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 |
|---|---|---|---|---|---|---|---|---|---|---|
mulfxC : commutative subfext_mul. | Proof.
by elim/quotW=> x; elim/quotW=> y; rewrite !piE /subfx_mul_rep /= mulrC.
Qed. | Fact | mulfxC | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrC",
"piE",
"quotW",
"subfext_mul",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1fx : left_id subfext1 subfext_mul. | Proof.
elim/quotW=> x; rewrite !piE /subfx_mul_rep poly_rV_K ?size_poly1 // mul1r.
by rewrite modp_small ?rVpolyK // (polySpred nz_p0) ltnS size_poly.
Qed. | Fact | mul1fx | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"ltnS",
"modp_small",
"mul1r",
"nz_p0",
"piE",
"polySpred",
"poly_rV_K",
"quotW",
"rVpolyK",
"size_poly",
"size_poly1",
"subfext1",
"subfext_mul",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulfx_addl : left_distributive subfext_mul subfext_add. | Proof.
elim/quotW=> x; elim/quotW=> y; elim/quotW=> w.
by rewrite !piE /subfx_mul_rep linearD /= mulrDl modpD linearD.
Qed. | Fact | mulfx_addl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"linearD",
"modpD",
"mulrDl",
"piE",
"quotW",
"subfext_add",
"subfext_mul",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonzero1fx : subfext1 != subfext0. | Proof.
rewrite !piE /equiv_subfext /iotaFz !linear0.
by rewrite poly_rV_K ?rmorph1 ?oner_eq0 // size_poly1.
Qed. | Fact | nonzero1fx | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"equiv_subfext",
"iotaFz",
"linear0",
"oner_eq0",
"piE",
"poly_rV_K",
"rmorph1",
"size_poly1",
"subfext0",
"subfext1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_poly_inv (q : {poly F}) : {poly F} | :=
if iotaPz q == 0 then 0 else
let r := gdcop q p0 in let: (u, v) := egcdp q r in
((u * q + v * r)`_0)^-1 *: u. | Definition | subfx_poly_inv | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"egcdp",
"gdcop",
"iotaPz",
"p0",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_poly_invE q : iotaPz (subfx_poly_inv q) = (iotaPz q)^-1. | Proof.
rewrite /subfx_poly_inv.
have [-> | nzq] := eqVneq; first by rewrite rmorph0 invr0.
rewrite [nth]lock -[_^-1]mul1r; apply: canRL (mulfK nzq) _; rewrite -rmorphM /=.
have rz0: iotaPz (gdcop q p0) = 0.
by apply/rootP; rewrite gdcop_map root_gdco ?map_poly_eq0 // p0z0 nzq.
do [case: gdcopP => r _; rewrite (negPf ... | Let | subfx_poly_invE | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"Da",
"addrK",
"apply",
"coefC",
"coprimep_size_gcd",
"coprimep_sym",
"egcdp",
"egcdpE",
"eqVneq",
"eqp_size",
"gdcop",
"gdcopP",
"gdcop_map",
"horner_morphC",
"invr0",
"iotaPz",
"linearB",
"linearZ",
"map_poly_eq0",
"mul1r",
"mulVf",
"mulfK",
"mulr0",
"nth",
"nz_p0",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inv_rep (x : 'rV[F]_n) : 'rV[F]_n | :=
poly_rV (subfx_poly_inv (rVpoly x) %% p0). | Definition | subfx_inv_rep | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"p0",
"poly_rV",
"rVpoly",
"subfx_poly_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfext_inv | := lift_op1 subFExtend subfx_inv_rep. | Definition | subfext_inv | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_op1",
"subFExtend",
"subfx_inv_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_inv : {morph \pi : x / subfx_inv_rep x >-> subfext_inv x}. | Proof.
unlock subfext_inv => x /=; apply/eqmodP/eqP; rewrite /iotaFz.
by rewrite 2!{1}poly_rV_modp_K 2!{1}iotaPz_modp !subfx_poly_invE iotaPz_repr.
Qed. | Fact | pi_subfext_inv | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"eqmodP",
"iotaFz",
"iotaPz_modp",
"iotaPz_repr",
"pi",
"poly_rV_modp_K",
"subfext_inv",
"subfx_inv_rep",
"subfx_poly_invE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_subfext_inv_morph | := PiMorph1 pi_subfext_inv. | Canonical | pi_subfext_inv_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiMorph1",
"pi_subfext_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_fieldAxiom : forall x, x != 0 -> subfext_inv x * x = 1. | Proof.
elim/quotW=> x; apply: contraNeq; rewrite !piE /equiv_subfext /iotaFz !linear0.
apply: contraR => nz_x; rewrite poly_rV_K ?size_poly1 // !poly_rV_modp_K.
by rewrite iotaPz_modp rmorph1 rmorphM /= iotaPz_modp subfx_poly_invE mulVf.
Qed. | Fact | subfx_fieldAxiom | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"contraNeq",
"equiv_subfext",
"iotaFz",
"iotaPz_modp",
"linear0",
"mulVf",
"piE",
"poly_rV_K",
"poly_rV_modp_K",
"quotW",
"rmorph1",
"rmorphM",
"size_poly1",
"subfext_inv",
"subfx_poly_invE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inv0 : subfext_inv (0 : subFExtend) = (0 : subFExtend). | Proof.
apply/eqP; rewrite !piE /equiv_subfext /iotaFz /subfx_inv_rep !linear0.
by rewrite /subfx_poly_inv rmorph0 eqxx mod0p !linear0.
Qed. | Fact | subfx_inv0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"equiv_subfext",
"eqxx",
"iotaFz",
"linear0",
"mod0p",
"piE",
"rmorph0",
"subFExtend",
"subfext_inv",
"subfx_inv_rep",
"subfx_poly_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj_is_zmod_morphism : zmod_morphism subfx_inj. | Proof.
by elim/quotW => x; elim/quotW => y; rewrite !piE /iotaFz linearB rmorphB.
Qed. | Fact | subfx_inj_is_zmod_morphism | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iotaFz",
"linearB",
"piE",
"quotW",
"rmorphB",
"subfx_inj",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj_is_additive | := subfx_inj_is_zmod_morphism. | Definition | subfx_inj_is_additive | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"subfx_inj_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj_is_monoid_morphism : monoid_morphism subfx_inj. | Proof.
split; first by rewrite piE /iotaFz poly_rV_K ?rmorph1 ?size_poly1.
elim/quotW=> x; elim/quotW=> y; rewrite !piE /subfx_mul_rep /iotaFz.
by rewrite poly_rV_modp_K iotaPz_modp rmorphM.
Qed. | Fact | subfx_inj_is_monoid_morphism | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iotaFz",
"iotaPz_modp",
"monoid_morphism",
"piE",
"poly_rV_K",
"poly_rV_modp_K",
"quotW",
"rmorph1",
"rmorphM",
"size_poly1",
"split",
"subfx_inj",
"subfx_mul_rep"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj_is_multiplicative | :=
(fun g => (g.2,g.1)) subfx_inj_is_monoid_morphism. | Definition | subfx_inj_is_multiplicative | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"subfx_inj_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_eval | := lift_embed subFExtend (fun q => poly_rV (q %% p0)). | Definition | subfx_eval | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"lift_embed",
"p0",
"poly_rV",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_eval_morph | := PiEmbed subfx_eval. | Canonical | subfx_eval_morph | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"PiEmbed",
"subfx_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_root | := subfx_eval 'X. | Definition | subfx_root | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"subfx_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_eval_is_zmod_morphism : zmod_morphism subfx_eval. | Proof. by move=> x y; apply/eqP; rewrite piE -linearB modpD modNp. Qed. | Lemma | subfx_eval_is_zmod_morphism | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"linearB",
"modNp",
"modpD",
"piE",
"subfx_eval",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_eval_is_additive | := subfx_eval_is_zmod_morphism. | Definition | subfx_eval_is_additive | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"subfx_eval_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_eval_is_monoid_morphism : monoid_morphism subfx_eval. | Proof.
split=> [|x y]; apply/eqP; rewrite piE.
by rewrite modp_small // size_poly1 -subn_gt0 subn1.
by rewrite /subfx_mul_rep !poly_rV_modp_K !(modp_mul, mulrC _ y).
Qed. | Lemma | subfx_eval_is_monoid_morphism | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"modp_mul",
"modp_small",
"monoid_morphism",
"mulrC",
"piE",
"poly_rV_modp_K",
"size_poly1",
"split",
"subfx_eval",
"subfx_mul_rep",
"subn1",
"subn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_eval_is_multiplicative | :=
(fun g => (g.2,g.1)) subfx_eval_is_monoid_morphism. | Definition | subfx_eval_is_multiplicative | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"subfx_eval_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_subfx | := (subfx_eval \o polyC). | Definition | inj_subfx | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"polyC",
"subfx_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfxE x: exists p, x = subfx_eval p. | Proof.
elim/quotW: x => x; exists (rVpoly x); apply/eqP; rewrite piE /equiv_subfext.
by rewrite /iotaFz poly_rV_modp_K iotaPz_modp.
Qed. | Lemma | subfxE | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"equiv_subfext",
"iotaFz",
"iotaPz_modp",
"piE",
"poly_rV_modp_K",
"quotW",
"rVpoly",
"subfx_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_scale a x | := inj_subfx a * x. | Definition | subfx_scale | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"inj_subfx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_scalerA a b x :
subfx_scale a (subfx_scale b x) = subfx_scale (a * b) x. | Proof. by rewrite /subfx_scale rmorphM mulrA. Qed. | Fact | subfx_scalerA | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrA",
"rmorphM",
"subfx_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_scaler1r : left_id 1 subfx_scale. | Proof. by move=> x; rewrite /subfx_scale rmorph1 mul1r. Qed. | Fact | subfx_scaler1r | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mul1r",
"rmorph1",
"subfx_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_scalerDr : right_distributive subfx_scale +%R. | Proof. by move=> a; apply: mulrDr. Qed. | Fact | subfx_scalerDr | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"mulrDr",
"subfx_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_scalerDl x : {morph subfx_scale^~ x : a b / a + b}. | Proof. by move=> a b; rewrite /subfx_scale rmorphD mulrDl. Qed. | Fact | subfx_scalerDl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrDl",
"rmorphD",
"subfx_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_scaleAl a u v : subfx_scale a (u * v) = (subfx_scale a u) * v. | Proof. exact: mulrA. Qed. | Fact | subfx_scaleAl | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mulrA",
"subfx_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_evalZ : scalable subfx_eval. | Proof. by move=> a q; rewrite -mul_polyC rmorphM. Qed. | Fact | subfx_evalZ | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"mul_polyC",
"rmorphM",
"scalable",
"subfx_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(pz0 : root p^iota z). | Hypothesis | pz0 | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iota",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nz_p : p != 0. | Hypothesis | nz_p | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
subfx_inj_eval q : subfx_inj (subfx_eval q) = (q^iota).[z]. | Proof.
by rewrite piE /iotaFz poly_rV_modp_K iotaPz_modp /iotaPz /z0 /wf_p nz_p pz0.
Qed. | Lemma | subfx_inj_eval | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iota",
"iotaFz",
"iotaPz",
"iotaPz_modp",
"nz_p",
"piE",
"poly_rV_modp_K",
"pz0",
"subfx_eval",
"subfx_inj",
"wf_p",
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj_root : subfx_inj subfx_root = z. | Proof. by rewrite subfx_inj_eval // map_polyX hornerX. Qed. | Lemma | subfx_inj_root | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"hornerX",
"map_polyX",
"subfx_inj",
"subfx_inj_eval",
"subfx_root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_injZ b x : subfx_inj (b *: x) = iota b * subfx_inj x. | Proof. by rewrite rmorphM /= subfx_inj_eval // map_polyC hornerC. Qed. | Lemma | subfx_injZ | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"hornerC",
"iota",
"map_polyC",
"rmorphM",
"subfx_inj",
"subfx_inj_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_inj_base b : subfx_inj b%:A = iota b. | Proof. by rewrite subfx_injZ rmorph1 mulr1. Qed. | Lemma | subfx_inj_base | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"iota",
"mulr1",
"rmorph1",
"subfx_inj",
"subfx_injZ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfxEroot x : {q | x = (map_poly (in_alg subFExtend) q).[subfx_root]}. | Proof.
have /sig_eqW[q ->] := subfxE x; exists q.
apply: (fmorph_inj subfx_inj).
rewrite -horner_map /= subfx_inj_root subfx_inj_eval //.
by rewrite -map_poly_comp (eq_map_poly subfx_inj_base).
Qed. | Lemma | subfxEroot | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"apply",
"eq_map_poly",
"fmorph_inj",
"horner_map",
"in_alg",
"map_poly",
"map_poly_comp",
"sig_eqW",
"subFExtend",
"subfxE",
"subfx_inj",
"subfx_inj_base",
"subfx_inj_eval",
"subfx_inj_root",
"subfx_root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subfx_irreducibleP :
(forall q, root q^iota z -> q != 0 -> size p <= size q) <-> irreducible_poly p. | Proof.
split=> [min_p | irr_p q qz0 nz_q].
split=> [|q nonC_q q_dv_p].
by rewrite -(size_map_poly iota) (root_size_gt1 _ pz0) ?map_poly_eq0.
have /dvdpP[r Dp] := q_dv_p; rewrite -dvdp_size_eqp // eqn_leq dvdp_leq //=.
have [nz_r nz_q]: r != 0 /\ q != 0 by apply/norP; rewrite -mulf_eq0 -Dp.
have: root r^iota... | Lemma | subfx_irreducibleP | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"addSnnS",
"addnC",
"apply",
"dvdpP",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_leq",
"dvdp_size_eqp",
"eq_sym",
"eqn_leq",
"eqp_size",
"gcdp",
"gcdp_eq0",
"gcdp_map",
"gtn_eqF",
"iota",
"irr_p",
"irreducible_poly",
"leqNgt",
"ltn_add2l",
"ltn_neqAle",
"ltn_subRL",
"map_poly_eq0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_p : irreducible_poly p. | Hypothesis | irr_p | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"irreducible_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nz_p : p != 0. | Proof. exact: irredp_neq0. Qed. | Let | nz_p | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"irredp_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
min_subfx_vect : Vector.axiom (size p).-1 subFExtend. | Proof.
move/subfx_irreducibleP: irr_p => /=/(_ nz_p) min_p; set d := (size p).-1.
have Dd: d.+1 = size p by rewrite polySpred.
pose Fz2v x : 'rV_d := poly_rV (sval (sig_eqW (subfxE x)) %% p).
pose vFz : 'rV_d -> subFExtend := subfx_eval \o rVpoly.
have FLinj: injective subfx_inj by apply: fmorph_inj.
have Fz2vK: cancel... | Lemma | min_subfx_vect | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"add0r",
"apply",
"axiom",
"can2_linear",
"divp_eq",
"fmorph_inj",
"hornerE",
"irr_p",
"leqNgt",
"linearB",
"ltnS",
"ltn_modpN0",
"mulr0",
"nz_p",
"polySpred",
"poly_rV",
"poly_rV_K",
"pz0",
"rVpoly",
"rVpolyK",
"raddfB",
"rmorphD",
"rmorphM",
"sig_eqW",
"size",
"si... | The Vector axiom requires irreducibility. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
SubfxVect | := Lmodule_hasFinDim.Build _ subFExtend min_subfx_vect. | Definition | SubfxVect | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"Build",
"min_subfx_vect",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SubVectType : vectType F | := HB.pack subFExtend SubfxVect. | Definition | SubVectType | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"SubfxVect",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SubFieldExtType : fieldExtType F | :=
HB.pack subFExtend SubfxVect (UnitAlgebra_isFalgebra.Build F SubVectType). | Definition | SubFieldExtType | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"Build",
"SubVectType",
"SubfxVect",
"subFExtend"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irredp_FAdjoin (F : fieldType) (p : {poly F}) :
irreducible_poly p ->
{L : fieldExtType F & \dim {:L} = (size p).-1 &
{z | root (map_poly (in_alg L) p) z & <<1; z>>%VS = fullv}}. | Proof.
case=> p_gt1 irr_p; set n := (size p).-1; pose vL : vectType F := 'rV_n.
have Dn: n.+1 = size p := ltn_predK p_gt1.
have nz_p: p != 0 by rewrite -size_poly_eq0 -Dn.
suffices [L dimL [toPF [toL toPF_K toL_K]]]:
{L : fieldExtType F & \dim {:L} = (size p).-1
& {toPF : {linear L -> {poly F}} & {toL : {lrmor... | Lemma | irredp_FAdjoin | field | field/fieldext.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"countalg",
"finalg",
"zmodp",
"matrix",
"vector",
"falgebra",
"poly",
"polydiv",
"mxpoly",
"generic_quotie... | [
"Bezout_eq1_coprimepP",
"Build",
"Fadjoin_polyP",
"add0r",
"alg_polyC",
"apply",
"can_eq",
"coef_map",
"coprimep",
"dim",
"dimvf",
"dvdp_gcdl",
"eqp_size",
"fullv",
"horner0",
"hornerMXaddC",
"in_alg",
"inj_eq",
"iota",
"irr_p",
"irreducible_poly",
"leq_gcdpr",
"linear",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finNzRing_nontrivial : [set: R] != 1%g. | Proof. by apply/trivgPn; exists 1; rewrite ?inE ?oner_neq0. Qed. | Lemma | finNzRing_nontrivial | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"inE",
"oner_neq0",
"trivgPn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finNzRing_gt1 : 1 < #|R|. | Proof. by rewrite -cardsT cardG_gt1 finNzRing_nontrivial. Qed. | Lemma | finNzRing_gt1 | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"cardG_gt1",
"cardsT",
"finNzRing_nontrivial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finRing_nontrivial | := (finNzRing_nontrivial) (only parsing). | Notation | finRing_nontrivial | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"finNzRing_nontrivial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finRing_gt1 | := (finNzRing_gt1) (only parsing). | Notation | finRing_gt1 | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"finNzRing_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_finField_unit : #|[set: {unit F}]| = #|F|.-1. | Proof.
by rewrite -(cardC1 0) cardsT card_sub; apply: eq_card => x; rewrite unitfE.
Qed. | Lemma | card_finField_unit | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"cardC1",
"card_sub",
"cardsT",
"eq_card",
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finField_unit x (nz_x : x != 0) | :=
FinRing.unit F (etrans (unitfE x) nz_x). | Definition | finField_unit | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expf_card x : x ^+ #|F| = x :> F. | Proof.
rewrite -[RHS]mulr1 -(ltn_predK (finNzRing_gt1 F)) exprS.
apply/eqP; rewrite -subr_eq0 -mulrBr mulf_eq0 subr_eq0 -implyNb -unitfE.
apply/implyP=> Ux; rewrite -(val_unitX _ (Sub x _)) -val_unit1 val_eqE.
by rewrite -order_dvdn -card_finField_unit order_dvdG ?inE.
Qed. | Lemma | expf_card | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"Sub",
"apply",
"card_finField_unit",
"exprS",
"finNzRing_gt1",
"inE",
"ltn_predK",
"mulf_eq0",
"mulr1",
"mulrBr",
"order_dvdG",
"order_dvdn",
"subr_eq0",
"unitfE",
"val_eqE",
"val_unit1",
"val_unitX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finField_genPoly : 'X^#|F| - 'X = \prod_x ('X - x%:P) :> {poly F}. | Proof.
set n := #|F|; set oppX := - 'X; set pF := LHS.
have le_oppX_n: size oppX <= n by rewrite size_polyN size_polyX finNzRing_gt1.
have: size pF = (size (enum F)).+1 by rewrite -cardE size_polyDl size_polyXn.
move/all_roots_prod_XsubC->; [|by rewrite uniq_rootsE enum_uniq|].
by apply/allP=> x _; rewrite rootE !hor... | Lemma | finField_genPoly | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"allP",
"all_roots_prod_XsubC",
"apply",
"big_enum",
"cardE",
"enum",
"enum_uniq",
"expf_card",
"finNzRing_gt1",
"hornerE",
"lead_coefDl",
"lead_coefXn",
"poly",
"rootE",
"scale1r",
"size",
"size_polyDl",
"size_polyN",
"size_polyX",
"size_polyXn",
"subrr",
"uniq_rootsE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finPcharP : {p | prime p & p \in [pchar F]}. | Proof.
pose e := exponent [set: F]; have e_gt0: e > 0 by apply: exponent_gt0.
have: e%:R == 0 :> F by rewrite -zmodXgE expg_exponent // inE.
by case/natf0_pchar/sigW=> // p pcharFp; exists p; rewrite ?(pcharf_prime pcharFp).
Qed. | Lemma | finPcharP | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"expg_exponent",
"exponent",
"exponent_gt0",
"inE",
"natf0_pchar",
"pchar",
"pcharFp",
"pcharf_prime",
"prime",
"sigW",
"zmodXgE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finField_is_abelem : is_abelem [set: F]. | Proof.
have [p pr_p pcharFp] := finPcharP.
by apply/is_abelemP; exists p; last apply: fin_ring_pchar_abelem.
Qed. | Lemma | finField_is_abelem | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"finPcharP",
"fin_ring_pchar_abelem",
"is_abelem",
"is_abelemP",
"last",
"pcharFp",
"pr_p"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_finPcharP p n : #|F| = (p ^ n)%N -> prime p -> p \in [pchar F]. | Proof.
move=> oF pr_p; rewrite inE pr_p -order_dvdn.
rewrite (abelem_order_p finField_is_abelem) ?inE ?oner_neq0 //=.
have n_gt0: n > 0 by rewrite -(ltn_exp2l _ _ (prime_gt1 pr_p)) -oF finNzRing_gt1.
by rewrite cardsT oF -(prednK n_gt0) pdiv_pfactor.
Qed. | Lemma | card_finPcharP | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"abelem_order_p",
"cardsT",
"finField_is_abelem",
"finNzRing_gt1",
"inE",
"ltn_exp2l",
"n_gt0",
"oner_neq0",
"order_dvdn",
"pchar",
"pdiv_pfactor",
"pr_p",
"prednK",
"prime",
"prime_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finCharP | := (finPcharP) (only parsing). | Notation | finCharP | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"finPcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_finCharP | := (card_finPcharP) (only parsing). | Notation | card_finCharP | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"card_finPcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vT | := Vector.Pack cvT. | Let | vT | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_vspace (V : {vspace vT}) : #|V| = (#|F| ^ \dim V)%N. | Proof.
set n := \dim V; pose V2rV v := \row_i coord (vbasis V) i v.
pose rV2V (rv : 'rV_n) := \sum_i rv 0 i *: (vbasis V)`_i.
have rV2V_K: cancel rV2V V2rV.
have freeV: free (vbasis V) := basis_free (vbasisP V).
by move=> rv; apply/rowP=> i; rewrite mxE coord_sum_free.
rewrite -[n]mul1n -card_mx -(card_imset _ (can... | Lemma | card_vspace | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"basis_free",
"card_imset",
"card_mx",
"coord",
"coord_sum_free",
"coord_vbasis",
"dim",
"eq_bigr",
"eq_card",
"free",
"imsetP",
"memt_nth",
"mul1n",
"mxE",
"rowP",
"rpredZ",
"rpred_sum",
"vT",
"vbasis",
"vbasisP",
"vbasis_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_vspacef : #|{: vT}%VS| = #|T|. | Proof. by apply: eq_card => v; rewrite (@memvf _ vT). Qed. | Lemma | card_vspacef | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"eq_card",
"memvf",
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aT | := Falgebra.Pack caT. | Let | aT | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_vspace1 : #|(1%VS : {vspace aT})| = #|F|. | Proof. by rewrite card_vspace (dimv1 aT). Qed. | Lemma | card_vspace1 | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"aT",
"card_vspace",
"dimv1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finvect_type (vT : Type) : predArgType | := vT. | Definition | finvect_type | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fvT | := (finvect_type vT). | Notation | fvT | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"finvect_type",
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffT | := (finvect_type fT). | Notation | ffT | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"fT",
"finvect_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffT_splitting_subproof : SplittingField.axiom ffT. | Proof.
exists ('X^#|ffT| - 'X).
by rewrite (@rpredB {poly fT}) 1?rpredX ?polyOverX.
exists (enum ffT); first by rewrite big_enum finField_genPoly eqpxx.
by apply/vspaceP=> x; rewrite memvf seqv_sub_adjoin ?mem_enum.
Qed. | Lemma | ffT_splitting_subproof | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"axiom",
"big_enum",
"enum",
"eqpxx",
"fT",
"ffT",
"finField_genPoly",
"mem_enum",
"memvf",
"poly",
"polyOverX",
"rpredB",
"rpredX",
"seqv_sub_adjoin",
"vspaceP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FinSplittingFieldType (F : finFieldType) (fT : fieldExtType F) | :=
HB.pack_for (splittingFieldType F) fT (SplittingField.on (finvect_type fT)). | Definition | FinSplittingFieldType | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"fT",
"finvect_type",
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FinFieldExtType (F : finFieldType) (fT : fieldExtType F) | :=
HB.pack_for finFieldType (FinSplittingFieldType fT)
(FinRing.Field.on (finvect_type fT)). | Definition | FinFieldExtType | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"FinSplittingFieldType",
"fT",
"finvect_type",
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pPrimeCharType & p \in [pchar R0] : predArgType | := R0. | Definition | pPrimeCharType | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcharRp : p \in [pchar R0]. | Hypothesis | pcharRp | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
R | := (pPrimeCharType pcharRp). | Notation | R | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pPrimeCharType",
"pcharRp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scale a x | := a%:R * x. | Definition | pprimeChar_scale | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrFp n : (inZp n : 'F_p)%:R = n%:R :> R. | Proof.
rewrite [in RHS](divn_eq n p) natrD mulrnA (mulrn_pchar pcharRp) add0r.
by rewrite /= (Fp_cast (pcharf_prime pcharRp)).
Qed. | Let | natrFp | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"Fp_cast",
"add0r",
"divn_eq",
"inZp",
"mulrnA",
"mulrn_pchar",
"natrD",
"pcharRp",
"pcharf_prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scaleA a b x : a *p': (b *p': x) = (a * b) *p': x. | Proof. by rewrite /pprimeChar_scale mulrA -natrM natrFp. Qed. | Lemma | pprimeChar_scaleA | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"mulrA",
"natrFp",
"natrM",
"pprimeChar_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scale1 : left_id 1 pprimeChar_scale. | Proof. by move=> x; rewrite /pprimeChar_scale mul1r. Qed. | Lemma | pprimeChar_scale1 | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"mul1r",
"pprimeChar_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scaleDr : right_distributive pprimeChar_scale +%R. | Proof. by move=> a x y /=; rewrite /pprimeChar_scale mulrDr. Qed. | Lemma | pprimeChar_scaleDr | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"mulrDr",
"pprimeChar_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scaleDl x : {morph pprimeChar_scale^~ x: a b / a + b}. | Proof. by move=> a b; rewrite /pprimeChar_scale natrFp natrD mulrDl. Qed. | Lemma | pprimeChar_scaleDl | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"mulrDl",
"natrD",
"natrFp",
"pprimeChar_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scaleAl (a : 'F_p) (u v : R) : a *: (u * v) = (a *: u) * v. | Proof. by apply: mulrA. Qed. | Lemma | pprimeChar_scaleAl | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_scaleAr (a : 'F_p) (x y : R) : a *: (x * y) = x * (a *: y). | Proof. by rewrite ![a *: _]mulr_natl mulrnAr. Qed. | Lemma | pprimeChar_scaleAr | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"mulr_natl",
"mulrnAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
type | := @pPrimeCharType. | Notation | type | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pPrimeCharType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R | := (type _ pcharRp). | Notation | R | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pcharRp",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pr_p : prime p. | Proof. exact: pcharf_prime pcharRp. Qed. | Let | pr_p | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pcharRp",
"pcharf_prime",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_abelem : p.-abelem [set: R]. | Proof. exact: fin_Fp_lmod_abelem. Qed. | Lemma | pprimeChar_abelem | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"abelem",
"fin_Fp_lmod_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_pgroup : p.-group [set: R]. | Proof. by case/and3P: pprimeChar_abelem. Qed. | Lemma | pprimeChar_pgroup | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"group",
"pprimeChar_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_pprimeChar x : x != 0 :> R -> #[x]%g = p. | Proof. by apply: (abelem_order_p pprimeChar_abelem); rewrite inE. Qed. | Lemma | order_pprimeChar | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"abelem_order_p",
"apply",
"inE",
"pprimeChar_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n | := logn p #|R|. | Let | n | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"logn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_pprimeChar : #|R| = (p ^ n)%N. | Proof. by rewrite /n -cardsT {1}(card_pgroup pprimeChar_pgroup). Qed. | Lemma | card_pprimeChar | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"card_pgroup",
"cardsT",
"pprimeChar_pgroup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_vectAxiom : Vector.axiom n R. | Proof.
have /isog_isom/=[f /isomP[injf im_f]]: [set: R] \isog [set: 'rV['F_p]_n].
rewrite (@isog_abelem_card _ _ p) fin_Fp_lmod_abelem //=.
by rewrite !cardsT card_pprimeChar card_mx mul1n card_Fp.
exists f; last by exists (invm injf) => x; rewrite ?invmE ?invmK ?im_f ?inE.
move=> a x y; rewrite [a *: _]mulr_natl m... | Lemma | pprimeChar_vectAxiom | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"axiom",
"card_Fp",
"card_mx",
"card_pprimeChar",
"cardsT",
"fin_Fp_lmod_abelem",
"inE",
"injf",
"invm",
"invmE",
"invmK",
"isog",
"isog_abelem_card",
"isog_isom",
"isomP",
"last",
"morphM",
"morphX",
"mul1n",
"mulr_natl",
"natr_Zp",
"scaler_nat",
"zmodXgE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pprimeChar_dimf : \dim {: R : vectType 'F_p } = n. | Proof. by rewrite dimvf. Qed. | Lemma | pprimeChar_dimf | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"dim",
"dimvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F | := (type _ pcharFp). | Notation | F | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pcharFp",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PrimeCharType | := (pPrimeCharType) (only parsing). | Notation | PrimeCharType | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pPrimeCharType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_scale | := (pprimeChar_scale) (only parsing). | Notation | primeChar_scale | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_scaleA | := (pprimeChar_scaleA) (only parsing). | Notation | primeChar_scaleA | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scaleA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_scale1 | := (pprimeChar_scale1) (only parsing). | Notation | primeChar_scale1 | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scale1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_scaleDr | := (pprimeChar_scaleDr) (only parsing). | Notation | primeChar_scaleDr | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scaleDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_scaleDl | := (pprimeChar_scaleDl) (only parsing). | Notation | primeChar_scaleDl | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scaleDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_scaleAl | := (pprimeChar_scaleAl) (only parsing). | Notation | primeChar_scaleAl | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scaleAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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