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 |
|---|---|---|---|---|---|---|---|---|---|---|
rat_algebraic_decidable (C : fieldType) (QtoC : Qmorphism C) :
integralRange QtoC -> decidable_embedding QtoC. | Proof.
have QtoCinj: injective QtoC by apply: fmorph_inj.
pose ZtoQ : int -> rat := intr; pose ZtoC : int -> C := intr.
have ZtoQinj: injective ZtoQ by apply: intr_inj.
have defZtoC: ZtoC =1 QtoC \o ZtoQ by move=> m; rewrite /= rmorph_int.
move=> algC x; have /sig2_eqW[q mon_q qx0] := algC x; pose d := (size q).-1.
hav... | Lemma | rat_algebraic_decidable | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"Da",
"Dx",
"Gauss_dvdzl",
"Qmorphism",
"QtoC",
"ZtoC",
"ZtoQ",
"abszE",
"add0n",
"addSnnS",
"addnn",
"addrK",
"algC",
"apply",
"archi_boundP",
"bound",
"coefB",
"coefXn",
"coef_map",
"coprime_num_den",
"coprimez",
"coprimezXl",
"coprimez_sym",
"decidable_embedding",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minPoly_decidable_closure
(F : fieldType) (L : closedFieldType) (FtoL : {rmorphism F -> L}) x :
decidable_embedding FtoL -> integralOver FtoL x ->
{p | [/\ p \is monic, root (p ^ FtoL) x & irreducible_poly p]}. | Proof.
move=> isF /sig2W[p /monicP mon_p px0].
have [r Dp] := closed_field_poly_normal (p ^ FtoL); pose n := size r.
rewrite lead_coef_map {}mon_p rmorph1 scale1r in Dp.
pose Fpx q := (q \is a polyOver isF) && root q x.
have FpxF q: Fpx (q ^ FtoL) = root (q ^ FtoL) x.
by rewrite /Fpx polyOver_poly // => j _; apply/su... | Lemma | minPoly_decidable_closure | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"addKn",
"allP",
"apply",
"big_enum",
"big_mkord",
"big_nth",
"closed_field_poly_normal",
"decidable_embedding",
"dvdpN0",
"dvdpP",
"dvdpZl",
"dvdp_map",
"dvdp_mulr",
"dvdp_prod_XsubC",
"dvdp_size_eqp",
"enum",
"enum_uniq",
"eqSS",
"eq_bigl",
"eqp_monic",
"ex_minset",
"inE"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
alg_integral (F : fieldType) (L : fieldExtType F) :
integralRange (in_alg L). | Proof.
move=> x; have [/polyOver1P[p Dp]] := (minPolyOver 1 x, monic_minPoly 1 x).
by rewrite Dp map_monic; exists p; rewrite // -Dp root_minPoly.
Qed. | Lemma | alg_integral | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"in_alg",
"integralRange",
"map_monic",
"minPolyOver",
"monic_minPoly",
"polyOver1P",
"root_minPoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Fundamental_Theorem_of_Algebraics :
{L : closedFieldType &
{conj : {rmorphism L -> L} | involutive conj & ~ conj =1 id}}. | Proof.
have maxn3 n1 n2 n3: {m | [/\ n1 <= m, n2 <= m & n3 <= m]%N}.
by exists (maxn n1 (maxn n2 n3)); apply/and3P; rewrite -!geq_max.
have [C [/= QtoC algC]] := countable_algebraic_closure rat.
exists C; have [i Di2] := GRing.imaginary_exists C.
pose Qfield := fieldExtType rat.
pose Cmorph (L : Qfield) := {rmorphism... | Theorem | Fundamental_Theorem_of_Algebraics | field | field/algebraics_fundamentals.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"ssrnat",
"eqtype",
"seq",
"choice",
"div",
"fintype",
"path",
"tuple",
"bigop",
"finset",
"prime",
"order",
"ssralg",
"poly",
"polydiv",
"mxpoly",
"countalg",
"closed_field",
"ssrnum",
"ssrint",
"a... | [
"AHom_lker0",
"Bezout_eq1_coprimepP",
"Build",
"Dx",
"Fadjoin1_polyP",
"FadjoinP",
"Fadjoin_idP",
"Fadjoin_poly",
"Fadjoin_polyOver",
"Fadjoin_polyP",
"Fadjoin_poly_eq",
"Fadjoin_seqP",
"Gg",
"LtoC",
"Poly",
"PolyK",
"QtoC",
"SubFieldExtType",
"Sylow_exists",
"a1",
"a2",
"a... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
alg_num_field (Qz : fieldExtType rat) a : a%:A = ratr a :> Qz. | Proof. by rewrite -in_algE fmorph_eq_rat. Qed. | Lemma | alg_num_field | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"fmorph_eq_rat",
"in_algE",
"rat",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rmorphZ_num (Qz : fieldExtType rat) rR (f : {rmorphism Qz -> rR}) a x :
f (a *: x) = ratr a * f x. | Proof. by rewrite -mulr_algl rmorphM alg_num_field fmorph_rat. Qed. | Lemma | rmorphZ_num | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"alg_num_field",
"fmorph_rat",
"mulr_algl",
"rat",
"ratr",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_numZ (Qz1 Qz2 : fieldExtType rat) (f : {rmorphism Qz1 -> Qz2}) :
scalable f. | Proof. by move=> a x; rewrite rmorphZ_num -alg_num_field mulr_algl. Qed. | Lemma | fmorph_numZ | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"alg_num_field",
"mulr_algl",
"rat",
"rmorphZ_num",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC_PET (s : seq algC) :
{z | exists a : nat ^ size s, z = \sum_(i < size s) s`_i *+ a i
& exists ps, s = [seq (pQtoC p).[z] | p <- ps]}. | Proof.
elim: s => [|x s [z /sig_eqW[a Dz] /sig_eqW[ps Ds]]].
by exists 0; [exists [ffun _ => 2%N]; rewrite big_ord0 | exists nil].
have r_exists (y : algC): {r | r != 0 & root (pQtoC r) y}.
have [r [_ mon_r] dv_r] := minCpolyP y.
by exists r; rewrite ?monic_neq0 ?dv_r.
suffices /sig_eqW[[n [|px [|pz []]]]// [Dpx ... | Lemma | algC_PET | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"algC",
"apply",
"big_ord0",
"big_ord_recl",
"eq_bigr",
"eq_map",
"ffunE",
"hornerN",
"horner_comp",
"liftK",
"map_comp",
"map_comp_poly",
"minCpolyP",
"monic_neq0",
"nat",
"opprK",
"ord0",
"pQtoC",
"pchar",
"pchar0_PET",
"pchar_num",
"raddfN",
"rat",
"root",
"seq",
... | Number fields and rational spans. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
num_field_exists (s : seq algC) :
{Qs : fieldExtType rat & {QsC : {rmorphism Qs -> algC}
& {s1 : seq Qs | map QsC s1 = s & <<1 & s1>>%VS = fullv}}}. | Proof.
have [z /sig_eqW[a Dz] /sig_eqW[ps Ds]] := algC_PET s.
suffices [Qs [QsC [z1 z1C z1gen]]]:
{Qs : fieldExtType rat & {QsC : {rmorphism Qs -> algC} &
{z1 : Qs | QsC z1 = z & forall xx, exists p, fieldExt_horner z1 p = xx}}}.
- set inQs := fieldExt_horner z1 in z1gen *; pose s1 := map inQs ps.
have inQsK p... | Lemma | num_field_exists | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"QtoC",
"SubFieldExtType",
"algC",
"algC_PET",
"alg_num_field",
"apply",
"dvdp_leq",
"eq_bigr",
"eq_map",
"eq_map_poly",
"fieldExt_horner",
"fieldExt_hornerC",
"fieldExt_hornerX",
"fmorph_inj",
"fmorph_rat",
"fullv",
"horner_map",
"irreducible_poly",
"map",
"map_comp",
"map_p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_Crat_span s x | :=
exists a : rat ^ size s, x = \sum_i QtoC (a i) * s`_i. | Definition | in_Crat_span | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"QtoC",
"rat",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_span_subproof s x : decidable (in_Crat_span s x). | Proof.
have [Qxs [QxsC [[|x1 s1] // [<- <-] {x s} _]]] := num_field_exists (x :: s).
apply: decP (x1 \in <<in_tuple s1>>%VS) _; rewrite /in_Crat_span size_map.
apply: (iffP idP) => [/coord_span-> | [a Dx]].
move: (coord _) => a; exists [ffun i => a i x1]; rewrite rmorph_sum /=.
by apply: eq_bigr => i _; rewrite ffu... | Fact | Crat_span_subproof | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Dx",
"apply",
"coord",
"coord_span",
"decidable",
"eq_bigr",
"ffunE",
"fmorph_inj",
"in_Crat_span",
"in_tuple",
"mem_nth",
"memvZ",
"memv_span",
"memv_suml",
"nth_map",
"num_field_exists",
"rmorphZ_num",
"rmorph_sum",
"s1",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_span s : pred algC | := Crat_span_subproof s. | Definition | Crat_span | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span_subproof",
"algC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_spanP s x : reflect (in_Crat_span s x) (x \in Crat_span s). | Proof. exact: sumboolP. Qed. | Lemma | Crat_spanP | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span",
"in_Crat_span"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_Crat_span s : {subset s <= Crat_span s}. | Proof.
move=> _ /(nthP 0)[ix ltxs <-]; pose i0 := Ordinal ltxs.
apply/Crat_spanP; exists [ffun i => (i == i0)%:R].
rewrite (bigD1_ord i0) //= ffunE eqxx // rmorph1 mul1r.
by rewrite big1 ?addr0 // => i; rewrite ffunE rmorph_nat mulr_natl lift_eqF.
Qed. | Lemma | mem_Crat_span | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span",
"Crat_spanP",
"addr0",
"apply",
"big1",
"bigD1_ord",
"eqxx",
"ffunE",
"i0",
"lift_eqF",
"mul1r",
"mulr_natl",
"nthP",
"rmorph1",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_span_zmod_closed s : zmod_closed (Crat_span s). | Proof.
split=> [|_ _ /Crat_spanP[x ->] /Crat_spanP[y ->]].
apply/Crat_spanP; exists 0.
by apply/esym/big1=> i _; rewrite ffunE rmorph0 mul0r.
apply/Crat_spanP; exists (x - y); rewrite -sumrB; apply: eq_bigr => i _.
by rewrite -mulrBl -rmorphB !ffunE.
Qed. | Fact | Crat_span_zmod_closed | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span",
"Crat_spanP",
"apply",
"big1",
"eq_bigr",
"ffunE",
"mul0r",
"mulrBl",
"rmorph0",
"rmorphB",
"split",
"sumrB",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_spanZ b a : {in Crat_span b, forall x, ratr a * x \in Crat_span b}. | Proof.
move=> _ /Crat_spanP[a1 ->]; apply/Crat_spanP; exists [ffun i => a * a1 i].
by rewrite mulr_sumr; apply: eq_bigr => i _; rewrite ffunE mulrA -rmorphM.
Qed. | Lemma | Crat_spanZ | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span",
"Crat_spanP",
"a1",
"apply",
"eq_bigr",
"ffunE",
"mulrA",
"mulr_sumr",
"ratr",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crat_spanM b : {in Crat & Crat_span b, forall a x, a * x \in Crat_span b}. | Proof. by move=> _ x /CratP[a ->]; apply: Crat_spanZ. Qed. | Lemma | Crat_spanM | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat",
"CratP",
"Crat_span",
"Crat_spanZ",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_field_proj : {CtoQn | CtoQn 0 = 0 & cancel QnC CtoQn}. | Proof.
pose b := vbasis {:Qn}.
have Qn_bC (u : {x | x \in Crat_span (map QnC b)}): {y | QnC y = sval u}.
case: u => _ /= /Crat_spanP/sig_eqW[a ->].
exists (\sum_i a i *: b`_i); rewrite rmorph_sum /=; apply: eq_bigr => i _.
by rewrite rmorphZ_num (nth_map 0) // -(size_map QnC).
pose CtoQn x := oapp (fun u => sval ... | Lemma | num_field_proj | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span",
"Crat_spanP",
"Crat_spanZ",
"apply",
"coord_vbasis",
"eq_bigr",
"fmorph_inj",
"insub",
"insubT",
"map",
"mem_Crat_span",
"mem_tnth",
"memvf",
"nth_map",
"rmorph0",
"rmorphZ_num",
"rmorph_sum",
"rpred_sum",
"sig_eqW",
"size_map",
"tnth_map",
"tnth_nth",
"vbasi... | would require a limit construction. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
restrict_aut_to_num_field (nu : {rmorphism algC -> algC}) :
(forall x, exists y, nu (QnC x) = QnC y) ->
{nu0 : {lrmorphism Qn -> Qn} | {morph QnC : x / nu0 x >-> nu x}}. | Proof.
move=> Qn_nu; pose nu0 x := sval (sig_eqW (Qn_nu x)).
have QnC_nu0: {morph QnC : x / nu0 x >-> nu x}.
by rewrite /nu0 => x; case: (sig_eqW _).
have nu0a : zmod_morphism nu0.
by move=> x y; apply: (fmorph_inj QnC); rewrite !(QnC_nu0, rmorphB).
have nu0m : monoid_morphism nu0.
split=> [|x y]; apply: (fmorph_... | Lemma | restrict_aut_to_num_field | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Build",
"algC",
"apply",
"fmorph_inj",
"fmorph_numZ",
"monoid_morphism",
"rat",
"rmorph1",
"rmorphB",
"rmorphM",
"sig_eqW",
"split",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_Qnum_poly (nu : {rmorphism algC -> algC}) p :
p \in polyOver 1%VS -> map_poly (nu \o QnC) p = (map_poly QnC p). | Proof.
move=> Qp; apply/polyP=> i; rewrite /= !coef_map /=.
have /vlineP[a ->]: p`_i \in 1%VS by apply: polyOverP.
by rewrite alg_num_field !fmorph_rat.
Qed. | Lemma | map_Qnum_poly | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"algC",
"alg_num_field",
"apply",
"coef_map",
"fmorph_rat",
"map_poly",
"polyOver",
"polyOverP",
"polyP",
"vlineP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
restrict_aut_to_normal_num_field (Qn : splittingFieldType rat)
(QnC : {rmorphism Qn -> algC})(nu : {rmorphism algC -> algC}) :
{nu0 : {lrmorphism Qn -> Qn} | {morph QnC : x / nu0 x >-> nu x}}. | Proof.
apply: restrict_aut_to_num_field => x.
case: (splitting_field_normal 1%AS x) => rs /eqP Hrs.
have: root (map_poly (nu \o QnC) (minPoly 1%AS x)) (nu (QnC x)).
by rewrite fmorph_root root_minPoly.
rewrite map_Qnum_poly ?minPolyOver // Hrs.
rewrite [map_poly _ _](_:_ = \prod_(y <- map QnC rs) ('X - y%:P)); last f... | Lemma | restrict_aut_to_normal_num_field | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"algC",
"apply",
"big_map",
"eq_bigr",
"fmorph_root",
"last",
"map",
"mapP",
"map_Qnum_poly",
"map_poly",
"map_polyXsubC",
"minPoly",
"minPolyOver",
"rat",
"restrict_aut_to_num_field",
"rmorph_prod",
"root",
"root_minPoly",
"root_prod_XsubC",
"splitting_field_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dec_Cint_span (V : vectType algC) m (s : m.-tuple V) v :
decidable (inIntSpan s v). | Proof.
have s_s (i : 'I_m): s`_i \in <<s>>%VS by rewrite memv_span ?memt_nth.
have s_Zs a: \sum_(i < m) s`_i *~ a i \in <<s>>%VS.
by rewrite memv_suml // => i _; rewrite -scaler_int memvZ.
case s_v: (v \in <<s>>%VS); last by right=> [[a Dv]]; rewrite Dv s_Zs in s_v.
pose IzT := {: 'I_m * 'I_(\dim <<s>>)}; pose Iz := ... | Lemma | dec_Cint_span | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat_span",
"Crat_spanP",
"algC",
"apply",
"big_mkcond",
"can2_eq",
"cardE",
"coord",
"coord_vbasis",
"dec_Qint_span",
"decidable",
"dim",
"enum_rank",
"enum_rankK",
"enum_val",
"enum_valK",
"eqVneq",
"eq_bigr",
"eqxx",
"exchange_big",
"ffunE",
"ffunMzE",
"ffunP",
"fmo... | Integral spans. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Cint_span (s : seq algC) : pred algC | :=
fun x => dec_Cint_span (in_tuple [seq \row_(i < 1) y | y <- s]) (\row_i x). | Definition | Cint_span | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"algC",
"dec_Cint_span",
"in_tuple",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_spanP n (s : n.-tuple algC) x :
reflect (inIntSpan s x) (x \in Cint_span s). | Proof.
rewrite unfold_in; case: (dec_Cint_span _ _) => [Zs_x | Zs'x] /=.
left; have{Zs_x} [] := Zs_x; rewrite /= size_map size_tuple => a /rowP/(_ 0).
rewrite !mxE => ->; exists a; rewrite summxE; apply: eq_bigr => i _.
by rewrite -scaler_int (nth_map 0) ?size_tuple // !mxE mulrzl.
right=> [[a Dx]]; have{Zs'x} []... | Lemma | Cint_spanP | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Cint_span",
"Dx",
"algC",
"apply",
"dec_Cint_span",
"eq_bigr",
"i0",
"inIntSpan",
"mulrzl",
"mxE",
"nth_map",
"rowP",
"scaler_int",
"size_map",
"size_tuple",
"summxE",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_Cint_span s : {subset s <= Cint_span s}. | Proof.
move=> _ /(nthP 0)[ix ltxs <-]; apply/(Cint_spanP (in_tuple s)).
exists [ffun i => i == Ordinal ltxs : int].
rewrite (bigD1 (Ordinal ltxs)) //= ffunE eqxx.
by rewrite big1 ?addr0 // => i; rewrite ffunE => /negbTE->.
Qed. | Lemma | mem_Cint_span | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Cint_span",
"Cint_spanP",
"addr0",
"apply",
"big1",
"bigD1",
"eqxx",
"ffunE",
"in_tuple",
"int",
"nthP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_span_zmod_closed s : zmod_closed (Cint_span s). | Proof.
have sP := Cint_spanP (in_tuple s); split=> [|_ _ /sP[x ->] /sP[y ->]].
by apply/sP; exists 0; rewrite big1 // => i; rewrite ffunE.
apply/sP; exists (x - y); rewrite -sumrB; apply: eq_bigr => i _.
by rewrite !ffunE raddfB.
Qed. | Lemma | Cint_span_zmod_closed | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Cint_span",
"Cint_spanP",
"apply",
"big1",
"eq_bigr",
"ffunE",
"in_tuple",
"raddfB",
"split",
"sumrB",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extend_algC_subfield_aut (Qs : fieldExtType rat)
(QsC : {rmorphism Qs -> algC}) (phi : {rmorphism Qs -> Qs}) :
{nu : {rmorphism algC -> algC} | {morph QsC : x / phi x >-> nu x}}. | Proof.
pose numF_inj (Qr : fieldExtType rat) := {rmorphism Qr -> algC}.
pose subAut := {Qr : _ & numF_inj Qr * {lrmorphism Qr -> Qr}}%type.
pose SubAut := existT _ _ (_, _) : subAut.
pose Sdom (mu : subAut) := projT1 mu.
pose Sinj (mu : subAut) : {rmorphism Sdom mu -> algC} := (projT2 mu).1.
pose Saut (mu : subAut) : {... | Lemma | extend_algC_subfield_aut | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Build",
"Dx",
"Sub",
"addSnnS",
"algC",
"algC_PET",
"alg_num_field",
"apply",
"big_map",
"closed_field_poly_normal",
"coef_map",
"coord",
"coord_vbasis",
"dvdpp",
"eq_bigr",
"eq_map_poly",
"eqpxx",
"f1",
"fmorph_inj",
"fullv",
"fun_of_lfun",
"geq_max",
"has_algid1",
"h... | Automorphism extensions. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Qn_aut_exists k n :
coprime k n ->
{u : {rmorphism algC -> algC} | forall z, z ^+ n = 1 -> u z = z ^+ k}. | Proof.
have [-> /eqnP | n_gt0 co_k_n] := posnP n.
by rewrite gcdn0 => ->; exists idfun.
have [z prim_z] := C_prim_root_exists n_gt0.
have [Qn [QnC [[|zn []] // [Dz]]] genQn] := num_field_exists [:: z].
pose phi := kHomExtend 1 \1 zn (zn ^+ k).
have homQn1: kHom 1 1 (\1%VF : 'End(Qn)) by rewrite kHom1.
have pzn_zk0: r... | Lemma | Qn_aut_exists | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Build",
"C_prim_root_exists",
"aP",
"algC",
"alg_num_field",
"apply",
"bigD1",
"coef_map",
"coef_poly",
"coprime",
"coprime_modl",
"cyclotomic",
"dvdpP",
"eq_map_poly",
"eqnP",
"eqxx",
"exprAC",
"extend_algC_subfield_aut",
"fmorph_rat",
"fmorph_root",
"fun_of_lfun",
"gcdn0... | Extended automorphisms of Q_n. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Aint : {pred algC} | := fun x => minCpoly x \is a polyOver Num.int. | Definition | Aint | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"algC",
"int",
"minCpoly",
"polyOver"
] | Algebraic integers. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
root_monic_Aint p x :
root p x -> p \is monic -> p \is a polyOver Num.int -> x \in Aint. | Proof.
have pZtoQtoC pz: pQtoC (pZtoQ pz) = pZtoC pz.
by rewrite -map_poly_comp; apply: eq_map_poly => b; rewrite /= rmorph_int.
move=> px0 mon_p /floorpP[pz Dp]; rewrite unfold_in.
move: px0; rewrite Dp -pZtoQtoC; have [q [-> mon_q] ->] := minCpolyP x.
case/dvdpP_rat_int=> qz [a nz_a Dq] [r].
move/(congr1 (fun q1 =>... | Lemma | root_monic_Aint | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"QtoC",
"apply",
"coefZ",
"coef_map",
"dvdpP_rat_int",
"eq_map_poly",
"floorpP",
"int",
"intr_inj",
"lead_coef",
"lead_coefM",
"lead_coefZ",
"lead_coef_map_inj",
"map_monic",
"map_poly_comp",
"minCpolyP",
"monic",
"monicP",
"mul1r",
"mulr1",
"pQtoC",
"pZtoC",
"p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_rat_Aint z : z \in Crat -> z \in Aint -> z \in Num.int. | Proof.
case/CratP=> a ->{z} /polyOverP/(_ 0).
have [p [Dp mon_p] dv_p] := minCpolyP (ratr a); rewrite Dp coef_map.
suffices /eqP->: p == 'X - a%:P by rewrite polyseqXsubC /= rmorphN rpredN.
rewrite -eqp_monic ?monicXsubC // irredp_XsubC //.
by rewrite -(size_map_poly QtoC) -Dp neq_ltn size_minCpoly orbT.
by rewrite -... | Lemma | Cint_rat_Aint | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Crat",
"CratP",
"QtoC",
"coef_map",
"eqp_monic",
"fmorph_root",
"int",
"irredp_XsubC",
"minCpolyP",
"monicXsubC",
"neq_ltn",
"polyOverP",
"polyseqXsubC",
"ratr",
"rmorphN",
"root_XsubC",
"rpredN",
"size_map_poly",
"size_minCpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_Cint : {subset Num.int <= Aint}. | Proof.
move=> x; rewrite -polyOverXsubC.
by apply: root_monic_Aint; rewrite ?monicXsubC ?root_XsubC.
Qed. | Lemma | Aint_Cint | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"apply",
"int",
"monicXsubC",
"polyOverXsubC",
"root_XsubC",
"root_monic_Aint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_int x : x%:~R \in Aint. | Proof. by rewrite Aint_Cint. Qed. | Lemma | Aint_int | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Aint_Cint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint0 : 0 \in Aint. | Proof. exact: Aint_int 0. Qed. | Lemma | Aint0 | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Aint_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint1 : 1 \in Aint. | Proof. exact: Aint_int 1. Qed. | Lemma | Aint1 | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Aint_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_unity_root n x : (n > 0)%N -> n.-unity_root x -> x \in Aint. | Proof.
move=> n_gt0 xn1; apply: root_monic_Aint xn1 (monicXnsubC _ n_gt0) _.
by apply/polyOverP=> i; rewrite coefB coefC -mulrb coefXn /= rpredB ?rpred_nat.
Qed. | Lemma | Aint_unity_root | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"apply",
"coefB",
"coefC",
"coefXn",
"monicXnsubC",
"mulrb",
"n_gt0",
"polyOverP",
"root_monic_Aint",
"rpredB",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_prim_root n z : n.-primitive_root z -> z \in Aint. | Proof.
move=> pr_z; apply/(Aint_unity_root (prim_order_gt0 pr_z))/unity_rootP.
exact: prim_expr_order.
Qed. | Lemma | Aint_prim_root | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Aint_unity_root",
"apply",
"prim_expr_order",
"prim_order_gt0",
"unity_rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_Cnat : {subset Num.nat <= Aint}. | Proof. by move=> z /intr_nat/Aint_Cint. Qed. | Lemma | Aint_Cnat | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Aint_Cint",
"intr_nat",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_subring_exists (X : seq algC) :
{subset X <= Aint} ->
{S : pred algC &
(*a*) subring_closed S
/\ (*b*) {subset X <= S}
& (*c*) {Y : {n : nat & n.-tuple algC} &
{subset tagged Y <= S}
& forall x, reflect (inIntSpan (tagged Y) x) (x \in S)}}. | Proof.
move=> AZ_X; pose m := (size X).+1.
pose n (i : 'I_m) := (size (minCpoly X`_i)).-2; pose N := (\max_i n i).+1.
pose IY := family (fun i => [pred e : 'I_N | e <= n i]%N).
have IY_0: 0 \in IY by apply/familyP=> // i; rewrite ffunE.
pose inIY := enum_rank_in IY_0.
pose Y := [seq \prod_(i < m) X`_i ^+ (f : 'I_N ^ m)... | Lemma | Aint_subring_exists | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Cint_span",
"Cint_spanP",
"addrC",
"addr_eq0",
"algC",
"apply",
"big1",
"bigD1",
"big_ord_recr",
"big_rec",
"bigmax_sup",
"coef_map",
"enum_rank_in",
"enum_val",
"enum_valP",
"eq_bigr",
"eqxx",
"exprS",
"fP",
"family",
"familyP",
"ffunE",
"floorpP",
"horner_c... | This is Isaacs, Lemma (3.3) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
fin_Csubring_Aint S n (Y : n.-tuple algC) :
mulr_closed S -> (forall x, reflect (inIntSpan Y x) (x \in S)) ->
{subset S <= Aint}. | Proof.
move=> mulS.
pose Sm := GRing.isMulClosed.Build _ _ mulS.
pose SC : mulrClosed _ := HB.pack S Sm.
have ZP_C c: (ZtoC c)%:P \is a polyOver Num.int_num_subdef.
by rewrite raddfMz rpred_int.
move=> S_P x Sx; pose v := \row_(i < n) Y`_i.
have [v0 | nz_v] := eqVneq v 0.
case/S_P: Sx => {}x ->; rewrite big1 ?isAlg... | Theorem | fin_Csubring_Aint | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Build",
"Cint_spanP",
"ZtoC",
"algC",
"apply",
"big1",
"char_poly",
"char_poly_monic",
"eigenvalueP",
"eigenvalue_root_char",
"eqVneq",
"eq_bigr",
"inIntSpan",
"map_mx",
"mem_Cint_span",
"memt_nth",
"mul0rz",
"mulr_closed",
"mulrzr",
"mxE",
"polyOver",
"polyOverX... | This is Isaacs, Theorem (3.4). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Aint_subring : subring_closed Aint. | Proof.
suff rAZ: {in Aint &, forall x y, (x - y \in Aint) * (x * y \in Aint)}.
by split=> // x y AZx AZy; rewrite rAZ.
move=> x y AZx AZy.
have [|S [ringS] ] := @Aint_subring_exists [:: x; y]; first exact/allP/and3P.
move=> /allP/and3P[Sx Sy _] [Y _ genYS].
have AZ_S := fin_Csubring_Aint ringS genYS.
by have [_ S_B S... | Corollary | Aint_subring | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"Aint_subring_exists",
"allP",
"fin_Csubring_Aint",
"split",
"subring_closed"
] | This is Isaacs, Corollary (3.5). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Aint_aut (nu : {rmorphism algC -> algC}) x :
(nu x \in Aint) = (x \in Aint). | Proof. by rewrite !unfold_in minCpoly_aut. Qed. | Lemma | Aint_aut | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"algC",
"minCpoly_aut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdA (e : Algebraics.divisor) : {pred algC} | :=
fun z => if e == 0 then z == 0 else z / e \in Aint. | Definition | dvdA | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"algC",
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"e %| x" | := (x \in dvdA e) : algC_expanded_scope. | Notation | e %| x | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"dvdA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"e %| x" | := (@in_mem Algebraics.divisor x (mem (dvdA e))) : algC_scope. | Notation | e %| x | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"divisor",
"dvdA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdA_zmod_closed e : zmod_closed (dvdA e). | Proof.
split=> [|x y]; first by rewrite unfold_in mul0r eqxx rpred0 ?if_same.
rewrite ![(e %| _)%A]unfold_in.
case: ifP => [_ x0 /eqP-> | _]; first by rewrite subr0.
by rewrite mulrBl; apply: rpredB.
Qed. | Fact | dvdA_zmod_closed | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"apply",
"dvdA",
"eqxx",
"mul0r",
"mulrBl",
"rpred0",
"rpredB",
"split",
"subr0",
"zmod_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod (e x y : Algebraics.divisor) | := (e %| x - y)%A. | Definition | eqAmod | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"divisor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x == y %[mod e ]" | := (eqAmod e x y) : algC_scope. | Notation | x == y %[mod e ] | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x != y %[mod e ]" | := (~~ (eqAmod e x y)) : algC_scope. | Notation | x != y %[mod e ] | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_refl e x : (x == x %[mod e])%A. | Proof. by rewrite /eqAmod subrr rpred0. Qed. | Lemma | eqAmod_refl | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod",
"rpred0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_sym e x y : ((x == y %[mod e]) = (y == x %[mod e]))%A. | Proof. by rewrite /eqAmod -opprB rpredN. Qed. | Lemma | eqAmod_sym | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod",
"opprB",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_trans e y x z :
(x == y %[mod e] -> y == z %[mod e] -> x == z %[mod e])%A. | Proof.
by move=> Exy Eyz; rewrite /eqAmod -[x](subrK y) -[_ - z]addrA rpredD.
Qed. | Lemma | eqAmod_trans | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"addrA",
"eqAmod",
"rpredD",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_transl e x y z :
(x == y %[mod e])%A -> (x == z %[mod e])%A = (y == z %[mod e])%A. | Proof. by move/(sym_left_transitive (eqAmod_sym e) (@eqAmod_trans e)). Qed. | Lemma | eqAmod_transl | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod_sym",
"eqAmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_transr e x y z :
(x == y %[mod e])%A -> (z == x %[mod e])%A = (z == y %[mod e])%A. | Proof. by move/(sym_right_transitive (eqAmod_sym e) (@eqAmod_trans e)). Qed. | Lemma | eqAmod_transr | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod_sym",
"eqAmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod0 e x : (x == 0 %[mod e])%A = (e %| x)%A. | Proof. by rewrite /eqAmod subr0. Qed. | Lemma | eqAmod0 | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodN e x y : (- x == y %[mod e])%A = (x == - y %[mod e])%A. | Proof. by rewrite eqAmod_sym /eqAmod !opprK addrC. Qed. | Lemma | eqAmodN | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"addrC",
"eqAmod",
"eqAmod_sym",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodDr e x y z : (y + x == z + x %[mod e])%A = (y == z %[mod e])%A. | Proof. by rewrite /eqAmod [z + x]addrC addrKA. Qed. | Lemma | eqAmodDr | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"addrC",
"addrKA",
"eqAmod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodDl e x y z : (x + y == x + z %[mod e])%A = (y == z %[mod e])%A. | Proof. by rewrite !(addrC x) eqAmodDr. Qed. | Lemma | eqAmodDl | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"addrC",
"eqAmodDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodD e x1 x2 y1 y2 :
(x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 + y1 == x2 + y2 %[mod e])%A. | Proof.
by rewrite -(eqAmodDl e x2 y1) -(eqAmodDr e y1); apply: eqAmod_trans.
Qed. | Lemma | eqAmodD | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"apply",
"eqAmodDl",
"eqAmodDr",
"eqAmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodm0 e : (e == 0 %[mod e])%A. | Proof. by rewrite /eqAmod subr0 unfold_in; case: ifPn => // /divff->. Qed. | Lemma | eqAmodm0 | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"divff",
"eqAmod",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodMr e :
{in Aint, forall z x y, x == y %[mod e] -> x * z == y * z %[mod e]}%A. | Proof.
move=> z Zz x y.
rewrite /eqAmod -mulrBl ![(e %| _)%A]unfold_in mulf_eq0 mulrAC.
by case: ifP => [_ -> // | _ Exy]; apply: rpredM.
Qed. | Lemma | eqAmodMr | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"apply",
"eqAmod",
"mulf_eq0",
"mulrAC",
"mulrBl",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodMl e :
{in Aint, forall z x y, x == y %[mod e] -> z * x == z * y %[mod e]}%A. | Proof. by move=> z Zz x y Exy; rewrite !(mulrC z) eqAmodMr. Qed. | Lemma | eqAmodMl | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"eqAmodMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodMl0 e : {in Aint, forall x, x * e == 0 %[mod e]}%A. | Proof. by move=> x Zx; rewrite -(mulr0 x) eqAmodMl. Qed. | Lemma | eqAmodMl0 | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"eqAmodMl",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodMr0 e : {in Aint, forall x, e * x == 0 %[mod e]}%A. | Proof. by move=> x Zx; rewrite /= mulrC eqAmodMl0. Qed. | Lemma | eqAmodMr0 | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"eqAmodMl0",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_addl_mul e : {in Aint, forall x y, x * e + y == y %[mod e]}%A. | Proof. by move=> x Zx y; rewrite -{2}[y]add0r eqAmodDr eqAmodMl0. Qed. | Lemma | eqAmod_addl_mul | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"add0r",
"eqAmodDr",
"eqAmodMl0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmodM e : {in Aint &, forall x1 y2 x2 y1,
x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 * y1 == x2 * y2 %[mod e]}%A. | Proof.
move=> x1 y2 Zx1 Zy2 x2 y1 eq_x /(eqAmodMl Zx1)/eqAmod_trans-> //.
exact: eqAmodMr.
Qed. | Lemma | eqAmodM | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint",
"eqAmodMl",
"eqAmodMr",
"eqAmod_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_rat :
{in Crat & &, forall e m n, (m == n %[mod e])%A = (m == n %[mod e])%C}. | Proof.
move=> e m n Qe Qm Qn; rewrite /eqCmod unfold_in /eqAmod unfold_in.
case: ifPn => // nz_e; apply/idP/idP=> [/Cint_rat_Aint | /Aint_Cint] -> //.
by rewrite rpred_div ?rpredB.
Qed. | Lemma | eqAmod_rat | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Aint_Cint",
"Cint_rat_Aint",
"Crat",
"apply",
"eqAmod",
"eqCmod",
"rpredB",
"rpred_div"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod0_rat : {in Crat &, forall e n, (n == 0 %[mod e])%A = (e %| n)%C}. | Proof. by move=> e n Qe Qn; rewrite /= eqAmod_rat /eqCmod ?subr0 ?Crat0. Qed. | Lemma | eqAmod0_rat | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"Crat",
"Crat0",
"eqAmod_rat",
"eqCmod",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod_nat (e m n : nat) : (m == n %[mod e])%A = (m == n %[mod e])%N. | Proof. by rewrite eqAmod_rat ?rpred_nat // eqCmod_nat. Qed. | Lemma | eqAmod_nat | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"eqAmod_rat",
"eqCmod_nat",
"nat",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqAmod0_nat (e m : nat) : (m == 0 %[mod e])%A = (e %| m)%N. | Proof. by rewrite eqAmod0_rat ?rpred_nat // dvdC_nat. Qed. | Lemma | eqAmod0_nat | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"dvdC_nat",
"eqAmod0_rat",
"nat",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderC x | :=
let p := minCpoly x in
oapp val 0 [pick n : 'I_(2 * size p ^ 2) | p == intrp 'Phi_n]. | Definition | orderC | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"intrp",
"minCpoly",
"pick",
"size",
"val"
] | Multiplicative order. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"#[ x ]" | := (orderC x) : C_scope. | Notation | #[ x ] | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"orderC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp_orderC x : x ^+ #[x]%C = 1. | Proof.
rewrite /orderC; case: pickP => //= [] [n _] /= /eqP Dp.
have n_gt0: (0 < n)%N.
rewrite lt0n; apply: contraTneq (size_minCpoly x) => n0.
by rewrite Dp n0 Cyclotomic0 rmorph1 size_poly1.
have [z prim_z] := C_prim_root_exists n_gt0.
rewrite prim_expr_order // -(root_cyclotomic prim_z).
by rewrite -Cintr_Cyclot... | Lemma | exp_orderC | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"C_prim_root_exists",
"Cintr_Cyclotomic",
"Cyclotomic0",
"apply",
"contraTneq",
"lt0n",
"n_gt0",
"orderC",
"pickP",
"prim_expr_order",
"prim_z",
"rmorph1",
"root_cyclotomic",
"root_minCpoly",
"size_minCpoly",
"size_poly1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_orderC x n : (#[x]%C %| n)%N = (x ^+ n == 1). | Proof.
apply/idP/eqP=> [|x_n_1]; first by apply: expr_dvd; apply: exp_orderC.
have [-> | n_gt0] := posnP n; first by rewrite dvdn0.
have [m prim_x m_dv_n] := prim_order_exists n_gt0 x_n_1.
have{n_gt0} m_gt0 := dvdn_gt0 n_gt0 m_dv_n; congr (_ %| n)%N: m_dv_n.
pose p := minCpoly x; have Dp: p = cyclotomic x m := minCpoly... | Lemma | dvdn_orderC | field | field/algnum.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"ssrnum",
"ssrint",
"archimedean",
"rat",
"finalg",
"zmodp",
"matrix"... | [
"C_prim_root_exists",
"Cintr_Cyclotomic",
"Cyclotomic0",
"add1n",
"addnAC",
"apply",
"big_filter",
"big_hasC",
"big_ind2",
"big_map",
"big_mkcond",
"big_seq",
"big_seq1",
"big_split",
"contraTneq",
"cyclotomic",
"dvdn0",
"dvdn_gt0",
"eqn_dvd",
"eqxx",
"exp_orderC",
"expnS",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fF | := (@GRing.formula F). | Notation | fF | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"formula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tF | := (@GRing.term F). | Notation | tF | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"term"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf f | := (GRing.qf_form f && GRing.rformula f). | Notation | qf | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"qf_form",
"rformula"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyF | := seq tF. | Definition | polyF | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"seq",
"tF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_simpl (f : fF) :
(qf f -> GRing.qf_form f) * (qf f -> GRing.rformula f). | Proof. by split=> /andP[]. Qed. | Lemma | qf_simpl | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"fF",
"qf",
"qf_form",
"rformula",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cps T | := ((T -> fF) -> fF). | Notation | cps | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"fF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ret T1 : T1 -> cps T1 | := fun x k => k x. | Definition | ret | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"cps"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bind T1 T2 (x : cps T1) (f : T1 -> cps T2) : cps T2 | :=
fun k => x (fun x => f x k). | Definition | bind | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"cps"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''let' x <- y ; z" | :=
(bind y (fun x => z)) (at level 99, x at level 0, z at level 200,
format "'[hv' ''let' x <- y ; '/' z ']'"). | Notation | ''let' x <- y ; z | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"bind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpsif T (c : fF) (t : T) (e : T) : cps T | :=
fun k => GRing.If c (k t) (k e). | Definition | cpsif | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"If",
"cps",
"fF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''if' c1 'then' c2 'else' c3" | := (cpsif c1%T c2%T c3%T)
(at level 200, right associativity, format
"'[hv ' ''if' c1 '/' '[' 'then' c2 ']' '/' '[' 'else' c3 ']' ']'"). | Notation | ''if' c1 'then' c2 'else' c3 | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"c1",
"c2",
"c3",
"cpsif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rterm | := GRing.rterm. | Notation | rterm | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eval_poly (e : seq F) pf | :=
if pf is c :: q then eval_poly e q * 'X + (eval e c)%:P else 0. | Fixpoint | eval_poly | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"eval",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpoly (p : polyF) | := all (@rterm F) p. | Definition | rpoly | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"all",
"polyF",
"rterm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sizeT : polyF -> cps nat | := (fix loop p :=
if p isn't c :: q then ret 0
else 'let n <- loop q;
if n is m.+1 then ret m.+2 else
'if (c == 0) then 0%N else 1%N). | Definition | sizeT | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"cps",
"nat",
"polyF",
"ret"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_red_cps T (x : cps T) (y : _ -> T) | :=
forall e k, qf_eval e (x k) = qf_eval e (k (y e)). | Definition | qf_red_cps | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"cps",
"qf_eval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ->_ e y" | := (qf_red_cps x (fun e => y))
(e name, at level 90, format "x ->_ e y"). | Notation | x ->_ e y | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"qf_red_cps"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_cps T D (x : cps T) | :=
forall k, (forall y, D y -> qf (k y)) -> qf (x k). | Definition | qf_cps | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"cps",
"qf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_cps_ret T D (x : T) : D x -> qf_cps D (ret x). | Proof. move=> ??; exact. Qed. | Lemma | qf_cps_ret | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"qf_cps",
"ret"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_cps_bind T1 D1 T2 D2 (x : cps T1) (f : T1 -> cps T2) :
qf_cps D1 x -> (forall x, D1 x -> qf_cps D2 (f x)) -> qf_cps D2 (bind x f). | Proof. by move=> xP fP k kP /=; apply: xP => y ?; apply: fP. Qed. | Lemma | qf_cps_bind | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"apply",
"bind",
"cps",
"fP",
"qf_cps"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qf_cps_if T D (c : fF) (t : T) (e : T) : qf c -> D t -> D e ->
qf_cps D ('if c then t else e). | Proof.
move=> qfc Dt De k kP /=; have [qft qfe] := (kP _ Dt, kP _ De).
by do !rewrite qf_simpl //.
Qed. | Lemma | qf_cps_if | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"fF",
"qf",
"qf_cps",
"qf_simpl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sizeTP (pf : polyF) : sizeT pf ->_e size (eval_poly e pf). | Proof.
elim: pf=> [|c qf qfP /=]; first by rewrite /= size_poly0.
move=> e k; rewrite size_MXaddC qfP -(size_poly_eq0 (eval_poly _ _)).
by case: (size (eval_poly e qf))=> //=; case: eqP; rewrite // orbF.
Qed. | Lemma | sizeTP | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"eval_poly",
"polyF",
"qf",
"size",
"sizeT",
"size_MXaddC",
"size_poly0",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sizeT_qf (p : polyF) : rpoly p -> qf_cps xpredT (sizeT p). | Proof.
elim: p => /= [_|c p ihp /andP[rc rq]]; first exact: qf_cps_ret.
apply: qf_cps_bind; first exact: ihp.
move=> [|n] //= _; last exact: qf_cps_ret.
by apply: qf_cps_if; rewrite //= rc.
Qed. | Lemma | sizeT_qf | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"apply",
"last",
"polyF",
"qf_cps",
"qf_cps_bind",
"qf_cps_if",
"qf_cps_ret",
"rpoly",
"sizeT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isnull (p : polyF) : cps bool | :=
'let n <- sizeT p; ret (n == 0). | Definition | isnull | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"cps",
"polyF",
"ret",
"sizeT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isnullP (p : polyF) : isnull p ->_e (eval_poly e p == 0). | Proof. by move=> e k; rewrite sizeTP size_poly_eq0. Qed. | Lemma | isnullP | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"eval_poly",
"isnull",
"polyF",
"sizeTP",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isnull_qf (p : polyF) : rpoly p -> qf_cps xpredT (isnull p). | Proof.
move=> rp; apply: qf_cps_bind; first exact: sizeT_qf.
by move=> ? _; apply: qf_cps_ret.
Qed. | Lemma | isnull_qf | field | field/closed_field.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"fintype",
"generic_quotient",
"bigop",
"ssralg",
"poly",
"polydiv",
"matrix",
"mxpoly",
"countalg",
"ring_quotient",
"GRing.Theory",
"Pdiv.Ring",
"PreClosedField",
"C... | [
"apply",
"isnull",
"polyF",
"qf_cps",
"qf_cps_bind",
"qf_cps_ret",
"rpoly",
"sizeT_qf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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