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 |
|---|---|---|---|---|---|---|---|---|---|---|
fullrankfun : 'I_m ^ n | := finfun (mxf \o cast_ord (esym (eqP rkA))). | Definition | fullrankfun | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cast_ord",
"mxf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
frf | := fullrankfun. | Notation | frf | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"fullrankfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fullrowsub_full : row_full (rowsub frf A). | Proof.
by rewrite mxsub_ffunl rowsub_comp rowsub_cast esymK row_full_castmx.
Qed. | Lemma | fullrowsub_full | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"frf",
"mxsub_ffunl",
"row_full",
"row_full_castmx",
"rowsub",
"rowsub_cast",
"rowsub_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fullrowsub_unit : rowsub frf A \in unitmx. | Proof. by rewrite -row_full_unit fullrowsub_full. Qed. | Lemma | fullrowsub_unit | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"frf",
"fullrowsub_full",
"row_full_unit",
"rowsub",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fullrowsub_free : row_free (rowsub frf A). | Proof. by rewrite row_free_unit fullrowsub_unit. Qed. | Lemma | fullrowsub_free | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"frf",
"fullrowsub_unit",
"row_free",
"row_free_unit",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_fullrowsub : \rank (rowsub frf A) = n. | Proof. exact/eqP/fullrowsub_full. Qed. | Lemma | mxrank_fullrowsub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"frf",
"fullrowsub_full",
"rank",
"rowsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_fullrowsub : (rowsub frf A :=: A)%MS. | Proof.
rewrite mxsub_ffunl rowsub_comp rowsub_cast esymK.
exact: (eqmx_trans (eqmx_cast _ _) eq_maxrowsub).
Qed. | Lemma | eq_fullrowsub | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eq_maxrowsub",
"eqmx_cast",
"eqmx_trans",
"frf",
"mxsub_ffunl",
"rowsub",
"rowsub_cast",
"rowsub_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fullrankfun_inj : injective frf. | Proof.
by move=> i j; rewrite !ffunE => /maxrankfun_inj /(congr1 val)/= /val_inj.
Qed. | Lemma | fullrankfun_inj | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ffunE",
"frf",
"maxrankfun_inj",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsum_spec n : forall m, 'M[F]_(m, n) -> nat -> Prop | :=
| TrivialMxsum m A
: @mxsum_spec n m A (\rank A)
| ProperMxsum m1 m2 T1 T2 r1 r2 of
@mxsum_spec n m1 T1 r1 & @mxsum_spec n m2 T2 r2
: mxsum_spec (T1 + T2)%MS (r1 + r2)%N. | Inductive | mxsum_spec | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"nat",
"r1",
"r2",
"rank"
] | constant, namely wrap. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mxsum_expr m n | := Mxsum {
mxsum_val :> wrapped 'M_(m, n);
mxsum_rank : wrapped nat;
_ : mxsum_spec (unwrap mxsum_val) (unwrap mxsum_rank)
}. | Structure | mxsum_expr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxsum_spec",
"nat",
"wrapped"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivial_mxsum m n A | :=
@Mxsum m n (Wrap A) (Wrap (\rank A)) (TrivialMxsum A). | Canonical | trivial_mxsum | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proper_mxsum_expr n | := ProperMxsumExpr {
proper_mxsum_val :> 'M_n;
proper_mxsum_rank : nat;
_ : mxsum_spec proper_mxsum_val proper_mxsum_rank
}. | Structure | proper_mxsum_expr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxsum_spec",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proper_mxsumP n (S : proper_mxsum_expr n) | :=
let: ProperMxsumExpr _ _ termS := S return mxsum_spec S (proper_mxsum_rank S)
in termS. | Definition | proper_mxsumP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxsum_spec",
"proper_mxsum_expr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_mxsum n (S : proper_mxsum_expr n) | :=
@Mxsum n n (wrap (S : 'M_n)) (wrap (proper_mxsum_rank S)) (proper_mxsumP S). | Canonical | sum_mxsum | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"proper_mxsumP",
"proper_mxsum_expr",
"wrap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
binary_mxsum_proof :
mxsum_spec (unwrap S1 + unwrap S2)
(unwrap (mxsum_rank S1) + unwrap (mxsum_rank S2)). | Proof. by case: S1 S2 => [A1 r1 A1P] [A2 r2 A2P]; right. Qed. | Fact | binary_mxsum_proof | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"S1",
"S2",
"mxsum_spec",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
binary_mxsum_expr | := ProperMxsumExpr binary_mxsum_proof. | Canonical | binary_mxsum_expr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"binary_mxsum_proof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nary_mxsum_proof :
mxsum_spec (\sum_(j <- r | P j) unwrap (S_ j))
(\sum_(j <- r | P j) unwrap (mxsum_rank (S_ j))). | Proof.
elim/big_rec2: _ => [|j]; first by rewrite -(mxrank0 n n); left.
by case: (S_ j); right.
Qed. | Fact | nary_mxsum_proof | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"big_rec2",
"mxrank0",
"mxsum_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nary_mxsum_expr | := ProperMxsumExpr nary_mxsum_proof. | Canonical | nary_mxsum_expr | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"nary_mxsum_proof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_def m n T & phantom 'M_(m, n) (unwrap (mxsum_val T)) | :=
\rank (unwrap T) == unwrap (mxsum_rank T). | Definition | mxdirect_def | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect A | := (mxdirect_def (Phantom 'M_(_,_) A%MS)). | Notation | mxdirect | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxdirect_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirectP n (S : proper_mxsum_expr n) :
reflect (\rank S = proper_mxsum_rank S) (mxdirect S). | Proof. exact: eqnP. Qed. | Lemma | mxdirectP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqnP",
"mxdirect",
"proper_mxsum_expr",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_trivial m n A : mxdirect (unwrap (@trivial_mxsum m n A)). | Proof. exact: eqxx. Qed. | Lemma | mxdirect_trivial | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqxx",
"mxdirect",
"trivial_mxsum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_sum_leqif m n (S : mxsum_expr m n) :
\rank (unwrap S) <= unwrap (mxsum_rank S) ?= iff mxdirect (unwrap S). | Proof.
rewrite /mxdirect_def; case: S => [[A] [r] /= defAr]; split=> //=.
elim: m A r / defAr => // m1 m2 A1 A2 r1 r2 _ leAr1 _ leAr2.
by apply: leq_trans (leq_add leAr1 leAr2); rewrite mxrank_adds_leqif.
Qed. | Lemma | mxrank_sum_leqif | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"leq_add",
"leq_trans",
"mxdirect",
"mxdirect_def",
"mxrank_adds_leqif",
"mxsum_expr",
"r1",
"r2",
"rank",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirectE m n (S : mxsum_expr m n) :
mxdirect (unwrap S) = (\rank (unwrap S) == unwrap (mxsum_rank S)). | Proof. by []. Qed. | Lemma | mxdirectE | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxdirect",
"mxsum_expr",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirectEgeq m n (S : mxsum_expr m n) :
mxdirect (unwrap S) = (\rank (unwrap S) >= unwrap (mxsum_rank S)). | Proof. by rewrite (geq_leqif (mxrank_sum_leqif S)). Qed. | Lemma | mxdirectEgeq | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"geq_leqif",
"mxdirect",
"mxrank_sum_leqif",
"mxsum_expr",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_addsE (S1 : mxsum_expr m1 n) (S2 : mxsum_expr m2 n) :
mxdirect (unwrap S1 + unwrap S2)
= [&& mxdirect (unwrap S1), mxdirect (unwrap S2)
& unwrap S1 :&: unwrap S2 == 0]%MS. | Proof.
rewrite (@mxdirectE n) /=.
have:= leqif_add (mxrank_sum_leqif S1) (mxrank_sum_leqif S2).
move/(leqif_trans (mxrank_adds_leqif (unwrap S1) (unwrap S2)))=> ->.
by rewrite andbC -andbA submx0.
Qed. | Lemma | mxdirect_addsE | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"S1",
"S2",
"leqif_add",
"leqif_trans",
"mxdirect",
"mxdirectE",
"mxrank_adds_leqif",
"mxrank_sum_leqif",
"mxsum_expr",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_addsP (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (A :&: B = 0)%MS (mxdirect (A + B)). | Proof. by rewrite mxdirect_addsE !mxdirect_trivial; apply: eqP. Qed. | Lemma | mxdirect_addsP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mxdirect",
"mxdirect_addsE",
"mxdirect_trivial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
TIsum A_ i | := (A_ i :&: (\sum_(j | P j && (j != i)) A_ j) = 0 :> 'M_n)%MS. | Let | TIsum | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sums_recP (S_ : I -> mxsum_expr n n) :
reflect (forall i, P i -> mxdirect (unwrap (S_ i)) /\ TIsum (unwrap \o S_) i)
(mxdirect (\sum_(i | P i) (unwrap (S_ i)))). | Proof.
rewrite /TIsum; apply: (iffP eqnP) => /= [dxS i Pi | dxS].
set Si' := (\sum_(j | _) unwrap (S_ j))%MS.
have: mxdirect (unwrap (S_ i) + Si') by apply/eqnP; rewrite /= -!(bigD1 i).
by rewrite mxdirect_addsE => /and3P[-> _ /eqP].
set Q := P; have [m] := ubnP #|Q|; have: Q \subset P by [].
elim: m Q => // m IH... | Let | mxdirect_sums_recP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"TIsum",
"apply",
"bigD1",
"big_pred0",
"capmxS",
"cardD1x",
"eqnP",
"last",
"mxdirect",
"mxdirect_addsE",
"mxrank0",
"mxrank_disjoint_sum",
"mxsum_expr",
"pickP",
"submx0",
"subsetP",
"sumsmx_subP",
"sumsmx_sup",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sumsP (A_ : I -> 'M_n) :
reflect (forall i, P i -> A_ i :&: (\sum_(j | P j && (j != i)) A_ j) = 0)%MS
(mxdirect (\sum_(i | P i) A_ i)). | Proof.
apply: (iffP (mxdirect_sums_recP _)) => dxA i /dxA; first by case.
by rewrite mxdirect_trivial.
Qed. | Lemma | mxdirect_sumsP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"mxdirect",
"mxdirect_sums_recP",
"mxdirect_trivial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sumsE (S_ : I -> mxsum_expr n n) (xunwrap := unwrap) :
reflect (and (forall i, P i -> mxdirect (unwrap (S_ i)))
(mxdirect (\sum_(i | P i) (xunwrap (S_ i)))))
(mxdirect (\sum_(i | P i) (unwrap (S_ i)))). | Proof.
apply: (iffP (mxdirect_sums_recP _)) => [dxS | [dxS_ dxS] i Pi].
by do [split; last apply/mxdirect_sumsP] => i; case/dxS.
by split; [apply: dxS_ | apply: mxdirect_sumsP Pi].
Qed. | Lemma | mxdirect_sumsE | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"last",
"mxdirect",
"mxdirect_sumsP",
"mxdirect_sums_recP",
"mxsum_expr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_daddsmx_spec : Prop | :=
SubDaddsmxSpec A1 A2 of (A1 <= B1)%MS & (A2 <= B2)%MS & A = A1 + A2
& forall C1 C2, (C1 <= B1)%MS -> (C2 <= B2)%MS ->
A = C1 + C2 -> C1 = A1 /\ C2 = A2. | Variant | sub_daddsmx_spec | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_daddsmx : (B1 :&: B2 = 0)%MS -> (A <= B1 + B2)%MS -> sub_daddsmx_spec. | Proof.
move=> dxB /sub_addsmxP[u defA].
exists (u.1 *m B1) (u.2 *m B2); rewrite ?submxMl // => C1 C2 sCB1 sCB2.
move/(canLR (addrK _)) => defC1.
suffices: (C2 - u.2 *m B2 <= B1 :&: B2)%MS.
by rewrite dxB submx0 subr_eq0 -defC1 defA; move/eqP->; rewrite addrK.
rewrite sub_capmx -opprB -{1}(canLR (addKr _) defA) -addrA... | Lemma | sub_daddsmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addKr",
"addmx_sub",
"addrA",
"addrK",
"eqmx_opp",
"opprB",
"sub_addsmxP",
"sub_capmx",
"sub_daddsmx_spec",
"submx0",
"submxMl",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_dsumsmx_spec : Prop | :=
SubDsumsmxSpec A_ of forall i, P i -> (A_ i <= B i)%MS
& A = \sum_(i | P i) A_ i
& forall C, (forall i, P i -> C i <= B i)%MS ->
A = \sum_(i | P i) C i -> {in SimplPred P, C =1 A_}. | Variant | sub_dsumsmx_spec | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_dsumsmx :
mxdirect (\sum_(i | P i) B i) -> (A <= \sum_(i | P i) B i)%MS ->
sub_dsumsmx_spec. | Proof.
move/mxdirect_sumsP=> dxB /sub_sumsmxP[u defA].
pose A_ i := u i *m B i.
exists A_ => //= [i _ | C sCB defAC i Pi]; first exact: submxMl.
apply/eqP; rewrite -subr_eq0 -submx0 -{dxB}(dxB i Pi) /=.
rewrite sub_capmx addmx_sub ?eqmx_opp ?submxMl ?sCB //=.
rewrite -(subrK A (C i)) -addrA -opprB addmx_sub ?eqmx_opp /... | Lemma | sub_dsumsmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addKr",
"addmx_sub",
"addrA",
"addrC",
"apply",
"bigD1",
"eqmx_opp",
"mxdirect",
"mxdirect_sumsP",
"opprB",
"sub_capmx",
"sub_dsumsmx_spec",
"sub_sumsmxP",
"submx0",
"submxMl",
"subrK",
"subr_eq0",
"summx_sub",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenspace a | := kermx (g - a%:M). | Definition | eigenspace | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"kermx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue : pred F | := fun a => eigenspace a != 0. | Definition | eigenvalue | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eigenspace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenspaceP a m (W : 'M_(m, n)) :
reflect (W *m g = a *: W) (W <= eigenspace a)%MS. | Proof. by rewrite sub_kermx mulmxBr subr_eq0 mul_mx_scalar; apply/eqP. Qed. | Lemma | eigenspaceP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eigenspace",
"mul_mx_scalar",
"mulmxBr",
"sub_kermx",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalueP a :
reflect (exists2 v : 'rV_n, v *m g = a *: v & v != 0) (eigenvalue a). | Proof. by apply: (iffP (rowV0Pn _)) => [] [v]; move/eigenspaceP; exists v. Qed. | Lemma | eigenvalueP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eigenspaceP",
"eigenvalue",
"rowV0Pn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvectorP {v : 'rV_n} :
reflect (exists a, (v <= eigenspace a)%MS) (stablemx v g). | Proof. by apply: (iffP (sub_rVP _ _)) => -[a] /eigenspaceP; exists a. Qed. | Lemma | eigenvectorP | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eigenspace",
"eigenspaceP",
"stablemx",
"sub_rVP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sum_eigenspace (P : pred I) a_ :
{in P &, injective a_} -> mxdirect (\sum_(i | P i) eigenspace (a_ i)). | Proof.
have [m] := ubnP #|P|; elim: m P => // m IHm P lePm inj_a.
apply/mxdirect_sumsP=> i Pi; apply/eqP/rowV0P => v.
rewrite sub_capmx => /andP[/eigenspaceP def_vg].
set Vi' := (\sum_(i | _) _)%MS => Vi'v.
have dxVi': mxdirect Vi'.
rewrite (cardD1x Pi) in lePm; apply: IHm => //.
by apply: sub_in2 inj_a => j /andP[... | Lemma | mxdirect_sum_eigenspace | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"big1",
"cardD1x",
"eigenspace",
"eigenspaceP",
"eq_bigr",
"eq_sym",
"eqmx_eq0",
"eqmx_scale",
"inj_in_eq",
"mulmx_suml",
"mxdirect",
"mxdirect_sumsP",
"rowV0P",
"scalemx_sub",
"scalerBl",
"scaler_sumr",
"sub0mx",
"sub_capmx",
"sub_dsumsmx",
"subr_eq0",
"subrr",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"<< A >>" | := (genmx A) : matrix_set_scope. | Notation | << A >> | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A <= B <= C" | := ((submx A B) && (submx B C)) : matrix_set_scope. | Notation | A <= B <= C | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A < B <= C" | := (ltmx A B && submx B C) : matrix_set_scope. | Notation | A < B <= C | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A <= B < C" | := (submx A B && ltmx B C) : matrix_set_scope. | Notation | A <= B < C | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A < B < C" | := (ltmx A B && ltmx B C) : matrix_set_scope. | Notation | A < B < C | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"ltmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A == B" | := ((submx A B) && (submx B A)) : matrix_set_scope. | Notation | A == B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :=: B" | := (eqmx A B) : matrix_set_scope. | Notation | A :=: B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect S | := (mxdirect_def (Phantom 'M_(_,_) S%MS)). | Notation | mxdirect | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxdirect_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i <- r | P ) B" | :=
(\big[addsmx/0%R]_(i <- r | P%B) B%MS) : matrix_set_scope. | Notation | \sum_ ( i <- r | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i <- r ) B" | :=
(\big[addsmx/0%R]_(i <- r) B%MS) : matrix_set_scope. | Notation | \sum_ ( i <- r ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( m <= i < n | P ) B" | :=
(\big[addsmx/0%R]_(m <= i < n | P%B) B%MS) : matrix_set_scope. | Notation | \sum_ ( m <= i < n | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
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"binomial",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( m <= i < n ) B" | :=
(\big[addsmx/0%R]_(m <= i < n) B%MS) : matrix_set_scope. | Notation | \sum_ ( m <= i < n ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i | P ) B" | :=
(\big[addsmx/0%R]_(i | P%B) B%MS) : matrix_set_scope. | Notation | \sum_ ( i | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"fingroup",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ i B" | :=
(\big[addsmx/0%R]_i B%MS) : matrix_set_scope. | Notation | \sum_ i B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i : t | P ) B" | :=
(\big[addsmx/0%R]_(i : t | P%B) B%MS) (only parsing) : matrix_set_scope. | Notation | \sum_ ( i : t | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i : t ) B" | :=
(\big[addsmx/0%R]_(i : t) B%MS) (only parsing) : matrix_set_scope. | Notation | \sum_ ( i : t ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i < n | P ) B" | :=
(\big[addsmx/0%R]_(i < n | P%B) B%MS) : matrix_set_scope. | Notation | \sum_ ( i < n | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i < n ) B" | :=
(\big[addsmx/0%R]_(i < n) B%MS) : matrix_set_scope. | Notation | \sum_ ( i < n ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i 'in' A | P ) B" | :=
(\big[addsmx/0%R]_(i in A | P%B) B%MS) : matrix_set_scope. | Notation | \sum_ ( i 'in' A | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
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"fingroup",
"perm",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i 'in' A ) B" | :=
(\big[addsmx/0%R]_(i in A) B%MS) : matrix_set_scope. | Notation | \sum_ ( i 'in' A ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i <- r | P ) B" | :=
(\big[capmx/1%:M]_(i <- r | P%B) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i <- r | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i <- r ) B" | :=
(\big[capmx/1%:M]_(i <- r) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i <- r ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( m <= i < n | P ) B" | :=
(\big[capmx/1%:M]_(m <= i < n | P%B) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( m <= i < n | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( m <= i < n ) B" | :=
(\big[capmx/1%:M]_(m <= i < n) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( m <= i < n ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i | P ) B" | :=
(\big[capmx/1%:M]_(i | P%B) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ i B" | :=
(\big[capmx/1%:M]_i B%MS) : matrix_set_scope. | Notation | \bigcap_ i B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i : t | P ) B" | :=
(\big[capmx/1%:M]_(i : t | P%B) B%MS) (only parsing) : matrix_set_scope. | Notation | \bigcap_ ( i : t | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
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"ssreflect",
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"binomial",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i : t ) B" | :=
(\big[capmx/1%:M]_(i : t) B%MS) (only parsing) : matrix_set_scope. | Notation | \bigcap_ ( i : t ) B | algebra | algebra/mxalgebra.v | [
"HB",
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"mathcomp",
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"fingroup",
"perm",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i < n | P ) B" | :=
(\big[capmx/1%:M]_(i < n | P%B) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i < n | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i < n ) B" | :=
(\big[capmx/1%:M]_(i < n) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i < n ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"binomial",
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"fingroup",
"perm",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i 'in' A | P ) B" | :=
(\big[capmx/1%:M]_(i in A | P%B) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i 'in' A | P ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i 'in' A ) B" | :=
(\big[capmx/1%:M]_(i in A) B%MS) : matrix_set_scope. | Notation | \bigcap_ ( i 'in' A ) B | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_stable m m' n (V : 'M[F]_(m, n)) (V' : 'M[F]_(m', n)) (f : 'M[F]_n) :
(V :=: V')%MS -> stablemx V f = stablemx V' f. | Proof. by move=> eqVV'; rewrite (eqmxMr _ eqVV') eqVV'. Qed. | Lemma | eqmx_stable | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqmxMr",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx_row_base : (stablemx (row_base V) f) = (stablemx V f). | Proof. by apply: eqmx_stable; apply: eq_row_base. Qed. | Lemma | stablemx_row_base | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"eq_row_base",
"eqmx_stable",
"row_base",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx_full : row_full V -> stablemx V f. | Proof. exact: submx_full. Qed. | Lemma | stablemx_full | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"ssrbool",
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"binomial",
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"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"row_full",
"stablemx",
"submx_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemxM : stablemx V f -> stablemx V g -> stablemx V (f *m g). | Proof. by move=> f_stab /(submx_trans _)->//; rewrite mulmxA submxMr. Qed. | Lemma | stablemxM | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mulmxA",
"stablemx",
"submxMr",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemxD : stablemx V f -> stablemx V g -> stablemx V (f + g). | Proof. by move=> f_stab g_stab; rewrite mulmxDr addmx_sub. Qed. | Lemma | stablemxD | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addmx_sub",
"mulmxDr",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemxN : stablemx V (- f) = stablemx V f. | Proof. by rewrite mulmxN eqmx_opp. Qed. | Lemma | stablemxN | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqmx_opp",
"mulmxN",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemxC x : stablemx V x%:M. | Proof. by rewrite mul_mx_scalar scalemx_sub. Qed. | Lemma | stablemxC | algebra | algebra/mxalgebra.v | [
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"structures",
"mathcomp",
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"divalg",
"finalg",
"zmodp"... | [
"mul_mx_scalar",
"scalemx_sub",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx0 : stablemx V 0. | Proof. by rewrite mulmx0 sub0mx. Qed. | Lemma | stablemx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mulmx0",
"stablemx",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stableDmx : stablemx V f -> stablemx W f -> stablemx (V + W)%MS f. | Proof. by move=> fV fW; rewrite addsmxMr addsmxS. Qed. | Lemma | stableDmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addsmxMr",
"addsmxS",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stableNmx : stablemx (- V) f = stablemx V f. | Proof. by rewrite mulNmx !eqmx_opp. Qed. | Lemma | stableNmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqmx_opp",
"mulNmx",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stable0mx : stablemx (0 : 'M_(m, n)) f. | Proof. by rewrite mul0mx. Qed. | Lemma | stable0mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mul0mx",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stableCmx (m n : nat) x (f : 'M[F]_(m, n)) : stablemx x%:M f. | Proof.
have [->|x_neq0] := eqVneq x 0; first by rewrite mul_scalar_mx scale0r sub0mx.
by rewrite -![x%:M]scalemx1 eqmx_scale// submx_full// -sub1mx.
Qed. | Lemma | stableCmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"eqVneq",
"eqmx_scale",
"mul_scalar_mx",
"nat",
"scale0r",
"scalemx1",
"stablemx",
"sub0mx",
"sub1mx",
"submx_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx_sums (n : nat) (I : finType) (V_ : I -> 'M[F]_n) (f : 'M_n) :
(forall i, stablemx (V_ i) f) -> stablemx (\sum_i V_ i)%MS f. | Proof.
by move=> fV; rewrite sumsmxMr; apply/sumsmx_subP => i; rewrite (sumsmx_sup i).
Qed. | Lemma | stablemx_sums | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"nat",
"stablemx",
"sumsmxMr",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx_unit (n : nat) (V f : 'M[F]_n) : V \in unitmx -> stablemx V f. | Proof. by move=> Vunit; rewrite submx_full ?row_full_unit. Qed. | Lemma | stablemx_unit | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"nat",
"row_full_unit",
"stablemx",
"submx_full",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_stable (f g : 'M[F]_n) : comm_mx f g -> stablemx f g. | Proof. by move=> comm_fg; rewrite [_ *m _]comm_fg mulmx_sub. Qed. | Lemma | comm_mx_stable | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"comm_mx",
"mulmx_sub",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_stable_ker (f g : 'M[F]_n) :
comm_mx f g -> stablemx (kermx f) g. | Proof.
move=> comm_fg; apply/sub_kermxP.
by rewrite -mulmxA -[g *m _]comm_fg mulmxA mulmx_ker mul0mx.
Qed. | Lemma | comm_mx_stable_ker | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"comm_mx",
"kermx",
"mul0mx",
"mulmxA",
"mulmx_ker",
"stablemx",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_stable_eigenspace (f g : 'M[F]_n) a :
comm_mx f g -> stablemx (eigenspace f a) g. | Proof.
move=> cfg; rewrite comm_mx_stable_ker//.
by apply/comm_mx_sym/comm_mxB => //; apply:comm_mx_scalar.
Qed. | Lemma | comm_mx_stable_eigenspace | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"comm_mx",
"comm_mxB",
"comm_mx_scalar",
"comm_mx_stable_ker",
"comm_mx_sym",
"eigenspace",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_delta n f : {in P &, injective f} ->
mxdirect (\sum_(i | P i) <<delta_mx 0 (f i) : 'rV[F]_n>>). | Proof.
pose fP := image f P => Uf; have UfP: uniq fP by apply/dinjectiveP.
suffices /mxdirectP : mxdirect (\sum_i <<delta_mx 0 i : 'rV[F]_n>>).
rewrite /= !(bigID [in fP] predT) -!big_uniq //= !big_map !big_enum.
by move/mxdirectP; rewrite mxdirect_addsE => /andP[].
apply/mxdirectP=> /=; transitivity (mxrank (1%:M ... | Lemma | mxdirect_delta | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"bigID",
"big_enum",
"big_map",
"big_uniq",
"card_ord",
"delta_mx",
"dinjectiveP",
"eq_bigr",
"eqmx_rank",
"fP",
"genmxE",
"image",
"mul_delta_mx",
"mx1_sum_delta",
"mxdirect",
"mxdirectP",
"mxdirect_addsE",
"mxrank",
"mxrank1",
"mxrank_delta",
"mxrank_gen",
"sub... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_GL n : n > 0 ->
#|'GL_n[F]| = (#|F| ^ 'C(n, 2) * \prod_(1 <= i < n.+1) (#|F| ^ i - 1))%N. | Proof.
case: n => // n' _; set n := n'.+1; set p := #|F|.
rewrite big_nat_rev big_add1 -bin2_sum expn_sum -big_split /=.
pose fr m := [pred A : 'M[F]_(m, n) | \rank A == m].
set m := n; rewrite [in m.+1]/m; transitivity #|fr m|.
by rewrite cardsT /= card_sub; apply: eq_card => A; rewrite -row_free_unit.
have: m <= n ... | Lemma | card_GL | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Ad",
"add1n",
"addnA",
"addnAC",
"addsmxE",
"apply",
"big_add1",
"big_geq",
"big_nat_recr",
"big_nat_rev",
"big_split",
"bin2_sum",
"capmxSl",
"card_imset",
"card_mx",
"card_ord",
"card_sub",
"cardsCs",
"cardsT",
"col_mx",
"col_mxKd",
"col_mxKu",
"dsubmx",
"eq_bigl",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
LUP_card_GL n : n > 0 ->
#|'GL_n[F]| = (#|F| ^ 'C(n, 2) * \prod_(1 <= i < n.+1) (#|F| ^ i - 1))%N. | Proof.
case: n => // n' _; set n := n'.+1; set p := #|F|.
rewrite cardsT /= card_sub /GRing.unit /= big_add1 /= -bin2_sum -/n /=.
elim: {n'}n => [|n IHn].
rewrite !big_geq // mul1n (@eq_card _ _ predT) ?card_mx //= => M.
by rewrite {1}[M]flatmx0 -(flatmx0 1%:M) unitmx1.
rewrite !big_nat_recr //= expnD mulnAC mulnA ... | Lemma | LUP_card_GL | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Schur",
"addKr",
"addr0",
"addrC",
"apply",
"big_add1",
"big_geq",
"big_nat_recr",
"bin2_sum",
"block_mx",
"block_mxEh",
"block_mxKdr",
"block_mxKur",
"cV0Pn",
"cardC1",
"card_mx",
"card_sub",
"cardsT",
"col_mx0",
"det1",
"det_lblock",
"det_mulmx",
"det_scalar1",
"det_... | row-space theory, but directly performs the LUP decomposition. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
card_GL_1 : #|'GL_1[F]| = #|F|.-1. | Proof. by rewrite card_GL // mul1n big_nat1 expn1 subn1. Qed. | Lemma | card_GL_1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"big_nat1",
"card_GL",
"expn1",
"mul1n",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_GL_2 : #|'GL_2[F]| = (#|F| * #|F|.-1 ^ 2 * #|F|.+1)%N. | Proof.
rewrite card_GL // big_ltn // big_nat1 expn1 -(addn1 #|F|) -subn1 -!mulnA.
by rewrite -subn_sqr.
Qed. | Lemma | card_GL_2 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"addn1",
"big_ltn",
"big_nat1",
"card_GL",
"expn1",
"mulnA",
"subn1",
"subn_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
logn_card_GL_p n p : prime p -> logn p #|'GL_n(p)| = 'C(n, 2). | Proof.
move=> p_pr; have p_gt1 := prime_gt1 p_pr.
have p_i_gt0: p ^ _ > 0 by move=> i; rewrite expn_gt0 ltnW.
have <- : #|'GL_n.-1.+1(p)| = #|'GL_n(p)| by [].
rewrite (card_GL _ (ltn0Sn n.-1)) card_ord Fp_cast // big_add1 /=.
pose p'gt0 m := m > 0 /\ logn p m = 0.
suffices [Pgt0 p'P]: p'gt0 (\prod_(0 <= i < n.-1.+1) (p... | Lemma | logn_card_GL_p | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"Fp_cast",
"addn0",
"apply",
"big_add1",
"big_ind",
"card_GL",
"card_ord",
"dvdn_exp",
"dvdn_subr",
"exp1n",
"expn_gt0",
"gtnNdvd",
"logn",
"logn1",
"lognE",
"lognM",
"ltn0Sn",
"ltnW",
"ltn_exp2r",
"muln_gt0",
"p_gt1",
"p_pr",
"pfactorK",
"prime",
"prime_gt1",
"subn... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A \in R" | := (@submx F _ _ _ (mxvec A) R). | Notation | A \in R | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxvec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem0mx m n (R : 'A_(m, n)) : 0 \in R. | Proof. by rewrite linear0 sub0mx. Qed. | Lemma | mem0mx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"linear0",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx0 n A : (A \in (0 : 'A_n)) -> A = 0. | Proof. by rewrite submx0 mxvec_eq0; move/eqP. Qed. | Lemma | memmx0 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxvec_eq0",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memmx1 n (A : 'M_n) : (A \in mxvec 1%:M) = is_scalar_mx A. | Proof.
apply/sub_rVP/is_scalar_mxP=> [[a] | [a ->]].
by rewrite -linearZ scale_scalar_mx mulr1 => /(can_inj mxvecK); exists a.
by exists a; rewrite -linearZ scale_scalar_mx mulr1.
Qed. | Lemma | memmx1 | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"is_scalar_mx",
"is_scalar_mxP",
"linearZ",
"mulr1",
"mxvec",
"mxvecK",
"scale_scalar_mx",
"sub_rVP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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