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 |
|---|---|---|---|---|---|---|---|---|---|---|
comm_horner_mx2 A p q : GRing.comm (horner_mx A p) (horner_mx A q). | Proof. exact/comm_mx_horner/comm_horner_mx. Qed. | Lemma | comm_horner_mx2 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"comm",
"comm_horner_mx",
"comm_mx_horner",
"horner_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_stable (K : fieldType) m n p
(V : 'M[K]_(n.+1, m.+1)) (f : 'M_m.+1) :
stablemx V f -> stablemx V (horner_mx f p). | Proof.
move=> V_fstab; elim/poly_ind: p => [|p c]; first by rewrite rmorph0 stablemx0.
move=> fp_stable; rewrite rmorphD rmorphM/= horner_mx_X horner_mx_C.
by rewrite stablemxD ?stablemxM ?fp_stable ?stablemxC.
Qed. | Lemma | horner_mx_stable | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"poly_ind",
"rmorph0",
"rmorphD",
"rmorphM",
"stablemx",
"stablemx0",
"stablemxC",
"stablemxD",
"stablemxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_poly_mx | := 'X%:M - map_mx (@polyC R) A. | Definition | char_poly_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"map_mx",
"polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_poly | := \det char_poly_mx. | Definition | char_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"char_poly_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diagA | := [seq A i i | i <- index_enum _ & true]. | Let | diagA | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"index_enum",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_diagA : size diagA = n. | Proof. by rewrite -[n]card_ord size_map; have [e _ _ []] := big_enumP. Qed. | Let | size_diagA | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"big_enumP",
"card_ord",
"diagA",
"size",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split_diagA :
exists2 q, \prod_(x <- diagA) ('X - x%:P) + q = char_poly & size q <= n.-1. | Proof.
rewrite [char_poly](bigD1 1%g) //=; set q := \sum_(s | _) _; exists q.
congr (_ + _); rewrite odd_perm1 mul1r big_map big_filter /=.
by apply: eq_bigr => i _; rewrite !mxE perm1 eqxx.
apply: leq_trans {q}(size_sum _ _ _) _; apply/bigmax_leqP=> s nt_s.
have{nt_s} [i nfix_i]: exists i, s i != i.
apply/exists... | Let | split_diagA | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addnA",
"addnS",
"apply",
"bigD1",
"big_filter",
"big_ind2",
"big_map",
"big_mkcond",
"bigmax_leqP",
"card2",
"cardC",
"card_ord",
"char_poly",
"contraNneq",
"diagA",
"eq_bigr",
"eq_sym",
"eqxx",
"existsP",
"forallP",
"inE",
"inj_eq",
"leq_add",
"leq_b1",
"leq_subLR"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_char_poly : size char_poly = n.+1. | Proof.
have [q <- lt_q_n] := split_diagA; have le_q_n := leq_trans lt_q_n (leq_pred n).
by rewrite size_polyDl size_prod_XsubC size_diagA.
Qed. | Lemma | size_char_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"char_poly",
"leq_pred",
"leq_trans",
"size",
"size_diagA",
"size_polyDl",
"size_prod_XsubC",
"split_diagA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_poly_monic : char_poly \is monic. | Proof.
rewrite monicE -(monicP (monic_prod_XsubC diagA xpredT id)).
rewrite !lead_coefE size_char_poly.
have [q <- lt_q_n] := split_diagA; have le_q_n := leq_trans lt_q_n (leq_pred n).
by rewrite size_prod_XsubC size_diagA coefD (nth_default 0 le_q_n) addr0.
Qed. | Lemma | char_poly_monic | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addr0",
"char_poly",
"coefD",
"diagA",
"id",
"lead_coefE",
"leq_pred",
"leq_trans",
"monic",
"monicE",
"monicP",
"monic_prod_XsubC",
"nth_default",
"size_char_poly",
"size_diagA",
"size_prod_XsubC",
"split_diagA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_poly_trace : n > 0 -> char_poly`_n.-1 = - \tr A. | Proof.
move=> n_gt0; have [q <- lt_q_n] := split_diagA; set p := \prod_(x <- _) _.
rewrite coefD {q lt_q_n}(nth_default 0 lt_q_n) addr0.
have{n_gt0} ->: p`_n.-1 = ('X * p)`_n by rewrite coefXM eqn0Ngt n_gt0.
have ->: \tr A = \sum_(x <- diagA) x by rewrite big_map big_filter.
rewrite -size_diagA {}/p; elim: diagA => [|x... | Lemma | char_poly_trace | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addr0",
"addrC",
"big_cons",
"big_filter",
"big_map",
"big_nil",
"char_poly",
"coefB",
"coefD",
"coefX",
"coefXM",
"coefZ",
"diagA",
"eqn0Ngt",
"id",
"lead_coefE",
"monicP",
"monic_prod_XsubC",
"mul_polyC",
"mulr1",
"mulrBl",
"n_gt0",
"nth_default",
"oppr0",
"opprD",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_poly_det : char_poly`_0 = (- 1) ^+ n * \det A. | Proof.
rewrite big_distrr coef_sum [0%N]lock /=; apply: eq_bigr => s _.
rewrite -{1}rmorphN -rmorphXn mul_polyC coefZ /=.
rewrite mulrA -exprD addnC exprD -mulrA -lock; congr (_ * _).
transitivity (\prod_(i < n) - A i (s i)); last by rewrite prodrN card_ord.
elim: (index_enum _) => [|i e IHe]; rewrite !(big_nil, big_co... | Lemma | char_poly_det | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addnC",
"apply",
"big_cons",
"big_distrr",
"big_nil",
"big_ord1",
"card_ord",
"char_poly",
"coef1",
"coefB",
"coefC",
"coefM",
"coefMn",
"coefX",
"coefZ",
"coef_sum",
"eq_bigr",
"exprD",
"index_enum",
"last",
"mul0rn",
"mul_polyC",
"mulrA",
"mxE",
"prodrN",
"rmorph... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_poly_ring_isom (R : nzSemiRingType) n' (n := n'.+1) :
exists phi : {rmorphism 'M[{poly R}]_n -> {poly 'M[R]_n}},
[/\ bijective phi,
forall p, phi p%:M = map_poly scalar_mx p,
forall A, phi (map_mx polyC A) = A%:P
& forall A i j k, (phi A)`_k i j = (A i j)`_k]. | Proof.
set M_RX := 'M[{poly R}]_n; set MR_X := ({poly 'M[R]_n}).
pose Msize (A : M_RX) := \max_i \max_j size (A i j).
pose phi (A : M_RX) := \poly_(k < Msize A) \matrix_(i, j) (A i j)`_k.
have coef_phi A i j k: (phi A)`_k i j = (A i j)`_k.
rewrite coef_poly; case: (ltnP k _) => le_m_k; rewrite mxE // nth_default //.
... | Lemma | mx_poly_ring_isom | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Build",
"apply",
"coef0",
"coefC",
"coefD",
"coefM",
"coefMn",
"coef_map",
"coef_poly",
"coef_sum",
"eq_bigr",
"exchange_big",
"last",
"leqP",
"leq_bigmax",
"leq_trans",
"ltnP",
"map_mx",
"map_poly",
"matrixP",
"monoid_morphism",
"mul0rn",
"mxE",
"n'",
"nmod_morphism... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cayley_Hamilton (R : comNzRingType) n' (A : 'M[R]_n'.+1) :
horner_mx A (char_poly A) = 0. | Proof.
have [phi [_ phiZ phiC _]] := mx_poly_ring_isom R n'.
apply/rootP/factor_theorem; rewrite -phiZ -mul_adj_mx rmorphM /=.
by move: (phi _) => q; exists q; rewrite rmorphB phiC phiZ map_polyX.
Qed. | Theorem | Cayley_Hamilton | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"char_poly",
"factor_theorem",
"horner_mx",
"map_polyX",
"mul_adj_mx",
"mx_poly_ring_isom",
"n'",
"rmorphB",
"rmorphM",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue_root_char (F : fieldType) n (A : 'M[F]_n) a :
eigenvalue A a = root (char_poly A) a. | Proof.
transitivity (\det (a%:M - A) == 0).
apply/eigenvalueP/det0P=> [[v Av_av v_nz] | [v v_nz Av_av]]; exists v => //.
by rewrite mulmxBr Av_av mul_mx_scalar subrr.
by apply/eqP; rewrite -mul_mx_scalar eq_sym -subr_eq0 -mulmxBr Av_av.
congr (_ == 0); rewrite horner_sum; apply: eq_bigr => s _.
rewrite hornerM ... | Lemma | eigenvalue_root_char | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"big_morph",
"char_poly",
"det0P",
"eigenvalue",
"eigenvalueP",
"eq_bigr",
"eq_sym",
"hornerC",
"hornerE",
"hornerM",
"hornerMn",
"horner_exp",
"horner_sum",
"mul_mx_scalar",
"mulmxBr",
"mxE",
"root",
"subr_eq0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_poly_trig {R : comNzRingType} n (A : 'M[R]_n) : is_trig_mx A ->
char_poly A = \prod_(i < n) ('X - (A i i)%:P). | Proof.
move=> /is_trig_mxP Atrig; rewrite /char_poly det_trig; last first.
by apply: eq_bigr => i; rewrite !mxE eqxx.
by apply/is_trig_mxP => i j lt_ij; rewrite !mxE -val_eqE ltn_eqF ?Atrig ?subrr.
Qed. | Lemma | char_poly_trig | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"char_poly",
"det_trig",
"eq_bigr",
"eqxx",
"is_trig_mx",
"is_trig_mxP",
"last",
"ltn_eqF",
"mxE",
"subrr",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
companionmx {R : nzRingType} (p : seq R) (d := (size p).-1) | :=
\matrix_(i < d, j < d)
if (i == d.-1 :> nat) then - p`_j else (i.+1 == j :> nat)%:R. | Definition | companionmx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"nat",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
companionmxK {R : comNzRingType} (p : {poly R}) :
p \is monic -> char_poly (companionmx p) = p. | Proof.
pose D n : 'M[{poly R}]_n := \matrix_(i, j)
('X *+ (i == j.+1 :> nat) - ((i == j)%:R)%:P).
have detD n : \det (D n) = (-1) ^+ n.
elim: n => [|n IHn]; first by rewrite det_mx00.
rewrite (expand_det_row _ ord0) big_ord_recl !mxE /= sub0r.
rewrite big1 ?addr0; first by move=> i _; rewrite !mxE /= subrr mul... | Lemma | companionmxK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"add0n",
"add0r",
"add1n",
"addn0",
"addr0",
"apply",
"big1",
"big_ord_recl",
"big_ord_recr",
"bump",
"char_poly",
"coefC",
"coefD",
"coefMX",
"cofactor",
"companionmx",
"det_mx00",
"det_scalar1",
"eqSS",
"eqVneq",
"eq_sym",
"eqxx",
"expand_det_col",
"expand_det_row",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmx_delta_companion (R : nzRingType) (p : seq R)
(i: 'I_(size p).-1) (i_small : i.+1 < (size p).-1):
delta_mx 0 i *m companionmx p = delta_mx 0 (Ordinal i_small) :> 'rV__. | Proof.
apply/rowP => j; rewrite !mxE (bigD1 i) //= ?(=^~val_eqE, mxE) /= eqxx mul1r.
rewrite ltn_eqF ?big1 ?addr0 1?eq_sym //.
by rewrite -ltnS prednK // (leq_trans _ i_small).
by move=> k /negPf ki_eqF; rewrite !mxE eqxx ki_eqF mul0r.
Qed. | Lemma | mulmx_delta_companion | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"addr0",
"apply",
"big1",
"bigD1",
"companionmx",
"delta_mx",
"eq_sym",
"eqxx",
"leq_trans",
"ltnS",
"ltn_eqF",
"mul0r",
"mul1r",
"mxE",
"prednK",
"rowP",
"seq",
"size",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row'_col'_char_poly_mx {R : nzRingType} m i (M : 'M[R]_m) :
row' i (col' i (char_poly_mx M)) = char_poly_mx (row' i (col' i M)). | Proof. by apply/matrixP => k l; rewrite !mxE (inj_eq lift_inj). Qed. | Lemma | row'_col'_char_poly_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"char_poly_mx",
"col'",
"inj_eq",
"lift_inj",
"matrixP",
"mxE",
"row'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_block_diag_mx {R : nzRingType} m n (A : 'M[R]_m) (B : 'M[R]_n) :
char_poly_mx (block_mx A 0 0 B) =
block_mx (char_poly_mx A) 0 0 (char_poly_mx B). | Proof.
rewrite /char_poly_mx map_block_mx/= !map_mx0.
by rewrite scalar_mx_block opp_block_mx add_block_mx !subr0.
Qed. | Lemma | char_block_diag_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"add_block_mx",
"block_mx",
"char_poly_mx",
"map_block_mx",
"map_mx0",
"opp_block_mx",
"scalar_mx_block",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
degree_mxminpoly_proof : exists d, \rank (powers_mx A d.+1) <= d. | Proof. by exists (n ^ 2)%N; rewrite rank_leq_col. Qed. | Fact | degree_mxminpoly_proof | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fintype",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"powers_mx",
"rank",
"rank_leq_col"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
degree_mxminpoly | := ex_minn degree_mxminpoly_proof. | Definition | degree_mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fintype",
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"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"degree_mxminpoly_proof",
"ex_minn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
d | := degree_mxminpoly. | Notation | d | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"degree_mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Ad | := (powers_mx A d). | Notation | Ad | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"powers_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_nonconstant : d > 0. | Proof.
rewrite /d; case: ex_minnP => -[] //; rewrite leqn0 mxrank_eq0; move/eqP.
by move/row_matrixP/(_ 0)/eqP; rewrite rowK row0 mxvec_eq0 -mxrank_eq0 mxrank1.
Qed. | Lemma | mxminpoly_nonconstant | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"ex_minnP",
"leqn0",
"mxrank1",
"mxrank_eq0",
"mxvec_eq0",
"row0",
"rowK",
"row_matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minpoly_mx1 : (1%:M \in Ad)%MS. | Proof.
by apply: (eq_row_sub (Ordinal mxminpoly_nonconstant)); rewrite rowK.
Qed. | Lemma | minpoly_mx1 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"apply",
"eq_row_sub",
"mxminpoly_nonconstant",
"rowK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minpoly_mx_free : row_free Ad. | Proof.
have:= mxminpoly_nonconstant; rewrite /d; case: ex_minnP => -[] // d' _ /(_ d').
by move/implyP; rewrite ltnn implybF -ltnS ltn_neqAle rank_leq_row andbT negbK.
Qed. | Lemma | minpoly_mx_free | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"ex_minnP",
"ltnS",
"ltn_neqAle",
"ltnn",
"mxminpoly_nonconstant",
"rank_leq_row",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mx_mem p : (horner_mx A p \in Ad)%MS. | Proof.
elim/poly_ind: p => [|p a IHp]; first by rewrite rmorph0 // linear0 sub0mx.
rewrite rmorphD rmorphM /= horner_mx_C horner_mx_X.
rewrite addrC -scalemx1 linearP /= -(mul_vec_lin (mulmxr A)).
case/submxP: IHp => u ->{p}.
have: (powers_mx A (1 + d) <= Ad)%MS.
rewrite -(geq_leqif (mxrank_leqif_sup _)); last first.... | Lemma | horner_mx_mem | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"addmx_sub",
"addnC",
"addrC",
"apply",
"eq_row_sub",
"eqnP",
"ex_minnP",
"exprSr",
"geq_leqif",
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"last",
"linear0",
"linearP",
"lshift",
"minpoly_mx_free",
"mul_vec_lin",
"mulmxA",
"mulmxE",
"mulmx_sub",
"mulmxr",
"mxrank... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_inv_horner B | := rVpoly (mxvec B *m pinvmx Ad). | Definition | mx_inv_horner | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"mxvec",
"pinvmx",
"rVpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_inv_horner0 : mx_inv_horner 0 = 0. | Proof. by rewrite /mx_inv_horner !(linear0, mul0mx). Qed. | Lemma | mx_inv_horner0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"linear0",
"mul0mx",
"mx_inv_horner"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_inv_hornerK B : (B \in Ad)%MS -> horner_mx A (mx_inv_horner B) = B. | Proof. by move=> sBAd; rewrite horner_rVpoly mulmxKpV ?mxvecK. Qed. | Lemma | mx_inv_hornerK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
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"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"horner_mx",
"horner_rVpoly",
"mulmxKpV",
"mx_inv_horner",
"mxvecK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minpoly_mxM B C : (B \in Ad -> C \in Ad -> B * C \in Ad)%MS. | Proof.
move=> AdB AdC; rewrite -(mx_inv_hornerK AdB) -(mx_inv_hornerK AdC).
by rewrite -rmorphM ?horner_mx_mem.
Qed. | Lemma | minpoly_mxM | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"horner_mx_mem",
"mx_inv_hornerK",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minpoly_mx_ring : mxring Ad. | Proof.
apply/andP; split; first exact/mulsmx_subP/minpoly_mxM.
apply/mxring_idP; exists 1%:M; split=> *; rewrite ?mulmx1 ?mul1mx //.
by rewrite -mxrank_eq0 mxrank1.
exact: minpoly_mx1.
Qed. | Lemma | minpoly_mx_ring | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Ad",
"apply",
"minpoly_mx1",
"minpoly_mxM",
"mul1mx",
"mulmx1",
"mulsmx_subP",
"mxrank1",
"mxrank_eq0",
"mxring",
"mxring_idP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly | := 'X^d - mx_inv_horner (A ^+ d). | Definition | mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"mx_inv_horner"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_A | := mxminpoly. | Notation | p_A | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mxminpoly : size p_A = d.+1. | Proof. by rewrite size_polyDl ?size_polyXn // size_polyN ltnS size_poly. Qed. | Lemma | size_mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"ltnS",
"p_A",
"size",
"size_poly",
"size_polyDl",
"size_polyN",
"size_polyXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_monic : p_A \is monic. | Proof.
rewrite monicE /lead_coef size_mxminpoly coefB coefXn eqxx /=.
by rewrite nth_default ?size_poly // subr0.
Qed. | Lemma | mxminpoly_monic | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"coefB",
"coefXn",
"eqxx",
"lead_coef",
"monic",
"monicE",
"nth_default",
"p_A",
"size_mxminpoly",
"size_poly",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mod_mxminpoly p : size (p %% p_A) <= d. | Proof.
by rewrite -ltnS -size_mxminpoly ltn_modp // -size_poly_eq0 size_mxminpoly.
Qed. | Lemma | size_mod_mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"ltnS",
"ltn_modp",
"p_A",
"size",
"size_mxminpoly",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_root_minpoly : horner_mx A p_A = 0. | Proof.
rewrite rmorphB -{3}(horner_mx_X A) -rmorphXn /=.
by rewrite mx_inv_hornerK ?subrr ?horner_mx_mem.
Qed. | Lemma | mx_root_minpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx",
"horner_mx_X",
"horner_mx_mem",
"mx_inv_hornerK",
"p_A",
"rmorphB",
"rmorphXn",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_rVpolyK (u : 'rV_d) :
mx_inv_horner (horner_mx A (rVpoly u)) = rVpoly u. | Proof.
congr rVpoly; rewrite horner_rVpoly vec_mxK.
by apply: (row_free_inj minpoly_mx_free); rewrite mulmxKpV ?submxMl.
Qed. | Lemma | horner_rVpolyK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"horner_mx",
"horner_rVpoly",
"minpoly_mx_free",
"mulmxKpV",
"mx_inv_horner",
"rVpoly",
"row_free_inj",
"submxMl",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mxK p : mx_inv_horner (horner_mx A p) = p %% p_A. | Proof.
rewrite {1}(Pdiv.IdomainMonic.divp_eq mxminpoly_monic p) rmorphD rmorphM /=.
rewrite mx_root_minpoly mulr0 add0r.
by rewrite -(poly_rV_K (size_mod_mxminpoly _)) horner_rVpolyK.
Qed. | Lemma | horner_mxK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"add0r",
"divp_eq",
"horner_mx",
"horner_rVpolyK",
"mulr0",
"mx_inv_horner",
"mx_root_minpoly",
"mxminpoly_monic",
"p_A",
"poly_rV_K",
"rmorphD",
"rmorphM",
"size_mod_mxminpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_min p : horner_mx A p = 0 -> p_A %| p. | Proof. by move=> pA0; rewrite /dvdp -horner_mxK pA0 mx_inv_horner0. Qed. | Lemma | mxminpoly_min | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"dvdp",
"horner_mx",
"horner_mxK",
"mx_inv_horner0",
"p_A"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_minP p : reflect (horner_mx A p = 0) (p_A %| p). | Proof.
apply: (iffP idP); last exact: mxminpoly_min.
by move=> /Pdiv.Field.dvdpP[q ->]; rewrite rmorphM/= mx_root_minpoly mulr0.
Qed. | Lemma | mxminpoly_minP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly... | [
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"last",
"mulr0",
"mx_root_minpoly",
"mxminpoly_min",
"p_A",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvd_mxminpoly p : (p_A %| p) = (horner_mx A p == 0). | Proof. exact/mxminpoly_minP/eqP. Qed. | Lemma | dvd_mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx",
"mxminpoly_minP",
"p_A"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_rVpoly_inj : injective (horner_mx A \o rVpoly : 'rV_d -> 'M_n). | Proof.
apply: can_inj (poly_rV \o mx_inv_horner) _ => u /=.
by rewrite horner_rVpolyK rVpolyK.
Qed. | Lemma | horner_rVpoly_inj | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"horner_mx",
"horner_rVpolyK",
"mx_inv_horner",
"poly_rV",
"rVpoly",
"rVpolyK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_linear_is_scalar : (d <= 1) = is_scalar_mx A. | Proof.
have scalP := has_non_scalar_mxP minpoly_mx1.
rewrite leqNgt -(eqnP minpoly_mx_free); apply/scalP/idP=> [|[[B]]].
case scalA: (is_scalar_mx A); [by right | left].
by exists A; rewrite ?scalA // -{1}(horner_mx_X A) horner_mx_mem.
move/mx_inv_hornerK=> <- nsB; case/is_scalar_mxP=> a defA; case/negP: nsB.
move:... | Lemma | mxminpoly_linear_is_scalar | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"eqnP",
"has_non_scalar_mxP",
"horner_mx_C",
"horner_mx_X",
"horner_mx_mem",
"is_scalar_mx",
"is_scalar_mxP",
"leqNgt",
"minpoly_mx1",
"minpoly_mx_free",
"mx0_is_scalar",
"mx_inv_hornerK",
"poly_ind",
"rmorph0",
"rmorphD",
"rmorphM",
"scalar_mx_is_scalar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_dvd_char : p_A %| char_poly A. | Proof. exact/mxminpoly_min/Cayley_Hamilton. Qed. | Lemma | mxminpoly_dvd_char | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Cayley_Hamilton",
"char_poly",
"mxminpoly_min",
"p_A"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue_root_min a : eigenvalue A a = root p_A a. | Proof.
apply/idP/idP=> Aa; last first.
rewrite eigenvalue_root_char !root_factor_theorem in Aa *.
exact: dvdp_trans Aa mxminpoly_dvd_char.
have{Aa} [v Av_av v_nz] := eigenvalueP Aa.
apply: contraR v_nz => pa_nz; rewrite -{pa_nz}(eqmx_eq0 (eqmx_scale _ pa_nz)).
apply/eqP; rewrite -(mulmx0 _ v) -mx_root_minpoly.
elim... | Lemma | eigenvalue_root_min | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"dvdp_trans",
"eigenvalue",
"eigenvalueP",
"eigenvalue_root_char",
"eqmx_eq0",
"eqmx_scale",
"horner0",
"hornerE",
"horner_mx_C",
"horner_mx_X",
"last",
"mul_mx_scalar",
"mulmx0",
"mulmxA",
"mulmxDr",
"mx_root_minpoly",
"mxminpoly_dvd_char",
"p_A",
"poly_ind",
"rmorp... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_mxminpoly a : root p_A a = root (char_poly A) a. | Proof. by rewrite -eigenvalue_root_min eigenvalue_root_char. Qed. | Lemma | root_mxminpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"char_poly",
"eigenvalue_root_char",
"eigenvalue_root_min",
"p_A",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_diag {F : fieldType} {n} (d : 'rV[F]_n.+1)
(u := undup [seq d 0 i | i <- enum 'I_n.+1]) :
mxminpoly (diag_mx d) = \prod_(r <- u) ('X - r%:P). | Proof.
apply/eqP; rewrite -eqp_monic ?mxminpoly_monic ?monic_prod_XsubC// /eqp.
rewrite mxminpoly_min/=.
rewrite horner_mx_diag; apply/matrixP => i j; rewrite !mxE horner_prod.
case: (altP (i =P j)) => [->|neq_ij//]; rewrite mulr1n.
rewrite (bigD1_seq (d 0 j)) ?undup_uniq ?mem_undup ?map_f// /=.
by rewrite horn... | Lemma | mxminpoly_diag | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"allP",
"apply",
"bigD1_seq",
"big_map",
"char_poly_trig",
"diag_mx",
"enum",
"eqp",
"eqp_monic",
"eqxx",
"hornerC",
"hornerD",
"hornerN",
"hornerX",
"horner_mx_diag",
"horner_prod",
"last",
"mapP",
"map_f",
"matrixP",
"mem_undup",
"monic_prod_XsubC",
"mul0r",
"mulr1n",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fp | := (map_poly f). | Notation | fp | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rVpoly (u : 'rV_d) : fp (rVpoly u) = rVpoly u^f. | Proof.
apply/polyP=> k; rewrite coef_map !coef_rVpoly.
by case: (insub k) => [i|]; rewrite /= ?rmorph0 // mxE.
Qed. | Lemma | map_rVpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coef_map",
"coef_rVpoly",
"fp",
"insub",
"mxE",
"polyP",
"rVpoly",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_rV p : (poly_rV p)^f = poly_rV (fp p) :> 'rV_d. | Proof. by apply/rowP=> j; rewrite !mxE coef_map. Qed. | Lemma | map_poly_rV | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coef_map",
"fp",
"mxE",
"poly_rV",
"rowP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_char_poly_mx : map_mx fp (char_poly_mx A) = char_poly_mx A^f. | Proof.
rewrite raddfB /= map_scalar_mx /= map_polyX; congr (_ - _).
by apply/matrixP=> i j; rewrite !mxE map_polyC.
Qed. | Lemma | map_char_poly_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"char_poly_mx",
"fp",
"map_mx",
"map_polyC",
"map_polyX",
"map_scalar_mx",
"matrixP",
"mxE",
"raddfB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_char_poly : fp (char_poly A) = char_poly A^f. | Proof. by rewrite -det_map_mx map_char_poly_mx. Qed. | Lemma | map_char_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"char_poly",
"det_map_mx",
"fp",
"map_char_poly_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_resultant (aR rR : nzRingType) (f : {rmorphism {poly aR} -> rR}) p q :
f (lead_coef p) != 0 -> f (lead_coef q) != 0 ->
f (resultant p q)= resultant (map_poly f p) (map_poly f q). | Proof.
move=> nz_fp nz_fq; rewrite /resultant /Sylvester_mx !size_map_poly_id0 //.
rewrite -det_map_mx /= map_col_mx; congr (\det (col_mx _ _));
by apply: map_lin1_mx => v; rewrite map_poly_rV rmorphM /= map_rVpoly.
Qed. | Lemma | map_resultant | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Sylvester_mx",
"apply",
"col_mx",
"det_map_mx",
"lead_coef",
"map_col_mx",
"map_lin1_mx",
"map_poly",
"map_poly_rV",
"map_rVpoly",
"poly",
"resultant",
"rmorphM",
"size_map_poly_id0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_powers_mx e : (powers_mx A e)^f = powers_mx A^f e. | Proof. by apply/row_matrixP=> i; rewrite -map_row !rowK map_mxvec rmorphXn. Qed. | Lemma | map_powers_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"map_mxvec",
"map_row",
"powers_mx",
"rmorphXn",
"rowK",
"row_matrixP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_horner_mx p : (horner_mx A p)^f = horner_mx A^f (fp p). | Proof.
rewrite -[p](poly_rV_K (leqnn _)) map_rVpoly.
by rewrite !horner_rVpoly map_vec_mx map_mxM map_powers_mx.
Qed. | Lemma | map_horner_mx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"fp",
"horner_mx",
"horner_rVpoly",
"leqnn",
"map_mxM",
"map_powers_mx",
"map_rVpoly",
"map_vec_mx",
"poly_rV_K"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_companion (e := congr1 predn (size_map_poly _ _)) :
(companionmx p)^f = castmx (e, e) (companionmx (fp p)). | Proof.
apply/matrixP => i j; rewrite !(castmxE, mxE) /= (fun_if f).
by rewrite rmorphN coef_map size_map_poly rmorph_nat.
Qed. | Lemma | map_mx_companion | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"castmx",
"castmxE",
"coef_map",
"companionmx",
"fp",
"matrixP",
"mxE",
"predn",
"rmorphN",
"rmorph_nat",
"size_map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
companion_map_poly (e := esym (congr1 predn (size_map_poly _ _))) :
companionmx (fp p) = castmx (e, e) (companionmx p)^f. | Proof. by rewrite map_mx_companion castmx_comp castmx_id. Qed. | Lemma | companion_map_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"castmx",
"castmx_comp",
"castmx_id",
"companionmx",
"fp",
"map_mx_companion",
"predn",
"size_map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
degree_mxminpoly_map : degree_mxminpoly A^f = degree_mxminpoly A. | Proof. by apply: eq_ex_minn => e; rewrite -map_powers_mx mxrank_map. Qed. | Lemma | degree_mxminpoly_map | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"degree_mxminpoly",
"eq_ex_minn",
"map_powers_mx",
"mxrank_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxminpoly_map : mxminpoly A^f = fp (mxminpoly A). | Proof.
rewrite rmorphB; congr (_ - _).
by rewrite /= map_polyXn degree_mxminpoly_map.
rewrite degree_mxminpoly_map -rmorphXn /=.
apply/polyP=> i; rewrite coef_map //= !coef_rVpoly degree_mxminpoly_map.
case/insub: i => [i|]; last by rewrite rmorph0.
by rewrite -map_powers_mx -map_pinvmx // -map_mxvec -map_mxM // mxE.... | Lemma | mxminpoly_map | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coef_map",
"coef_rVpoly",
"degree_mxminpoly_map",
"fp",
"insub",
"last",
"map_mxM",
"map_mxvec",
"map_pinvmx",
"map_polyXn",
"map_powers_mx",
"mxE",
"mxminpoly",
"polyP",
"rmorph0",
"rmorphB",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mx_inv_horner u : fp (mx_inv_horner A u) = mx_inv_horner A^f u^f. | Proof.
rewrite map_rVpoly map_mxM map_mxvec map_pinvmx map_powers_mx.
by rewrite /mx_inv_horner degree_mxminpoly_map.
Qed. | Lemma | map_mx_inv_horner | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"degree_mxminpoly_map",
"fp",
"map_mxM",
"map_mxvec",
"map_pinvmx",
"map_powers_mx",
"map_rVpoly",
"mx_inv_horner"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermxpoly n (g : 'M_n) (p : {poly K}) : 'M_n | :=
kermx ((if n is n.+1 then horner_mx^~ p : 'M_n.+1 -> 'M_n.+1 else \0) g). | Definition | kermxpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx",
"kermx",
"poly"
] | convertible to kermx (horner_mx g p) when n = n.+1 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
kermxpolyC n (g : 'M_n) c : c != 0 -> kermxpoly g c%:P = 0. | Proof.
move=> c_neq0; case: n => [|n] in g *; first by rewrite thinmx0.
apply/eqP; rewrite /kermxpoly horner_mx_C kermx_eq0 row_free_unit.
by rewrite -scalemx1 scaler_unit ?unitmx1// unitfE.
Qed. | Lemma | kermxpolyC | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finfun",
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"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"horner_mx_C",
"kermx_eq0",
"kermxpoly",
"row_free_unit",
"scalemx1",
"scaler_unit",
"thinmx0",
"unitfE",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermxpoly1 n (g : 'M_n) : kermxpoly g 1 = 0. | Proof. by rewrite kermxpolyC ?oner_eq0. Qed. | Lemma | kermxpoly1 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"kermxpoly",
"kermxpolyC",
"oner_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermxpolyX n (g : 'M_n) : kermxpoly g 'X = kermx g. | Proof.
case: n => [|n] in g *; first by rewrite !thinmx0.
by rewrite /kermxpoly horner_mx_X.
Qed. | Lemma | kermxpolyX | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"horner_mx_X",
"kermx",
"kermxpoly",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermxpoly_min n (g : 'M[K]_n.+1) p :
mxminpoly g %| p -> (kermxpoly g p :=: 1)%MS. | Proof. by rewrite /kermxpoly => /mxminpoly_minP ->; apply: kermx0. Qed. | Lemma | kermxpoly_min | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"kermx0",
"kermxpoly",
"mxminpoly",
"mxminpoly_minP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_stable_kermxpoly n (f g : 'M_n) (p : {poly K}) : comm_mx f g ->
stablemx (kermxpoly f p) g. | Proof.
case: n => [|n] in f g *; first by rewrite !thinmx0.
move=> fg; rewrite /kermxpoly; apply: comm_mx_stable_ker.
by apply/comm_mx_sym/comm_mx_horner/comm_mx_sym.
Qed. | Lemma | comm_mx_stable_kermxpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"comm_mx",
"comm_mx_horner",
"comm_mx_stable_ker",
"comm_mx_sym",
"kermxpoly",
"poly",
"stablemx",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_kermxpoly n (g : 'M_n) (p q : {poly K}) :
coprimep p q -> (kermxpoly g p :&: kermxpoly g q = 0)%MS. | Proof.
case: n => [|n] in g *; first by rewrite thinmx0 ?cap0mx ?submx_refl.
move=> /Bezout_eq1_coprimepP [[/= u v]]; rewrite mulrC [v * _]mulrC => cpq.
apply/eqP/rowV0P => x.
rewrite sub_capmx => /andP[/sub_kermxP xgp0 /sub_kermxP xgq0].
move: cpq => /(congr1 (mulmx x \o horner_mx g))/=.
rewrite !(rmorphM, rmorphD, rm... | Lemma | mxdirect_kermxpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"finfun",
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"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Bezout_eq1_coprimepP",
"add0r",
"apply",
"cap0mx",
"coprimep",
"horner_mx",
"kermxpoly",
"mul0mx",
"mulmx",
"mulmx1",
"mulmxA",
"mulmxDr",
"mulrC",
"poly",
"rmorph1",
"rmorphD",
"rmorphM",
"rowV0P",
"sub_capmx",
"sub_kermxP",
"submx_refl",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermxpolyM n (g : 'M_n) (p q : {poly K}) : coprimep p q ->
(kermxpoly g (p * q) :=: kermxpoly g p + kermxpoly g q)%MS. | Proof.
case: n => [|n] in g *; first by rewrite !thinmx0.
move=> /Bezout_eq1_coprimepP [[/= u v]]; rewrite mulrC [v * _]mulrC => cpq.
apply/eqmxP/andP; split; last first.
apply/sub_kermxP/eqmx0P; rewrite !addsmxMr [in X in (_ + X)%MS]mulrC.
by rewrite !rmorphM/= !mulmxA !mulmx_ker !mul0mx !addsmx0 submx_refl.
move:... | Lemma | kermxpolyM | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"Bezout_eq1_coprimepP",
"addmx_sub_adds",
"addrC",
"addsmx0",
"addsmxMr",
"apply",
"coprimep",
"eqmx0P",
"eqmxP",
"horner_mx",
"kermxpoly",
"last",
"mul0mx",
"mulmxA",
"mulmxE",
"mulmx_ker",
"mulr1",
"mulrA",
"mulrAC",
"mulrC",
"mulrDr",
"poly",
"rmorph1",
"rmorphD",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermxpoly_prod n (g : 'M_n)
(I : finType) (P : {pred I}) (p_ : I -> {poly K}) :
{in P &, forall i j, j != i -> coprimep (p_ i) (p_ j)} ->
(kermxpoly g (\prod_(i | P i) p_ i) :=: \sum_(i | P i) kermxpoly g (p_ i))%MS. | Proof.
move=> p_coprime; elim: index_enum (index_enum_uniq I).
by rewrite !big_nil ?kermxpoly1 ?submx_refl//.
move=> j js ihjs /= /andP[jNjs js_uniq]; apply/eqmxP.
rewrite !big_cons; case: ifP => [Pj|PNj]; rewrite ?ihjs ?submx_refl//.
suff cjjs: coprimep (p_ j) (\prod_(i <- js | P i) p_ i).
by rewrite !kermxpolyM//... | Lemma | kermxpoly_prod | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"adds_eqmx",
"allP",
"apply",
"big_all_cond",
"big_cons",
"big_morph",
"big_nil",
"contraNneq",
"coprimep",
"coprimep1",
"coprimepMr",
"eqmxP",
"eqmx_refl",
"index_enum",
"index_enum_uniq",
"kermxpoly",
"kermxpoly1",
"kermxpolyM",
"poly",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sum_kermx n (g : 'M_n)
(I : finType) (P : {pred I}) (p_ : I -> {poly K}) :
{in P &, forall i j, j != i -> coprimep (p_ i) (p_ j)} ->
mxdirect (\sum_(i | P i) kermxpoly g (p_ i))%MS. | Proof.
move=> p_coprime; apply/mxdirect_sumsP => i Pi; apply/eqmx0P.
have cpNi : {in [pred j | P j && (j != i)] &,
forall j k : I, k != j -> coprimep (p_ j) (p_ k)}.
by move=> j k /andP[Pj _] /andP[Pk _]; apply: p_coprime.
rewrite -!(cap_eqmx (eqmx_refl _) (kermxpoly_prod g _))//.
rewrite mxdirect_kermxpoly ?subm... | Lemma | mxdirect_sum_kermx | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"allP",
"apply",
"big_all_cond",
"big_morph",
"cap_eqmx",
"coprimep",
"coprimep1",
"coprimepMr",
"eqmx0P",
"eqmx_refl",
"kermxpoly",
"kermxpoly_prod",
"mxdirect",
"mxdirect_kermxpoly",
"mxdirect_sumsP",
"poly",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenspace_poly n a (f : 'M_n) :
eigenspace f a = kermxpoly f ('X - a%:P). | Proof.
case: n => [|m] in a f *; first by rewrite !thinmx0.
by congr (kermx _); rewrite rmorphB /= ?horner_mx_X ?horner_mx_C.
Qed. | Lemma | eigenspace_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"eigenspace",
"horner_mx_C",
"horner_mx_X",
"kermx",
"kermxpoly",
"rmorphB",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geigenspace n (g : 'M_n) a | := kermxpoly g (('X - a%:P) ^+ n). | Definition | geigenspace | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
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"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"kermxpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geigenspaceE n' (g : 'M_n'.+1) a :
geigenspace g a = kermx ((g - a%:M) ^+ n'.+1). | Proof.
by rewrite /geigenspace /kermxpoly rmorphXn/= rmorphB/= horner_mx_X horner_mx_C.
Qed. | Lemma | geigenspaceE | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"geigenspace",
"horner_mx_C",
"horner_mx_X",
"kermx",
"kermxpoly",
"n'",
"rmorphB",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenspace_sub_geigen n (g : 'M_n) a :
(eigenspace g a <= geigenspace g a)%MS. | Proof.
case: n => [|n] in g *; rewrite ?thinmx0 ?sub0mx// geigenspaceE.
by apply/sub_kermxP; rewrite exprS mulmxA mulmx_ker mul0mx.
Qed. | Lemma | eigenspace_sub_geigen | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"eigenspace",
"exprS",
"geigenspace",
"geigenspaceE",
"mul0mx",
"mulmxA",
"mulmx_ker",
"sub0mx",
"sub_kermxP",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxdirect_sum_geigenspace
(I : finType) (n : nat) (g : 'M_n) (P : {pred I}) (a_ : I -> K) :
{in P &, injective a_} -> mxdirect (\sum_(i | P i) geigenspace g (a_ i)). | Proof.
move=> /inj_in_eq eq_a; apply: mxdirect_sum_kermx => i j Pi Pj Nji.
by rewrite coprimep_expr ?coprimep_expl// coprimep_XsubC root_XsubC eq_a.
Qed. | Lemma | mxdirect_sum_geigenspace | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coprimep_XsubC",
"coprimep_expl",
"coprimep_expr",
"geigenspace",
"inj_in_eq",
"mxdirect",
"mxdirect_sum_kermx",
"nat",
"root_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenpoly n (g : 'M_n) : pred {poly K} | :=
(fun p => kermxpoly g p != 0). | Definition | eigenpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"kermxpoly",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenpolyP n (g : 'M_n) (p : {poly K}) :
reflect (exists2 v : 'rV_n, (v <= kermxpoly g p)%MS & v != 0) (eigenpoly g p). | Proof. exact: rowV0Pn. Qed. | Lemma | eigenpolyP | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"eigenpoly",
"kermxpoly",
"poly",
"rowV0Pn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenvalue_poly n a (f : 'M_n) : eigenvalue f a = eigenpoly f ('X - a%:P). | Proof. by rewrite /eigenpoly /eigenvalue eigenspace_poly. Qed. | Lemma | eigenvalue_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"eigenpoly",
"eigenspace_poly",
"eigenvalue"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_mx_stable_geigenspace n (f g : 'M_n) a : comm_mx f g ->
stablemx (geigenspace f a) g. | Proof. exact: comm_mx_stable_kermxpoly. Qed. | Lemma | comm_mx_stable_geigenspace | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"comm_mx",
"comm_mx_stable_kermxpoly",
"geigenspace",
"stablemx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_kermxpoly (n : nat) (g : 'M_n) (p : {poly aF}) :
map_mx f (kermxpoly g p) = kermxpoly (map_mx f g) (map_poly f p). | Proof.
by case: n => [|n] in g *; rewrite ?thinmx0// map_kermx map_horner_mx.
Qed. | Lemma | map_kermxpoly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"kermxpoly",
"map_horner_mx",
"map_kermx",
"map_mx",
"map_poly",
"nat",
"poly",
"thinmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_geigenspace (n : nat) (g : 'M_n) (a : aF) :
map_mx f (geigenspace g a) = geigenspace (map_mx f g) (f a). | Proof. by rewrite map_kermxpoly rmorphXn/= rmorphB /= map_polyX map_polyC. Qed. | Lemma | map_geigenspace | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"geigenspace",
"map_kermxpoly",
"map_mx",
"map_polyC",
"map_polyX",
"nat",
"rmorphB",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eigenpoly_map n (g : 'M_n) (p : {poly aF}) :
eigenpoly (map_mx f g) (map_poly f p) = eigenpoly g p. | Proof. by rewrite /eigenpoly -map_kermxpoly map_mx_eq0. Qed. | Lemma | eigenpoly_map | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"eigenpoly",
"map_kermxpoly",
"map_mx",
"map_mx_eq0",
"map_poly",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integralOver (R K : nzRingType) (RtoK : R -> K) (z : K) | :=
exists2 p, p \is monic & root (map_poly RtoK p) z. | Definition | integralOver | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"map_poly",
"monic",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integralRange R K RtoK | := forall z, @integralOver R K RtoK z. | Definition | integralRange | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integralOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_rmorph x :
integralOver BtoR x -> integralOver (RtoK \o BtoR) (RtoK x). | Proof. by case=> p; exists p; rewrite // map_poly_comp rmorph_root. Qed. | Lemma | integral_rmorph | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integralOver",
"map_poly_comp",
"rmorph_root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_id x : integralOver RtoK (RtoK x). | Proof. by exists ('X - x%:P); rewrite ?monicXsubC ?rmorph_root ?root_XsubC. Qed. | Lemma | integral_id | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integralOver",
"monicXsubC",
"rmorph_root",
"root_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_nat n : integralOver RtoK n%:R. | Proof. by rewrite -(rmorph_nat RtoK); apply: integral_id. Qed. | Lemma | integral_nat | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"integralOver",
"integral_id",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral0 : integralOver RtoK 0. | Proof. exact: (integral_nat 0). Qed. | Lemma | integral0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integralOver",
"integral_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral1 : integralOver RtoK 1. | Proof. exact: (integral_nat 1). Qed. | Lemma | integral1 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integralOver",
"integral_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_poly (p : {poly K}) :
(forall i, integralOver RtoK p`_i) <-> {in p : seq K, integralRange RtoK}. | Proof.
split=> intRp => [_ /(nthP 0)[i _ <-] // | i]; rewrite -[p]coefK coef_poly.
by case: ifP => [ltip | _]; [apply/intRp/mem_nth | apply: integral0].
Qed. | Lemma | integral_poly | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coefK",
"coef_poly",
"integral0",
"integralOver",
"integralRange",
"mem_nth",
"nthP",
"poly",
"seq",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_horner_root w (p q : {poly K}) :
p \is monic -> root p w ->
{in p : seq K, integralRange RtoK} -> {in q : seq K, integralRange RtoK} ->
integralOver RtoK q.[w]. | Proof.
move=> mon_p pw0 intRp intRq.
pose memR y := exists x, y = RtoK x.
have memRid x: memR (RtoK x) by exists x.
have memR_nat n: memR n%:R by rewrite -(rmorph_nat RtoK) /=.
have [memR0 memR1]: memR 0 * memR 1 := (memR_nat 0, memR_nat 1).
have memRN1: memR (- 1) by exists (- 1); rewrite rmorphN1.
pose rVin (E : K ->... | Lemma | integral_horner_root | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"S1",
"S2",
"True",
"a1",
"addnBA",
"addnC",
"addnS",
"addrC",
"all",
"allP",
"apply",
"big_ind",
"big_ord_recr",
"char_poly",
"char_poly_monic",
"coefC",
"coefD",
"coefK",
"coefM",
"coefX",
"coef_map",
"coef_poly",
"coef_rVpoly",
"coef_rVpoly_ord",
"delta_mx",
"det... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral_root_monic u p :
p \is monic -> root p u -> {in p : seq K, integralRange RtoK} ->
integralOver RtoK u. | Proof.
move=> mon_p pu0 intRp; rewrite -[u]hornerX.
apply: integral_horner_root mon_p pu0 intRp _.
by apply/integral_poly => i; rewrite coefX; apply: integral_nat.
Qed. | Lemma | integral_root_monic | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"apply",
"coefX",
"hornerX",
"integralOver",
"integralRange",
"integral_horner_root",
"integral_nat",
"integral_poly",
"monic",
"root",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral0_RtoK | := integral0 RtoK. | Let | integral0_RtoK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integral0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
integral1_RtoK | := integral1 RtoK. | Let | integral1_RtoK | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"integral1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monicXsubC_K | := @monicXsubC K. | Let | monicXsubC_K | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"monicXsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
XsubC0 (u : K) : root ('X - u%:P) u. | Proof. by rewrite root_XsubC. Qed. | Let | XsubC0 | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"root",
"root_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intR_XsubC u :
integralOver RtoK (- u) -> {in 'X - u%:P : seq K, integralRange RtoK}. | Proof. by move=> intRu v; rewrite polyseqXsubC !inE => /pred2P[]->. Qed. | Let | intR_XsubC | algebra | algebra/mxpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"fingroup",
"perm",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"matrix",
"mxalgebra",
"poly",
"poly... | [
"inE",
"integralOver",
"integralRange",
"polyseqXsubC",
"pred2P",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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