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Coq-MathComp

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invg_inj : injective (@inv G).
Proof. exact: can_inj invgK. Qed.
Lemma
invg_inj
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "invgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg1 : 1^-1 = 1 :> G.
Proof. by apply: invg_inj; rewrite -[1^-1 in LHS]mul1g invgM invgK mul1g. Qed.
Lemma
invg1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "invgK", "invgM", "invg_inj", "mul1g" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
invgF x y : (x / y)^-1 = y / x.
Proof. by rewrite invgM invgK. Qed.
Lemma
invgF
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invgK", "invgM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodgV I 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
prodgV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "big_cons", "big_nil", "big_rcons", "invg1", "invgM", "mulg1", "rev", "rev_cons" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqg_inv x y : (x^-1 == y^-1) = (x == y).
Proof. exact: can_eq invgK x y. Qed.
Lemma
eqg_inv
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "can_eq", "invgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqg_invLR x y : (x^-1 == y) = (x == y^-1).
Proof. exact: inv_eq invgK x y. Qed.
Lemma
eqg_invLR
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv_eq", "invgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_eq1 x : (x^-1 == 1) = (x == 1).
Proof. by rewrite eqg_invLR invg1. Qed.
Lemma
invg_eq1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "eqg_invLR", "invg1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
expVgn x n : x^-1 ^+ n = x ^- n.
Proof. by elim: n => [|n IHn]; rewrite ?invg1 // expgSr expgS invgM IHn. Qed.
Lemma
expVgn
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "expgS", "expgSr", "invg1", "invgM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgE x y : x ^ y = y^-1 * (x * y).
Proof. by []. Qed.
Lemma
conjgE
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgEl x y : [~ x, y] = x^-1 * x ^ y.
Proof. by []. Qed.
Lemma
commgEl
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgEr x y : [~ x, y] = y^-1 ^ x * y.
Proof. by rewrite -!mulgA. Qed.
Lemma
commgEr
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
invgK : involutive (@inv G).
Proof. by move=> x; rewrite -[LHS]mul1g -(mulgV x) -mulgA mulgV mulg1. Qed.
Fact
invgK
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "mul1g", "mulg1", "mulgA", "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulKg : @left_loop G G inv *%g.
Proof. by move=> x y; rewrite [LHS]mulgA mulVg mul1g. Qed.
Fact
mulKg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "mul1g", "mulVg", "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.
Fact
invgM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "inv", "mulKg", "mulg1", "mulgA", "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgV : right_inverse one inv (@mul G).
Proof. by move=> x; rewrite -{1}(invgK x) mulVg. Qed.
Lemma
mulgV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "invgK", "mul", "mulVg", "one" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgg
:= mulgV.
Definition
divgg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulKg : @left_loop G G (@inv G) *%g.
Proof. by move=> x y; rewrite mulgA mulVg mul1g. Qed.
Lemma
mulKg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "mul1g", "mulVg", "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVKg : @rev_left_loop G G (@inv G) *%g.
Proof. by move=> x y ; rewrite mulgA mulgV mul1g. Qed.
Lemma
mulVKg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "mul1g", "mulgA", "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgK : @right_loop G G (@inv G) *%g.
Proof. by move=> x y; rewrite -mulgA mulgV mulg1. Qed.
Lemma
mulgK
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "mulg1", "mulgA", "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgVK : @rev_right_loop G G (@inv G) *%g.
Proof. by move=> x y ; rewrite -mulgA mulVg mulg1. Qed.
Lemma
mulgVK
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv", "mulVg", "mulg1", "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgK
:= mulgVK.
Definition
divgK
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulgVK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgI : @right_injective G G G *%g.
Proof. by move=> x; apply: can_inj (mulKg x). Qed.
Lemma
mulgI
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "mulKg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulIg : @left_injective G G G *%g.
Proof. by move=> x; apply: can_inj (mulgK x). Qed.
Lemma
mulIg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "mulgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgI : @right_injective G G G (fun x y => x / y).
Proof. by move=> x y z /mulgI/invg_inj. Qed.
Lemma
divgI
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invg_inj", "mulgI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divIg : @left_injective G G G (fun x y => x / y).
Proof. by move=> x y z /mulIg. Qed.
Lemma
divIg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulIg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg1 x : x / 1 = x.
Proof. by rewrite invg1 mulg1. Qed.
Lemma
divg1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invg1", "mulg1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
div1g x : 1 / x = x^-1.
Proof. by rewrite mul1g. Qed.
Lemma
div1g
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mul1g" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divKg x y : commute x y -> x / (x / y) = y.
Proof. by move=> xyC; rewrite invgF mulgA xyC mulgK. Qed.
Lemma
divKg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commute", "invgF", "mulgA", "mulgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgKA z x y : (x * z) / (y * z) = x / y.
Proof. by rewrite invgM mulgA mulgK. Qed.
Lemma
mulgKA
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invgM", "mulgA", "mulgK" ]
TOTHINK : This does not have the same form as addrKA in ssralg.v
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgKA z x y : (x / z) * (z * y) = x * y.
Proof. by rewrite mulgA mulgVK. Qed.
Lemma
divgKA
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulgA", "mulgVK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg1_eq x y : x * y = 1 -> x^-1 = y.
Proof. by rewrite -[x^-1]mulg1 => <-; rewrite mulKg. Qed.
Lemma
mulg1_eq
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulKg", "mulg1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg1_eq x y : x / y = 1 -> x = y.
Proof. by move/mulg1_eq/invg_inj. Qed.
Lemma
divg1_eq
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invg_inj", "mulg1_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg_eq x y z : (x / z == y) = (x == y * z).
Proof. exact: can2_eq (divgK z) (mulgK z) x y. Qed.
Lemma
divg_eq
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "can2_eq", "divgK", "mulgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg_eq1 x y : (x / y == 1) = (x == y).
Proof. by rewrite divg_eq mul1g. Qed.
Lemma
divg_eq1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "divg_eq", "mul1g" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg_eq1 x y : (x * y == 1) = (x == y^-1).
Proof. by rewrite -[y in LHS]invgK divg_eq1. Qed.
Lemma
mulg_eq1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "divg_eq1", "invgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commuteV x y : commute x y -> commute x y^-1.
Proof. by move=> cxy; apply: (@mulIg y); rewrite mulgVK -mulgA cxy mulKg. Qed.
Lemma
commuteV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "commute", "mulIg", "mulKg", "mulgA", "mulgVK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgnFr x m n : n <= m -> x ^+ (m - n) = x ^+ m / x ^+ n.
Proof. by move=> lenm; rewrite -[in RHS](subnK lenm) expgnDr mulgK. Qed.
Lemma
expgnFr
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "expgnDr", "mulgK", "subnK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
expgnFl x y n : commute x y -> (x / y) ^+ n = x ^+ n / y ^+ n.
Proof. by move=> xyC; rewrite expgMn 1?expVgn; first exact/commuteV. Qed.
Lemma
expgnFl
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commute", "commuteV", "expVgn", "expgMn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgC x y : x * y = y * x ^ y.
Proof. by rewrite mulVKg. Qed.
Lemma
conjgC
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulVKg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgCV x y : x * y = y ^ x^-1 * x.
Proof. by rewrite -mulgA mulgVK invgK. Qed.
Lemma
conjgCV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invgK", "mulgA", "mulgVK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg1 x : x ^ 1 = x.
Proof. by rewrite conjgE commute1 mulKg. Qed.
Lemma
conjg1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commute1", "conjgE", "mulKg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conj1g x : 1 ^ x = 1.
Proof. by rewrite conjgE mul1g mulVg. Qed.
Lemma
conj1g
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjgE", "mul1g", "mulVg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjMg x y z : (x * y) ^ z = x ^ z * y ^ z.
Proof. by rewrite !conjgE !mulgA mulgK. Qed.
Lemma
conjMg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjgE", "mulgA", "mulgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgM x y z : x ^ (y * z) = (x ^ y) ^ z.
Proof. by rewrite !conjgE invgM !mulgA. Qed.
Lemma
conjgM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjgE", "invgM", "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjVg x y : x^-1 ^ y = (x ^ y)^-1.
Proof. by rewrite !conjgE !invgM invgK mulgA. Qed.
Lemma
conjVg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjgE", "invgK", "invgM", "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjJg x y z : (x ^ y) ^ z = (x ^ z) ^ y ^ z.
Proof. by rewrite 2!conjMg conjVg. Qed.
Lemma
conjJg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjMg", "conjVg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjXg x y n : (x ^+ n) ^ y = (x ^ y) ^+ n.
Proof. by elim: n => [|n IHn]; rewrite ?conj1g // !expgS conjMg IHn. Qed.
Lemma
conjXg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conj1g", "conjMg", "expgS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgK : @right_loop G G (@inv G) (@conjg G).
Proof. by move=> y x; rewrite -conjgM mulgV conjg1. Qed.
Lemma
conjgK
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjg", "conjg1", "conjgM", "inv", "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjgKV : @rev_right_loop G G (@inv G) (@conjg G).
Proof. by move=> y x; rewrite -conjgM mulVg conjg1. Qed.
Lemma
conjgKV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjg", "conjg1", "conjgM", "inv", "mulVg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_inj : @left_injective G G G (@conjg G).
Proof. by move=> y; apply: can_inj (conjgK y). Qed.
Lemma
conjg_inj
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "conjg", "conjgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_eq1 x y : (x ^ y == 1) = (x == 1).
Proof. by rewrite (can2_eq (conjgK _) (conjgKV _)) conj1g. Qed.
Lemma
conjg_eq1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "can2_eq", "conj1g", "conjgK", "conjgKV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_prod I 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
conjg_prod
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "big_morph", "conj1g", "conjMg", "conjg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgC x y : x * y = y * x * [~ x, y].
Proof. by rewrite -mulgA !mulVKg. Qed.
Lemma
commgC
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulVKg", "mulgA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgCV x y : x * y = [~ x^-1, y^-1] * (y * x).
Proof. by rewrite commgEl !mulgA !invgK !mulgVK. Qed.
Lemma
commgCV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgEl", "invgK", "mulgA", "mulgVK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjRg x y z : [~ x, y] ^ z = [~ x ^ z, y ^ z].
Proof. by rewrite !conjMg !conjVg. Qed.
Lemma
conjRg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "conjMg", "conjVg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
invgR x y : [~ x, y]^-1 = [~ y, x].
Proof. by rewrite commgEr conjVg invgM invgK. Qed.
Lemma
invgR
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgEr", "conjVg", "invgK", "invgM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgP x y : reflect (commute x y) ([~ x, y] == 1).
Proof. rewrite [[~ x, y]]mulgA -invgM mulg_eq1 eqg_inv eq_sym; apply: eqP. Qed.
Lemma
commgP
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "commute", "eq_sym", "eqg_inv", "invgM", "mulgA", "mulg_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_fix x y : (x ^ y == x) = ([~ x, y] == 1).
Proof. by rewrite mulg_eq1 eqg_inv. Qed.
Lemma
conjg_fix
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "eqg_inv", "mulg_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_fixP x y : reflect (x ^ y = x) ([~ x, y] == 1).
Proof. by rewrite -conjg_fix; apply: eqP. Qed.
Lemma
conjg_fixP
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "conjg_fix" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commg1_sym x y : ([~ x, y] == 1) = ([~ y, x] == 1).
Proof. by rewrite -invgR (inv_eq invgK) invg1. Qed.
Lemma
commg1_sym
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "inv_eq", "invg1", "invgK", "invgR" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commg1 x : [~ x, 1] = 1.
Proof. exact/eqP/commgP/commute1. Qed.
Lemma
commg1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgP", "commute1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm1g x : [~ 1, x] = 1.
Proof. by rewrite -invgR commg1 invg1. Qed.
Lemma
comm1g
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commg1", "invg1", "invgR" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgg x : [~ x, x] = 1.
Proof. exact/eqP/commgP. Qed.
Lemma
commgg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgXg x n : [~ x, x ^+ n] = 1.
Proof. exact/eqP/commgP/commuteX. Qed.
Lemma
commgXg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgP", "commuteX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgVg x : [~ x, x^-1] = 1.
Proof. exact/eqP/commgP/commuteV. Qed.
Lemma
commgVg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgP", "commuteV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
commgXVg x n : [~ x, x ^- n] = 1.
Proof. exact/eqP/commgP/commuteV/commuteX. Qed.
Lemma
commgXVg
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commgP", "commuteV", "commuteX" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_closedV : group_closed S -> invg_closed S.
Proof. by move=> [S1 SB] x /(SB 1)-/(_ S1); rewrite div1g. Qed.
Lemma
group_closedV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "S1", "div1g", "group_closed", "invg_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_closedM : group_closed S -> mulg_closed S.
Proof. move=> /[dup]-[S1 SB] /group_closedV SV x y xS /SV yS. rewrite -[y]invgK; exact: SB. Qed.
Lemma
group_closedM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "S1", "group_closed", "group_closedV", "invgK", "mulg_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gsimpl
:= autorewrite with gsimpl; try done.
Ltac
gsimpl
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gsimp
:= (@mulg1, @mul1g, (@invg1, @invgK), (@mulgV, @mulVg)).
Definition
gsimp
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "invg1", "invgK", "mul1g", "mulVg", "mulg1", "mulgV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gnorm
:= (gsimp, (@mulgK, @mulgVK, (@mulgA, @invgM))).
Definition
gnorm
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gsimp", "invgM", "mulgA", "mulgK", "mulgVK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
monoid_morphism (G H : baseUMagmaType) (f : G -> H) : Prop
:= (f 1 = 1) * {morph f : x y / x * y}.
Definition
monoid_morphism
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulf1 : apply 1 = 1.
Proof. by rewrite -[1]divg1 gmulfF divgg. Qed.
Lemma
gmulf1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "divg1", "divgg", "gmulfF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.
Lemma
gmulfM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "div1g", "gmulf1", "gmulfF", "invgK", "mul1g" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'multiplicative' U -> V }"
:= (Multiplicative.type U%type V%type) : type_scope.
Notation
{ 'multiplicative' U -> V }
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_fun (f g : T -> G) x
:= f x * g x.
Definition
mul_fun
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
one_fun : T -> G
:= fun=> 1.
Definition
one_fun
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\1"
:= (one_fun _) : function_scope.
Notation
\1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "one_fun" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_gmulfM f' : 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
can2_gmulfM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "fK", "gmulfM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulf_commute x y : commute x y -> commute (f x) (f y).
Proof. by move=> xy; rewrite /commute -!gmulfM xy. Qed.
Lemma
gmulf_commute
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "commute", "gmulfM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulf_eq1 x : injective f -> (f x == 1) = (x == 1).
Proof. by move=> /inj_eq <-; rewrite gmulf1. Qed.
Lemma
gmulf_eq1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulf1", "inj_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_gmulf1 f' : cancel f f' -> cancel f' f -> f' 1 = 1.
Proof. by move=> fK f'K; apply: (canLR fK); rewrite gmulf1. Qed.
Lemma
can2_gmulf1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "fK", "gmulf1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulfXn n : {morph f : x / x ^+ n}.
Proof. by elim: n => [|[|n] IHn] x /=; rewrite ?(gmulf1, gmulfM) // IHn. Qed.
Lemma
gmulfXn
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulf1", "gmulfM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulf_prod I 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
gmulf_prod
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "big_morph", "gmulf1", "gmulfM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulfV : {morph f : x / x^-1}.
Proof. by move=> x; apply/divg1_eq; rewrite invgK -gmulfM mulVg gmulf1. Qed.
Lemma
gmulfV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "divg1_eq", "gmulf1", "gmulfM", "invgK", "mulVg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulfF : {morph f : x y / x / y}.
Proof. by move=> x y; rewrite gmulfM gmulfV. Qed.
Lemma
gmulfF
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulfM", "gmulfV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
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
gmulf_inj
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "divg1_eq", "divgg", "gmulfF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulfXVn n : {morph f : x / x ^- n}.
Proof. by move=> x /=; rewrite gmulfV gmulfXn. Qed.
Lemma
gmulfXVn
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulfV", "gmulfXn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulfJ : {morph f : x y / x ^ y}.
Proof. by move=> x y; rewrite !gmulfM/= gmulfV. Qed.
Lemma
gmulfJ
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulfM", "gmulfV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gmulfR : {morph f : x y / [~ x, y]}.
Proof. by move=> x y; rewrite !gmulfM/= !gmulfV. Qed.
Lemma
gmulfR
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulfM", "gmulfV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_gmulfM : {morph @idfun G : x y / x * y}.
Proof. by []. Qed.
Fact
idfun_gmulfM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_gmulfM : {morph f \o h : x y / x * y}.
Proof. by move=> x y /=; rewrite !gmulfM. Qed.
Fact
comp_gmulfM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulfM" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_gmulf1 : idfun 1 = 1 :> H.
Proof. by []. Qed.
Fact
idfun_gmulf1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
one_fun_gmulfM : {morph @one_fun G H : x y / x * y}.
Proof. by move=> x y; rewrite mulg1. Qed.
Fact
one_fun_gmulfM
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "mulg1", "one_fun" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_gmulf1 : (f \o h) 1 = 1.
Proof. by rewrite /= !gmulf1. Qed.
Fact
comp_gmulf1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulf1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
one_fun_gmulf1 : @one_fun G H 1 = 1.
Proof. by []. Qed.
Fact
one_fun_gmulf1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "one_fun" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_fun_gmulf1 : (f \* g) 1 = 1.
Proof. by rewrite /= !gmulf1 mulg1. Qed.
Fact
mul_fun_gmulf1
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "gmulf1", "mulg1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gpred_prod I 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
gpred_prod
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "big_ind" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gpredXn n : {in S, forall u, u ^+ n \in S}.
Proof. by move=> x xS; elim: n => [|[//|n] IHn]; [exact: gpred1 | exact: gpredM]. Qed.
Lemma
gpredXn
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gpredV (S : invgClosed G) : {mono (@inv G): u / u \in S}.
Proof. by move=> u; apply/idP/idP=> /gpredVr; rewrite ?invgK; apply. Qed.
Lemma
gpredV
boot
boot/monoid.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun" ]
[ "apply", "inv", "invgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d