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 |
|---|---|---|---|---|---|---|---|---|---|---|
det_Vandermonde (R : comPzRingType) (n : nat) (a : 'rV[R]_n) :
\det (Vandermonde n a) = \prod_(i < n) \prod_(j < n | i < j) (a 0 j - a 0 i). | Proof.
set V := @Vandermonde R.
elim: n => [|n IHn] in a *; first by rewrite det_mx00 big1// => -[] [].
pose b : 'rV_n := \row_i a 0 (lift 0 i).
pose C : 'M_n := diag_mx (\row_(i < n) (b 0 i - a 0 0)).
pose D : 'M_n.+1 := 1 - a 0 0 *: \matrix_(i, j) (i == j.+1 :> nat)%:R.
have detD : \det D = 1.
rewrite det_trig ?bi... | Lemma | det_Vandermonde | algebra | algebra/matrix.v | [
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"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"Vandermonde",
"apply",
"big1",
"big_mkcond",
"big_ord1_eq",
"big_ord_recl",
"big_pred1_eq",
"block_mx",
"const_mx",
"det1",
"detM",
"det_diag",
"det_mulmx",
"det_mx00",
"det_trig",
"det_ublock",
"determinant",
"diag_mx",
"eqSS",
"eq_bigr",
"eq_sym",
"eqxx",
"expr0",
"e... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''A_' ( m , n )" | := 'M_(m, n ^ 2)
(format "''A_' ( m , n )") : type_scope. | Notation | ''A_' ( m , n ) | algebra | algebra/mxalgebra.v | [
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''A_' ( n )" | := 'A_(n ^ 2, n) (only parsing) : type_scope. | Notation | ''A_' ( n ) | algebra | algebra/mxalgebra.v | [
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''A_' n" | := 'A_(n)
(at level 0, n at level 0, format "''A_' n") : type_scope. | Notation | ''A_' n | algebra | algebra/mxalgebra.v | [
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''A' [ F ]_ ( m , n )" | := 'M[F]_(m, n ^ 2)
(only parsing) : type_scope. | Notation | ''A' [ F ]_ ( m , n ) | algebra | algebra/mxalgebra.v | [
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"nmodule",
"fingroup",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''A' [ F ]_ ( n )" | := 'A[F]_(n ^ 2, n) (only parsing) : type_scope. | Notation | ''A' [ F ]_ ( n ) | algebra | algebra/mxalgebra.v | [
"HB",
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"mathcomp",
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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 | |
"''A' [ F ]_ n" | := 'A[F]_(n) (n at level 2, only parsing) : type_scope. | Notation | ''A' [ F ]_ n | algebra | algebra/mxalgebra.v | [
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gaussian_elimination_ {F : fieldType} {m n} : 'M[F]_(m, n) -> 'M_m * 'M_n * nat | :=
match m, n with
| _.+1, _.+1 => fun A : 'M_(1 + _, 1 + _) =>
if [pick ij | A ij.1 ij.2 != 0] is Some (i, j) then
let a := A i j in let A1 := xrow i 0 (xcol j 0 A) in
let u := ursubmx A1 in let v := a^-1 *: dlsubmx A1 in
let: (L, U, r) := Gaussian_elimination_ (drsubmx A1 - v *m u) in
... | Fixpoint | Gaussian_elimination_ | algebra | algebra/mxalgebra.v | [
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"block_mx",
"dlsubmx",
"drsubmx",
"nat",
"pick",
"ursubmx",
"xcol",
"xrow"
] | images, kernels, and complements of a matrix. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Gaussian_elimination_unlockable | := Unlockable Gaussian_elimination.unlock. | Canonical | Gaussian_elimination_unlockable | algebra | algebra/mxalgebra.v | [
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"mathcomp",
"ssreflect",
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"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_unlockable | := Unlockable mxrank.unlock. | Canonical | mxrank_unlockable | algebra | algebra/mxalgebra.v | [
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxrank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M_' ( m , n )" | := 'M[F]_(m, n) : type_scope. | Notation | ''M_' ( m , n ) | algebra | algebra/mxalgebra.v | [
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"mathcomp",
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"eqtype",
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"binomial",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M_' n" | := 'M[F]_(n, n) : type_scope. | Notation | ''M_' n | algebra | algebra/mxalgebra.v | [
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"binomial",
"nmodule",
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"perm",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank | := (@mxrank F m n A). | Notation | mxrank | algebra | algebra/mxalgebra.v | [
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"mathcomp",
"ssreflect",
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"ssrbool",
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"binomial",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
LUr | := @Gaussian_elimination F m n A. | Let | LUr | algebra | algebra/mxalgebra.v | [
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_ebase | := LUr.1.1. | Definition | col_ebase | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"LUr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_ebase | := LUr.1.2. | Definition | row_ebase | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"LUr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_free | := mxrank == m. | Definition | row_free | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"mxrank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_full | := mxrank == n. | Definition | row_full | algebra | algebra/mxalgebra.v | [
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"finalg",
"zmodp"... | [
"mxrank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_base : 'M_(mxrank, n) | := pid_mx mxrank *m row_ebase. | Definition | row_base | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"mxrank",
"pid_mx",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_base : 'M_(m, mxrank) | := col_ebase *m pid_mx mxrank. | Definition | col_base | algebra | algebra/mxalgebra.v | [
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"mathcomp",
"ssreflect",
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"zmodp"... | [
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"mxrank",
"pid_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
complmx : 'M_n | := copid_mx mxrank *m row_ebase. | Definition | complmx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
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"mxrank",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermx : 'M_m | := copid_mx mxrank *m invmx col_ebase. | Definition | kermx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"col_ebase",
"copid_mx",
"invmx",
"mxrank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cokermx : 'M_n | := invmx row_ebase *m copid_mx mxrank. | Definition | cokermx | algebra | algebra/mxalgebra.v | [
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"structures",
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"ssrbool",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"copid_mx",
"invmx",
"mxrank",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pinvmx : 'M_(n, m) | :=
invmx row_ebase *m pid_mx mxrank *m invmx col_ebase. | Definition | pinvmx | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"col_ebase",
"invmx",
"mxrank",
"pid_mx",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_unlockable | := Unlockable submx.unlock. | Canonical | submx_unlockable | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\rank A" | := (mxrank A) : nat_scope. | Notation | \rank A | algebra | algebra/mxalgebra.v | [
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"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"binomial",
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"perm",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"mxrank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A ^C" | := (complmx A) : matrix_set_scope. | Notation | A ^C | algebra | algebra/mxalgebra.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"complmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A <= B" | := (submx A B) : matrix_set_scope. | Notation | A <= B | algebra | algebra/mxalgebra.v | [
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"mathcomp",
"ssreflect",
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"ssrbool",
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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 | |
"A <= B <= C" | := ((A <= B) && (B <= C))%MS : matrix_set_scope. | Notation | A <= B <= C | algebra | algebra/mxalgebra.v | [
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"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A == B" | := (A <= B <= A)%MS : matrix_set_scope. | Notation | A == B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmx F m1 m2 n (A : 'M[F]_(m1, n)) (B : 'M_(m2, n)) | :=
(A <= B)%MS && ~~ (B <= A)%MS. | Definition | ltmx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A < B" | := (ltmx A B) : matrix_set_scope. | Notation | A < B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
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] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx F m1 m2 n (A : 'M[F]_(m1, n)) (B : 'M_(m2, n)) | :=
prod (\rank A = \rank B)
(forall m3 (C : 'M_(m3, n)),
((A <= C) = (B <= C)) * ((C <= A) = (C <= B)))%MS. | Definition | eqmx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"prod",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :=: B" | := (eqmx A%MS B%MS) : matrix_set_scope. | Notation | A :=: B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"eqmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stablemx V f | := (V%MS *m f%R <= V%MS)%MS. | Notation | stablemx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmxE : (A < B)%MS = ((A <= B)%MS && ~~ (B <= A)%MS). | Proof. by []. Qed. | Lemma | ltmxE | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmxW : (A < B)%MS -> (A <= B)%MS. | Proof. by case/andP. Qed. | Lemma | ltmxW | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmxEneq : (A < B)%MS = (A <= B)%MS && ~~ (A == B)%MS. | Proof. by apply: andb_id2l => ->. Qed. | Lemma | ltmxEneq | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxElt : (A <= B)%MS = (A == B)%MS || (A < B)%MS. | Proof. by rewrite -andb_orr orbN andbT. Qed. | Lemma | submxElt | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qidmx F m n (A : 'M[F]_(m, n)) | :=
if m == n then A == pid_mx n else row_full A. | Definition | qidmx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"pid_mx",
"row_full"
] | row_base A padded with null rows. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
equivmx F m n (A : 'M[F]_(m, n)) idA (B : 'M_n) | :=
(B == A)%MS && (qidmx B == idA). | Definition | equivmx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
equivmx_spec F m n (A : 'M[F]_(m, n)) idA (B : 'M_n) | :=
prod (B :=: A)%MS (qidmx B = idA). | Definition | equivmx_spec | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"prod",
"qidmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_witness F m n (A : 'M[F]_(m, n)) : 'M_n | :=
if row_full A then 1%:M else pid_mx (\rank A) *m row_ebase A. | Definition | genmx_witness | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"pid_mx",
"rank",
"row_ebase",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_unlockable | := Unlockable genmx.unlock. | Canonical | genmx_unlockable | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"<< A >>" | := (genmx A%MS) : matrix_set_scope. | Notation | << A >> | algebra | algebra/mxalgebra.v | [
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"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_nop F m n (A : 'M[F]_(m, n)) | :=
conform_mx <<A>>%MS A. | Definition | addsmx_nop | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"conform_mx"
] | setwise sum is not quite strictly extensional. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
addsmx_unlockable | := Unlockable addsmx.unlock. | Canonical | addsmx_unlockable | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A + B" | := (addsmx A B) : matrix_set_scope. | Notation | A + B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i | P ) B" | := (\big[addsmx/0]_(i | P) B%MS)
: matrix_set_scope. | Notation | \sum_ ( i | P ) B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\sum_ ( i <- r | P ) B" | := (\big[addsmx/0]_(i <- r | P) B%MS)
: matrix_set_scope. | Notation | \sum_ ( i <- r | P ) B | algebra | algebra/mxalgebra.v | [
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"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
capmx_witness F m n (A : 'M[F]_(m, n)) | :=
if row_full A then conform_mx 1%:M A else <<A>>%MS. | Definition | capmx_witness | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
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capmx_norm F m n (A : 'M[F]_(m, n)) | :=
choose (equivmx A (qidmx A)) (capmx_witness A). | Definition | capmx_norm | algebra | algebra/mxalgebra.v | [
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capmx_nop F m n (A : 'M[F]_(m, n)) | :=
conform_mx (capmx_norm A) A. | Definition | capmx_nop | algebra | algebra/mxalgebra.v | [
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capmx_gen F m1 m2 n (A : 'M[F]_(m1, n)) (B : 'M_(m2, n)) | :=
lsubmx (kermx (col_mx A B)) *m A. | Definition | capmx_gen | algebra | algebra/mxalgebra.v | [
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capmx_unlockable | := Unlockable capmx.unlock. | Canonical | capmx_unlockable | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :&: B" | := (capmx A B) : matrix_set_scope. | Notation | A :&: B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\bigcap_ ( i | P ) B" | := (\big[capmx/1%:M]_(i | P) B)
: matrix_set_scope. | Notation | \bigcap_ ( i | P ) B | algebra | algebra/mxalgebra.v | [
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"finalg",
"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diffmx_unlockable | := Unlockable diffmx.unlock. | Canonical | diffmx_unlockable | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :\: B" | := (diffmx A B) : matrix_set_scope. | Notation | A :\: B | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_mx n (U V : 'M_n) : 'M_n | := pinvmx (col_mx U V) *m col_mx U 0. | Definition | proj_mx | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
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GaussE | := Gaussian_elimination_. | Notation | GaussE | algebra | algebra/mxalgebra.v | [
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"zmodp"... | [
"Gaussian_elimination_"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrankE m n (A : 'M_(m, n)) : \rank A = (GaussE A).2. | Proof. by rewrite mxrank.unlock unlock /=; case: m n A => [|m] [|n]. Qed. | Fact | mxrankE | 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"... | [
"GaussE",
"mxrank",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_leq_row m n (A : 'M_(m, n)) : \rank A <= m. | Proof.
rewrite mxrankE.
elim: m n A => [|m IHm] [|n] //= A; case: pickP => [[i j] _|] //=.
by move: (_ - _) => B; case: GaussE (IHm _ B) => [[L U] r] /=.
Qed. | Lemma | rank_leq_row | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"GaussE",
"mxrankE",
"pickP",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_leq_rank m n (A : 'M_(m, n)) : (m <= \rank A) = row_free A. | Proof. by rewrite /row_free eqn_leq rank_leq_row. Qed. | Lemma | row_leq_rank | 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"... | [
"eqn_leq",
"rank",
"rank_leq_row",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_leq_col m n (A : 'M_(m, n)) : \rank A <= n. | Proof.
rewrite mxrankE.
elim: m n A => [|m IHm] [|n] //= A; case: pickP => [[i j] _|] //=.
by move: (_ - _) => B; case: GaussE (IHm _ B) => [[L U] r] /=.
Qed. | Lemma | rank_leq_col | 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"... | [
"GaussE",
"mxrankE",
"pickP",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_leq_rank m n (A : 'M_(m, n)) : (n <= \rank A) = row_full A. | Proof. by rewrite /row_full eqn_leq rank_leq_col. Qed. | Lemma | col_leq_rank | 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"... | [
"eqn_leq",
"rank",
"rank_leq_col",
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_row_full m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :=: B)%MS -> row_full A = row_full B. | Proof. by rewrite /row_full => ->. Qed. | Lemma | eq_row_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"... | [
"row_full"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitmx1F | := @unitmx1 F. | Let | unitmx1F | 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"... | [
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_ebase_unit m n (A : 'M_(m, n)) : row_ebase A \in unitmx. | Proof.
rewrite /row_ebase unlock; elim: m n A => [|m IHm] [|n] //= A.
case: pickP => [[i j] /= nzAij | //=]; move: (_ - _) => B.
case: GaussE (IHm _ B) => [[L U] r] /= uU.
rewrite unitmxE xcolE det_mulmx (@det_ublock _ 1) det_scalar1 !unitrM.
by rewrite unitfE nzAij -!unitmxE uU unitmx_perm.
Qed. | Lemma | row_ebase_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"... | [
"GaussE",
"det_mulmx",
"det_scalar1",
"det_ublock",
"pickP",
"row_ebase",
"unitfE",
"unitmx",
"unitmxE",
"unitmx_perm",
"unitrM",
"xcolE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_ebase_unit m n (A : 'M_(m, n)) : col_ebase A \in unitmx. | Proof.
rewrite /col_ebase unlock; elim: m n A => [|m IHm] [|n] //= A.
case: pickP => [[i j] _|] //=; move: (_ - _) => B.
case: GaussE (IHm _ B) => [[L U] r] /= uL.
rewrite unitmxE xrowE det_mulmx (@det_lblock _ 1) det1 mul1r unitrM.
by rewrite -unitmxE unitmx_perm.
Qed. | Lemma | col_ebase_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"... | [
"GaussE",
"col_ebase",
"det1",
"det_lblock",
"det_mulmx",
"mul1r",
"pickP",
"unitmx",
"unitmxE",
"unitmx_perm",
"unitrM",
"xrowE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_ebase m n (A : 'M_(m, n)) :
col_ebase A *m pid_mx (\rank A) *m row_ebase A = A. | Proof.
rewrite mxrankE /col_ebase /row_ebase unlock.
elim: m n A => [n A | m IHm]; first by rewrite [A]flatmx0 [_ *m _]flatmx0.
case=> [A | n]; first by rewrite [_ *m _]thinmx0 [A]thinmx0.
rewrite -(add1n m) -?(add1n n) => A /=.
case: pickP => [[i0 j0] | A0] /=; last first.
apply/matrixP=> i j; rewrite pid_mx_0 mulmx... | Lemma | mulmx_ebase | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"GaussE",
"add0r",
"add1n",
"addr0",
"apply",
"block_mx",
"block_mxA",
"castmx_id",
"col_ebase",
"col_mx0",
"divff",
"flatmx0",
"i0",
"last",
"lshift0",
"matrixP",
"mul0mx",
"mul1mx",
"mul_mx_scalar",
"mulmx0",
"mulmx1",
"mulmxA",
"mulmx_block",
"mx11_scalar",
"mxE",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_base m n (A : 'M_(m, n)) : col_base A *m row_base A = A. | Proof. by rewrite mulmxA -[col_base A *m _]mulmxA pid_mx_id ?mulmx_ebase. Qed. | Lemma | mulmx_base | 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"... | [
"col_base",
"mulmxA",
"mulmx_ebase",
"pid_mx_id",
"row_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx1_min_rank r m n (A : 'M_(m, n)) M N :
M *m A *m N = 1%:M :> 'M_r -> r <= \rank A. | Proof. by rewrite -{1}(mulmx_base A) mulmxA -mulmxA; move/mulmx1_min. Qed. | Lemma | mulmx1_min_rank | 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"... | [
"mulmx1_min",
"mulmxA",
"mulmx_base",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_max_rank r m n (M : 'M_(m, r)) (N : 'M_(r, n)) :
\rank (M *m N) <= r. | Proof.
set MN := M *m N; set rMN := \rank _.
pose L : 'M_(rMN, m) := pid_mx rMN *m invmx (col_ebase MN).
pose U : 'M_(n, rMN) := invmx (row_ebase MN) *m pid_mx rMN.
suffices: L *m M *m (N *m U) = 1%:M by apply: mulmx1_min.
rewrite mulmxA -(mulmxA L) -[M *m N]mulmx_ebase -/MN.
by rewrite !mulmxA mulmxKV // mulmxK // !pi... | Lemma | mulmx_max_rank | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"finset",
"binomial",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"apply",
"col_ebase",
"invmx",
"mulmx1_min",
"mulmxA",
"mulmxK",
"mulmxKV",
"mulmx_ebase",
"pid_mx",
"pid_mx_1",
"pid_mx_id",
"rank",
"row_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_tr m n (A : 'M_(m, n)) : \rank A^T = \rank A. | Proof.
apply/eqP; rewrite eqn_leq -{3}[A]trmxK -{1}(mulmx_base A) -{1}(mulmx_base A^T).
by rewrite !trmx_mul !mulmx_max_rank.
Qed. | Lemma | mxrank_tr | 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",
"eqn_leq",
"mulmx_base",
"mulmx_max_rank",
"rank",
"trmxK",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_add m n (A B : 'M_(m, n)) : \rank (A + B)%R <= \rank A + \rank B. | Proof.
by rewrite -{1}(mulmx_base A) -{1}(mulmx_base B) -mul_row_col mulmx_max_rank.
Qed. | Lemma | mxrank_add | 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"... | [
"mul_row_col",
"mulmx_base",
"mulmx_max_rank",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrankM_maxl m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
\rank (A *m B) <= \rank A. | Proof. by rewrite -{1}(mulmx_base A) -mulmxA mulmx_max_rank. Qed. | Lemma | mxrankM_maxl | 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"... | [
"mulmxA",
"mulmx_base",
"mulmx_max_rank",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrankM_maxr m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
\rank (A *m B) <= \rank B. | Proof. by rewrite -mxrank_tr -(mxrank_tr B) trmx_mul mxrankM_maxl. Qed. | Lemma | mxrankM_maxr | 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"... | [
"mxrankM_maxl",
"mxrank_tr",
"rank",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_scale m n a (A : 'M_(m, n)) : \rank (a *: A) <= \rank A. | Proof. by rewrite -mul_scalar_mx mxrankM_maxr. Qed. | Lemma | mxrank_scale | 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"... | [
"mul_scalar_mx",
"mxrankM_maxr",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_scale_nz m n a (A : 'M_(m, n)) :
a != 0 -> \rank (a *: A) = \rank A. | Proof.
move=> nza; apply/eqP; rewrite eqn_leq -{3}[A]scale1r -(mulVf nza).
by rewrite -scalerA !mxrank_scale.
Qed. | Lemma | mxrank_scale_nz | 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",
"eqn_leq",
"mulVf",
"mxrank_scale",
"rank",
"scale1r",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_opp m n (A : 'M_(m, n)) : \rank (- A) = \rank A. | Proof. by rewrite -scaleN1r mxrank_scale_nz // oppr_eq0 oner_eq0. Qed. | Lemma | mxrank_opp | 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"... | [
"mxrank_scale_nz",
"oner_eq0",
"oppr_eq0",
"rank",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank0 m n : \rank (0 : 'M_(m, n)) = 0%N. | Proof. by apply/eqP; rewrite -leqn0 -(@mulmx0 _ m 0 n 0) mulmx_max_rank. Qed. | Lemma | mxrank0 | 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",
"leqn0",
"mulmx0",
"mulmx_max_rank",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_eq0 m n (A : 'M_(m, n)) : (\rank A == 0) = (A == 0). | Proof.
apply/eqP/eqP=> [rA0 | ->{A}]; last exact: mxrank0.
move: (col_base A) (row_base A) (mulmx_base A); rewrite rA0 => Ac Ar <-.
by rewrite [Ac]thinmx0 mul0mx.
Qed. | Lemma | mxrank_eq0 | 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",
"col_base",
"last",
"mul0mx",
"mulmx_base",
"mxrank0",
"rank",
"row_base",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_coker m n (A : 'M_(m, n)) : A *m cokermx A = 0. | Proof.
by rewrite -{1}[A]mulmx_ebase -!mulmxA mulKVmx // mul_pid_mx_copid ?mulmx0.
Qed. | Lemma | mulmx_coker | algebra | algebra/mxalgebra.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"finfun",
"bigop",
"prime",
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"binomial",
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"order",
"rings_modules_and_algebras",
"divalg",
"finalg",
"zmodp"... | [
"cokermx",
"mulKVmx",
"mul_pid_mx_copid",
"mulmx0",
"mulmxA",
"mulmx_ebase"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxE m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A <= B)%MS = (A *m cokermx B == 0). | Proof. by rewrite unlock. Qed. | Lemma | submxE | 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"... | [
"cokermx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxKpV m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A <= B)%MS -> A *m pinvmx B *m B = A. | Proof.
rewrite submxE !mulmxA mulmxBr mulmx1 subr_eq0 => /eqP defA.
rewrite -{4}[B]mulmx_ebase -!mulmxA mulKmx //.
by rewrite (mulmxA (pid_mx _)) pid_mx_id // !mulmxA -{}defA mulmxKV.
Qed. | Lemma | mulmxKpV | 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"... | [
"mulKmx",
"mulmx1",
"mulmxA",
"mulmxBr",
"mulmxKV",
"mulmx_ebase",
"pid_mx",
"pid_mx_id",
"pinvmx",
"submxE",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxVp m n (A : 'M[F]_(m, n)) : row_free A -> A *m pinvmx A = 1%:M. | Proof.
move=> fA; rewrite -[X in X *m _]mulmx_ebase !mulmxA mulmxK ?row_ebase_unit//.
rewrite -[X in X *m _]mulmxA mul_pid_mx !minnn (minn_idPr _) ?rank_leq_col//.
by rewrite (eqP fA) pid_mx_1 mulmx1 mulmxV ?col_ebase_unit.
Qed. | Lemma | mulmxVp | 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"... | [
"col_ebase_unit",
"fA",
"minn_idPr",
"minnn",
"mul_pid_mx",
"mulmx1",
"mulmxA",
"mulmxK",
"mulmxV",
"mulmx_ebase",
"pid_mx_1",
"pinvmx",
"rank_leq_col",
"row_ebase_unit",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxKp p m n (B : 'M[F]_(m, n)) : row_free B ->
cancel ((@mulmx _ p _ _)^~ B) (mulmx^~ (pinvmx B)). | Proof. by move=> ? A; rewrite -mulmxA mulmxVp ?mulmx1. Qed. | Lemma | mulmxKp | 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"... | [
"mulmx",
"mulmx1",
"mulmxA",
"mulmxVp",
"pinvmx",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxP m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
reflect (exists D, A = D *m B) (A <= B)%MS. | Proof.
apply: (iffP idP) => [/mulmxKpV | [D ->]]; first by exists (A *m pinvmx B).
by rewrite submxE -mulmxA mulmx_coker mulmx0.
Qed. | Lemma | submxP | 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",
"mulmx0",
"mulmxA",
"mulmxKpV",
"mulmx_coker",
"pinvmx",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_refl m n (A : 'M_(m, n)) : (A <= A)%MS. | Proof. by rewrite submxE mulmx_coker. Qed. | Lemma | submx_refl | 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"... | [
"mulmx_coker",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxMl m n p (D : 'M_(m, n)) (A : 'M_(n, p)) : (D *m A <= A)%MS. | Proof. by rewrite submxE -mulmxA mulmx_coker mulmx0. Qed. | Lemma | submxMl | 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",
"mulmxA",
"mulmx_coker",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submxMr m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) :
(A <= B)%MS -> (A *m C <= B *m C)%MS. | Proof. by case/submxP=> D ->; rewrite -mulmxA submxMl. Qed. | Lemma | submxMr | 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"... | [
"mulmxA",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_sub m n1 n2 p (C : 'M_(m, n1)) A (B : 'M_(n2, p)) :
(A <= B -> C *m A <= B)%MS. | Proof. by case/submxP=> D ->; rewrite mulmxA submxMl. Qed. | Lemma | mulmx_sub | 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"... | [
"mulmxA",
"submxMl",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_trans m1 m2 m3 n
(A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A <= B -> B <= C -> A <= C)%MS. | Proof. by case/submxP=> D ->{A}; apply: mulmx_sub. Qed. | Lemma | submx_trans | 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",
"mulmx_sub",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmx_sub_trans m1 m2 m3 n
(A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A < B)%MS -> (B <= C)%MS -> (A < C)%MS. | Proof.
case/andP=> sAB ltAB sBC; rewrite ltmxE (submx_trans sAB) //.
by apply: contra ltAB; apply: submx_trans.
Qed. | Lemma | ltmx_sub_trans | 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",
"ltmxE",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_ltmx_trans m1 m2 m3 n
(A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) :
(A <= B)%MS -> (B < C)%MS -> (A < C)%MS. | Proof.
move=> sAB /andP[sBC ltBC]; rewrite ltmxE (submx_trans sAB) //.
by apply: contra ltBC => sCA; apply: submx_trans sAB.
Qed. | Lemma | sub_ltmx_trans | 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",
"ltmxE",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmx_trans m n : transitive (@ltmx F m m n). | Proof. by move=> A B C; move/ltmxW; apply: sub_ltmx_trans. Qed. | Lemma | ltmx_trans | 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",
"ltmx",
"ltmxW",
"sub_ltmx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltmx_irrefl m n : irreflexive (@ltmx F m m n). | Proof. by move=> A; rewrite /ltmx submx_refl andbF. Qed. | Lemma | ltmx_irrefl | 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",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub0mx m1 m2 n (A : 'M_(m2, n)) : ((0 : 'M_(m1, n)) <= A)%MS. | Proof. by rewrite submxE mul0mx. Qed. | Lemma | sub0mx | 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",
"submxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx0null m1 m2 n (A : 'M[F]_(m1, n)) :
(A <= (0 : 'M_(m2, n)))%MS -> A = 0. | Proof. by case/submxP=> D; rewrite mulmx0. Qed. | Lemma | submx0null | 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",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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