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 |
|---|---|---|---|---|---|---|
expg2x : 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"
] | boot/monoid.v | expg2 | |
expgSSx 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"
] | boot/monoid.v | expgSS | |
expgbx (b : bool) : x ^+ b = (if b then x else 1).
Proof. by case: b. 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 | expgb | |
umagma_closed:= 1 \in S /\ mulg_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"
] | boot/monoid.v | umagma_closed | |
RecordBaseUMagma_isUMagma G of BaseUMagma G := {
mul1g : left_id one (@mul G);
mulg1 : right_id one (@mul G)
}.
HB.factory Record Magma_isUMagma G of Magma G := {
one : G;
mul1g : left_id one (@mul G);
mulg1 : right_id one (@mul G)
}.
HB.builders Context G of Magma_isUMagma G.
HB.instance Definition _ := hasOne.Build G one.
#[warning="-HB.no-new-instance"]
HB.instance Definition _ := BaseUMagma_isUMagma.Build G mul1g mulg1.
HB.end.
#[short(type="umagmaType")]
HB.structure Definition UMagma := {G of Magma_isUMagma G & ChoiceMagma G}.
Bind Scope group_scope with UMagma.sort. | 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 | |
expgSx n : x ^+ n.+1 = x * x ^+ n.
Proof. by case: n => //=; rewrite mulg1. 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 | expgS | |
expg1nn : 1 ^+ n = 1 :> G.
Proof. by elim: n => // n IHn; rewrite expgS 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 | expg1n | |
commute1x : commute x 1.
Proof. by rewrite /commute mulg1 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 | commute1 | |
DefinitionMonoid := {G of Magma_isUMagma G & Semigroup G}.
HB.factory Record Semigroup_isMonoid G of Semigroup G := {
one : G;
mul1g : left_id one mul;
mulg1 : right_id one mul
}.
HB.builders Context G of Semigroup_isMonoid G.
HB.instance Definition _ := Magma_isUMagma.Build G mul1g mulg1.
HB.end.
HB.factory Record UMagma_isMonoid G of UMagma G := {
mulgA : associative (@mul G)
}.
HB.builders Context G of UMagma_isMonoid G.
HB.instance Definition _ := Magma_isSemigroup.Build G mulgA.
HB.end.
HB.factory Record isMonoid G of Choice G := {
mul : G -> G -> G;
one : G;
mulgA : associative mul;
mul1g : left_id one mul;
mulg1 : right_id one mul
}.
HB.builders Context G of isMonoid G.
HB.instance Definition _ := hasMul.Build G mul.
HB.instance Definition _ := Magma_isSemigroup.Build G mulgA.
HB.instance Definition _ := Magma_isUMagma.Build G mul1g mulg1.
HB.end. | 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 | |
expgnDrx m n : x ^+ (m + n) = x ^+ m * x ^+ n.
Proof. by elim: m => [|m IHm]; rewrite ?mul1g // !expgS IHm mulgA. 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 | expgnDr | |
expgSrx n : x ^+ n.+1 = x ^+ n * x.
Proof. by rewrite -addn1 expgnDr expg1. 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 | expgSr | |
expgnAx m n : x ^+ (m * n) = x ^+ m ^+ n.
Proof. by rewrite mulnC; elim: n => //= n IHn; rewrite expgS expgnDr 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 | expgnA | |
expgnACx m n : x ^+ m ^+ n = x ^+ n ^+ m.
Proof. by rewrite -2!expgnA mulnC. 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 | expgnAC | |
iter_mulgn x y : iter n ( *%g x) y = x ^+ n * y.
Proof. by elim: n => [|n IHn]; rewrite ?mul1g //= IHn expgS mulgA. 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 | iter_mulg | |
iter_mulg_1n x : iter n ( *%g x) 1 = x ^+ n.
Proof. by rewrite iter_mulg mulg1. 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 | iter_mulg_1 | |
prodg_const(I : finType) (A : pred I) x : \prod_(i in A) x = x ^+ #|A|.
Proof. by rewrite big_const -Monoid.iteropE. 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 | prodg_const | |
prodg_const_natn m x : \prod_(n <= i < m) x = x ^+ (m - n).
Proof. by rewrite big_const_nat -Monoid.iteropE. 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 | prodg_const_nat | |
prodgXrx I r P (F : I -> nat) :
\prod_(i <- r | P i) x ^+ F i = x ^+ (\sum_(i <- r | P i) F i).
Proof. by rewrite (big_morph _ (expgnDr _) (erefl _)). 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 | prodgXr | |
commute_prod(I : Type) (s : seq I) (P : pred I) (F : I -> G) x :
(forall i, P i -> commute x (F i)) -> commute x (\prod_(i <- s | P i) F i).
Proof. exact: (big_ind _ (commute1 x) (@commuteM _ x)). 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 | commute_prod | |
prodgM_commute{I : eqType} r (P : pred I) (F H : I -> G) :
(forall i j, P i -> P j -> commute (F i) (H j)) ->
\prod_(i <- r | P i) (F i * H i) =
\prod_(i <- r | P i) F i * \prod_(i <- r | P i) H i.
Proof.
move=> FH; elim: r => [|i r IHr]; rewrite !(big_nil, big_cons) ?mulg1//.
case: ifPn => // Pi; rewrite IHr !mulgA; congr (_ * _); rewrite -!mulgA.
by rewrite commute_prod // => j Pj; apply/commute_sym/FH.
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 | prodgM_commute | |
prodgMl_commute{I : finType} (A : pred I) (x : G) F :
(forall i, A i -> commute x (F i)) ->
\prod_(i in A) (x * F i) = x ^+ #|A| * \prod_(i in A) F i.
Proof. by move=> xF; rewrite prodgM_commute ?prodg_const// => i j _ /xF. 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 | prodgMl_commute | |
prodgMr_commute{I : finType} (A : pred I) (x : G) F :
(forall i, A i -> commute x (F i)) ->
\prod_(i in A) (F i * x) = \prod_(i in A) F i * x ^+ #|A|.
Proof. by move=> xF; rewrite prodgM_commute ?prodg_const// => i j /xF. 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 | prodgMr_commute | |
commuteXx y n : commute x y -> commute x (y ^+ n).
Proof.
by move=> cxy; case: n; [apply: commute1 | elim=> // n; apply: commuteM].
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 | commuteX | |
commuteX2x y m n : commute x y -> commute (x ^+ m) (y ^+ n).
Proof. by move=> cxy; apply/commuteX/commute_sym/commuteX. 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 | commuteX2 | |
expgMnx y n : commute x y -> (x * y) ^+ n = x ^+ n * y ^+ n.
Proof.
move=> cxy; elim: n => [|n IHn]; first by rewrite mulg1.
by rewrite !expgS IHn -mulgA (mulgA y) (commuteX _ (commute_sym cxy)) !mulgA.
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 | expgMn | |
monoid_closed:= umagma_closed.
HB.mixin Record hasInv G := {
inv : G -> G
}.
#[short(type="baseGroupType")]
HB.structure Definition BaseGroup := {G of hasInv G & BaseUMagma G}.
Bind Scope group_scope with BaseGroup.sort.
Local Notation "x ^-1" := (inv x) : group_scope.
Local Notation "x / y" := (x * y^-1) : group_scope.
Local Notation "x ^- n" := ((x ^+ n)^-1) : group_scope. | 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 | monoid_closed | |
conjg(G : baseGroupType) (x y : G) := y^-1 * (x * y).
Local Notation "x ^ y" := (conjg x y) : group_scope. | 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 | conjg | |
commg(G : baseGroupType) (x y : G) := x^-1 * (conjg x y).
Local Notation "[ ~ x1 , x2 , .. , xn ]" := (commg .. (commg x1 x2) .. xn)
: group_scope.
HB.mixin Record Monoid_isStarMonoid G of BaseGroup G := {
invgK : involutive (@inv G);
invgM : {morph @inv G : x y / x * y >-> y * x}
}.
#[short(type="starMonoidType")]
HB.structure Definition StarMonoid :=
{ G of Monoid_isStarMonoid G & Monoid G & BaseGroup G }.
Prenex Implicits invgK.
Bind Scope group_scope with StarMonoid.sort.
HB.factory Record isStarMonoid G of Choice G := {
mul : G -> G -> G;
one : G;
inv : G -> G;
mulgA : associative mul;
mul1g : left_id one mul;
invgK : involutive inv;
invgM : {morph inv : x y / mul x y >-> mul y x}
}.
HB.builders Context G of isStarMonoid G. | 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 | commg | |
invg1: inv one = one.
Proof.
by apply: (can_inj invgK); rewrite -[inv one in LHS]mul1g invgM invgK 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 | invg1 | |
mulg1: right_id one mul.
Proof. by move=> x; apply: (can_inj invgK); rewrite invgM invg1 mul1g. Qed.
HB.instance Definition _ := isMonoid.Build G mulgA mul1g mulg1.
HB.instance Definition _ := hasInv.Build G inv.
HB.instance Definition _ := Monoid_isStarMonoid.Build G invgK invgM.
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"
] | boot/monoid.v | mulg1 | |
invg_inj: injective (@inv G). Proof. exact: can_inj invgK. 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 | invg_inj | |
invg1: 1^-1 = 1 :> G.
Proof. by apply: invg_inj; rewrite -[1^-1 in LHS]mul1g invgM invgK 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 | invg1 | |
invgFx y : (x / y)^-1 = y / x.
Proof. by rewrite invgM invgK. 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 | invgF | |
prodgVI r (P : pred I) (E : I -> G) :
\prod_(i <- r | P i) (E i)^-1 = (\prod_(i <- rev r | P i) E i)^-1.
Proof.
elim: r => [|x r IHr]; first by rewrite !big_nil invg1.
rewrite big_cons rev_cons big_rcons/= IHr.
by case: ifP => _; rewrite ?mulg1// invgM.
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 | prodgV | |
eqg_invx y : (x^-1 == y^-1) = (x == y).
Proof. exact: can_eq invgK x y. 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 | eqg_inv | |
eqg_invLRx y : (x^-1 == y) = (x == y^-1).
Proof. exact: inv_eq invgK x y. 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 | eqg_invLR | |
invg_eq1x : (x^-1 == 1) = (x == 1).
Proof. by rewrite eqg_invLR invg1. 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 | invg_eq1 | |
expVgnx n : x^-1 ^+ n = x ^- n.
Proof. by elim: n => [|n IHn]; rewrite ?invg1 // expgSr expgS invgM 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 | expVgn | |
conjgEx y : x ^ y = y^-1 * (x * y). 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"
] | boot/monoid.v | conjgE | |
commgElx y : [~ x, y] = x^-1 * x ^ y. 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"
] | boot/monoid.v | commgEl | |
commgErx y : [~ x, y] = y^-1 ^ x * y.
Proof. by rewrite -!mulgA. 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 | commgEr | |
RecordStarMonoid_isGroup G of BaseGroup G := {
mulVg : left_inverse one inv (@mul G);
}.
#[short(type="groupType")]
HB.structure Definition Group :=
{G of StarMonoid_isGroup G & BaseGroup G & StarMonoid G}.
HB.factory Record Monoid_isGroup G of Monoid G & BaseGroup G := {
mulVg : left_inverse one inv (@mul G);
mulgV : right_inverse one inv (@mul G);
}.
HB.builders Context G of Monoid_isGroup G.
Fact invgK : involutive (@inv G).
Proof. by move=> x; rewrite -[LHS]mul1g -(mulgV x) -mulgA mulgV mulg1. Qed.
Fact mulKg : @left_loop G G inv *%g.
Proof. by move=> x y; rewrite [LHS]mulgA mulVg mul1g. Qed.
Fact invgM : {morph inv : x y / x * y >-> y * x : G}.
Proof.
move=> x y; apply: (can_inj (mulKg (x * y))).
by rewrite [LHS]mulgV [RHS]mulgA -(mulgA x) mulgV mulg1 mulgV.
Qed.
HB.instance Definition _ := Monoid_isStarMonoid.Build G invgK invgM.
HB.instance Definition _ := StarMonoid_isGroup.Build G mulVg.
HB.end.
HB.factory Record isGroup G of Choice G := {
one : G;
inv : G -> G;
mul : G -> G -> G;
mulgA : associative mul;
mul1g : left_id one mul;
mulg1 : right_id one mul;
mulVg : left_inverse one inv mul; | 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 | |
mulgV: right_inverse one inv (@mul G).
Proof. by move=> x; rewrite -{1}(invgK x) 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 | mulgV | |
divgg:= mulgV. | 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 | divgg | |
mulKg: @left_loop G G (@inv G) *%g.
Proof. by move=> x y; rewrite mulgA mulVg 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 | mulKg | |
mulVKg: @rev_left_loop G G (@inv G) *%g.
Proof. by move=> x y ; rewrite mulgA mulgV 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 | mulVKg | |
mulgK: @right_loop G G (@inv G) *%g.
Proof. by move=> x y; rewrite -mulgA mulgV mulg1. 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 | mulgK | |
mulgVK: @rev_right_loop G G (@inv G) *%g.
Proof. by move=> x y ; rewrite -mulgA mulVg mulg1. 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 | mulgVK | |
divgK:= mulgVK. | 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 | divgK | |
mulgI: @right_injective G G G *%g.
Proof. by move=> x; apply: can_inj (mulKg x). 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 | mulgI | |
mulIg: @left_injective G G G *%g.
Proof. by move=> x; apply: can_inj (mulgK x). 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 | mulIg | |
divgI: @right_injective G G G (fun x y => x / y).
Proof. by move=> x y z /mulgI/invg_inj. 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 | divgI | |
divIg: @left_injective G G G (fun x y => x / y).
Proof. by move=> x y z /mulIg. 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 | divIg | |
divg1x : x / 1 = x. Proof. by rewrite invg1 mulg1. 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 | divg1 | |
div1gx : 1 / x = x^-1. Proof. by 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 | div1g | |
divKgx y : commute x y -> x / (x / y) = y.
Proof. by move=> xyC; rewrite invgF mulgA xyC mulgK. 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 | divKg | |
mulgKAz x y : (x * z) / (y * z) = x / y.
Proof. by rewrite invgM mulgA mulgK. 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 | mulgKA | |
divgKAz x y : (x / z) * (z * y) = x * y.
Proof. by rewrite mulgA mulgVK. 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 | divgKA | |
mulg1_eqx y : x * y = 1 -> x^-1 = y.
Proof. by rewrite -[x^-1]mulg1 => <-; rewrite mulKg. 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 | mulg1_eq | |
divg1_eqx y : x / y = 1 -> x = y.
Proof. by move/mulg1_eq/invg_inj. 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 | divg1_eq | |
divg_eqx y z : (x / z == y) = (x == y * z).
Proof. exact: can2_eq (divgK z) (mulgK z) x y. 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 | divg_eq | |
divg_eq1x y : (x / y == 1) = (x == y).
Proof. by rewrite divg_eq 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 | divg_eq1 | |
mulg_eq1x y : (x * y == 1) = (x == y^-1).
Proof. by rewrite -[y in LHS]invgK divg_eq1. 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 | mulg_eq1 | |
commuteVx y : commute x y -> commute x y^-1.
Proof. by move=> cxy; apply: (@mulIg y); rewrite mulgVK -mulgA cxy mulKg. 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 | commuteV | |
expgnFrx m n : n <= m -> x ^+ (m - n) = x ^+ m / x ^+ n.
Proof. by move=> lenm; rewrite -[in RHS](subnK lenm) expgnDr mulgK. 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 | expgnFr | |
expgnFlx y n : commute x y -> (x / y) ^+ n = x ^+ n / y ^+ n.
Proof. by move=> xyC; rewrite expgMn 1?expVgn; last exact/commuteV. 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 | expgnFl | |
conjgCx y : x * y = y * x ^ y.
Proof. by rewrite mulVKg. 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 | conjgC | |
conjgCVx y : x * y = y ^ x^-1 * x.
Proof. by rewrite -mulgA mulgVK invgK. 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 | conjgCV | |
conjg1x : x ^ 1 = x.
Proof. by rewrite conjgE commute1 mulKg. 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 | conjg1 | |
conj1gx : 1 ^ x = 1.
Proof. by rewrite conjgE mul1g 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 | conj1g | |
conjMgx y z : (x * y) ^ z = x ^ z * y ^ z.
Proof. by rewrite !conjgE !mulgA mulgK. 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 | conjMg | |
conjgMx y z : x ^ (y * z) = (x ^ y) ^ z.
Proof. by rewrite !conjgE invgM !mulgA. 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 | conjgM | |
conjVgx y : x^-1 ^ y = (x ^ y)^-1.
Proof. by rewrite !conjgE !invgM invgK mulgA. 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 | conjVg | |
conjJgx y z : (x ^ y) ^ z = (x ^ z) ^ y ^ z.
Proof. by rewrite 2!conjMg conjVg. 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 | conjJg | |
conjXgx y n : (x ^+ n) ^ y = (x ^ y) ^+ n.
Proof. by elim: n => [|n IHn]; rewrite ?conj1g // !expgS conjMg 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 | conjXg | |
conjgK: @right_loop G G (@inv G) (@conjg G).
Proof. by move=> y x; rewrite -conjgM mulgV conjg1. 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 | conjgK | |
conjgKV: @rev_right_loop G G (@inv G) (@conjg G).
Proof. by move=> y x; rewrite -conjgM mulVg conjg1. 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 | conjgKV | |
conjg_inj: @left_injective G G G (@conjg G).
Proof. by move=> y; apply: can_inj (conjgK y). 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 | conjg_inj | |
conjg_eq1x y : (x ^ y == 1) = (x == 1).
Proof. by rewrite (can2_eq (conjgK _) (conjgKV _)) conj1g. 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 | conjg_eq1 | |
conjg_prodI r (P : pred I) (F : I -> G) z :
(\prod_(i <- r | P i) F i) ^ z = \prod_(i <- r | P i) (F i ^ z).
Proof.
by apply: (big_morph ((@conjg G)^~ z)) => [x y|]; rewrite ?conj1g ?conjMg.
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 | conjg_prod | |
commgCx y : x * y = y * x * [~ x, y].
Proof. by rewrite -mulgA !mulVKg. 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 | commgC | |
commgCVx y : x * y = [~ x^-1, y^-1] * (y * x).
Proof. by rewrite commgEl !mulgA !invgK !mulgVK. 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 | commgCV | |
conjRgx y z : [~ x, y] ^ z = [~ x ^ z, y ^ z].
Proof. by rewrite !conjMg !conjVg. 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 | conjRg | |
invgRx y : [~ x, y]^-1 = [~ y, x].
Proof. by rewrite commgEr conjVg invgM invgK. 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 | invgR | |
commgPx y : reflect (commute x y) ([~ x, y] == 1).
Proof. rewrite [[~ x, y]]mulgA -invgM mulg_eq1 eqg_inv eq_sym; apply: eqP. 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 | commgP | |
conjg_fixx y : x ^ y == x = ([~ x, y] == 1).
Proof. by rewrite mulg_eq1 eqg_inv. 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 | conjg_fix | |
conjg_fixPx y : reflect (x ^ y = x) ([~ x, y] == 1).
Proof. by rewrite -conjg_fix; apply: eqP. 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 | conjg_fixP | |
commg1_symx y : ([~ x, y] == 1) = ([~ y, x] == 1).
Proof. by rewrite -invgR (inv_eq invgK) invg1. 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 | commg1_sym | |
commg1x : [~ x, 1] = 1.
Proof. exact/eqP/commgP/commute1. 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 | commg1 | |
comm1gx : [~ 1, x] = 1.
Proof. by rewrite -invgR commg1 invg1. 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 | comm1g | |
commggx : [~ x, x] = 1.
Proof. exact/eqP/commgP. 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 | commgg | |
commgXgx n : [~ x, x ^+ n] = 1.
Proof. exact/eqP/commgP/commuteX. 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 | commgXg | |
commgVgx : [~ x, x^-1] = 1.
Proof. exact/eqP/commgP/commuteV. 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 | commgVg | |
commgXVgx n : [~ x, x ^- n] = 1.
Proof. exact/eqP/commgP/commuteV/commuteX. 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 | commgXVg | |
invg_closed:= {in S, forall u, u^-1 \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"
] | boot/monoid.v | invg_closed | |
divg_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"
] | boot/monoid.v | divg_closed | |
group_closed:= 1 \in S /\ divg_closed. | 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 | group_closed | |
group_closedV: group_closed -> invg_closed.
Proof. by move=> [S1 SB] x /(SB 1)-/(_ S1); rewrite div1g. 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 | group_closedV | |
group_closedM: group_closed -> mulg_closed S.
Proof.
move=> /[dup]-[S1 SB] /group_closedV SV x y xS /SV yS.
rewrite -[y]invgK; exact: SB.
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 | group_closedM | |
gsimpl:= autorewrite with gsimpl; try done. | Ltac | 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 | gsimpl |
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