hx', by aesop ⟩;
   · intro a x hx y hy
-    have hx_not_in : x ∉ s ∩ f ⁻¹' Set.Iic y := by
-      simp [hy]
+    have hx_not_in : x ∉ s ∩ f ⁻¹' Set.Iic y := by simp [hy]
     rw [ eventually_nhdsWithin_iff ]
     filter_upwards [ IsOpen.mem_nhds ( isOpen_compl_iff.2 ( a y ) ) hx_not_in ] with z hz hzs
     exact lt_of_not_ge fun h => hz ⟨hzs, h⟩
 
 theorem segment.isConnected {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a b : E) :
     IsConnected (segment ℝ a b) := by
-  rw [← Path.range_segment a b]
-  exact isConnected_range (Path.segment a b).continuous
+  simpa [Path.range_segment a b] using isConnected_range (Path.segment a b).continuous
 
 theorem BddAbove.range_inf_of_image2 {M N α : Type*} {f : M → N → α} [ConditionallyCompleteLinearOrder α]
   {S : Set M} {T : Set N} (h_bddA : BddAbove (Set.image2 f S T)) (h_bddB : BddBelow (Set.image2 f S T)) :
@@ -269,8 +266,7 @@ theorem LeftOrdContinuous.comp_lowerSemicontinuousOn_strong_assumptions {α γ 
         by_contra h_contra
         simp only [not_exists, not_and, not_lt] at h_contra
         have h_exists_z : ∃ z, z < f x ∧ g z > y := by
-          have h_lub : IsLUB (Set.Iio (f x)) (f x) := by
-            exact isLUB_Iio
+          have h_lub : IsLUB (Set.Iio (f x)) (f x) := isLUB_Iio
           have := hg h_lub;
           have := this.exists_between hy
           simp_all only [Set.mem_image, Set.mem_Iio, exists_exists_and_eq_and, gt_iff_lt]
@@ -471,9 +467,7 @@ private lemma sion_exists_min_2 (y₁ y₂ : N) (hy₁ : y₁ ∈ T) (hy₂ : y
     simp only [Set.mem_setOf_eq, I, C] at hzI
     replace ⟨hzI, hzI2⟩ := And.intro (hzI · |>.left) (hzI · |>.right)
     have hz_mem : z ∈ segment ℝ y₁ y₂ :=
-      have cloL : IsClosed (segment ℝ y₁ y₂) := by
-        rw [← closure_openSegment]
-        exact isClosed_closure
+      have cloL : IsClosed (segment ℝ y₁ y₂) := by simp [← closure_openSegment]
       cloL.isSeqClosed hzI hzs
     let x := (hC_nonempty z hz_mem).some
     have hx : x ∈ C z := (hC_nonempty z hz_mem).some_mem
@@ -509,9 +503,7 @@ private lemma sion_exists_min_2 (y₁ y₂ : N) (hy₁ : y₁ ∈ T) (hy₂ : y
     simp only [Set.mem_setOf_eq, J, C] at hzI
     replace ⟨hzI, hzI2⟩ := And.intro (hzI · |>.left) (hzI · |>.right)
     have hz_mem : z ∈ segment ℝ y₁ y₂ :=
-      have cloL : IsClosed (segment ℝ y₁ y₂) := by
-        rw [← closure_openSegment]
-        exact isClosed_closure
+      have cloL : IsClosed (segment ℝ y₁ y₂) := by simp [← closure_openSegment]
       cloL.isSeqClosed hzI hzs
     let x := (hC_nonempty z hz_mem).some
     have hx : x ∈ C z := (hC_nonempty z hz_mem).some_mem
diff --git a/QuantumInfo/ForMathlib/ULift.lean b/QuantumInfo/ForMathlib/ULift.lean
index 82195f7de..355989d42 100644
--- a/QuantumInfo/ForMathlib/ULift.lean
+++ b/QuantumInfo/ForMathlib/ULift.lean
@@ -28,8 +28,7 @@ instance ULift.instStar {𝕜 : Type u} [Star 𝕜] : Star (ULift.{v,u} 𝕜) wh
   star x := .up (star x.down)
 
 @[simp]
-theorem ULift.star_eq {𝕜 : Type u} [Star 𝕜] (x : ULift.{v,u} 𝕜) : star x = .up (star x.down) := by
-  rfl
+theorem ULift.star_eq {𝕜 : Type u} [Star 𝕜] (x : ULift.{v,u} 𝕜) : star x = .up (star x.down) := rfl
 
 instance ULift.instInvolutiveStar {𝕜 : Type u} [InvolutiveStar 𝕜] :
     InvolutiveStar (ULift.{v,u} 𝕜) where
@@ -43,8 +42,7 @@ instance {𝕜 : Type u} [NonUnitalNonAssocSemiring 𝕜] [StarRing 𝕜] : Star
 
 @[simp]
 theorem ULift.starRingEnd_down {𝕜 : Type u} (x : ULift.{v,u} 𝕜) [CommSemiring 𝕜] [StarRing 𝕜] :
-    ((starRingEnd (ULift.{v, u} 𝕜)) x).down = star x.down := by
-  rfl
+    ((starRingEnd (ULift.{v, u} 𝕜)) x).down = star x.down := rfl
 
 instance {𝕜 : Type u} [NormedField 𝕜] : NormedField (ULift.{v,u} 𝕜) where
   dist_eq x y := NormedField.dist_eq x.down y.down
@@ -55,8 +53,7 @@ instance {𝕜 : Type u} [DenselyNormedField 𝕜] : DenselyNormedField (ULift.{
 
 @[simp]
 theorem AddEquiv.ulift_apply {α : Type u} [Add α] (x : ULift.{v, u} α) :
-    AddEquiv.ulift.{u, v} x = x.down := by
-  rfl
+    AddEquiv.ulift.{u, v} x = x.down := rfl
 
 noncomputable instance {𝕜 : Type u} [RCLike 𝕜] : RCLike (ULift.{v,u} 𝕜) where
   re := RCLike.re.comp AddEquiv.ulift.toAddMonoidHom (N := 𝕜)
diff --git a/QuantumInfo/ForMathlib/Unitary.lean b/QuantumInfo/ForMathlib/Unitary.lean
index 5b09a3d5b..58e3f84bd 100644
--- a/QuantumInfo/ForMathlib/Unitary.lean
+++ b/QuantumInfo/ForMathlib/Unitary.lean
@@ -25,13 +25,11 @@ open Module.End
 
 @[simp]
 theorem unitary_star_apply_eq (U : unitary (E →ₗ[𝕜] E)) (v : E) :
-    (star U.val) (U.val v) = v := by
-  rw [← mul_apply, (Unitary.mem_iff.mp U.prop).left, one_apply]
+    (star U.val) (U.val v) = v := by rw [← mul_apply, (Unitary.mem_iff.mp U.prop).left, one_apply]
 
 @[simp]
 theorem unitary_apply_star_eq (U : unitary (E →ₗ[𝕜] E)) (v : E) :
-    U.val ((star U.val) v) = v := by
-  rw [← mul_apply, (Unitary.mem_iff.mp U.prop).right, one_apply]
+    U.val ((star U.val) v) = v := by rw [← mul_apply, (Unitary.mem_iff.mp U.prop).right, one_apply]
 
 /-- Conjugating a linear map by a unitary operator gives a map whose μ-eigenspace is
   isomorphic (same dimension) as those of the original linear map. -/
diff --git a/QuantumInfo/Measurements/POVM.lean b/QuantumInfo/Measurements/POVM.lean
index 781d0e9ab..c2c7c32e0 100644
--- a/QuantumInfo/Measurements/POVM.lean
+++ b/QuantumInfo/Measurements/POVM.lean
@@ -161,8 +161,7 @@ noncomputable def measureDiscard (Λ : POVM X d) : CPTPMap d X :=
   CPTPMap.traceLeft ∘ₘ Λ.measurementMap
 
 theorem measureDiscard_apply (Λ : POVM X d) (ρ : MState d) :
-    Λ.measureDiscard ρ = MState.ofClassical (Λ.measure ρ) := by
-  simp [measureDiscard, traceLeft_measurementMap_eq_measure]
+    Λ.measureDiscard ρ = MState.ofClassical (Λ.measure ρ) := by simp [measureDiscard, traceLeft_measurementMap_eq_measure]
 
 /-- The action of measuring a state with the POVM `Λ`, forgetting the measurement outcome, and
 keeping the disturbed state. -/
diff --git a/QuantumInfo/Operators/Unitary.lean b/QuantumInfo/Operators/Unitary.lean
index 9a8fa46a0..10bc5eaf5 100644
--- a/QuantumInfo/Operators/Unitary.lean
+++ b/QuantumInfo/Operators/Unitary.lean
@@ -47,8 +47,7 @@ of a matrix are always canonically sorted, this is actually an equality.
 -/
 @[simp]
 theorem U_conj_spectrum_eq (ρ : MState d) (U : 𝐔[d]) :
-    (ρ.U_conj U).spectrum = ρ.spectrum := by
-  simp [spectrum, U_conj]
+    (ρ.U_conj U).spectrum = ρ.spectrum := by simp [spectrum, U_conj]
 
 @[simp]
 theorem inner_U_conj (ρ σ : MState d) (U : 𝐔[d]) : ⟪U ◃ ρ, U ◃ σ⟫_Prob = ⟪ρ, σ⟫_Prob := by
diff --git a/QuantumInfo/ResourceTheory/FreeState.lean b/QuantumInfo/ResourceTheory/FreeState.lean
index 1140fb184..d93598b96 100644
--- a/QuantumInfo/ResourceTheory/FreeState.lean
+++ b/QuantumInfo/ResourceTheory/FreeState.lean
@@ -117,8 +117,7 @@ open ComplexOrder in
 theorem PosDef.prod {ρ : MState (H i)} {σ : MState (H j)} (hρ : ρ.m.PosDef) (hσ : σ.m.PosDef)
     : (ρ ⊗ᵣ σ).m.PosDef := by
   have : (ρ ⊗ᴹ σ).m.PosDef := MState.PosDef.kron hρ hσ
-  rw [prodRelabel]
-  exact MState.PosDef.relabel this (prodEquiv i j)
+  simpa [prodRelabel] using MState.PosDef.relabel this (prodEquiv i j)
 
 --BAD old attempt at PNat powers
 -- /-- Powers of spaces. Defined for `PNat` so that we don't have zeroth powers. -/
@@ -153,13 +152,11 @@ theorem PosDef.prod {ρ : MState (H i)} {σ : MState (H j)} (hρ : ρ.m.PosDef)
 
 @[simp]
 theorem qRelEntropy_prodRelabel (ρ₁ ρ₂ : MState (H i)) (σ₁ σ₂ : MState (H j)):
-    𝐃(ρ₁ ⊗ᵣ σ₁‖ρ₂ ⊗ᵣ σ₂) = 𝐃(ρ₁‖ρ₂) + 𝐃(σ₁‖σ₂) := by
-  simp [prodRelabel]
+    𝐃(ρ₁ ⊗ᵣ σ₁‖ρ₂ ⊗ᵣ σ₂) = 𝐃(ρ₁‖ρ₂) + 𝐃(σ₁‖σ₂) := by simp [prodRelabel]
 
 @[simp]
 theorem sandwichedRelRentropy_prodRelabel {α : ℝ} (ρ₁ ρ₂ : MState (H i)) (σ₁ σ₂ : MState (H j)):
-    D̃_ α(ρ₁ ⊗ᵣ σ₁‖ρ₂ ⊗ᵣ σ₂) = D̃_ α(ρ₁‖ρ₂) + D̃_ α(σ₁‖σ₂) := by
-  simp [prodRelabel]
+    D̃_ α(ρ₁ ⊗ᵣ σ₁‖ρ₂ ⊗ᵣ σ₂) = D̃_ α(ρ₁‖ρ₂) + D̃_ α(σ₁‖σ₂) := by simp [prodRelabel]
 
 end ResourcePretheory
 
@@ -193,15 +190,12 @@ noncomputable def spacePow (i : ι) (n : ℕ) : ι :=
 scoped notation i "⊗^H[" n "]" => spacePow i n
 
 @[simp]
-theorem spacePow_zero (i : ι) : i ^ 0 = 1 := by
-  rfl
+theorem spacePow_zero (i : ι) : i ^ 0 = 1 := rfl
 
 @[simp]
-theorem spacePow_one (i : ι) : i ^ 1 = i := by
-  simp
+theorem spacePow_one (i : ι) : i ^ 1 = i := by simp
 
-theorem spacePow_succ (i : ι) (n : ℕ) : i ^ (n + 1) = (i ^ n) * i := by
-  rfl
+theorem spacePow_succ (i : ι) (n : ℕ) : i ^ (n + 1) = (i ^ n) * i := rfl
 
 theorem spacePow_add (m n : ℕ) :
     i ^ (m + n) = (i ^ m) * (i ^ n) := by
@@ -223,8 +217,7 @@ noncomputable def statePow (ρ : MState (H i)) (n : ℕ) : MState (H (i ^ n)) :=
 scoped notation ρ " ⊗ᵣ^[" n "]" => statePow ρ n
 
 @[simp]
-theorem statePow_zero (ρ : MState (H i)) : ρ ⊗ᵣ^[0] = default :=
-  rfl
+theorem statePow_zero (ρ : MState (H i)) : ρ ⊗ᵣ^[0] = default := rfl
 
 @[simp]
 theorem statePow_one (ρ : MState (H i)) : ρ ⊗ᵣ^[1] ≍ ρ := by
@@ -233,8 +226,7 @@ theorem statePow_one (ρ : MState (H i)) : ρ ⊗ᵣ^[1] ≍ ρ := by
   · rw [eq_cast_iff_heq, statePow]
     exact default_prod ρ
 
-theorem statePow_succ (ρ : MState (H i)) (n : ℕ) : ρ ⊗ᵣ^[n + 1] = ρ ⊗ᵣ^[n] ⊗ᵣ ρ := by
-  rfl
+theorem statePow_succ (ρ : MState (H i)) (n : ℕ) : ρ ⊗ᵣ^[n + 1] = ρ ⊗ᵣ^[n] ⊗ᵣ ρ := rfl
 
 theorem statePow_add (ρ : MState (H i)) (m n : ℕ) : ρ ⊗ᵣ^[m + n] ≍ ρ ⊗ᵣ^[m] ⊗ᵣ ρ ⊗ᵣ^[n] := by
   rw [← eq_cast_iff_heq]; swap
diff --git a/QuantumInfo/ResourceTheory/HypothesisTesting.lean b/QuantumInfo/ResourceTheory/HypothesisTesting.lean
index 9a9dbec0f..e6bec24a8 100644
--- a/QuantumInfo/ResourceTheory/HypothesisTesting.lean
+++ b/QuantumInfo/ResourceTheory/HypothesisTesting.lean
@@ -117,8 +117,7 @@ theorem le_of_subset (ρ : MState d) (ε : Prob) {S1 S2 : Set (MState d)} (h : S
 theorem of_singleton {ρ σ : MState d} {ε : Prob} :
     β_ ε(ρ‖{σ}) =
       ⨅ T : { m : HermitianMat d ℂ // ρ.exp_val (1 - m) ≤ ε ∧ 0 ≤ m ∧ m ≤ 1},
-        ⟨_, σ.exp_val_prob T.2.right⟩ := by
-  simp only [OptimalHypothesisRate, iSup_singleton]
+        ⟨_, σ.exp_val_prob T.2.right⟩ := by simp only [OptimalHypothesisRate, iSup_singleton]
 
 open scoped Prob in
 theorem negLog_le_singleton (ρ : MState d) (ε : Prob) (S : Set (MState d))
@@ -209,8 +208,7 @@ theorem exists_min (ρ : MState d) (ε : Prob) (S : Set (MState d)):
     gcongr
     · exact T.2.2.1
     · simp; positivity
-  have hρT' : ρ.exp_val (1 - T') = ε := by
-    simp [T', MState.exp_val_sub, δ, field]
+  have hρT' : ρ.exp_val (1 - T') = ε := by simp [T', MState.exp_val_sub, δ, field]
 
   have hT' : ρ.exp_val (1 - T') ≤ ε ∧ 0 ≤ T' ∧ T' ≤ 1 := by
     use hρT'.le
diff --git a/QuantumInfo/ResourceTheory/SteinsLemma.lean b/QuantumInfo/ResourceTheory/SteinsLemma.lean
index 6db4c8b62..b5d4edadd 100644
--- a/QuantumInfo/ResourceTheory/SteinsLemma.lean
+++ b/QuantumInfo/ResourceTheory/SteinsLemma.lean
@@ -620,9 +620,8 @@ private lemma f_image_bound (mineig : ℝ) (n : ℕ) (h : 0 < mineig) (hn : 0 <
         exact Real.log_nonpos ( by linarith [ pow_pos h n ] ) right_1;
       · have := h_f_le n 1
         simp_all only [Real.log_one, zero_div, Int.ceil_zero, Int.cast_zero, zero_mul, zero_add, lt_neg_add_iff_add_lt, add_zero]
-  have h_card_image : Set.ncard (Set.image (fun k : ℤ => Real.exp (k * (Real.log (1 / mineig) + Real.log 3 / (max n 1)))) (Set.Icc (⌈(n * Real.log mineig - Real.log 3) / (Real.log (1 / mineig) + Real.log 3 / (max n 1))⌉) 0)) ≤ Set.ncard (Set.Icc (⌈(n * Real.log mineig - Real.log 3) / (Real.log (1 / mineig) + Real.log 3 / (max n 1))⌉) 0) := by
-    apply Set.ncard_image_le;
-    exact Set.finite_Icc _ _;
+  have h_card_image : Set.ncard (Set.image (fun k : ℤ => Real.exp (k * (Real.log (1 / mineig) + Real.log 3 / (max n 1)))) (Set.Icc (⌈(n * Real.log mineig - Real.log 3) / (Real.log (1 / mineig) + Real.log 3 / (max n 1))⌉) 0)) ≤ Set.ncard (Set.Icc (⌈(n * Real.log mineig - Real.log 3) / (Real.log (1 / mineig) + Real.log 3 / (max n 1))⌉) 0) :=
+    Set.ncard_image_le (Set.finite_Icc _ _)
   simp_all +decide [ Set.ncard_eq_toFinset_card' ];
   refine le_trans h_card <| le_trans h_card_image ?_;
   rcases n with _ | n
@@ -781,14 +780,12 @@ private theorem log_le_f (n : ℕ) (lam : ℝ) : Real.log lam ≤ f_map i n lam
   _ ≤ (⌈Real.log lam / (σ₁_c i n)⌉) * σ₁_c i n := by
     rw [← mul_inv_le_iff₀ (σ₁_c_pos i n)]
     apply Int.le_ceil _
-  _ = _ := by
-    rfl
+  _ = _ := rfl
 
 /-- (S46), part 2 -/
 private theorem f_le_log (n : ℕ) (lam : ℝ) : f_map i n lam < Real.log lam + σ₁_c i n :=
   calc
-  _ = ⌈Real.log lam / σ₁_c i n⌉ * σ₁_c i n := by
-    rfl
+  _ = ⌈Real.log lam / σ₁_c i n⌉ * σ₁_c i n := rfl
   _ < (Real.log lam / σ₁_c i n + 1) * σ₁_c i n := by
     gcongr
     · exact σ₁_c_pos i n
diff --git a/QuantumInfo/States/Ensemble.lean b/QuantumInfo/States/Ensemble.lean
index 621e1fe4a..ec2a0ae1d 100644
--- a/QuantumInfo/States/Ensemble.lean
+++ b/QuantumInfo/States/Ensemble.lean
@@ -42,8 +42,7 @@ namespace Ensemble
 instance : Coe (PEnsemble d α) (MEnsemble d α) := ⟨toMEnsemble⟩
 
 @[simp]
-theorem toMEnsemble_mk : (toMEnsemble ⟨ps, distr⟩ : MEnsemble d α) = ⟨pure ∘ ps, distr⟩ :=
-  rfl
+theorem toMEnsemble_mk : (toMEnsemble ⟨ps, distr⟩ : MEnsemble d α) = ⟨pure ∘ ps, distr⟩ := rfl
 
 /-- A mixed-state ensemble comes from a pure-state ensemble if and only if all states are pure. -/
 theorem coe_PEnsemble_iff_pure_states (me : MEnsemble d α): (∃ pe : PEnsemble d α, ↑pe = me) ↔ (∃ ψ : α → Ket d, me.states = MState.pure ∘ ψ) := by
@@ -112,14 +111,12 @@ def pure_average_NNReal {d : Type _} [Fintype d] (f : Ket d → NNReal) (e : PEn
 is equal to averaging the restricted function over Kets `f ∘ pure : Ket d → T` on `e`. -/
 theorem average_of_pure_ensemble {T : Type _} {U : Type*} [AddCommGroup U] [Module ℝ U] [inst : Mixable U T]
   (f : MState d → T) (e : PEnsemble d α) :
-  average f (toMEnsemble e) = pure_average (f ∘ pure) e := by
-  simp only [average, pure_average, toMEnsemble, comp_map]
+  average f (toMEnsemble e) = pure_average (f ∘ pure) e := by simp only [average, pure_average, toMEnsemble, comp_map]
 
 variable {ψ : Ket d}
 
 @[simp]
-theorem distr_toMEnsemble (e : PEnsemble d α) : (toMEnsemble e).distr = e.distr := by
-  rfl
+theorem distr_toMEnsemble (e : PEnsemble d α) : (toMEnsemble e).distr = e.distr := rfl
 
 /-
 A pure-state ensemble mixes into a pure state if and only if
diff --git a/QuantumInfo/States/Entanglement.lean b/QuantumInfo/States/Entanglement.lean
index 7ada04aba..76667a7b0 100644
--- a/QuantumInfo/States/Entanglement.lean
+++ b/QuantumInfo/States/Entanglement.lean
@@ -84,8 +84,7 @@ theorem convex_roof_ne_top : ∀ ρ, convex_roof_ENNReal g ρ ≠ ∞ := fun ρ
   simp only [convex_roof_ENNReal, ne_eq, iInf_eq_top, coe_ne_top, imp_false, not_forall]
   use ⟨Fintype.card d, Fintype.card_pos⟩
   have ed : d ≃ Fin ↑(⟨Fintype.card d, Fintype.card_pos⟩ : ℕ+) := by
-    simp only
-    exact Fintype.equivFin d
+    simpa only using Fintype.equivFin d
   use (congrPEnsemble ed) <| spectral_ensemble ρ
   rw [mix_congrPEnsemble_eq_mix ed]
   push Not
@@ -126,8 +125,7 @@ theorem le_mixed_convex_roof (ρ : MState d) :
   unfold mixed_convex_roof
   rw [WithTop.le_untop_iff]
   apply le_iInf; intro ⟨n, hnpos⟩; apply le_iInf; intro e; apply le_iInf; intro hmix
-  rw [some_eq_coe', ENNReal.coe_le_coe]
-  exact h n hnpos e hmix
+  exact ENNReal.coe_le_coe.mpr (h n hnpos e hmix)
 
 set_option backward.isDefEq.respectTransparency false in
 theorem le_convex_roof (ρ : MState d) :
@@ -135,8 +133,7 @@ theorem le_convex_roof (ρ : MState d) :
   unfold convex_roof
   rw [WithTop.le_untop_iff]
   apply le_iInf; intro ⟨n, hnpos⟩; apply le_iInf; intro e; apply le_iInf; intro hmix
-  rw [some_eq_coe', ENNReal.coe_le_coe]
-  exact h n hnpos e hmix
+  exact ENNReal.coe_le_coe.mpr (h n hnpos e hmix)
 
 set_option backward.isDefEq.respectTransparency false in
 theorem convex_roof_le (ρ : MState d):
@@ -145,8 +142,7 @@ theorem convex_roof_le (ρ : MState d):
   unfold convex_roof
   rw [WithTop.untop_le_iff]
   apply iInf_le_of_le ⟨n, hnpos⟩; apply iInf_le_of_le e; apply iInf_le_of_le hmix
-  rw [some_eq_coe', ENNReal.coe_le_coe]
-  exact h
+  exact ENNReal.coe_le_coe.mpr h
 
 set_option backward.isDefEq.respectTransparency false in
 omit [Nonempty d] in
@@ -156,8 +152,7 @@ theorem mixed_convex_roof_le (ρ : MState d):
   unfold mixed_convex_roof
   rw [WithTop.untop_le_iff]
   apply iInf_le_of_le ⟨n, hnpos⟩; apply iInf_le_of_le e; apply iInf_le_of_le hmix
-  rw [some_eq_coe', ENNReal.coe_le_coe]
-  exact h
+  exact ENNReal.coe_le_coe.mpr h
 
 /-- The mixed convex roof extension of `f` is smaller than or equal to its convex roof extension, since
 the former minimizes over a larger set of ensembles. -/
@@ -267,15 +262,13 @@ The von Neumann entropy of a state is equal to the trace of `ρ log ρ` (technic
 theorem Sᵥₙ_eq_trace_cfc {d : Type*} [Fintype d] [DecidableEq d] (ρ : MState d) :
     Sᵥₙ ρ = (HermitianMat.cfc ρ.M Real.negMulLog).trace := by
   -- By definition of von Neumann entropy, we have Sᵥₙ ρ = Finset.sum Finset.univ (fun x ↦ Real.negMulLog (ρ.M.H.eigenvalues x)).
-  have h_def : Sᵥₙ ρ = Finset.sum Finset.univ (fun x ↦ Real.negMulLog (ρ.M.H.eigenvalues x)) := by
-    rfl
+  have h_def : Sᵥₙ ρ = Finset.sum Finset.univ (fun x ↦ Real.negMulLog (ρ.M.H.eigenvalues x)) := rfl
   -- By definition of trace, the trace of `cfc ρ.M Real.negMulLog` is the sum of its eigenvalues.
   have h_trace : (ρ.M.cfc Real.negMulLog).trace =
       ∑ x, (ρ.M.cfc Real.negMulLog).H.eigenvalues x := by
     exact (HermitianMat.sum_eigenvalues_eq_trace _).symm
   obtain ⟨e, he⟩ : ∃ e : d ≃ d, (ρ.M.cfc Real.negMulLog).H.eigenvalues =
-      Real.negMulLog ∘ ρ.M.H.eigenvalues ∘ e := by
-   exact Matrix.IsHermitian.cfc_eigenvalues _ _
+      Real.negMulLog ∘ ρ.M.H.eigenvalues ∘ e := Matrix.IsHermitian.cfc_eigenvalues _ _
   rw [h_def, h_trace, he]
   simp only [Function.comp_apply]
   conv_lhs => rw [ ← Equiv.sum_comp e ]
diff --git a/QuantumInfo/States/Mixed/MState.lean b/QuantumInfo/States/Mixed/MState.lean
index 12446857d..ff29ab560 100644
--- a/QuantumInfo/States/Mixed/MState.lean
+++ b/QuantumInfo/States/Mixed/MState.lean
@@ -130,9 +130,8 @@ theorem ext_m {ρ₁ ρ₂ : MState d} (h : ρ₁.m = ρ₂.m) : ρ₁ = ρ₂ :
 theorem m_inj : (MState.m (d := d)).Injective :=
   fun _ _ h ↦ by ext1; ext1; exact h
 
-theorem M_Injective : Function.Injective (MState.M (d := d)) := by
-  intro _ _
-  exact MState.ext
+theorem M_Injective : Function.Injective (MState.M (d := d)) :=
+  fun _ _ => MState.ext
 
 variable (d) in
 /-- The matrices corresponding to MStates are `Convex ℝ` -/
@@ -160,8 +159,7 @@ def nonempty : Nonempty d := by
 
 -- Could have used properties of ρ.spectrum
 theorem eigenvalue_nonneg : ∀ i, 0 ≤ ρ.Hermitian.eigenvalues i := by
-  rw [← Matrix.PosSemidef.nonneg_iff_eigenvalue_nonneg ρ.Hermitian]
-  exact ρ.nonneg
+  exact (zero_le_iff.mp ρ.nonneg).eigenvalues_nonneg
 
 set_option backward.isDefEq.respectTransparency false in
 -- Could have used properties of ρ.spectrum
@@ -188,11 +186,9 @@ scoped instance : Inner Prob (MState d) where
 
 theorem inner_def : ⟪ρ, σ⟫_Prob = ⟨⟪ρ.M, σ.M⟫,
     inner_ge_zero ρ.nonneg σ.nonneg,
-    (inner_le_mul_trace ρ.nonneg σ.nonneg).trans (by simp)⟩ := by
-  rfl
+    (inner_le_mul_trace ρ.nonneg σ.nonneg).trans (by simp)⟩ := rfl
 
-theorem val_inner : (⟪ρ, σ⟫_Prob : ℝ) = ⟪ρ.M, σ.M⟫ := by
-  rfl
+theorem val_inner : (⟪ρ, σ⟫_Prob : ℝ) = ⟪ρ.M, σ.M⟫ := rfl
 
 section exp_val
 
@@ -211,12 +207,10 @@ theorem exp_val_nonneg {T : HermitianMat d ℂ} (h : 0 ≤ T) : 0 ≤ ρ.exp_val
 -- to write first.)
 
 @[simp]
-theorem exp_val_zero : ρ.exp_val 0 = 0 := by
-  simp [MState.exp_val]
+theorem exp_val_zero : ρ.exp_val 0 = 0 := by simp [MState.exp_val]
 
 @[simp]
-theorem exp_val_one : ρ.exp_val 1 = 1 := by
-  simp [MState.exp_val]
+theorem exp_val_one : ρ.exp_val 1 = 1 := by simp [MState.exp_val]
 
 theorem exp_val_le_one {T : HermitianMat d ℂ} (h : T ≤ 1) : ρ.exp_val T ≤ 1 := by
   have hmono := inner_mono ρ.nonneg h
@@ -227,14 +221,12 @@ theorem exp_val_prob {T : HermitianMat d ℂ} (h : 0 ≤ T ∧ T ≤ 1) :
   ⟨ρ.exp_val_nonneg h.1, ρ.exp_val_le_one h.2⟩
 
 theorem exp_val_sub (A B : HermitianMat d ℂ) :
-    ρ.exp_val (A - B) = ρ.exp_val A - ρ.exp_val B := by
-  simp [exp_val, inner_sub_right]
+    ρ.exp_val (A - B) = ρ.exp_val A - ρ.exp_val B := by simp [exp_val, inner_sub_right]
 
 /-- If a PSD observable `A` has expectation value of 0 on a state `ρ`, it must entirely contain the
 support of `ρ` in its kernel. -/
 theorem exp_val_eq_zero_iff {A : HermitianMat d ℂ} (hA₁ : 0 ≤ A) :
-    ρ.exp_val A = 0 ↔ ρ.M.support ≤ A.ker := by
-  exact inner_zero_iff ρ.nonneg hA₁
+    ρ.exp_val A = 0 ↔ ρ.M.support ≤ A.ker := inner_zero_iff ρ.nonneg hA₁
 
 /-- If an observable `A` has expectation value of 1 on a state `ρ`, it must entirely contain the
 support of `ρ` in its 1-eigenspace. -/
@@ -245,13 +237,11 @@ theorem exp_val_eq_one_iff {A : HermitianMat d ℂ} (hA₂ : A ≤ 1) :
   rw [sub_eq_zero, eq_comm]
 
 theorem exp_val_add (A B : HermitianMat d ℂ) :
-    ρ.exp_val (A + B) = ρ.exp_val A + ρ.exp_val B := by
-  simp [exp_val, inner_add_right]
+    ρ.exp_val (A + B) = ρ.exp_val A + ρ.exp_val B := by simp [exp_val, inner_add_right]
 
 @[simp]
 theorem exp_val_smul (r : ℝ) (A : HermitianMat d ℂ) :
-    ρ.exp_val (r • A) = r * ρ.exp_val A := by
-  simp [MState.exp_val]
+    ρ.exp_val (r • A) = r * ρ.exp_val A := by simp [MState.exp_val]
 
 @[gcongr]
 theorem exp_val_le_exp_val (ρ : MState d) {A B : HermitianMat d ℂ} (h : A ≤ B) :
@@ -290,8 +280,7 @@ theorem pure_inner : ⟪pure ψ, pure φ⟫_Prob = ‖Braket.dot ψ φ‖^2 := b
     Complex.normSq_eq_norm_sq (((ψ : Bra d) : d → ℂ) ⬝ᵥ (φ : d → ℂ))
 
 @[simp]
-theorem pure_apply {i j : d} : (pure ψ).m i j = (ψ i) * conj (ψ j) := by
-  rfl
+theorem pure_apply {i j : d} : (pure ψ).m i j = (ψ i) * conj (ψ j) := rfl
 
 theorem pure_mul_self : (pure ψ).m * (pure ψ).m = (pure ψ : Matrix d d ℂ) := by
   dsimp [pure, MState.m]
@@ -529,8 +518,7 @@ def ofClassical (dist : ProbDistribution d) : MState d where
 
 @[simp]
 theorem coe_ofClassical (dist : ProbDistribution d) :
-    (ofClassical dist).M = diagonal ℂ (dist ·) := by
-  rfl
+    (ofClassical dist).M = diagonal ℂ (dist ·) := rfl
 
 theorem ofClassical_pow (dist : ProbDistribution d) (p : ℝ) :
     (ofClassical dist).M ^ p = diagonal ℂ (fun i ↦ (dist i) ^ p) := by
@@ -721,9 +709,7 @@ theorem eq_of_sum_eq_pure {d : Type*} [Fintype d] [DecidableEq d]
         exact ⟨ by assumption, h_eq _ ( ρs i |>.2 ) ( ρs i |>.3 ) ⟩;
       linarith [ h_trace i hi hpi, (ρ.M - (ρs i).M).inner_self_nonneg ];
     -- Since the inner product of a matrix with itself is zero if and only if the matrix is zero, we have ρ.M - (ρs i).M = 0.
-    have h_zero : ρ.M - (ρs i).M = 0 := by
-      apply inner_self_eq_zero.mp h_eq;
-    exact eq_of_sub_eq_zero h_zero;
+    exact eq_of_sub_eq_zero (inner_self_eq_zero.mp h_eq);
   exact MState.ext h_eq.symm
 
 theorem purity_prod {d₁ d₂ : Type*} [Fintype d₁] [Fintype d₂] [DecidableEq d₁] [DecidableEq d₂]
@@ -738,8 +724,7 @@ theorem pure_eq_pure_iff {d : Type*} [Fintype d] [DecidableEq d] (ψ φ : Ket d)
       intro i j;
       replace h := congr_arg ( fun ρ => ρ.M.mat i j ) h ; aesop;
     -- Let $k$ be such that $\varphi_k \neq 0$.
-    obtain ⟨k, hk⟩ : ∃ k, φ.vec k ≠ 0 := by
-      exact φ.exists_ne_zero;
+    obtain ⟨k, hk⟩ : ∃ k, φ.vec k ≠ 0 := φ.exists_ne_zero
     -- Let $z = \frac{\psi_k}{\varphi_k}$. Then $|z| = 1$.
     obtain ⟨z, hz⟩ : ∃ z : ℂ, ψ.vec k = z * φ.vec k ∧ ‖z‖ = 1 := by
       specialize h_eq k k
@@ -765,8 +750,7 @@ theorem pure_eq_pure_iff {d : Type*} [Fintype d] [DecidableEq d] (ψ φ : Ket d)
 
 /-- Two kets are phase-equivalent if and only if their pure states are equal. -/
 theorem PhaseEquiv_iff_pure_eq {d : Type*} [Fintype d] [DecidableEq d] (ψ φ : Ket d) :
-    Ket.PhaseEquiv.r ψ φ ↔ MState.pure ψ = MState.pure φ := by
-  exact (pure_eq_pure_iff ψ φ).symm
+    Ket.PhaseEquiv.r ψ φ ↔ MState.pure ψ = MState.pure φ := (pure_eq_pure_iff ψ φ).symm
 
 /-- `MState.pure` descends to the quotient `KetUpToPhase`. -/
 def pureQ {d : Type*} [Fintype d] [DecidableEq d] : KetUpToPhase d → MState d :=
@@ -788,8 +772,7 @@ theorem pure_separable_imp_IsProd {d₁ d₂ : Type*} [Fintype d₁] [Fintype d
   obtain ⟨ ρLRs, ps, hps ⟩ := h;
   -- Since `pure ψ` is pure (`purity = 1`), by `MState.eq_of_sum_eq_pure`, for any `k` with `p_k > 0`, we have `pure ψ = ρL_k ⊗ᴹ ρR_k`.
   obtain ⟨k, hk⟩ : ∃ k : { x : MState d₁ × MState d₂ // x ∈ ρLRs }, 0 < (ps k : ℝ) ∧ (MState.pure ψ).M = (k.val.1).M ⊗ₖ (k.val.2).M := by
-    have h_pure : (MState.pure ψ).purity = 1 := by
-      exact ( pure_iff_purity_one _ ).mp ⟨ ψ, rfl ⟩;
+    have h_pure : (MState.pure ψ).purity = 1 := ( pure_iff_purity_one _ ).mp ⟨ ψ, rfl ⟩;
     obtain ⟨k, hk⟩ : ∃ k : { x : MState d₁ × MState d₂ // x ∈ ρLRs }, 0 < (ps k : ℝ) := by
       exact ⟨ Classical.choose ( show ∃ k : ρLRs, 0 < ( ps k : ℝ ) from by exact not_forall_not.mp fun h => by have := ps.2; simp_all ), Classical.choose_spec ( show ∃ k : ρLRs, 0 < ( ps k : ℝ ) from by exact not_forall_not.mp fun h => by have := ps.2; simp_all) ⟩;
     refine' ⟨ k, hk, _ ⟩;
@@ -884,8 +867,7 @@ theorem pure_iff_rank_eq_one {d : Type*} [Fintype d] [DecidableEq d] (ρ : MStat
       -- Since ρ is Hermitian and has rank 1, it must be of the form |ψ⟩⟨ψ| for some ket ψ. Use this fact.
       have h_pure : ∃ ψ : d → ℂ, ρ.m = Matrix.of (fun i j => ψ i * star (ψ j)) := by
         have h_rank : ρ.m.rank = 1 := h
-        have h_herm : ρ.m.IsHermitian := by
-          exact ρ.M.property
+        have h_herm : ρ.m.IsHermitian := ρ.M.property
         have := h_herm.spectral_theorem;
         -- Since the rank of ρ.m is 1, the diagonal matrix in the spectral theorem must have exactly one non-zero entry.
         obtain ⟨i, hi⟩ : ∃ i : d, h_herm.eigenvalues i ≠ 0 ∧ ∀ j : d, j ≠ i → h_herm.eigenvalues j = 0 := by
@@ -1055,8 +1037,7 @@ def relabel (ρ : MState d₁) (e : d₂ ≃ d₁) : MState d₂ where
 
 @[simp]
 theorem relabel_m (ρ : MState d₁) (e : d₂ ≃ d₁) :
-    (ρ.relabel e).m = ρ.m.submatrix e e := by
-  rfl
+    (ρ.relabel e).m = ρ.m.submatrix e e := rfl
 
 @[simp]
 theorem relabel_refl {d : Type*} [Fintype d] [DecidableEq d] (ρ : MState d) :
@@ -1074,8 +1055,7 @@ theorem relabel_pure_exists (ψ : Ket d₁) (e : d₂ ≃ d₁) :
 @[simp]
 theorem relabel_relabel {d d₂ d₃ : Type*}
     [Fintype d] [DecidableEq d] [Fintype d₂] [DecidableEq d₂] [Fintype d₃] [DecidableEq d₃]
-    (ρ : MState d) (e : d₂ ≃ d) (e₂ : d₃ ≃ d₂) : (ρ.relabel e).relabel e₂ = ρ.relabel (e₂.trans e) := by
-  rfl
+    (ρ : MState d) (e : d₂ ≃ d) (e₂ : d₃ ≃ d₂) : (ρ.relabel e).relabel e₂ = ρ.relabel (e₂.trans e) := rfl
 
 theorem eq_relabel_iff {d₁ d₂ : Type u} [Fintype d₁] [DecidableEq d₁] [Fintype d₂] [DecidableEq d₂]
     (ρ : MState d₁) (σ : MState d₂) (h : d₁ ≃ d₂) :
@@ -1167,16 +1147,13 @@ def spectrum_SWAP (ρ : MState (d₁ × d₂)) : ∃ e, ρ.SWAP.spectrum.relabel
   rfl
 
 @[simp]
-theorem SWAP_SWAP (ρ : MState (d₁ × d₂)) : ρ.SWAP.SWAP = ρ :=
-  rfl
+theorem SWAP_SWAP (ρ : MState (d₁ × d₂)) : ρ.SWAP.SWAP = ρ := rfl
 
 @[simp]
-theorem traceLeft_SWAP (ρ : MState (d₁ × d₂)) : ρ.SWAP.traceLeft = ρ.traceRight :=
-  rfl
+theorem traceLeft_SWAP (ρ : MState (d₁ × d₂)) : ρ.SWAP.traceLeft = ρ.traceRight := rfl
 
 @[simp]
-theorem traceRight_SWAP (ρ : MState (d₁ × d₂)) : ρ.SWAP.traceRight = ρ.traceLeft :=
-  rfl
+theorem traceRight_SWAP (ρ : MState (d₁ × d₂)) : ρ.SWAP.traceRight = ρ.traceLeft := rfl
 
 /-- The associator that re-clusters the parts of a quantum system. -/
 def assoc (ρ : MState ((d₁ × d₂) × d₃)) : MState (d₁ × d₂ × d₃) :=
@@ -1187,12 +1164,10 @@ def assoc' (ρ : MState (d₁ × d₂ × d₃)) : MState ((d₁ × d₂) × d₃
   ρ.SWAP.assoc.SWAP.assoc.SWAP
 
 @[simp]
-theorem assoc_assoc' (ρ : MState (d₁ × d₂ × d₃)) : ρ.assoc'.assoc = ρ := by
-  rfl
+theorem assoc_assoc' (ρ : MState (d₁ × d₂ × d₃)) : ρ.assoc'.assoc = ρ := rfl
 
 @[simp]
-theorem assoc'_assoc (ρ : MState ((d₁ × d₂) × d₃)) : ρ.assoc.assoc' = ρ := by
-  rfl
+theorem assoc'_assoc (ρ : MState ((d₁ × d₂) × d₃)) : ρ.assoc.assoc' = ρ := rfl
 
 @[simp]
 theorem traceLeft_right_assoc (ρ : MState ((d₁ × d₂) × d₃)) :
@@ -1219,13 +1194,11 @@ theorem traceLeft_assoc' (ρ : MState (d₁ × d₂ × d₃)) :
 
 @[simp]
 theorem traceLeft_left_assoc (ρ : MState ((d₁ × d₂) × d₃)) :
-    ρ.assoc.traceLeft.traceLeft = ρ.traceLeft := by
-  simp [← traceLeft_assoc']
+    ρ.assoc.traceLeft.traceLeft = ρ.traceLeft := by simp [← traceLeft_assoc']
 
 @[simp]
 theorem traceRight_right_assoc' (ρ : MState (d₁ × d₂ × d₃)) :
-    ρ.assoc'.traceRight.traceRight = ρ.traceRight := by
-  simp [assoc']
+    ρ.assoc'.traceRight.traceRight = ρ.traceRight := by simp [assoc']
 
 @[simp]
 theorem traceNorm_eq_one (ρ : MState d) : ρ.m.traceNorm = 1 :=
@@ -1237,12 +1210,10 @@ theorem traceNorm_eq_one (ρ : MState d) : ρ.m.traceNorm = 1 :=
 --TODO: This naming is very inconsistent. Should be better about "prod" vs "kron"
 
 theorem relabel_kron (ρ : MState d₁) (σ : MState d₂) (e : d₃ ≃ d₁) :
-    ((ρ.relabel e) ⊗ᴹ σ) = (ρ ⊗ᴹ σ).relabel (e.prodCongr (Equiv.refl d₂)) := by
-  rfl --is this defeq abuse? I don't know
+    ((ρ.relabel e) ⊗ᴹ σ) = (ρ ⊗ᴹ σ).relabel (e.prodCongr (Equiv.refl d₂)) := rfl
 
 theorem kron_relabel (ρ : MState d₁) (σ : MState d₂) (e : d₃ ≃ d₂) :
-    (ρ ⊗ᴹ σ.relabel e) = (ρ ⊗ᴹ σ).relabel ((Equiv.refl d₁).prodCongr e) := by
-  rfl
+    (ρ ⊗ᴹ σ.relabel e) = (ρ ⊗ᴹ σ).relabel ((Equiv.refl d₁).prodCongr e) := rfl
 
 theorem prod_assoc (ρ : MState d₁) (σ : MState d₂) (τ : MState d₃) :
     (ρ ⊗ᴹ (σ ⊗ᴹ τ)) = (ρ ⊗ᴹ σ ⊗ᴹ τ).relabel (Equiv.prodAssoc d₁ d₂ d₃).symm := by
@@ -1282,8 +1253,7 @@ instance : CompactSpace (MState d) := by
 noncomputable instance : MetricSpace (MState d) :=
   MetricSpace.induced MState.M MState.M_Injective inferInstance
 
-theorem dist_eq (x y : MState d) : dist x y = dist x.M y.M := by
-  rfl
+theorem dist_eq (x y : MState d) : dist x y = dist x.M y.M := rfl
 
 set_option backward.isDefEq.respectTransparency false in
 instance : BoundedSpace (MState d) where
@@ -1315,9 +1285,7 @@ def piProd (ρi : (i:ι) → MState (dI i)) : MState ((i:ι) → dI i) where
     val := Matrix.piProd (fun i ↦ (ρi i).m)
     property := Matrix.IsHermitian.piProd (fun i ↦ (ρi i).Hermitian)
   }
-  nonneg := by
-    rw [zero_le_iff]
-    exact Matrix.PosSemidef.piProd (fun i => psd (ρi i))
+  nonneg := by simpa [zero_le_iff] using Matrix.PosSemidef.piProd (fun i => psd (ρi i))
   tr := by simp [trace, Matrix.trace_piProd]
 
 /-- The n-copy "power" of a mixed state, with the standard basis indexed by pi types. -/
@@ -1371,8 +1339,7 @@ theorem uniform_posDef {d : Type*} [Nonempty d] [Fintype d] [DecidableEq d] :
   exact Fintype.card_pos
 
 theorem posDef_of_unique {d : Type*} [Fintype d] [DecidableEq d] (ρ : MState d) [Unique d] : ρ.m.PosDef := by
-  rw [Subsingleton.allEq ρ uniform]
-  exact uniform_posDef
+  simpa [Subsingleton.allEq ρ uniform] using uniform_posDef
 
 end posdef
 
diff --git a/QuantumInfo/States/Pure/Braket.lean b/QuantumInfo/States/Pure/Braket.lean
index e23c1e2a3..b74b0ff18 100644
--- a/QuantumInfo/States/Pure/Braket.lean
+++ b/QuantumInfo/States/Pure/Braket.lean
@@ -85,8 +85,7 @@ def dot (ξ : Bra d) (ψ : Ket d) : ℂ := ∑ x, (ξ x) * (ψ x)
 
 scoped notation "〈" ξ:90 "‖" ψ:90 "〉" => dot (ξ : Bra _) (ψ : Ket _)
 
-theorem dot_eq_dotProduct (ψ : Bra d) (φ : Ket d) :〈ψ‖φ〉= dotProduct (m := d) ψ φ := by
-  rfl
+theorem dot_eq_dotProduct (ψ : Bra d) (φ : Ket d) :〈ψ‖φ〉= dotProduct (m := d) ψ φ := rfl
 
 end Braket
 
@@ -95,11 +94,9 @@ open Braket
 
 variable {d : Type*} [Fintype d]
 
-theorem Ket.apply (ψ : Ket d) (i : d) : ψ i = ψ.vec i :=
-  rfl
+theorem Ket.apply (ψ : Ket d) (i : d) : ψ i = ψ.vec i := rfl
 
-theorem Bra.apply (ψ : Bra d) (i : d) : ψ i = ψ.vec i :=
-  rfl
+theorem Bra.apply (ψ : Bra d) (i : d) : ψ i = ψ.vec i := rfl
 
 @[ext]
 theorem Ket.ext {ξ ψ : Ket d} (h : ∀ x, ξ x = ψ x) : ξ = ψ :=
@@ -134,11 +131,9 @@ instance instBraOfKet : Coe (Ket d) (Bra d) := ⟨Ket.to_bra⟩
 instance instKetOfBra : Coe (Bra d) (Ket d) := ⟨Bra.to_ket⟩
 
 @[simp]
-theorem Bra.eq_conj (ψ : Ket d) (x : d) :〈ψ∣ x = conj (∣ψ〉 x) :=
-  rfl
+theorem Bra.eq_conj (ψ : Ket d) (x : d) :〈ψ∣ x = conj (∣ψ〉 x) := rfl
 
-theorem Bra.apply' (ψ : Ket d) (i : d) : 〈ψ∣ i = conj (ψ.vec i) :=
-  rfl
+theorem Bra.apply' (ψ : Ket d) (i : d) : 〈ψ∣ i = conj (ψ.vec i) := rfl
 
 theorem Ket.exists_ne_zero (ψ : Ket d) : ∃ x, ψ x ≠ 0 := by
   have hzerolt : ∑ x : d, Complex.normSq (ψ x) > ∑ x : d, 0 := by rw [ψ.normalized, Finset.sum_const_zero]; exact zero_lt_one
@@ -343,8 +338,7 @@ is specifically the MES with an all-positive phase. For instance on `d := Fin 2`
 Bell state. -/
 def Ket.MES (d) [Fintype d] [Nonempty d] : Ket (d × d) where
   vec := fun (i,j) ↦ if i = j then 1 / Real.sqrt (Fintype.card (α := d)) else 0
-  normalized' := by
-    simp [apply_ite, Fintype.sum_prod_type]
+  normalized' := by simp [apply_ite, Fintype.sum_prod_type]
 
 /-- On any space of dimension at least two, the maximally entangled state `MES` is entangled. -/
 theorem Ket.MES_isEntangled [Nontrivial d] : (Ket.MES d).IsEntangled := by
diff --git a/QuantumInfo/States/Pure/Qubit.lean b/QuantumInfo/States/Pure/Qubit.lean
index 52d00715b..0fb9765c6 100644
--- a/QuantumInfo/States/Pure/Qubit.lean
+++ b/QuantumInfo/States/Pure/Qubit.lean
@@ -132,16 +132,13 @@ theorem H_mul_Z_eq_X_mul_H : H * Z = X * H := by
   matrix_expand [H, X, Z]
 
 @[simp]
-theorem S_Z_comm : Z * S = S * Z := by
-  simp [← S_sq, mul_assoc]
+theorem S_Z_comm : Z * S = S * Z := by simp [← S_sq, mul_assoc]
 
 @[simp]
-theorem T_Z_comm : Z * T = T * Z := by
-  simp [← S_sq, ← T_sq, mul_assoc]
+theorem T_Z_comm : Z * T = T * Z := by simp [← S_sq, ← T_sq, mul_assoc]
 
 @[simp]
-theorem S_T_comm : S * T = T * S := by
-  simp [← T_sq, mul_assoc]
+theorem S_T_comm : S * T = T * S := by simp [← T_sq, mul_assoc]
 
 /-- Given a unitary `U` on some Hilbert space `k`, we have the controllized version that acts on `Fin 2 ⊗ k`
 where `U` is conditionally applied if the first qubit is `1`. -/
@@ -180,20 +177,16 @@ lemma CNOT_matrix :
 variable (g : 𝐔[k]) (j₁ j₂ : k)
 
 @[simp]
-theorem controllize_apply_zero_zero : C[g] (0, j₁) (0, j₂) = (1 : 𝐔[k]) j₁ j₂ := by
-  rfl
+theorem controllize_apply_zero_zero : C[g] (0, j₁) (0, j₂) = (1 : 𝐔[k]) j₁ j₂ := rfl
 
 @[simp]
-theorem controllize_apply_zero_one : C[g] (0, j₁) (1, j₂) = 0 := by
-  rfl
+theorem controllize_apply_zero_one : C[g] (0, j₁) (1, j₂) = 0 := rfl
 
 @[simp]
-theorem controllize_apply_one_zero : C[g] (1, j₁) (0, j₂) = 0 := by
-  rfl
+theorem controllize_apply_one_zero : C[g] (1, j₁) (0, j₂) = 0 := rfl
 
 @[simp]
-theorem controllize_apply_one_one : C[g] (1, j₁) (1, j₂) = g j₁ j₂ := by
-  rfl
+theorem controllize_apply_one_one : C[g] (1, j₁) (1, j₂) = g j₁ j₂ := rfl
 
 @[simp]
 theorem controllize_mul (g₁ g₂ : 𝐔[k]) : C[g₁] * C[g₂] = C[g₁ * g₂] := by
@@ -204,8 +197,7 @@ theorem controllize_one : C[(1 : 𝐔[k])] = 1 := by
   matrix_expand
 
 @[simp]
-theorem controllize_mul_inv : C[g] * C[g⁻¹] = 1 := by
-  simp
+theorem controllize_mul_inv : C[g] * C[g⁻¹] = 1 := by simp
 
 open scoped Matrix in
 @[simp]