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 // add0r.
by rewrite proj_ortho_id ?proj_mx_sub// add_proj_mx.
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_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=> -[].
rewrite -addn1 => A leq_Sm_n.
have lemSm : (m <= m + 1)%N by rewrite addn1.
have ltmSm : (m < m + 1)%N by rewrite addn1.
have lemn : (m <= n)%N by rewrite ltnW // -addn1.
have [B Bortho] := IHm (usubmx A) lemn.
move=> /forallP /= subAB.
have [v /and4P [vBn v_neq0 dAv_ge0 dAsub]] :
exists v, [&& B '_|_ v, v != 0, '[dsubmx A, v] >= 0 & (dsubmx A <= B + v)%MS].
have := add_proj_ortho B (dsubmx A).
set BoSn := (_ *m proj_ortho _^!%MS) => pBE.
have [BoSn_eq0|BoSn_neq0] := eqVneq BoSn 0.
rewrite BoSn_eq0 addr0 in pBE.
have /rowV0Pn [v vBn v_neq0] : B^!%MS != 0.
rewrite -mxrank_eq0 rank_ortho -lt0n subn_gt0.
by rewrite mxrank_unitary // -addn1.
rewrite orthomx_sym in vBn.
exists v; rewrite vBn v_neq0 -pBE.
rewrite ['[_, _]](hermmx_eq0P _ _) ?lexx //=.
rewrite (submx_trans (proj_ortho_sub _ _)) //.
by rewrite -{1}[B]addr0 addmx_sub_adds ?sub0mx.
by rewrite (submx_trans _ vBn) // proj_ortho_sub.
pose c := (sqrtC '[BoSn])^-1; have c_gt0 : c > 0.
by rewrite invr_gt0 sqrtC_gt0 lt_def ?dnorm_eq0 ?dnorm_ge0 BoSn_neq0.
exists BoSn; apply/and4P; split => //.
- by rewrite orthomx_sym ?proj_ortho_sub // /gtr_eqF.
- rewrite
... | 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_sym.
by do 2!rewrite eqmx_schmidt_free ?eq_row_base ?row_base_free // orthomx_sym.
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 | 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 suff: False by []; move: i; rewrite n_eq0 => -[].
rewrite leq_eqVlt => /predU1P [n_eqSN|/IHN//].
have /andP [n_gt0 n_small] : (n > 0)%N && (n - 1 <= N)%N.
by rewrite n_eqSN /= subn1.
move=> As As_comm;
have [v vN0 /allP /= vP] := common_eigenvector n_gt0 As_comm.
suff: exists2 P : 'M[C]_(\rank v + \rank v^!, n), P \is unitarymx &
all (fun A => is_trig_mx (P *m A *m ( P^t* ))) As.
rewrite add_rank_ortho // => -[P P_unitary] /=; rewrite -invmx_unitary//.
by under eq_all do rewrite -conjumx ?unitarymx_unit//; exists P.
pose S := schmidt_complete v.
pose r A := S *m A *m S^t*.
have vSvo X : stablemx v X ->
schmidt (row_base v) *m X *m schmidt (row_base v^!%MS) ^t* = 0.
move=> /eigenvectorP [a v_in].
rewrite (eigenspaceP (_ : (_ <= _ a))%MS); last first.
by rewrite eqmx_schmidt_free ?row_base_free ?eq_row_base.
rewrite -scalemxAl (orthomx1P _) ?scaler0 //.
by do 2!rewrite eqmx_schmidt_free ?row_base_free ?eq_row_base // orthomx_sym.
have drrE X : drsubmx (r X) =
schmidt (row_base v^!%MS) *m X *m schmidt (row_base v^!%MS) ^t*.
by rewrite /r mul_col_mx tr_col_mx map_row_mx mul_col_row block_mxKdr.
have vSv X a : (v <= eigenspace X a)%MS ->
schmidt (row_base v
... | 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 rewrite !PP ?(mem_head, in_cons, orbT).
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 | 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_unitary //.
rewrite !trmxCK ![_ *m P *m _]mulmxtVK //.
by rewrite -[X in X *m P]mulmxA tr_diag_mx map_diag_mx diag_mxC mulmxA.
move=> /cotrigonalization2 [P Punitary /andP[]] PA PATC.
have Punit := unitarymx_unit Punitary.
suff: similar_diag P A.
move=> /similar_diagPex[D] PAD; exists (P, D) => //=.
by rewrite -conjVmx//; exact/similarLR.
apply/similar_diagPp => // i j; case: ltngtP => // [lt_ij|lt_ji] _.
by have /is_trig_mxP-> := PA.
have /is_trig_mxP -/(_ j i lt_ji)/eqP := PATC.
rewrite !conjumx// invmx_unitary// -[P as X in X *m _]trmxCK.
by rewrite -!map_mxM -!trmx_mul mulmxA 2!mxE conjC_eq0 => /eqP.
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 | 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; constructor; apply/eqP.
apply: contra Ann; rewrite invmx1 mul1mx mulmx1 => /eqP->.
suff -> : diag_mx 0 = 0 by rewrite qualifE trmx0 (map_mx0 conjC).
by move=> ? ?; apply/matrixP=> i j; rewrite !mxE mul0rn.
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 | 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_comp.
rewrite tr_diag_mx map_diag_mx (map_mx_id (@conjCK _)).
rewrite -[in RHS]invmx_unitary ?spectral_unitarymx //.
have := is_hermitianmxP _ _ _ Ahermi; rewrite expr0 scale1r => <-.
rewrite {1}A_eq mulmxA => /(congr1 (mulmx^~ (invmx (spectralmx A)))).
rewrite !mulmxK ?spectral_unit//.
move=> /(congr1 (mulmx (spectralmx A))); rewrite !mulKVmx ?spectral_unit//.
move=> eq_A_conjA; apply/mxOverP => i j; rewrite ord1 {i}.
have /matrixP /(_ j j) := eq_A_conjA; rewrite !mxE eqxx !mulr1n.
by move=> /esym/CrealP.
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 | 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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