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Add expmn and mulm_eq1 #127

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Merged
pi8027 merged 2 commits into from
Jun 16, 2026
Merged

Add expmn and mulm_eq1 #127

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db04fdb
Generalize expcmn to expmn, which works for any monomType
pi8027 Aug 26, 2025
78d1a3b
Generalize cmM_eq1 and fmM_eq1 to mulm_eq1
pi8027 Aug 27, 2025
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30 changes: 26 additions & 4 deletions src/monalg.v
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Original file line number Diff line number Diff line change
Expand Up @@ -11,7 +11,7 @@
From mathcomp Require Import tuple bigop ssralg ssrint ssrnum.
Unset SsrOldRewriteGoalsOrder. (* remove the line when requiring MathComp >= 2.6 *)

Require Import xfinmap.

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Set Implicit Arguments.
Unset Strict Implicit.
Expand Down Expand Up @@ -137,11 +137,30 @@
#[export]
HB.instance Definition _ := Monoid.isLaw.Build M 1 mmul mulmA mul1m mulm1.

Definition expmn (x : M) (n : nat) : M := iterop n *%M x 1%M.

Arguments expmn : simpl never.

Lemma unitmP (x y : M) : reflect (x == 1 /\ y == 1) (x * y == 1).
Proof.
by apply: (iffP eqP)=> [/unitm[-> ->]|[/eqP-> /eqP->]] //; rewrite mulm1.
Qed.

Lemma mulm_eq1 (x y : M) : (x * y == 1) = (x == 1) && (y == 1).
Proof. exact/unitmP/andP. Qed.

Lemma expmnS (x : M) (n : nat) : expmn x n.+1 = x * expmn x n.
Proof. by rewrite /expmn !Monoid.iteropE iterS. Qed.

Lemma expmnD (x : M) (n m : nat) : expmn x (n + m) = expmn x n * expmn x m.
Proof.
elim: n => [|n IHn]; first by rewrite mul1m.
by rewrite addSn !expmnS IHn mulmA.
Qed.

Lemma expmnSr (x : M) (n : nat) : expmn x n.+1 = expmn x n * x.
Proof. by rewrite -addn1 expmnD. Qed.

End Monomial.

#[export]
Expand Down Expand Up @@ -1414,8 +1433,6 @@

Definition divcm m1 m2 : cmonom I := [cmonom m1 i - m2 i | i in finsupp m1]%M.

Definition expcmn m n : cmonom I := iterop n mulcm m onecm.

Lemma onecmE i : onecm i = 0%N.
Proof. by rewrite fsfun_ffun insubF. Qed.

Expand Down Expand Up @@ -1454,6 +1471,9 @@
mulcmA mul0cm mulcm0 mulcm_eq0.
HB.instance Definition _ := MonomialDef_isConomialDef.Build (cmonom I) mulcmC.

#[deprecated(since="multinomials 2.5.0", use=expmn)]
Definition expcmn m n : cmonom I := expmn m n.

End CmonomCanonicals.

(* -------------------------------------------------------------------- *)
Expand Down Expand Up @@ -1552,8 +1572,9 @@
Lemma mdeg_eq0 m : (mdeg m == 0%N) = (m == 1%M).
Proof. exact/mf_eq0. Qed.

#[deprecated(since="multinomials 2.5.0", use=mulm_eq1)]
Lemma cmM_eq1 m1 m2 : (m1 * m2 == 1)%M = (m1 == 1%M) && (m2 == 1%M).
Proof. by rewrite -!mdeg_eq0 mdegM addn_eq0. Qed.
Proof. exact: mulm_eq1. Qed.

Lemma cm1_eq1 i : (U_(i) == 1)%M = false.
Proof. by rewrite -mdeg_eq0 mdegU. Qed.
Expand Down Expand Up @@ -1761,8 +1782,9 @@
Lemma fdeg_eq0 m : (fdeg m == 0%N) = (m == 1%M).
Proof. exact/mf_eq0. Qed.

#[deprecated(since="multinomials 2.5.0", use=mulm_eq1)]
Lemma fmM_eq1 m1 m2 : (m1 * m2 == 1)%M = (m1 == 1%M) && (m2 == 1%M).
Proof. by rewrite -!fdeg_eq0 fdegM addn_eq0. Qed.
Proof. exact: mulm_eq1. Qed.

Lemma fm1_eq1 i : (U_(i) == 1)%M = false.
Proof. by rewrite -fdeg_eq0 fdegU. Qed.
Expand Down
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