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 |
|---|---|---|---|---|---|---|---|---|---|---|
poly_mul_comm p q : p * q = q * p. | Proof.
apply/polyP=> i; rewrite coefM coefMr.
by apply: eq_bigr => j _; rewrite mulrC.
Qed. | Fact | poly_mul_comm | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"coefM",
"coefMr",
"eq_bigr",
"mulrC",
"polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hornerM p q x : (p * q).[x] = p.[x] * q.[x]. | Proof. by rewrite hornerM_comm //; apply: mulrC. Qed. | Lemma | hornerM | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"hornerM_comm",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_eval_is_monoid_morphism (x : R) : monoid_morphism (horner_eval x). | Proof. by split => [|p q]; rewrite /horner_eval (hornerC, hornerM). Qed. | Fact | horner_eval_is_monoid_morphism | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"hornerC",
"hornerM",
"horner_eval",
"monoid_morphism",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_eval_is_multiplicative x | :=
(fun g => (g.2, g.1)) (horner_eval_is_monoid_morphism x). | Definition | horner_eval_is_multiplicative | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"horner_eval_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_exp p x n : (p ^+ n).[x] = p.[x] ^+ n. | Proof. exact: (rmorphXn (horner_eval _)). Qed. | Lemma | horner_exp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"horner_eval",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_prod I r (P : pred I) (F : I -> {poly R}) x :
(\prod_(i <- r | P i) F i).[x] = \prod_(i <- r | P i) (F i).[x]. | Proof. exact: (rmorph_prod (horner_eval _)). Qed. | Lemma | horner_prod | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"horner_eval",
"poly",
"rmorph_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hornerE | :=
(hornerD, hornerN, hornerX, hornerC, horner_exp,
simp, hornerCM, hornerZ, hornerM, horner_cons). | Definition | hornerE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"hornerC",
"hornerCM",
"hornerD",
"hornerM",
"hornerN",
"hornerX",
"hornerZ",
"horner_cons",
"horner_exp",
"simp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_poly_is_monoid_morphism q : monoid_morphism (comp_poly q). | Proof.
split=> [|p1 p2]; first by rewrite comp_polyC.
by rewrite /comp_poly rmorphM hornerM_comm //; apply: mulrC.
Qed. | Fact | comp_poly_is_monoid_morphism | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"comp_poly",
"comp_polyC",
"hornerM_comm",
"monoid_morphism",
"mulrC",
"rmorphM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_poly_multiplicative q | :=
(fun g => (g.2, g.1)) (comp_poly_is_monoid_morphism q). | Definition | comp_poly_multiplicative | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"comp_poly_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_polyM p q r : (p * q) \Po r = (p \Po r) * (q \Po r). | Proof. exact: rmorphM. Qed. | Lemma | comp_polyM | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_polyA p q r : p \Po (q \Po r) = (p \Po q) \Po r. | Proof.
elim/poly_ind: p => [|p c IHp]; first by rewrite !comp_polyC.
by rewrite !comp_polyD !comp_polyM !comp_polyX IHp !comp_polyC.
Qed. | Lemma | comp_polyA | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"comp_polyC",
"comp_polyD",
"comp_polyM",
"comp_polyX",
"poly_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_comp p q x : (p \Po q).[x] = p.[q.[x]]. | Proof. by apply: polyC_inj; rewrite -!comp_polyCr comp_polyA. Qed. | Lemma | horner_comp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"comp_polyA",
"comp_polyCr",
"polyC_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_comp p q x : root (p \Po q) x = root p (q.[x]). | Proof. by rewrite !rootE horner_comp. Qed. | Lemma | root_comp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"horner_comp",
"root",
"rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deriv_comp p q : (p \Po q) ^`() = (p ^`() \Po q) * q^`(). | Proof.
elim/poly_ind: p => [|p c IHp]; first by rewrite !(deriv0, comp_poly0) mul0r.
rewrite comp_poly_MXaddC derivD derivC derivM IHp derivMXaddC comp_polyD.
by rewrite comp_polyM comp_polyX addr0 addrC mulrAC -mulrDl.
Qed. | Lemma | deriv_comp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addr0",
"addrC",
"comp_poly0",
"comp_polyD",
"comp_polyM",
"comp_polyX",
"comp_poly_MXaddC",
"deriv0",
"derivC",
"derivD",
"derivM",
"derivMXaddC",
"mul0r",
"mulrAC",
"mulrDl",
"poly_ind"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deriv_exp p n : (p ^+ n)^`() = p^`() * p ^+ n.-1 *+ n. | Proof.
elim: n => [|n IHn]; first by rewrite expr0 mulr0n derivC.
by rewrite exprS derivM {}IHn (mulrC p) mulrnAl -mulrA -exprSr mulrS; case n.
Qed. | Lemma | deriv_exp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"derivC",
"derivM",
"expr0",
"exprS",
"exprSr",
"mulr0n",
"mulrA",
"mulrC",
"mulrS",
"mulrnAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derivCE | := (derivE, deriv_exp). | Definition | derivCE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"derivE",
"deriv_exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef_prod_XsubC (ps : seq R) (n : nat) :
(n <= size ps)%N ->
(\prod_(p <- ps) ('X - p%:P))`_n =
(-1) ^+ (size ps - n)%N *
\sum_(I in {set 'I_(size ps)} | #|I| == (size ps - n)%N)
\prod_(i in I) ps`_i. | Proof.
move=> nle.
under eq_bigr => i _ do rewrite addrC -raddfN/=.
rewrite -{1}(in_tupleE ps) -(map_tnth_enum (_ ps)) big_map.
rewrite enumT bigA_distr /= coef_sum.
transitivity (\sum_(I in {set 'I_(size ps)}) if #|I| == (size ps - n)%N then
\prod_(i < size ps | i \in I) - ps`_i else 0).
apply eq_b... | Lemma | coef_prod_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addrC",
"apply",
"bigA_distr",
"big_const",
"big_if",
"big_map",
"big_mkcond",
"cardC",
"card_ord",
"coefCM",
"coefXn",
"coef_sum",
"enumT",
"eq_bigr",
"eqn_add2l",
"eqn_add2r",
"in_tupleE",
"iter_mulr_1",
"map_tnth_enum",
"mulr0",
"mulr1",
"mulr_sumr",
"nat",
"prodrN"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coefPn_prod_XsubC (ps : seq R) : size ps != 0 ->
(\prod_(p <- ps) ('X - p%:P))`_((size ps).-1) = - \sum_(p <- ps) p. | Proof.
rewrite coef_prod_XsubC ?leq_pred// => ps0.
have -> : (size ps - (size ps).-1 = 1)%N.
by move: ps0; case: (size ps) => // n _; exact: subSnn.
rewrite expr1 mulN1r; congr GRing.opp.
set f : 'I_(size ps) -> {set 'I_(size ps)} := fun a => [set a].
transitivity (\sum_(I in imset f (mem setT)) \prod_(i in I) ps`_i)... | Lemma | coefPn_prod_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"big_imset",
"big_map",
"big_set1",
"cards1P",
"coef_prod_XsubC",
"congr_big",
"enumT",
"expr1",
"imsetP",
"in_setT",
"in_tupleE",
"leq_pred",
"map_tnth_enum",
"mulN1r",
"opp",
"seq",
"set11",
"set1P",
"setT",
"size",
"subSnn",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coef0_prod_XsubC (ps : seq R) :
(\prod_(p <- ps) ('X - p%:P))`_0 = (-1) ^+ (size ps) * \prod_(p <- ps) p. | Proof.
rewrite coef_prod_XsubC// subn0; congr GRing.mul.
transitivity (\sum_(I in [set setT : {set 'I_(size ps)}]) \prod_(i in I) ps`_i).
apply: congr_big =>// i/=.
apply/idP/set1P => [/eqP cardE | ->]; last by rewrite cardsT card_ord.
by apply/eqP; rewrite eqEcard subsetT cardsT card_ord cardE leqnn.
rewrite big... | Lemma | coef0_prod_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"big_map",
"big_set1",
"cardE",
"card_ord",
"cardsT",
"coef_prod_XsubC",
"congr_big",
"enumT",
"eqEcard",
"in_setT",
"in_tupleE",
"last",
"leqnn",
"map_tnth_enum",
"mul",
"seq",
"set1P",
"setT",
"size",
"subn0",
"subsetT",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_eval_is_linear x : linear_for *%R (@horner_eval R x). | Proof. exact: linearP. Qed. | Fact | horner_eval_is_linear | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"horner_eval",
"linearP",
"linear_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mul p q : p != 0 -> q != 0 -> size (p * q) = (size p + size q).-1. | Proof.
by move=> nz_p nz_q; rewrite -size_proper_mul ?mulf_neq0 ?lead_coef_eq0.
Qed. | Lemma | size_mul | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"lead_coef_eq0",
"mulf_neq0",
"nz_p",
"size",
"size_proper_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_idomainAxiom p q : p * q = 0 -> (p == 0) || (q == 0). | Proof.
move=> pq0; apply/norP=> [[p_nz q_nz]]; move/eqP: (size_mul p_nz q_nz).
by rewrite eq_sym pq0 size_poly0 (polySpred p_nz) (polySpred q_nz) addnS.
Qed. | Fact | poly_idomainAxiom | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addnS",
"apply",
"eq_sym",
"polySpred",
"size_mul",
"size_poly0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_unit : pred {poly R} | :=
fun p => (size p == 1) && (p`_0 \in GRing.unit). | Definition | poly_unit | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"poly",
"size",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_inv p | := if p \in poly_unit then (p`_0)^-1%:P else p. | Definition | poly_inv | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"poly_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_mulVp : {in poly_unit, left_inverse 1 poly_inv *%R}. | Proof.
move=> p Up; rewrite /poly_inv Up.
by case/andP: Up => /size_poly1P[c _ ->]; rewrite coefC -polyCM => /mulVr->.
Qed. | Fact | poly_mulVp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"Up",
"coefC",
"mulVr",
"polyCM",
"poly_inv",
"poly_unit",
"size_poly1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_intro_unit p q : q * p = 1 -> p \in poly_unit. | Proof.
move=> pq1; apply/andP; split; last first.
apply/unitrP; exists q`_0.
by rewrite 2!mulrC -!/(coefp 0 _) -rmorphM pq1 rmorph1.
have: size (q * p) == 1 by rewrite pq1 size_poly1.
have [-> | nz_p] := eqVneq p 0; first by rewrite mulr0 size_poly0.
have [-> | nz_q] := eqVneq q 0; first by rewrite mul0r size_poly0... | Fact | poly_intro_unit | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addSn",
"addnS",
"addn_eq0",
"apply",
"coefp",
"eqSS",
"eqVneq",
"last",
"mul0r",
"mulr0",
"mulrC",
"nz_p",
"polySpred",
"poly_unit",
"rmorph1",
"rmorphM",
"size",
"size_mul",
"size_poly0",
"size_poly1",
"split",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_inv_out : {in [predC poly_unit], poly_inv =1 id}. | Proof. by rewrite /poly_inv => p /negbTE/= ->. Qed. | Fact | poly_inv_out | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"id",
"poly_inv",
"poly_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_unitE p :
(p \in GRing.unit) = (size p == 1) && (p`_0 \in GRing.unit). | Proof. by []. Qed. | Lemma | poly_unitE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"size",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_invE p : p ^-1 = if p \in GRing.unit then (p`_0)^-1%:P else p. | Proof. by []. Qed. | Lemma | poly_invE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyCV c : c%:P^-1 = (c^-1)%:P. | Proof.
have [/rmorphV-> // | nUc] := boolP (c \in GRing.unit).
by rewrite !invr_out // poly_unitE coefC (negbTE nUc) andbF.
Qed. | Lemma | polyCV | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"coefC",
"invr_out",
"poly_unitE",
"rmorphV",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootM p q x : root (p * q) x = root p x || root q x. | Proof. by rewrite !rootE hornerM mulf_eq0. Qed. | Lemma | rootM | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"hornerM",
"mulf_eq0",
"root",
"rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootZ x a p : a != 0 -> root (a *: p) x = root p x. | Proof. by move=> nz_a; rewrite -mul_polyC rootM rootC (negPf nz_a). Qed. | Lemma | rootZ | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"mul_polyC",
"root",
"rootC",
"rootM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_exp p n a: comm_poly p a -> (0 < n)%N -> root (p ^+ n) a = root p a. | Proof. by move=> ? n0; rewrite !rootE horner_exp_comm// expf_eq0 n0. Qed. | Lemma | root_exp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"comm_poly",
"expf_eq0",
"horner_exp_comm",
"root",
"rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_scale a p : a != 0 -> size (a *: p) = size p. | Proof. by move/lregP/lreg_size->. Qed. | Lemma | size_scale | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"lregP",
"lreg_size",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_Cmul a p : a != 0 -> size (a%:P * p) = size p. | Proof. by rewrite mul_polyC => /size_scale->. Qed. | Lemma | size_Cmul | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"mul_polyC",
"size",
"size_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coefM p q : lead_coef (p * q) = lead_coef p * lead_coef q. | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite !(mul0r, lead_coef0).
have [-> | nz_q] := eqVneq q 0; first by rewrite !(mulr0, lead_coef0).
by rewrite lead_coef_proper_mul // mulf_neq0 ?lead_coef_eq0.
Qed. | Lemma | lead_coefM | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"eqVneq",
"lead_coef",
"lead_coef0",
"lead_coef_eq0",
"lead_coef_proper_mul",
"mul0r",
"mulf_neq0",
"mulr0",
"nz_p"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coef_prod I rI (P : {pred I}) (p : I -> {poly R}) :
lead_coef (\prod_(i <- rI | P i) p i) = \prod_(i <- rI | P i) lead_coef (p i). | Proof. by apply/big_morph/lead_coef1; apply: lead_coefM. Qed. | Lemma | lead_coef_prod | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"big_morph",
"lead_coef",
"lead_coef1",
"lead_coefM",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coefZ a p : lead_coef (a *: p) = a * lead_coef p. | Proof. by rewrite -mul_polyC lead_coefM lead_coefC. Qed. | Lemma | lead_coefZ | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"lead_coef",
"lead_coefC",
"lead_coefM",
"mul_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_poly_eq0 a p : (a *: p == 0) = (a == 0) || (p == 0). | Proof. by rewrite -mul_polyC mulf_eq0 polyC_eq0. Qed. | Lemma | scale_poly_eq0 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"mul_polyC",
"mulf_eq0",
"polyC_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_prod (I : finType) (P : pred I) (F : I -> {poly R}) :
(forall i, P i -> F i != 0) ->
size (\prod_(i | P i) F i) = ((\sum_(i | P i) size (F i)).+1 - #|P|)%N. | Proof.
move=> nzF; transitivity (\sum_(i | P i) (size (F i)).-1).+1; last first.
apply: canRL (addKn _) _; rewrite addnS -sum1_card -big_split /=.
by congr _.+1; apply: eq_bigr => i /nzF/polySpred.
elim/big_rec2: _ => [|i d p /nzF nzFi IHp]; first by rewrite size_poly1.
by rewrite size_mul // -?size_poly_eq0 IHp //... | Lemma | size_prod | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addKn",
"addnS",
"apply",
"big_rec2",
"big_split",
"eq_bigr",
"last",
"poly",
"polySpred",
"size",
"size_mul",
"size_poly1",
"size_poly_eq0",
"sum1_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_prod_seq (I : eqType) (s : seq I) (F : I -> {poly R}) :
(forall i, i \in s -> F i != 0) ->
size (\prod_(i <- s) F i) = ((\sum_(i <- s) size (F i)).+1 - size s)%N. | Proof.
move=> nzF; rewrite big_tnth size_prod; first by move=> i; rewrite nzF ?mem_tnth.
by rewrite cardT /= size_enum_ord [in RHS]big_tnth.
Qed. | Lemma | size_prod_seq | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"big_tnth",
"cardT",
"mem_tnth",
"poly",
"seq",
"size",
"size_enum_ord",
"size_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mul_eq1 p q : (size (p * q) == 1) = ((size p == 1) && (size q == 1)). | Proof.
have [->|pNZ] := eqVneq p 0; first by rewrite mul0r size_poly0.
have [->|qNZ] := eqVneq q 0; first by rewrite mulr0 size_poly0 andbF.
rewrite size_mul //.
by move: pNZ qNZ; rewrite -!size_poly_gt0; (do 2 case: size) => //= n [|[|]].
Qed. | Lemma | size_mul_eq1 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"eqVneq",
"mul0r",
"mulr0",
"size",
"size_mul",
"size_poly0",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_prod_seq_eq1 (I : eqType) (s : seq I) (P : pred I) (F : I -> {poly R}) :
reflect (forall i, P i && (i \in s) -> size (F i) = 1)
(size (\prod_(i <- s | P i) F i) == 1%N). | Proof.
rewrite (big_morph _ (id1:=true) size_mul_eq1) ?size_polyC ?oner_neq0//.
rewrite big_all_cond; apply/(iffP allP).
by move=> h i /andP[Pi ins]; apply/eqP/(implyP (h i ins) Pi).
by move=> h i ins; apply/implyP => Pi; rewrite h ?Pi.
Qed. | Lemma | size_prod_seq_eq1 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"allP",
"apply",
"big_all_cond",
"big_morph",
"id1",
"oner_neq0",
"poly",
"seq",
"size",
"size_mul_eq1",
"size_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_prod_eq1 (I : finType) (P : pred I) (F : I -> {poly R}) :
reflect (forall i, P i -> size (F i) = 1)
(size (\prod_(i | P i) F i) == 1). | Proof.
apply: (iffP (size_prod_seq_eq1 _ _ _)) => Hi i.
by move=> Pi; apply: Hi; rewrite Pi /= mem_index_enum.
by rewrite mem_index_enum andbT; apply: Hi.
Qed. | Lemma | size_prod_eq1 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"mem_index_enum",
"poly",
"size",
"size_prod_seq_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_exp p n : (size (p ^+ n)).-1 = ((size p).-1 * n)%N. | Proof.
elim: n => [|n IHn]; first by rewrite size_poly1 muln0.
have [-> | nz_p] := eqVneq p 0; first by rewrite exprS mul0r size_poly0.
rewrite exprS size_mul ?expf_neq0 // mulnS -{}IHn.
by rewrite polySpred // [size (p ^+ n)]polySpred ?expf_neq0 ?addnS.
Qed. | Lemma | size_exp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addnS",
"eqVneq",
"expf_neq0",
"exprS",
"mul0r",
"muln0",
"mulnS",
"nz_p",
"polySpred",
"size",
"size_mul",
"size_poly0",
"size_poly1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coef_exp p n : lead_coef (p ^+ n) = lead_coef p ^+ n. | Proof.
elim: n => [|n IHn]; first by rewrite !expr0 lead_coef1.
by rewrite !exprS lead_coefM IHn.
Qed. | Lemma | lead_coef_exp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"expr0",
"exprS",
"lead_coef",
"lead_coef1",
"lead_coefM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_prod_XsubC rs x :
root (\prod_(a <- rs) ('X - a%:P)) x = (x \in rs). | Proof.
elim: rs => [|a rs IHrs]; first by rewrite rootE big_nil hornerC oner_eq0.
by rewrite big_cons rootM IHrs root_XsubC.
Qed. | Lemma | root_prod_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"big_cons",
"big_nil",
"hornerC",
"oner_eq0",
"root",
"rootE",
"rootM",
"root_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_exp_XsubC n a x : root (('X - a%:P) ^+ n.+1) x = (x == a). | Proof. by rewrite rootE horner_exp expf_eq0 [_ == 0]root_XsubC. Qed. | Lemma | root_exp_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"expf_eq0",
"horner_exp",
"root",
"rootE",
"root_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_comp_poly p q :
(size (p \Po q)).-1 = ((size p).-1 * (size q).-1)%N. | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite comp_poly0 size_poly0.
have [/size1_polyC-> | nc_q] := leqP (size q) 1.
by rewrite comp_polyCr !size_polyC -!sub1b -!subnS muln0.
have nz_q: q != 0 by rewrite -size_poly_eq0 -(subnKC nc_q).
rewrite mulnC comp_polyE (polySpred nz_p) /= big_ord_recr /= addrC.
rewr... | Lemma | size_comp_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addrC",
"apply",
"big_ord_recr",
"bigmax_leqP",
"comp_poly0",
"comp_polyCr",
"comp_polyE",
"eqVneq",
"expf_neq0",
"lead_coef_eq0",
"leqP",
"leq_trans",
"ltnRHS",
"ltnS",
"ltn_pmul2l",
"muln0",
"mulnC",
"nz_p",
"polySpred",
"size",
"size1_polyC",
"size_exp",
"size_poly0",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coef_comp p q : size q > 1 ->
lead_coef (p \Po q) = (lead_coef p) * lead_coef q ^+ (size p).-1. | Proof.
move=> q_gt1; rewrite !lead_coefE coef_comp_poly size_comp_poly.
have [->|nz_p] := eqVneq p 0; first by rewrite size_poly0 big_ord0 coef0 mul0r.
rewrite polySpred //= big_ord_recr /= big1 ?add0r => [i _|]; last first.
by rewrite -!lead_coefE -lead_coef_exp !lead_coefE size_exp mulnC.
rewrite [X in _ * X]nth_de... | Lemma | lead_coef_comp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"add0r",
"big1",
"big_ord0",
"big_ord_recr",
"coef0",
"coef_comp_poly",
"eqVneq",
"last",
"lead_coef",
"lead_coefE",
"lead_coef_exp",
"leq_trans",
"ltn_mul2r",
"mul0r",
"mulnC",
"mulr0",
"nth_default",
"nz_p",
"polySpred",
"q_gt1",
"size",
"size_comp_poly",
"size_exp",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_poly_eq0 p q : size q > 1 -> (p \Po q == 0) = (p == 0). | Proof.
move=> sq_gt1; rewrite -!lead_coef_eq0 lead_coef_comp //.
rewrite mulf_eq0 expf_eq0 !lead_coef_eq0 -[q == 0]size_poly_leq0.
by rewrite [_ <= 0]leqNgt (leq_ltn_trans _ sq_gt1) ?andbF ?orbF.
Qed. | Lemma | comp_poly_eq0 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"expf_eq0",
"lead_coef_comp",
"lead_coef_eq0",
"leqNgt",
"leq_ltn_trans",
"mulf_eq0",
"size",
"size_poly_leq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_comp_poly2 p q : size q = 2 -> size (p \Po q) = size p. | Proof.
move=> sq2; have [->|pN0] := eqVneq p 0; first by rewrite comp_polyC.
by rewrite polySpred ?size_comp_poly ?comp_poly_eq0 ?sq2 // muln1 polySpred.
Qed. | Lemma | size_comp_poly2 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"comp_polyC",
"comp_poly_eq0",
"eqVneq",
"muln1",
"polySpred",
"size",
"size_comp_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_poly2_eq0 p q : size q = 2 -> (p \Po q == 0) = (p == 0). | Proof. by rewrite -!size_poly_eq0 => /size_comp_poly2->. Qed. | Lemma | comp_poly2_eq0 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"size",
"size_comp_poly2",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_poly_roots p rs :
p != 0 -> all (root p) rs -> uniq rs -> size rs < size p. | Proof.
elim: rs p => [p pn0 _ _ | r rs ihrs p pn0] /=; first by rewrite size_poly_gt0.
case/andP => rpr arrs /andP [rnrs urs]; case/factor_theorem: rpr => q epq.
have [q0 | ?] := eqVneq q 0; first by move: pn0; rewrite epq q0 mul0r eqxx.
have -> : size p = (size q).+1.
by rewrite epq size_Mmonic ?monicXsubC // size_... | Theorem | max_poly_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addnC",
"all",
"apply",
"eqVneq",
"eq_in_all",
"eqxx",
"factor_theorem",
"monicXsubC",
"mul0r",
"root",
"rootM",
"root_XsubC",
"size",
"size_Mmonic",
"size_XsubC",
"size_poly_gt0",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
roots_geq_poly_eq0 p (rs : seq R) : all (root p) rs -> uniq rs ->
(size rs >= size p)%N -> p = 0. | Proof. by move=> ??; apply: contraTeq => ?; rewrite leqNgt max_poly_roots. Qed. | Lemma | roots_geq_poly_eq0 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"all",
"apply",
"contraTeq",
"leqNgt",
"max_poly_roots",
"root",
"seq",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_map_poly p : size p^f = size p. | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite rmorph0 !size_poly0.
by rewrite size_poly_eq // fmorph_eq0 // lead_coef_eq0.
Qed. | Lemma | size_map_poly | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"eqVneq",
"fmorph_eq0",
"lead_coef_eq0",
"nz_p",
"rmorph0",
"size",
"size_poly0",
"size_poly_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lead_coef_map p : lead_coef p^f = f (lead_coef p). | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite !(rmorph0, lead_coef0).
by rewrite lead_coef_map_eq // fmorph_eq0 // lead_coef_eq0.
Qed. | Lemma | lead_coef_map | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"eqVneq",
"fmorph_eq0",
"lead_coef",
"lead_coef0",
"lead_coef_eq0",
"lead_coef_map_eq",
"nz_p",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_eq0 p : (p^f == 0) = (p == 0). | Proof. by rewrite -!size_poly_eq0 size_map_poly. Qed. | Lemma | map_poly_eq0 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"size_map_poly",
"size_poly_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_inj : injective (map_poly f). | Proof.
move=> p q eqfpq; apply/eqP; rewrite -subr_eq0 -map_poly_eq0.
by rewrite rmorphB /= eqfpq subrr.
Qed. | Lemma | map_poly_inj | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"map_poly",
"map_poly_eq0",
"rmorphB",
"subr_eq0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_monic p : (p^f \is monic) = (p \is monic). | Proof. by rewrite [in LHS]monicE lead_coef_map fmorph_eq1. Qed. | Lemma | map_monic | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"fmorph_eq1",
"lead_coef_map",
"monic",
"monicE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_com p x : comm_poly p^f (f x). | Proof. exact: map_comm_poly (mulrC x _). Qed. | Lemma | map_poly_com | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"comm_poly",
"map_comm_poly",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_root p x : root p^f (f x) = root p x. | Proof. by rewrite rootE horner_map // fmorph_eq0. Qed. | Lemma | fmorph_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"fmorph_eq0",
"horner_map",
"root",
"rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_unity_root n z : n.-unity_root (f z) = n.-unity_root z. | Proof. by rewrite !unity_rootE -(inj_eq (fmorph_inj f)) rmorphXn ?rmorph1. Qed. | Lemma | fmorph_unity_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"fmorph_inj",
"inj_eq",
"rmorph1",
"rmorphXn",
"unity_rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_primitive_root n z :
n.-primitive_root (f z) = n.-primitive_root z. | Proof.
by congr (_ && _); apply: eq_forallb => i; rewrite fmorph_unity_root.
Qed. | Lemma | fmorph_primitive_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"apply",
"eq_forallb",
"fmorph_unity_root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diff_roots (x y : R) | := (x * y == y * x) && (y - x \in GRing.unit). | Definition | diff_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_roots rs | :=
if rs is x :: rs' then all (diff_roots x) rs' && uniq_roots rs' else true. | Fixpoint | uniq_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"all",
"diff_roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_roots_prod_XsubC p rs :
all (root p) rs -> uniq_roots rs ->
exists q, p = q * \prod_(z <- rs) ('X - z%:P). | Proof.
elim: rs => [|z rs IHrs] /=; first by rewrite big_nil; exists p; rewrite mulr1.
case/andP=> rpz rprs /andP[drs urs]; case: IHrs => {urs rprs}// q def_p.
have [|q' def_q] := factor_theorem q z _; last first.
by exists q'; rewrite big_cons mulrA -def_q.
rewrite {p}def_p in rpz.
elim/last_ind: rs drs rpz => [|rs ... | Lemma | uniq_roots_prod_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"all",
"all_rcons",
"big_cat",
"big_cons",
"big_nil",
"big_seq1",
"cats1",
"comm_poly",
"def_p",
"def_q",
"factor_theorem",
"hornerM_comm",
"hornerXsubC",
"inj_eq",
"last",
"last_ind",
"mul0r",
"mulIr",
"mulr1",
"mulrA",
"mulrBl",
"mulrBr",
"mulrN",
"opprB",
"oppr_eq0... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_ring_poly_roots p rs :
p != 0 -> all (root p) rs -> uniq_roots rs -> size rs < size p. | Proof.
move=> nz_p _ /(@uniq_roots_prod_XsubC p)[// | q def_p]; rewrite def_p in nz_p *.
have nz_q: q != 0 by apply: contraNneq nz_p => ->; rewrite mul0r.
rewrite size_Mmonic ?monic_prod_XsubC // (polySpred nz_q) addSn /=.
by rewrite size_prod_XsubC leq_addl.
Qed. | Theorem | max_ring_poly_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"addSn",
"all",
"apply",
"contraNneq",
"def_p",
"leq_addl",
"monic_prod_XsubC",
"mul0r",
"nz_p",
"polySpred",
"root",
"size",
"size_Mmonic",
"size_prod_XsubC",
"uniq_roots",
"uniq_roots_prod_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_roots_prod_XsubC p rs :
size p = (size rs).+1 -> all (root p) rs -> uniq_roots rs ->
p = lead_coef p *: \prod_(z <- rs) ('X - z%:P). | Proof.
move=> size_p /uniq_roots_prod_XsubC def_p Urs.
case/def_p: Urs => q -> {p def_p} in size_p *.
have [q0 | nz_q] := eqVneq q 0; first by rewrite q0 mul0r size_poly0 in size_p.
have{q nz_q size_p} /size_poly1P[c _ ->]: size q == 1.
rewrite -(eqn_add2r (size rs)) add1n -size_p.
by rewrite size_Mmonic ?monic_pro... | Lemma | all_roots_prod_XsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"add1n",
"addnS",
"all",
"def_p",
"eqVneq",
"eqn_add2r",
"lead_coef",
"lead_coefC",
"lead_coef_Mmonic",
"monic_prod_XsubC",
"mul0r",
"mul_polyC",
"root",
"size",
"size_Mmonic",
"size_poly0",
"size_poly1P",
"size_prod_XsubC",
"uniq_roots",
"uniq_roots_prod_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly2_root p : size p = 2 -> {r | root p r}. | Proof.
case: p => [[|p0 [|p1 []]] //= nz_p1]; exists (- p0 / p1).
by rewrite /root addr_eq0 /= mul0r add0r mulrC divfK ?opprK.
Qed. | Lemma | poly2_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"add0r",
"addr_eq0",
"divfK",
"mul0r",
"mulrC",
"opprK",
"p0",
"root",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_rootsE rs : uniq_roots rs = uniq rs. | Proof.
elim: rs => //= r rs ->; congr (_ && _); rewrite -has_pred1 -all_predC.
by apply: eq_all => t; rewrite /diff_roots mulrC eqxx unitfE subr_eq0.
Qed. | Lemma | uniq_rootsE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"all_predC",
"apply",
"diff_roots",
"eq_all",
"eqxx",
"has_pred1",
"mulrC",
"subr_eq0",
"uniq",
"uniq_roots",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_ZXsubC (a b r : F) : a != 0 ->
root (a *: 'X - b%:P) r = (r == b / a). | Proof.
move=> a0; rewrite rootE !hornerE.
by rewrite -[r in RHS]divr1 eqr_div ?oner_neq0// mulr1 mulrC subr_eq0.
Qed. | Lemma | root_ZXsubC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"divr1",
"eqr_div",
"hornerE",
"mulr1",
"mulrC",
"oner_neq0",
"root",
"rootE",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_unity_roots rs :
n > 0 -> all n.-unity_root rs -> uniq rs -> size rs <= n. | Proof.
move=> n_gt0 rs_n_1 Urs; have szPn := size_XnsubC (1 : F) n_gt0.
by rewrite -ltnS -szPn max_poly_roots -?size_poly_eq0 ?szPn.
Qed. | Lemma | max_unity_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"all",
"ltnS",
"max_poly_roots",
"n_gt0",
"size",
"size_XnsubC",
"size_poly_eq0",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_unity_roots rs :
n > 0 -> all n.-unity_root rs -> uniq rs -> size rs = n ->
n.-unity_root =i rs. | Proof.
move=> n_gt0 rs_n_1 Urs sz_rs_n x; rewrite -topredE /=.
apply/idP/idP=> xn1; last exact: (allP rs_n_1).
apply: contraFT (ltnn n) => not_rs_x.
by rewrite -{1}sz_rs_n (@max_unity_roots (x :: rs)) //= ?xn1 ?not_rs_x.
Qed. | Lemma | mem_unity_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"all",
"allP",
"apply",
"last",
"ltnn",
"max_unity_roots",
"n_gt0",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zn | := [seq z ^+ i | i <- index_iota 0 n]. | Let | zn | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"index_iota",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factor_Xn_sub_1 : \prod_(0 <= i < n) ('X - (z ^+ i)%:P) = 'X^n - 1. | Proof.
transitivity (\prod_(w <- zn) ('X - w%:P)); first by rewrite big_map.
have n_gt0: n > 0 := prim_order_gt0 prim_z.
rewrite (@all_roots_prod_XsubC _ ('X^n - 1) zn).
- by rewrite size_XnsubC // size_map size_iota subn0.
- apply/allP=> _ /mapP[i _ ->] /=; rewrite rootE !hornerE.
by rewrite exprAC (prim_expr_order ... | Lemma | factor_Xn_sub_1 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"allP",
"all_roots_prod_XsubC",
"apply",
"big_map",
"eq_prim_root_expr",
"expr1n",
"exprAC",
"hornerE",
"iota_uniq",
"mapP",
"map_inj_in_uniq",
"mem_index_iota",
"modn_small",
"monicP",
"monicXnsubC",
"n_gt0",
"prim_expr_order",
"prim_order_gt0",
"prim_z",
"rootE",
"scale1r",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_rootP x : x ^+ n = 1 -> {i : 'I_n | x = z ^+ i}. | Proof.
move=> xn1; pose logx := [pred i : 'I_n | x == z ^+ i].
case: (pickP logx) => [i /eqP-> | no_i]; first by exists i.
case: notF; suffices{no_i}: x \in zn.
case/mapP=> i; rewrite mem_index_iota => lt_i_n def_x.
by rewrite -(no_i (Ordinal lt_i_n)) /= -def_x.
rewrite -root_prod_XsubC big_map factor_Xn_sub_1.
by ... | Lemma | prim_rootP | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"big_map",
"factor_Xn_sub_1",
"mapP",
"mem_index_iota",
"pickP",
"root",
"root_prod_XsubC",
"unity_rootE",
"zn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_diff_roots x y : diff_roots (f x) (f y) = (x != y). | Proof.
rewrite /diff_roots -rmorphB // fmorph_unit // subr_eq0 //.
by rewrite rmorph_comm // eqxx eq_sym.
Qed. | Lemma | map_diff_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"diff_roots",
"eq_sym",
"eqxx",
"fmorph_unit",
"rmorphB",
"rmorph_comm",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_uniq_roots s : uniq_roots (map f s) = uniq s. | Proof.
elim: s => //= x s ->; congr (_ && _); elim: s => //= y s ->.
by rewrite map_diff_roots -negb_or.
Qed. | Lemma | map_uniq_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"map",
"map_diff_roots",
"uniq",
"uniq_roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_prim_rootP u z n :
n.-primitive_root z -> {k | coprime k n & u z = z ^+ k}. | Proof.
move=> prim_z; have:= prim_z; rewrite -(fmorph_primitive_root u) => prim_uz.
have [[k _] /= def_uz] := prim_rootP prim_z (prim_expr_order prim_uz).
by exists k; rewrite // -(prim_root_exp_coprime _ prim_z) -def_uz.
Qed. | Lemma | aut_prim_rootP | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"coprime",
"fmorph_primitive_root",
"prim_expr_order",
"prim_rootP",
"prim_root_exp_coprime",
"prim_z"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_unity_rootP u z n : n > 0 -> z ^+ n = 1 -> {k | u z = z ^+ k}. | Proof.
by move=> _ /prim_order_exists[// | m /(aut_prim_rootP u)[k]]; exists k.
Qed. | Lemma | aut_unity_rootP | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"aut_prim_rootP",
"prim_order_exists"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_unity_rootC u v z n : n > 0 -> z ^+ n = 1 -> u (v z) = v (u z). | Proof.
move=> n_gt0 /(aut_unity_rootP _ n_gt0) def_z.
have [[i def_uz] [j def_vz]] := (def_z u, def_z v).
by rewrite def_vz def_uz !rmorphXn /= def_vz def_uz exprAC.
Qed. | Lemma | aut_unity_rootC | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"aut_unity_rootP",
"exprAC",
"n_gt0",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .-unity_root" | := (root_of_unity n) : unity_root_scope. | Notation | n .-unity_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"root_of_unity"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .-primitive_root" | := (primitive_root_of_unity n) : unity_root_scope. | Notation | n .-primitive_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"primitive_root_of_unity"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unity_rootE | := unity_rootE. | Definition | unity_rootE | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unity_rootP | := @unity_rootP. | Definition | unity_rootP | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_order_exists | := prim_order_exists. | Definition | prim_order_exists | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_order_gt0 | := prim_order_gt0. | Notation | prim_order_gt0 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_expr_order | := prim_expr_order. | Notation | prim_expr_order | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_expr_mod | := prim_expr_mod. | Definition | prim_expr_mod | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_order_dvd | := prim_order_dvd. | Definition | prim_order_dvd | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_prim_root_expr | := eq_prim_root_expr. | Definition | eq_prim_root_expr | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rmorph_unity_root | := rmorph_unity_root. | Definition | rmorph_unity_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_unity_root | := fmorph_unity_root. | Definition | fmorph_unity_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_primitive_root | := fmorph_primitive_root. | Definition | fmorph_primitive_root | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_unity_roots | := max_unity_roots. | Definition | max_unity_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_unity_roots | := mem_unity_roots. | Definition | mem_unity_roots | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prim_rootP | := prim_rootP. | Definition | prim_rootP | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz2 : 2 != 0 :> F. | Hypothesis | nz2 | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
degp : size p = 3. | Hypothesis | degp | algebra | algebra/poly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"tuple",
"div",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"countalg",
"GRing.Theory",
"prime"
] | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.