fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
aspaceOverE := ASpace (aspaceOver_suproof E). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | aspaceOver | |
dim_vspaceOverM : (F * M <= M)%VS -> \dim (vspaceOver M) = \dim_F M.
Proof.
move=> modM; have [] := field_module_semisimple modM.
set n := \dim_F M => b [Mb nz_b] [defM dx_b].
suff: basis_of (vspaceOver M) b by apply: size_basis.
apply/andP; split.
rewrite eqEsubv; apply/andP; split; apply/span_subvP=> u.
by rewr... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_vspaceOver | |
dim_aspaceOverE : (F <= E)%VS -> \dim (vspaceOver E) = \dim_F E.
Proof. by rewrite -sup_field_module; apply: dim_vspaceOver. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_aspaceOver | |
vspaceOverPV_F :
{V | [/\ V_F = vspaceOver V, (F * V <= V)%VS & V_F =i V]}.
Proof.
pose V := (F * <<vbasis V_F : seq L>>)%VS.
have idV: (F * V)%VS = V by rewrite prodvA prodv_id.
suffices defVF: V_F = vspaceOver V.
by exists V; split=> [||u]; rewrite ?defVF ?mem_vspaceOver ?idV.
apply/vspaceP=> v; rewrite mem_vspac... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | vspaceOverP | |
aspaceOverP(E_F : {subfield L_F}) :
{E | [/\ E_F = aspaceOver E, (F <= E)%VS & E_F =i E]}.
Proof.
have [V [defEF modV memV]] := vspaceOverP E_F.
have algE: has_algid V && (V * V <= V)%VS.
rewrite has_algid1; last by rewrite -memV mem1v.
by apply/prodvP=> u v; rewrite -!memV; apply: memvM.
by exists (ASpace algE);... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | aspaceOverP | |
baseFieldType: Type := L. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | baseFieldType | |
L0:= baseFieldType.
HB.instance Definition _ := GRing.Field.on L0. | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | L0 | |
baseField_scale(a : F0) (u : L0) : L0 := in_alg F a *: u.
Local Infix "*F0:" := baseField_scale (at level 40).
Fact baseField_scaleA a b u : a *F0: (b *F0: u) = (a * b) *F0: u.
Proof. by rewrite [_ *F0: _]scalerA -rmorphM. Qed.
Fact baseField_scale1 u : 1 *F0: u = u.
Proof. by rewrite /(1 *F0: u) rmorph1 scale1r. Qed.
... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | baseField_scale | |
baseField_scaleEa (u : L) : a *: (u : L0) = a%:A *: u.
Proof. by []. Qed.
Fact baseField_scaleAl a (u v : L0) : a *F0: (u * v) = (a *F0: u) * v.
Proof. exact: scalerAl. Qed.
HB.instance Definition _ := GRing.Lmodule_isLalgebra.Build _ L0
baseField_scaleAl.
Fact baseField_scaleAr a u v : a *F0: (u * v) = u * (a *F0: v... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | baseField_scaleE | |
Definition_ := baseField_vectMixin.
Let F0ZEZ a x v : a *: ((x *: v : L) : L0) = (a *: x) *: v.
Proof. by rewrite [a *: _]scalerA -scalerAl mul1r. Qed.
Let baseVspace_basis V : seq L0 :=
[seq tnth bF ij.2 *: tnth (vbasis V) ij.1 | ij : 'I_(\dim V) * 'I_n]. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | Definition | |
baseVspaceV := <<baseVspace_basis V>>%VS. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | baseVspace | |
mem_baseVspaceV : baseVspace V =i V.
Proof.
move=> y; apply/idP/idP=> [/coord_span->|/coord_vbasis->]; last first.
apply: memv_suml => i _; rewrite (coordF (coord _ i (y : L))) scaler_suml -/n.
apply: memv_suml => j _; rewrite -/bF -F0ZEZ memvZ ?memv_span // -!tnth_nth.
by apply/imageP; exists (i, j). | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | mem_baseVspace | |
dim_baseVspaceV : \dim (baseVspace V) = (\dim V * n)%N.
Proof.
pose bV0 := baseVspace_basis V; set m := \dim V in bV0 *.
suffices /size_basis->: basis_of (baseVspace V) bV0.
by rewrite card_prod !card_ord.
rewrite /basis_of eqxx.
apply/freeP=> s sb0 k; rewrite -(enum_valK k); case/enum_val: k => i j.
have free_baseP ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_baseVspace | |
baseAspaceE := ASpace (baseAspace_suproof E). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | baseAspace | |
refBaseField_unlockable:= Unlockable refBaseField.unlock. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | refBaseField_unlockable | |
F1:= (refBaseField L). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | F1 | |
F1unlock:= refBaseField.unlock. | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | F1unlock | |
L0:= (baseFieldType L).
Let n := \dim {:F}.
Let bF : n.-tuple F := vbasis {:F}.
Let coordF (x : F) := (coord_vbasis (memvf x)). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | L0 | |
dim_refBaseField: \dim F1 = n.
Proof. by rewrite F1unlock dim_baseVspace dimv1 mul1n. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | dim_refBaseField | |
baseVspace_moduleV (V0 := baseVspace V) : (F1 * V0 <= V0)%VS.
Proof.
apply/prodvP=> u v; rewrite F1unlock !mem_baseVspace => /vlineP[x ->] Vv.
by rewrite -(@scalerAl F L) mul1r; apply: memvZ.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | baseVspace_module | |
sub_baseField(E : {subfield L}) : (F1 <= baseVspace E)%VS.
Proof. by rewrite -sup_field_module baseVspace_module. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | sub_baseField | |
vspaceOver_refBaseV : vspaceOver F1 (baseVspace V) =i V.
Proof.
move=> v; rewrite mem_vspaceOver field_module_eq ?baseVspace_module //.
by rewrite mem_baseVspace.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | vspaceOver_refBase | |
module_baseVspaceM0 :
(F1 * M0 <= M0)%VS -> {V | M0 = baseVspace V & M0 =i V}.
Proof.
move=> modM0; pose V := <<vbasis (vspaceOver F1 M0) : seq L>>%VS.
suffices memM0: M0 =i V.
by exists V => //; apply/vspaceP=> v; rewrite mem_baseVspace memM0.
move=> v; rewrite -{1}(field_module_eq modM0) -(mem_vspaceOver M0) {}/V... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | module_baseVspace | |
module_baseAspace(E0 : {subfield L0}) :
(F1 <= E0)%VS -> {E | E0 = baseAspace E & E0 =i E}.
Proof.
rewrite -sup_field_module => /module_baseVspace[E defE0 memE0].
suffices algE: is_aspace E by exists (ASpace algE); first apply: val_inj.
rewrite /is_aspace has_algid1 -?memE0 ?mem1v //.
by apply/prodvP=> u v; rewrite -... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | module_baseAspace | |
base_vspaceOverV : baseVspace (vspaceOver F V) =i (F * V)%VS.
Proof. by move=> v; rewrite mem_baseVspace mem_vspaceOver. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | base_vspaceOver | |
base_moduleOverV : (F * V <= V)%VS -> baseVspace (vspaceOver F V) =i V.
Proof. by move=> /field_module_eq defV v; rewrite base_vspaceOver defV. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | base_moduleOver | |
base_aspaceOver(E : {subfield L}) :
(F <= E)%VS -> baseVspace (vspaceOver F E) =i E.
Proof. by rewrite -sup_field_module; apply: base_moduleOver. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | base_aspaceOver | |
equiv_subfextx y := (iotaFz x == iotaFz y).
Fact equiv_subfext_is_equiv : equiv_class_of equiv_subfext.
Proof. by rewrite /equiv_subfext; split=> x // y w /eqP->. Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | equiv_subfext | |
equiv_subfext_equiv:= EquivRelPack equiv_subfext_is_equiv. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | equiv_subfext_equiv | |
equiv_subfext_encModRel:= defaultEncModRel equiv_subfext. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | equiv_subfext_encModRel | |
subFExtend:= {eq_quot equiv_subfext}.
HB.instance Definition _ := Choice.on subFExtend.
HB.instance Definition _ := Quotient.on subFExtend.
HB.instance Definition _ : EqQuotient 'M[F]_(1, n) equiv_subfext subFExtend :=
EqQuotient.on subFExtend. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subFExtend | |
subfx_inj:= lift_fun1 subFExtend iotaFz.
Fact pi_subfx_inj : {mono \pi : x / iotaFz x >-> subfx_inj x}.
Proof.
unlock subfx_inj => x; apply/eqP; rewrite -/(equiv_subfext _ x).
by rewrite -eqmodE reprK.
Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inj | |
pi_subfx_inj_morph:= PiMono1 pi_subfx_inj.
Let iotaPz_repr x : iotaPz (rVpoly (repr (\pi_(subFExtend) x))) = iotaFz x.
Proof. by rewrite -/(iotaFz _) -!pi_subfx_inj reprK. Qed. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | pi_subfx_inj_morph | |
subfext0:= lift_cst subFExtend 0. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext0 | |
subfext0_morph:= PiConst subfext0. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext0_morph | |
subfext_add:= lift_op2 subFExtend +%R.
Fact pi_subfext_add : {morph \pi : x y / x + y >-> subfext_add x y}.
Proof.
unlock subfext_add => x y /=; apply/eqmodP/eqP.
by rewrite /iotaFz !linearD /= !iotaPz_repr.
Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext_add | |
pi_subfx_add_morph:= PiMorph2 pi_subfext_add. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | pi_subfx_add_morph | |
subfext_opp:= lift_op1 subFExtend -%R.
Fact pi_subfext_opp : {morph \pi : x / - x >-> subfext_opp x}.
Proof.
unlock subfext_opp => y /=; apply/eqmodP/eqP.
by rewrite /iotaFz !linearN /= !iotaPz_repr.
Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext_opp | |
pi_subfext_opp_morph:= PiMorph1 pi_subfext_opp.
Fact addfxA : associative subfext_add.
Proof. by move=> x y t; rewrite -[x]reprK -[y]reprK -[t]reprK !piE addrA. Qed.
Fact addfxC : commutative subfext_add.
Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE addrC. Qed.
Fact add0fx : left_id subfext0 subfext_add.
Proo... | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | pi_subfext_opp_morph | |
subfx_mul_rep(x y : 'rV[F]_n) : 'rV[F]_n :=
poly_rV ((rVpoly x) * (rVpoly y) %% p0). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_mul_rep | |
subfext_mul:= lift_op2 subFExtend subfx_mul_rep.
Fact pi_subfext_mul :
{morph \pi : x y / subfx_mul_rep x y >-> subfext_mul x y}.
Proof.
unlock subfext_mul => x y /=; apply/eqmodP/eqP.
by rewrite /iotaFz !poly_rV_modp_K !iotaPz_modp !rmorphM /= !iotaPz_repr.
Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext_mul | |
pi_subfext_mul_morph:= PiMorph2 pi_subfext_mul. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | pi_subfext_mul_morph | |
subfext1:= lift_cst subFExtend (poly_rV 1). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext1 | |
subfext1_morph:= PiConst subfext1.
Fact mulfxA : associative (subfext_mul).
Proof.
elim/quotW=> x; elim/quotW=> y; elim/quotW=> w; rewrite !piE /subfx_mul_rep.
by rewrite !poly_rV_modp_K [_ %% p0 * _]mulrC !modp_mul // mulrA mulrC.
Qed.
Fact mulfxC : commutative subfext_mul.
Proof.
by elim/quotW=> x; elim/quotW=> y; re... | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext1_morph | |
subfx_poly_inv(q : {poly F}) : {poly F} :=
if iotaPz q == 0 then 0 else
let r := gdcop q p0 in let: (u, v) := egcdp q r in
((u * q + v * r)`_0)^-1 *: u.
Let subfx_poly_invE q : iotaPz (subfx_poly_inv q) = (iotaPz q)^-1.
Proof.
rewrite /subfx_poly_inv.
have [-> | nzq] := eqVneq; first by rewrite rmorph0 invr0.
rew... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_poly_inv | |
subfx_inv_rep(x : 'rV[F]_n) : 'rV[F]_n :=
poly_rV (subfx_poly_inv (rVpoly x) %% p0). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inv_rep | |
subfext_inv:= lift_op1 subFExtend subfx_inv_rep.
Fact pi_subfext_inv : {morph \pi : x / subfx_inv_rep x >-> subfext_inv x}.
Proof.
unlock subfext_inv => x /=; apply/eqmodP/eqP; rewrite /iotaFz.
by rewrite 2!{1}poly_rV_modp_K 2!{1}iotaPz_modp !subfx_poly_invE iotaPz_repr.
Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfext_inv | |
pi_subfext_inv_morph:= PiMorph1 pi_subfext_inv.
Fact subfx_fieldAxiom : forall x, x != 0 -> subfext_inv x * x = 1.
Proof.
elim/quotW=> x; apply: contraNeq; rewrite !piE /equiv_subfext /iotaFz !linear0.
apply: contraR => nz_x; rewrite poly_rV_K ?size_poly1 // !poly_rV_modp_K.
by rewrite iotaPz_modp rmorph1 rmorphM /= io... | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | pi_subfext_inv_morph | |
subfx_inj_is_additive:= subfx_inj_is_zmod_morphism.
Fact subfx_inj_is_monoid_morphism : monoid_morphism subfx_inj.
Proof.
split; first by rewrite piE /iotaFz poly_rV_K ?rmorph1 ?size_poly1.
elim/quotW=> x; elim/quotW=> y; rewrite !piE /subfx_mul_rep /iotaFz.
by rewrite poly_rV_modp_K iotaPz_modp rmorphM.
Qed.
#[warning... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inj_is_additive | |
subfx_inj_is_multiplicative:=
(fun g => (g.2,g.1)) subfx_inj_is_monoid_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build subFExtend L subfx_inj
subfx_inj_is_zmod_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build subFExtend L subfx_inj
subfx_inj_is_monoid_morphism. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inj_is_multiplicative | |
subfx_eval:= lift_embed subFExtend (fun q => poly_rV (q %% p0)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_eval | |
subfx_eval_morph:= PiEmbed subfx_eval. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_eval_morph | |
subfx_root:= subfx_eval 'X. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_root | |
subfx_eval_is_zmod_morphism: zmod_morphism subfx_eval.
Proof. by move=> x y; apply/eqP; rewrite piE -linearB modpD modNp. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `subfx_eval_is_zmod_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_eval_is_zmod_morphism | |
subfx_eval_is_additive:= subfx_eval_is_zmod_morphism. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_eval_is_additive | |
subfx_eval_is_monoid_morphism: monoid_morphism subfx_eval.
Proof.
split=> [|x y]; apply/eqP; rewrite piE.
by rewrite modp_small // size_poly1 -subn_gt0 subn1.
by rewrite /subfx_mul_rep !poly_rV_modp_K !(modp_mul, mulrC _ y).
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
n... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_eval_is_monoid_morphism | |
subfx_eval_is_multiplicative:=
(fun g => (g.2,g.1)) subfx_eval_is_monoid_morphism.
HB.instance Definition _ :=
GRing.isZmodMorphism.Build {poly F} subFExtend subfx_eval subfx_eval_is_zmod_morphism.
HB.instance Definition _ :=
GRing.isMonoidMorphism.Build {poly F} subFExtend subfx_eval
subfx_eval_is_monoid_mor... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_eval_is_multiplicative | |
inj_subfx:= (subfx_eval \o polyC).
HB.instance Definition _ := GRing.RMorphism.on inj_subfx. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | inj_subfx | |
subfxEx: exists p, x = subfx_eval p.
Proof.
elim/quotW: x => x; exists (rVpoly x); apply/eqP; rewrite piE /equiv_subfext.
by rewrite /iotaFz poly_rV_modp_K iotaPz_modp.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfxE | |
subfx_scalea x := inj_subfx a * x.
Fact subfx_scalerA a b x :
subfx_scale a (subfx_scale b x) = subfx_scale (a * b) x.
Proof. by rewrite /subfx_scale rmorphM mulrA. Qed.
Fact subfx_scaler1r : left_id 1 subfx_scale.
Proof. by move=> x; rewrite /subfx_scale rmorph1 mul1r. Qed.
Fact subfx_scalerDr : right_distributive s... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_scale | |
subfx_inj_evalq : subfx_inj (subfx_eval q) = (q^iota).[z].
Proof.
by rewrite piE /iotaFz poly_rV_modp_K iotaPz_modp /iotaPz /z0 /wf_p nz_p pz0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inj_eval | |
subfx_inj_root: subfx_inj subfx_root = z.
Proof. by rewrite subfx_inj_eval // map_polyX hornerX. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inj_root | |
subfx_injZb x : subfx_inj (b *: x) = iota b * subfx_inj x.
Proof. by rewrite rmorphM /= subfx_inj_eval // map_polyC hornerC. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_injZ | |
subfx_inj_baseb : subfx_inj b%:A = iota b.
Proof. by rewrite subfx_injZ rmorph1 mulr1. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_inj_base | |
subfxErootx : {q | x = (map_poly (in_alg subFExtend) q).[subfx_root]}.
Proof.
have /sig_eqW[q ->] := subfxE x; exists q.
apply: (fmorph_inj subfx_inj).
rewrite -horner_map /= subfx_inj_root subfx_inj_eval //.
by rewrite -map_poly_comp (eq_map_poly subfx_inj_base).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfxEroot | |
subfx_irreducibleP:
(forall q, root q^iota z -> q != 0 -> size p <= size q) <-> irreducible_poly p.
Proof.
split=> [min_p | irr_p q qz0 nz_q].
split=> [|q nonC_q q_dv_p].
by rewrite -(size_map_poly iota) (root_size_gt1 _ pz0) ?map_poly_eq0.
have /dvdpP[r Dp] := q_dv_p; rewrite -dvdp_size_eqp // eqn_leq dvdp_le... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | subfx_irreducibleP | |
min_subfx_vect: Vector.axiom (size p).-1 subFExtend.
Proof.
move/subfx_irreducibleP: irr_p => /=/(_ nz_p) min_p; set d := (size p).-1.
have Dd: d.+1 = size p by rewrite polySpred.
pose Fz2v x : 'rV_d := poly_rV (sval (sig_eqW (subfxE x)) %% p).
pose vFz : 'rV_d -> subFExtend := subfx_eval \o rVpoly.
have FLinj: injecti... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | min_subfx_vect | |
SubfxVect:= Lmodule_hasFinDim.Build _ subFExtend min_subfx_vect. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | SubfxVect | |
SubFieldExtType: fieldExtType F := HB.pack subFExtend SubfxVect. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | SubFieldExtType | |
irredp_FAdjoin(F : fieldType) (p : {poly F}) :
irreducible_poly p ->
{L : fieldExtType F & \dim {:L} = (size p).-1 &
{z | root (map_poly (in_alg L) p) z & <<1; z>>%VS = fullv}}.
Proof.
case=> p_gt1 irr_p; set n := (size p).-1; pose vL : vectType F := 'rV_n.
have Dn: n.+1 = size p := ltn_predK p_gt1.
have nz_p... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg countalg",
"From mathcomp Require Import finalg zmodp matrix vector falgebra poly polydiv",
"From mathcomp Require Import... | field/fieldext.v | irredp_FAdjoin | |
finNzRing_nontrivial: [set: R] != 1%g.
Proof. by apply/trivgPn; exists 1; rewrite ?inE ?oner_neq0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finNzRing_nontrivial | |
finNzRing_gt1: 1 < #|R|.
Proof. by rewrite -cardsT cardG_gt1 finNzRing_nontrivial. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finNzRing_gt1 | |
finRing_nontrivial:= (finNzRing_nontrivial) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Use finNzRing_gt1 instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finRing_nontrivial | |
finRing_gt1:= (finNzRing_gt1) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finRing_gt1 | |
card_finField_unit: #|[set: {unit F}]| = #|F|.-1.
Proof.
by rewrite -(cardC1 0) cardsT card_sub; apply: eq_card => x; rewrite unitfE.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_finField_unit | |
finField_unitx (nz_x : x != 0) :=
FinRing.unit F (etrans (unitfE x) nz_x). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finField_unit | |
expf_cardx : x ^+ #|F| = x :> F.
Proof.
rewrite -[RHS]mulr1 -(ltn_predK (finNzRing_gt1 F)) exprS.
apply/eqP; rewrite -subr_eq0 -mulrBr mulf_eq0 subr_eq0 -implyNb -unitfE.
apply/implyP=> Ux; rewrite -(val_unitX _ (Sub x _)) -val_unit1 val_eqE.
by rewrite -order_dvdn -card_finField_unit order_dvdG ?inE.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | expf_card | |
finField_genPoly: 'X^#|F| - 'X = \prod_x ('X - x%:P) :> {poly F}.
Proof.
set n := #|F|; set oppX := - 'X; set pF := LHS.
have le_oppX_n: size oppX <= n by rewrite size_polyN size_polyX finNzRing_gt1.
have: size pF = (size (enum F)).+1 by rewrite -cardE size_polyDl size_polyXn.
move/all_roots_prod_XsubC->; last by rewri... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finField_genPoly | |
finPcharP: {p | prime p & p \in [pchar F]}.
Proof.
pose e := exponent [set: F]; have e_gt0: e > 0 by apply: exponent_gt0.
have: e%:R == 0 :> F by rewrite -zmodXgE expg_exponent // inE.
by case/natf0_pchar/sigW=> // p pcharFp; exists p; rewrite ?(pcharf_prime pcharFp).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finPcharP | |
finField_is_abelem: is_abelem [set: F].
Proof.
have [p pr_p pcharFp] := finPcharP.
by apply/is_abelemP; exists p; last apply: fin_ring_pchar_abelem.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finField_is_abelem | |
card_finPcharPp n : #|F| = (p ^ n)%N -> prime p -> p \in [pchar F].
Proof.
move=> oF pr_p; rewrite inE pr_p -order_dvdn.
rewrite (abelem_order_p finField_is_abelem) ?inE ?oner_neq0 //=.
have n_gt0: n > 0 by rewrite -(ltn_exp2l _ _ (prime_gt1 pr_p)) -oF finNzRing_gt1.
by rewrite cardsT oF -(prednK n_gt0) pdiv_pfactor.
Q... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_finPcharP | |
finCharP:= (finPcharP) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use card_finPcharP instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finCharP | |
card_finCharP:= (card_finPcharP) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_finCharP | |
card_vspace(V : {vspace vT}) : #|V| = (#|F| ^ \dim V)%N.
Proof.
set n := \dim V; pose V2rV v := \row_i coord (vbasis V) i v.
pose rV2V (rv : 'rV_n) := \sum_i rv 0 i *: (vbasis V)`_i.
have rV2V_K: cancel rV2V V2rV.
have freeV: free (vbasis V) := basis_free (vbasisP V).
by move=> rv; apply/rowP=> i; rewrite mxE coord... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_vspace | |
card_vspacef: #|{: vT}%VS| = #|T|.
Proof. by apply: eq_card => v; rewrite (@memvf _ vT). Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_vspacef | |
card_vspace1: #|(1%VS : {vspace aT})| = #|F|.
Proof. by rewrite card_vspace (dimv1 aT). Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_vspace1 | |
finvect_type(vT : Type) : predArgType := vT. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finvect_type | |
Definition_ := Vector.on fvT.
HB.instance Definition _ := isCountable.Build fvT
(pcan_pickleK (can_pcan VectorInternalTheory.v2rK)).
HB.instance Definition _ := isFinite.Build fvT
(pcan_enumP (can_pcan (VectorInternalTheory.v2rK : @cancel _ fvT _ _))). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
Definition_ (F : finFieldType) (aT : falgType F) :=
Falgebra.on (finvect_type aT). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
Definition_ := FieldExt.on ffT. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
ffT_splitting_subproof: SplittingField.axiom ffT.
Proof.
exists ('X^#|ffT| - 'X).
by rewrite (@rpredB {poly fT}) 1?rpredX ?polyOverX.
exists (enum ffT); first by rewrite big_enum finField_genPoly eqpxx.
by apply/vspaceP=> x; rewrite memvf seqv_sub_adjoin ?mem_enum.
Qed.
HB.instance Definition _ := FieldExt_isSplittin... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | ffT_splitting_subproof | |
FinSplittingFieldType(F : finFieldType) (fT : fieldExtType F) :=
HB.pack_for (splittingFieldType F) fT (SplittingField.on (finvect_type fT)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | FinSplittingFieldType | |
FinFieldExtType(F : finFieldType) (fT : fieldExtType F) :=
HB.pack_for finFieldType (FinSplittingFieldType fT)
(FinRing.Field.on (finvect_type fT)).
Arguments FinSplittingFieldType : clear implicits. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | FinFieldExtType | |
pPrimeCharTypeof p \in [pchar R0] : predArgType := R0.
Hypothesis pcharRp : p \in [pchar R0].
Local Notation R := (pPrimeCharType pcharRp).
Implicit Types (a b : 'F_p) (x y : R).
HB.instance Definition _ := GRing.NzRing.on R. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pPrimeCharType | |
pprimeChar_scalea x := a%:R * x.
Local Infix "*p':" := pprimeChar_scale (at level 40).
Let natrFp n : (inZp n : 'F_p)%:R = n%:R :> R.
Proof.
rewrite [in RHS](divn_eq n p) natrD mulrnA (mulrn_pchar pcharRp) add0r.
by rewrite /= (Fp_cast (pcharf_prime pcharRp)).
Qed. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scale | |
pprimeChar_scaleAa b x : a *p': (b *p': x) = (a * b) *p': x.
Proof. by rewrite /pprimeChar_scale mulrA -natrM natrFp. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scaleA | |
pprimeChar_scale1: left_id 1 pprimeChar_scale.
Proof. by move=> x; rewrite /pprimeChar_scale mul1r. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scale1 | |
pprimeChar_scaleDr: right_distributive pprimeChar_scale +%R.
Proof. by move=> a x y /=; rewrite /pprimeChar_scale mulrDr. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scaleDr | |
pprimeChar_scaleDlx : {morph pprimeChar_scale^~ x: a b / a + b}.
Proof. by move=> a b; rewrite /pprimeChar_scale natrFp natrD mulrDl. Qed.
HB.instance Definition _ := GRing.Zmodule_isLmodule.Build 'F_p R
pprimeChar_scaleA pprimeChar_scale1 pprimeChar_scaleDr pprimeChar_scaleDl. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scaleDl | |
pprimeChar_scaleAl(a : 'F_p) (u v : R) : a *: (u * v) = (a *: u) * v.
Proof. by apply: mulrA. Qed.
HB.instance Definition _ := GRing.Lmodule_isLalgebra.Build 'F_p R
pprimeChar_scaleAl. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scaleAl |
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