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 |
|---|---|---|---|---|---|---|
fin_Csubring_AintS n (Y : n.-tuple algC) :
mulr_closed S -> (forall x, reflect (inIntSpan Y x) (x \in S)) ->
{subset S <= Aint}.
Proof.
move=> mulS.
pose Sm := GRing.isMulClosed.Build _ _ mulS.
pose SC : mulrClosed _ := HB.pack S Sm.
have ZP_C c: (ZtoC c)%:P \is a polyOver Num.int_num_subdef.
by rewrite raddfMz rpred_int.
move=> S_P x Sx; pose v := \row_(i < n) Y`_i.
have [v0 | nz_v] := eqVneq v 0.
case/S_P: Sx => {}x ->; rewrite big1 ?isAlgInt0 // => i _.
by have /rowP/(_ i)/[!mxE] -> := v0; rewrite mul0rz.
have sYS (i : 'I_n): x * Y`_i \in SC.
by rewrite rpredM //; apply/S_P/Cint_spanP/mem_Cint_span/memt_nth.
pose A := \matrix_(i, j < n) sval (sig_eqW (S_P _ (sYS j))) i.
pose p := char_poly (map_mx ZtoC A).
have: p \is a polyOver Num.int_num_subdef.
rewrite rpred_sum // => s _; rewrite rpredMsign rpred_prod // => j _.
by rewrite !mxE /= rpredB ?rpredMn ?polyOverX.
apply: root_monic_Aint (char_poly_monic _).
rewrite -eigenvalue_root_char; apply/eigenvalueP; exists v => //.
apply/rowP=> j; case dAj: (sig_eqW (S_P _ (sYS j))) => [a DxY].
by rewrite !mxE DxY; apply: eq_bigr => i _; rewrite !mxE dAj /= mulrzr.
Qed. | Theorem | 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 | fin_Csubring_Aint | |
Definition_ := GRing.isSubringClosed.Build _ Aint Aint_subring. | HB.instance | 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 | Definition | |
Aint_aut(nu : {rmorphism algC -> algC}) x :
(nu x \in Aint) = (x \in Aint).
Proof. by rewrite !unfold_in minCpoly_aut. 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_aut | |
dvdA(e : Algebraics.divisor) : {pred algC} :=
fun z => if e == 0 then z == 0 else z / e \in Aint.
Delimit Scope algC_scope with A.
Delimit Scope algC_expanded_scope with Ax. | 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 | dvdA | |
Definition_ e := GRing.isZmodClosed.Build _ (dvdA e)
(dvdA_zmod_closed e). | HB.instance | 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 | Definition | |
eqAmod(e x y : Algebraics.divisor) := (e %| x - y)%A. | 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 | eqAmod | |
eqAmod_refle x : (x == x %[mod e])%A.
Proof. by rewrite /eqAmod subrr rpred0. Qed.
#[global] Hint Resolve eqAmod_refl : 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 | eqAmod_refl | |
eqAmod_syme x y : ((x == y %[mod e]) = (y == x %[mod e]))%A.
Proof. by rewrite /eqAmod -opprB rpredN. 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 | eqAmod_sym | |
eqAmod_transe y x z :
(x == y %[mod e] -> y == z %[mod e] -> x == z %[mod e])%A.
Proof.
by move=> Exy Eyz; rewrite /eqAmod -[x](subrK y) -[_ - z]addrA rpredD.
Qed. | Lemma | 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 | eqAmod_trans | |
eqAmod_transle x y z :
(x == y %[mod e])%A -> (x == z %[mod e])%A = (y == z %[mod e])%A.
Proof. by move/(sym_left_transitive (eqAmod_sym e) (@eqAmod_trans e)). Qed. | Lemma | 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 | eqAmod_transl | |
eqAmod_transre x y z :
(x == y %[mod e])%A -> (z == x %[mod e])%A = (z == y %[mod e])%A.
Proof. by move/(sym_right_transitive (eqAmod_sym e) (@eqAmod_trans e)). Qed. | Lemma | 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 | eqAmod_transr | |
eqAmod0e x : (x == 0 %[mod e])%A = (e %| x)%A.
Proof. by rewrite /eqAmod subr0. 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 | eqAmod0 | |
eqAmodNe x y : (- x == y %[mod e])%A = (x == - y %[mod e])%A.
Proof. by rewrite eqAmod_sym /eqAmod !opprK addrC. 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 | eqAmodN | |
eqAmodDre x y z : (y + x == z + x %[mod e])%A = (y == z %[mod e])%A.
Proof. by rewrite /eqAmod addrAC opprD !addrA subrK. 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 | eqAmodDr | |
eqAmodDle x y z : (x + y == x + z %[mod e])%A = (y == z %[mod e])%A.
Proof. by rewrite !(addrC x) eqAmodDr. 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 | eqAmodDl | |
eqAmodDe x1 x2 y1 y2 :
(x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 + y1 == x2 + y2 %[mod e])%A.
Proof.
by rewrite -(eqAmodDl e x2 y1) -(eqAmodDr e y1); apply: eqAmod_trans.
Qed. | Lemma | 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 | eqAmodD | |
eqAmodm0e : (e == 0 %[mod e])%A.
Proof. by rewrite /eqAmod subr0 unfold_in; case: ifPn => // /divff->. Qed.
#[global] Hint Resolve eqAmodm0 : 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 | eqAmodm0 | |
eqAmodMre :
{in Aint, forall z x y, x == y %[mod e] -> x * z == y * z %[mod e]}%A.
Proof.
move=> z Zz x y.
rewrite /eqAmod -mulrBl ![(e %| _)%A]unfold_in mulf_eq0 mulrAC.
by case: ifP => [_ -> // | _ Exy]; apply: rpredM.
Qed. | Lemma | 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 | eqAmodMr | |
eqAmodMle :
{in Aint, forall z x y, x == y %[mod e] -> z * x == z * y %[mod e]}%A.
Proof. by move=> z Zz x y Exy; rewrite !(mulrC z) eqAmodMr. Qed. | Lemma | 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 | eqAmodMl | |
eqAmodMl0e : {in Aint, forall x, x * e == 0 %[mod e]}%A.
Proof. by move=> x Zx; rewrite -(mulr0 x) eqAmodMl. 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 | eqAmodMl0 | |
eqAmodMr0e : {in Aint, forall x, e * x == 0 %[mod e]}%A.
Proof. by move=> x Zx; rewrite /= mulrC eqAmodMl0. 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 | eqAmodMr0 | |
eqAmod_addl_mule : {in Aint, forall x y, x * e + y == y %[mod e]}%A.
Proof. by move=> x Zx y; rewrite -{2}[y]add0r eqAmodDr eqAmodMl0. Qed. | Lemma | 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 | eqAmod_addl_mul | |
eqAmodMe : {in Aint &, forall x1 y2 x2 y1,
x1 == x2 %[mod e] -> y1 == y2 %[mod e] -> x1 * y1 == x2 * y2 %[mod e]}%A.
Proof.
move=> x1 y2 Zx1 Zy2 x2 y1 eq_x /(eqAmodMl Zx1)/eqAmod_trans-> //.
exact: eqAmodMr.
Qed. | Lemma | 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 | eqAmodM | |
eqAmod_rat:
{in Crat & &, forall e m n, (m == n %[mod e])%A = (m == n %[mod e])%C}.
Proof.
move=> e m n Qe Qm Qn; rewrite /eqCmod unfold_in /eqAmod unfold_in.
case: ifPn => // nz_e; apply/idP/idP=> [/Cint_rat_Aint | /Aint_Cint] -> //.
by rewrite rpred_div ?rpredB.
Qed. | Lemma | 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 | eqAmod_rat | |
eqAmod0_rat: {in Crat &, forall e n, (n == 0 %[mod e])%A = (e %| n)%C}.
Proof. by move=> e n Qe Qn; rewrite /= eqAmod_rat /eqCmod ?subr0 ?Crat0. Qed. | Lemma | 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 | eqAmod0_rat | |
eqAmod_nat(e m n : nat) : (m == n %[mod e])%A = (m == n %[mod e])%N.
Proof. by rewrite eqAmod_rat ?rpred_nat // eqCmod_nat. Qed. | Lemma | 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 | eqAmod_nat | |
eqAmod0_nat(e m : nat) : (m == 0 %[mod e])%A = (e %| m)%N.
Proof. by rewrite eqAmod0_rat ?rpred_nat // dvdC_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 | eqAmod0_nat | |
orderCx :=
let p := minCpoly x in
oapp val 0 [pick n : 'I_(2 * size p ^ 2) | p == intrp 'Phi_n]. | 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 | orderC | |
exp_orderCx : x ^+ #[x]%C = 1.
Proof.
rewrite /orderC; case: pickP => //= [] [n _] /= /eqP Dp.
have n_gt0: (0 < n)%N.
rewrite lt0n; apply: contraTneq (size_minCpoly x) => n0.
by rewrite Dp n0 Cyclotomic0 rmorph1 size_poly1.
have [z prim_z] := C_prim_root_exists n_gt0.
rewrite prim_expr_order // -(root_cyclotomic prim_z).
by rewrite -Cintr_Cyclotomic // -Dp root_minCpoly.
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 | exp_orderC | |
dvdn_orderCx n : (#[x]%C %| n)%N = (x ^+ n == 1).
Proof.
apply/idP/eqP=> [|x_n_1]; first by apply: expr_dvd; apply: exp_orderC.
have [-> | n_gt0] := posnP n; first by rewrite dvdn0.
have [m prim_x m_dv_n] := prim_order_exists n_gt0 x_n_1.
have{n_gt0} m_gt0 := dvdn_gt0 n_gt0 m_dv_n; congr (_ %| n)%N: m_dv_n.
pose p := minCpoly x; have Dp: p = cyclotomic x m := minCpoly_cyclotomic prim_x.
rewrite /orderC; case: pickP => /= [k /eqP Dp_k | no_k]; last first.
suffices lt_m_2p: (m < 2 * size p ^ 2)%N.
have /eqP[] := no_k (Ordinal lt_m_2p).
by rewrite /= -/p Dp -Cintr_Cyclotomic.
rewrite Dp size_cyclotomic (sqrnD 1) addnAC mulnDr -add1n leq_add //.
suffices: (m <= \prod_(q <- primes m | q == 2) q * totient m ^ 2)%N.
have [m_even | m_odd] := boolP (2%N \in primes m).
by rewrite -big_filter filter_pred1_uniq ?primes_uniq // big_seq1.
by rewrite big_hasC ?has_pred1 // => /leq_trans-> //; apply: leq_addl.
rewrite big_mkcond totientE // -mulnn -!big_split /=.
rewrite {1}[m]prod_prime_decomp // prime_decompE big_map /= !big_seq.
elim/big_ind2: _ => // [n1 m1 n2 m2 | q]; first exact: leq_mul.
rewrite mem_primes => /and3P[q_pr _ q_dv_m].
rewrite lognE q_pr m_gt0 q_dv_m /=; move: (logn q _) => k.
rewrite !mulnA expnS leq_mul //.
case: (ltngtP q 2) (prime_gt1 q_pr) => // [q_gt2|->] _.
rewrite mul1n mulnAC mulnn -{1}[q]muln1 leq_mul ?expn_gt0 ?prime_gt0 //.
by rewrite -(subnKC q_gt2) (ltn_exp2l 1).
by rewrite !muln1 -expnS (ltn_exp2l 0).
have k_prim_x:
... | 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 | dvdn_orderC | |
fF:= (@GRing.formula F). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | fF | |
tF:= (@GRing.term F). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | tF | |
qff := (GRing.qf_form f && GRing.rformula f). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf | |
polyF:= seq tF. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | polyF | |
qf_simpl(f : fF) :
(qf f -> GRing.qf_form f) * (qf f -> GRing.rformula f).
Proof. by split=> /andP[]. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_simpl | |
cpsT := ((T -> fF) -> fF). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | cps | |
retT1 : T1 -> cps T1 := fun x k => k x.
Arguments ret {T1} x k /. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ret | |
bindT1 T2 (x : cps T1) (f : T1 -> cps T2) : cps T2 :=
fun k => x (fun x => f x k).
Arguments bind {T1 T2} x f k /. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | bind | |
cpsifT (c : fF) (t : T) (e : T) : cps T :=
fun k => GRing.If c (k t) (k e).
Arguments cpsif {T} c t e k /. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | cpsif | |
eval:= GRing.eval. | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval | |
rterm:= GRing.rterm. | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rterm | |
qf_eval:= GRing.qf_eval. | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_eval | |
eval_poly(e : seq F) pf :=
if pf is c :: q then eval_poly e q * 'X + (eval e c)%:P else 0. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_poly | |
rpoly(p : polyF) := all (@rterm F) p. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rpoly | |
sizeT: polyF -> cps nat := (fix loop p :=
if p isn't c :: q then ret 0
else 'let n <- loop q;
if n is m.+1 then ret m.+2 else
'if (c == 0) then 0%N else 1%N). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | sizeT | |
qf_red_cpsT (x : cps T) (y : _ -> T) :=
forall e k, qf_eval e (x k) = qf_eval e (k (y e)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_red_cps | |
qf_cpsT D (x : cps T) :=
forall k, (forall y, D y -> qf (k y)) -> qf (x k). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_cps | |
qf_cps_retT D (x : T) : D x -> qf_cps D (ret x).
Proof. move=> ??; exact. Qed.
Hint Resolve qf_cps_ret : core. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_cps_ret | |
qf_cps_bindT1 D1 T2 D2 (x : cps T1) (f : T1 -> cps T2) :
qf_cps D1 x -> (forall x, D1 x -> qf_cps D2 (f x)) -> qf_cps D2 (bind x f).
Proof. by move=> xP fP k kP /=; apply: xP => y ?; apply: fP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_cps_bind | |
qf_cps_ifT D (c : fF) (t : T) (e : T) : qf c -> D t -> D e ->
qf_cps D ('if c then t else e).
Proof.
move=> qfc Dt De k kP /=; have [qft qfe] := (kP _ Dt, kP _ De).
by do !rewrite qf_simpl //.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | qf_cps_if | |
sizeTP(pf : polyF) : sizeT pf ->_e size (eval_poly e pf).
Proof.
elim: pf=> [|c qf qfP /=]; first by rewrite /= size_poly0.
move=> e k; rewrite size_MXaddC qfP -(size_poly_eq0 (eval_poly _ _)).
by case: (size (eval_poly e qf))=> //=; case: eqP; rewrite // orbF.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | sizeTP | |
sizeT_qf(p : polyF) : rpoly p -> qf_cps xpredT (sizeT p).
Proof.
elim: p => /= [_|c p ihp /andP[rc rq]]; first exact: qf_cps_ret.
apply: qf_cps_bind; first exact: ihp.
move=> [|n] //= _; last exact: qf_cps_ret.
by apply: qf_cps_if; rewrite //= rc.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | sizeT_qf | |
isnull(p : polyF) : cps bool :=
'let n <- sizeT p; ret (n == 0). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | isnull | |
isnullP(p : polyF) : isnull p ->_e (eval_poly e p == 0).
Proof. by move=> e k; rewrite sizeTP size_poly_eq0. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | isnullP | |
isnull_qf(p : polyF) : rpoly p -> qf_cps xpredT (isnull p).
Proof.
move=> rp; apply: qf_cps_bind; first exact: sizeT_qf.
by move=> ? _; apply: qf_cps_ret.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | isnull_qf | |
lt_sizeT(p q : polyF) : cps bool :=
'let n <- sizeT p; 'let m <- sizeT q; ret (n < m). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | lt_sizeT | |
lift(p : {poly F}) := map GRing.Const p. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | lift | |
eval_lift(e : seq F) (p : {poly F}) : eval_poly e (lift p) = p.
Proof.
elim/poly_ind: p => [|p c]; first by rewrite /lift polyseq0.
rewrite -cons_poly_def /lift polyseq_cons /nilp.
case pn0: (_ == _) => /=; last by move->; rewrite -cons_poly_def.
move=> _; rewrite polyseqC.
case c0: (_==_)=> /=.
move: pn0; rewrite (eqP c0) size_poly_eq0; move/eqP->.
by apply: val_inj=> /=; rewrite polyseq_cons // polyseq0.
by rewrite mul0r add0r; apply: val_inj=> /=; rewrite polyseq_cons // /nilp pn0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_lift | |
lead_coefTp : cps tF :=
if p is c :: q then
'let l <- lead_coefT q; 'if (l == 0) then c else l
else ret 0%T. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | lead_coefT | |
lead_coefTP(k : tF -> fF) :
(forall x e, qf_eval e (k x) = qf_eval e (k (eval e x)%:T%T)) ->
forall (p : polyF) (e : seq F),
qf_eval e (lead_coefT p k) = qf_eval e (k (lead_coef (eval_poly e p))%:T%T).
Proof.
move=> kP p e; elim: p => [|a p IHp]/= in k kP e *.
by rewrite lead_coef0 kP.
rewrite IHp; last by move=> *; rewrite //= -kP.
rewrite GRing.eval_If /= lead_coef_eq0.
case p'0: (_ == _); first by rewrite (eqP p'0) mul0r add0r lead_coefC -kP.
rewrite lead_coefDl ?lead_coefMX // polyseqC size_mul ?p'0 //; last first.
by rewrite -size_poly_eq0 size_polyX.
rewrite size_polyX addnC /=; case: (_ == _)=> //=.
by rewrite ltnS lt0n size_poly_eq0 p'0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | lead_coefTP | |
lead_coefT_qf(p : polyF) : rpoly p -> qf_cps (@rterm _) (lead_coefT p).
Proof.
elim: p => [_|c q ihp //= /andP[rc rq]]; first by apply: qf_cps_ret.
apply: qf_cps_bind => [|y ty]; first exact: ihp.
by apply: qf_cps_if; rewrite //= ty.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | lead_coefT_qf | |
amulXnT(a : tF) (n : nat) : polyF :=
if n is n'.+1 then 0%T :: (amulXnT a n') else [:: a]. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | amulXnT | |
eval_amulXnT(a : tF) (n : nat) (e : seq F) :
eval_poly e (amulXnT a n) = (eval e a)%:P * 'X^n.
Proof.
elim: n=> [|n] /=; first by rewrite expr0 mulr1 mul0r add0r.
by move->; rewrite addr0 -mulrA -exprSr.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_amulXnT | |
ramulXnT: forall a n, rterm a -> rpoly (amulXnT a n).
Proof. by move=> a n; elim: n a=> [a /= -> //|n ihn a ra]; apply: ihn. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | ramulXnT | |
sumpT(p q : polyF) :=
match p, q with a :: p, b :: q => (a + b)%T :: sumpT p q
| [::], q => q | p, [::] => p end.
Arguments sumpT : simpl nomatch. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | sumpT | |
eval_sumpT(p q : polyF) (e : seq F) :
eval_poly e (sumpT p q) = (eval_poly e p) + (eval_poly e q).
Proof.
elim: p q => [|a p Hp] q /=; first by rewrite add0r.
case: q => [|b q] /=; first by rewrite addr0.
rewrite Hp mulrDl -!addrA; congr (_ + _); rewrite polyCD addrC -addrA.
by congr (_ + _); rewrite addrC.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_sumpT | |
rsumpT(p q : polyF) : rpoly p -> rpoly q -> rpoly (sumpT p q).
Proof.
elim: p q=> [|a p ihp] q rp rq //; move: rp; case/andP=> ra rp.
case: q rq => [|b q]; rewrite /= ?ra ?rp //=.
by case/andP=> -> rq //=; apply: ihp.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rsumpT | |
mulpT(p q : polyF) :=
if p isn't a :: p then [::]
else sumpT [seq (a * x)%T | x <- q] (0%T :: mulpT p q). | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | mulpT | |
eval_mulpT(p q : polyF) (e : seq F) :
eval_poly e (mulpT p q) = (eval_poly e p) * (eval_poly e q).
Proof.
elim: p q=> [|a p Hp] q /=; first by rewrite mul0r.
rewrite eval_sumpT /= Hp addr0 mulrDl addrC mulrAC; congr (_ + _).
by elim: q=> [|b q Hq] /=; rewrite ?mulr0 // Hq polyCM mulrDr mulrA.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_mulpT | |
rpoly_map_mul(t : tF) (p : polyF) (rt : rterm t) :
rpoly [seq (t * x)%T | x <- p] = rpoly p.
Proof. by rewrite /rpoly all_map; apply/eq_all => x; rewrite /= rt. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rpoly_map_mul | |
rmulpT(p q : polyF) : rpoly p -> rpoly q -> rpoly (mulpT p q).
Proof.
elim: p q=> [|a p ihp] q rp rq //=; move: rp; case/andP=> ra rp /=.
apply: rsumpT; last exact: ihp.
by rewrite rpoly_map_mul.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rmulpT | |
opppT: polyF -> polyF := map (GRing.Mul (- 1%T)%T). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | opppT | |
eval_opppT(p : polyF) (e : seq F) :
eval_poly e (opppT p) = - eval_poly e p.
Proof.
by elim: p; rewrite /= ?oppr0 // => ? ? ->; rewrite !mulNr opprD polyCN mul1r.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_opppT | |
natmulpTn : polyF -> polyF := map (GRing.Mul n%:R%T). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | natmulpT | |
eval_natmulpT(p : polyF) (n : nat) (e : seq F) :
eval_poly e (natmulpT n p) = (eval_poly e p) *+ n.
Proof.
elim: p; rewrite //= ?mul0rn // => c p ->.
rewrite mulrnDl mulr_natl polyCMn; congr (_ + _).
by rewrite -mulr_natl mulrAC -mulrA mulr_natl mulrC.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | eval_natmulpT | |
redivp_rec_loopT(q : polyF) sq cq (c : nat) (qq r : polyF)
(n : nat) {struct n} : cps (nat * polyF * polyF) :=
'let sr <- sizeT r;
if sr < sq then ret (c, qq, r) else
'let lr <- lead_coefT r;
let m := amulXnT lr (sr - sq) in
let qq1 := sumpT (mulpT qq [::cq]) m in
let r1 := sumpT (mulpT r ([::cq])) (opppT (mulpT m q)) in
if n is n1.+1 then redivp_rec_loopT q sq cq c.+1 qq1 r1 n1
else ret (c.+1, qq1, r1). | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivp_rec_loopT | |
redivp_rec_loop(q : {poly F}) sq cq
(k : nat) (qq r : {poly F}) (n : nat) {struct n} :=
if size r < sq then (k, qq, r) else
let m := (lead_coef r) *: 'X^(size r - sq) in
let qq1 := qq * cq%:P + m in
let r1 := r * cq%:P - m * q in
if n is n1.+1 then redivp_rec_loop q sq cq k.+1 qq1 r1 n1 else
(k.+1, qq1, r1). | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivp_rec_loop | |
redivp_rec_loopTP(k : nat * polyF * polyF -> fF) :
(forall c qq r e, qf_eval e (k (c,qq,r))
= qf_eval e (k (c, lift (eval_poly e qq), lift (eval_poly e r))))
-> forall q sq cq c qq r n e
(d := redivp_rec_loop (eval_poly e q) sq (eval e cq)
c (eval_poly e qq) (eval_poly e r) n),
qf_eval e (redivp_rec_loopT q sq cq c qq r n k)
= qf_eval e (k (d.1.1, lift d.1.2, lift d.2)).
Proof.
move=> Pk q sq cq c qq r n e /=.
elim: n c qq r k Pk e => [|n Pn] c qq r k Pk e; rewrite sizeTP.
case ltrq : (_ < _); first by rewrite /= ltrq /= -Pk.
rewrite lead_coefTP => [|a p]; rewrite [in LHS]Pk; [|symmetry].
rewrite ?(eval_mulpT,eval_amulXnT,eval_sumpT,eval_opppT) //=.
by rewrite ltrq //= !mul_polyC ?(mul0r,add0r,scale0r).
by rewrite [in LHS]Pk ?(eval_mulpT,eval_amulXnT,eval_sumpT, eval_opppT).
case ltrq : (_<_); first by rewrite /= ltrq Pk.
rewrite lead_coefTP.
rewrite Pn ?(eval_mulpT,eval_amulXnT,eval_sumpT,eval_opppT) //=.
by rewrite ltrq //= !mul_polyC ?(mul0r,add0r,scale0r).
rewrite -/redivp_rec_loopT => x e'.
rewrite Pn; last by move=> *; rewrite Pk.
symmetry; rewrite Pn; last by move=> *; rewrite Pk.
rewrite Pk ?(eval_lift,eval_mulpT,eval_amulXnT,eval_sumpT,eval_opppT).
by rewrite mul_polyC ?(mul0r,add0r).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivp_rec_loopTP | |
redivp_rec_loopT_qf(q : polyF) (sq : nat) (cq : tF)
(c : nat) (qq r : polyF) (n : nat) :
rpoly q -> rterm cq -> rpoly qq -> rpoly r ->
qf_cps (fun x => [&& rpoly x.1.2 & rpoly x.2])
(redivp_rec_loopT q sq cq c qq r n).
Proof.
do ![move=>x/(pair x){x}] => rw; elim: n => [|n IHn]//= in q sq cq c qq r rw *;
apply: qf_cps_bind; do ?[by apply: sizeT_qf; rewrite !rw] => sr _;
case: ifPn => // _; do ?[by apply: qf_cps_ret; rewrite //= ?rw];
apply: qf_cps_bind; do ?[by apply: lead_coefT_qf; rewrite !rw] => lr /= rlr;
[apply: qf_cps_ret|apply: IHn];
by do !rewrite ?(rsumpT,rmulpT,ramulXnT,rpoly_map_mul,rlr,rw) //=.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivp_rec_loopT_qf | |
redivpT(p : polyF) (q : polyF) : cps (nat * polyF * polyF) :=
'let b <- isnull q;
if b then ret (0, [::0%T], p) else
'let sq <- sizeT q; 'let sp <- sizeT p;
'let lq <- lead_coefT q;
redivp_rec_loopT q sq lq 0 [::0%T] p sp. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivpT | |
redivp_rec_loopP(q : {poly F}) (c : nat) (qq r : {poly F}) (n : nat) :
redivp_rec q c qq r n = redivp_rec_loop q (size q) (lead_coef q) c qq r n.
Proof. by elim: n c qq r => [| n Pn] c qq r //=; rewrite Pn. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivp_rec_loopP | |
redivpTP(k : nat * polyF * polyF -> fF) :
(forall c qq r e,
qf_eval e (k (c,qq,r)) =
qf_eval e (k (c, lift (eval_poly e qq), lift (eval_poly e r)))) ->
forall p q e (d := redivp (eval_poly e p) (eval_poly e q)),
qf_eval e (redivpT p q k) = qf_eval e (k (d.1.1, lift d.1.2, lift d.2)).
Proof.
move=> Pk p q e /=; rewrite isnullP unlock /=.
case q0 : (eval_poly e q == 0) => /=; first by rewrite Pk /= mul0r add0r polyC0.
rewrite !sizeTP lead_coefTP /=; last by move=> *; rewrite !redivp_rec_loopTP.
rewrite redivp_rec_loopTP /=; last by move=> *; rewrite Pk.
by rewrite mul0r add0r polyC0 redivp_rec_loopP.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivpTP | |
redivpT_qf(p : polyF) (q : polyF) : rpoly p -> rpoly q ->
qf_cps (fun x => [&& rpoly x.1.2 & rpoly x.2]) (redivpT p q).
Proof.
move=> rp rq; apply: qf_cps_bind => [|[] _]; first exact: isnull_qf.
by apply: qf_cps_ret.
apply: qf_cps_bind => [|sp _]; first exact: sizeT_qf.
apply: qf_cps_bind => [|sq _]; first exact: sizeT_qf.
apply: qf_cps_bind => [|lq rlq]; first exact: lead_coefT_qf.
by apply: redivp_rec_loopT_qf => //=.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | redivpT_qf | |
rmodpT(p : polyF) (q : polyF) : cps polyF :=
'let d <- redivpT p q; ret d.2. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rmodpT | |
rdivpT(p : polyF) (q : polyF) : cps polyF :=
'let d <- redivpT p q; ret d.1.2. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rdivpT | |
rscalpT(p : polyF) (q : polyF) : cps nat :=
'let d <- redivpT p q; ret d.1.1. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rscalpT | |
rdvdpT(p : polyF) (q : polyF) : cps bool :=
'let d <- rmodpT p q; isnull d. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rdvdpT | |
rgcdp_loopn (pp qq : {poly F}) {struct n} :=
let rr := rmodp pp qq in if rr == 0 then qq
else if n is n1.+1 then rgcdp_loop n1 qq rr else rr. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdp_loop | |
rgcdp_loopTn (pp : polyF) (qq : polyF) : cps polyF :=
'let rr <- rmodpT pp qq; 'let nrr <- isnull rr; if nrr then ret qq
else if n is n1.+1 then rgcdp_loopT n1 qq rr else ret rr. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdp_loopT | |
rgcdp_loopP(k : polyF -> fF) :
(forall p e, qf_eval e (k p) = qf_eval e (k (lift (eval_poly e p)))) ->
forall n p q e,
qf_eval e (rgcdp_loopT n p q k) =
qf_eval e (k (lift (rgcdp_loop n (eval_poly e p) (eval_poly e q)))).
Proof.
move=> Pk n p q e; elim: n => /= [| m IHm] in p q e *;
rewrite redivpTP /==> *; rewrite ?isnullP ?eval_lift -/(rmodp _ _);
by case: (_ == _); do ?by rewrite -?Pk ?IHm ?eval_lift.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdp_loopP | |
rgcdp_loopT_qf(n : nat) (p : polyF) (q : polyF) :
rpoly p -> rpoly q -> qf_cps rpoly (rgcdp_loopT n p q).
Proof.
elim: n => [|n IHn] in p q * => rp rq /=;
(apply: qf_cps_bind=> [|rr rrr]; [
apply: qf_cps_bind => [|[[a u] v]]; do ?exact: redivpT_qf;
by move=> /andP[/= ??]; apply: (@qf_cps_ret _ rpoly)|
apply: qf_cps_bind => [|[] _];
by [apply: isnull_qf|apply: qf_cps_ret|apply: IHn]]).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdp_loopT_qf | |
rgcdpT(p : polyF) (q : polyF) : cps polyF :=
let aux p1 q1 : cps polyF :=
'let b <- isnull p1; if b then ret q1
else 'let n <- sizeT p1; rgcdp_loopT n p1 q1 in
'let b <- lt_sizeT p q; if b then aux q p else aux p q. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdpT | |
rgcdpTP(k : polyF -> fF) :
(forall p e, qf_eval e (k p) = qf_eval e (k (lift (eval_poly e p)))) ->
forall p q e, qf_eval e (rgcdpT p q k) =
qf_eval e (k (lift (rgcdp (eval_poly e p) (eval_poly e q)))).
Proof.
move=> Pk p q e; rewrite /rgcdpT /rgcdp !sizeTP /=.
case: (_ < _); rewrite !isnullP /=; case: (_ == _); rewrite -?Pk ?sizeTP;
by rewrite ?rgcdp_loopP.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdpTP | |
rgcdpT_qf(p : polyF) (q : polyF) :
rpoly p -> rpoly q -> qf_cps rpoly (rgcdpT p q).
Proof.
move=> rp rq k kP; rewrite /rgcdpT /=; do ![rewrite sizeT_qf => // ? _].
case: (_ < _); rewrite ?isnull_qf // => -[]; rewrite ?kP // => _;
by rewrite sizeT_qf => // ? _; rewrite rgcdp_loopT_qf.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdpT_qf | |
rgcdpTs(ps : seq polyF) : cps polyF :=
if ps is p :: pr then 'let pr <- rgcdpTs pr; rgcdpT p pr else ret [::0%T]. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdpTs | |
rgcdpTsP(k : polyF -> fF) :
(forall p e, qf_eval e (k p) = qf_eval e (k (lift (eval_poly e p)))) ->
forall ps e,
qf_eval e (rgcdpTs ps k) =
qf_eval e (k (lift (\big[@rgcdp _/0%:P]_(i <- ps)(eval_poly e i)))).
Proof.
move=> Pk ps e; elim: ps k Pk => [|p ps Pps] /= k Pk.
by rewrite /= big_nil Pk /= mul0r add0r.
by rewrite big_cons Pps => *; rewrite !rgcdpTP // !eval_lift -?Pk.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdpTsP | |
rgcdpTs_qf(ps : seq polyF) :
all rpoly ps -> qf_cps rpoly (rgcdpTs ps).
Proof.
elim: ps => [_|c p ihp /andP[rc rp]] //=; first exact: qf_cps_ret.
by apply: qf_cps_bind => [|r rr]; [apply: ihp|apply: rgcdpT_qf].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgcdpTs_qf | |
rgdcop_recTn (q : polyF) (p : polyF) :=
if n is m.+1 then
'let g <- rgcdpT p q; 'let sg <- sizeT g;
if sg == 1 then ret p
else 'let r <- rdivpT p g;
rgdcop_recT m q r
else 'let b <- isnull q; ret [::b%:R%T]. | Fixpoint | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgdcop_recT | |
rgdcop_recTP(k : polyF -> fF) :
(forall p e, qf_eval e (k p) = qf_eval e (k (lift (eval_poly e p))))
-> forall p q n e, qf_eval e (rgdcop_recT n p q k)
= qf_eval e (k (lift (rgdcop_rec (eval_poly e p) (eval_poly e q) n))).
Proof.
move=> Pk p q n e; elim: n => [|n Pn] /= in k Pk p q e *.
rewrite isnullP /=.
by case: (_ == _); rewrite Pk /= mul0r add0r ?(polyC0, polyC1).
rewrite /rcoprimep rgcdpTP ?sizeTP ?eval_lift => * /=.
case: (_ == _);
by do ?[rewrite /= ?(=^~Pk, redivpTP, rgcdpTP, sizeTP, Pn, eval_lift) //==> *].
do ?[rewrite /= ?(=^~Pk, redivpTP, rgcdpTP, sizeTP, Pn, eval_lift) //==> *].
case: (_ == _);
by do ?[rewrite /= ?(=^~Pk, redivpTP, rgcdpTP, sizeTP, Pn, eval_lift) //==> *].
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgdcop_recTP | |
rgdcop_recT_qf(n : nat) (p : polyF) (q : polyF) :
rpoly p -> rpoly q -> qf_cps rpoly (rgdcop_recT n p q).
Proof.
elim: n => [|n ihn] in p q * => k kP rp rq /=.
by rewrite isnull_qf => //*; rewrite rq.
rewrite rgcdpT_qf=> //*; rewrite sizeT_qf=> //*.
case: (_ == _); rewrite ?kP ?rq //= redivpT_qf=> //= ? /andP[??].
by rewrite ihn.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import fintype generic_quotient bigop ssralg poly",
"From mathcomp Require Import polydiv matrix mxpoly countalg ring_quotient"
] | field/closed_field.v | rgdcop_recT_qf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.