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 |
|---|---|---|---|---|---|---|
mxof(h : 'Hom(uT, vT)) := lin1_mx (rVof e' \o h \o vecof e). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mxof | |
mxof_linear: linear mxof.
Proof.
move=> x h1 h2; apply/matrixP=> i j; do !rewrite ?lfunE/= ?mxE.
by rewrite linearP.
Qed.
HB.instance Definition _ := GRing.isSemilinear.Build F _ _ _ mxof
(GRing.semilinear_linear mxof_linear). | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mxof_linear | |
funmx(M : 'M[F]_(m, n)) u := vecof e' (rVof e u *m M). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | funmx | |
funmx_linearM : linear (funmx M).
Proof.
by rewrite /funmx => x u v; rewrite linearP mulmxDl -scalemxAl linearP.
Qed.
HB.instance Definition _ M := GRing.isSemilinear.Build F _ _ _ (funmx M)
(GRing.semilinear_linear (funmx_linear M)). | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | funmx_linear | |
hommxM : 'Hom(uT, vT) := linfun (funmx M). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommx | |
hommx_linear: linear hommx.
Proof.
rewrite /hommx; move=> x A B; apply/lfunP=> u; do !rewrite lfunE/=.
by rewrite /funmx mulmxDr -scalemxAr linearP.
Qed.
HB.instance Definition _ M := GRing.isSemilinear.Build F _ _ _ hommx
(GRing.semilinear_linear hommx_linear).
Hypothesis e_basis: basis_of {:uT} e.
Hypothesis f_basis: basis_of {:vT} e'. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommx_linear | |
mxofK: cancel mxof hommx.
Proof.
by move=> h; apply/lfunP=> u; rewrite lfunE/= /funmx mul_rV_lin1/= !rVofK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mxofK | |
hommxK: cancel hommx mxof.
Proof.
move=> M; apply/matrixP => i j; rewrite !mxE/= lfunE/=.
by rewrite /funmx vecofK// -rowE coord_vecof// mxE.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommxK | |
mul_mxofphi u : u *m mxof phi = rVof e' (phi (vecof e u)).
Proof. by rewrite mul_rV_lin1/=. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mul_mxof | |
hommxEM u : hommx M u = vecof e' (rVof e u *m M).
Proof. by rewrite -[M in RHS]hommxK mul_mxof !rVofK//. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommxE | |
rVof_mulM u : rVof e u *m M = rVof e' (hommx M u).
Proof. by rewrite hommxE vecofK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | rVof_mul | |
hom_vecof(phi : 'Hom(uT, vT)) u :
phi (vecof e u) = vecof e' (u *m mxof phi).
Proof. by rewrite mul_mxof rVofK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hom_vecof | |
rVof_app(phi : 'Hom(uT, vT)) u :
rVof e' (phi u) = rVof e u *m mxof phi.
Proof. by rewrite mul_mxof !rVofK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | rVof_app | |
vecof_mulM u : vecof e' (u *m M) = hommx M (vecof e u).
Proof. by rewrite hommxE vecofK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | vecof_mul | |
mxof_eq0phi : (mxof phi == 0) = (phi == 0).
Proof. by rewrite -(inj_eq (can_inj hommxK)) mxofK linear0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mxof_eq0 | |
hommx_eq0M : (hommx M == 0) = (M == 0).
Proof. by rewrite -(inj_eq (can_inj mxofK)) hommxK linear0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommx_eq0 | |
mxof_comp(phi : 'Hom(uT, vT)) (psi : 'Hom(vT, wT)) :
mxof e g (psi \o phi)%VF = mxof e f phi *m mxof f g psi.
Proof.
apply/matrixP => i k; rewrite !(mxE, comp_lfunE, lfunE) /=.
rewrite [phi _](coord_basis f_basis) ?memvf// 2!linear_sum/=.
by apply: eq_bigr => j _ /=; rewrite !mxE !linearZ/= !vecof_delta.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mxof_comp | |
hommx_mul(A : 'M_(m,n)) (B : 'M_(n, p)) :
hommx e g (A *m B) = (hommx f g B \o hommx e f A)%VF.
Proof.
by apply: (can_inj (mxofK e_basis g_basis)); rewrite mxof_comp !hommxK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommx_mul | |
msof(V : {vspace vT}) : 'M_n := mxof e e (projv V). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | msof | |
vsof(M : 'M[F]_n) := limg (hommx e e M). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | vsof | |
mxof1: free e -> mxof e e \1 = 1%:M.
Proof.
by move=> eF; apply/matrixP=> i j; rewrite !mxE vecof_delta lfunE coord_free.
Qed.
Hypothesis e_basis : basis_of {:vT} e. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mxof1 | |
hommx1: hommx e e 1%:M = \1%VF.
Proof. by rewrite -mxof1 ?(basis_free e_basis)// mxofK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | hommx1 | |
msofK: cancel msof vsof.
Proof. by rewrite /msof /vsof; move=> V; rewrite mxofK// limg_proj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | msofK | |
mem_vecofu (V : {vspace vT}) : (vecof e u \in V) = (u <= msof V)%MS.
Proof.
apply/idP/submxP=> [|[v ->{u}]]; last by rewrite -hom_vecof// memv_proj.
rewrite -[V in X in X -> _]msofK => /memv_imgP[v _].
by move=> /(canRL (vecofK _)) ->//; rewrite -rVof_mul//; eexists.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | mem_vecof | |
rVof_subu M : (rVof e u <= M)%MS = (u \in vsof M).
Proof.
apply/submxP/memv_imgP => [[v /(canRL (rVofK _)) ->//]|[v _ ->]]{u}.
by exists (vecof e v); rewrite ?memvf// -vecof_mul.
by exists (rVof e v); rewrite -rVof_mul.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | rVof_sub | |
vsof_subM V : (vsof M <= V)%VS = (M <= msof V)%MS.
Proof.
apply/subvP/rV_subP => [MsubV _/submxP[u ->]|VsubM _/memv_imgP[u _ ->]].
by rewrite -mem_vecof MsubV// -rVof_sub vecofK// submxMl.
by rewrite -[V]msofK -rVof_sub VsubM// -rVof_mul// submxMl.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | vsof_sub | |
msof_subV M : (msof V <= M)%MS = (V <= vsof M)%VS.
Proof.
apply/rV_subP/subvP => [VsubM v vV|MsubV _/submxP[u ->]].
by rewrite -rVof_sub VsubM// -mem_vecof rVofK.
by rewrite mul_mxof rVof_sub MsubV// memv_proj.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | msof_sub | |
vsofKM : (msof (vsof M) == M)%MS.
Proof. by rewrite msof_sub -vsof_sub subvv. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | vsofK | |
sub_msof: {mono msof : V V' / (V <= V')%VS >-> (V <= V')%MS}.
Proof. by move=> V V'; rewrite msof_sub msofK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | sub_msof | |
sub_vsof: {mono vsof : M M' / (M <= M')%MS >-> (M <= M')%VS}.
Proof. by move=> M M'; rewrite vsof_sub (eqmxP (vsofK _)). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | sub_vsof | |
msof0: msof 0 = 0.
Proof.
apply/eqP; rewrite -submx0; apply/rV_subP => v.
by rewrite -mem_vecof memv0 vecof_eq0// => /eqP->; rewrite sub0mx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | msof0 | |
vsof0: vsof 0 = 0%VS.
Proof. by apply/vspaceP=> v; rewrite memv0 -rVof_sub submx0 rVof_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | vsof0 | |
msof_eq0V : (msof V == 0) = (V == 0%VS).
Proof. by rewrite -(inj_eq (can_inj msofK)) msof0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | msof_eq0 | |
vsof_eq0M : (vsof M == 0%VS) = (M == 0).
Proof.
rewrite (sameP eqP eqmx0P) -!(eqmxP (vsofK M)) (sameP eqmx0P eqP) -msof0.
by rewrite (inj_eq (can_inj msofK)).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | vsof_eq0 | |
leigenspace(phi : 'End(uT)) a := lker (phi - a *: \1%VF). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | leigenspace | |
leigenvaluephi a := leigenspace phi a != 0%VS.
Local Notation m := (\dim {:uT}).
Variables (e : m.-tuple uT).
Hypothesis e_basis: basis_of {:uT} e.
Let e_free := basis_free e_basis. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | leigenvalue | |
lker_kerphi : lker phi = vsof e (kermx (mxof e e phi)).
Proof.
apply/vspaceP => v; rewrite memv_ker -rVof_sub// (sameP sub_kermxP eqP).
by rewrite -rVof_app// rVof_eq0.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | lker_ker | |
limgEphi : limg phi = vsof e (mxof e e phi).
Proof.
apply/vspaceP => v; rewrite -rVof_sub//.
apply/memv_imgP/submxP => [[u _ ->]|[u /(canRL (rVofK _)) ->//]].
by exists (rVof e u); rewrite -rVof_app.
by exists (vecof e u); rewrite ?memvf// -hom_vecof.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | limgE | |
leigenspaceEf a : leigenspace f a = vsof e (eigenspace (mxof e e f) a).
Proof. by rewrite [LHS]lker_ker linearB linearZ/= mxof1// scalemx1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype bigop finfun tuple",
"From mathcomp Require Import ssralg matrix mxalgebra zmodp"
] | algebra/vector.v | leigenspaceE | |
Zp_opp_subproofi : (p - i) %% p < p.
Proof. by case: p i => [[]//|k] i; apply/ltn_pmod. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_opp_subproof | |
Zp_oppi := Ordinal (Zp_opp_subproof i). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_opp | |
Zp_add_subproofi j : (i + j) %% p < p.
Proof. by case: p i j => [[]//|k] i j; apply/ltn_pmod. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_add_subproof | |
Zp_addi j := Ordinal (Zp_add_subproof i j). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_add | |
Zp_mul_subproofi j : (i * j) %% p < p.
Proof. by case: p i j => [[]//|k] i j; apply/ltn_pmod. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mul_subproof | |
Zp_muli j := Ordinal (Zp_mul_subproof i j). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mul | |
Zp_inv_subproofi : (egcdn i p).1 %% p < p.
Proof. by case: p i => [[]//|k] i; apply/ltn_pmod. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_inv_subproof | |
Zp_invi := if coprime p i then Ordinal (Zp_inv_subproof i) else i. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_inv | |
Zp_addA: associative Zp_add.
Proof.
by move=> x y z; apply: val_inj; rewrite /= modnDml modnDmr addnA.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_addA | |
Zp_addC: commutative Zp_add.
Proof. by move=> x y; apply: val_inj; rewrite /= addnC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_addC | |
Zp_mulC: commutative Zp_mul.
Proof. by move=> x y; apply: val_inj; rewrite /= mulnC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulC | |
Zp_mulA: associative Zp_mul.
Proof.
by move=> x y z; apply: val_inj; rewrite /= modnMml modnMmr mulnA.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulA | |
Zp_mul_addr: right_distributive Zp_mul Zp_add.
Proof.
by move=> x y z; apply: val_inj; rewrite /= modnMmr modnDm mulnDr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mul_addr | |
Zp_mul_addl: left_distributive Zp_mul Zp_add.
Proof. by move=> x y z; rewrite -!(Zp_mulC z) Zp_mul_addr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mul_addl | |
Zp_inv_outi : ~~ coprime p i -> Zp_inv i = i.
Proof. by rewrite /Zp_inv => /negPf->. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_inv_out | |
inZpi := Ordinal (ltn_pmod i (ltn0Sn p')). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | inZp | |
modZpx : x %% p = x.
Proof. by rewrite modn_small ?ltn_ord. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | modZp | |
valZpKx : inZp x = x.
Proof. by apply: val_inj; rewrite /= modZp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | valZpK | |
Zp0: 'I_p := ord0. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp0 | |
Zp1:= inZp 1. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp1 | |
Zp_add0z: left_id Zp0 Zp_add.
Proof. by move=> x; apply: val_inj; rewrite /= modZp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_add0z | |
Zp_addNz: left_inverse Zp0 Zp_opp Zp_add.
Proof.
by move=> x; apply: val_inj; rewrite /= modnDml subnK ?modnn // ltnW.
Qed.
HB.instance Definition _ :=
GRing.isZmodule.Build 'I_p (@Zp_addA _) (@Zp_addC _) Zp_add0z Zp_addNz. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_addNz | |
Definition_ := [finGroupMixin of 'I_p for +%R]. | HB.instance | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Definition | |
Zp_mul1z: left_id Zp1 Zp_mul.
Proof. by move=> x; apply: val_inj; rewrite /= modnMml mul1n modZp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mul1z | |
Zp_mulz1: right_id Zp1 Zp_mul.
Proof. by move=> x; rewrite Zp_mulC Zp_mul1z. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulz1 | |
Zp_mulVzx : coprime p x -> Zp_mul (Zp_inv x) x = Zp1.
Proof.
move=> co_p_x; apply: val_inj; rewrite /Zp_inv co_p_x /= modnMml.
by rewrite -(chinese_modl co_p_x 1 0) /chinese addn0 mul1n mulnC.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulVz | |
Zp_mulzVx : coprime p x -> Zp_mul x (Zp_inv x) = Zp1.
Proof. by move=> Ux; rewrite /= Zp_mulC Zp_mulVz. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulzV | |
Zp_intro_unitx y : Zp_mul y x = Zp1 -> coprime p x.
Proof.
case=> yx1; have:= coprimen1 p.
by rewrite -coprime_modr -yx1 coprime_modr coprimeMr; case/andP.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_intro_unit | |
Zp_mulrnx n : x *+ n = inZp (x * n).
Proof.
apply: val_inj => /=; elim: n => [|n IHn]; first by rewrite muln0 modn_small.
by rewrite !GRing.mulrS /= IHn modnDmr mulnS.
Qed.
Local Open Scope group_scope. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulrn | |
Zp_mulgC: @commutative 'I_p _ mul.
Proof. exact: Zp_addC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_mulgC | |
Zp_abelian: abelian [set: 'I_p].
Proof. exact: FinRing.zmod_abelian. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_abelian | |
Zp_expgx n : x ^+ n = inZp (x * n).
Proof. exact: Zp_mulrn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_expg | |
Zp1_expgzx : Zp1 ^+ x = x.
Proof.
rewrite Zp_expg; apply/val_inj.
by move: (Zp_mul1z x) => /(congr1 val).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp1_expgz | |
Zp_cycle: setT = <[Zp1]>.
Proof. by apply/setP=> x; rewrite -[x]Zp1_expgz inE groupX ?mem_gen ?set11. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_cycle | |
order_Zp1: #[Zp1] = p.
Proof. by rewrite orderE -Zp_cycle cardsT card_ord. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | order_Zp1 | |
ord1: all_equal_to (0 : 'I_1).
Proof. exact: ord1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | ord1 | |
lshift0m n : lshift m (0 : 'I_n.+1) = (0 : 'I_(n + m).+1).
Proof. exact: val_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | lshift0 | |
rshift1n : @rshift 1 n =1 lift (0 : 'I_n.+1).
Proof. by move=> i; apply: val_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | rshift1 | |
split1n i :
split (i : 'I_(1 + n)) = oapp (@inr _ _) (inl _ 0) (unlift 0 i).
Proof.
case: unliftP => [i'|] -> /=.
by rewrite -rshift1 (unsplitK (inr _ _)).
by rewrite -(lshift0 n 0) (unsplitK (inl _ _)).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | split1 | |
big_ord1:= big_ord1 (only parsing).
#[deprecated(since="mathcomp 2.3.0", note="Use bigop.big_ord1_cond instead.")] | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | big_ord1 | |
big_ord1_cond:= big_ord1_cond (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | big_ord1_cond | |
Zp_nontrivial: Zp1 != 0 :> 'I_p. Proof. by []. Qed.
HB.instance Definition _ :=
GRing.Zmodule_isComNzRing.Build 'I_p
(@Zp_mulA _) (@Zp_mulC _) (@Zp_mul1z _) (@Zp_mul_addl _) Zp_nontrivial.
HB.instance Definition _ :=
GRing.ComNzRing_hasMulInverse.Build 'I_p
(@Zp_mulVz _) (@Zp_intro_unit _) (@Zp_inv_out _). | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_nontrivial | |
Zp_natn : n%:R = inZp n :> 'I_p.
Proof. by apply: val_inj; rewrite [n%:R]Zp_mulrn /= modnMml mul1n. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_nat | |
natr_Zp(x : 'I_p) : x%:R = x.
Proof. by rewrite Zp_nat valZpK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | natr_Zp | |
natr_negZp(x : 'I_p) : (- x)%:R = - x.
Proof. by apply: val_inj; rewrite /= Zp_nat /= modn_mod. Qed.
Local Open Scope group_scope. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | natr_negZp | |
unit_Zp_mulgC: @commutative {unit 'I_p} _ mul.
Proof. by move=> u v; apply: val_inj; rewrite /= GRing.mulrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | unit_Zp_mulgC | |
unit_Zp_expg(u : {unit 'I_p}) n :
val (u ^+ n) = inZp (val u ^ n) :> 'I_p.
Proof.
apply: val_inj => /=; elim: n => [|n IHn] //.
by rewrite expgS /= IHn expnS modnMmr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | unit_Zp_expg | |
Zp_truncp := p.-2. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_trunc | |
add_1_Zpp (x : 'Z_p) : 1 + x = ordS x.
Proof. by case: p => [|[|p]] in x *; apply/val_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | add_1_Zp | |
add_Zp_1p (x : 'Z_p) : x + 1 = ordS x.
Proof. by rewrite addrC add_1_Zp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | add_Zp_1 | |
sub_Zp_1p (x : 'Z_p) : x - 1 = ord_pred x.
Proof. by apply: (addIr 1); rewrite addrNK add_Zp_1 ord_predK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | sub_Zp_1 | |
add_N1_Zpp (x : 'Z_p) : -1 + x = ord_pred x.
Proof. by rewrite addrC sub_Zp_1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | add_N1_Zp | |
Zp:= if p > 1 then [set: 'Z_p] else 1%g. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp | |
units_Zp:= [set: {unit 'Z_p}]. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | units_Zp | |
Zp_cast: p > 1 -> (Zp_trunc p).+2 = p.
Proof. by case: p => [|[]]. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_cast | |
val_Zp_nat(p_gt1 : p > 1) n : (n%:R : 'Z_p) = (n %% p)%N :> nat.
Proof. by rewrite Zp_nat /= Zp_cast. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | val_Zp_nat | |
Zp_nat_mod(p_gt1 : p > 1)m : (m %% p)%:R = m%:R :> 'Z_p.
Proof. by apply: ord_inj; rewrite !val_Zp_nat // modn_mod. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_nat_mod | |
pchar_Zp: p > 1 -> p%:R = 0 :> 'Z_p.
Proof. by move=> p_gt1; rewrite -Zp_nat_mod ?modnn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | pchar_Zp | |
unitZpEx : p > 1 -> ((x%:R : 'Z_p) \is a GRing.unit) = coprime p x.
Proof.
move=> p_gt1; rewrite qualifE /=.
by rewrite val_Zp_nat ?Zp_cast ?coprime_modr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | unitZpE | |
Zp_group_set: group_set Zp.
Proof. by rewrite /Zp; case: (p > 1); apply: groupP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_group_set | |
Zp_group:= Group Zp_group_set. | Canonical | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool choice eqtype ssrnat seq",
"From mathcomp Require Import div fintype bigop finset prime fingroup perm",
"From mathcomp Require Import ssralg finalg countalg"
] | algebra/zmodp.v | Zp_group |
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