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 |
|---|---|---|---|---|---|---|
gsimp:= (@mulg1, @mul1g, (@invg1, @invgK), (@mulgV, @mulVg)). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gsimp | |
gnorm:= (gsimp, (@mulgK, @mulgVK, (@mulgA, @invgM))).
Arguments mulgI [G].
Arguments mulIg [G].
Arguments conjg_inj {G} x [x1 x2].
Arguments commgP {G x y}.
Arguments conjg_fixP {G x y}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gnorm | |
RecordisMultiplicative (G H : magmaType) (apply : G -> H) := {
gmulfM : {morph apply : x y / x * y}
}.
HB.structure Definition Multiplicative (G H : magmaType) :=
{f of isMultiplicative G H f}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
RecordMultiplicative_isUMagmaMorphism (G H : baseUMagmaType)
(f : G -> H) := {
gmulf1 : f 1 = 1
}.
HB.structure Definition UMagmaMorphism (G H : baseUMagmaType) :=
{f of Multiplicative_isUMagmaMorphism G H f & isMultiplicative G H f}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
monoid_morphism(G H : baseUMagmaType) (f : G -> H) : Prop :=
(f 1 = 1) * {morph f : x y / x * y}.
HB.factory Record isUMagmaMorphism (G H : baseUMagmaType) (f : G -> H) := {
monoid_morphism_subproof : monoid_morphism f
}.
HB.builders Context G H apply of isUMagmaMorphism G H apply.
HB.instance Definition _ :=
isMultiplicative.Build G H apply monoid_morphism_subproof.2.
HB.instance Definition _ :=
Multiplicative_isUMagmaMorphism.Build G H apply monoid_morphism_subproof.1.
HB.end.
HB.factory Record isGroupMorphism (G H : groupType) (f : G -> H) := {
gmulfF : {morph f : x y / x / y}
}.
HB.builders Context G H apply of isGroupMorphism G H apply.
Local Lemma gmulf1 : apply 1 = 1.
Proof. by rewrite -[1]divg1 gmulfF divgg. Qed.
Local Lemma gmulfM : {morph apply : x y / x * y}.
Proof.
move=> x y; rewrite -[y in LHS] invgK -[y^-1]mul1g.
by rewrite !gmulfF gmulf1 div1g invgK.
Qed.
HB.instance Definition _ := isMultiplicative.Build G H apply gmulfM.
HB.instance Definition _ :=
Multiplicative_isUMagmaMorphism.Build G H apply gmulf1.
HB.end. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | monoid_morphism | |
Definition_ G H (f : UMagmaMorphism.type G H) :=
UMagmaMorphism.on (Multiplicative.sort f). | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
mul_fun(f g : T -> G) x := f x * g x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mul_fun | |
one_fun: T -> G := fun=> 1. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | one_fun | |
can2_gmulfMf' : cancel f f' -> cancel f' f -> {morph f' : x y / x * y}.
Proof. by move=> fK f'K x y; apply: (canLR fK); rewrite gmulfM !f'K. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | can2_gmulfM | |
gmulf_commutex y : commute x y -> commute (f x) (f y).
Proof. by move=> xy; rewrite /commute -!gmulfM xy. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulf_commute | |
gmulf_eq1x : injective f -> (f x == 1) = (x == 1).
Proof. by move=> /inj_eq <-; rewrite gmulf1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulf_eq1 | |
can2_gmulf1f' : cancel f f' -> cancel f' f -> f' 1 = 1.
Proof. by move=> fK f'K; apply: (canLR fK); rewrite gmulf1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | can2_gmulf1 | |
gmulfXnn : {morph f : x / x ^+ n}.
Proof. by elim: n => [|[|n] IHn] x /=; rewrite ?(gmulf1, gmulfM) // IHn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulfXn | |
gmulf_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 gmulfM gmulf1). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulf_prod | |
gmulfV: {morph f : x / x^-1}.
Proof. by move=> x; apply/divg1_eq; rewrite invgK -gmulfM mulVg gmulf1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulfV | |
gmulfF: {morph f : x y / x / y}.
Proof. by move=> x y; rewrite gmulfM gmulfV. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulfF | |
gmulf_inj: (forall x, f x = 1 -> x = 1) -> injective f.
Proof. by move=> fI x y xy; apply/divg1_eq/fI; rewrite gmulfF xy divgg. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulf_inj | |
gmulfXVnn : {morph f : x / x ^- n}.
Proof. by move=> x /=; rewrite gmulfV gmulfXn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulfXVn | |
gmulfJ: {morph f : x y / x ^ y}.
Proof. by move=> x y; rewrite !gmulfM/= gmulfV. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulfJ | |
gmulfR: {morph f : x y / [~ x, y]}.
Proof. by move=> x y; rewrite !gmulfM/= !gmulfV. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gmulfR | |
Definition_ := isMultiplicative.Build G G idfun idfun_gmulfM.
Fact comp_gmulfM : {morph f \o h : x y / x * y}.
Proof. by move=> x y /=; rewrite !gmulfM. Qed.
HB.instance Definition _ := isMultiplicative.Build G K (f \o h) comp_gmulfM. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
Definition_ :=
Multiplicative_isUMagmaMorphism.Build H H idfun idfun_gmulf1.
Fact one_fun_gmulfM : {morph @one_fun G H : x y / x * y}.
Proof. by move=> x y; rewrite mulg1. Qed.
HB.instance Definition _ :=
isMultiplicative.Build G H (@one_fun G H) one_fun_gmulfM. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
Definition_ :=
Multiplicative_isUMagmaMorphism.Build G K (f \o h) comp_gmulf1.
Fact one_fun_gmulf1 : @one_fun G H 1 = 1.
Proof. by []. Qed.
HB.instance Definition _ :=
Multiplicative_isUMagmaMorphism.Build G H (@one_fun G H) one_fun_gmulf1.
Fact mul_fun_gmulf1 : (f \* g) 1 = 1.
Proof. by rewrite /= !gmulf1 mulg1. Qed. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
RecordisMulClosed (G : magmaType) (S : {pred G}) := {
gpredM : mulg_closed S
}.
HB.mixin Record isMul1Closed (G : baseUMagmaType) (S : {pred G}) := {
gpred1 : 1 \in S
}.
HB.mixin Record isInvClosed (G : groupType) (S : {pred G}) := {
gpredVr : invg_closed S
}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
DefinitionMulClosed G := {S of isMulClosed G S}.
#[short(type="umagmaClosed")]
HB.structure Definition UMagmaClosed G :=
{S of isMul1Closed G S & isMulClosed G S}.
#[short(type="invgClosed")]
HB.structure Definition InvClosed G := {S of isInvClosed G S}.
#[short(type="groupClosed")]
HB.structure Definition GroupClosed G :=
{S of isInvClosed G S & isMul1Closed G S & isMulClosed G S}. | HB.structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
gpred_prodI r (P : pred I) F :
(forall i, P i -> F i \in S) -> \prod_(i <- r | P i) F i \in S.
Proof. by move=> IH; elim/big_ind: _; [apply: gpred1 | apply: gpredM |]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpred_prod | |
gpredXnn : {in S, forall u, u ^+ n \in S}.
Proof.
by move=> x xS; elim: n => [|[//|n] IHn]; [exact: gpred1 | exact: gpredM].
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredXn | |
gpredV(S : invgClosed G) : {mono (@inv G): u / u \in S}.
Proof. by move=> u; apply/idP/idP=> /gpredVr; rewrite ?invgK; apply. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredV | |
gpredF: {in S &, forall u v, u / v \in S}.
Proof. by move=> x y xS yS; rewrite gpredM// gpredV. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredF | |
gpredFCu v : u / v \in S = (v / u \in S).
Proof. by rewrite -gpredV invgF. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredFC | |
gpredXNnn: {in S, forall u, u ^- n \in S}.
Proof. by move=> x xS; apply/gpredVr/gpredXn. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredXNn | |
gpredMrx y : x \in S -> (y * x \in S) = (y \in S).
Proof.
move=> Sx; apply/idP/idP => [Sxy|/gpredM-> //].
by rewrite -(mulgK x y) gpredF.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredMr | |
gpredMlx y : x \in S -> (x * y \in S) = (y \in S).
Proof.
move=> Sx; apply/idP/idP => [Sxy|/(gpredM x y Sx)//].
by rewrite -(mulKg x y) gpredM// gpredV.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredMl | |
gpredFrx y : x \in S -> (y / x \in S) = (y \in S).
Proof. by rewrite -gpredV; apply: gpredMr. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredFr | |
gpredFlx y : x \in S -> (x / y \in S) = (y \in S).
Proof. by rewrite -(gpredV S y); apply: gpredMl. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredFl | |
gpredJx y : x \in S -> y \in S -> x ^ y \in S.
Proof. by move=> xS yS; apply/gpredM; [apply/gpredVr|apply/gpredM]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredJ | |
gpredRx y : x \in S -> y \in S -> [~ x, y] \in S.
Proof. by move=> xS yS; apply/gpredM; [apply/gpredVr|apply/gpredJ]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | gpredR | |
RecordisSubMagma (G : magmaType) (S : pred G) H
of SubType G S H & Magma H := {
valM_subproof : {morph (val : H -> G) : x y / x * y}
}.
#[short(type="subMagmaType")]
HB.structure Definition SubMagma (G : magmaType) S :=
{ H of SubChoice G S H & Magma H & isSubMagma G S H }. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
val:= (val : H -> G).
HB.instance Definition _ := isMultiplicative.Build H G val valM_subproof. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | val | |
valM: {morph val : x y / x * y}. Proof. exact: gmulfM. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | valM | |
RecordSubChoice_isSubMagma (G : magmaType) S H
of SubChoice G S H := {
mulg_closed_subproof : mulg_closed S
}.
HB.builders Context G S H of SubChoice_isSubMagma G S H.
HB.instance Definition _ := isMulClosed.Build G S mulg_closed_subproof.
Let inH v Sv : H := Sub v Sv.
Let mulH (u1 u2 : H) := inH (gpredM _ _ (valP u1) (valP u2)).
HB.instance Definition _ := hasMul.Build H mulH. | HB.factory | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
valM: {morph (val : H -> G) : x y / x * y}.
Proof. by move=> x y; rewrite SubK. Qed.
HB.instance Definition _ := isSubMagma.Build G S H valM.
HB.end.
#[short(type="subSemigroupType")]
HB.structure Definition SubSemigroup (G : semigroupType) S :=
{ H of SubMagma G S H & Semigroup H}.
HB.factory Record SubChoice_isSubSemigroup (G : semigroupType) S H
of SubChoice G S H := {
mulg_closed_subproof : mulg_closed S
}.
HB.builders Context G S H of SubChoice_isSubSemigroup G S H.
HB.instance Definition _ :=
SubChoice_isSubMagma.Build G S H mulg_closed_subproof. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | valM | |
mulgA: associative (@mul H).
Proof. by move=> x y z; apply/val_inj; rewrite !valM mulgA. Qed.
HB.instance Definition _ := isSemigroup.Build H mulgA.
HB.end.
HB.mixin Record isSubBaseUMagma (G : baseUMagmaType) (S : pred G) H
of SubMagma G S H & BaseUMagma H := {
val1_subproof : (val : H -> G) 1 = 1
}.
#[short(type="subBaseUMagmaType")]
HB.structure Definition SubBaseUMagma (G : umagmaType) S :=
{ H of SubMagma G S H & BaseUMagma H & isSubBaseUMagma G S H}.
#[short(type="subUMagmaType")]
HB.structure Definition SubUMagma (G : umagmaType) S :=
{ H of SubMagma G S H & UMagma H & isSubBaseUMagma G S H}. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mulgA | |
val:= (val : H -> G).
HB.instance Definition _ :=
Multiplicative_isUMagmaMorphism.Build H G val val1_subproof. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | val | |
val1: val 1 = 1. Proof. exact: gmulf1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | val1 | |
RecordSubChoice_isSubUMagma (G : umagmaType) S H
of SubChoice G S H := {
umagma_closed_subproof : umagma_closed S
}.
HB.builders Context G S H of SubChoice_isSubUMagma G S H.
HB.instance Definition _ :=
SubChoice_isSubMagma.Build G S H (snd umagma_closed_subproof).
Let inH v Sv : H := Sub v Sv.
Let oneH := inH (fst umagma_closed_subproof).
HB.instance Definition _ := hasOne.Build H oneH. | HB.factory | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
val1: (val : H -> G) 1 = 1.
Proof. exact/SubK. Qed.
HB.instance Definition _ := isSubBaseUMagma.Build G S H val1. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | val1 | |
mul1g: left_id 1 (@mul H).
Proof. by move=> x; apply/val_inj; rewrite valM val1 mul1g. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mul1g | |
mulg1: right_id 1 (@mul H).
Proof. by move=> x; apply/val_inj; rewrite valM val1 mulg1. Qed.
HB.instance Definition _ := BaseUMagma_isUMagma.Build H mul1g mulg1.
HB.end.
#[short(type="subMonoidType")]
HB.structure Definition SubMonoid (G : monoidType) S :=
{ H of SubUMagma G S H & Monoid H}.
HB.factory Record SubChoice_isSubMonoid (G : monoidType) S H
of SubChoice G S H := {
monoid_closed_subproof : monoid_closed S
}.
HB.builders Context G S H of SubChoice_isSubMonoid G S H.
HB.instance Definition _ :=
SubChoice_isSubUMagma.Build G S H monoid_closed_subproof.
HB.instance Definition _ :=
SubChoice_isSubSemigroup.Build G S H (snd monoid_closed_subproof).
HB.end.
#[short(type="subGroupType")]
HB.structure Definition SubGroup (G : groupType) S :=
{ H of SubUMagma G S H & Group H}.
HB.factory Record SubChoice_isSubGroup (G : groupType) S H
of SubChoice G S H := {
group_closed_subproof : group_closed S
}.
HB.builders Context G S H of SubChoice_isSubGroup G S H. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mulg1 | |
umagma_closed: umagma_closed S.
Proof.
split; first exact/(fst group_closed_subproof).
exact/group_closedM/group_closed_subproof.
Qed.
HB.instance Definition _ := SubChoice_isSubMonoid.Build G S H umagma_closed.
HB.instance Definition _ :=
isInvClosed.Build G S (group_closedV group_closed_subproof).
Let inH v Sv : H := Sub v Sv.
Let invH (u : H) := inH (gpredVr _ (valP u)).
HB.instance Definition _ := hasInv.Build H invH. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | umagma_closed | |
mulVg: left_inverse 1%g invH *%g.
Proof. by move=> x; apply/val_inj; rewrite valM SubK mulVg val1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mulVg | |
mulgV: right_inverse 1%g invH *%g.
Proof. by move=> x; apply/val_inj; rewrite valM SubK mulgV val1. Qed.
HB.instance Definition _ := Monoid_isGroup.Build H mulVg mulgV.
HB.end.
Prenex Implicits mul inv natexp conjg commg. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mulgV | |
ffun_mulf g := [ffun a => f a * g a].
HB.instance Definition _ := hasMul.Build {ffun aT -> rT} ffun_mul. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | ffun_mul | |
Definition_ (aT : finType) (rT : ChoiceMagma.type) :=
Magma.on {ffun aT -> rT}. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
Definition_ := isSemigroup.Build {ffun aT -> rT} ffun_mulgA. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
ffun_one:= [ffun a : aT => (1 : rT)].
HB.instance Definition _ :=
hasOne.Build {ffun aT -> rT} ffun_one. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | ffun_one | |
Definition_ (aT : finType) (rT : ChoiceBaseUMagma.type) :=
BaseUMagma.on {ffun aT -> rT}. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
Definition_ :=
Magma_isUMagma.Build {ffun aT -> rT} ffun_mul1g ffun_mulg1. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
Definition_ (aT : finType) (rT : monoidType) :=
UMagma.on {ffun aT -> rT}. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
ffun_invf := [ffun a => (f a)^-1].
HB.instance Definition _ := hasInv.Build {ffun aT -> rT} ffun_inv. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | ffun_inv | |
Definition_ := Monoid_isGroup.Build {ffun aT -> rT}
ffun_mulVg ffun_mulgV. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
mul_pair(x y : G * H) := (x.1 * y.1, x.2 * y.2).
HB.instance Definition _ := hasMul.Build (G * H)%type mul_pair.
Fact fst_is_multiplicative : {morph fst : x y / x * y}. Proof. by []. Qed.
HB.instance Definition _ :=
isMultiplicative.Build _ _ fst fst_is_multiplicative.
Fact snd_is_multiplicative : {morph snd : x y / x * y}. Proof. by []. Qed.
HB.instance Definition _ :=
isMultiplicative.Build _ _ snd snd_is_multiplicative. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mul_pair | |
pair_mulgA: associative (@mul (G * H)%type).
Proof. by move=> x y z; congr (_, _); apply/mulgA. Qed.
HB.instance Definition _ := Magma_isSemigroup.Build (G * H)%type pair_mulgA. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | pair_mulgA | |
one_pair: G * H := (1, 1).
HB.instance Definition _ := hasOne.Build (G * H)%type one_pair.
Fact fst_is_umagma_morphism : fst (1 : G * H) = 1. Proof. by []. Qed.
HB.instance Definition _ :=
Multiplicative_isUMagmaMorphism.Build _ _ fst fst_is_umagma_morphism.
Fact snd_is_umagma_morphism : snd (1 : G * H) = 1. Proof. by []. Qed.
HB.instance Definition _ :=
Multiplicative_isUMagmaMorphism.Build _ _ snd snd_is_umagma_morphism. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | one_pair | |
pair_mul1g: left_id (@one_pair G H) *%g.
Proof. by move=> [x y]; congr (_, _); rewrite mul1g. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | pair_mul1g | |
pair_mulg1: right_id (@one_pair G H) *%g.
Proof. by move=> [x y]; congr (_, _); rewrite mulg1. Qed.
HB.instance Definition _ :=
BaseUMagma_isUMagma.Build (G * H)%type pair_mul1g pair_mulg1. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | pair_mulg1 | |
Definition_ (G H : ChoiceMagma.type) := Magma.on (G * H)%type.
HB.instance Definition _ (G H : ChoiceBaseUMagma.type) :=
BaseUMagma.on (G * H)%type.
HB.instance Definition _ (G H : monoidType) := Semigroup.on (G * H)%type. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Definition | |
inv_pair(u : G * H) := (u.1 ^-1, u.2 ^-1).
HB.instance Definition _ := hasInv.Build (G * H)%type inv_pair. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | inv_pair | |
pair_mulVg: left_inverse one (@inv_pair G H) mul.
Proof. by move=> x; congr (_, _); apply/mulVg. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | pair_mulVg | |
pair_mulgV: right_inverse one (@inv_pair G H) mul.
Proof. by move=> x; congr (_, _); apply/mulgV. Qed.
HB.instance Definition _ :=
Monoid_isGroup.Build (G * H)%type pair_mulVg pair_mulgV. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | pair_mulgV | |
RecordhasAdd V := {
add : V -> V -> V
}.
#[short(type="baseAddMagmaType")]
HB.structure Definition BaseAddMagma := {V of hasAdd V}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
DefinitionChoiceBaseAddMagma := {V of BaseAddMagma V & Choice V}. | HB.structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
to_multiplicative:= @id Type.
#[export]
HB.instance Definition _ (V : choiceType) := Choice.on (to_multiplicative V).
#[export]
HB.instance Definition _ (V : baseAddMagmaType) :=
hasMul.Build (to_multiplicative V) (@add V). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | to_multiplicative | |
Definition_ (V : ChoiceBaseAddMagma.type) :=
Magma.on (to_multiplicative V). | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
addr_closed:= {in S &, forall u v, u + v \in S}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addr_closed | |
RecordBaseAddMagma_isAddMagma V of BaseAddMagma V := {
addrC : commutative (@add V)
}.
#[short(type="addMagmaType")]
HB.structure Definition AddMagma :=
{V of BaseAddMagma_isAddMagma V & ChoiceBaseAddMagma V}.
HB.factory Record isAddMagma V of Choice V := {
add : V -> V -> V;
addrC : commutative add
}.
HB.builders Context V of isAddMagma V.
HB.instance Definition _ := hasAdd.Build V add.
HB.instance Definition _ := BaseAddMagma_isAddMagma.Build V addrC.
HB.end. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
commuteTx y : @commute (to_multiplicative V) x y.
Proof. exact/addrC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | commuteT | |
RecordAddMagma_isAddSemigroup V of AddMagma V := {
addrA : associative (@add V)
}.
#[short(type="addSemigroupType")]
HB.structure Definition AddSemigroup :=
{V of AddMagma_isAddSemigroup V & AddMagma V}.
HB.factory Record isAddSemigroup V of Choice V := {
add : V -> V -> V;
addrC : commutative add;
addrA : associative add
}.
HB.builders Context V of isAddSemigroup V.
HB.instance Definition _ := isAddMagma.Build V addrC.
HB.instance Definition _ := AddMagma_isAddSemigroup.Build V addrA.
HB.end. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
Definition_ (V : addSemigroupType) :=
Magma_isSemigroup.Build (to_multiplicative V) addrA. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
addrCA: @left_commutative V V +%R.
Proof. by move=> x y z; rewrite !addrA [x + _]addrC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrCA | |
addrAC: @right_commutative V V +%R.
Proof. by move=> x y z; rewrite -!addrA [y + _]addrC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrAC | |
addrACA: @interchange V +%R +%R.
Proof. by move=> x y z t; rewrite -!addrA [y + (z + t)]addrCA. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrACA | |
RecordhasZero V := {
zero : V
}.
#[short(type="baseAddUMagmaType")]
HB.structure Definition BaseAddUMagma :=
{V of hasZero V & BaseAddMagma V}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
DefinitionChoiceBaseAddUMagma :=
{V of BaseAddUMagma V & Choice V}. | HB.structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
natmul(V : baseAddUMagmaType) (x : V) n : V := iterop n +%R x 0.
Arguments natmul : simpl never.
Local Notation "x *+ n" := (natmul x n) : ring_scope.
#[export]
HB.instance Definition _ (V : baseAddUMagmaType) :=
hasOne.Build (to_multiplicative V) (@zero V). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | natmul | |
Definition_ (V : ChoiceBaseAddUMagma.type) :=
BaseUMagma.on (to_multiplicative V). | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
mulr0nx : x *+ 0 = 0. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulr0n | |
mulr1nx : x *+ 1 = x. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulr1n | |
mulr2nx : x *+ 2 = x + x. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulr2n | |
mulrbx (b : bool) : x *+ b = (if b then x else 0).
Proof. exact: (@expgb (to_multiplicative V)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrb | |
mulrSSx n : x *+ n.+2 = x + x *+ n.+1. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrSS | |
addumagma_closed:= 0 \in S /\ addr_closed S. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addumagma_closed | |
RecordBaseAddUMagma_isAddUMagma V of BaseAddUMagma V := {
add0r : left_id zero (@add V)
}.
HB.factory Record isAddUMagma V of Choice V := {
add : V -> V -> V;
zero : V;
addrC : commutative add;
add0r : left_id zero add
}.
HB.builders Context V of isAddUMagma V.
HB.instance Definition _ := isAddMagma.Build V addrC.
HB.instance Definition _ := hasZero.Build V zero.
#[warning="-HB.no-new-instance"]
HB.instance Definition _ := BaseAddUMagma_isAddUMagma.Build V add0r.
HB.end.
#[short(type="addUMagmaType")]
HB.structure Definition AddUMagma := {V of isAddUMagma V & Choice V}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
addr0(V : addUMagmaType) : right_id (@zero V) add.
Proof. by move=> x; rewrite addrC add0r. Qed.
Local Notation "\sum_ ( i <- r | P ) F" := (\big[+%R/0]_(i <- r | P) F).
Local Notation "\sum_ ( m <= i < n ) F" := (\big[+%R/0]_(m <= i < n) F).
Local Notation "\sum_ ( i < n ) F" := (\big[+%R/0]_(i < n) F).
Local Notation "\sum_ ( i 'in' A ) F" := (\big[+%R/0]_(i in A) F).
Import Monoid.Theory.
#[export]
HB.instance Definition _ (V : addUMagmaType) :=
Magma_isUMagma.Build (to_multiplicative V) add0r (@addr0 V).
HB.factory Record isNmodule V of Choice V := {
zero : V;
add : V -> V -> V;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add
}.
HB.builders Context V of isNmodule V.
HB.instance Definition _ := isAddUMagma.Build V addrC add0r.
HB.instance Definition _ := AddMagma_isAddSemigroup.Build V addrA.
HB.end. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addr0 | |
DefinitionNmodule := {V of isNmodule V & Choice V}. | HB.structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
Definition_ (V : nmodType) :=
UMagma_isMonoid.Build (to_multiplicative V) addrA.
#[export]
HB.instance Definition _ (V : nmodType) :=
Monoid.isComLaw.Build V 0%R +%R addrA addrC add0r. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
mulrSx n : x *+ n.+1 = x + (x *+ n).
Proof. exact: (@expgS G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrS | |
mulrSrx n : x *+ n.+1 = x *+ n + x.
Proof. exact: (@expgSr G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrSr | |
mul0rnn : 0 *+ n = 0 :> V.
Proof. exact: (@expg1n G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mul0rn | |
mulrnDln : {morph (fun x => x *+ n) : x y / x + y}.
Proof. by move=> x y; apply/(@expgMn G)/commuteT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrnDl |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.