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auto-task(exists_boost_mul_rotation): prove restricted Lorentz elements are a boost times a rotation #1350

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auto-task(exists_boost_mul_rotation): prove restricted Lorentz elements are a boost times a rotation #1350

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auto-task(exists_boost_mul_rotation): prove restricted Lorentz elemen…
jstoobysmith Jul 1, 2026
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Update Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean
jstoobysmith Jul 3, 2026
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3 changes: 0 additions & 3 deletions Physlib/Relativity/LorentzGroup/Restricted/Basic.lean
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Expand Up @@ -16,9 +16,6 @@ This file is currently a stub.

@[expose] public section

TODO "Prove that every member of the restricted Lorentz group is
combination of a boost and a rotation."

namespace LorentzGroup

open Matrix
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10 changes: 10 additions & 0 deletions Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean
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Expand Up @@ -102,6 +102,16 @@ def toBoostRotation {d} : LorentzGroup.restricted d ≃ₜ Lorentz.Velocity d ×
continuous_toFun := by fun_prop
continuous_invFun := by fun_prop

/-- Every element of the restricted Lorentz group can be written as a product of a
generalized boost and a rotation. -/
theorem exists_boost_mul_rotation {d} (Λ : LorentzGroup.restricted d) :
∃ (v : Lorentz.Velocity d) (R : Rotations d),
(Λ : LorentzGroup d) = generalizedBoost 0 v * (R : LorentzGroup d) := by
refine ⟨toVelocity Λ, toRotation Λ, ?_⟩
have h : (toRotation Λ : LorentzGroup d) =
(generalizedBoost 0 (toVelocity Λ))⁻¹ * (Λ : LorentzGroup d) := rfl
rw [h, mul_inv_cancel_left]
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end LorentzGroup

end
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