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 |
|---|---|---|---|---|---|---|
orthomx_disjn p q (A : 'M[C]_(p, n)) (B :'M_(q, n)) :
A '_|_ B -> (A :&: B = 0)%MS.
Proof.
move=> nAB; apply/eqP/rowV0Pn => [[v]]; rewrite sub_capmx => /andP [vA vB].
apply/negP; rewrite negbK.
by rewrite -(dnorm_eq0 (@dotmx n)) -orthomxE (orthomxP _ _ _ nAB).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | orthomx_disj | |
orthomx_ortho_disjn p (A : 'M[C]_(p, n)) : (A :&: A^! = 0)%MS.
Proof. exact/orthomx_disj/ortho_mx_ortho. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | orthomx_ortho_disj | |
rank_orthop n (A : 'M[C]_(p, n)) : \rank A^! = (n - \rank A)%N.
Proof. by rewrite mxrank_ker mul1mx mxrank_map mxrank_tr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | rank_ortho | |
add_rank_orthop n (A : 'M[C]_(p, n)) : (\rank A + \rank A^!)%N = n.
Proof. by rewrite rank_ortho subnKC ?rank_leq_col. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | add_rank_ortho | |
addsmx_orthop n (A : 'M[C]_(p, n)) : (A + A^! :=: 1%:M)%MS.
Proof.
apply/eqmxP/andP; rewrite submx1; split=> //.
rewrite -mxrank_leqif_sup ?submx1 ?mxrank1 ?(mxdirectP _) /= ?add_rank_ortho //.
by rewrite mxdirect_addsE /= ?mxdirectE ?orthomx_ortho_disj !eqxx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | addsmx_ortho | |
ortho_idp n (A : 'M[C]_(p, n)) : (A^!^! :=: A)%MS.
Proof.
apply/eqmx_sym/eqmxP.
by rewrite -mxrank_leqif_eq 1?orthomx_sym // !rank_ortho subKn // ?rank_leq_col.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | ortho_id | |
submx_orthop m n (U : 'M[C]_(p, n)) (V : 'M_(m, n)) :
(U^! <= V^!)%MS = (V <= U)%MS.
Proof. by rewrite orthomx_sym ortho_id. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | submx_ortho | |
proj_orthop n (U : 'M[C]_(p, n)) := proj_mx <<U>>%MS U^!%MS. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_ortho | |
sub_adds_genmx_ortho(p m n : nat) (U : 'M[C]_(p, n)) (W : 'M_(m, n)) :
(W <= <<U>> + U^!)%MS.
Proof.
by rewrite !(adds_eqmx (genmxE _) (eqmx_refl _)) addsmx_ortho submx1.
Qed.
Local Hint Resolve sub_adds_genmx_ortho : core. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | sub_adds_genmx_ortho | |
cap_genmx_orthop n (U : 'M[C]_(p, n)) : (<<U>> :&: U^!)%MS = 0.
Proof.
apply/eqmx0P; rewrite !(cap_eqmx (genmxE _) (eqmx_refl _)).
by rewrite orthomx_ortho_disj; exact/eqmx0P.
Qed.
Local Hint Resolve cap_genmx_ortho : core. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | cap_genmx_ortho | |
proj_ortho_subp m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W *m proj_ortho U <= U)%MS.
Proof. by rewrite (submx_trans (proj_mx_sub _ _ _)) // genmxE. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_ortho_sub | |
proj_ortho_compl_subp m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W - W *m proj_ortho U <= U^!)%MS.
Proof. by rewrite proj_mx_compl_sub // addsmx_ortho submx1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_ortho_compl_sub | |
proj_ortho_idp m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W <= U)%MS -> W *m proj_ortho U = W.
Proof. by move=> WU; rewrite proj_mx_id ?genmxE. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_ortho_id | |
proj_ortho_0p m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W <= U^!)%MS -> W *m proj_ortho U = 0.
Proof. by move=> WUo; rewrite proj_mx_0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_ortho_0 | |
add_proj_orthop m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
W *m proj_ortho U + W *m proj_ortho U^!%MS = W.
Proof.
rewrite -[W in LHS](@add_proj_mx _ _ _ <<U>>%MS U^!%MS W)//.
rewrite !mulmxDl proj_ortho_id ?proj_ortho_sub //.
rewrite proj_ortho_0 ?proj_mx_sub // addr0.
rewrite proj_ortho_0 ?ortho_id ?proj_ortho_sub // add... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | add_proj_ortho | |
proj_ortho_projm n (U : 'M_(m, n)) : let P := proj_ortho U in P *m P = P.
Proof. by rewrite /= proj_mx_proj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_ortho_proj | |
proj_orthoEp n (U : 'M_(p, n)) : (proj_ortho U :=: U)%MS.
Proof.
apply/eqmxP/andP; split; first by rewrite -proj_ortho_proj proj_ortho_sub.
by rewrite -[X in (X <= _)%MS](proj_ortho_id (submx_refl U)) mulmx_sub.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | proj_orthoE | |
orthomx_proj_mx_orthop p' m m' n
(A : 'M_(p, n)) (A' : 'M_(p', n))
(W : 'M_(m, n)) (W' : 'M_(m', n)) :
A '_|_ A' -> W *m proj_ortho A '_|_ W' *m proj_ortho A'.
Proof.
rewrite orthomx_sym => An.
rewrite mulmx_sub // orthomx_sym (eqmx_ortho _ (proj_orthoE _)).
by rewrite (submx_trans _ An) // proj_ortho_sub.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | orthomx_proj_mx_ortho | |
schmidt_subproofm n (A : 'M[C]_(m, n)) : (m <= n)%N ->
exists2 B : 'M_(m, n), B \is unitarymx & [forall i : 'I_m,
(row i A <= (\sum_(k < m | (k <= i)%N) <<row k B>>))%MS
&& ('[row i A, row i B] >= 0) ].
Proof.
elim: m A => [|m IHm].
exists (pid_mx n); first by rewrite qualifE !thinmx0.
by apply/forallP=> -[... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | schmidt_subproof | |
schmidtm n (A : 'M[C]_(m, n)) :=
if (m <= n)%N =P true is ReflectT le_mn
then projT1 (sig2_eqW (schmidt_subproof A le_mn))
else A. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | schmidt | |
schmidt_unitarymxm n (A : 'M[C]_(m, n)) : (m <= n)%N ->
schmidt A \is unitarymx.
Proof. by rewrite /schmidt; case: eqP => // ?; case: sig2_eqW. Qed.
Hint Resolve schmidt_unitarymx : core. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | schmidt_unitarymx | |
row_schmidt_subm n (A : 'M[C]_(m, n)) i :
(row i A <= (\sum_(k < m | (k <= i)%N) <<row k (schmidt A)>>))%MS.
Proof.
rewrite /schmidt; case: eqP => // ?.
by case: sig2_eqW => ? ? /= /forallP /(_ i) /andP[].
by apply/(sumsmx_sup i) => //; rewrite genmxE.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | row_schmidt_sub | |
form1_row_schmidtm n (A : 'M[C]_(m, n)) i :
'[row i A, row i (schmidt A)] >= 0.
Proof.
rewrite /schmidt; case: eqP => // ?; rewrite ?dnorm_ge0 //.
by case: sig2_eqW => ? ? /= /forallP /(_ i) /andP[].
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | form1_row_schmidt | |
schmidt_subm n (A : 'M[C]_(m, n)) : (A <= schmidt A)%MS.
Proof.
apply/row_subP => i; rewrite (submx_trans (row_schmidt_sub _ _)) //.
by apply/sumsmx_subP => /= j le_ji; rewrite genmxE row_sub.
Qed.
Hint Resolve schmidt_sub : core. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | schmidt_sub | |
eqmx_schmidt_fullm n (A : 'M[C]_(m, n)) :
row_full A -> (schmidt A :=: A)%MS.
Proof.
move=> Afull; apply/eqmx_sym/eqmxP; rewrite -mxrank_leqif_eq //.
by rewrite eqn_leq mxrankS //= (@leq_trans n) ?rank_leq_col ?col_leq_rank.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | eqmx_schmidt_full | |
eqmx_schmidt_freem n (A : 'M[C]_(m, n)) :
row_free A -> (schmidt A :=: A)%MS.
Proof.
move=> Afree; apply/eqmx_sym/eqmxP; rewrite -mxrank_leqif_eq //.
by rewrite eqn_leq mxrankS //= (@leq_trans m) ?rank_leq_row // ?row_leq_rank.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | eqmx_schmidt_free | |
schmidt_completem n (V : 'M[C]_(m, n)) :=
col_mx (schmidt (row_base V)) (schmidt (row_base V^!%MS)). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | schmidt_complete | |
schmidt_complete_unitarymxm n (V : 'M[C]_(m, n)) :
schmidt_complete V \is unitarymx.
Proof.
apply/unitarymxP; rewrite tr_col_mx map_row_mx mul_col_row.
rewrite !(unitarymxP _) ?schmidt_unitarymx ?rank_leq_col //.
move=> [:nsV]; rewrite !(orthomx1P _) -?scalar_mx_block //;
[abstract: nsV|]; last by rewrite orthomx_s... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | schmidt_complete_unitarymx | |
cotrigonalizationn (As : seq 'M[C]_n) :
{in As &, forall A B, comm_mx A B} ->
cotrigonalizable_in (@unitarymx C n n) As.
Proof.
elim: n {-2}n (leqnn n) As => [|N IHN] n.
rewrite leqn0 => /eqP n_eq0.
exists 1%:M; first by rewrite qualifE mul1mx trmx1 map_mx1.
apply/allP => ? ?; apply/is_trig_mxP => i j.
by s... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | cotrigonalization | |
Schurn (A : 'M[C]_n) : (n > 0)%N ->
trigonalizable_in (@unitarymx C n n) A.
Proof.
case: n => [//|n] in A * => _; have [] := @cotrigonalization _ [:: A].
by move=> ? ? /=; rewrite !in_cons !orbF => /eqP-> /eqP->.
by move=> P P_unitary /=; rewrite andbT=> A_trigo; exists P.
Qed. | Theorem | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | Schur | |
cotrigonalization2n (A B : 'M[C]_n) : A *m B = B *m A ->
exists2 P : 'M[C]_n, P \is unitarymx &
similar_trig P A && similar_trig P B.
Proof.
move=> AB_comm; have [] := @cotrigonalization _ [:: A; B].
by move=> ??; rewrite !inE => /orP[]/eqP->/orP[]/eqP->.
move=> P Punitary /allP /= PP; exists P => //.
by rewrit... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | cotrigonalization2 | |
orthomx_spectral_subproofn {A : 'M[C]_n} : reflect
(exists2 sp : 'M_n * 'rV_n,
sp.1 \is unitarymx &
A = invmx sp.1 *m diag_mx sp.2 *m sp.1)
(A \is normalmx).
Proof.
apply: (iffP normalmxP); last first.
move=> [[/= P D] P_unitary ->].
rewrite !trmx_mul !map_mxM !mulmxA invmx_unita... | Theorem | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | orthomx_spectral_subproof | |
spectralmxn (A : 'M[C]_n) : 'M[C]_n :=
if @orthomx_spectral_subproof _ A is ReflectT P
then (projT1 (sig2_eqW P)).1 else 1%:M. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | spectralmx | |
spectral_diagn (A : 'M[C]_n) : 'rV_n :=
if @orthomx_spectral_subproof _ A is ReflectT P
then (projT1 (sig2_eqW P)).2 else 0. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | spectral_diag | |
spectral_unitarymxn (A : 'M[C]_n) : spectralmx A \is unitarymx.
Proof.
rewrite /spectralmx; case: orthomx_spectral_subproof; last first.
by move=> _; apply/unitarymxP; rewrite trmx1 map_mx1 mulmx1.
by move=> ?; case: sig2_eqW.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | spectral_unitarymx | |
spectral_unitn (A : 'M[C]_n) : spectralmx A \in unitmx.
Proof. exact/unitarymx_unit/spectral_unitarymx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | spectral_unit | |
orthomx_spectralP{n} {A : 'M[C]_n}
(P := spectralmx A) (sp := spectral_diag A) :
reflect (A = invmx P *m diag_mx sp *m P) (A \is normalmx).
Proof.
rewrite /P /sp /spectralmx /spectral_diag.
case: orthomx_spectral_subproof.
by move=> Psp; case: sig2_eqW => //=; constructor.
move=> /orthomx_spectral_subproof Ann; c... | Theorem | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | orthomx_spectralP | |
hermitian_spectral_diag_realn (A : 'M[C]_n) : A \is hermsymmx ->
spectral_diag A \is a realmx.
Proof.
move=> Ahermi; have /hermitian_normalmx /orthomx_spectralP A_eq := Ahermi.
have /(congr1 ( fun X => X^t* )) := A_eq.
rewrite invmx_unitary ?spectral_unitarymx //.
rewrite !trmx_mul !map_mxM map_trmx trmxK -map_mx_com... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat",
"From mathcomp Require Import seq div fintype bigop ssralg finset fingroup zmodp",
"From mathcomp Require Import poly polydiv order ssrnum matrix mxalgebra vector",
"From mathcomp Require Impor... | algebra/spectral.v | hermitian_spectral_diag_real | |
addrA: associative (@add V).
Proof. exact: addrA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrA | |
addrC: commutative (@add V).
Proof. exact: addrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrC | |
add0r: left_id (@zero V) add.
Proof. exact: add0r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | add0r | |
addr0: right_id (@zero V) add.
Proof. exact: addr0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addr0 | |
addrCA: @left_commutative V V +%R. Proof. exact: addrCA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrCA | |
addrAC: @right_commutative V V +%R. Proof. exact: addrAC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrAC | |
addrACA: @interchange V +%R +%R. Proof. exact: addrACA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrACA | |
mulr0nx : x *+ 0 = 0. Proof. exact: mulr0n. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr0n | |
mulr1nx : x *+ 1 = x. Proof. exact: mulr1n. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr1n | |
mulr2nx : x *+ 2 = x + x. Proof. exact: mulr2n. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr2n | |
mulrSx n : x *+ n.+1 = x + (x *+ n). Proof. exact: mulrS. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrS | |
mulrSrx n : x *+ n.+1 = x *+ n + x. Proof. exact: mulrSr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrSr | |
mulrbx (b : bool) : x *+ b = (if b then x else 0).
Proof. exact: mulrb. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrb | |
mul0rnn : 0 *+ n = 0 :> V. Proof. exact: mul0rn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mul0rn | |
mulrnDln : {morph (fun x => x *+ n) : x y / x + y}.
Proof. exact: mulrnDl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrnDl | |
mulrnDrx m n : x *+ (m + n) = x *+ m + x *+ n.
Proof. exact: mulrnDr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrnDr | |
mulrnAx m n : x *+ (m * n) = x *+ m *+ n. Proof. exact: mulrnA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrnA | |
mulrnACx m n : x *+ m *+ n = x *+ n *+ m. Proof. exact: mulrnAC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrnAC | |
iter_addrn x y : iter n (+%R x) y = x *+ n + y.
Proof. exact: iter_addr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | iter_addr | |
iter_addr_0n x : iter n (+%R x) 0 = x *+ n.
Proof. exact: iter_addr_0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | iter_addr_0 | |
sumrMnlI r P (F : I -> V) n :
\sum_(i <- r | P i) F i *+ n = (\sum_(i <- r | P i) F i) *+ n.
Proof. exact: sumrMnl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sumrMnl | |
sumrMnrx I r P (F : I -> nat) :
\sum_(i <- r | P i) x *+ F i = x *+ (\sum_(i <- r | P i) F i).
Proof. exact: sumrMnr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sumrMnr | |
sumr_const(I : finType) (A : pred I) x : \sum_(i in A) x = x *+ #|A|.
Proof. exact: sumr_const. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sumr_const | |
sumr_const_natm n x : \sum_(n <= i < m) x = x *+ (m - n).
Proof. exact: sumr_const_nat. Qed.
#[deprecated(since="mathcomp 2.4.0",
note="Use Algebra.nmod_closed instead.")] | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sumr_const_nat | |
addr_closed:= nmod_closed. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addr_closed | |
addNr: @left_inverse V V V 0 -%R +%R. Proof. exact: addNr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addNr | |
addrN: @right_inverse V V V 0 -%R +%R. Proof. exact: addrN. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrN | |
subrr:= addrN. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subrr | |
addKr: @left_loop V V -%R +%R. Proof. exact: addKr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addKr | |
addNKr: @rev_left_loop V V -%R +%R. Proof. exact: addNKr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addNKr | |
addrK: @right_loop V V -%R +%R. Proof. exact: addrK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrK | |
addrNK: @rev_right_loop V V -%R +%R. Proof. exact: addrNK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrNK | |
subrK:= addrNK. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subrK | |
subrKCx y : x + (y - x) = y. Proof. by rewrite addrC subrK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subrKC | |
subKrx : involutive (fun y => x - y). Proof. exact: subKr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subKr | |
addrI: @right_injective V V V +%R. Proof. exact: addrI. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrI | |
addIr: @left_injective V V V +%R. Proof. exact: addIr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addIr | |
subrI: right_injective (fun x y => x - y). Proof. exact: subrI. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subrI | |
subIr: left_injective (fun x y => x - y). Proof. exact: subIr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subIr | |
opprK: @involutive V -%R. Proof. exact: opprK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | opprK | |
oppr_inj: @injective V V -%R. Proof. exact: oppr_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | oppr_inj | |
oppr0: -0 = 0 :> V. Proof. exact: oppr0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | oppr0 | |
oppr_eq0x : (- x == 0) = (x == 0). Proof. exact: oppr_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | oppr_eq0 | |
subr0x : x - 0 = x. Proof. exact: subr0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subr0 | |
sub0rx : 0 - x = - x. Proof. exact: sub0r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sub0r | |
opprBx y : - (x - y) = y - x. Proof. exact: opprB. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | opprB | |
opprD: {morph -%R: x y / x + y : V}. Proof. exact: opprD. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | opprD | |
addrKAz x y : (x + z) - (z + y) = x - y. Proof. exact: addrKA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addrKA | |
subrKAz x y : (x - z) + (z + y) = x + y. Proof. exact: subrKA. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subrKA | |
addr0_eqx y : x + y = 0 -> - x = y. Proof. exact: addr0_eq. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addr0_eq | |
subr0_eqx y : x - y = 0 -> x = y. Proof. exact: subr0_eq. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subr0_eq | |
subr_eqx y z : (x - z == y) = (x == y + z). Proof. exact: subr_eq. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subr_eq | |
subr_eq0x y : (x - y == 0) = (x == y). Proof. exact: subr_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subr_eq0 | |
addr_eq0x y : (x + y == 0) = (x == - y). Proof. exact: addr_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | addr_eq0 | |
eqr_oppx y : (- x == - y) = (x == y). Proof. exact: eqr_opp. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eqr_opp | |
eqr_oppLRx y : (- x == y) = (x == - y). Proof. exact: eqr_oppLR. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | eqr_oppLR | |
mulNrnx n : (- x) *+ n = x *- n. Proof. exact: mulNrn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulNrn | |
mulrnBln : {morph (fun x => x *+ n) : x y / x - y}.
Proof. exact: mulrnBl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrnBl | |
mulrnBrx m n : n <= m -> x *+ (m - n) = x *+ m - x *+ n.
Proof. exact: mulrnBr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrnBr | |
sumrNI r P (F : I -> V) :
(\sum_(i <- r | P i) - F i = - (\sum_(i <- r | P i) F i)).
Proof. exact: sumrN. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sumrN | |
sumrBI r (P : pred I) (F1 F2 : I -> V) :
\sum_(i <- r | P i) (F1 i - F2 i)
= \sum_(i <- r | P i) F1 i - \sum_(i <- r | P i) F2 i.
Proof. exact: sumrB. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sumrB | |
telescope_sumrn m (f : nat -> V) : n <= m ->
\sum_(n <= k < m) (f k.+1 - f k) = f m - f n.
Proof. exact: telescope_sumr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | telescope_sumr |
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