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 |
|---|---|---|---|---|---|---|---|---|---|---|
normalField_cast_morphism | := Morphism normalField_castM. | Canonical | normalField_cast_morphism | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"normalField_castM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_ker : 'ker normalField_cast = 'Gal(E / M). | Proof.
have [sKM sME] := andP sKME.
apply/setP=> x; apply/idP/idP=> [kerMx | galEMx].
rewrite gal_kHom //; apply/kAHomP=> a Ma.
by rewrite -normalField_cast_eq ?(dom_ker kerMx) // (mker kerMx) gal_id.
have galEM: x \in 'Gal(E / K) := subsetP (galS E sKM) x galEMx.
apply/kerP=> //; apply/eqP/gal_eqP=> a Ma.
by rewri... | Lemma | normalField_ker | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"dom_ker",
"fixed_gal",
"galS",
"gal_eqP",
"gal_id",
"gal_kHom",
"kAHomP",
"ker",
"kerP",
"mker",
"normalField_cast",
"normalField_cast_eq",
"sKME",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_normal : 'Gal(E / M) <| 'Gal(E / K). | Proof. by rewrite -normalField_ker ker_normal. Qed. | Lemma | normalField_normal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"ker_normal",
"normalField_ker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_img : normalField_cast @* 'Gal(E / K) = 'Gal(M / K). | Proof.
have [[sKM sME] [sKE _ nKE]] := (andP sKME, and3P galKE).
apply/setP=> x; apply/idP/idP=> [/morphimP[{}x galEx _ ->] | galMx].
rewrite gal_kHom //; apply/kAHomP=> a Ka; have Ma := subvP sKM a Ka.
by rewrite normalField_cast_eq // (fixed_gal sKE).
have /(kHom_to_gal sKME nKE)[y galEy eq_xy]: kHom K M x by rew... | Lemma | normalField_img | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"fixed_gal",
"galKE",
"gal_eqP",
"gal_kHom",
"kAHomP",
"kHom",
"kHom_to_gal",
"morphimP",
"normalField_cast",
"normalField_cast_eq",
"sKE",
"sKME",
"setP",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_isom :
{f : {morphism ('Gal(E / K) / 'Gal(E / M)) >-> gal_of M} |
isom ('Gal(E / K) / 'Gal (E / M)) 'Gal(M / K) f
& (forall A, f @* (A / 'Gal(E / M)) = normalField_cast @* A)
/\ {in 'Gal(E / K) & M, forall x, f (coset 'Gal (E / M) x) =1 x} }%g. | Proof.
have:= first_isom normalField_cast_morphism; rewrite normalField_ker.
case=> f injf Df; exists f; first by apply/isomP; rewrite Df normalField_img.
split=> [//|x a galEx /normalField_cast_eq<- //]; congr ((_ : gal_of M) a).
apply: set1_inj; rewrite -!morphim_set1 ?mem_quotient ?Df //.
by rewrite (subsetP (normal... | Lemma | normalField_isom | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"coset",
"first_isom",
"gal_of",
"injf",
"isom",
"isomP",
"mem_quotient",
"morphim_set1",
"morphism",
"normalField_cast",
"normalField_cast_eq",
"normalField_cast_morphism",
"normalField_img",
"normalField_ker",
"normalField_normal",
"normal_norm",
"set1_inj",
"split",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_isog : 'Gal(E / K) / 'Gal(E / M) \isog 'Gal(M / K). | Proof. by rewrite -normalField_ker -normalField_img first_isog. Qed. | Lemma | normalField_isog | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"first_isog",
"isog",
"normalField_img",
"normalField_ker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nsGgalE : G <| 'Gal(E / K). | Hypothesis | nsGgalE | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
normal_fixedField_galois : galois K (fixedField G). | Proof.
have [[sKE sepKE nKE] [sGgal nGgal]] := (and3P galKE, andP nsGgalE).
rewrite /galois -(galois_connection _ sKE) sGgal.
rewrite (separableSr _ sepKE) ?capvSl //; apply/forall_inP=> f autKf.
rewrite eqEdim limg_dim_eq ?(eqP (AEnd_lker0 _)) ?capv0 // leqnn andbT.
apply/subvP => _ /memv_imgP[a /mem_fixedFieldP[Ea cG... | Lemma | normal_fixedField_galois | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"AEnd_lker0",
"apply",
"capv0",
"capvSl",
"conjgCV",
"eqEdim",
"fixedField",
"fixedFieldP",
"forall_inP",
"galKE",
"galM",
"galois",
"galois_connection",
"groupV",
"inE",
"kAHomP",
"kAut",
"kAut_to_gal",
"kAutfE",
"kHomP_tmp",
"leqnn",
"limg_dim_eq",
"memJ_norm",
"mem_f... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monic_irreducible_poly (p : {poly R}) | :=
((irreducible_poly p) * (p \is monic))%type. | Definition | monic_irreducible_poly | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"irreducible_poly",
"monic",
"poly",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hI : monic_irreducible_poly h. | Hypothesis | hI | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"monic_irreducible_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
qfpoly : monic_irreducible_poly h -> predArgType | :=
fun=> {poly %/ h}. | Definition | qfpoly | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"monic_irreducible_poly",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'poly' '%/' p 'with' hi }" | := (@qfpoly _ p hi). | Notation | { 'poly' '%/' p 'with' hi } | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"qfpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk_monicE : mk_monic h = h. | Proof. by rewrite /mk_monic !hI. Qed. | Lemma | mk_monicE | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"hI",
"mk_monic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_unit (p : {poly %/ h}) : p != 0%R -> coprimep hQ p. | Proof.
move=> pNZ.
rewrite irreducible_poly_coprime //.
by case: hI; rewrite mk_monicE.
apply: contra pNZ => H; case: eqP => // /eqP /dvdp_leq /(_ H).
by rewrite leqNgt size_mk_monic.
Qed. | Lemma | coprimep_unit | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"coprimep",
"dvdp_leq",
"hI",
"hQ",
"irreducible_poly_coprime",
"leqNgt",
"mk_monicE",
"poly",
"size_mk_monic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mulVp (p : {poly %/ h}) : p != 0%R -> (qpoly_inv p * p = 1)%R. | Proof. by move=> pNZ; apply/qpoly_mulVz/coprimep_unit. Qed. | Lemma | qpoly_mulVp | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"coprimep_unit",
"poly",
"qpoly_inv",
"qpoly_mulVz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_inv0 : qpoly_inv 0%R = 0%R :> {poly %/ h}. | Proof.
rewrite /qpoly_inv /= coprimep0 -size_poly_eq1.
rewrite [in X in X == _]mk_monicE.
by have [[]] := hI; case: size => [|[]].
Qed. | Lemma | qpoly_inv0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"coprimep0",
"hI",
"mk_monicE",
"poly",
"qpoly_inv",
"size",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_qfpoly : #|{poly %/ h with hI}| = #|R| ^ (size h).-1. | Proof. by rewrite card_monic_qpoly ?hI. Qed. | Lemma | card_qfpoly | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"card_monic_qpoly",
"hI",
"poly",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_qfpoly_gt1 : 1 < #|{poly %/ h with hI}|. | Proof. by have := card_finNzRing_gt1 {poly %/ h with hI}. Qed. | Lemma | card_qfpoly_gt1 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"card_finNzRing_gt1",
"hI",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly_comp_horner (p q : {poly R}) :
in_qpoly h (p \Po q) =
(map_poly (qpolyC h) p).[in_qpoly h q]. | Proof.
have hQM := monic_mk_monic h.
rewrite comp_polyE /map_poly poly_def horner_sum /=.
apply: val_inj.
rewrite /= rmodp_sum // poly_of_qpoly_sum.
apply: eq_bigr => i _.
rewrite !hornerE /in_qpoly /=.
rewrite mul_polyC // !rmodpZ //=.
by rewrite poly_of_qpolyX /= rmodp_id // rmodpX // rmodp_id.
Qed. | Lemma | in_qpoly_comp_horner | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"comp_polyE",
"eq_bigr",
"hornerE",
"horner_sum",
"in_qpoly",
"map_poly",
"monic_mk_monic",
"mul_polyC",
"poly",
"poly_def",
"poly_of_qpolyX",
"poly_of_qpoly_sum",
"qpolyC",
"rmodpX",
"rmodpZ",
"rmodp_id",
"rmodp_sum",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_div_inj : injective (map_poly (qpolyC h)). | Proof.
apply: map_inj_poly => [x y /val_eqP /eqP /polyC_inj //|].
by rewrite qpolyC0.
Qed. | Lemma | map_poly_div_inj | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"map_inj_poly",
"map_poly",
"polyC_inj",
"qpolyC",
"qpolyC0",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qfpoly_const (R : idomainType) (h : {poly R})
(hMI : monic_irreducible_poly h) : R -> {poly %/ h with hMI} | :=
qpolyC h. | Definition | qfpoly_const | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"monic_irreducible_poly",
"poly",
"qpolyC"
] | Unfortunately we need some duplications so inference
propagates qfpoly :-( ) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
map_fpoly_div_inj (R : idomainType) (h : {poly R})
(hMI : monic_irreducible_poly h) :
injective (map_poly (qfpoly_const hMI)). | Proof. by apply: (@map_poly_div_inj R h). Qed. | Lemma | map_fpoly_div_inj | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"map_poly",
"map_poly_div_inj",
"monic_irreducible_poly",
"poly",
"qfpoly_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qfpoly_splitting_field_type | :=
FinSplittingFieldType F {poly %/ h with hI}. | Definition | qfpoly_splitting_field_type | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"FinSplittingFieldType",
"hI",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sh_gt2 : 2 < size h. | Hypothesis | sh_gt2 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sh_gt1 : 1 < size h. | Proof. by apply: leq_ltn_trans sh_gt2. Qed. | Let | sh_gt1 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"leq_ltn_trans",
"sh_gt2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primitive_poly (p: {poly F}) | :=
let v := #|{poly %/ p}|.-1 in
[&& p \is monic,
irreducibleb p,
p %| 'X^v - 1 &
[forall n : 'I_v, (p %| 'X^n - 1) ==> (n == 0%N :> nat)]]. | Definition | primitive_poly | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"irreducibleb",
"monic",
"nat",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primitive_polyP (p : {poly F}) :
reflect
(let v := #|{poly %/ p}|.-1 in
[/\ monic_irreducible_poly p,
p %| 'X^v - 1 &
forall n, 0 < n < v -> ~~ (p %| 'X^n - 1)])
(primitive_poly p). | Proof.
apply: (iffP and4P) => [[H1 H2 H3 /forallP H4] v|[[H1 H2] H3 H4]]; split => //.
- by split => //; apply/irreducibleP.
- move=> n /andP[n_gt0 nLv]; apply/negP => /(implyP (H4 (Ordinal nLv))) /=.
by rewrite eqn0Ngt n_gt0.
- by apply/irreducibleP.
apply/forallP=> [] [[|n] Hn] /=; apply/implyP => pDX //.
by case/n... | Lemma | primitive_polyP | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"eqn0Ngt",
"forallP",
"irreducibleP",
"monic_irreducible_poly",
"n_gt0",
"poly",
"primitive_poly",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hh : primitive_poly h. | Hypothesis | Hh | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"primitive_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
primitive_mi : monic_irreducible_poly h. | Proof. by case/primitive_polyP: Hh. Qed. | Lemma | primitive_mi | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hh",
"monic_irreducible_poly",
"primitive_polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primitive_poly_in_qpoly_eq0 p : (in_qpoly h p == 0) = (h %| p). | Proof.
have hM : h \is monic by case/and4P:Hh.
have hMi : monic_irreducible_poly h by apply: primitive_mi.
apply/eqP/idP => [/val_eqP /= | hDp].
by rewrite -Pdiv.IdomainMonic.modpE mk_monicE.
by apply/val_eqP; rewrite /= -Pdiv.IdomainMonic.modpE mk_monicE.
Qed. | Lemma | primitive_poly_in_qpoly_eq0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hh",
"apply",
"in_qpoly",
"mk_monicE",
"modpE",
"monic",
"monic_irreducible_poly",
"primitive_mi",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qT | := {poly %/ h with primitive_mi}. | Notation | qT | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"poly",
"primitive_mi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_primitive_qpoly : #|{poly %/ h}|= #|F| ^ (size h).-1. | Proof. by rewrite card_monic_qpoly ?primitive_mi. Qed. | Lemma | card_primitive_qpoly | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"card_monic_qpoly",
"poly",
"primitive_mi",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_neq0 : 'qX != 0 :> qT. | Proof.
apply/eqP => /val_eqP/=.
by rewrite [rmodp _ _]qpolyXE ?polyX_eq0 //; case: primitive_mi.
Qed. | Lemma | qX_neq0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"polyX_eq0",
"primitive_mi",
"qT",
"qpolyXE",
"rmodp",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_in_unit : ('qX : qT) \in GRing.unit. | Proof. by rewrite unitfE /= qX_neq0. Qed. | Lemma | qX_in_unit | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"qT",
"qX_neq0",
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gX : {unit qT} | := FinRing.unit _ qX_in_unit. | Definition | gX | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"qT",
"qX_in_unit",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_order n : (h %| 'X^n - 1) = (gX ^+ n == 1)%g. | Proof.
have [hM hI] := primitive_mi.
have eqr_add2r (r : nzRingType) (a b c : r) : (a + c == b + c) = (a == b).
by apply/eqP/eqP => [H|->//]; rewrite -(addrK c a) H addrK.
rewrite -val_eqE /= val_unitX /= -val_eqE /=.
rewrite (poly_of_qpolyX) qpolyXE // mk_monicE //.
rewrite -[in RHS](subrK 1 'X^n) rmodpD //.
rewrite... | Lemma | dvdp_order | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"add0r",
"addrK",
"apply",
"dvdpE",
"gX",
"hI",
"mk_monicE",
"polyC1",
"poly_of_qpolyX",
"primitive_mi",
"qpolyXE",
"rmodp",
"rmodpD",
"rmodp_eq0P",
"rmodp_small",
"size_poly1",
"subrK",
"val_eqE",
"val_unitX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gX_order : #[gX]%g = (#|qT|).-1. | Proof.
have /primitive_polyP[Hp1 Hp2 Hp3] := Hh.
set n := _.-1 in Hp2 Hp3 *.
have n_gt0 : 0 < n by rewrite ltn_predRL card_qfpoly_gt1.
have [hM hI] := primitive_mi.
have gX_neq1 : gX != 1%g.
apply/eqP/val_eqP/eqP/val_eqP=> /=.
rewrite [X in X != _]qpolyXE /= //.
by apply/eqP=> Hx1; have := @size_polyX F; rewrite ... | Lemma | gX_order | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hf",
"Hh",
"apply",
"card_qfpoly_gt1",
"card_uniqP",
"cycle_traject",
"dvdp_order",
"eq_card",
"expgS",
"expg_order",
"gX",
"hI",
"iterSr",
"leqNgt",
"ltn_predRL",
"ltngtP",
"mul",
"n_gt0",
"order_gt0",
"path",
"prednK",
"primitive_mi",
"primitive_polyP",
"qT",
"qpol... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gX_all : <[gX]>%g = [set: {unit qT}]%G. | Proof.
apply/eqP; rewrite eqEcard; apply/andP; split.
by apply/subsetP=> i; rewrite inE.
rewrite leq_eqVlt; apply/orP; left; apply/eqP.
rewrite -orderE gX_order card_qfpoly -[in RHS](mk_monicE primitive_mi).
rewrite -card_qpoly -(cardC1 (0 : {poly %/ h with primitive_mi})).
rewrite cardsT card_sub.
by apply: eq_card ... | Lemma | gX_all | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"cardC1",
"card_qfpoly",
"card_qpoly",
"card_sub",
"cardsT",
"eqEcard",
"eq_card",
"gX",
"gX_order",
"inE",
"leq_eqVlt",
"mk_monicE",
"orderE",
"poly",
"primitive_mi",
"qT",
"split",
"subsetP",
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred_card_qT_gt0 : 0 < #|qT|.-1. | Proof. by rewrite ltn_predRL card_qfpoly_gt1. Qed. | Let | pred_card_qT_gt0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"card_qfpoly_gt1",
"ltn_predRL",
"qT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogp (p : qT) : nat | :=
odflt (Ordinal pred_card_qT_gt0) (pick [pred i in 'I_ _ | ('qX ^+ i == p)]). | Definition | qlogp | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"nat",
"pick",
"pred_card_qT_gt0",
"qT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogp_lt p : qlogp p < #|qT|.-1. | Proof. by rewrite /qlogp; case: pickP. Qed. | Lemma | qlogp_lt | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"pickP",
"qT",
"qlogp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogp_qX (p : qT) : p != 0 -> 'qX ^+ (qlogp p) = p. | Proof.
move=> p_neq0.
have Up : p \in GRing.unit by rewrite unitfE.
pose gp : {unit qT}:= FinRing.unit _ Up.
have /cyclePmin[i iLc iX] : gp \in <[gX]>%g by rewrite gX_all inE.
rewrite gX_order in iLc.
rewrite /qlogp; case: pickP => [j /eqP//|/(_ (Ordinal iLc))] /eqP[].
by have /val_eqP/eqP/= := iX; rewrite FinRing.val_... | Lemma | qlogp_qX | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Up",
"cyclePmin",
"gX",
"gX_all",
"gX_order",
"inE",
"pickP",
"qT",
"qlogp",
"unit",
"unitfE",
"val_eqP",
"val_unitX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_order_card : 'qX ^+ (#|qT|).-1 = 1 :> qT. | Proof.
have /primitive_polyP [_ Hd _] := Hh.
rewrite dvdp_order in Hd.
have -> : 1 = val (1%g : {unit qT}) by [].
by rewrite -(eqP Hd) val_unitX.
Qed. | Lemma | qX_order_card | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hh",
"dvdp_order",
"primitive_polyP",
"qT",
"unit",
"val",
"val_unitX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_order_dvd (i : nat) : 'qX ^+ i = 1 :> qT -> (#|qT|.-1 %| i)%N. | Proof.
rewrite -gX_order cyclic.order_dvdn => Hd.
by apply/eqP/val_inj; rewrite /= -Hd val_unitX.
Qed. | Lemma | qX_order_dvd | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"cyclic",
"gX_order",
"nat",
"order_dvdn",
"qT",
"val_inj",
"val_unitX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogp0 : qlogp 0 = 0%N. | Proof.
rewrite /qlogp; case: pickP => //= x.
by rewrite (expf_eq0 ('qX : qT)) (negPf qX_neq0) andbF.
Qed. | Lemma | qlogp0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"expf_eq0",
"pickP",
"qT",
"qX_neq0",
"qlogp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogp1 : qlogp 1 = 0%N. | Proof.
case: (qlogp 1 =P 0%N) => // /eqP log1_neq0.
have := qlogp_lt 1; rewrite ltnNge => /negP[].
apply: dvdn_leq; first by rewrite lt0n.
by rewrite qX_order_dvd // qlogp_qX ?oner_eq0.
Qed. | Lemma | qlogp1 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"dvdn_leq",
"lt0n",
"ltnNge",
"oner_eq0",
"qX_order_dvd",
"qlogp",
"qlogp_lt",
"qlogp_qX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogp_eq0 (q : qT) : (qlogp q == 0%N) = (q == 0) || (q == 1). | Proof.
case: (q =P 0) => [->|/eqP q_neq0]/=; first by rewrite qlogp0.
case: (q =P 1) => [->|/eqP q_neq1]/=; first by rewrite qlogp1.
rewrite /qlogp; case: pickP => [x|/(_ (Ordinal (qlogp_lt q)))] /=.
by case: ((x : nat) =P 0%N) => // ->; rewrite expr0 eq_sym (negPf q_neq1).
by rewrite qlogp_qX // eqxx.
Qed. | Lemma | qlogp_eq0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"eq_sym",
"eqxx",
"expr0",
"nat",
"pickP",
"qT",
"qlogp",
"qlogp0",
"qlogp1",
"qlogp_lt",
"qlogp_qX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_exp_neq0 i : 'qX ^+ i != 0 :> qT. | Proof. by rewrite expf_eq0 negb_and qX_neq0 orbT. Qed. | Lemma | qX_exp_neq0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"expf_eq0",
"qT",
"qX_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_exp_inj i j :
i < #|qT|.-1 -> j < #|qT|.-1 -> 'qX ^+ i = 'qX ^+ j :> qT -> i = j. | Proof.
wlog iLj : i j / (i <= j)%N => [Hw|] iL jL Hqx.
case: (ltngtP i j)=> // /ltnW iLj; first by apply: Hw.
by apply/sym_equal/Hw.
suff ji_eq0 : (j - i = 0)%N by rewrite -(subnK iLj) ji_eq0.
case: ((j - i)%N =P 0%N) => // /eqP ji_neq0.
have : j - i < #|qT|.-1 by apply: leq_ltn_trans (leq_subr _ _) jL.
rewrite ltn... | Lemma | qX_exp_inj | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"dvdn_leq",
"expf_eq0",
"exprD",
"leq_ltn_trans",
"leq_subr",
"lt0n",
"ltnNge",
"ltnW",
"ltngtP",
"mul1r",
"mulIf",
"qT",
"qX_neq0",
"qX_order_dvd",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
powX_eq_mod i j : i = j %[mod #|qT|.-1] -> 'qX ^+ i = 'qX ^+ j :> qT. | Proof.
set n := _.-1 => iEj.
rewrite [i](divn_eq i n) [j](divn_eq j n) !exprD ![(_ * n)%N]mulnC.
by rewrite !exprM !qX_order_card !expr1n !mul1r iEj.
Qed. | Lemma | powX_eq_mod | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"divn_eq",
"expr1n",
"exprD",
"exprM",
"mul1r",
"mulnC",
"qT",
"qX_order_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qX_expK i : i < #|qT|.-1 -> qlogp ('qX ^+ i) = i. | Proof.
move=> iLF; apply: qX_exp_inj => //; first by apply: qlogp_lt.
by rewrite qlogp_qX // expf_eq0 (negPf qX_neq0) andbF.
Qed. | Lemma | qX_expK | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"expf_eq0",
"qT",
"qX_exp_inj",
"qX_neq0",
"qlogp",
"qlogp_lt",
"qlogp_qX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qlogpD (q1 q2 : qT) :
q1 != 0 -> q2 != 0 ->qlogp (q1 * q2) = ((qlogp q1 + qlogp q2) %% #|qT|.-1)%N. | Proof.
move=> q1_neq0 q2_neq0.
apply: qX_exp_inj; [apply: qlogp_lt => // | rewrite ltn_mod // |].
rewrite -[RHS]mul1r -(expr1n _ ((qlogp q1 + qlogp q2) %/ #|qT|.-1)).
rewrite -qX_order_card -exprM mulnC -exprD -divn_eq exprD !qlogp_qX //.
by rewrite mulf_eq0 negb_or q1_neq0.
Qed. | Lemma | qlogpD | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"divn_eq",
"expr1n",
"exprD",
"exprM",
"ltn_mod",
"mul1r",
"mulf_eq0",
"mulnC",
"qT",
"qX_exp_inj",
"qX_order_card",
"qlogp",
"qlogp_lt",
"qlogp_qX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogp (p q : {poly F}) | :=
if boolP (primitive_poly p) is AltTrue Hh then
qlogp ((in_qpoly p q) : {poly %/ p with primitive_mi Hh})
else 0%N. | Definition | plogp | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hh",
"in_qpoly",
"poly",
"primitive_mi",
"primitive_poly",
"qlogp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogp_lt (p q : {poly F}) : 2 < size p -> plogp p q < #|{poly %/ p}|.-1. | Proof.
move=> /ltnW size_gt1.
rewrite /plogp.
case (boolP (primitive_poly p)) => // Hh; first by apply: qlogp_lt.
by rewrite ltn_predRL (card_finNzRing_gt1 {poly %/ p}).
Qed. | Lemma | plogp_lt | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hh",
"apply",
"card_finNzRing_gt1",
"ltnW",
"ltn_predRL",
"plogp",
"poly",
"primitive_poly",
"qlogp_lt",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogp_X (p q : {poly F}) :
2 < size p -> primitive_poly p -> ~~ (p %| q) -> p %| q - 'X ^+ plogp p q. | Proof.
move=> sp_gt2 Hh pNDq.
rewrite /plogp.
case (boolP (primitive_poly p)) => // Hh'; last by case/negP: Hh'.
have pM : p \is monic by case/and4P: Hh'.
have pMi : monic_irreducible_poly p by apply: primitive_mi.
set q' : {poly %/ p with primitive_mi Hh'} := in_qpoly p q.
apply/modp_eq0P; rewrite modpD modpN; apply/e... | Lemma | plogp_X | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"Hh",
"apply",
"dvdpE",
"in_qpoly",
"last",
"mk_monicE",
"modpD",
"modpE",
"modpN",
"modp_eq0P",
"monic",
"monic_irreducible_poly",
"plogp",
"poly",
"poly_of_qpolyX",
"primitive_mi",
"primitive_poly",
"qlogp",
"qlogp_qX",
"rmodp",
"rmodp_eq0P",
"rmodp_small",
"size",
"s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogp0 (p : {poly F}) : 2 < size p -> plogp p 0 = 0%N. | Proof.
move=> sp_gt2; rewrite /plogp; case (boolP (primitive_poly p)) => // i.
by rewrite in_qpoly0 qlogp0.
Qed. | Lemma | plogp0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"in_qpoly0",
"plogp",
"poly",
"primitive_poly",
"qlogp0",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogp1 (p : {poly F}) : 2 < size p -> plogp p 1 = 0%N. | Proof.
move=> sp_gt2; rewrite /plogp; case (boolP (primitive_poly p)) => // i.
suff->: in_qpoly p 1 = 1 by apply: qlogp1.
apply/val_eqP/eqP; apply: in_qpoly_small.
rewrite mk_monicE ?size_poly1 ?(leq_trans _ sp_gt2) //.
by apply: primitive_mi.
Qed. | Lemma | plogp1 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"in_qpoly",
"in_qpoly_small",
"leq_trans",
"mk_monicE",
"plogp",
"poly",
"primitive_mi",
"primitive_poly",
"qlogp1",
"size",
"size_poly1",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogp_div_eq0 (p q : {poly F}) :
2 < size p -> (p %| q) -> plogp p q = 0%N. | Proof.
move=> sp_gt2; rewrite /plogp; case (boolP (primitive_poly p)) => // i pDq.
suff-> : in_qpoly p q = 0 by apply: qlogp0.
by apply/eqP; rewrite primitive_poly_in_qpoly_eq0.
Qed. | Lemma | plogp_div_eq0 | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"apply",
"in_qpoly",
"plogp",
"poly",
"primitive_poly",
"primitive_poly_in_qpoly_eq0",
"qlogp0",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
plogpD (p q1 q2 : {poly F}) :
2 < size p -> primitive_poly p -> ~~ (p %| q1) -> ~~ (p %| q2) ->
plogp p (q1 * q2) = ((plogp p q1 + plogp p q2) %% #|{poly %/ p}|.-1)%N. | Proof.
move=> sp_gt2 Pp pNDq1 pNDq2.
rewrite /plogp; case (boolP (primitive_poly p)) => [|/negP//] i /=.
have pmi := primitive_mi i.
by rewrite rmorphM qlogpD //= primitive_poly_in_qpoly_eq0.
Qed. | Lemma | plogpD | field | field/qfpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"div",
"bigop",
"binomial",
"finset",
"finfun",
"ssralg",
"countalg",
"finalg",
"poly",
"polydiv",
"qpoly",
"perm",
"fingroup",
"falgebra",
"... | [
"plogp",
"poly",
"primitive_mi",
"primitive_poly",
"primitive_poly_in_qpoly_eq0",
"qlogpD",
"rmorphM",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_poly_unlockable | := Unlockable separable_poly.unlock. | Canonical | separable_poly_unlockable | 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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable | := (@separable_poly R). | Notation | 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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lcn_neq0 | := (Pdiv.Idomain.lc_expn_scalp_neq0 _). | Notation | lcn_neq0 | 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",
... | [
"lc_expn_scalp_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_poly_neq0 p : separable p -> p != 0. | Proof.
by apply: contraTneq => ->; rewrite unlock deriv0 coprime0p eqp01.
Qed. | Lemma | separable_poly_neq0 | 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",
"contraTneq",
"coprime0p",
"deriv0",
"eqp01",
"separable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_square_freeP p :
(forall u v, u * v %| p -> coprimep u v)
<-> (forall u, size u != 1 -> ~~ (u ^+ 2 %| p)). | Proof.
split=> [sq'p u | sq'p u v dvd_uv_p].
by apply: contra => /sq'p; rewrite coprimepp.
rewrite coprimep_def (contraLR (sq'p _)) // (dvdp_trans _ dvd_uv_p) //.
by rewrite dvdp_mul ?dvdp_gcdl ?dvdp_gcdr.
Qed. | Lemma | poly_square_freeP | 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",
"coprimep",
"coprimep_def",
"coprimepp",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mul",
"dvdp_trans",
"size",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_polyP {p} :
reflect [/\ forall u v, u * v %| p -> coprimep u v
& forall u, u %| p -> 1 < size u -> u^`() != 0]
(separable p). | Proof.
apply: (iffP idP) => [sep_p | [sq'p nz_der1p]].
split=> [u v | u u_dv_p]; last first.
apply: contraTneq => u'0; rewrite unlock in sep_p; rewrite -leqNgt -(eqnP sep_p).
rewrite dvdp_leq -?size_poly_eq0 ?(eqnP sep_p) // dvdp_gcd u_dv_p.
have /dvdpZr <-: lead_coef u ^+ scalp p u != 0 by rewrite lcn_ne... | Lemma | separable_polyP | 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",
... | [
"Gauss_dvdpr",
"addr0",
"apply",
"contraTneq",
"coprimep",
"coprimepMl",
"coprimepMr",
"coprimepZl",
"coprimepZr",
"coprimep_addl_mul",
"coprimep_def",
"coprimep_sym",
"derivM",
"derivZ",
"divpK",
"dvd0p",
"dvdpZl",
"dvdpZr",
"dvdp_addr",
"dvdp_eq",
"dvdp_gcd",
"dvdp_gcdl",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_coprime p u v : separable p -> u * v %| p -> coprimep u v. | Proof. by move=> /separable_polyP[sq'p _] /sq'p. Qed. | Lemma | separable_coprime | 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",
... | [
"coprimep",
"separable",
"separable_polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_nosquare p u k :
separable p -> 1 < k -> size u != 1 -> (u ^+ k %| p) = false. | Proof.
move=> /separable_polyP[/poly_square_freeP sq'p _] /subnKC <- /sq'p.
by apply: contraNF; apply: dvdp_trans; rewrite exprD dvdp_mulr.
Qed. | Lemma | separable_nosquare | 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",
"dvdp_mulr",
"dvdp_trans",
"exprD",
"poly_square_freeP",
"separable",
"separable_polyP",
"size",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_deriv_eq0 p u :
separable p -> u %| p -> 1 < size u -> (u^`() == 0) = false. | Proof. by move=> /separable_polyP[_ nz_der1p] u_p /nz_der1p/negPf->. Qed. | Lemma | separable_deriv_eq0 | 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",
"separable_polyP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_separable p q : q %| p -> separable p -> separable q. | Proof.
move=> /(dvdp_trans _)q_dv_p /separable_polyP[sq'p nz_der1p].
by apply/separable_polyP; split=> [u v /q_dv_p/sq'p | u /q_dv_p/nz_der1p].
Qed. | Lemma | dvdp_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",
... | [
"apply",
"dvdp_trans",
"separable",
"separable_polyP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_mul p q :
separable (p * q) = [&& separable p, separable q & coprimep p q]. | Proof.
apply/idP/and3P => [sep_pq | [sep_p sep_q co_pq]].
rewrite !(dvdp_separable _ sep_pq) ?dvdp_mulIr ?dvdp_mulIl //.
by rewrite (separable_coprime sep_pq).
rewrite unlock in sep_p sep_q *.
rewrite derivM coprimepMl {1}addrC mulrC !coprimep_addl_mul.
by rewrite !coprimepMr (coprimep_sym q p) co_pq !andbT; apply/... | Lemma | separable_mul | 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",
... | [
"addrC",
"apply",
"coprimep",
"coprimepMl",
"coprimepMr",
"coprimep_addl_mul",
"coprimep_sym",
"derivM",
"dvdp_mulIl",
"dvdp_mulIr",
"dvdp_separable",
"mulrC",
"separable",
"separable_coprime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_separable p q : p %= q -> separable p = separable q. | Proof. by case/andP=> p_q q_p; apply/idP/idP=> /dvdp_separable->. Qed. | Lemma | eqp_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",
... | [
"apply",
"dvdp_separable",
"separable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_root p x :
separable (p * ('X - x%:P)) = separable p && ~~ root p x. | Proof.
rewrite separable_mul; apply: andb_id2l => seq_p.
by rewrite unlock derivXsubC coprimep1 coprimep_XsubC.
Qed. | Lemma | separable_root | 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",
"coprimep1",
"coprimep_XsubC",
"derivXsubC",
"root",
"separable",
"separable_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_prod_XsubC (r : seq R) :
separable (\prod_(x <- r) ('X - x%:P)) = uniq r. | Proof.
elim: r => [|x r IH]; first by rewrite big_nil unlock /separable_poly coprime1p.
by rewrite big_cons mulrC separable_root IH root_prod_XsubC andbC.
Qed. | Lemma | separable_prod_XsubC | 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",
... | [
"big_cons",
"big_nil",
"coprime1p",
"mulrC",
"root_prod_XsubC",
"separable",
"separable_root",
"seq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
make_separable p : p != 0 -> separable (p %/ gcdp p p^`()). | Proof.
set g := gcdp p p^`() => nz_p; apply/separable_polyP.
have max_dvd_u (u : {poly R}): 1 < size u -> exists k, ~~ (u ^+ k %| p).
move=> u_gt1; exists (size p); rewrite gtNdvdp // polySpred //.
by rewrite -(ltn_subRL 1) subn1 size_exp leq_pmull // -(subnKC u_gt1).
split=> [|u u_pg u_gt1]; last first.
apply/eq... | Lemma | make_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",
... | [
"add2n",
"addr0",
"apply",
"derivCE",
"derivZ",
"divpK",
"divp_dvd",
"dvd0p",
"dvd1p",
"dvdpZr",
"dvdp_add",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_mul",
"dvdp_mull",
"dvdp_mulr",
"dvdp_trans",
"expr0n",
"exprD",
"exprS",
"gcdp",
"gtNdvdp",
"last",
"lcn_neq0",
"leq_pmull",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_map (F : fieldType) (R : idomainType)
(f : {rmorphism F -> R}) (p : {poly F}) :
separable_poly (map_poly f p) = separable_poly p. | Proof.
by rewrite unlock deriv_map /coprimep -gcdp_map size_map_poly.
Qed. | Lemma | separable_map | 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",
... | [
"coprimep",
"deriv_map",
"gcdp_map",
"map_poly",
"poly",
"size_map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(nz_p : p != 0) (px_0 : root (p ^ iota) x). | Hypotheses | nz_p | 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",
... | [
"iota",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
inFz z w | := exists q, (q ^ iota).[z] = w. | Let | inFz | 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",
... | [
"iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
large_field_PET q :
root (q ^ iota) y -> separable_poly q ->
exists2 r, r != 0
& forall t (z := iota t * y - x), ~~ root r (iota t) -> inFz z x /\ inFz z y. | Proof.
move=> qy_0 sep_q; have nz_q := separable_poly_neq0 sep_q.
have /factor_theorem[q0 Dq] := qy_0.
set p1 := p ^ iota \Po ('X + x%:P); set q1 := q0 \Po ('X + y%:P).
have nz_p1: p1 != 0.
apply: contraNneq nz_p => /(canRL (fun r => comp_polyXaddC_K r _))/eqP.
by rewrite comp_poly0 map_poly_eq0.
have{sep_q} nz_q10... | Lemma | large_field_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",
... | [
"Bezout_coprimepP",
"Gauss_gcdpr",
"addrA",
"addrAC",
"addrC",
"addrNK",
"algebraicOver",
"algebraic_id",
"algebraic_mul",
"algebraic_sub",
"apply",
"coefZ",
"coef_map",
"comp_poly0",
"comp_polyA",
"comp_polyC",
"comp_polyD",
"comp_polyM",
"comp_polyX",
"comp_polyXaddC_K",
"c... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pchar0_PET (q : {poly F}) :
q != 0 -> root (q ^ iota) y -> [pchar F] =i pred0 ->
exists n, let z := y *+ n - x in inFz z x /\ inFz z y. | Proof.
move=> nz_q qy_0 /pcharf0P pcharF0.
without loss{nz_q} sep_q: q qy_0 / separable_poly q.
move=> IHq; apply: IHq (make_separable nz_q).
have /dvdpP[q1 Dq] := dvdp_gcdl q q^`().
rewrite {1}Dq mulpK ?gcdp_eq0; first by apply/nandP; left.
have [n [r nz_ry Dr]] := multiplicity_XsubC (q ^ iota) y.
rewrite ma... | Lemma | pchar0_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",
... | [
"Gauss_dvdpl",
"allPn",
"all_map",
"apply",
"coprimep_XsubC",
"coprimep_sym",
"derivM",
"derivXsubC",
"deriv_exp",
"deriv_map",
"dvdpP",
"dvdpZr",
"dvdp_XsubCl",
"dvdp_addr",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_mul2r",
"dvdp_mulIr",
"eqn_leq",
"eqp_dvdr",
"expf_neq0",
"exprS",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char0_PET | := (pchar0_PET) (only parsing). | Notation | char0_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",
... | [
"pchar0_PET"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation : bool | :=
all2rel (fun u v => D (u * v) == D u * v + u * D v) (vbasis K). | Definition | Derivation | 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",
... | [
"all2rel",
"vbasis"
] | the Derivation predicate for linear endomorphisms. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
derD : Derivation. | 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 | ||
Derivation_mul : {in K &, forall u v, D (u * v) = D u * v + u * D v}. | Proof.
move=> u v /coord_vbasis-> /coord_vbasis->.
rewrite !(mulr_sumr, linear_sum) -big_split; apply: eq_bigr => /= j _.
rewrite !mulr_suml linear_sum -big_split; apply: eq_bigr => /= i _.
rewrite !(=^~ scalerAl, linearZZ) -!scalerAr linearZZ -!scalerDr !scalerA /=.
by congr (_ *: _); apply/eqP/(allrelP derD); exact: ... | Lemma | Derivation_mul | 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",
... | [
"allrelP",
"apply",
"big_split",
"coord_vbasis",
"derD",
"eq_bigr",
"linearZZ",
"linear_sum",
"memt_nth",
"mulr_suml",
"mulr_sumr",
"scalerA",
"scalerAl",
"scalerAr",
"scalerDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation_mul_poly (Dp := map_poly D) :
{in polyOver K &, forall p q, Dp (p * q) = Dp p * q + p * Dp q}. | Proof.
move=> p q Kp Kq; apply/polyP=> i; rewrite {}/Dp coefD coef_map /= !coefM.
rewrite linear_sum -big_split; apply: eq_bigr => /= j _.
by rewrite !{1}coef_map Derivation_mul ?(polyOverP _).
Qed. | Lemma | Derivation_mul_poly | 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",
"apply",
"big_split",
"coefD",
"coefM",
"coef_map",
"eq_bigr",
"linear_sum",
"map_poly",
"polyOver",
"polyOverP",
"polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DerivationS E K D : (K <= E)%VS -> Derivation E D -> Derivation K D. | Proof.
move/subvP=> sKE derD; apply/allrelP=> x y Kx Ky; apply/eqP.
by rewrite (Derivation_mul derD) ?sKE // vbasis_mem.
Qed. | Lemma | DerivationS | 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",
"allrelP",
"apply",
"derD",
"sKE",
"subvP",
"vbasis_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derD : Derivation E 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 | ||
Derivation1 : D 1 = 0. | Proof.
apply: (addIr (D (1 * 1))); rewrite add0r {1}mul1r.
by rewrite (Derivation_mul derD) ?mem1v // mulr1 mul1r.
Qed. | Lemma | Derivation1 | 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",
"add0r",
"addIr",
"apply",
"derD",
"mem1v",
"mul1r",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation_scalar x : x \in 1%VS -> D x = 0. | Proof. by case/vlineP=> y ->; rewrite linearZ /= Derivation1 scaler0. Qed. | Lemma | Derivation_scalar | 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",
... | [
"Derivation1",
"linearZ",
"scaler0",
"vlineP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation_exp x m : x \in E -> D (x ^+ m) = x ^+ m.-1 *+ m * D x. | Proof.
move=> Ex; case: m; first by rewrite expr0 mulr0n mul0r Derivation1.
elim=> [|m IHm]; first by rewrite mul1r.
rewrite exprS (Derivation_mul derD) //; first by apply: rpredX.
by rewrite mulrC IHm mulrA mulrnAr -exprS -mulrDl.
Qed. | Lemma | Derivation_exp | 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",
... | [
"Derivation1",
"Derivation_mul",
"apply",
"derD",
"expr0",
"exprS",
"mul0r",
"mul1r",
"mulr0n",
"mulrA",
"mulrC",
"mulrDl",
"mulrnAr",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation_horner p x :
p \is a polyOver E -> x \in E ->
D p.[x] = (map_poly D p).[x] + p^`().[x] * D x. | Proof.
move=> Ep Ex; elim/poly_ind: p Ep => [|p c IHp] /polyOverP EpXc.
by rewrite !(raddf0, horner0) mul0r add0r.
have Ep: p \is a polyOver E.
by apply/polyOverP=> i; have:= EpXc i.+1; rewrite coefD coefMX coefC addr0.
have->: map_poly D (p * 'X + c%:P) = map_poly D p * 'X + (D c)%:P.
apply/polyP=> i; rewrite !(... | Lemma | Derivation_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",
"add0r",
"addr0",
"addrAC",
"addrC",
"addrCA",
"apply",
"coefC",
"coefD",
"coefMX",
"coef_map",
"derD",
"derivMXaddC",
"horner0",
"hornerE",
"linear0",
"linearD",
"map_poly",
"mul0r",
"mulrAC",
"mulrDl",
"polyOver",
"polyOverP",
"polyP",
"poly_ind",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_element U x | := separable_poly (minPoly U x). | Definition | separable_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",
... | [
"minPoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sKxK : (K <= <<K; x>>)%VS | := subv_adjoin K x. | Let | sKxK | 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",
... | [
"subv_adjoin"
] | begin hide | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Kx_x : x \in <<K; x>>%VS | := memv_adjoin K x. | Let | Kx_x | 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",
... | [
"memv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_elementP :
reflect (exists f, [/\ f \is a polyOver K, root f x & separable_poly f])
(separable_element K x). | Proof.
apply: (iffP idP) => [sep_x | [f [Kf /(minPoly_dvdp Kf)/dvdpP[g ->]]]].
by exists (minPoly K x); rewrite minPolyOver root_minPoly.
by rewrite separable_mul => /and3P[].
Qed. | Lemma | separable_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",
... | [
"apply",
"dvdpP",
"minPoly",
"minPolyOver",
"minPoly_dvdp",
"polyOver",
"root",
"root_minPoly",
"separable_element",
"separable_mul"
] | end hide | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
base_separable : x \in K -> separable_element K x. | Proof.
move=> Kx; apply/separable_elementP; exists ('X - x%:P).
by rewrite polyOverXsubC root_XsubC unlock !derivCE coprimep1.
Qed. | Lemma | base_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",
... | [
"apply",
"coprimep1",
"derivCE",
"polyOverXsubC",
"root_XsubC",
"separable_element",
"separable_elementP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_nz_der : separable_element K x = ((minPoly K x)^`() != 0). | Proof.
rewrite /separable_element unlock.
apply/idP/idP=> [|nzPx'].
by apply: contraTneq => ->; rewrite coprimep0 -size_poly_eq1 size_minPoly.
have gcdK : gcdp (minPoly K x) (minPoly K x)^`() \in polyOver K.
by rewrite gcdp_polyOver ?polyOver_deriv // minPolyOver.
rewrite -gcdp_eqp1 -size_poly_eq1 -dvdp1.
have /orP... | Lemma | separable_nz_der | 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",
"contraTneq",
"coprimep0",
"dvdp1",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_leq",
"dvdpp",
"gcdp",
"gcdp_eqp1",
"gcdp_polyOver",
"leq_trans",
"ltnn",
"minPoly",
"minPolyOver",
"minPoly_irr",
"polyOver",
"polyOver_deriv",
"separable_element",
"size_minPoly",
"size_poly",
"si... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separablePn_pchar :
reflect (exists2 p, p \in [pchar L] &
exists2 g, g \is a polyOver K & minPoly K x = g \Po 'X^p)
(~~ separable_element K x). | Proof.
rewrite separable_nz_der negbK; set f := minPoly K x.
apply: (iffP eqP) => [f'0 | [p Hp [g _ ->]]]; last first.
by rewrite deriv_comp derivXn -scaler_nat (pcharf0 Hp) scale0r mulr0.
pose n := adjoin_degree K x; have sz_f: size f = n.+1 := size_minPoly K x.
have fn1: f`_n = 1 by rewrite -(monicP (monic_minPoly ... | Lemma | separablePn_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",
... | [
"Sub",
"addr0",
"adjoin_degree",
"apply",
"big1",
"bigD1",
"coef0",
"coefXn",
"coefZ",
"coef_deriv",
"coef_poly",
"coef_sum",
"comp_polyE",
"derivXn",
"deriv_comp",
"dvdnP",
"dvdn_mulr",
"dvdn_pcharf",
"eqn_pmul2l",
"eqxx",
"exprM",
"gtn_eqF",
"ifN_eqC",
"last",
"lead... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
separable_root_der : separable_element K x (+) root (minPoly K x)^`() x. | Proof.
have KpKx': _^`() \is a polyOver K := polyOver_deriv (minPolyOver K x).
rewrite separable_nz_der addNb (root_small_adjoin_poly KpKx') ?addbb //.
by rewrite (leq_trans (size_poly _ _)) ?size_minPoly.
Qed. | Lemma | separable_root_der | 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",
... | [
"leq_trans",
"minPoly",
"minPolyOver",
"polyOver",
"polyOver_deriv",
"root",
"root_small_adjoin_poly",
"separable_element",
"separable_nz_der",
"size_minPoly",
"size_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Derivation_separable D :
Derivation <<K; x>> D -> separable_element K x ->
D x = - (map_poly D (minPoly K x)).[x] / (minPoly K x)^`().[x]. | Proof.
move=> derD sepKx; have:= separable_root_der; rewrite {}sepKx -sub0r => nzKx'x.
apply: canRL (mulfK nzKx'x) (canRL (addrK _) _); rewrite mulrC addrC.
rewrite -(Derivation_horner derD) ?minPolyxx ?linear0 //.
exact: polyOverSv sKxK _ (minPolyOver _ _).
Qed. | Lemma | Derivation_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",
"Derivation_horner",
"addrC",
"addrK",
"apply",
"derD",
"linear0",
"map_poly",
"minPoly",
"minPolyOver",
"minPolyxx",
"mulfK",
"mulrC",
"polyOverSv",
"sKxK",
"separable_element",
"separable_root_der",
"sub0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Dx E | := - (map_poly D (minPoly E x)).[x] / ((minPoly E x)^`()).[x]. | Let | Dx | 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",
... | [
"map_poly",
"minPoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.