statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
EquivRel r er es et | := (@EquivRelPack _ r (EquivClass er es et)). | Notation | EquivRel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'equiv_rel' 'of' e ]" | := (@equiv_pack _ _ e _ id)
(format "[ 'equiv_rel' 'of' e ]") : form_scope. | Notation | [ 'equiv_rel' 'of' e ] | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"equiv_pack",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encModRel_class_of (r : rel D) | :=
EncModRelClassPack of (forall x, r x x -> r (ED (DE x)) x) & (r =2 e). | Variant | encModRel_class_of | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"ED",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encModRel | := EncModRelPack {
enc_mod_rel :> rel D;
_ : encModRel_class_of enc_mod_rel
}. | Record | encModRel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encModRel_class_of",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encModRelClass | :=
let: EncModRelPack _ c as r' := r return encModRel_class_of r' in c. | Definition | encModRelClass | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encModRel_class_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encModRelP (x : D) : r x x -> r (ED (DE x)) x. | Proof. by case: r => [] ? [] /= he _ /he. Qed. | Definition | encModRelP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"ED"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encModRelE : r =2 e. | Proof. by case: r => [] ? []. Qed. | Definition | encModRelE | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encoded_equiv : rel E | := [rel x y | r (ED x) (ED y)]. | Definition | encoded_equiv | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"ED",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
EncModRelClass m | :=
(EncModRelClassPack (fun x _ => m x) (fun _ _ => erefl _)). | Notation | EncModRelClass | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
EncModRel r m | := (@EncModRelPack _ _ _ _ _ r (EncModRelClass m)). | Notation | EncModRel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"EncModRelClass"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enc_mod_rel_is_equiv : equiv_class_of (enc_mod_rel r). | Proof.
split => [x|x y|y x z]; rewrite !encModRelE //; first by rewrite equiv_sym.
by move=> exy /(equiv_trans exy).
Qed. | Lemma | enc_mod_rel_is_equiv | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encModRelE",
"equiv_class_of",
"equiv_sym",
"equiv_trans",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enc_mod_rel_equiv_rel | := EquivRelPack enc_mod_rel_is_equiv. | Definition | enc_mod_rel_equiv_rel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"enc_mod_rel_is_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encModEquivP (x : D) : r (ED (DE x)) x. | Proof. by rewrite encModRelP ?encModRelE. Qed. | Definition | encModEquivP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"ED",
"encModRelE",
"encModRelP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
e' | := (encoded_equiv r). | Notation | e' | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encoded_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encoded_equivE : e' =2 [rel x y | e (ED x) (ED y)]. | Proof. by move=> x y; rewrite /encoded_equiv /= encModRelE. Qed. | Lemma | encoded_equivE | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"ED",
"e'",
"encModRelE",
"encoded_equiv",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
e'E | := encoded_equivE. | Notation | e'E | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encoded_equivE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encoded_equiv_is_equiv : equiv_class_of e'. | Proof.
split => [x|x y|y x z]; rewrite !e'E //=; first by rewrite equiv_sym.
by move=> exy /(equiv_trans exy).
Qed. | Lemma | encoded_equiv_is_equiv | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"e'",
"e'E",
"equiv_class_of",
"equiv_sym",
"equiv_trans",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encoded_equiv_equiv_rel | := EquivRelPack encoded_equiv_is_equiv. | Canonical | encoded_equiv_equiv_rel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encoded_equiv_is_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encoded_equivP x : e' (DE (ED x)) x. | Proof. by rewrite /encoded_equiv /= encModEquivP. Qed. | Lemma | encoded_equivP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"ED",
"e'",
"encModEquivP",
"encoded_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eC | := (encoded_equiv encD). | Notation | eC | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encoded_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
canon x | := choose (eC x) (x). | Definition | canon | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"choose",
"eC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equivQuotient | := EquivQuotient {
erepr : C;
_ : (frel canon) erepr erepr
}. | Record | equivQuotient | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"canon",
"frel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
type_of & (phantom (rel _) encD) | := equivQuotient. | Definition | type_of | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"equivQuotient",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
canon_id : forall x, (invariant canon canon) x. | Proof.
move=> x /=; rewrite /canon (@eq_choose _ _ (eC x)).
by move=> y; apply: equiv_ltrans; rewrite equiv_sym /= chooseP.
by rewrite (@choose_id _ (eC x) _ x) ?chooseP ?equiv_refl.
Qed. | Lemma | canon_id | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"canon",
"chooseP",
"choose_id",
"eC",
"eq_choose",
"equiv_ltrans",
"equiv_refl",
"equiv_sym",
"invariant"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi | := locked (fun x => EquivQuotient (canon_id x)). | Definition | pi | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"canon_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ereprK : cancel erepr pi. | Proof.
pose T : subType _ := HB.pack equivQuotient [isSub for erepr].
by unlock pi; case=> x hx; apply/(@val_inj _ _ T)/eqP.
Qed. | Lemma | ereprK | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"equivQuotient",
"pi",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encDE | := (encModRelE encD). | Notation | encDE | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encModRelE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encDP | := (encModEquivP encD). | Notation | encDP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"encModEquivP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
encD_equiv_rel | := EquivRelPack (enc_mod_rel_is_equiv encD). | Canonical | encD_equiv_rel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"enc_mod_rel_is_equiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_CD (x y : C) : reflect (pi x = pi y) (eC x y). | Proof.
apply: (iffP idP) => hxy.
apply: (can_inj ereprK); unlock pi canon => /=.
rewrite -(@eq_choose _ (eC x) (eC y)).
by move=> z; rewrite /eC /=; apply: equiv_ltrans.
by apply: choose_id; rewrite ?equiv_refl //.
rewrite (equiv_trans (chooseP (equiv_refl _ _))) //=.
move: hxy => /(f_equal erepr) /=; unlock ... | Lemma | pi_CD | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"canon",
"chooseP",
"choose_id",
"eC",
"eq_choose",
"equiv_ltrans",
"equiv_refl",
"equiv_sym",
"equiv_trans",
"ereprK",
"pi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_DC (x y : D) :
reflect (pi (DC x) = pi (DC y)) (eD x y). | Proof.
apply: (iffP idP)=> hxy.
apply/pi_CD; rewrite /eC /=.
by rewrite (equiv_ltrans (encDP _)) (equiv_rtrans (encDP _)) /= encDE.
rewrite -encDE -(equiv_ltrans (encDP _)) -(equiv_rtrans (encDP _)) /=.
exact/pi_CD.
Qed. | Lemma | pi_DC | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"eC",
"encDE",
"encDP",
"equiv_ltrans",
"equiv_rtrans",
"pi",
"pi_CD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equivQTP : cancel (CD \o erepr) (pi \o DC). | Proof. by move=> x; rewrite /= (pi_CD _ (erepr x) _) ?ereprK /eC /= ?encDP. Qed. | Lemma | equivQTP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"eC",
"encDP",
"ereprK",
"pi",
"pi_CD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qT | := (type_of (Phantom (rel D) encD)). | Notation | qT | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"rel",
"type_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmodP x y : reflect (x = y %[mod qT]) (eD x y). | Proof. by apply: (iffP (pi_DC _ _)); rewrite !unlock. Qed. | Lemma | eqmodP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"pi_DC",
"qT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmodE x y : (x == y %[mod qT]) = eD x y. | Proof. exact: sameP eqP (@eqmodP _ _). Qed. | Lemma | eqmodE | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"eqmodP",
"qT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{eq_quot e }" | :=
(@EquivQuot.type_of _ _ _ _ _ _ (Phantom (rel _) e)) : quotient_scope. | Notation | {eq_quot e } | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"rel",
"type_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x == y %[mod_eq r ]" | := (x == y %[mod {eq_quot r}]) : quotient_scope. | Notation | x == y %[mod_eq r ] | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x = y %[mod_eq r ]" | := (x = y %[mod {eq_quot r}]) : quotient_scope. | Notation | x = y %[mod_eq r ] | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x != y %[mod_eq r ]" | := (x != y %[mod {eq_quot r}]) : quotient_scope. | Notation | x != y %[mod_eq r ] | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x <> y %[mod_eq r ]" | := (x <> y %[mod {eq_quot r}]) : quotient_scope. | Notation | x <> y %[mod_eq r ] | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defaultEncModRelClass | :=
@EncModRelClassPack D D id id r r (fun _ rxx => rxx) (fun _ _ => erefl _). | Definition | defaultEncModRelClass | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defaultEncModRel | := EncModRelPack defaultEncModRelClass. | Canonical | defaultEncModRel | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"defaultEncModRelClass"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmodE x y : (x == y %[mod_eq e]) = e x y. | Proof. by rewrite pi_eq_quot. Qed. | Lemma | eqmodE | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmodP x y : reflect (x = y %[mod_eq e]) (e x y). | Proof. by rewrite -eqmodE; apply/eqP. Qed. | Lemma | eqmodP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"eqmodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqquotE x y : (x == y %[mod Q]) = e x y. | Proof. by rewrite pi_eq_quot. Qed. | Lemma | eqquotE | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqquotP x y : reflect (x = y %[mod Q]) (e x y). | Proof. by rewrite -eqquotE; apply/eqP. Qed. | Lemma | eqquotP | boot | boot/generic_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"QuotSubType.Exports",
"EquivQuot.Exports"
] | [
"apply",
"eqquotE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"*%g" | := (@mul _) : function_scope. | Notation | *%g | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x * y" | := (mul x y) : group_scope. | Notation | x * y | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commute x y | := x * y = y * x. | Definition | commute | 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 | |
commute_refl x : commute x x. | Proof. by []. Qed. | Lemma | commute_refl | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commute_sym x y : commute x y -> commute y x. | Proof. by []. Qed. | Lemma | commute_sym | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulg_closed | := {in S &, forall u v, u * v \in S}. | Definition | mulg_closed | 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 | |
commuteM x y z : commute x y -> commute x z -> commute x (y * z). | Proof. by move=> cxy cxz; rewrite /commute -mulgA -cxz !mulgA cxy. Qed. | Lemma | commuteM | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute",
"cxz",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"1" | := (@one _) : group_scope. | Notation | 1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"s `_ i" | := (nth 1 s i) : group_scope. | Notation | s `_ i | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\prod_ ( i <- r | P ) F" | := (\big[*%g/1]_(i <- r | P) F). | Notation | \prod_ ( i <- r | P ) F | 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 | |
"\prod_ ( i | P ) F" | := (\big[*%g/1]_(i | P) F). | Notation | \prod_ ( i | P ) F | 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 | |
"\prod_ ( i 'in' A ) F" | := (\big[*%g/1]_(i in A) F). | Notation | \prod_ ( i 'in' A ) F | 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 | |
"\prod_ ( m <= i < n ) F" | := (\big[*%g/1%g]_(m <= i < n) F%g). | Notation | \prod_ ( m <= i < n ) F | 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 | |
natexp (G : baseUMagmaType) (x : G) n : G | := iterop n *%g x 1. | Definition | natexp | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"iterop"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ^+ n" | := (natexp x n) : group_scope. | Notation | x ^+ n | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"natexp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgnE x n : x ^+ n = iterop n mul x 1. | Proof. by []. Qed. | Lemma | expgnE | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"iterop",
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expg0 x : x ^+ 0 = 1. | Proof. by []. Qed. | Lemma | expg0 | 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 | |
expg1 x : x ^+ 1 = x. | Proof. by []. Qed. | Lemma | expg1 | 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 | |
expg2 x : x ^+ 2 = x * x. | Proof. by []. Qed. | Lemma | expg2 | 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 | |
expgSS x n : x ^+ n.+2 = x * x ^+ n.+1. | Proof. by []. Qed. | Lemma | expgSS | 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 | |
expgb x (b : bool) : x ^+ b = (if b then x else 1). | Proof. by case: b. Qed. | Lemma | expgb | 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 | |
umagma_closed | := 1 \in S /\ mulg_closed S. | Definition | umagma_closed | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"mulg_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgS x n : x ^+ n.+1 = x * x ^+ n. | Proof. by case: n => //=; rewrite mulg1. Qed. | Lemma | expgS | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"mulg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expg1n n : 1 ^+ n = 1 :> G. | Proof. by elim: n => // n IHn; rewrite expgS mul1g. Qed. | Lemma | expg1n | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"expgS",
"mul1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commute1 x : commute x 1. | Proof. by rewrite /commute mulg1 mul1g. Qed. | Lemma | commute1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute",
"mul1g",
"mulg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgnDr x m n : x ^+ (m + n) = x ^+ m * x ^+ n. | Proof. by elim: m => [|m IHm]; rewrite ?mul1g // !expgS IHm mulgA. Qed. | Lemma | expgnDr | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"expgS",
"mul1g",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgSr x n : x ^+ n.+1 = x ^+ n * x. | Proof. by rewrite -addn1 expgnDr expg1. Qed. | Lemma | expgSr | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"addn1",
"expg1",
"expgnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgnA x m n : x ^+ (m * n) = x ^+ m ^+ n. | Proof. by rewrite mulnC; elim: n => //= n IHn; rewrite expgS expgnDr IHn. Qed. | Lemma | expgnA | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"expgS",
"expgnDr",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgnAC x m n : x ^+ m ^+ n = x ^+ n ^+ m. | Proof. by rewrite -2!expgnA mulnC. Qed. | Lemma | expgnAC | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"expgnA",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_mulg n x y : iter n ( *%g x) y = x ^+ n * y. | Proof. by elim: n => [|n IHn]; rewrite ?mul1g //= IHn expgS mulgA. Qed. | Lemma | iter_mulg | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"expgS",
"iter",
"mul1g",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_mulg_1 n x : iter n ( *%g x) 1 = x ^+ n. | Proof. by rewrite iter_mulg mulg1. Qed. | Lemma | iter_mulg_1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"iter",
"iter_mulg",
"mulg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodg_const (I : finType) (A : pred I) x : \prod_(i in A) x = x ^+ #|A|. | Proof. by rewrite big_const -Monoid.iteropE. Qed. | Lemma | prodg_const | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"big_const",
"iteropE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodg_const_nat n m x : \prod_(n <= i < m) x = x ^+ (m - n). | Proof. by rewrite big_const_nat -Monoid.iteropE. Qed. | Lemma | prodg_const_nat | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"big_const_nat",
"iteropE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodgXr x 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 | prodgXr | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"big_morph",
"expgnDr",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | commute_prod | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"big_ind",
"commute",
"commute1",
"commuteM",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | prodgM_commute | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"big_cons",
"big_nil",
"commute",
"commute_prod",
"commute_sym",
"mulg1",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | prodgMl_commute | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute",
"prodgM_commute",
"prodg_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | prodgMr_commute | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute",
"prodgM_commute",
"prodg_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commuteX x y n : commute x y -> commute x (y ^+ n). | Proof.
by move=> cxy; case: n; [apply: commute1 | elim=> // n; apply: commuteM].
Qed. | Lemma | commuteX | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"commute",
"commute1",
"commuteM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commuteX2 x y m n : commute x y -> commute (x ^+ m) (y ^+ n). | Proof. by move=> cxy; apply/commuteX/commute_sym/commuteX. Qed. | Lemma | commuteX2 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"commute",
"commuteX",
"commute_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expgMn x 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 | expgMn | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commute",
"commuteX",
"commute_sym",
"expgS",
"mulg1",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monoid_closed | := umagma_closed. | Notation | monoid_closed | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"umagma_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ^-1" | := (inv x) : group_scope. | Notation | x ^-1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x / y" | := (x * y^-1) : group_scope. | Notation | x / y | 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 | |
"x ^- n" | := ((x ^+ n)^-1) : group_scope. | Notation | x ^- n | 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 | |
conjg (G : baseGroupType) (x y : G) | := y^-1 * (x * y). | Definition | conjg | 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 | |
"x ^ y" | := (conjg x y) : group_scope. | Notation | x ^ y | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"conjg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commg (G : baseGroupType) (x y : G) | := x^-1 * (conjg x y). | Definition | commg | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"conjg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ ~ x1 , x2 , .. , xn ]" | := (commg .. (commg x1 x2) .. xn)
: group_scope. | Notation | [ ~ x1 , x2 , .. , xn ] | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"commg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invg_closed | := {in S, forall u, u^-1 \in S}. | Definition | invg_closed | 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 | |
divg_closed | := {in S &, forall u v, u / v \in S}. | Definition | divg_closed | 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 | |
group_closed | := 1 \in S /\ divg_closed. | Definition | group_closed | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"divg_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invg1 : inv one = one. | Proof.
by apply: (can_inj invgK); rewrite -[inv one 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",
"inv",
"invgK",
"invgM",
"mul1g",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulg1 : right_id one mul. | Proof. by move=> x; apply: (can_inj invgK); rewrite invgM invg1 mul1g. Qed. | Lemma | mulg1 | boot | boot/monoid.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun"
] | [
"apply",
"invg1",
"invgK",
"invgM",
"mul",
"mul1g",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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