statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
simmxRL {P A B} : P \in unitmx ->
reflect (B = P *m A *m invmx P) (A ~_P B). | Proof. by move=> ?; apply: (iffP eqP); rewrite conjumx. Qed. | Lemma | simmxRL | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjumx",
"invmx",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmxLR {P A B} : P \in unitmx ->
reflect (A = conjmx (invmx P) B) (A ~_P B). | Proof.
by move=> Pu; rewrite conjVmx//; apply: (iffP (simmxRL Pu)) => ->;
rewrite !mulmxA ?(mulmxK, mulmxKV, mulVmx, mulmxV, mul1mx, mulmx1).
Qed. | Lemma | simmxLR | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjVmx",
"conjmx",
"invmx",
"mul1mx",
"mulVmx",
"mulmx1",
"mulmxA",
"mulmxK",
"mulmxKV",
"mulmxV",
"simmxRL",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simmx_minpoly {n} {P A B : 'M[F]_n.+1} : P \in unitmx ->
A ~_P B -> mxminpoly A = mxminpoly B. | Proof. by move=> Pu /eqP<-; rewrite mxminpoly_uconj. Qed. | Lemma | simmx_minpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"mxminpoly",
"mxminpoly_uconj",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_for_row_base m n (P : 'M[F]_(m, n)) (A : 'M_n) :
diagonalizable_for (row_base P) A = is_diag_mx (restrictmx P A). | Proof. by []. Qed. | Lemma | diagonalizable_for_row_base | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"diagonalizable_for",
"is_diag_mx",
"restrictmx",
"row_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_forPp m n (P : 'M[F]_(m, n)) A :
reflect (forall i j : 'I__, i != j :> nat -> conjmx P A i j = 0)
(diagonalizable_for P A). | Proof. exact: @is_diag_mxP. Qed. | Lemma | diagonalizable_forPp | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"diagonalizable_for",
"is_diag_mxP",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_forP n (P : 'M[F]_n) A : P \in unitmx ->
reflect (forall i j : 'I__, i != j :> nat -> (P *m A *m invmx P) i j = 0)
(diagonalizable_for P A). | Proof. by move=> Pu; rewrite -conjumx//; exact: is_diag_mxP. Qed. | Lemma | diagonalizable_forP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjumx",
"diagonalizable_for",
"invmx",
"is_diag_mxP",
"nat",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_forPex {m} {n} {P : 'M[F]_(m, n)} {A} :
reflect (exists D, A ~_P (diag_mx D)) (diagonalizable_for P A). | Proof. by apply: (iffP (diag_mxP _)) => -[D]/eqP; exists D. Qed. | Lemma | diagonalizable_forPex | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"diag_mx",
"diag_mxP",
"diagonalizable_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_forLR n {P : 'M[F]_n} {A} : P \in unitmx ->
reflect (exists D, A = conjmx (invmx P) (diag_mx D)) (diagonalizable_for P A). | Proof.
by move=> Punit; apply: (iffP diagonalizable_forPex) => -[D /(simmxLR Punit)]; exists D.
Qed. | Lemma | diagonalizable_forLR | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"diag_mx",
"diagonalizable_for",
"diagonalizable_forPex",
"invmx",
"simmxLR",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_for_mxminpoly {n} {P A : 'M[F]_n.+1}
(rs := undup [seq conjmx P A i i | i <- enum 'I_n.+1]) :
P \in unitmx -> diagonalizable_for P A ->
mxminpoly A = \prod_(r <- rs) ('X - r%:P). | Proof.
rewrite /rs => pu /(diagonalizable_forLR pu)[d {A rs}->].
rewrite mxminpoly_uconj ?unitmx_inv// mxminpoly_diag.
by rewrite [in X in _ = X](@eq_map _ _ _ (d 0))// => i; rewrite conjmxVK// mxE eqxx.
Qed. | Lemma | diagonalizable_for_mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conjmx",
"conjmxVK",
"diagonalizable_for",
"diagonalizable_forLR",
"enum",
"eq_map",
"eqxx",
"mxE",
"mxminpoly",
"mxminpoly_diag",
"mxminpoly_uconj",
"seq",
"undup",
"unitmx",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_for_sum (F : fieldType) (m n : nat) (p_ : 'I_n -> nat)
(V_ : forall i, 'M[F]_(p_ i, m)) (A : 'M[F]_m) :
mxdirect (\sum_i <<V_ i>>) ->
(forall i, stablemx (V_ i) A) ->
(forall i, row_free (V_ i)) ->
diagonalizable_for (\mxcol_i V_ i) A = [forall i, diagonalizable_for (V_ i) A]. | Proof.
move=> Vd VA rAV; have aVA : stablemx (\mxcol_i V_ i) A.
rewrite (eqmx_stable _ (eqmx_col _)) stablemx_sums//.
by move=> i; rewrite (eqmx_stable _ (genmxE _)).
apply/diagonalizable_forPex/'forall_diagonalizable_forPex => /=
[[D /(simmxPp aVA) +] i|/(_ _)/sigW DoA].
rewrite mxcol_mul -[D]submxrowK diag_... | Lemma | diagonalizable_for_sum | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"diag_mxrow",
"diagonalizable_for",
"diagonalizable_forPex",
"eq_bigr",
"eq_mxcol",
"eq_mxcolP",
"eqmx_col",
"eqmx_stable",
"genmxE",
"leqRHS",
"mul_mxdiag_mxcol",
"mxcol_mul",
"mxdirect",
"mxdirectP",
"nat",
"row_free",
"row_leq_rank",
"sigW",
"simmxPp",
"simmxW",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizable1 n (A : 'M[F]_n) :
codiagonalizable [:: A] <-> diagonalizable A. | Proof. by split=> -[P Punit PA]; exists P; move: PA; rewrite //= andbT. Qed. | Lemma | codiagonalizable1 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"codiagonalizable",
"diagonalizable",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizablePfull n (As : seq 'M[F]_n) :
codiagonalizable As
<-> exists m, exists2 P : 'M_(m, n), row_full P &
all [pred A | diagonalizable_for P A] As. | Proof.
split => [[P Punit SPA]|[m [P Pfull SPA]]].
by exists n => //; exists P; rewrite ?row_full_unit.
have Qfull := fullrowsub_unit Pfull.
exists (rowsub (fullrankfun Pfull) P) => //; apply/allP => A AAs/=.
have /allP /(_ _ AAs)/= /diagonalizable_forPex[d /simmxPp] := SPA.
rewrite submx_full// => /(_ isT) PA_eq.
ap... | Definition | codiagonalizablePfull | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"all",
"allP",
"apply",
"codiagonalizable",
"colsub",
"diagonalizable_for",
"diagonalizable_forPex",
"fullrankfun",
"fullrowsub_unit",
"mxE",
"row",
"rowE",
"row_diag_mx",
"row_full",
"row_full_unit",
"row_matrixP",
"row_mul",
"row_rowsub",
"rowsub",
"scalemxAl",
"seq",
"si... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizable_on m n (V_ : 'I_n -> 'M[F]_m) (As : seq 'M[F]_m) :
(\sum_i V_ i :=: 1%:M)%MS -> mxdirect (\sum_i V_ i) ->
(forall i, all (fun A => stablemx (V_ i) A) As) ->
(forall i, codiagonalizable (map (restrictmx (V_ i)) As)) ->
codiagonalizable As. | Proof.
move=> V1 Vdirect /(_ _)/allP AV /(_ _) /sig2W/= Pof.
pose P_ i := tag (Pof i).
have P_unit i : P_ i \in unitmx by rewrite /P_; case: {+}Pof.
have P_diag i A : A \in As -> diagonalizable_for (P_ i *m row_base (V_ i)) A.
move=> AAs; rewrite /P_; case: {+}Pof => /= P Punit.
rewrite all_map => /allP/(_ A AAs); ... | Lemma | codiagonalizable_on | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"all",
"allP",
"all_map",
"apply",
"codiagonalizable",
"codiagonalizablePfull",
"conjuMmx",
"diagonalizable_for",
"diagonalizable_for_sum",
"eq_bigr",
"eq_genmx",
"eq_row_base",
"eqmxMfull",
"eqmx_col",
"eqmx_stable",
"eqmx_trans",
"forallP",
"genmxE",
"genmx_sums",
"last",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_diag {n} (d : 'rV[F]_n) : diagonalizable (diag_mx d). | Proof.
exists 1%:M; rewrite ?unitmx1// /(diagonalizable_for _ _).
by rewrite conj1mx diag_mx_is_diag.
Qed. | Lemma | diagonalizable_diag | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"conj1mx",
"diag_mx",
"diag_mx_is_diag",
"diagonalizable",
"diagonalizable_for",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_scalar {n} (a : F) : diagonalizable (a%:M : 'M_n). | Proof. by rewrite -diag_const_mx. Qed. | Lemma | diagonalizable_scalar | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"diag_const_mx",
"diagonalizable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable0 {n} : diagonalizable (0 : 'M[F]_n). | Proof.
by rewrite (_ : 0 = 0%:M)//; apply/matrixP => i j; rewrite !mxE// mul0rn.
Qed. | Lemma | diagonalizable0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"diagonalizable",
"matrixP",
"mul0rn",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizablePeigen {n} {A : 'M[F]_n} :
diagonalizable A <->
exists2 rs, uniq rs & (\sum_(r <- rs) eigenspace A r :=: 1%:M)%MS. | Proof.
split=> [df|[rs urs rsP]].
suff [rs rsP] : exists rs, (\sum_(r <- rs) eigenspace A r :=: 1%:M)%MS.
exists (undup rs); rewrite ?undup_uniq//; apply: eqmx_trans rsP.
elim: rs => //= r rs IHrs; rewrite big_cons.
case: ifPn => in_rs; rewrite ?big_cons; last exact: adds_eqmx.
apply/(eqmx_trans IHrs)... | Lemma | diagonalizablePeigen | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"adds_eqmx",
"addsmx_idPr",
"apply",
"big_cons",
"big_image",
"big_mkord",
"big_nth",
"codiagonalizable1",
"codiagonalizable_on",
"comm_mx_stable_eigenspace",
"conjVmx",
"conjmx_eigenvalue",
"diagonalizable",
"diagonalizable_forLR",
"diagonalizable_scalar",
"eigenspace",
"eigenspaceP... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizableP n' (n := n'.+1) (A : 'M[F]_n) :
diagonalizable A <->
exists2 rs, uniq rs & mxminpoly A %| \prod_(x <- rs) ('X - x%:P). | Proof.
split=> [[P Punit /diagonalizable_forPex[d /(simmxLR Punit)->]]|].
rewrite mxminpoly_uconj ?unitmx_inv// mxminpoly_diag.
by eexists; [|by []]; rewrite undup_uniq.
move=> + /ltac:(apply/diagonalizablePeigen) => -[rs rsu rsP]; exists rs => //.
rewrite (big_nth 0) [X in (X :=: _)%MS](big_nth 0) !big_mkord in rs... | Lemma | diagonalizableP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"big_mkord",
"big_nth",
"coprimep_XsubC",
"diagonalizable",
"diagonalizablePeigen",
"diagonalizable_forPex",
"eigenspace_poly",
"eq_bigr",
"eqmx_sym",
"eqmx_trans",
"kermxpoly_min",
"kermxpoly_prod",
"mxminpoly",
"mxminpoly_diag",
"mxminpoly_uconj",
"n'",
"nth_uniq",
"ro... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_conj_diag m n (V : 'M[F]_(m, n)) (d : 'rV[F]_n) :
stablemx V (diag_mx d) -> row_free V -> diagonalizable (conjmx V (diag_mx d)). | Proof.
case: m n => [|m] [|n] in V d * => Vd rdV; rewrite ?thinmx0.
- by [].
- by [].
- by exfalso; move: rdV; rewrite /row_free mxrank.unlock eqxx orbT.
apply/diagonalizableP; pose u := undup [seq d 0 i | i <- enum 'I_n.+1].
exists u; first by rewrite undup_uniq.
by rewrite (dvdp_trans (mxminpoly_conj (f:=diag_mx d) _... | Lemma | diagonalizable_conj_diag | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"conjmx",
"diag_mx",
"diagonalizable",
"diagonalizableP",
"dvdp_trans",
"enum",
"eqxx",
"mxminpoly_conj",
"mxminpoly_diag",
"mxrank",
"row_free",
"seq",
"stablemx",
"thinmx0",
"undup",
"undup_uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizableP n (As : seq 'M[F]_n) :
{in As &, forall A B, comm_mx A B} /\ {in As, forall A, diagonalizable A}
<-> codiagonalizable As. | Proof.
split => [cdAs|[P Punit /allP/= AsD]]/=; last first.
split; last by exists P; rewrite // AsD.
move=> A B AAs BAs; move=> /(_ _ _)/diagonalizable_forPex/sigW in AsD.
have [[dA /simmxLR->//] [dB /simmxLR->//]] := (AsD _ AAs, AsD _ BAs).
by rewrite /comm_mx -!conjmxM 1?diag_mxC// inE stablemx_unit ?unitmx_i... | Lemma | codiagonalizableP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"allP",
"all_comm_mxP",
"all_comm_mx_cons",
"apply",
"big_mkord",
"big_nth",
"codiagonalizable",
"codiagonalizable_on",
"comm_mx",
"comm_mx_stable_eigenspace",
"conjMumx",
"conjmxM",
"conjmx_eigenvalue",
"conjmx_scalar",
"diag_mxC",
"diagonalizable",
"diagonalizablePeigen",
"diagon... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_conj m n p (B : 'M[F]_(n.+1, m.+1)) (f : 'M_m.+1) :
row_free B -> stablemx B f ->
horner_mx (conjmx B f) p = conjmx B (horner_mx f p). | Proof.
move=> B_free B_stab; rewrite/conjmx; elim/poly_ind: p => [|p c].
by rewrite !rmorph0 mulmx0 mul0mx.
rewrite !(rmorphD, rmorphM)/= !(horner_mx_X, horner_mx_C) => ->.
rewrite [_ * _]mulmxA [_ *m (B *m _)]mulmxA mulmxKpV ?horner_mx_stable//.
apply: (row_free_inj B_free); rewrite [_ *m B]mulmxDl.
pose stablemxE :... | Lemma | horner_mx_conj | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"conjmx",
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"horner_mx_stable",
"mul0mx",
"mulmx0",
"mulmxA",
"mulmxDl",
"mulmxDr",
"mulmxKpV",
"poly_ind",
"rmorph0",
"rmorphD",
"rmorphM",
"row_free",
"row_free_inj",
"scalar_mxC",
"stablemx",
"stablemxC",
"stablemxD",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_uconj n p (B : 'M[F]_(n.+1)) (f : 'M_n.+1) :
B \is a GRing.unit ->
horner_mx (B *m f *m invmx B) p = B *m horner_mx f p *m invmx B. | Proof.
move=> B_unit; rewrite -!conjumx//.
by rewrite horner_mx_conj ?row_free_unit ?stablemx_unit.
Qed. | Lemma | horner_mx_uconj | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjumx",
"horner_mx",
"horner_mx_conj",
"invmx",
"row_free_unit",
"stablemx_unit",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_uconjC n p (B : 'M[F]_(n.+1)) (f : 'M_n.+1) :
B \is a GRing.unit ->
horner_mx (invmx B *m f *m B) p = invmx B *m horner_mx f p *m B. | Proof.
move=> B_unit; rewrite -[X in _ *m X](invmxK B).
by rewrite horner_mx_uconj ?invmxK ?unitmx_inv.
Qed. | Lemma | horner_mx_uconjC | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"horner_mx",
"horner_mx_uconj",
"invmx",
"invmxK",
"unit",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_kermxpoly_conjmx V f p W : stablemx V f -> row_free V ->
(W <= kermxpoly (conjmx V f) p)%MS = (W *m V <= kermxpoly f p)%MS. | Proof.
move: n m => [|n'] [|m']// in V f W *; rewrite ?thinmx0// => fV rfV.
- by rewrite /row_free mxrank0 in rfV.
- by rewrite mul0mx !sub0mx.
- apply/sub_kermxP/sub_kermxP; rewrite horner_mx_conj//; last first.
by move=> /(congr1 (mulmxr (pinvmx V)))/=; rewrite mul0mx !mulmxA.
move=> /(congr1 (mulmxr V))/=; rew... | Lemma | sub_kermxpoly_conjmx | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"conjmx",
"horner_mx_conj",
"horner_mx_stable",
"kermxpoly",
"last",
"mul0mx",
"mulmxA",
"mulmxKpV",
"mulmx_sub",
"mulmxr",
"mxrank0",
"n'",
"pinvmx",
"row_free",
"stablemx",
"sub0mx",
"sub_kermxP",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjmx_eigenvalue a V f : (V <= eigenspace f a)%MS -> row_free V ->
conjmx V f = a%:M. | Proof.
by move=> /eigenspaceP Vfa rfV; rewrite /conjmx Vfa -mul_scalar_mx mulmxKp.
Qed. | Lemma | conjmx_eigenvalue | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjmx",
"eigenspace",
"eigenspaceP",
"mul_scalar_mx",
"mulmxKp",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_to {F : fieldType} {m n} (P : 'M_(m, n)) A
(C : {pred 'M[F]_m}) | := C (conjmx P A). | Definition | similar_to | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar P A B | := (similar_to P A (PredOfSimpl.coerce (pred1 B))). | Notation | similar | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"pred1",
"similar_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_in D A B | := (exists2 P, P \in D & similar P A B). | Notation | similar_in | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"similar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_in_to D A C | := (exists2 P, P \in D & similar_to P A C). | Notation | similar_in_to | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"similar_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_similar_to D As C | := (exists2 P, P \in D & all [pred A | similar_to P A C] As). | Notation | all_similar_to | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"all",
"similar_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diag P A | := (similar_to P A is_diag_mx). | Notation | similar_diag | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"is_diag_mx",
"similar_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_in D A | := (similar_in_to D A is_diag_mx). | Notation | diagonalizable_in | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"is_diag_mx",
"similar_in_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizable_in D As | := (all_similar_to D As is_diag_mx). | Notation | codiagonalizable_in | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"all_similar_to",
"is_diag_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_trig P A | := (similar_to P A is_trig_mx). | Notation | similar_trig | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"is_trig_mx",
"similar_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trigonalizable_in D A | := (similar_in_to D A is_trig_mx). | Notation | trigonalizable_in | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"is_trig_mx",
"similar_in_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trigonalizable A | := (trigonalizable_in unitmx A). | Notation | trigonalizable | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"trigonalizable_in",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cotrigonalizable_in D As | := (all_similar_to D As is_trig_mx). | Notation | cotrigonalizable_in | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"all_similar_to",
"is_trig_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cotrigonalizable As | := (cotrigonalizable_in unitmx As). | Notation | cotrigonalizable | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"cotrigonalizable_in",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similarPp m n {P : 'M[F]_(m, n)} {A B} :
stablemx P A -> similar P A B -> P *m A = B *m P. | Proof. by move=> stablemxPA /eqP <-; rewrite mulmxKpV. Qed. | Lemma | similarPp | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"mulmxKpV",
"similar",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similarW m n {P : 'M[F]_(m, n)} {A B} : row_free P ->
P *m A = B *m P -> similar P A B. | Proof. by rewrite /similar_to/= /conjmx => fP ->; rewrite mulmxKp. Qed. | Lemma | similarW | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjmx",
"fP",
"mulmxKp",
"row_free",
"similar",
"similar_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similarP {p f g} : p \in unitmx ->
reflect (p *m f = g *m p) (similar p f g). | Proof.
move=> p_unit; apply: (iffP idP); first exact/similarPp/stablemx_unit.
by apply: similarW; rewrite row_free_unit.
Qed. | Lemma | similarP | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"row_free_unit",
"similar",
"similarPp",
"similarW",
"stablemx_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similarRL {p f g} : p \in unitmx ->
reflect (g = p *m f *m invmx p) (similar p f g). | Proof. by move=> ?; apply: (iffP eqP); rewrite conjumx. Qed. | Lemma | similarRL | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"conjumx",
"invmx",
"similar",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similarLR {p f g} : p \in unitmx ->
reflect (f = conjmx (invmx p) g) (similar p f g). | Proof.
by move=> pu; rewrite conjVmx//; apply: (iffP (similarRL pu)) => ->;
rewrite !mulmxA ?(mulmxK, mulmxKV, mulVmx, mulmxV, mul1mx, mulmx1).
Qed. | Lemma | similarLR | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"conjVmx",
"conjmx",
"invmx",
"mul1mx",
"mulVmx",
"mulmx1",
"mulmxA",
"mulmxK",
"mulmxKV",
"mulmxV",
"similar",
"similarRL",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_mxminpoly {n} {p f g : 'M[F]_n.+1} : p \in unitmx ->
similar p f g -> mxminpoly f = mxminpoly g. | Proof. by move=> pu /eqP<-; rewrite mxminpoly_uconj. Qed. | Lemma | similar_mxminpoly | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"mxminpoly",
"mxminpoly_uconj",
"similar",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diag_row_base m n (P : 'M[F]_(m, n)) (A : 'M_n) :
similar_diag (row_base P) A = is_diag_mx (restrictmx P A). | Proof. by []. Qed. | Lemma | similar_diag_row_base | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"is_diag_mx",
"restrictmx",
"row_base",
"similar_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diagPp m n (P : 'M[F]_(m, n)) A :
reflect (forall i j : 'I__, i != j :> nat -> conjmx P A i j = 0)
(similar_diag P A). | Proof. exact: @is_diag_mxP. Qed. | Lemma | similar_diagPp | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjmx",
"is_diag_mxP",
"nat",
"similar_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diagP n (P : 'M[F]_n) A : P \in unitmx ->
reflect (forall i j : 'I__, i != j :> nat -> (P *m A *m invmx P) i j = 0)
(similar_diag P A). | Proof. by move=> Pu; rewrite -conjumx//; exact: is_diag_mxP. Qed. | Lemma | similar_diagP | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjumx",
"invmx",
"is_diag_mxP",
"nat",
"similar_diag",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diagPex {m} {n} {P : 'M[F]_(m, n)} {A} :
reflect (exists D, similar P A (diag_mx D)) (similar_diag P A). | Proof. by apply: (iffP (diag_mxP _)) => -[D]/eqP; exists D. Qed. | Lemma | similar_diagPex | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"diag_mx",
"diag_mxP",
"similar",
"similar_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diagLR n {P : 'M[F]_n} {A} : P \in unitmx ->
reflect (exists D, A = conjmx (invmx P) (diag_mx D)) (similar_diag P A). | Proof.
by move=> Punit; apply: (iffP similar_diagPex) => -[D /(similarLR Punit)]; exists D.
Qed. | Lemma | similar_diagLR | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"conjmx",
"diag_mx",
"invmx",
"similarLR",
"similar_diag",
"similar_diagPex",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diag_mxminpoly {n} {p f : 'M[F]_n.+1}
(rs := undup [seq conjmx p f i i | i <- enum 'I_n.+1]) :
p \in unitmx -> similar_diag p f ->
mxminpoly f = \prod_(r <- rs) ('X - r%:P). | Proof.
rewrite /rs => pu /(similar_diagLR pu)[d {f rs}->].
rewrite mxminpoly_uconj ?unitmx_inv// mxminpoly_diag.
by rewrite [in RHS](@eq_map _ _ _ (d 0))// => i; rewrite conjmxVK// mxE eqxx.
Qed. | Lemma | similar_diag_mxminpoly | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conjmx",
"conjmxVK",
"enum",
"eq_map",
"eqxx",
"mxE",
"mxminpoly",
"mxminpoly_diag",
"mxminpoly_uconj",
"seq",
"similar_diag",
"similar_diagLR",
"undup",
"unitmx",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
similar_diag_sum (F : fieldType) (m n : nat) (p_ : 'I_n -> nat)
(V_ : forall i, 'M[F]_(p_ i, m)) (f : 'M[F]_m) :
mxdirect (\sum_i <<V_ i>>) ->
(forall i, stablemx (V_ i) f) ->
(forall i, row_free (V_ i)) ->
similar_diag (\mxcol_i V_ i) f = [forall i, similar_diag (V_ i) f]. | Proof.
move=> Vd Vf rfV; have aVf : stablemx (\mxcol_i V_ i) f.
rewrite (eqmx_stable _ (eqmx_col _)) stablemx_sums//.
by move=> i; rewrite (eqmx_stable _ (genmxE _)).
apply/similar_diagPex/'forall_similar_diagPex => /=
[[D /(similarPp aVf) +] i|/(_ _)/sigW Dof].
rewrite mxcol_mul -[D]submxrowK diag_mxrow mul_... | Lemma | similar_diag_sum | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"diag_mxrow",
"eq_bigr",
"eq_mxcol",
"eq_mxcolP",
"eqmx_col",
"eqmx_stable",
"genmxE",
"mul_mxdiag_mxcol",
"mxcol_mul",
"mxdirect",
"mxdirectP",
"nat",
"row_free",
"row_leq_rank",
"sigW",
"similarPp",
"similarW",
"similar_diag",
"similar_diagPex",
"stablemx",
"stab... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizablePfull n (As : seq 'M[F]_n) :
codiagonalizable As <-> exists m,
exists2 P : 'M_(m, n), row_full P & all [pred A | similar_diag P A] As. | Proof.
split => [[P Punit SPA]|[m [P Pfull SPA]]].
by exists n => //; exists P; rewrite ?row_full_unit.
have Qfull := fullrowsub_unit Pfull.
exists (rowsub (fullrankfun Pfull) P) => //; apply/allP => A AAs/=.
have /allP /(_ _ AAs)/= /similar_diagPex[d /similarPp] := SPA.
rewrite submx_full// => /(_ isT) PA_eq.
apply/... | Lemma | codiagonalizablePfull | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"all",
"allP",
"apply",
"codiagonalizable",
"colsub",
"fullrankfun",
"fullrowsub_unit",
"mxE",
"row",
"rowE",
"row_diag_mx",
"row_full",
"row_full_unit",
"row_matrixP",
"row_mul",
"row_rowsub",
"rowsub",
"scalemxAl",
"seq",
"similarP",
"similarPp",
"similar_diag",
"simila... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizable_on m n (V_ : 'I_n -> 'M[F]_m) (As : seq 'M[F]_m) :
(\sum_i V_ i :=: 1%:M)%MS -> mxdirect (\sum_i V_ i) ->
(forall i, all (fun A => stablemx (V_ i) A) As) ->
(forall i, codiagonalizable (map (restrictmx (V_ i)) As)) -> codiagonalizable As. | Proof.
move=> V1 Vdirect /(_ _)/allP AV /(_ _) /sig2W/= Pof.
pose P_ i := tag (Pof i).
have P_unit i : P_ i \in unitmx by rewrite /P_; case: {+}Pof.
have P_diag i A : A \in As -> similar_diag (P_ i *m row_base (V_ i)) A.
move=> AAs; rewrite /P_; case: {+}Pof => /= P Punit.
rewrite all_map => /allP/(_ A AAs); rewrit... | Lemma | codiagonalizable_on | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"all",
"allP",
"all_map",
"apply",
"codiagonalizable",
"codiagonalizablePfull",
"conjuMmx",
"eq_bigr",
"eq_genmx",
"eq_row_base",
"eqmxMfull",
"eqmx_col",
"eqmx_stable",
"eqmx_trans",
"forallP",
"genmxE",
"genmx_sums",
"last",
"map",
"mxdirect",
"mxdirectE",
"mxdirectP",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_diag {n} (d : 'rV[F]_n) : diagonalizable (diag_mx d). | Proof.
by exists 1%:M; rewrite ?unitmx1// /similar_to conj1mx diag_mx_is_diag.
Qed. | Lemma | diagonalizable_diag | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"conj1mx",
"diag_mx",
"diag_mx_is_diag",
"diagonalizable",
"similar_to",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizablePeigen {n} {f : 'M[F]_n} :
diagonalizable f <->
exists2 rs, uniq rs & (\sum_(r <- rs) eigenspace f r :=: 1%:M)%MS. | Proof.
split=> [df|[rs urs rsP]].
suff [rs rsP] : exists rs, (\sum_(r <- rs) eigenspace f r :=: 1%:M)%MS.
exists (undup rs); rewrite ?undup_uniq//; apply: eqmx_trans rsP.
elim: rs => //= r rs IHrs; rewrite big_cons.
case: ifPn => in_rs; rewrite ?big_cons; last exact: adds_eqmx.
apply/(eqmx_trans IHrs)... | Lemma | diagonalizablePeigen | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"adds_eqmx",
"addsmx_idPr",
"apply",
"big_cons",
"big_image",
"big_mkord",
"big_nth",
"codiagonalizable1",
"codiagonalizable_on",
"comm_mx_stable_eigenspace",
"conjVmx",
"conjmx_eigenvalue",
"diagonalizable",
"eigenspace",
"eigenspaceP",
"enum",
"eq_row_base",
"eqmxP",
"eqmx_sym"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizableP n' (n := n'.+1) (f : 'M[F]_n) :
diagonalizable f <->
exists2 rs, uniq rs & mxminpoly f %| \prod_(x <- rs) ('X - x%:P). | Proof.
split=> [[P Punit /similar_diagPex[d /(similarLR Punit)->]]|].
rewrite mxminpoly_uconj ?unitmx_inv// mxminpoly_diag.
by eexists; [|by []]; rewrite undup_uniq.
move=> [rs rsU rsP]; apply: diagonalizablePeigen.2.
exists rs => //.
rewrite (big_nth 0) big_mkord (eq_bigr _ (fun _ _ => eigenspace_poly _ _)).
apply... | Lemma | diagonalizableP | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"big_mkord",
"big_nth",
"coprimep_XsubC",
"diagonalizable",
"diagonalizablePeigen",
"eigenspace_poly",
"eq_bigr",
"eqmx_sym",
"eqmx_trans",
"kermxpoly_min",
"kermxpoly_prod",
"mxminpoly",
"mxminpoly_diag",
"mxminpoly_uconj",
"n'",
"nth_uniq",
"root_XsubC",
"similarLR",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagonalizable_conj_diag m n (V : 'M[F]_(m, n)) (d : 'rV[F]_n) :
stablemx V (diag_mx d) -> row_free V -> diagonalizable (conjmx V (diag_mx d)). | Proof.
(move: m n => [|m] [|n] in V d *; rewrite ?thinmx0; [by []|by []| |]) => Vd rdV.
- by rewrite /row_free mxrank0 in rdV.
- apply/diagonalizableP; pose u := undup [seq d 0 i | i <- enum 'I_n.+1].
exists u; first by rewrite undup_uniq.
by rewrite (dvdp_trans (mxminpoly_conj rdV _))// mxminpoly_diag.
Qed. | Lemma | diagonalizable_conj_diag | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"apply",
"conjmx",
"diag_mx",
"diagonalizable",
"diagonalizableP",
"dvdp_trans",
"enum",
"mxminpoly_conj",
"mxminpoly_diag",
"mxrank0",
"row_free",
"seq",
"stablemx",
"thinmx0",
"undup",
"undup_uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codiagonalizableP n (fs : seq 'M[F]_n) :
{in fs &, forall f g, comm_mx f g} /\ (forall f, f \in fs -> diagonalizable f)
<-> codiagonalizable fs. | Proof.
split => [cdfs|[P Punit /allP/= fsD]]/=; last first.
split; last by exists P; rewrite // fsD.
move=> f g ffs gfs; move=> /(_ _ _)/similar_diagPex/sigW in fsD.
have [[df /similarLR->//] [dg /similarLR->//]] := (fsD _ ffs, fsD _ gfs).
by rewrite /comm_mx -!conjmxM 1?diag_mxC// inE stablemx_unit ?unitmx_inv... | Lemma | codiagonalizableP | algebra | algebra/mxred.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"mxpoly",
"GRing.Theory",
"Monoid.Theory"
] | [
"allP",
"all_comm_mxP",
"all_comm_mx_cons",
"apply",
"big_mkord",
"big_nth",
"codiagonalizable",
"codiagonalizable_on",
"comm_mx",
"comm_mx_stable_eigenspace",
"conjMumx",
"conjmxM",
"conjmx_eigenvalue",
"conjmx_scalar",
"diag_mxC",
"diagonalizable",
"diagonalizablePeigen",
"diagon... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simp | := Monoid.simpm. | Notation | simp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"simpm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polynomial | := Polynomial {polyseq :> seq R; _ : last 1 polyseq != 0}. | Record | polynomial | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"last",
"seq"
] | Defines a polynomial as a sequence with <> 0 last element | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
poly_inj : injective polyseq. | Proof. exact: val_inj. Qed. | Lemma | poly_inj | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coefp i (p : polynomial) | := p`_i. | Definition | coefp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"polynomial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'poly' T }" | := (polynomial T) : type_scope. | Notation | { 'poly' T } | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"polynomial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coef p | := p`_(size p).-1. | Definition | lead_coef | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coefE p : lead_coef p = p`_(size p).-1. | Proof. by []. Qed. | Lemma | lead_coefE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"lead_coef",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_nil | := @Polynomial R [::] (oner_neq0 R). | Definition | poly_nil | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"oner_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyC c : {poly R} | := insubd poly_nil [:: c]. | Definition | polyC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"insubd",
"poly",
"poly_nil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"c %:P" | := (polyC c). | Notation | c %:P | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyseqC c : c%:P = nseq (c != 0) c :> seq R. | Proof. by rewrite val_insubd /=; case: (c == 0). Qed. | Lemma | polyseqC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"nseq",
"seq",
"val_insubd"
] | Remember the boolean (c != 0) is coerced to 1 if true and 0 if false | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
size_polyC c : size c%:P = (c != 0). | Proof. by rewrite polyseqC size_nseq. Qed. | Lemma | size_polyC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"polyseqC",
"size",
"size_nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coefC c i : c%:P`_i = if i == 0 then c else 0. | Proof. by rewrite polyseqC; case: i => [|[]]; case: eqP. Qed. | Lemma | coefC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"polyseqC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyCK : cancel polyC (coefp 0). | Proof. by move=> c; rewrite [coefp 0 _]coefC. Qed. | Lemma | polyCK | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"coefC",
"coefp",
"polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyC_inj : injective polyC. | Proof. exact: can_inj polyCK. Qed. | Lemma | polyC_inj | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"polyC",
"polyCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coefC c : lead_coef c%:P = c. | Proof. by rewrite /lead_coef polyseqC; case: eqP. Qed. | Lemma | lead_coefC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"lead_coef",
"polyseqC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyP p q : nth 0 p =1 nth 0 q <-> p = q. | Proof.
split=> [eq_pq | -> //]; apply: poly_inj.
without loss lt_pq: p q eq_pq / size p < size q.
move=> IH; case: (ltngtP (size p) (size q)); try by move/IH->.
by move/(@eq_from_nth _ 0); apply.
case: q => q nz_q /= in lt_pq eq_pq *; case/eqP: nz_q.
by rewrite (last_nth 0) -(subnKC lt_pq) /= -eq_pq nth_default ?le... | Lemma | polyP | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"eq_from_nth",
"last_nth",
"leq_addr",
"ltngtP",
"nth",
"nth_default",
"poly_inj",
"size",
"split",
"subnKC"
] | Extensional interpretation (poly <=> nat -> R) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
size1_polyC p : size p <= 1 -> p = (p`_0)%:P. | Proof.
move=> le_p_1; apply/polyP=> i; rewrite coefC.
by case: i => // i; rewrite nth_default // (leq_trans le_p_1).
Qed. | Lemma | size1_polyC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"coefC",
"leq_trans",
"nth_default",
"polyP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cons_poly c p : {poly R} | :=
if p is Polynomial ((_ :: _) as s) ns then
@Polynomial R (c :: s) ns
else c%:P. | Definition | cons_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"poly"
] | Builds a polynomial by extension. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
polyseq_cons c p :
cons_poly c p = (if ~~ nilp p then c :: p else c%:P) :> seq R. | Proof. by case: p => [[]]. Qed. | Lemma | polyseq_cons | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"cons_poly",
"nilp",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_cons_poly c p :
size (cons_poly c p) = (if nilp p && (c == 0) then 0 else (size p).+1). | Proof. by case: p => [[|c' s] _] //=; rewrite size_polyC; case: eqP. Qed. | Lemma | size_cons_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"cons_poly",
"nilp",
"size",
"size_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef_cons c p i : (cons_poly c p)`_i = if i == 0 then c else p`_i.-1. | Proof.
by case: p i => [[|c' s] _] [] //=; rewrite polyseqC; case: eqP => //= _ [].
Qed. | Lemma | coef_cons | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"cons_poly",
"polyseqC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Poly | := foldr cons_poly 0%:P. | Definition | Poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"cons_poly",
"foldr"
] | Build a polynomial directly from a list of coefficients. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
PolyK c s : last c s != 0 -> Poly s = s :> seq R. | Proof.
case: s => {c}/= [_ |c s]; first by rewrite polyseqC eqxx.
elim: s c => /= [|a s IHs] c nz_c; rewrite polyseq_cons ?{}IHs //.
by rewrite !polyseqC !eqxx nz_c.
Qed. | Lemma | PolyK | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"Poly",
"eqxx",
"last",
"polyseqC",
"polyseq_cons",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyseqK p : Poly p = p. | Proof. by apply: poly_inj; apply: PolyK (valP p). Qed. | Lemma | polyseqK | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"Poly",
"PolyK",
"apply",
"poly_inj",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_Poly s : size (Poly s) <= size s. | Proof.
elim: s => [|c s IHs] /=; first by rewrite polyseqC eqxx.
by rewrite size_cons_poly; case: ifP.
Qed. | Lemma | size_Poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"Poly",
"eqxx",
"polyseqC",
"size",
"size_cons_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef_Poly s i : (Poly s)`_i = s`_i. | Proof.
by elim: s i => [|c s IHs] /= [|i]; rewrite !(coefC, eqxx, coef_cons) /=.
Qed. | Lemma | coef_Poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"Poly",
"coefC",
"coef_cons",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_expanded_def n E | := Poly (mkseq E n). | Definition | poly_expanded_def | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"Poly",
"mkseq"
] | Build a polynomial from an infinite sequence of coefficients and a bound. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
poly_key : unit. | Proof. by []. Qed. | Fact | poly_key | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly | := locked_with poly_key poly_expanded_def. | Definition | poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"poly_expanded_def",
"poly_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_unlockable | := [unlockable fun poly]. | Canonical | poly_unlockable | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\poly_ ( i < n ) E" | := (poly n (fun i : nat => E)). | Notation | \poly_ ( i < n ) E | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"nat",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyseq_poly n E :
E n.-1 != 0 -> \poly_(i < n) E i = mkseq [eta E] n :> seq R. | Proof.
rewrite unlock; case: n => [|n] nzEn; first by rewrite polyseqC eqxx.
by rewrite (@PolyK 0) // -nth_last nth_mkseq size_mkseq.
Qed. | Lemma | polyseq_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"PolyK",
"eqxx",
"mkseq",
"nth_last",
"nth_mkseq",
"polyseqC",
"seq",
"size_mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_poly n E : size (\poly_(i < n) E i) <= n. | Proof. by rewrite unlock (leq_trans (size_Poly _)) ?size_mkseq. Qed. | Lemma | size_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"leq_trans",
"size",
"size_Poly",
"size_mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_poly_eq n E : E n.-1 != 0 -> size (\poly_(i < n) E i) = n. | Proof. by move/polyseq_poly->; apply: size_mkseq. Qed. | Lemma | size_poly_eq | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"polyseq_poly",
"size",
"size_mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef_poly n E k : (\poly_(i < n) E i)`_k = (if k < n then E k else 0). | Proof.
rewrite unlock coef_Poly.
have [lt_kn | le_nk] := ltnP k n; first by rewrite nth_mkseq.
by rewrite nth_default // size_mkseq.
Qed. | Lemma | coef_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"coef_Poly",
"ltnP",
"nth_default",
"nth_mkseq",
"size_mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coef_poly n E :
n > 0 -> E n.-1 != 0 -> lead_coef (\poly_(i < n) E i) = E n.-1. | Proof.
by case: n => // n _ nzE; rewrite /lead_coef size_poly_eq // coef_poly leqnn.
Qed. | Lemma | lead_coef_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"coef_poly",
"lead_coef",
"leqnn",
"size_poly_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coefK p : \poly_(i < size p) p`_i = p. | Proof.
by apply/polyP=> i; rewrite coef_poly; case: ltnP => // /(nth_default 0)->.
Qed. | Lemma | coefK | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"coef_poly",
"ltnP",
"nth_default",
"polyP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_poly_def p q | := \poly_(i < maxn (size p) (size q)) (p`_i + q`_i). | Definition | add_poly_def | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"maxn",
"size"
] | Nmodule structure for polynomial | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
add_poly_key : unit. | Proof. by []. Qed. | Fact | add_poly_key | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_poly | := locked_with add_poly_key add_poly_def. | Definition | add_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"add_poly_def",
"add_poly_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_poly_unlockable | := [unlockable fun add_poly]. | Canonical | add_poly_unlockable | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"add_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.