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 |
|---|---|---|---|---|---|---|---|---|---|---|
extendDerivation_zmod_morphism_subproof E (adjEx := Fadjoin_poly E x) :
let body y (p := adjEx y) := (map_poly D p).[x] + p^`().[x] * Dx E in
zmod_morphism body. | Proof.
move: Dx => C /= u v; rewrite /adjEx.
rewrite raddfB /= derivB -/adjEx !hornerE /= raddfB /= !hornerE.
by rewrite mulrBl addrACA opprD.
Qed. | Fact | extendDerivation_zmod_morphism_subproof | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Dx",
"Fadjoin_poly",
"addrACA",
"body",
"derivB",
"hornerE",
"map_poly",
"mulrBl",
"opprD",
"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extendDerivation_scalable_subproof E (adjEx := Fadjoin_poly E x) :
let body y (p := adjEx y) := (map_poly D p).[x] + p^`().[x] * Dx E in
scalable body. | Proof.
move: Dx => C /= a u; rewrite /adjEx linearZ /= derivZ -/adjEx.
rewrite hornerE -[RHS]mulr_algl mulrDr mulrA -[in RHS]hornerZ.
congr (_.[x] + _); apply/polyP=> i.
by rewrite coefZ !coef_map coefZ !mulr_algl /= linearZ.
Qed. | Fact | extendDerivation_scalable_subproof | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Dx",
"Fadjoin_poly",
"apply",
"body",
"coefZ",
"coef_map",
"derivZ",
"hornerE",
"hornerZ",
"linearZ",
"map_poly",
"mulrA",
"mulrDr",
"mulr_algl",
"polyP",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
body (y : L) (p := Fadjoin_poly E x y) : L | :=
(map_poly D p).[x] + p^`().[x] * Dx E. | Let | body | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Dx",
"Fadjoin_poly",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extendDerivationLinear | := Eval hnf in (body : {linear _ -> _}). | Let | extendDerivationLinear | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"body",
"linear"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extendDerivation : 'End(L) | := linfun extendDerivationLinear. | Definition | extendDerivation | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"extendDerivationLinear",
"linfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derD : Derivation K D. | Hypothesis | derD | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Derivation"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
extendDerivation_id y : y \in K -> extendDerivation K y = D y. | Proof.
move=> yK; rewrite lfunE /= Fadjoin_polyC // derivC map_polyC hornerC.
by rewrite horner0 mul0r addr0.
Qed. | Lemma | extendDerivation_id | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Fadjoin_polyC",
"addr0",
"derivC",
"extendDerivation",
"horner0",
"hornerC",
"lfunE",
"map_polyC",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extendDerivation_horner p :
p \is a polyOver K -> separable_element K x ->
extendDerivation K p.[x] = (map_poly D p).[x] + p^`().[x] * Dx K. | Proof.
move=> Kp sepKx; have:= separable_root_der; rewrite {}sepKx /= => nz_pKx'x.
rewrite [in RHS](divp_eq p (minPoly K x)) lfunE /= Fadjoin_poly_mod ?raddfD //=.
rewrite (Derivation_mul_poly derD) ?divp_polyOver ?minPolyOver //.
rewrite derivM !{1}hornerD !{1}hornerM minPolyxx !{1}mulr0 !{1}add0r.
rewrite mulrDl addr... | Lemma | extendDerivation_horner | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Derivation_mul_poly",
"Dx",
"Fadjoin_poly_mod",
"add0r",
"addKr",
"addrA",
"addrC",
"derD",
"derivM",
"divp_eq",
"divp_polyOver",
"extendDerivation",
"hornerD",
"hornerM",
"lfunE",
"map_poly",
"minPoly",
"minPolyOver",
"minPolyxx",
"mulVKf",
"mulr0",
"mulrA",
"mulrC",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extendDerivationP :
separable_element K x -> Derivation <<K; x>> (extendDerivation K). | Proof.
move=> sep; apply/allrelP=> u v /vbasis_mem Hu /vbasis_mem Hv; apply/eqP.
rewrite -(Fadjoin_poly_eq Hu) -(Fadjoin_poly_eq Hv) -hornerM.
rewrite !{1}extendDerivation_horner ?{1}rpredM ?Fadjoin_polyOver //.
rewrite (Derivation_mul_poly derD) ?Fadjoin_polyOver //.
rewrite derivM !{1}hornerD !{1}hornerM !{1}mulrDl !... | Lemma | extendDerivationP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Derivation",
"Derivation_mul_poly",
"Dx",
"Fadjoin_polyOver",
"Fadjoin_poly_eq",
"addrA",
"addrC",
"allrelP",
"apply",
"derD",
"derivM",
"extendDerivation",
"extendDerivation_horner",
"hornerD",
"hornerM",
"mulrA",
"mulrC",
"mulrDl",
"mulrDr",
"rpredM",
"separable_element",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation_separableP :
reflect
(forall D, Derivation <<K; x>> D -> K <= lker D -> <<K; x>> <= lker D)%VS
(separable_element K x). | Proof.
apply: (iffP idP) => [sepKx D derD /subvP DK_0 | derKx_0].
have{} DK_0 q: q \is a polyOver K -> map_poly D q = 0.
move=> /polyOverP Kq; apply/polyP=> i; apply/eqP.
by rewrite coef0 coef_map -memv_ker DK_0.
apply/subvP=> _ /Fadjoin_polyP[p Kp ->]; rewrite memv_ker.
rewrite (Derivation_horner derD) ?... | Lemma | Derivation_separableP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Build",
"Derivation",
"Derivation_horner",
"Derivation_separable",
"Fadjoin_poly",
"Fadjoin_polyC",
"Fadjoin_polyOver",
"Fadjoin_polyP",
"Fadjoin_polyX",
"Fadjoin_poly_eq",
"Fadjoin_poly_mod",
"Kx_x",
"Px",
"add0r",
"addr0",
"allrelP",
"apply",
"base_separable",
"coef0",
"coef... | Reference:
http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/separable2.pdf | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
separablePn | := (separablePn_pchar) (only parsing). | Notation | separablePn | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separablePn_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_elementS K E x :
(K <= E)%VS -> separable_element K x -> separable_element E x. | Proof.
move=> sKE /separable_elementP[f [fK rootf sepf]]; apply/separable_elementP.
by exists f; rewrite (polyOverSv sKE).
Qed. | Lemma | separable_elementS | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"fK",
"polyOverSv",
"sKE",
"separable_element",
"separable_elementP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjoin_separableP {K x} :
reflect (forall y, y \in <<K; x>>%VS -> separable_element K y)
(separable_element K x). | Proof.
apply: (iffP idP) => [sepKx | -> //]; last exact: memv_adjoin.
move=> _ /Fadjoin_polyP[q Kq ->]; apply/Derivation_separableP=> D derD DK_0.
apply/subvP=> _ /Fadjoin_polyP[p Kp ->].
rewrite memv_ker -(extendDerivation_id x D (mempx_Fadjoin _ Kp)).
have sepFyx: (separable_element <<K; q.[x]>> x).
by apply: (sepa... | Lemma | adjoin_separableP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Derivation_horner",
"Derivation_separable",
"Derivation_separableP",
"FadjoinP",
"Fadjoin_polyP",
"add0r",
"adjoinSl",
"apply",
"coef0",
"coef_map",
"derD",
"eqEsubv",
"extendDerivationP",
"extendDerivation_id",
"horner0",
"horner_comp",
"last",
"map_poly",
"mempx_Fadjoin",
"m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_exponent_pchar K x :
exists n, [pchar L].-nat n && separable_element K (x ^+ n). | Proof.
pose d := adjoin_degree K x; move: {2}d.+1 (ltnSn d) => n.
elim: n => // n IHn in x @d *; rewrite ltnS => le_d_n.
have [[p pcharLp]|] := altP (separablePn_pchar K x); last by rewrite negbK; exists 1.
case=> g Kg defKx; have p_pr := pcharf_prime pcharLp.
suffices /IHn[m /andP[pcharLm sepKxpm]]: adjoin_degree K (x... | Lemma | separable_exponent_pchar | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"adjoin_degree",
"apply",
"comp_poly0",
"comp_polyC",
"contra_eqN",
"contra_eqT",
"dvdp_leq",
"exprM",
"hornerXn",
"horner_comp",
"last",
"leqNgt",
"leq_ltn_trans",
"leq_trans",
"ltnS",
"ltnSn",
"ltn_Pmulr",
"minPoly",
"minPoly_dvdp",
"minPolyxx",
"monic_minPoly",
"monic_ne... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_exponent | := (separable_exponent_pchar) (only parsing). | Notation | separable_exponent | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separable_exponent_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcharf0_separable K : [pchar L] =i pred0 -> forall x, separable_element K x. | Proof.
move=> pcharL0 x; have [n /andP[pcharLn]] := separable_exponent_pchar K x.
by rewrite (pnat_1 pcharLn (sub_in_pnat _ pcharLn)) // => p _; rewrite pcharL0.
Qed. | Lemma | pcharf0_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"pchar",
"pnat_1",
"separable_element",
"separable_exponent_pchar",
"sub_in_pnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
charf0_separable | := (pcharf0_separable) (only parsing). | Notation | charf0_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"pcharf0_separable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcharf_p_separable K x e p :
p \in [pchar L] -> separable_element K x = (x \in <<K; x ^+ (p ^ e.+1)>>%VS). | Proof.
move=> pcharLp; apply/idP/idP=> [sepKx | /Fadjoin_poly_eq]; last first.
set m := p ^ _; set f := Fadjoin_poly K _ x => Dx; apply/separable_elementP.
have mL0: m%:R = 0 :> L by apply/eqP; rewrite -(dvdn_pcharf pcharLp) dvdn_exp.
exists ('X - (f \Po 'X^m)); split.
- by rewrite rpredB ?polyOver_comp ?rpredX... | Lemma | pcharf_p_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Dx",
"FadjoinP",
"Fadjoin_poly",
"Fadjoin_polyOver",
"Fadjoin_poly_eq",
"adjoin_separableP",
"apply",
"cons_poly_def",
"coprimep",
"coprimep1",
"coprimep_expr",
"derivE",
"deriv_comp",
"dvdn_exp",
"dvdn_pcharf",
"dvdpP",
"dvdp_XsubCl",
"dvdp_gcd_idl",
"dvdp_mulIl",
"dvdp_trans... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
charf_p_separable | := (pcharf_p_separable) (only parsing). | Notation | charf_p_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"pcharf_p_separable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcharf_n_separable K x n :
[pchar L].-nat n -> 1 < n -> separable_element K x = (x \in <<K; x ^+ n>>%VS). | Proof.
rewrite -pi_pdiv; set p := pdiv n => pcharLn pi_n_p.
have pcharLp: p \in [pchar L] := pnatPpi pcharLn pi_n_p.
have <-: (n`_p)%N = n by rewrite -(eq_partn n (pcharf_eq pcharLp)) part_pnat_id.
by rewrite p_part lognE -mem_primes pi_n_p -pcharf_p_separable.
Qed. | Lemma | pcharf_n_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"eq_partn",
"lognE",
"mem_primes",
"nat",
"p_part",
"part_pnat_id",
"pchar",
"pcharf_eq",
"pcharf_p_separable",
"pdiv",
"pi_pdiv",
"pnatPpi",
"separable_element"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
charf_n_separable | := (pcharf_n_separable) (only parsing). | Notation | charf_n_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"pcharf_n_separable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparable_element U x | :=
x ^+ ex_minn (separable_exponent_pchar <<U>> x) \in U. | Definition | purely_inseparable_element | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"ex_minn",
"separable_exponent_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparable_elementP_pchar {K x} :
reflect (exists2 n, [pchar L].-nat n & x ^+ n \in K)
(purely_inseparable_element K x). | Proof.
rewrite /purely_inseparable_element.
case: ex_minnP => n /andP[pcharLn /=]; rewrite subfield_closed => sepKxn min_xn.
apply: (iffP idP) => [Kxn | [m pcharLm Kxm]]; first by exists n.
have{min_xn}: n <= m by rewrite min_xn ?pcharLm ?base_separable.
rewrite leq_eqVlt => /predU1P[-> // | ltnm]; pose p := pdiv m.
ha... | Lemma | purely_inseparable_elementP_pchar | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Fadjoin_idP",
"addSnnS",
"apply",
"base_separable",
"eq_pnat",
"ex_minnP",
"expnD",
"exprM",
"leq_eqVlt",
"leq_ltn_trans",
"ltn_exp2l",
"nat",
"p_natP",
"pchar",
"pcharf_eq",
"pcharf_p_separable",
"pdiv",
"pdiv_prime",
"pi_pdiv",
"pnatPpi",
"predU1P",
"prime_gt1",
"purel... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparable_elementP | := (purely_inseparable_elementP_pchar) (only parsing). | Notation | purely_inseparable_elementP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"purely_inseparable_elementP_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_inseparable_element K x :
separable_element K x && purely_inseparable_element K x = (x \in K). | Proof.
rewrite /purely_inseparable_element; case: ex_minnP => [[|m]] //=.
rewrite subfield_closed; case: m => /= [-> //| m _ /(_ 1)/implyP/= insepKx].
by rewrite (negPf insepKx) (contraNF (@base_separable K x) insepKx).
Qed. | Lemma | separable_inseparable_element | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"base_separable",
"ex_minnP",
"purely_inseparable_element",
"separable_element",
"subfield_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
base_inseparable K x : x \in K -> purely_inseparable_element K x. | Proof. by rewrite -separable_inseparable_element => /andP[]. Qed. | Lemma | base_inseparable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"purely_inseparable_element",
"separable_inseparable_element"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_inseparable K E x :
(K <= E)%VS -> purely_inseparable_element K x ->
purely_inseparable_element E x. | Proof.
move/subvP=> sKE /purely_inseparable_elementP_pchar[n pcharLn /sKE Exn].
by apply/purely_inseparable_elementP_pchar; exists n.
Qed. | Lemma | sub_inseparable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"purely_inseparable_element",
"purely_inseparable_elementP_pchar",
"sKE",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
K_is_large | := exists s, [/\ uniq s, {subset s <= K} & N < size s]. | Let | K_is_large | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_or_large (z : L) : z != 0 -> K_is_large \/ exists a, z ^+ a.+1 = 1. | Proof.
move=> nz_z; pose d := adjoin_degree K z.
pose h0 (i : 'I_(N ^ d).+1) (j : 'I_d) := (Fadjoin_poly K z (z ^+ i))`_j.
pose s := undup [seq h0 i j | i <- enum 'I_(N ^ d).+1, j <- enum 'I_d].
have s_h0 i j: h0 i j \in s.
by rewrite mem_undup; apply/allpairsP; exists (i, j); rewrite !mem_enum.
pose h i := [ffun j =... | Let | cyclic_or_large | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Fadjoin_poly",
"Fadjoin_polyOver",
"Fadjoin_poly_eq",
"K_is_large",
"adjoin_degree",
"allpairsP",
"apply",
"card_ffun",
"card_image",
"card_ord",
"divff",
"enum",
"eq_bigr",
"expfB",
"expf_neq0",
"ffunE",
"horner_coef_wide",
"index_mem",
"injectiveP",
"injectivePn",
"ltn_exp... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finite_PET : K_is_large \/ exists z, (<< <<K; y>>; x>> = <<K; z>>)%VS. | Proof.
have [-> | /cyclic_or_large[|[a Dxa]]] := eqVneq x 0; first 2 [by left].
by rewrite addv0 subfield_closed; right; exists y.
have [-> | /cyclic_or_large[|[b Dyb]]] := eqVneq y 0; first 2 [by left].
by rewrite addv0 subfield_closed; right; exists x.
pose h0 (ij : 'I_a.+1 * 'I_b.+1) := x ^+ ij.1 * y ^+ ij.2.
po... | Lemma | finite_PET | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"FadjoinP",
"K_is_large",
"Zp_mulgC",
"addv0",
"apply",
"centP",
"cent_sub",
"coset",
"cosetP",
"coset_id",
"coset_of",
"cycleP",
"cyclic",
"cyclicP",
"cyclic_or_large",
"eqEsubv",
"eqVneq",
"expgS",
"exprD",
"expr_mod",
"field_mul_group_cyclic",
"gen_set_id",
"group_setP... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sepKy : separable_element K y. | Hypothesis | sepKy | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separable_element"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Primitive_Element_Theorem : exists z, (<< <<K; y>>; x>> = <<K; z>>)%VS. | Proof.
have /polyOver_subvs[p Dp]: minPoly K x \is a polyOver K := minPolyOver K x.
have nz_pKx: minPoly K x != 0 by rewrite monic_neq0 ?monic_minPoly.
have{nz_pKx} nz_p: p != 0 by rewrite Dp map_poly_eq0 in nz_pKx.
have{Dp} px0: root (map_poly vsval p) x by rewrite -Dp root_minPoly.
have [q0 [Kq0 q0y0 sepKq0]] := sepa... | Lemma | Primitive_Element_Theorem | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Dx",
"FadjoinP",
"all",
"allPn",
"apply",
"dvdp_separable",
"eqEsubv",
"finite_PET",
"large_field_PET",
"ltnNge",
"ltnW",
"map_poly",
"map_poly_eq0",
"max_poly_roots",
"memv_adjoin",
"minPoly",
"minPolyOver",
"minPoly_dvdp",
"monic_minPoly",
"monic_neq0",
"nz_p",
"poly",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjoin_separable : separable_element <<K; y>> x -> separable_element K x. | Proof.
have /Derivation_separableP derKy := sepKy => /Derivation_separableP derKy_x.
have [z defKz] := Primitive_Element_Theorem.
suffices /adjoin_separableP: separable_element K z.
by apply; rewrite -defKz memv_adjoin.
apply/Derivation_separableP=> D; rewrite -defKz => derKxyD DK_0.
suffices derKyD: Derivation <<K; ... | Lemma | adjoin_separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Derivation",
"DerivationS",
"Derivation_separableP",
"Primitive_Element_Theorem",
"adjoin_separableP",
"apply",
"memv_adjoin",
"sepKy",
"separable_element",
"subv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
strong_Primitive_Element_Theorem K x y :
separable_element <<K; x>> y ->
exists2 z : L, (<< <<K; y>>; x>> = <<K; z>>)%VS
& separable_element K x -> separable_element K y. | Proof.
move=> sepKx_y; have [n /andP[pcharLn sepKyn]] := separable_exponent_pchar K y.
have adjK_C z t: (<<<<K; z>>; t>> = <<<<K; t>>; z>>)%VS.
by rewrite !agenv_add_id -!addvA (addvC <[_]>%VS).
have [z defKz] := Primitive_Element_Theorem x sepKyn.
exists z => [|/adjoin_separable->]; rewrite ?sepKx_y // -defKz.
have ... | Lemma | strong_Primitive_Element_Theorem | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"FadjoinP",
"Primitive_Element_Theorem",
"addvA",
"addvC",
"adjoin_separable",
"agenv_add_id",
"apply",
"eqEsubv",
"ltngtP",
"memv_adjoin",
"pcharf_n_separable",
"rpredX",
"separable_element",
"separable_exponent_pchar",
"split",
"subv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable U W : bool | :=
all (separable_element U) (vbasis W). | Definition | separable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"all",
"separable_element",
"vbasis"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparable U W : bool | :=
all (purely_inseparable_element U) (vbasis W). | Definition | purely_inseparable | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"all",
"purely_inseparable_element",
"vbasis"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_add K x y :
separable_element K x -> separable_element K y -> separable_element K (x + y). | Proof.
move/(separable_elementS (subv_adjoin K y))=> sepKy_x sepKy.
have [z defKz] := Primitive_Element_Theorem x sepKy.
have /(adjoin_separableP _): x + y \in <<K; z>>%VS.
by rewrite -defKz rpredD ?memv_adjoin // subvP_adjoin ?memv_adjoin.
apply; apply: adjoin_separable sepKy (adjoin_separable sepKy_x _).
by rewrite... | Lemma | separable_add | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"Primitive_Element_Theorem",
"adjoin_separable",
"adjoin_separableP",
"apply",
"base_separable",
"memv_adjoin",
"rpredD",
"sepKy",
"separable_element",
"separable_elementS",
"subvP_adjoin",
"subv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_sum I r (P : pred I) (v_ : I -> L) K :
(forall i, P i -> separable_element K (v_ i)) ->
separable_element K (\sum_(i <- r | P i) v_ i). | Proof.
move=> sepKi.
by elim/big_ind: _; [apply/base_separable/mem0v | apply: separable_add |].
Qed. | Lemma | separable_sum | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"base_separable",
"big_ind",
"mem0v",
"separable_add",
"separable_element"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inseparable_add K x y :
purely_inseparable_element K x -> purely_inseparable_element K y ->
purely_inseparable_element K (x + y). | Proof.
have insepP := purely_inseparable_elementP_pchar.
move=> /insepP[n pcharLn Kxn] /insepP[m pcharLm Kym]; apply/insepP.
have pcharLnm: [pchar L].-nat (n * m)%N by rewrite pnatM pcharLn.
by exists (n * m)%N; rewrite ?exprDn_pchar // {2}mulnC !exprM memvD // rpredX.
Qed. | Lemma | inseparable_add | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"exprDn_pchar",
"exprM",
"memvD",
"mulnC",
"nat",
"pchar",
"pnatM",
"purely_inseparable_element",
"purely_inseparable_elementP_pchar",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inseparable_sum I r (P : pred I) (v_ : I -> L) K :
(forall i, P i -> purely_inseparable_element K (v_ i)) ->
purely_inseparable_element K (\sum_(i <- r | P i) v_ i). | Proof.
move=> insepKi.
by elim/big_ind: _; [apply/base_inseparable/mem0v | apply: inseparable_add |].
Qed. | Lemma | inseparable_sum | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"base_inseparable",
"big_ind",
"inseparable_add",
"mem0v",
"purely_inseparable_element"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separableP {K E} :
reflect (forall y, y \in E -> separable_element K y) (separable K E). | Proof.
apply/(iffP idP)=> [/allP|] sepK_E; last by apply/allP=> x /vbasis_mem/sepK_E.
move=> y /coord_vbasis->; apply/separable_sum=> i _.
have: separable_element K (vbasis E)`_i by apply/sepK_E/memt_nth.
by move/adjoin_separableP; apply; rewrite rpredZ ?memv_adjoin.
Qed. | Lemma | separableP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"adjoin_separableP",
"allP",
"apply",
"coord_vbasis",
"last",
"memt_nth",
"memv_adjoin",
"rpredZ",
"separable",
"separable_element",
"separable_sum",
"vbasis",
"vbasis_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparableP {K E} :
reflect (forall y, y \in E -> purely_inseparable_element K y)
(purely_inseparable K E). | Proof.
apply/(iffP idP)=> [/allP|] sep'K_E; last by apply/allP=> x /vbasis_mem/sep'K_E.
move=> y /coord_vbasis->; apply/inseparable_sum=> i _.
have: purely_inseparable_element K (vbasis E)`_i by apply/sep'K_E/memt_nth.
case/purely_inseparable_elementP_pchar=> n pcharLn K_Ein.
by apply/purely_inseparable_elementP_pchar;... | Lemma | purely_inseparableP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"allP",
"apply",
"coord_vbasis",
"exprZn",
"inseparable_sum",
"last",
"memt_nth",
"purely_inseparable",
"purely_inseparable_element",
"purely_inseparable_elementP_pchar",
"rpredZ",
"vbasis",
"vbasis_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjoin_separable_eq K x : separable_element K x = separable K <<K; x>>%VS. | Proof. exact: sameP adjoin_separableP separableP. Qed. | Lemma | adjoin_separable_eq | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"adjoin_separableP",
"separable",
"separableP",
"separable_element"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_inseparable_decomposition E K :
{x | x \in E /\ separable_element K x & purely_inseparable <<K; x>> E}. | Proof.
without loss sKE: K / (K <= E)%VS.
case/(_ _ (capvSr K E)) => x [Ex sepKEx] /purely_inseparableP sep'KExE.
exists x; first by split; last exact/(separable_elementS _ sepKEx)/capvSl.
apply/purely_inseparableP=> y /sep'KExE; apply: sub_inseparable.
exact/adjoinSl/capvSl.
pose E_ i := (vbasis E)`_i; pose fP... | Lemma | separable_inseparable_decomposition | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"E_",
"FadjoinP",
"Primitive_Element_Theorem",
"addv0",
"adjoinSl",
"adjoin_nil",
"adjoin_rcons",
"adjoin_separable",
"adjoin_separableP",
"all",
"allP",
"all_map",
"all_nthP",
"all_rcons",
"apply",
"base_separable",
"capvSl",
"capvSr",
"dim",
"ex_minn",
"ex_minnP",
"fP",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_generator K E : L | :=
s2val (locked (separable_inseparable_decomposition E K)). | Definition | separable_generator | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separable_inseparable_decomposition"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_generator_mem E K : separable_generator K E \in E. | Proof. by rewrite /separable_generator; case: (locked _) => ? []. Qed. | Lemma | separable_generator_mem | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separable_generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_generatorP E K : separable_element K (separable_generator K E). | Proof. by rewrite /separable_generator; case: (locked _) => ? []. Qed. | Lemma | separable_generatorP | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separable_element",
"separable_generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_generator_maximal E K :
purely_inseparable <<K; separable_generator K E>> E. | Proof. by rewrite /separable_generator; case: (locked _). Qed. | Lemma | separable_generator_maximal | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"purely_inseparable",
"separable_generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_adjoin_separable_generator E K :
separable K E -> (E <= <<K; separable_generator K E>>)%VS. | Proof.
move/separableP=> sepK_E; apply/subvP=> v Ev.
rewrite -separable_inseparable_element.
have /purely_inseparableP-> // := separable_generator_maximal E K.
by rewrite (separable_elementS _ (sepK_E _ Ev)) // subv_adjoin.
Qed. | Lemma | sub_adjoin_separable_generator | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"purely_inseparableP",
"separable",
"separableP",
"separable_elementS",
"separable_generator",
"separable_generator_maximal",
"separable_inseparable_element",
"subvP",
"subv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_adjoin_separable_generator E K :
separable K E -> (K <= E)%VS ->
E = <<K; separable_generator K E>>%VS :> {vspace _}. | Proof.
move=> sepK_E sKE; apply/eqP; rewrite eqEsubv sub_adjoin_separable_generator //.
by apply/FadjoinP/andP; rewrite sKE separable_generator_mem.
Qed. | Lemma | eq_adjoin_separable_generator | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"FadjoinP",
"apply",
"eqEsubv",
"sKE",
"separable",
"separable_generator",
"separable_generator_mem",
"sub_adjoin_separable_generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_refl K : separable K K. | Proof. exact/separableP/base_separable. Qed. | Lemma | separable_refl | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"base_separable",
"separable",
"separableP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_trans M K E : separable K M -> separable M E -> separable K E. | Proof.
move/sub_adjoin_separable_generator.
set x := separable_generator K M => sMKx /separableP sepM_E.
apply/separableP => w /sepM_E/(separable_elementS sMKx).
case/strong_Primitive_Element_Theorem => _ _ -> //.
exact: separable_generatorP.
Qed. | Lemma | separable_trans | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"separable",
"separableP",
"separable_elementS",
"separable_generator",
"separable_generatorP",
"strong_Primitive_Element_Theorem",
"sub_adjoin_separable_generator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separableS K1 K2 E2 E1 :
(K1 <= K2)%VS -> (E2 <= E1)%VS -> separable K1 E1 -> separable K2 E2. | Proof.
move=> sK12 /subvP sE21 /separableP sepK1_E1.
by apply/separableP=> y /sE21/sepK1_E1/(separable_elementS sK12).
Qed. | Lemma | separableS | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"separable",
"separableP",
"separable_elementS",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separableSl K M E : (K <= M)%VS -> separable K E -> separable M E. | Proof. by move/separableS; apply. Qed. | Lemma | separableSl | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"separable",
"separableS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separableSr K M E : (M <= E)%VS -> separable K E -> separable K M. | Proof. exact: separableS. Qed. | Lemma | separableSr | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"separable",
"separableS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_Fadjoin_seq K rs :
all (separable_element K) rs -> separable K <<K & rs>>. | Proof.
elim/last_ind: rs => [|s x IHs] in K *.
by rewrite adjoin_nil subfield_closed separable_refl.
rewrite all_rcons adjoin_rcons => /andP[sepKx /IHs/separable_trans-> //].
by rewrite -adjoin_separable_eq (separable_elementS _ sepKx) ?subv_adjoin_seq.
Qed. | Lemma | separable_Fadjoin_seq | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"adjoin_nil",
"adjoin_rcons",
"adjoin_separable_eq",
"all",
"all_rcons",
"last_ind",
"separable",
"separable_element",
"separable_elementS",
"separable_refl",
"separable_trans",
"subfield_closed",
"subv_adjoin_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparable_refl K : purely_inseparable K K. | Proof. by apply/purely_inseparableP; apply: base_inseparable. Qed. | Lemma | purely_inseparable_refl | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"base_inseparable",
"purely_inseparable",
"purely_inseparableP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
purely_inseparable_trans M K E :
purely_inseparable K M -> purely_inseparable M E -> purely_inseparable K E. | Proof.
have insepP := purely_inseparableP => /insepP insepK_M /insepP insepM_E.
have insepPe := purely_inseparable_elementP_pchar.
apply/insepP=> x /insepM_E/insepPe[n pcharLn /insepK_M/insepPe[m pcharLm Kxnm]].
by apply/insepPe; exists (n * m)%N; rewrite ?exprM // pnatM pcharLn pcharLm.
Qed. | Lemma | purely_inseparable_trans | field | field/separable.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"quotient",
"gproduct",
"ssralg",
"finalg",
"zmodp",
... | [
"apply",
"exprM",
"pnatM",
"purely_inseparable",
"purely_inseparableP",
"purely_inseparable_elementP_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_morph to x | := forall a b, to x (a * b) = to (to x a) b. | Definition | act_morph | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_action to | :=
left_injective to /\ forall x, {in D &, act_morph to x}. | Definition | is_action | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"act_morph",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
action | := Action {act :> rT -> aT -> rT; _ : is_action act}. | Record | action | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"aT",
"act",
"is_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
clone_action to | :=
let: Action _ toP := to return {type of Action for to} -> action in
fun k => k toP. | Definition | clone_action | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"action",
"to",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'action' aT &-> T }" | := (action [set: aT] T)
(format "{ 'action' aT &-> T }") : type_scope. | Notation | { 'action' aT &-> T } | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"aT",
"action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'action' 'of' to ]" | := (clone_action (@Action _ _ _ to))
(format "[ 'action' 'of' to ]") : form_scope. | Notation | [ 'action' 'of' to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"clone_action",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_dom aT D rT & @action aT D rT | := D. | Definition | act_dom | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"aT",
"action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(to1 : to^~ 1 =1 id) (toM : forall x, act_morph to x). | Hypotheses | to1 | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"act_morph",
"id",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
is_total_action : is_action setT to. | Proof.
split=> [a | x a b _ _] /=; last by rewrite toM.
by apply: can_inj (to^~ a^-1) _ => x; rewrite -toM ?mulgV.
Qed. | Lemma | is_total_action | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"apply",
"is_action",
"last",
"mulgV",
"setT",
"split",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
TotalAction | := Action is_total_action. | Definition | TotalAction | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"is_total_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_act rT rT' (to : action D rT) (to' : action D' rT') f fA | :=
forall x a, f (to x a) = to' (f x) (fA a). | Definition | morph_act | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"action",
"fA",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actm to a | := if a \in D then to^~ a else id. | Definition | actm | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"id",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setact to S a | := [set to x a | x in S]. | Definition | setact | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbit to A x | := to x @: A. | Definition | orbit | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
amove to A x y | := [set a in A | to x a == y]. | Definition | amove | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
afix to A | := [set x | A \subset [set a | to x a == x]]. | Definition | afix | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
astab S to | := D :&: [set a | S \subset [set x | to x a == x]]. | Definition | astab | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
astabs S to | := D :&: [set a | S \subset to^~ a @^-1: S]. | Definition | astabs | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
acts_on A S to | := {in A, forall a x, (to x a \in S) = (x \in S)}. | Definition | acts_on | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
atrans A S to | := S \in orbit to A @: S. | Definition | atrans | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"orbit",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
faithful A S to | := A :&: astab S to \subset [1]. | Definition | faithful | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"astab",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"to ^*" | := (setact to) : function_scope. | Notation | to ^* | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"setact",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Fix_' to ( A )" | := (afix to A)
(to at level 2, format "''Fix_' to ( A )") : group_scope. | Notation | ''Fix_' to ( A ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"afix",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Fix_' ( to ) ( A )" | := 'Fix_to(A) (only parsing) : group_scope. | Notation | ''Fix_' ( to ) ( A ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [] | camlp4 grammar factoring | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"''Fix_' ( S | to ) ( A )" | := (S :&: 'Fix_to(A))
(format "''Fix_' ( S | to ) ( A )") : group_scope. | Notation | ''Fix_' ( S | to ) ( A ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Fix_' to [ a ]" | := ('Fix_to([set a]))
(to at level 2, format "''Fix_' to [ a ]") : group_scope. | Notation | ''Fix_' to [ a ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Fix_' ( S | to ) [ a ]" | := (S :&: 'Fix_to[a])
(format "''Fix_' ( S | to ) [ a ]") : group_scope. | Notation | ''Fix_' ( S | to ) [ a ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C' ( S | to )" | := (astab S to) : group_scope. | Notation | ''C' ( S | to ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"astab",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C_' A ( S | to )" | := (A :&: 'C(S | to)) : group_scope. | Notation | ''C_' A ( S | to ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C_' ( A ) ( S | to )" | := 'C_A(S | to) (only parsing) : group_scope. | Notation | ''C_' ( A ) ( S | to ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C' [ x | to ]" | := ('C([set x] | to)) : group_scope. | Notation | ''C' [ x | to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C_' A [ x | to ]" | := (A :&: 'C[x | to]) : group_scope. | Notation | ''C_' A [ x | to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''C_' ( A ) [ x | to ]" | := 'C_A[x | to] (only parsing) : group_scope. | Notation | ''C_' ( A ) [ x | to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''N' ( S | to )" | := (astabs S to)
(format "''N' ( S | to )") : group_scope. | Notation | ''N' ( S | to ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"astabs",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''N_' A ( S | to )" | := (A :&: 'N(S | to))
(A at level 2, format "''N_' A ( S | to )") : group_scope. | Notation | ''N_' A ( S | to ) | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'acts' A , 'on' S | to ]" | := (A \subset pred_of_set 'N(S | to))
(format "[ 'acts' A , 'on' S | to ]") : form_scope. | Notation | [ 'acts' A , 'on' S | to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"pred_of_set",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'acts' A , 'on' S | to }" | := (acts_on A S to)
(format "{ 'acts' A , 'on' S | to }") : type_scope. | Notation | { 'acts' A , 'on' S | to } | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"acts_on",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'transitive' A , 'on' S | to ]" | := (atrans A S to)
(format "[ 'transitive' A , 'on' S | to ]") : form_scope. | Notation | [ 'transitive' A , 'on' S | to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"atrans",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'faithful' A , 'on' S | to ]" | := (faithful A S to)
(format "[ 'faithful' A , 'on' S | to ]") : form_scope. | Notation | [ 'faithful' A , 'on' S | to ] | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"faithful",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_inj : left_injective to. | Proof. by case: to => ? []. Qed. | Lemma | act_inj | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actMin x : {in D &, act_morph to x}. | Proof. by case: to => ? []. Qed. | Lemma | actMin | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"act_morph",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actmEfun a : a \in D -> actm to a = to^~ a. | Proof. by rewrite /actm => ->. Qed. | Lemma | actmEfun | finite_group | finite_group/action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnotations",
"eqtype",
"ssrnat",
"div",
"seq",
"prime",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient"
] | [
"actm",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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