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 |
|---|---|---|---|---|---|---|---|---|---|---|
mem_npoly_enum p : p \in npoly_enum. | Proof.
rewrite /npoly_enum; case: n => [|k] // in p *.
case: p => [p sp] /=.
by rewrite in_cons -val_eqE /= -size_poly_leq0 [size _ <= _]sp.
rewrite mem_pmap_sub; apply/mapP.
eexists [ffun i : 'I__ => p`_i]; first by rewrite mem_enum.
apply/polyP => i; rewrite coef_poly.
have [i_small|i_big] := ltnP; first by rewri... | Lemma | mem_npoly_enum | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"coef_poly",
"ffunE",
"in_cons",
"inordK",
"leq_trans",
"ltnP",
"mapP",
"mem_enum",
"mem_pmap_sub",
"npoly_enum",
"nth_default",
"polyP",
"size",
"size_npoly",
"size_poly_leq0",
"sp",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_npoly : #|{poly_n R}| = (#|R| ^ n)%N. | Proof.
rewrite -(card_imset _ (can_inj (@npoly_rV_K _ _))) eq_cardT; last first.
by rewrite -cardT /= card_mx mul1n.
by move=> v; apply/imsetP; exists (rVnpoly v); rewrite ?rVnpolyK //.
Qed. | Lemma | card_npoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"cardT",
"card_imset",
"card_mx",
"eq_cardT",
"imsetP",
"last",
"mul1n",
"npoly_rV_K",
"rVnpoly",
"rVnpolyK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irreducibleb | :=
((1 < size p) &&
[forall q : {poly_((size p).-1) R}, (rdvdp q p)%R ==> (size q <= 1)])%N. | Definition | irreducibleb | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"rdvdp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irreducibleP : reflect (irreducible_poly p) irreducibleb. | Proof.
rewrite /irreducibleb /irreducible_poly.
apply: (iffP idP) => [/andP[sp /'forall_implyP /= Fp]|[sp Fpoly]].
have sp_gt0 : size p > 0 by case: size sp.
have p_neq0 : p != 0 by rewrite -size_poly_eq0; case: size sp.
split => // q sq_neq1 dvd_qp; rewrite -dvdp_size_eqp // eqn_leq dvdp_leq //=.
apply: contra... | Lemma | irreducibleP | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"dvd0p",
"dvdpE",
"dvdp_leq",
"dvdp_size_eqp",
"eqn_leq",
"eqp_size",
"irreducible_poly",
"irreducibleb",
"leq_eqVlt",
"ltnNge",
"ltnS",
"ltnn",
"prednK",
"size",
"size_npoly",
"size_poly_eq0",
"size_poly_leq0",
"sp",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_polyn : \dim (fullv : {vspace {poly_n K}}) = n. | Proof. by rewrite [LHS]mxrank_gen mxrank1. Qed. | Lemma | dim_polyn | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"dim",
"fullv",
"mxrank1",
"mxrank_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
npolyX : n.-tuple {poly_n K} | := [tuple npolyp n 'X^i | i < n]. | Definition | npolyX | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"npolyp",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''nX^' i" | := (tnth npolyX i). | Notation | ''nX^' i | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"npolyX",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
npolyXE (i : 'I_n) : 'nX^i = 'X^i :> {poly _}. | Proof. by rewrite tnth_map tnth_ord_tuple npolypK // size_polyXn. Qed. | Lemma | npolyXE | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"npolypK",
"poly",
"size_polyXn",
"tnth_map",
"tnth_ord_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_npolyX (i : 'I_n) : npolyX`_i = 'nX^i. | Proof. by rewrite -tnth_nth. Qed. | Lemma | nth_npolyX | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"npolyX",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
npolyX_free : free npolyX. | Proof.
apply/freeP=> u /= sum_uX_eq0 i; have /npolyP /(_ i) := sum_uX_eq0.
rewrite (@big_morph _ _ _ 0%R +%R) // coef_sum coef0.
rewrite (bigD1 i) ?big1 /= ?addr0 ?coefZ ?(nth_map 0%N) ?size_iota //;
last by rewrite nth_npolyX npolyXE coefXn eqxx mulr1.
move=> j; rewrite -val_eqE /= => neq_ji.
by rewrite nth_npolyX n... | Lemma | npolyX_free | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"addr0",
"apply",
"big1",
"bigD1",
"big_morph",
"coef0",
"coefXn",
"coefZ",
"coef_sum",
"eq_sym",
"eqxx",
"free",
"freeP",
"last",
"mulr0",
"mulr1",
"npolyP",
"npolyX",
"npolyXE",
"nth_map",
"nth_npolyX",
"size_iota",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
npolyX_full : basis_of fullv npolyX. | Proof.
by rewrite basisEfree npolyX_free subvf size_map size_enum_ord dim_polyn /=.
Qed. | Lemma | npolyX_full | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"basisEfree",
"basis_of",
"dim_polyn",
"fullv",
"npolyX",
"npolyX_free",
"size_enum_ord",
"size_map",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
npolyX_coords (p : {poly_n K}) i : coord npolyX i p = p`_i. | Proof.
rewrite [p in RHS](coord_basis npolyX_full) ?memvf // coefn_sum.
rewrite (bigD1 i) //= coefZ nth_npolyX npolyXE coefXn eqxx mulr1 big1 ?addr0//.
move=> j; rewrite -val_eqE => /= neq_ji.
by rewrite coefZ nth_npolyX npolyXE coefXn eq_sym (negPf neq_ji) mulr0.
Qed. | Lemma | npolyX_coords | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"addr0",
"big1",
"bigD1",
"coefXn",
"coefZ",
"coefn_sum",
"coord",
"coord_basis",
"eq_sym",
"eqxx",
"memvf",
"mulr0",
"mulr1",
"npolyX",
"npolyXE",
"npolyX_full",
"nth_npolyX",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
npolyX_gen (p : {poly K}) : (size p <= n)%N ->
p = \sum_(i < n) p`_i *: 'nX^i. | Proof.
move=> sp; rewrite -[p](@npolypK _ n) //.
rewrite [npolyp _ _ in LHS](coord_basis npolyX_full) ?memvf //.
rewrite (@big_morph _ _ _ 0%R +%R) // !raddf_sum.
by apply: eq_bigr=> i _; rewrite npolyX_coords //= nth_npolyX npolyXE.
Qed. | Lemma | npolyX_gen | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"big_morph",
"coord_basis",
"eq_bigr",
"memvf",
"npolyXE",
"npolyX_coords",
"npolyX_full",
"npolyp",
"npolypK",
"nth_npolyX",
"poly",
"raddf_sum",
"size",
"sp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_def | := (fun i :'I_n =>
let k := i in let p := \prod_(j < n | j != k) ('X - (x j)%:P)
in (p.[x k]^-1)%:P * p). | Notation | lagrange_def | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_key : unit. | Proof. exact: tt. Qed. | Fact | lagrange_key | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange | := locked_with lagrange_key
[tuple npolyp n (lagrange_def i) | i < n]. | Definition | lagrange | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"lagrange_def",
"lagrange_key",
"npolyp",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_ | := (tnth lagrange). | Notation | lagrange_ | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"lagrange",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_gt0 : (0 < n)%N. | Hypothesis | n_gt0 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
x_inj : injective x. | Hypothesis | x_inj | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
lagrange_def_sample (i j : 'I_n) : (lagrange_def i).[x j] = (i == j)%:R. | Proof.
clear n_gt0; rewrite hornerM hornerC; set p := (\prod_(_ < _ | _) _).
have [<-|neq_ij] /= := altP eqP.
rewrite mulVf // horner_prod; apply/prodf_neq0 => k neq_ki.
by rewrite hornerXsubC subr_eq0 inj_eq // eq_sym.
rewrite [X in _ * X]horner_prod (bigD1 j) 1?eq_sym //=.
by rewrite hornerXsubC subrr mul0r mulr0... | Let | lagrange_def_sample | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"bigD1",
"eq_sym",
"hornerC",
"hornerM",
"hornerXsubC",
"horner_prod",
"inj_eq",
"lagrange_def",
"mul0r",
"mulVf",
"mulr0",
"n_gt0",
"prodf_neq0",
"subr_eq0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_lagrange_def i : size (lagrange_def i) = n. | Proof.
rewrite size_Cmul.
suff : (lagrange_def i).[x i] != 0.
by rewrite hornerE mulf_eq0 => /norP [].
by rewrite lagrange_def_sample ?eqxx ?oner_eq0.
rewrite size_prod /=.
by move=> j neq_ji; rewrite polyXsubC_eq0.
rewrite (eq_bigr (fun=> (2 * 1)%N)).
by move=> j neq_ji; rewrite size_XsubC.
rewrite -big_di... | Let | size_lagrange_def | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"addSn",
"addnK",
"addnn",
"big_distrr",
"cardC1",
"card_ord",
"eq_bigr",
"eqxx",
"hornerE",
"lagrange_def",
"lagrange_def_sample",
"mul2n",
"mulf_eq0",
"n_gt0",
"oner_eq0",
"polyXsubC_eq0",
"size",
"size_Cmul",
"size_XsubC",
"size_prod",
"sum1_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrangeE i : lagrange_ i = lagrange_def i :> {poly _}. | Proof.
rewrite [lagrange]unlock tnth_map.
by rewrite [val _]npolypK tnth_ord_tuple // size_lagrange_def.
Qed. | Lemma | lagrangeE | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"lagrange",
"lagrange_",
"lagrange_def",
"npolypK",
"poly",
"size_lagrange_def",
"tnth_map",
"tnth_ord_tuple",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_lagrange (i : 'I_n) : lagrange`_i = lagrange_ i. | Proof. by rewrite -tnth_nth. Qed. | Lemma | nth_lagrange | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"lagrange",
"lagrange_",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_lagrange_ i : size (lagrange_ i) = n. | Proof. by rewrite lagrangeE size_lagrange_def. Qed. | Lemma | size_lagrange_ | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly",
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"... | [
"lagrangeE",
"lagrange_",
"size",
"size_lagrange_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_lagrange : size lagrange = n. | Proof. by rewrite size_tuple. Qed. | Lemma | size_lagrange | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
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"... | [
"lagrange",
"size",
"size_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_sample (i j : 'I_n) : (lagrange_ i).[x j] = (i == j)%:R. | Proof. by rewrite lagrangeE lagrange_def_sample. Qed. | Lemma | lagrange_sample | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"lagrangeE",
"lagrange_",
"lagrange_def_sample"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_free : free lagrange. | Proof.
apply/freeP=> lambda eq_l i.
have /(congr1 (fun p : {poly__ _} => p.[x i])) := eq_l.
rewrite (@big_morph _ _ _ 0%R +%R) // horner_sum horner0.
rewrite (bigD1 i) // big1 => [j /= /negPf ji|] /=;
by rewrite ?hornerE nth_lagrange lagrange_sample ?eqxx ?ji ?mulr1 ?mulr0.
Qed. | Lemma | lagrange_free | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
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"bigD1",
"big_morph",
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"free",
"freeP",
"horner0",
"hornerE",
"horner_sum",
"lagrange",
"lagrange_sample",
"mulr0",
"mulr1",
"nth_lagrange"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_full : basis_of fullv lagrange. | Proof.
by rewrite basisEfree lagrange_free subvf size_lagrange dim_polyn /=.
Qed. | Lemma | lagrange_full | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"matrix",
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"... | [
"basisEfree",
"basis_of",
"dim_polyn",
"fullv",
"lagrange",
"lagrange_free",
"size_lagrange",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_coords (p : {poly_n K}) i : coord lagrange i p = p.[x i]. | Proof.
rewrite [p in RHS](coord_basis lagrange_full) ?memvf //.
rewrite (@big_morph _ _ _ 0%R +%R) // horner_sum.
rewrite (bigD1 i) // big1 => [j /= /negPf ji|] /=;
by rewrite ?hornerE nth_lagrange lagrange_sample ?eqxx ?ji ?mulr1 ?mulr0.
Qed. | Lemma | lagrange_coords | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"big1",
"bigD1",
"big_morph",
"coord",
"coord_basis",
"eqxx",
"hornerE",
"horner_sum",
"lagrange",
"lagrange_full",
"lagrange_sample",
"memvf",
"mulr0",
"mulr1",
"nth_lagrange"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lagrange_gen (p : {poly K}) : (size p <= n)%N ->
p = \sum_(i < n) p.[x i]%:P * lagrange_ i. | Proof.
move=> sp; rewrite -[p](@npolypK _ n) //.
rewrite [npolyp _ _ in LHS](coord_basis lagrange_full) ?memvf //.
rewrite (@big_morph _ _ _ 0%R +%R) //; apply: eq_bigr=> i _.
by rewrite lagrange_coords mul_polyC nth_lagrange.
Qed. | Lemma | lagrange_gen | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"... | [
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"lagrange_full",
"memvf",
"mul_polyC",
"npolyp",
"npolypK",
"nth_lagrange",
"poly",
"size",
"sp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''nX^' i" | := (tnth (npolyX _) i) : ring_scope. | Notation | ''nX^' i | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"npolyX",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x .-lagrange" | := (lagrange x) : ring_scope. | Notation | x .-lagrange | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"lagrange"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x .-lagrange_" | := (tnth x.-lagrange) : ring_scope. | Notation | x .-lagrange_ | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
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"matrix",
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"... | [
"lagrange",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk_monic | :=
if (1 < size h)%N && (h \is monic) then h else 'X. | Definition | mk_monic | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"monic",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly | := {poly_(size mk_monic).-1 R}. | Definition | qpoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"... | [
"mk_monic",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'poly' '%/' p }" | := (qpoly p) : type_scope. | Notation | { 'poly' '%/' p } | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monic_mk_monic : (mk_monic h) \is monic. | Proof.
rewrite /mk_monic; case: leqP=> [_|/=]; first by apply: monicX.
by case E : (h \is monic) => [->//|] => _; apply: monicX.
Qed. | Lemma | monic_mk_monic | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly",
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"matrix",
"mxalgebra",
"... | [
"apply",
"leqP",
"mk_monic",
"monic",
"monicX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mk_monic_gt1 : (1 < size (mk_monic h))%N. | Proof.
by rewrite !fun_if size_polyX; case: leqP => //=; rewrite if_same.
Qed. | Lemma | size_mk_monic_gt1 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"poly",
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"matrix",
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"... | [
"leqP",
"mk_monic",
"size",
"size_polyX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mk_monic_gt0 : (0 < size (mk_monic h))%N. | Proof. by rewrite (leq_trans _ size_mk_monic_gt1). Qed. | Lemma | size_mk_monic_gt0 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"leq_trans",
"mk_monic",
"size",
"size_mk_monic_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk_monic_neq0 : mk_monic h != 0. | Proof. by rewrite -size_poly_gt0 size_mk_monic_gt0. Qed. | Lemma | mk_monic_neq0 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"mk_monic",
"size_mk_monic_gt0",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mk_monic (p : {poly %/ h}) : size p < size (mk_monic h). | Proof.
have: (p : {poly R}) \is a poly_of_size (size (mk_monic h)).-1 by case: p.
by rewrite qualifE/= -ltnS prednK // size_mk_monic_gt0.
Qed. | Lemma | size_mk_monic | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"... | [
"ltnS",
"mk_monic",
"poly",
"poly_of_size",
"prednK",
"size",
"size_mk_monic_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_of_size_mod p :
rmodp p (mk_monic h) \is a poly_of_size (size (mk_monic h)).-1. | Proof.
rewrite qualifE/= -ltnS prednK ?size_mk_monic_gt0 //.
by apply: ltn_rmodpN0; rewrite mk_monic_neq0.
Qed. | Lemma | poly_of_size_mod | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"... | [
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"mk_monic",
"mk_monic_neq0",
"poly_of_size",
"prednK",
"rmodp",
"size",
"size_mk_monic_gt0"
] | standard inject | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
in_qpoly p : {poly %/ h} | := NPoly (poly_of_size_mod p). | Definition | in_qpoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"matrix",
"mxalgebra",
"... | [
"poly",
"poly_of_size_mod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly_small (p : {poly R}) :
size p < size (mk_monic h) -> in_qpoly p = p :> {poly R}. | Proof. exact: rmodp_small. Qed. | Lemma | in_qpoly_small | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"matrix",
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"... | [
"in_qpoly",
"mk_monic",
"poly",
"rmodp_small",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly0 : in_qpoly 0 = 0. | Proof. by apply/val_eqP; rewrite /= rmod0p. Qed. | Lemma | in_qpoly0 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"matrix",
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"... | [
"apply",
"in_qpoly",
"rmod0p",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpolyD p q : in_qpoly (p + q) = in_qpoly p + in_qpoly q. | Proof. by apply/val_eqP=> /=; rewrite rmodpD ?monic_mk_monic. Qed. | Lemma | in_qpolyD | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"matrix",
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"... | [
"apply",
"in_qpoly",
"monic_mk_monic",
"rmodpD",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpolyZ a p : in_qpoly (a *: p) = a *: in_qpoly p. | Proof. apply/val_eqP=> /=; rewrite rmodpZ ?monic_mk_monic //. Qed. | Lemma | in_qpolyZ | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"... | [
"apply",
"in_qpoly",
"monic_mk_monic",
"rmodpZ",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly_is_linear : linear in_qpoly. | Proof. by move=> k p q; rewrite in_qpolyD in_qpolyZ. Qed. | Fact | in_qpoly_is_linear | algebra | algebra/qpoly.v | [
"HB",
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"mathcomp",
"ssreflect",
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"eqtype",
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"matrix",
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"... | [
"in_qpoly",
"in_qpolyD",
"in_qpolyZ",
"linear"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC_proof k :
(k%:P : {poly R}) \is a poly_of_size (size (mk_monic h)).-1. | Proof.
rewrite qualifE/= -ltnS size_polyC prednK ?size_mk_monic_gt0 //.
by rewrite (leq_ltn_trans _ size_mk_monic_gt1) //; case: eqP.
Qed. | Lemma | qpolyC_proof | algebra | algebra/qpoly.v | [
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"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"matrix",
"mxalgebra",
"... | [
"leq_ltn_trans",
"ltnS",
"mk_monic",
"poly",
"poly_of_size",
"prednK",
"size",
"size_mk_monic_gt0",
"size_mk_monic_gt1",
"size_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC k : {poly %/ h} | := NPoly (qpolyC_proof k). | Definition | qpolyC | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"poly",
"qpolyC_proof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyCE k : qpolyC k = k%:P :> {poly R}. | Proof. by []. Qed. | Lemma | qpolyCE | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"poly",
"qpolyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC0 : qpolyC 0 = 0. | Proof. by apply/val_eqP/eqP. Qed. | Lemma | qpolyC0 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpolyC",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly1 | := qpolyC 1. | Definition | qpoly1 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpolyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mul (q1 q2 : {poly %/ h}) : {poly %/ h} | :=
in_qpoly ((q1 : {poly R}) * q2). | Definition | qpoly_mul | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"in_qpoly",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mul1z : left_id qpoly1 qpoly_mul. | Proof.
by move=> x; apply: val_inj; rewrite /= mul1r rmodp_small // size_mk_monic.
Qed. | Lemma | qpoly_mul1z | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"mul1r",
"qpoly1",
"qpoly_mul",
"rmodp_small",
"size_mk_monic",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mulz1 : right_id qpoly1 qpoly_mul. | Proof.
by move=> x; apply: val_inj; rewrite /= mulr1 rmodp_small // size_mk_monic.
Qed. | Lemma | qpoly_mulz1 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"mulr1",
"qpoly1",
"qpoly_mul",
"rmodp_small",
"size_mk_monic",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_nontrivial : qpoly1 != 0. | Proof. by apply/eqP/val_eqP; rewrite /= oner_eq0. Qed. | Lemma | qpoly_nontrivial | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"oner_eq0",
"qpoly1",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyX | := in_qpoly 'X. | Definition | qpolyX | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"in_qpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'qX" | := qpolyX. | Notation | 'qX | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpolyX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyXE : 2 < size h -> h \is monic -> 'qX = 'X :> {poly R}. | Proof.
move=> sh_gt2 h_mo.
by rewrite in_qpoly_small // size_polyX /mk_monic ifT // (ltn_trans _ sh_gt2).
Qed. | Lemma | qpolyXE | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"in_qpoly_small",
"ltn_trans",
"mk_monic",
"monic",
"poly",
"sh_gt2",
"size",
"size_polyX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'qX" | := (qpolyX _) : ring_scope. | Notation | 'qX | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpolyX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk_monic_X (R : nzSemiRingType) : mk_monic 'X = 'X :> {poly R}. | Proof. by rewrite /mk_monic size_polyX monicX. Qed. | Lemma | mk_monic_X | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"mk_monic",
"monicX",
"poly",
"size_polyX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk_monic_Xn (R : nzSemiRingType) n :
mk_monic 'X^n = 'X^(n.-1.+1) :> {poly R}. | Proof. by case: n => [|n]; rewrite /mk_monic size_polyXn monicXn /= ?expr1. Qed. | Lemma | mk_monic_Xn | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"expr1",
"mk_monic",
"monicXn",
"poly",
"size_polyXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_qpoly (R : finNzSemiRingType) (h : {poly R}):
#|{poly %/ h}| = #|R| ^ (size (mk_monic h)).-1. | Proof. by rewrite card_npoly. Qed. | Lemma | card_qpoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"card_npoly",
"mk_monic",
"poly",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_monic_qpoly (R : finNzSemiRingType) (h : {poly R}):
1 < size h -> h \is monic -> #|{poly %/ h}| = #|R| ^ (size h).-1. | Proof. by move=> sh_gt1 hM; rewrite card_qpoly /mk_monic sh_gt1 hM. Qed. | Lemma | card_monic_qpoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"card_qpoly",
"mk_monic",
"monic",
"poly",
"sh_gt1",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mulC : commutative (@qpoly_mul A h). | Proof. by move=> p q; apply: val_inj; rewrite /= mulrC. Qed. | Lemma | qpoly_mulC | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"mulrC",
"qpoly_mul",
"val_inj"
] | Ring operations | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
qpoly_mulA : associative (@qpoly_mul A h). | Proof.
have rPM := monic_mk_monic h; move=> p q r; apply: val_inj.
by rewrite /= rmodp_mulml // rmodp_mulmr // mulrA.
Qed. | Lemma | qpoly_mulA | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"monic_mk_monic",
"mulrA",
"qpoly_mul",
"rmodp_mulml",
"rmodp_mulmr",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mul_addr : right_distributive (@qpoly_mul A h) +%R. | Proof.
have rPM := monic_mk_monic h; move=> p q r; apply: val_inj.
by rewrite /= !(mulrDr, rmodp_mulmr, rmodpD).
Qed. | Lemma | qpoly_mul_addr | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"monic_mk_monic",
"mulrDr",
"qpoly_mul",
"rmodpD",
"rmodp_mulmr",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mul_addl : left_distributive (@qpoly_mul A h) +%R. | Proof. by move=> p q r; rewrite -!(qpoly_mulC r) qpoly_mul_addr. Qed. | Lemma | qpoly_mul_addl | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpoly_mul",
"qpoly_mulC",
"qpoly_mul_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly1 : in_qpoly h 1 = 1. | Proof.
apply/val_eqP/eqP/in_qpoly_small.
by rewrite size_polyC oner_eq0 /= size_mk_monic_gt1.
Qed. | Lemma | in_qpoly1 | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"in_qpoly",
"in_qpoly_small",
"oner_eq0",
"size_mk_monic_gt1",
"size_polyC",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpolyM q1 q2 : in_qpoly h (q1 * q2) = in_qpoly h q1 * in_qpoly h q2. | Proof.
apply/val_eqP => /=.
by rewrite rmodp_mulml ?rmodp_mulmr // monic_mk_monic.
Qed. | Lemma | in_qpolyM | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"in_qpoly",
"monic_mk_monic",
"rmodp_mulml",
"rmodp_mulmr",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly_monoid_morphism : monoid_morphism (in_qpoly h). | Proof. by split; [ apply: in_qpoly1 | apply: in_qpolyM]. Qed. | Fact | in_qpoly_monoid_morphism | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"in_qpoly",
"in_qpoly1",
"in_qpolyM",
"monoid_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_qpoly_is_multiplicative | :=
(fun g => (g.2,g.1)) in_qpoly_monoid_morphism. | Definition | in_qpoly_is_multiplicative | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"in_qpoly_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_of_qpoly_sum I (r : seq I) (P1 : pred I) (F : I -> {poly %/ h}) :
((\sum_(i <- r | P1 i) F i) =
\sum_(p <- r | P1 p) ((F p) : {poly A}) :> {poly A})%R. | Proof. exact: raddf_sum. Qed. | Lemma | poly_of_qpoly_sum | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"P1",
"poly",
"raddf_sum",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_of_qpolyD (p q : {poly %/ h}) :
p + q= (p : {poly A}) + q :> {poly A}. | Proof. by []. Qed. | Lemma | poly_of_qpolyD | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC_natr p : (p%:R : {poly %/ h}) = p%:R :> {poly A}. | Proof. by elim: p => //= p IH; rewrite !mulrS poly_of_qpolyD IH. Qed. | Lemma | qpolyC_natr | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"mulrS",
"poly",
"poly_of_qpolyD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pchar_qpoly : [pchar {poly %/ h}] =i [pchar A]. | Proof.
move=> p; rewrite !inE; congr (_ && _).
apply/eqP/eqP=> [/(congr1 val) /=|pE]; last first.
by apply: val_inj => //=; rewrite qpolyC_natr /= -polyC_natr pE.
rewrite !qpolyC_natr -!polyC_natr => /(congr1 val) /=.
by rewrite polyseqC polyseq0; case: eqP.
Qed. | Lemma | pchar_qpoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"inE",
"last",
"pE",
"pchar",
"poly",
"polyC_natr",
"polyseq0",
"polyseqC",
"qpolyC_natr",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_of_qpolyM (p q : {poly %/ h}) :
p * q = rmodp ((p : {poly A}) * q) (mk_monic h) :> {poly A}. | Proof. by []. Qed. | Lemma | poly_of_qpolyM | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"mk_monic",
"poly",
"rmodp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_of_qpolyX (p : {poly %/ h}) n :
p ^+ n = rmodp ((p : {poly A}) ^+ n) (mk_monic h) :> {poly A}. | Proof.
have HhQ := monic_mk_monic h.
elim: n => //= [|n IH].
rewrite rmodp_small // size_polyC ?(leq_ltn_trans _ (size_mk_monic_gt1 _)) //.
by case: eqP.
by rewrite exprS /= IH // rmodp_mulmr // -exprS.
Qed. | Lemma | poly_of_qpolyX | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"exprS",
"leq_ltn_trans",
"mk_monic",
"monic_mk_monic",
"poly",
"rmodp",
"rmodp_mulmr",
"rmodp_small",
"size_mk_monic_gt1",
"size_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyCN (a : A) : qpolyC h (- a) = -(qpolyC h a). | Proof. by apply: val_inj; rewrite /= raddfN //= raddfN. Qed. | Lemma | qpolyCN | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpolyC",
"raddfN",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyCD : {morph (qpolyC h) : a b / a + b >-> a + b}%R. | Proof. by move=> a b; apply/val_eqP/eqP=> /=; rewrite -!raddfD. Qed. | Lemma | qpolyCD | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpolyC",
"raddfD",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyCM : {morph (qpolyC h) : a b / a * b >-> a * b}%R. | Proof.
move=> a b; apply/val_eqP/eqP=> /=; rewrite -polyCM rmodp_small //=.
have := qpolyC_proof h (a * b).
by rewrite qualifE/= -ltnS prednK // size_mk_monic_gt0.
Qed. | Lemma | qpolyCM | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"ltnS",
"polyCM",
"prednK",
"qpolyC",
"qpolyC_proof",
"rmodp_small",
"size_mk_monic_gt0",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC_is_zmod_morphism : zmod_morphism (qpolyC h). | Proof. by move=> x y; rewrite qpolyCD qpolyCN. Qed. | Lemma | qpolyC_is_zmod_morphism | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpolyC",
"qpolyCD",
"qpolyCN",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC_is_additive | := qpolyC_is_zmod_morphism. | Definition | qpolyC_is_additive | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpolyC_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC_is_monoid_morphism : monoid_morphism (qpolyC h). | Proof. by split=> // x y; rewrite qpolyCM. Qed. | Lemma | qpolyC_is_monoid_morphism | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"monoid_morphism",
"qpolyC",
"qpolyCM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpolyC_is_multiplicative | :=
(fun g => (g.2,g.1)) qpolyC_is_monoid_morphism. | Definition | qpolyC_is_multiplicative | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"qpolyC_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scale k (p : {poly %/ h}) : {poly %/ h} | := (k *: p)%R. | Definition | qpoly_scale | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scaleA a b p :
qpoly_scale a (qpoly_scale b p) = qpoly_scale (a * b) p. | Proof. by apply/val_eqP; rewrite /= scalerA. Qed. | Fact | qpoly_scaleA | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpoly_scale",
"scalerA",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scale1l : left_id 1%R qpoly_scale. | Proof. by move=> p; apply/val_eqP; rewrite /= scale1r. Qed. | Fact | qpoly_scale1l | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpoly_scale",
"scale1r",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scaleDr a : {morph qpoly_scale a : p q / (p + q)%R}. | Proof. by move=> p q; apply/val_eqP; rewrite /= scalerDr. Qed. | Fact | qpoly_scaleDr | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpoly_scale",
"scalerDr",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scaleDl p : {morph qpoly_scale^~ p : a b / a + b}%R. | Proof. by move=> a b; apply/val_eqP; rewrite /= scalerDl. Qed. | Fact | qpoly_scaleDl | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"qpoly_scale",
"scalerDl",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scaleAl a p q : qpoly_scale a (p * q) = (qpoly_scale a p * q). | Proof. by apply/val_eqP; rewrite /= -scalerAl rmodpZ // monic_mk_monic. Qed. | Fact | qpoly_scaleAl | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"monic_mk_monic",
"qpoly_scale",
"rmodpZ",
"scalerAl",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_scaleAr a p q : qpoly_scale a (p * q) = p * (qpoly_scale a q). | Proof. by apply/val_eqP; rewrite /= -scalerAr rmodpZ // monic_mk_monic. Qed. | Fact | qpoly_scaleAr | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"monic_mk_monic",
"qpoly_scale",
"rmodpZ",
"scalerAr",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_of_qpolyZ (p : {poly %/ h}) a :
a *: p = a *: (p : {poly A}) :> {poly A}. | Proof. by []. Qed. | Lemma | poly_of_qpolyZ | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_qpoly | := (pchar_qpoly) (only parsing). | Notation | char_qpoly | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"pchar_qpoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hQ | := (mk_monic h). | Notation | hQ | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"mk_monic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_inv (p : {poly %/ h}) | :=
if coprimep hQ p then let v : {poly %/ h} := in_qpoly h (egcdp hQ p).2 in
((lead_coef (v * p)) ^-1 *: v) else p. | Definition | qpoly_inv | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"coprimep",
"egcdp",
"hQ",
"in_qpoly",
"lead_coef",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_mulVz (p : {poly %/ h}) : coprimep hQ p -> (qpoly_inv p * p = 1)%R. | Proof.
have hQM := monic_mk_monic h.
move=> hCp; apply: val_inj; rewrite /qpoly_inv /in_qpoly hCp /=.
have p_neq0 : p != 0%R.
apply/eqP=> pZ; move: hCp; rewrite pZ.
rewrite coprimep0 -size_poly_eq1.
by case: size (size_mk_monic_gt1 h) => [|[]].
have F : (egcdp hQ p).1 * hQ + (egcdp hQ p).2 * p %= 1.
apply: eqp_... | Lemma | qpoly_mulVz | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"add0r",
"alg_polyC",
"apply",
"coprimep",
"coprimep0",
"divr1",
"egcdp",
"egcdpP",
"eq_sym",
"eqp_modpl",
"eqp_sym",
"eqp_trans",
"eqpfP",
"eqxx",
"gcdp",
"hQ",
"in_qpoly",
"lead_coefC",
"lead_coef_eq0",
"mk_monic_neq0",
"modpD",
"modpE",
"modp_mull",
"modp_small",
"... | Ugly | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
qpoly_mulzV (p : {poly %/ h}) :
coprimep hQ p -> (p * (qpoly_inv p) = 1)%R. | Proof. by move=> hCp; rewrite /= mulrC qpoly_mulVz. Qed. | Lemma | qpoly_mulzV | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"coprimep",
"hQ",
"mulrC",
"poly",
"qpoly_inv",
"qpoly_mulVz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
qpoly_intro_unit (p q : {poly %/ h}) : (q * p = 1)%R -> coprimep hQ p. | Proof.
have hQM := monic_mk_monic h.
case; rewrite -[rmodp]/rmodp -!Pdiv.IdomainMonic.modpE // => qp1.
have:= coprimep1 hQ.
rewrite -coprimep_modr -[1%R]qp1 !coprimep_modr coprimepMr; by case/andP.
Qed. | Lemma | qpoly_intro_unit | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"coprimep",
"coprimep1",
"coprimepMr",
"coprimep_modr",
"hQ",
"modpE",
"monic_mk_monic",
"poly",
"rmodp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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