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 |
|---|---|---|---|---|---|---|---|---|---|---|
PEeval_default_isIn l e1 p1 e2 p2 n e3 :
default_isIn e1 p1 e2 p2 = Some (n, e3) ->
((Pos.to_nat p1 > N.to_nat n)%N
/\ PEeval l (PEpow e2 (Npos p2))
= PEeval l (PEmul (PEpow e1 (N.sub (Npos p1) n)) e3)). | Proof.
rewrite /default_isIn; case: PExpr_eqP => [->|//].
case: Z.pos_sub (Zint_pos_sub (pos_nat_Pos_to_nat p1) (pos_nat_Pos_to_nat p2)).
- rewrite /Zint eq_sym subr_eq0 => /eqP[/Pos_to_natI<-] [<- <-]/=.
by rewrite Pos_to_nat_gt0 rmorph1 mulr1.
- move=> p pp1p2.
have p1p2 : (Pos.to_nat p2 < Pos.to_nat p1)%N.
b... | Lemma | PEeval_default_isIn | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"N_to_natB",
"NegzE",
"PEeval",
"PEeval_NPEpow",
"PExpr_eqP",
"Pos_to_natB",
"Pos_to_natI",
"Pos_to_nat_gt0",
"PoszD",
"Zint",
"Zint_pos_sub",
"add0r",
"addrA",
"apply",
"default_isIn",
"eq_sym",
"eqbLHS",
"eqz_nat",
"exprD",
"int",
"ltnW",
"ltn_subrL",
"ltz_nat",
"mulr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_isIn l e1 p1 e2 p2 n e3 :
isIn e1 p1 e2 p2 = Some (n, e3) ->
((Pos.to_nat p1 > N.to_nat n)%N
/\ PEeval l (PEpow e2 (Npos p2))
= PEeval l (PEmul (PEpow e1 (N.sub (Npos p1) n)) e3)). | Proof.
(elim: e2 p1 p2 n e3 => [||?|?|????|????|e3 IH e4 IH'|??|e2 IH n'] p1 p2 n e5'';
do ?[exact: PEeval_default_isIn]; last first)=> /=.
case: n' => [//|p3].
case: isIn (IH p1 (Pos.mul p3 p2)) => [[n' e5'''] /(_ _ _ erefl)/[swap]|//].
by move=> [{n'}-> {e5'''}->] [np1 e]; rewrite -exprM/= -Pos_to_natM.
cas... | Lemma | PEeval_isIn | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"NPEpow",
"N_to_natB",
"PEeval",
"PEeval_NPEmul",
"PEeval_NPEpow",
"PEeval_default_isIn",
"Pos_to_natM",
"addnBA",
"exprD",
"exprM",
"exprMn",
"isIn",
"last",
"ltnW",
"ltn_trans",
"mul",
"mulrA",
"mulrACA",
"mulrCA",
"n'",
"split",
"sub",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_split_aux l e1 p e2 :
PEeval l (PEpow e1 (Npos p))
= PEeval l (PEmul (rsplit_left (split_aux e1 p e2))
(rsplit_common (split_aux e1 p e2)))
/\ PEeval l e2
= PEeval l (PEmul (rsplit_right (split_aux e1 p e2))
(rsplit_common (split_aux e1 p e2))). | Proof.
have main_case e1' p' e2' :
let res :=
match isIn e1' p' e2' xH with
| Some (N0, e3) => mk_rsplit (PEc 1) (NPEpow e1' (Npos p')) e3
| Some (Npos q, e3) =>
mk_rsplit (NPEpow e1' (Npos q)) (NPEpow e1' (Npos (Pos.sub p' q))) e3
| None => mk_rsplit (NPEpow e1' (Npos p')) (PEc 1)... | Lemma | PEeval_split_aux | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"NPEpow",
"N_to_natB",
"PEeval",
"PEeval_NPEmul",
"PEeval_NPEpow",
"PEeval_isIn",
"Pos_to_natB",
"Pos_to_natM",
"expr0",
"expr1",
"expr1n",
"exprD",
"exprM",
"exprMn",
"isIn",
"ltnW",
"mul",
"mul1r",
"mulr1",
"mulrA",
"mulrACA",
"mulrC",
"n'",
"r1",
"r2",
"rmorph1",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_split_l l e1 e2 :
PEeval l e1
= PEeval l (PEmul (rsplit_left (split e1 e2)) (rsplit_common (split e1 e2))). | Proof. by case: (PEeval_split_aux l e1 xH e2) => /=/[!expr1]. Qed. | Lemma | PEeval_split_l | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PEeval",
"PEeval_split_aux",
"expr1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_split_r l e1 e2 :
PEeval l e2
= PEeval l (PEmul (rsplit_right (split e1 e2)) (rsplit_common (split e1 e2))). | Proof. by case: (PEeval_split_aux l e1 xH e2). Qed. | Lemma | PEeval_split_r | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PEeval",
"PEeval_split_aux",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split_neq0_l l e1 e2 :
PEeval l e1 != 0 -> PEeval l (rsplit_left (split e1 e2)) != 0. | Proof.
by apply: contraNN; rewrite (PEeval_split_l l e1 e2)/= mulf_eq0 => ->.
Qed. | Lemma | split_neq0_l | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PEeval",
"PEeval_split_l",
"apply",
"mulf_eq0",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split_neq0_r l e1 e2 :
PEeval l e2 != 0 -> PEeval l (rsplit_right (split e1 e2)) != 0. | Proof.
by apply: contraNN; rewrite (PEeval_split_r l e1 e2)/= mulf_eq0 => ->.
Qed. | Lemma | split_neq0_r | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PEeval",
"PEeval_split_r",
"apply",
"mulf_eq0",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_Fnorm l e :
PCond l (condition (Fnorm e)) -> PEeval l (denum (Fnorm e)) != 0. | Proof.
elim: e => [||c|i|e IH e' IH'|e IH e' IH'|e IH e' IH'|//|e IH|e IH e' IH'
|e IH n]/=; do ?[by rewrite rmorph1 oner_neq0];
rewrite ?PEeval_NPEmul/= ?PCond_cons ?PCond_app.
- move=> /andP[/IH en0 /IH' e'n0].
by rewrite PEeval_NPEmul -PEeval_split_r mulf_neq0// split_neq0_l.
- move=> /andP[/IH en0 /IH' e'... | Lemma | PCond_Fnorm | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fnorm",
"PCond",
"PCond_app",
"PCond_cons",
"PEeval",
"PEeval_NPEmul",
"PEeval_NPEpow",
"PEeval_split_r",
"e'",
"expf_neq0",
"mulf_neq0",
"oner_neq0",
"rmorph1",
"split_neq0_l",
"split_neq0_r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addf_div_common (F : fieldType) (c x1 y1 x2 y2 : F) :
y1 * c != 0 -> y2 * c != 0 ->
x1 / (y1 * c) + x2 / (y2 * c) = (x1 * y2 + x2 * y1) / (y1 * c * y2). | Proof.
move=> y1c y2c; rewrite addf_div// !mulrA -mulrDl -mulf_div divff ?mulr1//.
by apply: contraNN y1c; rewrite mulf_eq0 => ->; rewrite orbT.
Qed. | Lemma | addf_div_common | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"addf_div",
"apply",
"divff",
"mulf_div",
"mulf_eq0",
"mulr1",
"mulrA",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulf_div_common (F : fieldType) (c1 c2 x1 y1 x2 y2 : F) :
y1 * c1 != 0 -> y2 * c2 != 0 ->
(x1 * c2) / (y1 * c1) * ((x2 * c1) / (y2 * c2)) = (x1 * x2) / (y1 * y2). | Proof.
move=> y1c1 y2c2; rewrite mulf_div mulrACA [X in _ / X]mulrACA.
rewrite -[LHS]mulf_div [c2 * c1]mulrC divff ?mulr1// mulf_neq0//.
by apply: contraNN y1c1; rewrite mulf_eq0 => ->; rewrite orbT.
by apply: contraNN y2c2; rewrite mulf_eq0 => ->; rewrite orbT.
Qed. | Lemma | mulf_div_common | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"apply",
"c1",
"c2",
"divff",
"mulf_div",
"mulf_eq0",
"mulf_neq0",
"mulr1",
"mulrACA",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_Fnorm l e : PCond l (condition (Fnorm e)) ->
FEeval l e = PEeval l (num (Fnorm e)) / PEeval l (denum (Fnorm e)). | Proof.
elim: e => [||c|i|e IH e' IH'|e IH e' IH'|e IH e' IH'|e IH|e IH|e IH e' IH'
|e IH n]/=; rewrite ?PCond_cons ?PCond_app.
- by rewrite rmorph0 mul0r.
- by rewrite rmorph1 invr1 mulr1.
- by rewrite rmorph1 invr1 mulr1.
- by rewrite rmorph1 invr1 mulr1.
- move=> /andP[ce ce']; rewrite IH// IH'//.
rewrite PEeva... | Lemma | PEeval_Fnorm | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"FEeval",
"Fnorm",
"PCond",
"PCond_Fnorm",
"PCond_app",
"PCond_cons",
"PEeval",
"PEeval_NPEadd",
"PEeval_NPEmul",
"PEeval_NPEopp",
"PEeval_NPEpow",
"PEeval_NPEsub",
"PEeval_split_l",
"PEeval_split_r",
"addf_div_common",
"e'",
"eqbLHS",
"expr_div_n",
"invf_div",
"invr1",
"mul0... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cfield_checkerT
cond_norm (cond_normP : forall l el, PCond l (cond_norm el) -> PCond l el)
n l lpe fe1 fe2 lc :
PEeval_eqs l lpe ->
field_checker cond_norm n lpe fe1 fe2 = Some lc ->
PCond l lc ->
FEeval l fe1 = FEeval l fe2. | Proof.
rewrite /field_checker; set pe1 := PEmul _ _; set pe2 := PEmul _ _.
move/(Cring_checkerT cdivP) => /(_ n pe1 pe2); rewrite /ring_checker.
case: Peq => [/(_ erefl) epe12 [{lc}<-] /cond_normP/[!PCond_app]/andP[P P']|//].
by apply/eqP; rewrite !PEeval_Fnorm// eqr_div ?PCond_Fnorm//; apply/eqP.
Qed. | Lemma | Cfield_checkerT | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Cring_checkerT",
"FEeval",
"PCond",
"PCond_Fnorm",
"PCond_app",
"PEeval_Fnorm",
"PEeval_eqs",
"Peq",
"apply",
"cond_norm",
"eqr_div",
"field_checker",
"ring_checker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_Fcons0 l a el :
PCond l (Fcons0 a el) -> PEeval l a != 0 /\ PCond l el. | Proof.
elim: el => [//|e el IH] /=; set pa := _ a; set pe := _ e.
case: Peq (@Peval_Peq _ _ R_of_C l pa pe) => [/(_ erefl)|_].
by rewrite !Peval_Pol_of_PExpr PCond_cons => -> /andP[-> ->].
by rewrite !PCond_cons => /andP[-> /IH[->]].
Qed. | Lemma | PCond_Fcons0 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons0",
"PCond",
"PCond_cons",
"PEeval",
"Peq",
"Peval_Peq",
"Peval_Pol_of_PExpr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_Fcons00 l a el :
PCond l (Fcons00 a el) -> PEeval l a != 0 /\ PCond l el. | Proof.
elim: a el => [||c|i|a IH a' IH'|a IH a' IH'|a IH a' IH'|a IH| a IH n] el /=;
do ?[by move/PCond_Fcons0].
- by move=> /IH[an0 /IH'[a'n0 ->]]; rewrite mulf_neq0.
- by move=> /IH[an0 ->]; rewrite expf_neq0.
Qed. | Lemma | PCond_Fcons00 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons00",
"PCond",
"PCond_Fcons0",
"PEeval",
"expf_neq0",
"mulf_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_Fapp Fcons l el el' :
(forall l e el, PCond l (Fcons e el) -> PEeval l e != 0 /\ PCond l el) ->
PCond l (Fapp Fcons el el') -> PCond l el /\ PCond l el'. | Proof.
move=> FcondP; elim: el el' => [//|e el IH] el' /=.
by rewrite PCond_cons => /FcondP[-> /IH[-> ->]].
Qed. | Lemma | PCond_Fapp | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PCond",
"PCond_cons",
"PEeval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_cond_norm Fcons l el :
(forall l e el, PCond l (Fcons e el) -> PEeval l e != 0 /\ PCond l el) ->
PCond l (cond_norm Fcons el) -> PCond l el. | Proof. by move=> /PCond_Fapp /[apply] -[]. Qed. | Lemma | PCond_cond_norm | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PCond",
"PCond_Fapp",
"PEeval",
"apply",
"cond_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEsimp | := (PEsimp 0 1 +%R *%R sub -%R pow_pos eq_op). | Notation | PEsimp | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"pow_pos",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_Fcons1 l a el :
PCond l (Fcons1 a el) -> PEeval l a != 0 /\ PCond l el. | Proof.
elim: a el => [||c|i|a IH a' IH'|a IH a' IH'|a IH a' IH'|a IH| a IH n] el /=;
do ?[by move/PCond_Fcons0].
- case: eqP => [->|/eqP]; rewrite /= ?rmorph0 ?eqxx// => c0 ->; split=> [|//].
by apply: contraNN c0; rewrite !eq_le -oppr_ge0 -rmorphN !R_of_C_ge0 oppr_ge0.
- by move=> /IH[an0 /IH'[a'n0 ->]]; rewrite... | Lemma | PCond_Fcons1 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons1",
"PCond",
"PCond_Fcons0",
"PEeval",
"R_of_C_ge0",
"apply",
"c0",
"eq_le",
"eqxx",
"expf_neq0",
"mulf_neq0",
"oner_eq0",
"oppr_eq0",
"oppr_ge0",
"rmorph0",
"rmorphN",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_PEsimp l e : PEeval l (PEsimp e) = PEeval l e. | Proof.
elim: e => [||c|i|a IH a' IH'|a IH a' IH'|a IH a' IH'|a IH| a IH n] //=.
- by rewrite PEeval_NPEadd/= IH IH'.
- by rewrite PEeval_NPEsub/= IH IH'.
- by rewrite PEeval_NPEmul/= IH IH'.
- by rewrite PEeval_NPEopp/= IH.
- by rewrite PEeval_NPEpow/= IH.
Qed. | Lemma | PEeval_PEsimp | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PEeval",
"PEeval_NPEadd",
"PEeval_NPEmul",
"PEeval_NPEopp",
"PEeval_NPEpow",
"PEeval_NPEsub",
"PEsimp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_Fcons2 l a el :
PCond l (Fcons2 a el) -> PEeval l a != 0 /\ PCond l el. | Proof. by move=> /PCond_Fcons1; rewrite PEeval_PEsimp. Qed. | Lemma | PCond_Fcons2 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons2",
"PCond",
"PCond_Fcons1",
"PEeval",
"PEeval_PEsimp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
option_R_eq T (s s' : option T) : option_R eq s s' -> s = s'. | Proof. by case=> // ? ? ->. Qed. | Lemma | option_R_eq | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [] | a bunch of helper lemmas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
PExpr_R_eq T (e e' : PExpr T) : PExpr_R eq e e' -> e = e'. | Proof.
by elim=> [||??->|??/positive_R_eq->|?? _-> ?? _->|?? _-> ?? _->|?? _-> ?? _->
|?? _->|?? _-> ??/N_R_eq->]//.
Qed. | Lemma | PExpr_R_eq | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"N_R_eq",
"e'",
"positive_R_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
option_R_omap2 A B (RAB : A -> B -> Type) C D (RCD : C -> D -> Type)
(f : A -> C) (g : B -> D) : (forall a b, RAB a b -> RCD (f a) (g b)) ->
forall a b, option_R RAB a b -> option_R RCD (omap f a) (omap g b). | Proof. by move=> fg _ _ [a b ab|]; constructor; apply: fg. Qed. | Lemma | option_R_omap2 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
list_R_map2 A B (RAB : A -> B -> Type) C D (RCD : C -> D -> Type)
(f : A -> C) (g : B -> D) : (forall a b, RAB a b -> RCD (f a) (g b)) ->
forall x y, list_R RAB x y -> list_R RCD (map f x) (map g y). | Proof.
by move=> fg ? ?; elim=> [|? ? ? ? ? ? IH]; constructor; [apply: fg|apply: IH].
Qed. | Lemma | list_R_map2 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"apply",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PExpr_R_PEmap2 A B (RAB : A -> B -> Type) C D (RCD : C -> D -> Type)
(f : A -> C) (g : B -> D) : (forall a b, RAB a b -> RCD (f a) (g b)) ->
forall x y, PExpr_R RAB x y -> PExpr_R RCD (PEmap f x) (PEmap g y). | Proof.
move=> fg ? ?; elim=> [||a b ab|a b ab||||a b ab|a b ab IH na nb nab];
[| | | |move=> a1 b1 ab1 IH1 a2 b2 ab2 IH2..| |]; constructor=> //; exact: fg.
Qed. | Lemma | PExpr_R_PEmap2 | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FExpr_R_map A B (RAB : A -> B -> Type) (f : A -> B) :
(forall a, RAB a (f a)) ->
forall g : FExpr A, FExpr_R RAB g (FEmap f g). | Proof.
move=> rf; elim=> [||c|p||||g IH|g IH|f1 IH1 f2 IH2|g IH n];
[| | | |move=> f1 IH1 f2 IH2..| | | |].
- exact: FEO_R.
- exact: FEI_R.
- exact: FEc_R.
- exact/FEX_R/positive_Rxx.
- by apply: FEadd_R; [apply: IH1 | apply: IH2].
- by apply: FEsub_R; [apply: IH1 | apply: IH2].
- by apply: FEmul_R; [apply: IH1 | a... | Lemma | FExpr_R_map | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"N_Rxx",
"apply",
"f1",
"f2",
"positive_Rxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEmap_id T (e : PExpr T) : PEmap id e = e. | Proof. by elim: e => /=[||||?->?->|?->?->|?->?->|?->|?->]. Qed. | Lemma | PEmap_id | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_of_Z | := (R_of_Z R). | Notation | R_of_Z | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zint_pow_pos_pos i n (Zin : Zint i n) p p' (pp' : positive_R p p') :
Zint (Z.pow_pos i p) (n ^+ Pos.to_nat p'). | Proof. by apply: Zint_pow_pos => //; rewrite (positive_R_eq pp'). Qed. | Lemma | Zint_pow_pos_pos | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Zint",
"Zint_pow_pos",
"apply",
"positive_R_eq",
"pow_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_checker_map_int_of_Z Zcn icn
(cnR : forall (Zc : seq (PExpr Z)) (ic : seq (PExpr int)),
list_R (PExpr_R Zint) Zc ic -> list_R (PExpr_R Zint) (Zcn Zc) (icn ic))
n lpe fe1 fe2 :
omap (map (PEmap int_of_Z))
(field_checker Z0 (Zpos xH) Z.add Z.mul Z.sub Z.opp Z.pow_pos
Z.eqb (triv_div Z0 ... | Proof.
set fc1 := _ fe1 fe2; set fc2 := _ (FEmap _ fe1) (FEmap _ fe2).
have fc12 : option_R (list_R (PExpr_R Zint)) fc1 fc2.
apply: (field_checker_R _ _ ZintD ZintM ZintB ZintN Zint_pow_pos_pos
(eq_bool_R2 Zint_eq) (triv_div_R _ _ (eq_bool_R2 Zint_eq)) cnR (nat_Rxx n));
move=> //; [|exact/FExpr_R_map/Zint_int... | Lemma | field_checker_map_int_of_Z | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"FExpr_R_map",
"PEmap_id",
"PExpr_R_PEmap2",
"PExpr_R_eq",
"PExpr_R_map",
"Zint",
"ZintB",
"ZintD",
"ZintM",
"ZintN",
"Zint_eq",
"Zint_int_of_Z",
"Zint_pow_pos_pos",
"add",
"apply",
"eq_bool_R2",
"eqb",
"field_checker",
"int",
"int_of_Z",
"list_R_eq",
"list_R_map",
"list_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FEeval_map_int_of_Z l (fe : FExpr Z) :
FEeval 0 1 +%R *%R (fun x y => x - y) -%R (fun x y => x / y) (@GRing.inv R)
N.to_nat (GRing.exp (R:=R)) intr (env_nth 0) l (FEmap int_of_Z fe)
= FEeval 0 1 +%R *%R (fun x y => x - y) -%R (fun x y => x / y) (@GRing.inv R)
N.to_nat (GRing.exp (R:=R)) R_of_Z (env_nth 0)... | Proof. by elim: fe => //= ? -> // ? ->. Qed. | Lemma | FEeval_map_int_of_Z | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"FEeval",
"R_of_Z",
"env_nth",
"exp",
"int_of_Z",
"intr",
"inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond_map_int_of_Z l lpe :
PCond true negb andb 0 1 +%R *%R (fun x y => x - y) -%R
N.to_nat (GRing.exp (R:=R)) eq_op R_of_Z (env_nth 0) l lpe ->
PCond true negb andb 0 1 +%R *%R (fun x y => x - y) -%R
N.to_nat (GRing.exp (R:=R)) eq_op intr (env_nth 0) l
(map (PEmap int_of_Z) lpe). | Proof.
elim: lpe => [//|pe lpe IH] /=.
by rewrite !PCond_cons PEeval_map_int_of_Z => /andP[-> /IH].
Qed. | Lemma | PCond_map_int_of_Z | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"PCond",
"PCond_cons",
"PEeval_map_int_of_Z",
"R_of_Z",
"env_nth",
"exp",
"int_of_Z",
"intr",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cond_norm00_map_int_of_Z Zc ic : list_R (PExpr_R Zint) Zc ic ->
list_R (PExpr_R Zint)
(cond_norm (Fcons00 Z0 (Zpos xH) Z.add Z.mul Z.sub Z.opp Z.eqb) Zc)
(cond_norm (Fcons00 0 1 +%R *%R (fun x y => x - y) -%R eq_op) ic). | Proof.
apply: cond_norm_R => Ze ie Zeie {}Zc {}ic Zcic.
exact: (Fcons00_R _ _ ZintD ZintM ZintB ZintN (eq_bool_R2 Zint_eq)).
Qed. | Lemma | cond_norm00_map_int_of_Z | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons00",
"Zint",
"ZintB",
"ZintD",
"ZintM",
"ZintN",
"Zint_eq",
"add",
"apply",
"cond_norm",
"eq_bool_R2",
"eqb",
"mul",
"opp",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FEeval | := (FEeval 0 1 +%R *%R (fun x y => x - y) -%R
(fun x y => x / y) (@GRing.inv R) N.to_nat (@GRing.exp R) R_of_Z (env_nth 0)). | Notation | FEeval | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"R_of_Z",
"env_nth",
"exp",
"inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PEeval_eqs | := (PEeval_eqs true andb 0 1 +%R *%R
(fun x y => x - y) -%R N.to_nat (@GRing.exp R) eq_op R_of_Z (env_nth 0)). | Notation | PEeval_eqs | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"R_of_Z",
"env_nth",
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PCond | := (PCond true negb andb 0 1 +%R *%R (fun x y => x - y)
-%R N.to_nat (@GRing.exp R) eq_op R_of_Z (env_nth 0)). | Notation | PCond | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"R_of_Z",
"env_nth",
"exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cond_norm | := (cond_norm (Fcons00
Z0 (Zpos xH) Z.add Z.mul Z.sub Z.opp Z.eqb)). | Notation | cond_norm | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons00",
"add",
"eqb",
"mul",
"opp",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_checker | := (field_checker Z0 (Zpos xH) Z.add Z.mul
Z.sub Z.opp Z.pow_pos Z.eqb (triv_div Z0 (Zpos xH) Z.eqb) cond_norm). | Notation | field_checker | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"add",
"cond_norm",
"eqb",
"mul",
"opp",
"pow_pos",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zfield_correct n l lpe fe1 fe2 lc :
PEeval_eqs l lpe ->
field_checker n lpe fe1 fe2 = Some lc ->
PCond l lc ->
FEeval l fe1 = FEeval l fe2. | Proof.
move=> /PEeval_eqs_map_int_of_Z + /(congr1 (omap (map (PEmap int_of_Z)))).
move=> + + /PCond_map_int_of_Z; rewrite -!FEeval_map_int_of_Z.
rewrite (field_checker_map_int_of_Z cond_norm00_map_int_of_Z).
apply: (Cfield_checkerT (@Ctriv_divP int)) => el el'.
exact/PCond_cond_norm/PCond_Fcons00.
Qed. | Lemma | Zfield_correct | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Cfield_checkerT",
"Ctriv_divP",
"FEeval",
"FEeval_map_int_of_Z",
"PCond",
"PCond_Fcons00",
"PCond_cond_norm",
"PCond_map_int_of_Z",
"PEeval_eqs",
"PEeval_eqs_map_int_of_Z",
"apply",
"cond_norm00_map_int_of_Z",
"field_checker",
"field_checker_map_int_of_Z",
"int",
"int_of_Z",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cond_norm2_map_int_of_Z Zc ic : list_R (PExpr_R Zint) Zc ic ->
list_R (PExpr_R Zint)
(cond_norm (Fcons2 Z0 (Zpos xH) Z.add Z.mul Z.sub Z.opp Z.pow_pos Z.eqb) Zc)
(cond_norm (Fcons2 0 1 +%R *%R (fun x y => x - y) -%R
(fun x n => x ^+ Pos.to_nat n) eq_op) ic). | Proof.
apply: cond_norm_R => Ze ie Zeie {}Zc {}ic Zcic.
exact: (Fcons2_R _ _ ZintD ZintM ZintB ZintN Zint_pow_pos_pos
(eq_bool_R2 Zint_eq)).
Qed. | Lemma | cond_norm2_map_int_of_Z | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons2",
"Zint",
"ZintB",
"ZintD",
"ZintM",
"ZintN",
"Zint_eq",
"Zint_pow_pos_pos",
"add",
"apply",
"cond_norm",
"eq_bool_R2",
"eqb",
"mul",
"opp",
"pow_pos",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cond_norm | := (cond_norm (Fcons2
Z0 (Zpos xH) Z.add Z.mul Z.sub Z.opp Z.pow_pos Z.eqb)). | Notation | cond_norm | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Fcons2",
"add",
"eqb",
"mul",
"opp",
"pow_pos",
"sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ZnumField_correct n l lpe fe1 fe2 lc :
PEeval_eqs l lpe ->
field_checker n lpe fe1 fe2 = Some lc ->
PCond l lc ->
FEeval l fe1 = FEeval l fe2. | Proof.
move=> /PEeval_eqs_map_int_of_Z + /(congr1 (omap (map (PEmap int_of_Z)))).
move=> + + /PCond_map_int_of_Z; rewrite -!FEeval_map_int_of_Z.
rewrite (field_checker_map_int_of_Z cond_norm2_map_int_of_Z).
apply: (Cfield_checkerT (@Ctriv_divP int)) => el el'.
exact/PCond_cond_norm/PCond_Fcons2/ler0z.
Qed. | Lemma | ZnumField_correct | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"Cfield_checkerT",
"Ctriv_divP",
"FEeval",
"FEeval_map_int_of_Z",
"PCond",
"PCond_Fcons2",
"PCond_cond_norm",
"PCond_map_int_of_Z",
"PEeval_eqs",
"PEeval_eqs_map_int_of_Z",
"apply",
"cond_norm2_map_int_of_Z",
"field_checker",
"field_checker_map_int_of_Z",
"int",
"int_of_Z",
"ler0z",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_correct (F : fieldType) n env
(lpe : seq ((RExpr F * RExpr F) * (PExpr Z * PExpr Z)))
(re1 re2 : RExpr F) (fe1 fe2 : FExpr Z) lc :
Reval_eqs lpe ->
(forall R_of_Z zero one add opp mul exp inv,
let R_of_N b n := let n := R_of_Z (Z.of_N n) in if b then inv n else n in
let opp_intr := S... | Proof.
pose R_of_N b n : Field F :=
let n := R_of_Z F (Z.of_N n) in if b then n^-1 else n.
have R_of_NE : R_of_N =2 fun b n => @invi (Field F) b (N.to_nat n)%:R.
by case=> [] [].
rewrite !(@Rnorm_correct (Field F) false _ erefl) -lock.
rewrite -!(Rnorm_eq_F_of_N R_of_NE) => elpe.
move=> /( _ (R_of_Z F) 0 1 +%R -%R ... | Lemma | field_correct | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"FEeval",
"Fcons00",
"PCond",
"PEeval",
"RExpr",
"R_of_N",
"R_of_Z",
"Reval",
"Reval_eqs",
"Rnorm",
"Rnorm_correct",
"Rnorm_eq_F_of_N",
"Zfield_correct",
"add",
"apply",
"cond_norm",
"env",
"env_nth",
"eqb",
"eqxx",
"eval",
"exp",
"field_checker",
"id",
"intr",
"inv... | Everything below is essentially imported form algebra-tactics | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
numField_correct (F : numFieldType) n env
(lpe : seq ((RExpr F * RExpr F) * (PExpr Z * PExpr Z)))
(re1 re2 : RExpr F) (fe1 fe2 : FExpr Z) lc :
Reval_eqs lpe ->
(forall R_of_Z zero one add opp mul exp inv,
let R_of_N b n := let n := R_of_Z (Z.of_N n) in if b then inv n else n in
let opp_int... | Proof.
pose R_of_N b n : Field F :=
let n := R_of_Z F (Z.of_N n) in if b then n^-1 else n.
have R_of_NE : R_of_N =2 fun b n => @invi (Field F) b (N.to_nat n)%:R.
by case=> [] [].
rewrite !(@Rnorm_correct (Field F) false _ erefl) -lock.
rewrite -!(Rnorm_eq_F_of_N R_of_NE) => elpe.
move=> /( _ (R_of_Z F) 0 1 +%R -%R ... | Lemma | numField_correct | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"FEeval",
"Fcons2",
"PCond",
"PEeval",
"RExpr",
"R_of_N",
"R_of_Z",
"Reval",
"Reval_eqs",
"Rnorm",
"Rnorm_correct",
"Rnorm_eq_F_of_N",
"ZnumField_correct",
"add",
"apply",
"cond_norm",
"env",
"env_nth",
"eqb",
"eqxx",
"eval",
"exp",
"field_checker",
"id",
"intr",
"i... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_reflection Lem F VarMap Lpe RE1 RE2 FE1 FE2 LpeProofs | :=
let nvarmap := fresh "__varmap" in
let nlpe := fresh "__lpe" in
let nre1 := fresh "__re1" in
let nre2 := fresh "__re2" in
let nfe1 := fresh "__fe1" in
let nfe2 := fresh "__fe2" in
let nlpeproofs := fresh "__lpeproofs" in
pose nvarmap := VarMap;
pose nlpe := Lpe;
pose nre1 := RE1;
pose nre2 := R... | Ltac | field_reflection | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [
"R_of_Z",
"apply",
"env_nth",
"eval",
"nth",
"pmulrn",
"split"
] | Main tactics, called from the elpi parser (c.f., ring.elpi) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
field_term t | :=
solve [ t ; done ]
|| fail 1 "There are remaining goals, use ""field?"" to inspect them". | Ltac | field_term | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
field_tactic | := done. | Ltac | field_tactic | algebra | algebra/field_tactic.v | [
"elpi",
"derive.std",
"param2",
"micromega_plugin",
"NatDef",
"Corelib",
"IntDef",
"formula",
"witness",
"checker",
"eval",
"field_checker",
"field_eval",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"order",
"ssralg",
"ssrnum",
"ssrint",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'finGroupMixin' 'of' R 'for' +%R ]" | :=
(Finite_isGroup.Build R (@addrA _) (@add0r _) (@addNr _))
(format "[ 'finGroupMixin' 'of' R 'for' +%R ]") : form_scope. | Notation | [ 'finGroupMixin' 'of' R 'for' +%R ] | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"Build",
"add0r",
"addNr",
"addrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Lalgebra R | := (NzLalgebra R) (only parsing). | Notation | Lalgebra | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort | := (NzLalgebra.sort) (only parsing). | Notation | sort | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
on R | := (NzLalgebra.on R) (only parsing). | Notation | on | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
copy T U | := (NzLalgebra.copy T U) (only parsing). | Notation | copy | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Algebra R | := (NzAlgebra R) (only parsing). | Notation | Algebra | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sort | := (NzAlgebra.sort) (only parsing). | Notation | sort | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
on R | := (NzAlgebra.on R) (only parsing). | Notation | on | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
copy T U | := (NzAlgebra.copy T U) (only parsing). | Notation | copy | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zmodule_to_baseFinGroup (R : finZmodType) | := FinStarMonoid.clone R _. | Coercion | Zmodule_to_baseFinGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zmodule_to_finGroup (R : finZmodType) | := FinGroup.clone R _. | Coercion | Zmodule_to_finGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod1gE : 1%g = 0 :> U. | Proof. by []. Qed. | Lemma | zmod1gE | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmodVgE x : x^-1%g = - x. | Proof. by []. Qed. | Lemma | zmodVgE | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmodMgE x y : (x * y)%g = x + y. | Proof. by []. Qed. | Lemma | zmodMgE | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmodXgE n x : (x ^+ n)%g = x *+ n. | Proof. by []. Qed. | Lemma | zmodXgE | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_mulgC x y : commute x y. | Proof. exact: addrC. Qed. | Lemma | zmod_mulgC | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"addrC",
"commute"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zmod_abelian (A : {set U}) : abelian A. | Proof. by apply/centsP=> x _ y _; apply: zmod_mulgC. Qed. | Lemma | zmod_abelian | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"abelian",
"apply",
"centsP",
"zmod_mulgC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NzRing_to_baseFinGroup (R : finNzRingType) | := FinStarMonoid.clone R _. | Coercion | NzRing_to_baseFinGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NzRing_to_finGroup (R : finNzRingType) | := FinGroup.clone R _. | Coercion | NzRing_to_finGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Build R | := (isNzRing.Build R) (only parsing). | Notation | Build | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isRing R | := (isNzRing R) (only parsing). | Notation | isRing | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_inv (x y : R) | := (x * y == 1) && (y * x == 1). | Definition | is_inv | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit | := [qualify a x : R | [exists y, is_inv x y]]. | Definition | unit | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"is_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv x | := odflt x (pick (is_inv x)). | Definition | inv | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"is_inv",
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVr : {in unit, left_inverse 1 inv *%R}. | Proof.
rewrite /inv => x Ux; case: pickP => [y | no_y]; last by case/pred0P: Ux.
by case/andP=> _; move/eqP.
Qed. | Lemma | mulVr | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"inv",
"last",
"pickP",
"pred0P",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrV : {in unit, right_inverse 1 inv *%R}. | Proof.
rewrite /inv => x Ux; case: pickP => [y | no_y]; last by case/pred0P: Ux.
by case/andP; move/eqP.
Qed. | Lemma | mulrV | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"inv",
"last",
"pickP",
"pred0P",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intro_unit x y : y * x = 1 /\ x * y = 1 -> x \is a unit. | Proof.
by case=> yx1 xy1; apply/existsP; exists y; rewrite /is_inv xy1 yx1 !eqxx.
Qed. | Lemma | intro_unit | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"apply",
"eqxx",
"existsP",
"is_inv",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_out : {in [predC unit], inv =1 id}. | Proof.
rewrite /inv => x nUx; case: pickP => // y invxy.
by case/existsP: nUx; exists y.
Qed. | Lemma | invr_out | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"existsP",
"id",
"inv",
"pickP",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ComNzRing_to_baseFinGroup (R : finComNzRingType) | :=
FinStarMonoid.clone R _. | Coercion | ComNzRing_to_baseFinGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ComNzRing_to_finGroup (R : finComNzRingType) | := FinGroup.clone R _. | Coercion | ComNzRing_to_finGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
UnitRing_to_baseFinGroup (R : finUnitRingType) | :=
FinStarMonoid.clone R _. | Coercion | UnitRing_to_baseFinGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
UnitRing_to_finGroup (R : finUnitRingType) | := FinGroup.clone R _. | Coercion | UnitRing_to_finGroup | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"clone"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_of | := Unit (x : R) of x \is a GRing.unit. | Inductive | unit_of | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uval u | := let: Unit x _ := u in x. | Definition | uval | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit1 | := Unit (@GRing.unitr1 _). | Definition | unit1 | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_inv_proof u : (val u)^-1 \is a GRing.unit. | Proof. by rewrite unitrV ?(valP u). Qed. | Lemma | unit_inv_proof | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit",
"unitrV",
"val",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_inv u | := Unit (unit_inv_proof u). | Definition | unit_inv | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_inv_proof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_mul_proof u v : val u * val v \is a GRing.unit. | Proof. by rewrite (unitrMr _ (valP u)) ?(valP v). Qed. | Lemma | unit_mul_proof | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit",
"unitrMr",
"val",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_mul u v | := Unit (unit_mul_proof u v). | Definition | unit_mul | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_mul_proof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_muluA : associative unit_mul. | Proof. by move=> u v w; apply/val_inj/mulrA. Qed. | Lemma | unit_muluA | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"apply",
"mulrA",
"unit_mul",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_mul1u : left_id unit1 unit_mul. | Proof. by move=> u; apply/val_inj/mul1r. Qed. | Lemma | unit_mul1u | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"apply",
"mul1r",
"unit1",
"unit_mul",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_mulVu : left_inverse unit1 unit_inv unit_mul. | Proof. by move=> u; apply/val_inj/(mulVr (valP u)). Qed. | Lemma | unit_mulVu | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"apply",
"mulVr",
"unit1",
"unit_inv",
"unit_mul",
"valP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_unit1 : val (1%g : unit_of) = 1. | Proof. by []. Qed. | Lemma | val_unit1 | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_of",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_unitM x y : val (x * y : unit_of)%g = val x * val y. | Proof. by []. Qed. | Lemma | val_unitM | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_of",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_unitV x : val (x^-1 : unit_of)%g = (val x)^-1. | Proof. by []. Qed. | Lemma | val_unitV | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_of",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_unitX n x : val (x ^+ n : unit_of)%g = val x ^+ n. | Proof. by case: n; last by elim=> //= n ->. Qed. | Lemma | val_unitX | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"last",
"unit_of",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_act x u | := x * val u. | Definition | unit_act | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
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"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_actE x u : unit_act x u = x * val u. | Proof. by []. Qed. | Lemma | unit_actE | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_act",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_action | :=
@TotalAction _ _ unit_act (@mulr1 _) (fun _ _ _ => mulrA _ _ _). | Canonical | unit_action | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"TotalAction",
"mulr1",
"mulrA",
"unit_act"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_is_groupAction : @is_groupAction _ R setT setT unit_action. | Proof.
move=> u _ /[1!inE]; apply/andP; split; first by apply/subsetP=> x /[1!inE].
by apply/morphicP=> x y _ _; rewrite !actpermE /= [_ u]mulrDl.
Qed. | Lemma | unit_is_groupAction | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"actpermE",
"apply",
"inE",
"is_groupAction",
"morphicP",
"mulrDl",
"setT",
"split",
"subsetP",
"unit_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_groupAction | := GroupAction unit_is_groupAction. | Canonical | unit_groupAction | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [
"unit_is_groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unit_action. | Canonical | unit_action | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
unit_groupAction. | Canonical | unit_groupAction | algebra | algebra/finalg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finset",
"fingroup",
"morphism",
"perm",
"action",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"FinRing"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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