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 |
|---|---|---|---|---|---|---|
eqCmod_addl_mule : {in Num.int, forall x y, x * e + y == y %[mod e]}%C.
Proof. by move=> x Zx y; rewrite -{2}[y]add0r eqCmodDr eqCmodMl0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | eqCmod_addl_mul | |
eqCmodMe : {in Num.int & Num.int, forall x1 y2 x2 y1,
x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 * y1 == x2 * y2 %[mod e]}%C.
Proof.
move=> x1 y2 Zx1 Zy2 x2 y1 eq_x /(eqCmodMl Zx1)/eqCmod_trans-> //.
exact: eqCmodMr.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | eqCmodM | |
ratCK: cancel QtoC CtoQ.
Proof. by rewrite /getCrat; case: getCrat_subproof. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | ratCK | |
getCratK: {in Crat, cancel CtoQ QtoC}.
Proof. by move=> x /eqP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | getCratK | |
Crat_rat(a : rat) : QtoC a \in Crat.
Proof. by rewrite unfold_in ratCK. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Crat_rat | |
CratPx : reflect (exists a, x = QtoC a) (x \in Crat).
Proof.
by apply: (iffP eqP) => [<- | [a ->]]; [exists (CtoQ x) | rewrite ratCK].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | CratP | |
Crat0: 0 \in Crat. Proof. by apply/CratP; exists 0; rewrite rmorph0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Crat0 | |
Crat1: 1 \in Crat. Proof. by apply/CratP; exists 1; rewrite rmorph1. Qed.
#[local] Hint Resolve Crat0 Crat1 : core.
Fact Crat_divring_closed : divring_closed Crat.
Proof.
split=> // _ _ /CratP[x ->] /CratP[y ->].
by rewrite -rmorphB Crat_rat.
by rewrite -fmorph_div Crat_rat.
Qed.
HB.instance Definition _ := GRing.isD... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Crat1 | |
rpred_Crat(S : divringClosed algC) : {subset Crat <= S}.
Proof. by move=> _ /CratP[a ->]; apply: rpred_rat. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | rpred_Crat | |
conj_Cratz : z \in Crat -> z^* = z.
Proof. by move/getCratK <-; rewrite fmorph_div !rmorph_int. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | conj_Crat | |
Creal_Crat: {subset Crat <= Creal}.
Proof. by move=> x /conj_Crat/CrealP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Creal_Crat | |
Cint_rata : (QtoC a \in Num.int) = (a \in Num.int).
Proof.
apply/idP/idP=> [Za | /numqK <-]; last by rewrite rmorph_int.
apply/intrP; exists (Num.floor (QtoC a)); apply: (can_inj ratCK).
by rewrite rmorph_int floorK.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Cint_rat | |
minCpolyPx :
{p : {poly rat} | minCpoly x = pQtoC p /\ p \is monic
& forall q, root (pQtoC q) x = (p %| q)%R}.
Proof. by rewrite /minCpoly; case: (minCpoly_subproof x) => p; exists p. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | minCpolyP | |
minCpoly_monicx : minCpoly x \is monic.
Proof. by have [p [-> mon_p] _] := minCpolyP x; rewrite map_monic. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | minCpoly_monic | |
minCpoly_eq0x : (minCpoly x == 0) = false.
Proof. exact/negbTE/monic_neq0/minCpoly_monic. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | minCpoly_eq0 | |
root_minCpolyx : root (minCpoly x) x.
Proof. by have [p [-> _] ->] := minCpolyP x. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | root_minCpoly | |
size_minCpolyx : (1 < size (minCpoly x))%N.
Proof. by apply: root_size_gt1 (root_minCpoly x); rewrite ?minCpoly_eq0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | size_minCpoly | |
aut_Cratnu : {in Crat, nu =1 id}.
Proof. by move=> _ /CratP[a ->]; apply: fmorph_rat. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | aut_Crat | |
Crat_autnu x : (nu x \in Crat) = (x \in Crat).
Proof.
apply/idP/idP=> /CratP[a] => [|->]; last by rewrite fmorph_rat Crat_rat.
by rewrite -(fmorph_rat nu) => /fmorph_inj->; apply: Crat_rat.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Crat_aut | |
algC_invaut_subproofnu x : {y | nu y = x}.
Proof.
have [r Dp] := closed_field_poly_normal (minCpoly x).
suffices /mapP/sig2_eqW[y _ ->]: x \in map nu r by exists y.
rewrite -root_prod_XsubC; congr (root _ x): (root_minCpoly x).
have [q [Dq _] _] := minCpolyP x; rewrite Dq -(eq_map_poly (fmorph_rat nu)).
rewrite (map_po... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_invaut_subproof | |
algC_invautnu x := sval (algC_invaut_subproof nu x). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_invaut | |
algC_invautKnu : cancel (algC_invaut nu) nu.
Proof. by move=> x; rewrite /algC_invaut; case: algC_invaut_subproof. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_invautK | |
algC_autKnu : cancel nu (algC_invaut nu).
Proof. exact: inj_can_sym (algC_invautK nu) (fmorph_inj nu). Qed.
Fact algC_invaut_is_zmod_morphism nu : zmod_morphism (algC_invaut nu).
Proof. exact: can2_zmod_morphism (algC_autK nu) (algC_invautK nu). Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathc... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_autK | |
algC_invaut_is_additive:= algC_invaut_is_zmod_morphism.
Fact algC_invaut_is_monoid_morphism nu : monoid_morphism (algC_invaut nu).
Proof. exact: can2_monoid_morphism (algC_autK nu) (algC_invautK nu). Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `algC_invaut_is_mo... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_invaut_is_additive | |
algC_invaut_is_multiplicativenu :=
(fun g => (g.2,g.1)) (algC_invaut_is_monoid_morphism nu).
HB.instance Definition _ (nu : {rmorphism algC -> algC}) :=
GRing.isZmodMorphism.Build algC algC (algC_invaut nu)
(algC_invaut_is_zmod_morphism nu).
HB.instance Definition _ (nu : {rmorphism algC -> algC}) :=
GRing.is... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_invaut_is_multiplicative | |
minCpoly_autnu x : minCpoly (nu x) = minCpoly x.
Proof.
wlog suffices dvd_nu: nu x / (minCpoly x %| minCpoly (nu x))%R.
apply/eqP; rewrite -eqp_monic ?minCpoly_monic //; apply/andP; split=> //.
by rewrite -{2}(algC_autK nu x) dvd_nu.
have [[q [Dq _] min_q] [q1 [Dq1 _] _]] := (minCpolyP x, minCpolyP (nu x)).
rewrite... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | minCpoly_aut | |
Cchar:= (Cpchar) (only parsing).
#[global] Hint Resolve Crat0 Crat1 dvdC0 dvdC_refl eqCmod_refl eqCmodm0 : core.
Local Notation "p ^^ f" := (map_poly f p)
(at level 30, f at level 30, format "p ^^ f"). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Cchar | |
algR:= in_algR {algRval :> algC; algRvalP : algRval \is Creal}.
HB.instance Definition _ := [isSub for algRval].
HB.instance Definition _ := [Countable of algR by <:].
HB.instance Definition _ := [SubChoice_isSubIntegralDomain of algR by <:].
HB.instance Definition _ := [SubIntegralDomain_isSubField of algR by <:].
HB.... | Record | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR | |
total_algR: total (<=%O : rel (algR : porderType _)).
Proof. by move=> x y; apply/real_leVge/valP/valP. Qed.
HB.instance Definition _ := Order.POrder_isTotal.Build _ algR total_algR. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | total_algR | |
algRval_is_zmod_morphism: zmod_morphism algRval. Proof. by []. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `algRval_is_zmod_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algRval_is_zmod_morphism | |
algRval_is_additive:= algRval_is_zmod_morphism. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algRval_is_additive | |
algRval_is_monoid_morphism: monoid_morphism algRval. Proof. by []. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `algRval_is_monoid_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algRval_is_monoid_morphism | |
algRval_is_multiplicative:=
(fun g => (g.2,g.1)) algRval_is_monoid_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build algR algC algRval
algRval_is_zmod_morphism.
HB.instance Definition _ := GRing.isMonoidMorphism.Build algR algC algRval
algRval_is_monoid_morphism. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algRval_is_multiplicative | |
algR_norm(x : algR) : algR := in_algR (normr_real (val x)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_norm | |
algR_ler_normDx y : algR_norm (x + y) <= (algR_norm x + algR_norm y).
Proof. exact: ler_normD. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_ler_normD | |
algR_normr0_eq0x : algR_norm x = 0 -> x = 0.
Proof. by move=> /(congr1 val)/normr0_eq0 ?; apply/val_inj. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_normr0_eq0 | |
algR_normrMnx n : algR_norm (x *+ n) = algR_norm x *+ n.
Proof. by apply/val_inj; rewrite /= !rmorphMn/= normrMn. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_normrMn | |
algR_normrNx : algR_norm (- x) = algR_norm x.
Proof. by apply/val_inj; apply: normrN. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_normrN | |
algR_addr_gt0(x y : algR) : z < x -> z < y -> z < x + y.
Proof. exact: addr_gt0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_addr_gt0 | |
algR_ger_leVge(x y : algR) : z <= x -> z <= y -> (x <= y) || (y <= x).
Proof. exact: ger_leVge. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_ger_leVge | |
algR_normrM: {morph algR_norm : x y / x * y}.
Proof. by move=> *; apply/val_inj; apply: normrM. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_normrM | |
algR_ler_def(x y : algR) : (x <= y) = (algR_norm (y - x) == y - x).
Proof. by apply: ler_def. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_ler_def | |
Definition_ := Num.Zmodule_isNormed.Build _ algR
algR_ler_normD algR_normr0_eq0 algR_normrMn algR_normrN.
HB.instance Definition _ := Num.isNumRing.Build algR
algR_addr_gt0 algR_ger_leVge algR_normrM algR_ler_def. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | Definition | |
algR_archiFieldMixin: Num.archimedean_axiom algR.
Proof.
move=> /= x; have := real_floorD1_gt (valP `|x|).
set n := Num.floor _ + 1 => x_lt.
exists (`|(n + 1)%R|%N); apply: (lt_le_trans x_lt _).
by rewrite /= rmorphMn/= pmulrn ler_int (le_trans _ (lez_abs _))// lerDl.
Qed.
HB.instance Definition _ := Num.NumDomain_boun... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_archiFieldMixin | |
algR_pfactor(x : algC) : {poly algR} :=
if x \is Creal =P true is ReflectT xR then 'X - (in_algR xR)%:P else
'X^2 - (in_algR (Creal_Re x) *+ 2) *: 'X + ((in_algR (normr_real x))^+2)%:P. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_pfactor | |
algC_pfactorx := (algR_pfactor x ^^ algRval). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_pfactor | |
algR_pfactorRE(x : algC) (xR : x \is Creal) :
algR_pfactor x = 'X - (in_algR xR)%:P.
Proof.
rewrite /algR_pfactor; case: eqP xR => //= p1 p2.
by rewrite (bool_irrelevance p1 p2).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_pfactorRE | |
algC_pfactorRE(x : algC) : x \is Creal ->
algC_pfactor x = 'X - x%:P.
Proof. by move=> xR; rewrite algR_pfactorRE map_polyXsubC. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_pfactorRE | |
algR_pfactorCE(x : algC) : x \isn't Creal ->
algR_pfactor x =
'X^2 - (in_algR (Creal_Re x) *+ 2) *: 'X + ((in_algR (normr_real x))^+2)%:P.
Proof. by rewrite /algR_pfactor; case: eqP => // p; rewrite p. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_pfactorCE | |
algC_pfactorCE(x : algC) : x \isn't Creal ->
algC_pfactor x = ('X - x%:P) * ('X - x^*%:P).
Proof.
move=> xNR; rewrite algR_pfactorCE//=.
rewrite rmorphD /= rmorphB/= !map_polyZ !map_polyXn/= map_polyX.
rewrite (map_polyC algRval)/=.
rewrite mulrBl !mulrBr -!addrA; congr (_ + _).
rewrite opprD addrA opprK -opprD -rmor... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_pfactorCE | |
algC_pfactorEx :
algC_pfactor x = ('X - x%:P) * ('X - x^*%:P) ^+ (x \isn't Creal).
Proof.
by have [/algC_pfactorRE|/algC_pfactorCE] := boolP (_ \is _); rewrite ?mulr1.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_pfactorE | |
size_algC_pfactorx : size (algC_pfactor x) = (x \isn't Creal).+2.
Proof.
have [xR|xNR] := boolP (_ \is _); first by rewrite algC_pfactorRE// size_XsubC.
by rewrite algC_pfactorCE// size_mul ?size_XsubC ?polyXsubC_eq0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | size_algC_pfactor | |
size_algR_pfactorx : size (algR_pfactor x) = (x \isn't Creal).+2.
Proof. by have := size_algC_pfactor x; rewrite size_map_poly. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | size_algR_pfactor | |
algC_pfactor_eq0x : (algC_pfactor x == 0) = false.
Proof. by rewrite -size_poly_eq0 size_algC_pfactor. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_pfactor_eq0 | |
algR_pfactor_eq0x : (algR_pfactor x == 0) = false.
Proof. by rewrite -size_poly_eq0 size_algR_pfactor. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_pfactor_eq0 | |
algC_pfactorCgt0x y : x \isn't Creal -> y \is Creal ->
(algC_pfactor x).[y] > 0.
Proof.
move=> xNR yR; rewrite algC_pfactorCE// hornerM !hornerXsubC.
rewrite [x]algCrect conjC_rect ?Creal_Re ?Creal_Im// !opprD !addrA opprK.
rewrite -subr_sqr exprMn sqrCi mulN1r opprK ltr_wpDl//.
- by rewrite real_exprn_even_ge0// ?rp... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algC_pfactorCgt0 | |
algR_pfactorR_mul_gt0(x a b : algC) :
x \is Creal -> a \is Creal -> b \is Creal ->
a <= b ->
((algC_pfactor x).[a] * (algC_pfactor x).[b] <= 0) =
(a <= x <= b).
Proof.
move=> xR aR bR ab; rewrite !algC_pfactorRE// !hornerXsubC.
have [lt_xa|lt_ax|->]/= := real_ltgtP xR aR; last first.
- by rewrite subrr mu... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_pfactorR_mul_gt0 | |
monic_algC_pfactorx : algC_pfactor x \is monic.
Proof. by rewrite algC_pfactorE rpredM ?rpredX ?monicXsubC. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | monic_algC_pfactor | |
monic_algR_pfactorx : algR_pfactor x \is monic.
Proof. by have := monic_algC_pfactor x; rewrite map_monic. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | monic_algR_pfactor | |
poly_algR_pfactor(p : {poly algR}) :
{ r : seq algC |
p ^^ algRval = val (lead_coef p) *: \prod_(z <- r) algC_pfactor z }.
Proof.
wlog p_monic : p / p \is monic => [hwlog|].
have [->|pN0] := eqVneq p 0.
by exists [::]; rewrite lead_coef0/= rmorph0 scale0r.
have [|r] := hwlog ((lead_coef p)^-1 *: p).
b... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | poly_algR_pfactor | |
algR_rcfMixin: Num.real_closed_axiom algR.
Proof.
move=> p a b le_ab /andP[pa_le0 pb_ge0]/=.
case: ltgtP pa_le0 => //= pa0 _; last first.
by exists a; rewrite ?lexx// rootE pa0.
case: ltgtP pb_ge0 => //= pb0 _; last first.
by exists b; rewrite ?lexx ?andbT// rootE -pb0.
have p_neq0 : p != 0 by apply: contraTneq pa0... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path bigop finset prime order ssralg",
"From mathcomp Require Import poly polydiv mxpoly generic_quotient countalg",
"From mathcomp Require Import ... | field/algC.v | algR_rcfMixin | |
rat_algebraic_archimedean(C : numFieldType) (QtoC : Qmorphism C) :
integralRange QtoC -> Num.archimedean_axiom C.
Proof.
move=> algC x.
without loss x_ge0: x / 0 <= x by rewrite -normr_id; apply.
have [-> | nz_x] := eqVneq x 0; first by exists 1; rewrite normr0.
have [p mon_p px0] := algC x; exists (\sum_(j < size p)... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path tuple bigop finset prime order",
"From mathcomp Require Import ssralg poly polydiv mxpoly countalg closed_field",
"From mathcomp Require Impor... | field/algebraics_fundamentals.v | rat_algebraic_archimedean | |
decidable_embeddingsT T (f : sT -> T) :=
forall y, decidable (exists x, y = f x). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path tuple bigop finset prime order",
"From mathcomp Require Import ssralg poly polydiv mxpoly countalg closed_field",
"From mathcomp Require Impor... | field/algebraics_fundamentals.v | decidable_embedding | |
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 m... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path tuple bigop finset prime order",
"From mathcomp Require Import ssralg poly polydiv mxpoly countalg closed_field",
"From mathcomp Require Impor... | field/algebraics_fundamentals.v | rat_algebraic_decidable | |
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 ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path tuple bigop finset prime order",
"From mathcomp Require Import ssralg poly polydiv mxpoly countalg closed_field",
"From mathcomp Require Impor... | field/algebraics_fundamentals.v | minPoly_decidable_closure | |
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.
Prenex Implicits alg_integral.
Arguments map_poly_inj {F R} f [p1 p2]. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path tuple bigop finset prime order",
"From mathcomp Require Import ssralg poly polydiv mxpoly countalg closed_field",
"From mathcomp Require Impor... | field/algebraics_fundamentals.v | alg_integral | |
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 ... | Theorem | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq choice",
"From mathcomp Require Import div fintype path tuple bigop finset prime order",
"From mathcomp Require Import ssralg poly polydiv mxpoly countalg closed_field",
"From mathcomp Require Impor... | field/algebraics_fundamentals.v | Fundamental_Theorem_of_Algebraics | |
alg_num_field(Qz : fieldExtType rat) a : a%:A = ratr a :> Qz.
Proof. by rewrite -in_algE fmorph_eq_rat. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | alg_num_field | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | rmorphZ_num | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | fmorph_numZ | |
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 & ro... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | algC_PET | |
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 :... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | num_field_exists | |
in_Crat_spans x :=
exists a : rat ^ size s, x = \sum_i QtoC (a i) * s`_i.
Fact 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) => [/c... | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | in_Crat_span | |
Crat_spans : pred algC := Crat_span_subproof s. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Crat_span | |
Crat_spanPs x : reflect (in_Crat_span s x) (x \in Crat_span s).
Proof. exact: sumboolP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Crat_spanP | |
mem_Crat_spans : {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.
Fact Crat_span_zmod_clo... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | mem_Crat_span | |
Crat_spanZb 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Crat_spanZ | |
Crat_spanMb : {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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Crat_spanM | |
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... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | num_field_proj | |
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_e... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | restrict_aut_to_num_field | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | map_Qnum_poly | |
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 (... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | restrict_aut_to_normal_num_field | |
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=> ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | dec_Cint_span | |
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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Cint_span | |
Cint_spanPn (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 ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Cint_spanP | |
mem_Cint_spans : {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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | mem_Cint_span | |
Cint_span_zmod_closeds : 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.
HB.instance Definition _ s := G... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Cint_span_zmod_closed | |
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... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | extend_algC_subfield_aut | |
Qn_aut_existsk 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 ... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Qn_aut_exists | |
Aint: {pred algC} := fun x => minCpoly x \is a polyOver Num.int. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint | |
root_monic_Aintp 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 [-> m... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | root_monic_Aint | |
Cint_rat_Aintz : 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... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Cint_rat_Aint | |
Aint_Cint: {subset Num.int <= Aint}.
Proof.
move=> x; rewrite -polyOverXsubC.
by apply: root_monic_Aint; rewrite ?monicXsubC ?root_XsubC.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint_Cint | |
Aint_intx : x%:~R \in Aint. Proof. by rewrite Aint_Cint. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint_int | |
Aint0: 0 \in Aint. Proof. exact: Aint_int 0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint0 | |
Aint1: 1 \in Aint. Proof. exact: Aint_int 1. Qed.
#[global] Hint Resolve Aint0 Aint1 : core. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint1 | |
Aint_unity_rootn 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint_unity_root | |
Aint_prim_rootn 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint_prim_root | |
Aint_Cnat: {subset Num.nat <= Aint}.
Proof. by move=> z /intr_nat/Aint_Cint. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint_Cnat | |
Aint_subring_exists(X : seq algC) :
{subset X <= Aint} ->
{S : pred algC & | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv ssrnum ssrint archimedean rat",
"From mathcomp Require Import fin... | field/algnum.v | Aint_subring_exists |
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