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 |
|---|---|---|---|---|---|---|---|---|---|---|
modpN p : (- p) %% d = - (p %% d). | Proof. by rewrite -mulN1r -[RHS]mulN1r -polyCN !mul_polyC modpZl. Qed. | Lemma | modpN | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"modpZl",
"mulN1r",
"mul_polyC",
"polyCN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpN p : (- p) %/ d = - (p %/ d). | Proof. by rewrite -mulN1r -[RHS]mulN1r -polyCN !mul_polyC divpZl. Qed. | Lemma | divpN | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divpZl",
"mulN1r",
"mul_polyC",
"polyCN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpD p q : (p + q) %% d = p %% d + q %% d. | Proof.
have/edivpP [] // : (p + q) = (p %/ d + q %/ d) * d + (p %% d + q %% d).
by rewrite mulrDl addrACA -!divp_eq.
apply: leq_ltn_trans (size_polyD _ _) _.
rewrite gtn_max !ltn_modp andbb -lead_coef_eq0.
by apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | modpD | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addrACA",
"apply",
"contraTneq",
"divp_eq",
"edivpP",
"gtn_max",
"lead_coef_eq0",
"leq_ltn_trans",
"ltn_modp",
"mulrDl",
"size_polyD",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpD p q : (p + q) %/ d = p %/ d + q %/ d. | Proof.
have/edivpP [] // : (p + q) = (p %/ d + q %/ d) * d + (p %% d + q %% d).
by rewrite mulrDl addrACA -!divp_eq.
apply: leq_ltn_trans (size_polyD _ _) _.
rewrite gtn_max !ltn_modp andbb -lead_coef_eq0.
by apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | divpD | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
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"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addrACA",
"apply",
"contraTneq",
"divp_eq",
"edivpP",
"gtn_max",
"lead_coef_eq0",
"leq_ltn_trans",
"ltn_modp",
"mulrDl",
"size_polyD",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulpK q : (q * d) %/ d = q. | Proof.
case/esym/edivpP: (addr0 (q * d)); rewrite // size_poly0 size_poly_gt0.
by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0.
Qed. | Lemma | mulpK | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addr0",
"apply",
"contraTneq",
"edivpP",
"lead_coef_eq0",
"size_poly0",
"size_poly_gt0",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKp q : (d * q) %/ d = q. | Proof. by rewrite mulrC; apply: mulpK. Qed. | Lemma | mulKp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"mulpK",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_addl_mul_small q r : size r < size d -> (q * d + r) %/ d = q. | Proof. by move=> srd; rewrite divpD (divp_small srd) addr0 mulpK. Qed. | Lemma | divp_addl_mul_small | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addr0",
"divpD",
"divp_small",
"mulpK",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modp_addl_mul_small q r : size r < size d -> (q * d + r) %% d = r. | Proof. by move=> srd; rewrite modpD modp_mull add0r modp_small. Qed. | Lemma | modp_addl_mul_small | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"modpD",
"modp_mull",
"modp_small",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_addl_mul q r : (q * d + r) %/ d = q + r %/ d. | Proof. by rewrite divpD mulpK. Qed. | Lemma | divp_addl_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
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"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divpD",
"mulpK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpp : d %/ d = 1. | Proof. by rewrite -[d in d %/ _]mul1r mulpK. Qed. | Lemma | divpp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
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"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"mul1r",
"mulpK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_divMp m : size (m %/ d * d) <= size m. | Proof.
case: (eqVneq d 0) ulcd => [->|dn0 _]; first by rewrite lead_coef0 unitr0.
have [->|q0] := eqVneq (m %/ d) 0; first by rewrite mul0r size_poly0 leq0n.
rewrite {2}(divp_eq m) size_polyDl // size_mul // (polySpred q0) addSn /=.
by rewrite ltn_addl // ltn_modp.
Qed. | Lemma | leq_divMp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addSn",
"divp_eq",
"eqVneq",
"lead_coef0",
"leq0n",
"ltn_addl",
"ltn_modp",
"mul0r",
"polySpred",
"size",
"size_mul",
"size_poly0",
"size_polyDl",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpP p : reflect (exists q, p = q * d) (d %| p). | Proof.
apply: (iffP idP) => [| [k ->]]; last by apply/eqP; rewrite modp_mull.
by rewrite dvdp_eq; move/eqP->; exists (p %/ d).
Qed. | Lemma | dvdpP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"dvdp_eq",
"last",
"modp_mull"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpK p : d %| p -> p %/ d * d = p. | Proof. by rewrite dvdp_eq; move/eqP. Qed. | Lemma | divpK | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"dvdp_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpKC p : d %| p -> d * (p %/ d) = p. | Proof. by move=> ?; rewrite mulrC divpK. Qed. | Lemma | divpKC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divpK",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq_div p q : d %| p -> (q == p %/ d) = (q * d == p). | Proof.
move/divpK=> {2}<-; apply/eqP/eqP; first by move->.
apply/mulIf; rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->.
by rewrite unitr0.
Qed. | Lemma | dvdp_eq_div | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"contraTneq",
"divpK",
"lead_coef_eq0",
"mulIf",
"ulcd",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq_mul p q : d %| p -> (p == q * d) = (p %/ d == q). | Proof. by move=> dv_d_p; rewrite eq_sym -dvdp_eq_div // eq_sym. Qed. | Lemma | dvdp_eq_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"dvdp_eq_div",
"eq_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_mulA p q : d %| q -> p * (q %/ d) = p * q %/ d. | Proof.
move=> hdm; apply/eqP; rewrite eq_sym -dvdp_eq_mul; last first.
by rewrite -mulrA divpK.
by move/divpK: hdm<-; rewrite mulrA dvdp_mull // dvdpp.
Qed. | Lemma | divp_mulA | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"divpK",
"dvdp_eq_mul",
"dvdp_mull",
"dvdpp",
"eq_sym",
"last",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_mulAC m n : d %| m -> m %/ d * n = m * n %/ d. | Proof. by move=> hdm; rewrite mulrC (mulrC m); apply: divp_mulA. Qed. | Lemma | divp_mulAC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"divp_mulA",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_mulCA p q : d %| p -> d %| q -> p * (q %/ d) = q * (p %/ d). | Proof. by move=> hdp hdq; rewrite mulrC divp_mulAC // divp_mulA. Qed. | Lemma | divp_mulCA | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_mulA",
"divp_mulAC",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modp_mul p q : (p * (q %% d)) %% d = (p * q) %% d. | Proof. by rewrite [q in RHS]divp_eq mulrDr modpD mulrA modp_mull add0r. Qed. | Lemma | modp_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"divp_eq",
"modpD",
"modp_mull",
"mulrA",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_trunc_divp | := leq_divMp (only parsing). | Notation | leq_trunc_divp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"leq_divMp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expp_sub m n : n <= m -> (d ^+ (m - n))%N = d ^+ m %/ d ^+ n. | Proof. by move/subnK=> {2}<-; rewrite exprD mulpK // lead_coef_exp unitrX. Qed. | Lemma | expp_sub | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"exprD",
"lead_coef_exp",
"mulpK",
"subnK",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_pmul2l p q : lead_coef q \in GRing.unit -> d * p %/ (d * q) = p %/ q. | Proof.
move=> uq; rewrite {1}(divp_eq uq p) mulrDr mulrCA divp_addl_mul //.
by rewrite lead_coefM unitrM_comm ?ulcd //; red; rewrite mulrC.
have dn0 : d != 0.
by rewrite -lead_coef_eq0; apply: contraTneq ulcd => ->; rewrite unitr0.
have qn0 : q != 0.
by rewrite -lead_coef_eq0; apply: contraTneq uq => ->; rewrite ... | Lemma | divp_pmul2l | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addSn",
"addr0",
"apply",
"contraTneq",
"divpN0",
"divp_addl_mul",
"divp_eq",
"eqVneq",
"lead_coef",
"lead_coefM",
"lead_coef_eq0",
"ltnNge",
"ltn_add2l",
"ltn_modp",
"mulf_eq0",
"mulr0",
"mulrC",
"mulrCA",
"mulrDr",
"polySpred",
"size",
"size_mul",
"size_poly0",
"size... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_pmul2r p q : lead_coef p \in GRing.unit -> q * d %/ (p * d) = q %/ p. | Proof. by move=> uq; rewrite -!(mulrC d) divp_pmul2l. Qed. | Lemma | divp_pmul2r | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_pmul2l",
"lead_coef",
"mulrC",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_divl r p q :
lead_coef r \in GRing.unit -> lead_coef p \in GRing.unit ->
q %/ p %/ r = q %/ (p * r). | Proof.
move=> ulcr ulcp.
have e : q = (q %/ p %/ r) * (p * r) + ((q %/ p) %% r * p + q %% p).
by rewrite addrA (mulrC p) mulrA -mulrDl; rewrite -divp_eq //; apply: divp_eq.
have pn0 : p != 0.
by rewrite -lead_coef_eq0; apply: contraTneq ulcp => ->; rewrite unitr0.
have rn0 : r != 0.
by rewrite -lead_coef_eq0; app... | Lemma | divp_divl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"addSn",
"addnS",
"addrA",
"apply",
"contraTneq",
"divp_eq",
"edivpP",
"eqVneq",
"last",
"lead_coef",
"lead_coefM",
"lead_coef_eq0",
"leq_addr",
"leq_trans",
"ltn_add2l",
"ltn_modp",
"mul0r",
"mulrA",
"mulrC",
"mulrDl",
"polySpred",
"size",
"size_mul",
"size_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpAC p q : lead_coef p \in GRing.unit -> q %/ d %/ p = q %/ p %/ d. | Proof. by move=> ulcp; rewrite !divp_divl // mulrC. Qed. | Lemma | divpAC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_divl",
"lead_coef",
"mulrC",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpZr c p : c \in GRing.unit -> p %% (c *: d) = (p %% d). | Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; first by rewrite scaler0 !modp0.
have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d).
by rewrite scalerCA scalerA mulVr // scale1r -(divp_eq ulcd).
suff s : size (p %% d) < size (c *: d).
by rewrite (modpP _ e s) // -mul_polyC lead_coefM lead_coefC unitrM cn0.
by rewri... | Lemma | modpZr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"contraTneq",
"divp_eq",
"eqVneq",
"lead_coefC",
"lead_coefM",
"ltn_modp",
"modp0",
"modpP",
"mulVr",
"mul_polyC",
"scale1r",
"scaler0",
"scalerA",
"scalerCA",
"size",
"size_scale",
"ulcd",
"unit",
"unitr0",
"unitrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpZr c p : c \in GRing.unit -> p %/ (c *: d) = c^-1 *: (p %/ d). | Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; first by rewrite scaler0 !divp0 scaler0.
have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d).
by rewrite scalerCA scalerA mulVr // scale1r -(divp_eq ulcd).
suff s : size (p %% d) < size (c *: d).
by rewrite (divpP _ e s) // -mul_polyC lead_coefM lead_coefC unitrM cn0.
... | Lemma | divpZr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"contraTneq",
"divp0",
"divpP",
"divp_eq",
"eqVneq",
"lead_coefC",
"lead_coefM",
"ltn_modp",
"mulVr",
"mul_polyC",
"scale1r",
"scaler0",
"scalerA",
"scalerCA",
"size",
"size_scale",
"ulcd",
"unit",
"unitr0",
"unitrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_eq p q : p = (p %/ q) * q + (p %% q). | Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite modp0 mulr0 add0r.
by apply: IdomainUnit.divp_eq; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divp_eq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"apply",
"eqVneq",
"lead_coef_eq0",
"modp0",
"mulr0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_modpP p q d r : p = q * d + r -> size r < size d ->
q = (p %/ d) /\ r = p %% d. | Proof.
move=> he hs; apply: IdomainUnit.edivpP => //; rewrite unitfE lead_coef_eq0.
by rewrite -size_poly_gt0; apply: leq_trans hs.
Qed. | Lemma | divp_modpP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"edivpP",
"lead_coef_eq0",
"leq_trans",
"size",
"size_poly_gt0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpP p q d r : p = q * d + r -> size r < size d ->
q = (p %/ d). | Proof. by move/divp_modpP=> h; case/h. Qed. | Lemma | divpP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_modpP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpP p q d r : p = q * d + r -> size r < size d -> r = (p %% d). | Proof. by move/divp_modpP=> h; case/h. Qed. | Lemma | modpP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_modpP",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqpfP p q : p %= q -> p = (lead_coef p / lead_coef q) *: q. | Proof.
have [->|nz_q] := eqVneq q 0; first by rewrite eqp0 scaler0 => /eqP ->.
by apply/IdomainUnit.ucl_eqp_eq; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | eqpfP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"eqVneq",
"eqp0",
"lead_coef",
"lead_coef_eq0",
"scaler0",
"ucl_eqp_eq",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq q p : (q %| p) = (p == p %/ q * q). | Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite dvd0p mulr0 eq_sym.
by apply: IdomainUnit.dvdp_eq; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | dvdp_eq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"dvd0p",
"eqVneq",
"eq_sym",
"lead_coef_eq0",
"mulr0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqpf_eq p q : reflect (exists2 c, c != 0 & p = c *: q) (p %= q). | Proof.
apply: (iffP idP); last first.
case=> c nz_c ->; apply/eqpP.
by exists (1, c); rewrite ?scale1r ?oner_eq0.
have [->|nz_q] := eqVneq q 0.
by rewrite eqp0=> /eqP ->; exists 1; rewrite ?scale1r ?oner_eq0.
case/IdomainUnit.ulc_eqpP; first by rewrite unitfE lead_coef_eq0.
by move=> c nz_c ->; exists c.
Qed. | Lemma | eqpf_eq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"eqVneq",
"eqp0",
"eqpP",
"last",
"lead_coef_eq0",
"oner_eq0",
"scale1r",
"ulc_eqpP",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpZl c p q : (c *: p) %% q = c *: (p %% q). | Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite !modp0.
by apply: IdomainUnit.modpZl; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | modpZl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"eqVneq",
"lead_coef_eq0",
"modp0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulpK p q : q != 0 -> p * q %/ q = p. | Proof. by move=> qn0; rewrite IdomainUnit.mulpK // unitfE lead_coef_eq0. Qed. | Lemma | mulpK | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"lead_coef_eq0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulKp p q : q != 0 -> q * p %/ q = p. | Proof. by rewrite mulrC; apply: mulpK. Qed. | Lemma | mulKp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"mulpK",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpZl c p q : (c *: p) %/ q = c *: (p %/ q). | Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite !divp0 scaler0.
by apply: IdomainUnit.divpZl; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divpZl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"divp0",
"eqVneq",
"lead_coef_eq0",
"scaler0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpZr c p d : c != 0 -> p %% (c *: d) = (p %% d). | Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; first by rewrite scaler0 !modp0.
have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d).
by rewrite scalerCA scalerA mulVf // scale1r -divp_eq.
suff s : size (p %% d) < size (c *: d) by rewrite (modpP e s).
by rewrite size_scale ?ltn_modp.
Qed. | Lemma | modpZr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_eq",
"eqVneq",
"ltn_modp",
"modp0",
"modpP",
"mulVf",
"scale1r",
"scaler0",
"scalerA",
"scalerCA",
"size",
"size_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpZr c p d : c != 0 -> p %/ (c *: d) = c^-1 *: (p %/ d). | Proof.
case: (eqVneq d 0) => [-> | dn0 cn0]; first by rewrite scaler0 !divp0 scaler0.
have e : p = (c^-1 *: (p %/ d)) * (c *: d) + (p %% d).
by rewrite scalerCA scalerA mulVf // scale1r -divp_eq.
suff s : size (p %% d) < size (c *: d) by rewrite (divpP e s).
by rewrite size_scale ?ltn_modp.
Qed. | Lemma | divpZr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp0",
"divpP",
"divp_eq",
"eqVneq",
"ltn_modp",
"mulVf",
"scale1r",
"scaler0",
"scalerA",
"scalerCA",
"size",
"size_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_modpl d p q : p %= q -> (p %% d) %= (q %% d). | Proof.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e].
by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!modpZl e.
Qed. | Lemma | eqp_modpl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"c1",
"c2",
"eqpP",
"modpZl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_divl d p q : p %= q -> (p %/ d) %= (q %/ d). | Proof.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0 e].
by apply/eqpP; exists (c1, c2); rewrite ?c1n0 // -!divpZl e.
Qed. | Lemma | eqp_divl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"c1",
"c2",
"divpZl",
"eqpP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_modpr d p q : p %= q -> (d %% p) %= (d %% q). | Proof.
case/eqpP=> [[c1 c2]] /andP [c1n0 c2n0 e].
have -> : p = (c1^-1 * c2) *: q by rewrite -scalerA -e scalerA mulVf // scale1r.
by rewrite modpZr ?eqpxx // mulf_eq0 negb_or invr_eq0 c1n0.
Qed. | Lemma | eqp_modpr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"c1",
"c2",
"eqpP",
"eqpxx",
"invr_eq0",
"modpZr",
"mulVf",
"mulf_eq0",
"scale1r",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_mod p1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> p1 %% q1 %= p2 %% q2. | Proof. move=> e1 e2; exact: eqp_trans (eqp_modpl _ e1) (eqp_modpr _ e2). Qed. | Lemma | eqp_mod | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"eqp_modpl",
"eqp_modpr",
"eqp_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_divr (d m n : {poly F}) : m %= n -> (d %/ m) %= (d %/ n). | Proof.
case/eqpP=> [[c1 c2]] /andP [c1n0 c2n0 e].
have -> : m = (c1^-1 * c2) *: n by rewrite -scalerA -e scalerA mulVf // scale1r.
by rewrite divpZr ?eqp_scale // ?invr_eq0 mulf_eq0 negb_or invr_eq0 c1n0.
Qed. | Lemma | eqp_divr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"c1",
"c2",
"divpZr",
"eqpP",
"eqp_scale",
"invr_eq0",
"mulVf",
"mulf_eq0",
"poly",
"scale1r",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_div p1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> p1 %/ q1 %= p2 %/ q2. | Proof. move=> e1 e2; exact: eqp_trans (eqp_divl _ e1) (eqp_divr _ e2). Qed. | Lemma | eqp_div | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"eqp_divl",
"eqp_divr",
"eqp_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_gdcor p q r : q %= r -> gdcop p q %= gdcop p r. | Proof.
move=> eqr; rewrite /gdcop (eqp_size eqr).
move: (size r)=> n; elim: n p q r eqr => [|n ihn] p q r; first by rewrite eqpxx.
move=> eqr /=; rewrite (eqp_coprimepl p eqr); case: ifP => _ //.
exact/ihn/eqp_div/eqp_gcdl.
Qed. | Lemma | eqp_gdcor | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"eqp_coprimepl",
"eqp_div",
"eqp_gcdl",
"eqp_size",
"eqpxx",
"gdcop",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_gdcol p q r : q %= r -> gdcop q p %= gdcop r p. | Proof.
move=> eqr; rewrite /gdcop; move: (size p)=> n.
elim: n p q r eqr {1 3}p (eqpxx p) => [|n ihn] p q r eqr s esp /=.
case: (eqVneq q 0) eqr => [-> | nq0 eqr] /=.
by rewrite eqp_sym eqp0 => ->; rewrite eqpxx.
by case: (eqVneq r 0) eqr nq0 => [->|]; rewrite ?eqpxx // eqp0 => ->.
rewrite (eqp_coprimepr _ eqr)... | Lemma | eqp_gdcol | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"eqVneq",
"eqp0",
"eqp_coprimepl",
"eqp_coprimepr",
"eqp_div",
"eqp_gcd",
"eqp_sym",
"eqpxx",
"gdcop",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_rgdco_gdco q p : rgdcop q p %= gdcop q p. | Proof.
rewrite /rgdcop /gdcop; move: (size p)=> n.
elim: n p q {1 3}p {1 3}q (eqpxx p) (eqpxx q) => [|n ihn] p q s t /= sp tq.
case: (eqVneq t 0) tq => [-> | nt0 etq].
by rewrite eqp_sym eqp0 => ->; rewrite eqpxx.
by case: (eqVneq q 0) etq nt0 => [->|]; rewrite ?eqpxx // eqp0 => ->.
rewrite rcoprimep_coprimep (... | Lemma | eqp_rgdco_gdco | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"eqVneq",
"eqp0",
"eqp_coprimepl",
"eqp_coprimepr",
"eqp_div",
"eqp_gcd",
"eqp_rdiv_div",
"eqp_rgcd_gcd",
"eqp_sym",
"eqp_trans",
"eqpxx",
"gdcop",
"rcoprimep_coprimep",
"rgdcop",
"size",
"sp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpD d p q : (p + q) %% d = p %% d + q %% d. | Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite !modp0.
by apply: IdomainUnit.modpD; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | modpD | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"eqVneq",
"lead_coef_eq0",
"modp0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpN p q : (- p) %% q = - (p %% q). | Proof. by apply/eqP; rewrite -addr_eq0 -modpD addNr mod0p. Qed. | Lemma | modpN | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addNr",
"addr_eq0",
"apply",
"mod0p",
"modpD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modNp p q : (- p) %% q = - (p %% q). | Proof. exact: modpN. Qed. | Lemma | modNp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"modpN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpD d p q : (p + q) %/ d = p %/ d + q %/ d. | Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite !divp0 addr0.
by apply: IdomainUnit.divpD; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divpD | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addr0",
"apply",
"divp0",
"eqVneq",
"lead_coef_eq0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpN p q : (- p) %/ q = - (p %/ q). | Proof. by apply/eqP; rewrite -addr_eq0 -divpD addNr div0p. Qed. | Lemma | divpN | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addNr",
"addr_eq0",
"apply",
"div0p",
"divpD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_addl_mul_small d q r : size r < size d -> (q * d + r) %/ d = q. | Proof.
move=> srd; rewrite divpD (divp_small srd) addr0 mulpK // -size_poly_gt0.
exact: leq_trans srd.
Qed. | Lemma | divp_addl_mul_small | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"addr0",
"divpD",
"divp_small",
"leq_trans",
"mulpK",
"size",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modp_addl_mul_small d q r : size r < size d -> (q * d + r) %% d = r. | Proof. by move=> srd; rewrite modpD modp_mull add0r modp_small. Qed. | Lemma | modp_addl_mul_small | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"modpD",
"modp_mull",
"modp_small",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_addl_mul d q r : d != 0 -> (q * d + r) %/ d = q + r %/ d. | Proof. by move=> dn0; rewrite divpD mulpK. Qed. | Lemma | divp_addl_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divpD",
"mulpK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpp d : d != 0 -> d %/ d = 1. | Proof.
by move=> dn0; apply: IdomainUnit.divpp; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divpp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"lead_coef_eq0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_divMp d m : size (m %/ d * d) <= size m. | Proof.
have [-> | dn0] := eqVneq d 0; first by rewrite mulr0 size_poly0.
by apply: IdomainUnit.leq_divMp; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | leq_divMp | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"eqVneq",
"lead_coef_eq0",
"mulr0",
"size",
"size_poly0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpK d p : d %| p -> p %/ d * d = p. | Proof.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite mulr0.
by apply: IdomainUnit.divpK; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divpK | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"dvd0pP",
"eqVneq",
"lead_coef_eq0",
"mulr0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpKC d p : d %| p -> d * (p %/ d) = p. | Proof. by move=> ?; rewrite mulrC divpK. Qed. | Lemma | divpKC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divpK",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq_div d p q : d != 0 -> d %| p -> (q == p %/ d) = (q * d == p). | Proof.
by move=> dn0; apply: IdomainUnit.dvdp_eq_div; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | dvdp_eq_div | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"lead_coef_eq0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eq_mul d p q : d != 0 -> d %| p -> (p == q * d) = (p %/ d == q). | Proof. by move=> dn0 dv_d_p; rewrite eq_sym -dvdp_eq_div // eq_sym. Qed. | Lemma | dvdp_eq_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"dvdp_eq_div",
"eq_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_mulA d p q : d %| q -> p * (q %/ d) = p * q %/ d. | Proof.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite !divp0 mulr0.
by apply: IdomainUnit.divp_mulA; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divp_mulA | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"divp0",
"dvd0pP",
"eqVneq",
"lead_coef_eq0",
"mulr0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_mulAC d m n : d %| m -> m %/ d * n = m * n %/ d. | Proof. by move=> hdm; rewrite mulrC (mulrC m); apply: divp_mulA. Qed. | Lemma | divp_mulAC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"divp_mulA",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_mulCA d p q : d %| p -> d %| q -> p * (q %/ d) = q * (p %/ d). | Proof. by move=> hdp hdq; rewrite mulrC divp_mulAC // divp_mulA. Qed. | Lemma | divp_mulCA | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_mulA",
"divp_mulAC",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expp_sub d m n : d != 0 -> m >= n -> (d ^+ (m - n))%N = d ^+ m %/ d ^+ n. | Proof. by move=> dn0 /subnK=> {2}<-; rewrite exprD mulpK // expf_neq0. Qed. | Lemma | expp_sub | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"expf_neq0",
"exprD",
"mulpK",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_pmul2l d q p : d != 0 -> q != 0 -> d * p %/ (d * q) = p %/ q. | Proof.
by move=> dn0 qn0; apply: IdomainUnit.divp_pmul2l; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divp_pmul2l | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"lead_coef_eq0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_pmul2r d p q : d != 0 -> p != 0 -> q * d %/ (p * d) = q %/ p. | Proof. by move=> dn0 qn0; rewrite -!(mulrC d) divp_pmul2l. Qed. | Lemma | divp_pmul2r | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_pmul2l",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divp_divl r p q : q %/ p %/ r = q %/ (p * r). | Proof.
have [-> | rn0] := eqVneq r 0; first by rewrite mulr0 !divp0.
have [-> | pn0] := eqVneq p 0; first by rewrite mul0r !divp0 div0p.
by apply: IdomainUnit.divp_divl; rewrite unitfE lead_coef_eq0.
Qed. | Lemma | divp_divl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"div0p",
"divp0",
"eqVneq",
"lead_coef_eq0",
"mul0r",
"mulr0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpAC d p q : q %/ d %/ p = q %/ p %/ d. | Proof. by rewrite !divp_divl // mulrC. Qed. | Lemma | divpAC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp_divl",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
edivp_def p q : edivp p q = (0, p %/ q, p %% q). | Proof.
rewrite Idomain.edivp_def; congr (_, _, _); rewrite /scalp 2!unlock /=.
have [-> | qn0] := eqVneq; first by rewrite lead_coef0 unitr0.
by rewrite unitfE lead_coef_eq0 qn0 /=; case: (redivp_rec _ _ _ _) => [[]].
Qed. | Lemma | edivp_def | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"edivp",
"eqVneq",
"lead_coef0",
"lead_coef_eq0",
"redivp_rec",
"scalp",
"unitfE",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divpE p q : p %/ q = (lead_coef q)^-(rscalp p q) *: (rdivp p q). | Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite rdivp0 divp0 scaler0.
by rewrite Idomain.divpE unitfE lead_coef_eq0 qn0.
Qed. | Lemma | divpE | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"divp0",
"eqVneq",
"lead_coef",
"lead_coef_eq0",
"rdivp",
"rdivp0",
"rscalp",
"scaler0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modpE p q : p %% q = (lead_coef q)^-(rscalp p q) *: (rmodp p q). | Proof.
have [-> | qn0] := eqVneq q 0.
by rewrite rmodp0 modp0 /rscalp unlock eqxx lead_coef0 expr0 invr1 scale1r.
by rewrite Idomain.modpE unitfE lead_coef_eq0 qn0.
Qed. | Lemma | modpE | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"eqVneq",
"eqxx",
"expr0",
"invr1",
"lead_coef",
"lead_coef0",
"lead_coef_eq0",
"modp0",
"rmodp",
"rmodp0",
"rscalp",
"scale1r",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalpE p q : scalp p q = 0. | Proof.
have [-> | qn0] := eqVneq q 0; first by rewrite scalp0.
by rewrite Idomain.scalpE unitfE lead_coef_eq0 qn0.
Qed. | Lemma | scalpE | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"eqVneq",
"lead_coef_eq0",
"scalp",
"scalp0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpE p q : (p %| q) = rdvdp p q. | Proof. exact: Idomain.dvdpE. Qed. | Lemma | dvdpE | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"rdvdp"
] | Just to have it without importing the weak theory | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
edivp_spec m d : nat * {poly F} * {poly F} -> Type | :=
EdivpSpec n q r of
m = q * d + r & (d != 0) ==> (size r < size d) : edivp_spec m d (n, q, r). | Variant | edivp_spec | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"nat",
"poly",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
edivpP m d : edivp_spec m d (edivp m d). | Proof.
rewrite edivp_def; constructor; first exact: divp_eq.
by apply/implyP=> dn0; rewrite ltn_modp.
Qed. | Lemma | edivpP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"divp_eq",
"edivp",
"edivp_def",
"edivp_spec",
"ltn_modp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
edivp_eq d q r : size r < size d -> edivp (q * d + r) d = (0, q, r). | Proof.
move=> srd; apply: Idomain.edivp_eq; rewrite // unitfE lead_coef_eq0.
by rewrite -size_poly_gt0; apply: leq_trans srd.
Qed. | Lemma | edivp_eq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"edivp",
"lead_coef_eq0",
"leq_trans",
"size",
"size_poly_gt0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modp_mul p q m : (p * (q %% m)) %% m = (p * q) %% m. | Proof. by rewrite [in RHS](divp_eq q m) mulrDr modpD mulrA modp_mull add0r. Qed. | Lemma | modp_mul | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"divp_eq",
"modpD",
"modp_mull",
"mulrA",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_mod p q x : root q x -> (p %% q).[x] = p.[x]. | Proof.
by rewrite [in RHS](divp_eq p q) !hornerE => /eqP->; rewrite mulr0 add0r.
Qed. | Lemma | horner_mod | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add0r",
"divp_eq",
"hornerE",
"mulr0",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpP p q : reflect (exists qq, p = qq * q) (q %| p). | Proof.
have [-> | qn0] := eqVneq q 0; last first.
by apply: IdomainUnit.dvdpP; rewrite unitfE lead_coef_eq0.
by rewrite dvd0p; apply: (iffP eqP) => [->| [? ->]]; [exists 1|]; rewrite mulr0.
Qed. | Lemma | dvdpP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"dvd0p",
"eqVneq",
"last",
"lead_coef_eq0",
"mulr0",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Bezout_eq1_coprimepP p q :
reflect (exists u, u.1 * p + u.2 * q = 1) (coprimep p q). | Proof.
apply: (iffP idP)=> [hpq|]; last first.
by case=> -[u v] /= e; apply/Bezout_coprimepP; exists (u, v); rewrite e eqpxx.
case/Bezout_coprimepP: hpq => [[u v]] /=.
case/eqpP=> [[c1 c2]] /andP /= [c1n0 c2n0] e.
exists (c2^-1 *: (c1 *: u), c2^-1 *: (c1 *: v)); rewrite /= -!scalerAl.
by rewrite -!scalerDr e scalerA ... | Lemma | Bezout_eq1_coprimepP | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"Bezout_coprimepP",
"apply",
"c1",
"c2",
"coprimep",
"eqpP",
"eqpxx",
"last",
"mulVf",
"scale1r",
"scalerA",
"scalerAl",
"scalerDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gdcor p q : q != 0 -> p %| (gdcop q p) * (q ^+ size p). | Proof.
rewrite /gdcop => nz_q; have [n hsp] := ubnPleq (size p).
elim: n => [|n IHn] /= in p hsp *; first by rewrite (negPf nz_q) mul0r dvdp0.
have [_ | ncop_pq] := ifPn; first by rewrite dvdp_mulr.
have g_gt1: 1 < size (gcdp p q).
rewrite ltn_neqAle eq_sym ncop_pq size_poly_gt0 gcdp_eq0.
by rewrite negb_and nz_q o... | Lemma | dvdp_gdcor | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"add2n",
"div0p",
"divpK",
"dvdp0",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mul",
"dvdp_mulr",
"eqVneq",
"eq_sym",
"exprSr",
"gcdp",
"gcdp_eq0",
"gdcop",
"leq_subr",
"leq_trans",
"ltnS",
"ltn_neqAle",
"mul0r",
"mulrA",
"nz_p",
"polySpred",
"size",
"size_divp",
"size_poly0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reducible_cubic_root p q :
size p <= 4 -> 1 < size q < size p -> q %| p -> {r | root p r}. | Proof.
move=> p_le4 /andP[]; rewrite leq_eqVlt eq_sym.
have [/poly2_root[x qx0] _ _ | _ /= q_gt2 p_gt_q] := size q =P 2.
by exists x; rewrite -!dvdp_XsubCl in qx0 *; apply: (dvdp_trans qx0).
case/dvdpP/sig_eqW=> r def_p; rewrite def_p.
suffices /poly2_root[x rx0]: size r = 2 by exists x; rewrite rootM rx0.
have /norP... | Lemma | reducible_cubic_root | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"def_p",
"dvdpP",
"dvdp_XsubCl",
"dvdp_trans",
"eq_sym",
"eqn_add2r",
"eqn_leq",
"leq_eqVlt",
"leq_ltn_trans",
"leq_subLR",
"leq_trans",
"ltn_subRL",
"mulf_eq0",
"poly2_root",
"root",
"rootM",
"sig_eqW",
"size",
"size_mul",
"size_poly_gt0",
"subn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cubic_irreducible p :
1 < size p <= 4 -> (forall x, ~~ root p x) -> irreducible_poly p. | Proof.
move=> /andP[p_gt1 p_le4] root'p; split=> // q sz_q_neq1 q_dv_p.
have nz_p: p != 0 by rewrite -size_poly_gt0 ltnW.
have nz_q: q != 0 by apply: contraTneq q_dv_p => ->; rewrite dvd0p.
have q_gt1: size q > 1 by rewrite ltn_neqAle eq_sym sz_q_neq1 size_poly_gt0.
rewrite -dvdp_size_eqp // eqn_leq dvdp_leq //= leqNgt... | Lemma | cubic_irreducible | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"contraTneq",
"dvd0p",
"dvdp_leq",
"dvdp_size_eqp",
"eq_sym",
"eqn_leq",
"irreducible_poly",
"leqNgt",
"ltnW",
"ltn_neqAle",
"nz_p",
"p_gt1",
"q_gt1",
"reducible_cubic_root",
"root",
"size",
"size_poly_gt0",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mup x q | :=
[arg max_(n > (ord0 : 'I_(size q).+1) | ('X - x%:P) ^+ n %| q) n] : nat. | Definition | mup | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"nat",
"ord0",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mup_geq x q n : q != 0 -> (n <= mup x q)%N = (('X - x%:P) ^+ n %| q). | Proof.
move=> q_neq0; rewrite /mup; symmetry.
case: arg_maxnP; rewrite ?expr0 ?dvd1p//= => i i_dvd gti.
case: ltnP => [|/dvdp_exp2l/dvdp_trans]; last exact.
apply: contraTF => dvdq; rewrite -leqNgt.
suff n_small : (n < (size q).+1)%N by exact: (gti (Ordinal n_small)).
by rewrite ltnS ltnW// -(size_exp_XsubC _ x) dvdp_l... | Lemma | mup_geq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"arg_maxnP",
"dvd1p",
"dvdp_exp2l",
"dvdp_leq",
"dvdp_trans",
"expr0",
"last",
"leqNgt",
"ltnP",
"ltnS",
"ltnW",
"mup",
"size",
"size_exp_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mup_leq x q n : q != 0 ->
(mup x q <= n)%N = ~~ (('X - x%:P) ^+ n.+1 %| q). | Proof. by move=> qN0; rewrite leqNgt mup_geq. Qed. | Lemma | mup_leq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"leqNgt",
"mup",
"mup_geq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mup_ltn x q n : q != 0 -> (mup x q < n)%N = ~~ (('X - x%:P) ^+ n %| q). | Proof. by move=> qN0; rewrite ltnNge mup_geq. Qed. | Lemma | mup_ltn | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"ltnNge",
"mup",
"mup_geq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
XsubC_dvd x q : q != 0 -> ('X - x%:P %| q) = (0 < mup x q)%N. | Proof. by move=> /mup_geq-/(_ _ 1%N)/esym; apply. Qed. | Lemma | XsubC_dvd | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"mup",
"mup_geq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mup_XsubCX n x y :
mup x (('X - y%:P) ^+ n) = (if (y == x) then n else 0)%N. | Proof.
have Xxn0 : ('X - y%:P) ^+ n != 0 by rewrite ?expf_neq0 ?polyXsubC_eq0.
apply/eqP; rewrite eqn_leq mup_leq ?mup_geq//.
have [->|Nxy] := eqVneq x y.
by rewrite /= dvdpp ?dvdp_Pexp2l ?size_XsubC ?ltnn.
by rewrite dvd1p dvdp_XsubCl /root horner_exp !hornerE expf_neq0// subr_eq0.
Qed. | Lemma | mup_XsubCX | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"dvd1p",
"dvdp_Pexp2l",
"dvdp_XsubCl",
"dvdpp",
"eqVneq",
"eqn_leq",
"expf_neq0",
"hornerE",
"horner_exp",
"ltnn",
"mup",
"mup_geq",
"mup_leq",
"polyXsubC_eq0",
"root",
"size_XsubC",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mupNroot x q : ~~ root q x -> mup x q = 0%N. | Proof.
move=> qNx; have qN0 : q != 0 by apply: contraNneq qNx => ->; rewrite root0.
by move: qNx; rewrite -dvdp_XsubCl XsubC_dvd// lt0n negbK => /eqP.
Qed. | Lemma | mupNroot | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"XsubC_dvd",
"apply",
"contraNneq",
"dvdp_XsubCl",
"lt0n",
"mup",
"root",
"root0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mupMr x q1 q2 : ~~ root q1 x -> mup x (q1 * q2) = mup x q2. | Proof.
move=> q1Nx; have q1N0 : q1 != 0 by apply: contraNneq q1Nx => ->; rewrite root0.
have [->|q2N0] := eqVneq q2 0; first by rewrite mulr0.
apply/esym/eqP; rewrite eqn_leq mup_geq ?mulf_neq0// dvdp_mull -?mup_geq//=.
rewrite mup_leq ?mulf_neq0// Gauss_dvdpr -?mup_ltn//.
by rewrite coprimep_expl// coprimep_sym coprim... | Lemma | mupMr | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"Gauss_dvdpr",
"apply",
"contraNneq",
"coprimep_XsubC",
"coprimep_expl",
"coprimep_sym",
"dvdp_mull",
"eqVneq",
"eqn_leq",
"mulf_neq0",
"mulr0",
"mup",
"mup_geq",
"mup_leq",
"mup_ltn",
"root",
"root0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mupMl x q1 q2 : ~~ root q2 x -> mup x (q1 * q2) = mup x q1. | Proof. by rewrite mulrC; apply/mupMr. Qed. | Lemma | mupMl | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"mulrC",
"mup",
"mupMr",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mupM x q1 q2 : q1 != 0 -> q2 != 0 ->
mup x (q1 * q2) = (mup x q1 + mup x q2)%N. | Proof.
move=> q1N0 q2N0; apply/eqP; rewrite eqn_leq mup_leq ?mulf_neq0//.
rewrite mup_geq ?mulf_neq0// exprD ?dvdp_mul; do ?by rewrite -mup_geq.
have [m1 [r1]] := multiplicity_XsubC q1 x; rewrite q1N0 /= => r1Nx ->.
have [m2 [r2]] := multiplicity_XsubC q2 x; rewrite q2N0 /= => r2Nx ->.
rewrite !mupMr// ?mup_XsubCX eqxx... | Lemma | mupM | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"apply",
"dvdp_XsubCl",
"dvdp_mul",
"dvdp_mul2r",
"eqn_leq",
"eqxx",
"expf_neq0",
"exprD",
"exprS",
"mulf_neq0",
"mulrACA",
"multiplicity_XsubC",
"mup",
"mupMr",
"mup_XsubCX",
"mup_geq",
"mup_leq",
"polyXsubC_eq0",
"r1",
"r2",
"rootM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mu_prod_XsubC x (s : seq F) :
mup x (\prod_(y <- s) ('X - y%:P)) = count_mem x s. | Proof.
elim: s => [|y s IHs]; rewrite (big_cons, big_nil)/=.
by rewrite mupNroot// root1.
rewrite mupM ?polyXsubC_eq0// ?monic_neq0 ?monic_prod_XsubC//.
by rewrite IHs (@mup_XsubCX 1).
Qed. | Lemma | mu_prod_XsubC | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"big_cons",
"big_nil",
"count_mem",
"monic_neq0",
"monic_prod_XsubC",
"mup",
"mupM",
"mupNroot",
"mup_XsubCX",
"polyXsubC_eq0",
"root1",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_XsubC_eq (s t : seq F) :
\prod_(x <- s) ('X - x%:P) = \prod_(x <- t) ('X - x%:P) -> perm_eq s t. | Proof.
move=> eq_prod; apply/allP => x _ /=; apply/eqP.
by have /(congr1 (mup x)) := eq_prod; rewrite !mu_prod_XsubC.
Qed. | Lemma | prod_XsubC_eq | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"allP",
"apply",
"mu_prod_XsubC",
"mup",
"perm_eq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
redivp_map a b :
redivp a^f b^f = (rscalp a b, (rdivp a b)^f, (rmodp a b)^f). | Proof.
rewrite /rdivp /rscalp /rmodp !unlock map_poly_eq0 size_map_poly.
have [// | q_nz] := ifPn; rewrite -(rmorph0 (map_poly f)) //.
have [m _] := ubnPeq (size a); elim: m 0%N 0 a => [|m IHm] qq r a /=.
rewrite -!mul_polyC !size_map_poly !lead_coef_map // -(map_polyXn f).
by rewrite -!(map_polyC f) -!rmorphM -rmo... | Lemma | redivp_map | algebra | algebra/polydiv.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"GRing.Theory",
"CommonRing",
"RingComRreg",
"RingMonic",
"Ring",
"ComRing",
"UnitRing",
"IdomainD... | [
"lead_coef_map",
"map_poly",
"map_polyC",
"map_polyXn",
"map_poly_eq0",
"mul_polyC",
"rdivp",
"redivp",
"rmodp",
"rmorph0",
"rmorphB",
"rmorphD",
"rmorphM",
"rscalp",
"size",
"size_map_poly",
"ubnPeq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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