fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
mulIr0_rregx : (forall y, y * x = 0 -> y = 0) -> rreg x.
Proof. exact: (@mulrI0_lreg R^c). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulIr0_rreg
| |
rregNx : rreg x -> rreg (- x). Proof. exact: (@lregN R^c). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rregN
| |
RecordNmodule_isLSemiModule (R : pzSemiRingType) V of Nmodule V := {
scale : R -> V -> V;
scalerA : forall a b v, scale a (scale b v) = scale (a * b) v;
scale0r : forall v, scale 0 v = 0;
scale1r : left_id 1 scale;
scalerDr : right_distributive scale +%R;
scalerDl : forall v, {morph scale^~ v: a b / a + b}
}.
#[short(type="lSemiModType")]
HB.structure Definition LSemiModule (R : pzSemiRingType) :=
{M of Nmodule M & Nmodule_isLSemiModule R M}.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
DefinitionLmodule (R : pzRingType) :=
{M of Zmodule M & Nmodule_isLSemiModule R M}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
RecordZmodule_isLmodule (R : pzRingType) V of Zmodule V := {
scale : R -> V -> V;
scalerA : forall a b v, scale a (scale b v) = scale (a * b) v;
scale1r : left_id 1 scale;
scalerDr : right_distributive scale +%R;
scalerDl : forall v, {morph scale^~ v: a b / a + b}
}.
HB.builders Context R V of Zmodule_isLmodule R V.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
scale0rv : scale 0 v = 0.
Proof. by apply: (addIr (scale 1 v)); rewrite -scalerDl !add0r. Qed.
HB.instance Definition _ :=
Nmodule_isLSemiModule.Build R V scalerA scale0r scale1r scalerDr scalerDl.
HB.end.
HB.factory Record LSemiModule_isLmodule (R : pzRingType) V
of LSemiModule R V := {}.
HB.builders Context R V of LSemiModule_isLmodule R V.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scale0r
| |
opp: V -> V := scale (- 1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
opp
| |
addNr: left_inverse 0 opp +%R.
Proof.
move=> v; suff : scale (-1 + 1) v = 0 by rewrite scalerDl scale1r.
by rewrite addNr scale0r.
Qed.
HB.instance Definition _ := Nmodule_isZmodule.Build V addNr.
HB.end.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
addNr
| |
scaler0a : a *: 0 = 0 :> V.
Proof. by rewrite -[0 in LHS](scale0r 0) scalerA mulr0 scale0r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaler0
| |
scaler_natn v : n%:R *: v = v *+ n.
Proof.
elim: n => /= [|n]; first by rewrite scale0r.
by rewrite !mulrS scalerDl ?scale1r => ->.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaler_nat
| |
scalerMnla v n : a *: v *+ n = (a *+ n) *: v.
Proof.
elim: n => [|n IHn]; first by rewrite !mulr0n scale0r.
by rewrite !mulrSr IHn scalerDl.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalerMnl
| |
scalerMnra v n : a *: v *+ n = a *: (v *+ n).
Proof.
elim: n => [|n IHn]; first by rewrite !mulr0n scaler0.
by rewrite !mulrSr IHn scalerDr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalerMnr
| |
scaler_sumlv I r (P : pred I) F :
(\sum_(i <- r | P i) F i) *: v = \sum_(i <- r | P i) F i *: v.
Proof. exact: (big_morph _ (scalerDl v) (scale0r v)). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaler_suml
| |
scaler_sumra I r (P : pred I) (F : I -> V) :
a *: (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) a *: F i.
Proof. exact: big_endo (scalerDr a) (scaler0 a) I r P F. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaler_sumr
| |
scaler_closed:= forall a, {in S, forall v, a *: v \in S}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaler_closed
| |
subsemimod_closed:= nmod_closed S /\ scaler_closed.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemimod_closed
| |
subsemimod_closedD: subsemimod_closed -> nmod_closed S.
Proof. by case. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemimod_closedD
| |
subsemimod_closedZ: subsemimod_closed -> scaler_closed.
Proof. by case. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemimod_closedZ
| |
scaleNra v : - a *: v = - (a *: v).
Proof. by apply: (addIr (a *: v)); rewrite -scalerDl !addNr scale0r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaleNr
| |
scaleN1rv : - 1 *: v = - v.
Proof. by rewrite scaleNr scale1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaleN1r
| |
scalerNa v : a *: - v = - (a *: v).
Proof. by apply: (addIr (a *: v)); rewrite -scalerDr !addNr scaler0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalerN
| |
scalerBla b v : (a - b) *: v = a *: v - b *: v.
Proof. by rewrite scalerDl scaleNr. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalerBl
| |
scalerBra u v : a *: (u - v) = a *: u - a *: v.
Proof. by rewrite scalerDr scalerN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalerBr
| |
scaler_sign(b : bool) v : (-1) ^+ b *: v = (if b then - v else v).
Proof. by case: b; rewrite ?scaleNr scale1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scaler_sign
| |
signrZKn : @involutive V ( *:%R ((-1) ^+ n)).
Proof. by move=> u; rewrite scalerA -expr2 sqrr_sign scale1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
signrZK
| |
linear_closed:= forall a, {in S &, forall u v, a *: u + v \in S}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linear_closed
| |
submod_closed:= 0 \in S /\ linear_closed.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
submod_closed
| |
linear_closedB: linear_closed -> subr_2closed S.
Proof. by move=> Slin u v Su Sv; rewrite addrC -scaleN1r Slin. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
linear_closedB
| |
submod_closedB: submod_closed -> zmod_closed S.
Proof. by case=> S0 /linear_closedB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
submod_closedB
| |
submod_closed_semi: submod_closed -> subsemimod_closed S.
Proof.
move=> /[dup] /submod_closedB /zmod_closedD SD [S0 Slin]; split => // a v Sv.
by rewrite -[a *: v]addr0 Slin.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
submod_closed_semi
| |
subsemimod_closed_submod: subsemimod_closed S -> submod_closed.
Proof. by case=> [[S0 SD] SZ]; split => // a u v Su Sv; apply/SD/Sv/SZ. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemimod_closed_submod
| |
subsemimod_closedB: subsemimod_closed S -> zmod_closed S.
Proof. by move/subsemimod_closed_submod/submod_closedB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemimod_closedB
| |
RecordLSemiModule_isLSemiAlgebra R V
of NzSemiRing V & LSemiModule R V := {
scalerAl : forall (a : R) (u v : V), a *: (u * v) = (a *: u) * v
}.
#[short(type="lSemiAlgType")]
HB.structure Definition LSemiAlgebra R :=
{A of LSemiModule R A & NzSemiRing A & LSemiModule_isLSemiAlgebra R A}.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
DefinitionLalgebra R :=
{A of Lmodule R A & NzRing A & LSemiModule_isLSemiAlgebra R A}.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
RecordLmodule_isLalgebra R V of NzRing V & Lmodule R V := {
scalerAl : forall (a : R) (u v : V), a *: (u * v) = (a *: u) * v
}.
HB.builders Context R V of Lmodule_isLalgebra R V.
HB.instance Definition _ := LSemiModule_isLSemiAlgebra.Build R V scalerAl.
HB.end.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
regularR : Type := R.
Local Notation "R ^o" := (regular R) : type_scope.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
regular
| |
Definition_ (V : nmodType) := Nmodule.on V^o.
#[export]
HB.instance Definition _ (V : zmodType) := Zmodule.on V^o.
#[export]
HB.instance Definition _ (R : pzSemiRingType) := PzSemiRing.on R^o.
#[export]
HB.instance Definition _ (R : nzSemiRingType) := NzSemiRing.on R^o.
#[export]
HB.instance Definition _ (R : pzSemiRingType) :=
@Nmodule_isLSemiModule.Build R R^o
mul mulrA mul0r mul1r mulrDr (fun v a b => mulrDl a b v).
#[export]
HB.instance Definition _ (R : nzSemiRingType) :=
LSemiModule_isLSemiAlgebra.Build R R^o mulrA.
#[export]
HB.instance Definition _ (R : pzRingType) := PzRing.on R^o.
#[export]
HB.instance Definition _ (R : nzRingType) := NzRing.on R^o.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
mulr_algl(a : R) (x : A) : (a *: 1) * x = a *: x.
Proof. by rewrite -scalerAl mul1r. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr_algl
| |
subsemialg_closed:=
[/\ 1 \in S, nmod_closed S, scaler_closed S & mulr_2closed S].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemialg_closed
| |
subsemialg_closedZ: subsemialg_closed -> subsemimod_closed S.
Proof. by case. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemialg_closedZ
| |
subsemialg_closedM: subsemialg_closed -> semiring_closed S.
Proof. by case. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemialg_closedM
| |
subalg_closed:= [/\ 1 \in S, linear_closed S & mulr_2closed S].
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subalg_closed
| |
subalg_closedZ: subalg_closed -> submod_closed S.
Proof. by case=> S1 Slin _; split; rewrite // -(subrr 1) linear_closedB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subalg_closedZ
| |
subalg_closedBM: subalg_closed -> subring_closed S.
Proof. by case=> S1 Slin SM; split=> //; apply: linear_closedB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subalg_closedBM
| |
subalg_closed_semi: subalg_closed -> subsemialg_closed S.
Proof.
move=> /[dup] /subalg_closedZ /submod_closedB /zmod_closedD.
by move=> [S0 SD] [S1 Slin SM]; split => // a u Su; rewrite -[a *: u]addr0 Slin.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subalg_closed_semi
| |
subsemialg_closed_subalg: subsemialg_closed S -> subalg_closed.
Proof. by case=> S1 [S0 SD] SZ SM; split => // a u v Su Sv; apply/SD/Sv/SZ. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemialg_closed_subalg
| |
subsemialg_closedBM: subsemialg_closed S -> subring_closed S.
Proof. by move/subsemialg_closed_subalg/subalg_closedBM. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
subsemialg_closedBM
| |
mull_funa f x := a * f x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mull_fun
| |
mulr_funa f x := f x * a.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mulr_fun
| |
mul_funf g x := f x * g x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
mul_fun
| |
scale_funa (f : U -> V) x := a *: f x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scale_fun
| |
in_algk : A := k%:A.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
in_alg
| |
raddfMnatn x : f (n%:R * x) = n%:R * f x.
Proof. by rewrite !mulr_natl raddfMn. Qed.
Variables (U : lSemiModType R) (V : lSemiModType S) (h : {additive U -> V}).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfMnat
| |
raddfZnatn u : h (n%:R *: u) = n%:R *: h u.
Proof. by rewrite !scaler_nat raddfMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfZnat
| |
Definition_ := isNmodMorphism.Build U R (a \*o f)
mull_fun_is_nmod_morphism.
Fact mulr_fun_is_nmod_morphism : nmod_morphism (a \o* f).
Proof. by split=> [|x y]; rewrite /= ?raddf0 ?mul0r// raddfD mulrDl. Qed.
#[export]
HB.instance Definition _ := isNmodMorphism.Build U R (a \o* f)
mulr_fun_is_nmod_morphism.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
raddfN: {morph f : x / - x}. Proof. exact: raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfN
| |
raddfB: {morph f : x y / x - y}. Proof. exact: raddfB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfB
| |
raddf_inj: (forall x, f x = 0 -> x = 0) -> injective f.
Proof. exact: raddf_inj. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddf_inj
| |
raddfMNnn : {morph f : x / x *- n}. Proof. exact: raddfMNn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfMNn
| |
raddfMsignn x : f ((-1) ^+ n * x) = (-1) ^+ n * f x.
Proof. by rewrite !(mulr_sign, =^~ signr_odd) (fun_if f) raddfN. Qed.
Variables (U : lmodType R) (V : lmodType S) (h : {additive U -> V}).
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfMsign
| |
raddfZsignn u : h ((-1) ^+ n *: u) = (-1) ^+ n *: h u.
Proof. by rewrite !(scaler_sign, =^~ signr_odd) (fun_if h) raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
raddfZsign
| |
Definition_ :=
isNmodMorphism.Build V V ( *:%R a) (conj (scaler0 _ a) (scalerDr a)).
#[export]
HB.instance Definition _ := Additive.copy (a \*: f) (f \; *:%R a).
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
multiplicative(R S : pzSemiRingType) (f : R -> S) : Prop :=
{morph f : x y / x * y}%R * (f 1 = 1).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
multiplicative
| |
monoid_morphism(R S : pzSemiRingType) (f : R -> S) : Prop :=
(f 1 = 1) * {morph f : x y / x * y}%R.
HB.mixin Record isMonoidMorphism (R S : pzSemiRingType) (f : R -> S) := {
monoid_morphism_subproof : monoid_morphism f
}.
HB.structure Definition RMorphism (R S : pzSemiRingType) :=
{f of @isNmodMorphism R S f & isMonoidMorphism R S f}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
monoid_morphism
| |
RecordisMultiplicative (R S : pzSemiRingType) (f : R -> S) := {
rmorphism_subproof : multiplicative f
}.
HB.builders Context R S f of isMultiplicative R S f.
#[warning="-HB.no-new-instance"]
HB.instance Definition _ := isMonoidMorphism.Build R S f
(rmorphism_subproof.2, rmorphism_subproof.1).
HB.end.
|
HB.factory
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
rmorph0: f 0 = 0. Proof. exact: raddf0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph0
| |
rmorphD: {morph f : x y / x + y}. Proof. exact: raddfD. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphD
| |
rmorphMnn : {morph f : x / x *+ n}. Proof. exact: raddfMn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphMn
| |
rmorph_sumI r (P : pred I) E :
f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).
Proof. exact: raddf_sum. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_sum
| |
rmorphism_monoidP: monoid_morphism f.
Proof. exact: monoid_morphism_subproof. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `rmorphism_monoidP` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphism_monoidP
| |
rmorphismMP: multiplicative f :=
(fun p => (p.2, p.1)) rmorphism_monoidP.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphismMP
| |
rmorph1: f 1 = 1. Proof. by case: rmorphism_monoidP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph1
| |
rmorphM: {morph f: x y / x * y}. Proof. by case: rmorphism_monoidP. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphM
| |
rmorph_prodI r (P : pred I) E :
f (\prod_(i <- r | P i) E i) = \prod_(i <- r | P i) f (E i).
Proof. exact: (big_morph f rmorphM rmorph1). Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_prod
| |
rmorphXnn : {morph f : x / x ^+ n}.
Proof. by elim: n => [|n IHn] x; rewrite ?rmorph1 // !exprS rmorphM IHn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphXn
| |
rmorph_natn : f n%:R = n%:R. Proof. by rewrite rmorphMn rmorph1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_nat
| |
rmorph_eq_natx n : injective f -> (f x == n%:R) = (x == n%:R).
Proof. by move/inj_eq <-; rewrite rmorph_nat. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_eq_nat
| |
rmorph_eq1x : injective f -> (f x == 1) = (x == 1).
Proof. exact: rmorph_eq_nat 1%N. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_eq1
| |
can2_monoid_morphismf' : cancel f f' -> cancel f' f -> monoid_morphism f'.
Proof.
move=> fK f'K.
by split=> [|x y]; apply: (canLR fK); rewrite /= (rmorph1, rmorphM) ?f'K.
Qed.
#[deprecated(since="mathcomp 2.5.0",
note="use `can2_monoid_morphism` instead")]
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
can2_monoid_morphism
| |
can2_rmorphismf' (cff' : cancel f f') :=
(fun p => (p.2, p.1)) \o (can2_monoid_morphism cff').
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
can2_rmorphism
| |
rmorph_pchar(R S : nzSemiRingType) (f : {rmorphism R -> S}) p :
p \in pchar R -> p \in pchar S.
Proof.
by rewrite !inE -(rmorph_nat f) => /andP[-> /= /eqP->]; rewrite rmorph0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_pchar
| |
Definition_ := isMonoidMorphism.Build R R idfun
idfun_is_monoid_morphism.
Fact comp_is_monoid_morphism : monoid_morphism (f \o g).
Proof. by split=> [|x y] /=; rewrite ?rmorph1 ?rmorphM. Qed.
#[export]
HB.instance Definition _ := isMonoidMorphism.Build R T (f \o g)
comp_is_monoid_morphism.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
rmorphN: {morph f : x / - x}. Proof. exact: raddfN. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphN
| |
rmorphB: {morph f: x y / x - y}. Proof. exact: raddfB. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphB
| |
rmorphMNnn : {morph f : x / x *- n}. Proof. exact: raddfMNn. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphMNn
| |
rmorphMsignn : {morph f : x / (- 1) ^+ n * x}.
Proof. exact: raddfMsign. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphMsign
| |
rmorphN1: f (- 1) = (- 1). Proof. by rewrite rmorphN rmorph1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorphN1
| |
rmorph_signn : f ((- 1) ^+ n) = (- 1) ^+ n.
Proof. by rewrite rmorphXn /= rmorphN1. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_sign
| |
Definition_ :=
isNmodMorphism.Build R A (in_alg A) in_alg_is_nmod_morphism.
Fact in_alg_is_monoid_morphism : monoid_morphism (in_alg A).
Proof. by split=> [|x y]; rewrite /= ?scale1r // mulr_algl scalerA. Qed.
#[export]
HB.instance Definition _ := isMonoidMorphism.Build R A (in_alg A)
in_alg_is_monoid_morphism.
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
in_algEa : in_alg A a = a%:A. Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
in_algE
| |
rmorph_char:= rmorph_pchar (only parsing).
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
rmorph_char
| |
RecordisPreLaw
(R : pzSemiRingType) (V : nmodType) (op : R -> V -> V) := {
op_nmod_morphism : forall a, nmod_morphism (op a);
}.
#[export]
HB.structure Definition PreLaw R V := {op of isPreLaw R V op}.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Record
| |
preLaw:= PreLaw.type.
HB.mixin Record isSemiLaw
(R : pzSemiRingType) (V : nmodType) (op : R -> V -> V) := {
op0v : forall v, op 0 v = 0;
op1v : op 1 =1 id;
opA : forall a b v, op a (op b v) = op (a * b) v;
}.
#[export]
HB.structure Definition SemiLaw R V :=
{op of isPreLaw R V op & isSemiLaw R V op}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
preLaw
| |
semiLaw:= SemiLaw.type.
HB.mixin Record isLaw (R : pzRingType) (V : zmodType) (op : R -> V -> V) :=
{ N1op : op (-1) =1 -%R }.
#[export]
HB.structure Definition Law (R : pzRingType) (V : zmodType) :=
{op of isPreLaw R V op & isLaw R V op}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semiLaw
| |
law:= Law.type.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
law
| |
Definition_ (R : pzSemiRingType) :=
Scale.isPreLaw.Build R R *%R (fun => mull_fun_is_nmod_morphism _ idfun).
#[export]
HB.instance Definition _ (R : pzSemiRingType) :=
Scale.isSemiLaw.Build R R *%R mul0r mul1r mulrA.
#[export]
HB.instance Definition _ (R : pzRingType) :=
Scale.isLaw.Build R R *%R (@mulN1r R).
#[export]
HB.instance Definition _ (R : pzSemiRingType) (V : lSemiModType R) :=
Scale.isPreLaw.Build R V *:%R (fun => (scaler0 _ _, scalerDr _)).
#[export]
HB.instance Definition _ (R : pzSemiRingType) (V : lSemiModType R) :=
Scale.isSemiLaw.Build R V *:%R scale0r scale1r (@scalerA _ _).
#[export]
HB.instance Definition _ (R : pzRingType) (U : lmodType R) :=
Scale.isLaw.Build R U *:%R (@scaleN1r R U).
#[export]
HB.instance Definition _
(R : pzSemiRingType) (V : nmodType) (s : Scale.preLaw R V)
(aR : pzSemiRingType) (nu : {rmorphism aR -> R}) :=
Scale.isPreLaw.Build aR V (nu \; s) (fun => Scale.op_nmod_morphism _).
#[export]
HB.instance Definition _
(R : pzSemiRingType) (V : nmodType) (s : Scale.semiLaw R V)
(aR : pzSemiRingType) (nu : {rmorphism aR -> R}) :=
Scale.isSemiLaw.Build aR V (nu \; s)
(Scale.comp_op0v s nu) (Scale.comp_op1v s nu) (Scale.comp_opA s nu).
#[export]
HB.instance Definition _
(R : pzRingType) (V : zmodType) (s : Scale.law R V)
(aR : pzRingType) (nu : {rmorphism aR -> R}) :=
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
Definition
| |
scalable_for(R : pzSemiRingType) (U : lSemiModType R) (V : nmodType)
(s : R -> V -> V) (f : U -> V) :=
forall a, {morph f : u / a *: u >-> s a u}.
HB.mixin Record isScalable (R : pzSemiRingType) (U : lSemiModType R)
(V : nmodType) (s : R -> V -> V) (f : U -> V) := {
semi_linear_subproof : scalable_for s f;
}.
HB.structure Definition Linear (R : pzSemiRingType)
(U : lSemiModType R) (V : nmodType) (s : R -> V -> V) :=
{f of @isNmodMorphism U V f & isScalable R U V s f}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
scalable_for
| |
semilinear_for(R : pzSemiRingType)
(U : lSemiModType R) (V : nmodType) (s : R -> V -> V) (f : U -> V) : Type :=
scalable_for s f * {morph f : x y / x + y}.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
semilinear_for
| |
nmod_morphism_semilinear(R : pzSemiRingType)
(U : lSemiModType R) (V : nmodType) (s : Scale.semiLaw R V) (f : U -> V) :
semilinear_for s f -> nmod_morphism f.
Proof.
by case=> sf Df; split => //; rewrite -[0 in LHS](scale0r 0) sf Scale.op0v.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
nmod_morphism_semilinear
| |
additive_semilinear:= nmod_morphism_semilinear.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] |
algebra/ssralg.v
|
additive_semilinear
|
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