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 |
|---|---|---|---|---|---|---|---|---|---|---|
dvdp_mul d1 d2 m1 m2 : d1 %| m1 -> d2 %| m2 -> d1 * d2 %| m1 * m2. | Proof.
case: (eqVneq d1 0) => [-> /dvd0pP -> | d1n0]; first by rewrite !mul0r dvdpp.
case: (eqVneq d2 0) => [-> _ /dvd0pP -> | d2n0]; first by rewrite !mulr0.
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Hq1.
rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _; move/eqP=> Hq2.
apply: (@eq_dvdp (c1 * ... | Lemma | dvdp_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... | [
"apply",
"c1",
"c2",
"dvd0pP",
"dvdp_eq",
"dvdpp",
"eqVneq",
"eq_dvdp",
"expf_neq0",
"lead_coef_eq0",
"mul0r",
"mulf_neq0",
"mulr0",
"mulrA",
"mulrCA",
"scalerA",
"scalerAl",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_addr m d n : d %| m -> (d %| m + n) = (d %| n). | Proof.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite add0r.
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Eq1.
apply/idP/idP; rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _.
have sn0 : c1 * c2 != 0.
by rewrite !mulf_neq0 // expf_eq0 lead_coef_eq0 (negPf dn0) andbF.
move/eq... | Lemma | dvdp_addr | 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... | [
"Eq1",
"add0r",
"addKr",
"addrC",
"apply",
"c1",
"c2",
"dvd0pP",
"dvdp_eq",
"eqVneq",
"eq_dvdp",
"expf_eq0",
"lead_coef_eq0",
"mulNr",
"mulf_neq0",
"mulrC",
"mulrDl",
"scaleNr",
"scalerA",
"scalerAl",
"scalerBr",
"scalerDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_addl n d m : d %| n -> (d %| m + n) = (d %| m). | Proof. by rewrite addrC; apply: dvdp_addr. Qed. | Lemma | dvdp_addl | 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... | [
"addrC",
"apply",
"dvdp_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_add d m n : d %| m -> d %| n -> d %| m + n. | Proof. by move/dvdp_addr->. Qed. | Lemma | dvdp_add | 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_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_add_eq d m n : d %| m + n -> (d %| m) = (d %| n). | Proof. by move=> ?; apply/idP/idP; [move/dvdp_addr <-| move/dvdp_addl <-]. Qed. | Lemma | dvdp_add_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",
"dvdp_addl",
"dvdp_addr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_subr d m n : d %| m -> (d %| m - n) = (d %| n). | Proof. by move=> ?; apply: dvdp_add_eq; rewrite -addrA addNr simp. Qed. | Lemma | dvdp_subr | 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",
"addrA",
"apply",
"dvdp_add_eq",
"simp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_subl d m n : d %| n -> (d %| m - n) = (d %| m). | Proof. by move/dvdp_addl<-; rewrite subrK. Qed. | Lemma | dvdp_subl | 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_addl",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_sub d m n : d %| m -> d %| n -> d %| m - n. | Proof. by move=> *; rewrite dvdp_subl. Qed. | Lemma | dvdp_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... | [
"dvdp_subl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_mod d n m : d %| n -> (d %| m) = (d %| m %% n). | Proof.
have [-> | nn0] := eqVneq n 0; first by rewrite modp0.
case: (eqVneq d 0) => [-> /dvd0pP -> | dn0]; first by rewrite modp0.
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Eq1.
apply/idP/idP; rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _.
have sn0 : c1 * c2 != 0.
by rewrite !mulf_neq0... | Lemma | dvdp_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... | [
"Eq1",
"addKr",
"addrC",
"apply",
"c1",
"c2",
"divp_eq",
"dvd0pP",
"dvdp_eq",
"eqVneq",
"eq_dvdp",
"expf_eq0",
"lead_coef",
"lead_coef_eq0",
"modp0",
"mulNr",
"mulf_neq0",
"mulrA",
"mulrC",
"mulrDl",
"scalerA",
"scalerAl",
"scalerAr",
"scalerBr",
"scalerDr",
"scalp"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_trans : transitive (@dvdp R). | Proof.
move=> n d m.
case: (eqVneq d 0) => [-> /dvd0pP -> // | dn0].
case: (eqVneq n 0) => [-> _ /dvd0pP -> // | nn0].
rewrite dvdp_eq; set c1 := _ ^+ _; set q1 := _ %/ _; move/eqP=> Hq1.
rewrite dvdp_eq; set c2 := _ ^+ _; set q2 := _ %/ _; move/eqP=> Hq2.
have sn0 : c1 * c2 != 0 by rewrite mulf_neq0 // expf_neq0 // le... | Lemma | dvdp_trans | 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",
"dvd0pP",
"dvdp",
"dvdp_eq",
"eqVneq",
"eq_dvdp",
"expf_neq0",
"lead_coef_eq0",
"mulf_neq0",
"mulrA",
"scalerA",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_mulIl p q : p %| p * q. | Proof. exact/dvdp_mulr/dvdpp. Qed. | Lemma | dvdp_mulIl | 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_mulr",
"dvdpp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_mulIr p q : q %| p * q. | Proof. exact/dvdp_mull/dvdpp. Qed. | Lemma | dvdp_mulIr | 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_mull",
"dvdpp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_mul2r r p q : r != 0 -> (p * r %| q * r) = (p %| q). | Proof.
move=> nzr.
have [-> | pn0] := eqVneq p 0.
by rewrite mul0r !dvd0p mulf_eq0 (negPf nzr) orbF.
have [-> | qn0] := eqVneq q 0; first by rewrite mul0r !dvdp0.
apply/idP/idP; last by move=> ?; rewrite dvdp_mul ?dvdpp.
rewrite dvdp_eq; set c := _ ^+ _; set x := _ %/ _; move/eqP=> Hx.
apply: (@eq_dvdp c x); first by... | Lemma | dvdp_mul2r | 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",
"dvdp0",
"dvdp_eq",
"dvdp_mul",
"dvdpp",
"eqVneq",
"eq_dvdp",
"expf_neq0",
"last",
"lead_coef_eq0",
"mul0r",
"mulIf",
"mulf_eq0",
"mulf_neq0",
"mulrA",
"scalerAl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_mul2l r p q: r != 0 -> (r * p %| r * q) = (p %| q). | Proof. by rewrite ![r * _]mulrC; apply: dvdp_mul2r. Qed. | Lemma | dvdp_mul2l | 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_mul2r",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_divpr d p q :
d %| q -> (size p < size (q %/ d)) = (size (p * d) < size q). | Proof. by move=> dv_d_q; rewrite !ltnNge leq_divpl. Qed. | Lemma | ltn_divpr | 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_divpl",
"ltnNge",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_exp d k p : 0 < k -> d %| p -> d %| (p ^+ k). | Proof. by case: k => // k _ d_dv_m; rewrite exprS dvdp_mulr. Qed. | Lemma | dvdp_exp | 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_mulr",
"exprS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_exp2l d k l : k <= l -> d ^+ k %| d ^+ l. | Proof. by move/subnK <-; rewrite exprD dvdp_mull // ?lead_coef_exp ?unitrX. Qed. | Lemma | dvdp_exp2l | 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_mull",
"exprD",
"lead_coef_exp",
"subnK",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_Pexp2l d k l : 1 < size d -> (d ^+ k %| d ^+ l) = (k <= l). | Proof.
move=> sd; case: leqP => [|gt_n_m]; first exact: dvdp_exp2l.
have dn0 : d != 0 by rewrite -size_poly_gt0; apply: ltn_trans sd.
rewrite gtNdvdp ?expf_neq0 // polySpred ?expf_neq0 // size_exp /=.
rewrite [size (d ^+ k)]polySpred ?expf_neq0 // size_exp ltnS ltn_mul2l.
by move: sd; rewrite -subn_gt0 subn1; move->.
Q... | Lemma | dvdp_Pexp2l | 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_exp2l",
"expf_neq0",
"gtNdvdp",
"leqP",
"ltnS",
"ltn_mul2l",
"ltn_trans",
"polySpred",
"size",
"size_exp",
"size_poly_gt0",
"subn1",
"subn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_exp2r p q k : p %| q -> p ^+ k %| q ^+ k. | Proof.
case: (eqVneq p 0) => [-> /dvd0pP -> // | pn0].
rewrite dvdp_eq; set c := _ ^+ _; set t := _ %/ _; move/eqP=> e.
apply: (@eq_dvdp (c ^+ k) (t ^+ k)); first by rewrite !expf_neq0 ?lead_coef_eq0.
by rewrite -exprMn -exprZn; congr (_ ^+ k).
Qed. | Lemma | dvdp_exp2r | 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",
"dvdp_eq",
"eqVneq",
"eq_dvdp",
"expf_neq0",
"exprMn",
"exprZn",
"lead_coef_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_exp_sub p q k l: p != 0 ->
(p ^+ k %| q * p ^+ l) = (p ^+ (k - l) %| q). | Proof.
move=> pn0; case: (leqP k l)=> [|/ltnW] hkl.
move: (hkl); rewrite -subn_eq0; move/eqP->; rewrite expr0 dvd1p.
exact/dvdp_mull/dvdp_exp2l.
by rewrite -[in LHS](subnK hkl) exprD dvdp_mul2r // expf_eq0 (negPf pn0) andbF.
Qed. | Lemma | dvdp_exp_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... | [
"dvd1p",
"dvdp_exp2l",
"dvdp_mul2r",
"dvdp_mull",
"expf_eq0",
"expr0",
"exprD",
"leqP",
"ltnW",
"subnK",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_XsubCl p x : (('X - x%:P) %| p) = root p x. | Proof. by rewrite dvdpE; apply: Ring.rdvdp_XsubCl. Qed. | Lemma | dvdp_XsubCl | 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... | [
"Ring",
"apply",
"dvdpE",
"rdvdp_XsubCl",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_dvdp p q x : p %| q -> root p x -> root q x. | Proof. by rewrite -!dvdp_XsubCl => /[swap]; exact: dvdp_trans. Qed. | Lemma | root_dvdp | 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_XsubCl",
"dvdp_trans",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyXsubCP p x : reflect (p.[x] = 0) (('X - x%:P) %| p). | Proof. by rewrite dvdpE; apply: Ring.polyXsubCP. Qed. | Lemma | polyXsubCP | 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... | [
"Ring",
"apply",
"dvdpE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_div_XsubC p c :
(p == (p %/ ('X - c%:P)) * ('X - c%:P)) = ('X - c%:P %| p). | Proof. by rewrite dvdp_eq lead_coefXsubC expr1n scale1r. Qed. | Lemma | eqp_div_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... | [
"dvdp_eq",
"expr1n",
"lead_coefXsubC",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_factor_theorem p x : root p x = (('X - x%:P) %| p). | Proof. by rewrite dvdp_XsubCl. Qed. | Lemma | root_factor_theorem | 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_XsubCl",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_roots_dvdp p rs : all (root p) rs -> uniq_roots rs ->
(\prod_(z <- rs) ('X - z%:P)) %| p. | Proof.
move=> rrs; case/(uniq_roots_prod_XsubC rrs)=> q ->.
by apply: dvdp_mull; rewrite // (eqP (monic_prod_XsubC _)) unitr1.
Qed. | Lemma | uniq_roots_dvdp | 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... | [
"all",
"apply",
"dvdp_mull",
"monic_prod_XsubC",
"root",
"uniq_roots",
"uniq_roots_prod_XsubC",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_bigmul x (ps : seq {poly R}) :
~~root (\big[*%R/1]_(p <- ps) p) x = all (fun p => ~~ root p x) ps. | Proof.
elim: ps => [|p ps ihp]; first by rewrite big_nil root1.
by rewrite big_cons /= rootM negb_or ihp.
Qed. | Lemma | root_bigmul | 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... | [
"all",
"big_cons",
"big_nil",
"poly",
"root",
"root1",
"rootM",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqpP m n :
reflect (exists2 c12, (c12.1 != 0) && (c12.2 != 0) & c12.1 *: m = c12.2 *: n)
(m %= n). | Proof.
apply: (iffP idP) => [| [[c1 c2]/andP[nz_c1 nz_c2 eq_cmn]]]; last first.
rewrite /eqp (@eq_dvdp c2 c1%:P) -?eq_cmn ?mul_polyC // (@eq_dvdp c1 c2%:P)//.
by rewrite eq_cmn mul_polyC.
case: (eqVneq m 0) => [-> /andP [/dvd0pP -> _] | m_nz].
by exists (1, 1); rewrite ?scaler0 // oner_eq0.
case: (eqVneq n 0) => ... | Lemma | eqpP | 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",
"dvd0pP",
"dvdp_eq",
"eqVneq",
"eq_dvdp",
"eqp",
"expf_eq0",
"last",
"lead_coef_eq0",
"leq_addl",
"leq_eqVlt",
"ltnS",
"mulIf",
"mul_polyC",
"mulf_eq0",
"mulf_neq0",
"mulrAC",
"mulrC",
"oner_eq0",
"polySpred",
"scaler0",
"scalerA",
"scalerAr",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_eq p q: p %= q -> (lead_coef q) *: p = (lead_coef p) *: q. | Proof.
move=> /eqpP [[c1 c2] /= /andP [nz_c1 nz_c2]] eq.
have/(congr1 lead_coef) := eq; rewrite !lead_coefZ.
move=> eqC; apply/(@mulfI _ c2%:P); rewrite ?polyC_eq0 //.
by rewrite !mul_polyC scalerA -eqC mulrC -scalerA eq !scalerA mulrC.
Qed. | Lemma | eqp_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",
"c1",
"c2",
"eqC",
"eqpP",
"lead_coef",
"lead_coefZ",
"mul_polyC",
"mulfI",
"mulrC",
"polyC_eq0",
"scalerA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqpxx : reflexive (@eqp R). | Proof. by move=> p; rewrite /eqp dvdpp. Qed. | Lemma | eqpxx | 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... | [
"dvdpp",
"eqp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqpW p q : p = q -> p %= q. | Proof. by move->; rewrite eqpxx. Qed. | Lemma | eqpW | 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... | [
"eqpxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_sym : symmetric (@eqp R). | Proof. by move=> p q; rewrite /eqp andbC. Qed. | Lemma | eqp_sym | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_trans : transitive (@eqp R). | Proof.
move=> p q r; case/andP=> Dp pD; case/andP=> Dq qD.
by rewrite /eqp (dvdp_trans Dp) // (dvdp_trans qD).
Qed. | Lemma | eqp_trans | 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_trans",
"eqp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_ltrans : left_transitive (@eqp R). | Proof. exact: sym_left_transitive eqp_sym eqp_trans. Qed. | Lemma | eqp_ltrans | 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",
"eqp_sym",
"eqp_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_rtrans : right_transitive (@eqp R). | Proof. exact: sym_right_transitive eqp_sym eqp_trans. Qed. | Lemma | eqp_rtrans | 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",
"eqp_sym",
"eqp_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp0 p : (p %= 0) = (p == 0). | Proof. by apply/idP/eqP => [/andP [_ /dvd0pP] | -> //]. Qed. | Lemma | eqp0 | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp01 : (0 %= (1 : {poly R})) = false. | Proof. by rewrite eqp_sym eqp0 oner_eq0. Qed. | Lemma | eqp01 | 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... | [
"eqp0",
"eqp_sym",
"oner_eq0",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_scale p c : c != 0 -> c *: p %= p. | Proof.
move=> c0; apply/eqpP; exists (1, c); first by rewrite c0 oner_eq0.
by rewrite scale1r.
Qed. | Lemma | eqp_scale | 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",
"c0",
"eqpP",
"oner_eq0",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_size p q : p %= q -> size p = size q. | Proof.
have [->|Eq] := eqVneq q 0; first by rewrite eqp0; move/eqP->.
rewrite eqp_sym; have [->|Ep] := eqVneq p 0; first by rewrite eqp0; move/eqP->.
by case/andP => Dp Dq; apply: anti_leq; rewrite !dvdp_leq.
Qed. | Lemma | eqp_size | 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... | [
"anti_leq",
"apply",
"dvdp_leq",
"eqVneq",
"eqp0",
"eqp_sym",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_poly_eq1 p : (size p == 1) = (p %= 1). | Proof.
apply/size_poly1P/idP=> [[c cn0 ep] |].
by apply/eqpP; exists (1, c); rewrite ?oner_eq0 // alg_polyC scale1r.
by move/eqp_size; rewrite size_poly1; move/eqP/size_poly1P.
Qed. | Lemma | size_poly_eq1 | 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... | [
"alg_polyC",
"apply",
"eqpP",
"eqp_size",
"oner_eq0",
"scale1r",
"size",
"size_poly1",
"size_poly1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyXsubC_eqp1 (x : R) : ('X - x%:P %= 1) = false. | Proof. by rewrite -size_poly_eq1 size_XsubC. Qed. | Lemma | polyXsubC_eqp1 | 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... | [
"size_XsubC",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_eqp1 p q : p %| q -> q %= 1 -> p %= 1. | Proof.
move=> dpq hq.
have sizeq : size q == 1 by rewrite size_poly_eq1.
have n0q : q != 0 by case: eqP hq => // ->; rewrite eqp01.
rewrite -size_poly_eq1 eqn_leq -{1}(eqP sizeq) dvdp_leq //= size_poly_gt0.
by apply/eqP => p0; move: dpq n0q; rewrite p0 dvd0p => ->.
Qed. | Lemma | dvdp_eqp1 | 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",
"dvdp_leq",
"eqn_leq",
"eqp01",
"p0",
"size",
"size_poly_eq1",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_dvdr q p d: p %= q -> (d %| p) = (d %| q). | Proof.
suff Hmn m n: m %= n -> (d %| m) -> (d %| n).
by move=> mn; apply/idP/idP; apply: Hmn=> //; rewrite eqp_sym.
by rewrite /eqp; case/andP=> pq qp dp; apply: (dvdp_trans dp).
Qed. | Lemma | eqp_dvdr | 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_trans",
"eqp",
"eqp_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_dvdl d2 d1 p : d1 %= d2 -> (d1 %| p) = (d2 %| p). | suff Hmn m n: m %= n -> (m %| p) -> (n %| p).
by move=> ?; apply/idP/idP; apply: Hmn; rewrite // eqp_sym.
by rewrite /eqp; case/andP=> dd' d'd dp; apply: (dvdp_trans d'd).
Qed. | Lemma | eqp_dvdl | 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_trans",
"eqp",
"eqp_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpZr c m n : c != 0 -> (m %| c *: n) = (m %| n). | Proof. by move=> cn0; exact/eqp_dvdr/eqp_scale. Qed. | Lemma | dvdpZr | 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_dvdr",
"eqp_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpZl c m n : c != 0 -> (c *: m %| n) = (m %| n). | Proof. by move=> cn0; exact/eqp_dvdl/eqp_scale. Qed. | Lemma | dvdpZl | 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_dvdl",
"eqp_scale"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpNl d p : ((- d) %| p) = (d %| p). | Proof.
by rewrite -scaleN1r; apply/eqp_dvdl/eqp_scale; rewrite oppr_eq0 oner_neq0.
Qed. | Lemma | dvdpNl | 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",
"eqp_dvdl",
"eqp_scale",
"oner_neq0",
"oppr_eq0",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpNr d p : (d %| (- p)) = (d %| p). | Proof. by apply: eqp_dvdr; rewrite -scaleN1r eqp_scale ?oppr_eq0 ?oner_eq0. Qed. | Lemma | dvdpNr | 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",
"eqp_dvdr",
"eqp_scale",
"oner_eq0",
"oppr_eq0",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_mul2r r p q : r != 0 -> (p * r %= q * r) = (p %= q). | Proof. by move=> nz_r; rewrite /eqp !dvdp_mul2r. Qed. | Lemma | eqp_mul2r | 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_mul2r",
"eqp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_mul2l r p q: r != 0 -> (r * p %= r * q) = (p %= q). | Proof. by move=> nz_r; rewrite /eqp !dvdp_mul2l. Qed. | Lemma | eqp_mul2l | 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_mul2l",
"eqp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_mull r p q: q %= r -> p * q %= p * r. | Proof.
case/eqpP=> [[c d]] /andP [c0 d0 e]; apply/eqpP; exists (c, d); rewrite ?c0 //.
by rewrite scalerAr e -scalerAr.
Qed. | Lemma | eqp_mull | 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",
"c0",
"eqpP",
"scalerAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_mulr q p r : p %= q -> p * r %= q * r. | Proof. by move=> epq; rewrite ![_ * r]mulrC eqp_mull. Qed. | Lemma | eqp_mulr | 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_mull",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_exp p q k : p %= q -> p ^+ k %= q ^+ k. | Proof.
move=> pq; elim: k=> [|k ihk]; first by rewrite !expr0 eqpxx.
by rewrite !exprS (@eqp_trans (q * p ^+ k)) // (eqp_mulr, eqp_mull).
Qed. | Lemma | eqp_exp | 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_mull",
"eqp_mulr",
"eqp_trans",
"eqpxx",
"expr0",
"exprS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyC_eqp1 (c : R) : (c%:P %= 1) = (c != 0). | Proof.
apply/eqpP/idP => [[[x y]] |nc0] /=.
case: (eqVneq c) => [->|] //= /andP [_] /negPf <- /eqP.
by rewrite alg_polyC scaler0 eq_sym polyC_eq0.
exists (1, c); first by rewrite nc0 /= oner_neq0.
by rewrite alg_polyC scale1r.
Qed. | Lemma | polyC_eqp1 | 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... | [
"alg_polyC",
"apply",
"eqVneq",
"eq_sym",
"eqpP",
"oner_neq0",
"polyC_eq0",
"scale1r",
"scaler0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdUp d p: d %= 1 -> d %| p. | Proof. by move/eqp_dvdl->; rewrite dvd1p. Qed. | Lemma | dvdUp | 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... | [
"dvd1p",
"eqp_dvdl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_size_eqp p q : p %| q -> (size p == size q) = (p %= q). | Proof.
move=> pq; apply/idP/idP; last by move/eqp_size->.
have [->|Hq] := eqVneq q 0; first by rewrite size_poly0 size_poly_eq0 eqp0.
have [->|Hp] := eqVneq p 0.
by rewrite size_poly0 eq_sym size_poly_eq0 eqp_sym eqp0.
move: pq; rewrite dvdp_eq; set c := _ ^+ _; set x := _ %/ _; move/eqP=> eqpq.
have /= := congr1 (si... | Lemma | dvdp_size_eqp | 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... | [
"add1n",
"addnS",
"apply",
"dvdp_eq",
"eqVneq",
"eq_sym",
"eqn_add2r",
"eqp0",
"eqpP",
"eqp_scale",
"eqp_size",
"eqp_sym",
"expf_neq0",
"last",
"lead_coef_eq0",
"mul0r",
"mul_polyC",
"p0",
"scale_poly_eq0",
"size",
"size_mul",
"size_poly0",
"size_poly1P",
"size_poly_eq0... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_root p q : p %= q -> root p =1 root q. | Proof.
move/eqpP=> [[c d]] /andP [c0 d0 e] x; move/negPf:c0=>c0; move/negPf:d0=>d0.
by rewrite rootE -[_==_]orFb -c0 -mulf_eq0 -hornerZ e hornerZ mulf_eq0 d0.
Qed. | Lemma | eqp_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... | [
"c0",
"eqpP",
"hornerZ",
"mulf_eq0",
"root",
"rootE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_rmod_mod p q : rmodp p q %= modp p q. | Proof.
rewrite modpE eqp_sym; case: ifP => ulcq //.
apply: eqp_scale; rewrite invr_eq0 //.
by apply: expf_neq0; apply: contraTneq ulcq => ->; rewrite unitr0.
Qed. | Lemma | eqp_rmod_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... | [
"apply",
"contraTneq",
"eqp_scale",
"eqp_sym",
"expf_neq0",
"invr_eq0",
"modp",
"modpE",
"rmodp",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_rdiv_div p q : rdivp p q %= divp p q. | Proof.
rewrite divpE eqp_sym; case: ifP=> ulcq//; apply: eqp_scale; rewrite invr_eq0//.
by apply: expf_neq0; apply: contraTneq ulcq => ->; rewrite unitr0.
Qed. | Lemma | eqp_rdiv_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",
"divp",
"divpE",
"eqp_scale",
"eqp_sym",
"expf_neq0",
"invr_eq0",
"rdivp",
"unitr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvd_eqp_divl d p q (dvd_dp : d %| q) (eq_pq : p %= q) :
p %/ d %= q %/ d. | Proof.
case: (eqVneq q 0) eq_pq=> [->|q_neq0]; first by rewrite eqp0=> /eqP->.
have d_neq0: d != 0 by apply: contraTneq dvd_dp=> ->; rewrite dvd0p.
move=> eq_pq; rewrite -(@eqp_mul2r d) // !divpK // ?(eqp_dvdr _ eq_pq) //.
rewrite (eqp_ltrans (eqp_scale _ _)) ?lc_expn_scalp_neq0 //.
by rewrite (eqp_rtrans (eqp_scale _ ... | Lemma | dvd_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",
"contraTneq",
"divpK",
"dvd0p",
"eqVneq",
"eqp0",
"eqp_dvdr",
"eqp_ltrans",
"eqp_mul2r",
"eqp_rtrans",
"eqp_scale",
"lc_expn_scalp_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp p q | :=
let: (p1, q1) := if size p < size q then (q, p) else (p, q) in
if p1 == 0 then q1 else
let fix loop (n : nat) (pp qq : {poly R}) {struct n} :=
let rr := modp pp qq in
if rr == 0 then qq else
if n is n1.+1 then loop n1 qq rr else rr in
loop (size p1) p1 q1. | Definition | gcdp | 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... | [
"modp",
"nat",
"poly",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcd0p : left_id 0 gcdp. | Proof.
move=> p; rewrite /gcdp size_poly0 size_poly_gt0 if_neg.
case: ifP => /= [_ | nzp]; first by rewrite eqxx.
by rewrite polySpred !(modp0, nzp) //; case: _.-1 => [|m]; rewrite mod0p eqxx.
Qed. | Lemma | gcd0p | 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... | [
"eqxx",
"gcdp",
"mod0p",
"modp0",
"polySpred",
"size_poly0",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp0 : right_id 0 gcdp. | Proof.
move=> p; have:= gcd0p p; rewrite /gcdp size_poly0 size_poly_gt0.
by case: eqVneq => //= ->; rewrite eqxx.
Qed. | Lemma | gcdp0 | 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",
"gcd0p",
"gcdp",
"size_poly0",
"size_poly_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdpE p q :
gcdp p q = if size p < size q
then gcdp (modp q p) p else gcdp (modp p q) q. | Proof.
pose gcdpE_rec := fix gcdpE_rec (n : nat) (pp qq : {poly R}) {struct n} :=
let rr := modp pp qq in
if rr == 0 then qq else
if n is n1.+1 then gcdpE_rec n1 qq rr else rr.
have Irec: forall k l p q, size q <= k -> size q <= l
-> size q < size p -> gcdpE_rec k p q = gcdpE_rec l p q.
+ elim=> [|m Hrec... | Lemma | gcdpE | 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",
"eqxx",
"gcd0p",
"gcdp",
"gcdp0",
"last",
"leqW",
"leq_trans",
"ltnP",
"ltnS",
"ltnW",
"ltn_modp",
"mod0p",
"modp",
"modp0",
"nat",
"poly",
"polySpred",
"size",
"size_poly0",
"size_poly_gt0",
"size_poly_leq0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_gcd1p p : size (gcdp 1 p) = 1. | Proof.
rewrite gcdpE size_polyC oner_eq0 /= modp1; have [|/size1_polyC ->] := ltnP.
by rewrite gcd0p size_polyC oner_eq0.
have [->|p00] := eqVneq p`_0 0; first by rewrite modp0 gcdp0 size_poly1.
by rewrite modpC // gcd0p size_polyC p00.
Qed. | Lemma | size_gcd1p | 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",
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"ltnP",
"modp0",
"modp1",
"modpC",
"oner_eq0",
"size",
"size1_polyC",
"size_poly1",
"size_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_gcdp1 p : size (gcdp p 1) = 1. | Proof.
rewrite gcdpE size_polyC oner_eq0 /= modp1 ltnS; case: leqP.
by move/size_poly_leq0P->; rewrite gcdp0 modp0 size_polyC oner_eq0.
by rewrite gcd0p size_polyC oner_eq0.
Qed. | Lemma | size_gcdp1 | 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... | [
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"leqP",
"ltnS",
"modp0",
"modp1",
"oner_eq0",
"size",
"size_polyC",
"size_poly_leq0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdpp : idempotent_op gcdp. | Proof. by move=> p; rewrite gcdpE ltnn modpp gcd0p. Qed. | Lemma | gcdpp | 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... | [
"gcd0p",
"gcdp",
"gcdpE",
"idempotent_op",
"ltnn",
"modpp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gcdlr p q : (gcdp p q %| p) && (gcdp p q %| q). | Proof.
have [r] := ubnP (minn (size q) (size p)); elim: r => // r IHr in p q *.
have [-> | nz_p] := eqVneq p 0; first by rewrite gcd0p dvdpp andbT.
have [-> | nz_q] := eqVneq q 0; first by rewrite gcdp0 dvdpp /=.
rewrite ltnS gcdpE; case: leqP => [le_pq | lt_pq] le_qr.
suffices /IHr/andP[E1 E2]: minn (size q) (size (... | Lemma | dvdp_gcdlr | 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_mod",
"dvdpp",
"eqVneq",
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"gtn_min",
"leqP",
"leq_trans",
"ltnS",
"ltn_modp",
"minn",
"nz_p",
"size",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gcdl p q : gcdp p q %| p. | Proof. by case/andP: (dvdp_gcdlr p q). Qed. | Lemma | dvdp_gcdl | 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_gcdlr",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gcdr p q :gcdp p q %| q. | Proof. by case/andP: (dvdp_gcdlr p q). Qed. | Lemma | dvdp_gcdr | 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_gcdlr",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_gcdpl p q : p != 0 -> size (gcdp p q) <= size p. | Proof. by move=> pn0; move: (dvdp_gcdl p q); apply: dvdp_leq. Qed. | Lemma | leq_gcdpl | 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_gcdl",
"dvdp_leq",
"gcdp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_gcdpr p q : q != 0 -> size (gcdp p q) <= size q. | Proof. by move=> qn0; move: (dvdp_gcdr p q); apply: dvdp_leq. Qed. | Lemma | leq_gcdpr | 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_gcdr",
"dvdp_leq",
"gcdp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gcd p m n : (p %| gcdp m n) = (p %| m) && (p %| n). | Proof.
apply/idP/andP=> [dv_pmn | []].
by rewrite ?(dvdp_trans dv_pmn) ?dvdp_gcdl ?dvdp_gcdr.
have [r] := ubnP (minn (size n) (size m)); elim: r => // r IHr in m n *.
have [-> | nz_m] := eqVneq m 0; first by rewrite gcd0p.
have [-> | nz_n] := eqVneq n 0; first by rewrite gcdp0.
rewrite gcdpE ltnS; case: leqP => [le_n... | Lemma | dvdp_gcd | 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_gcdl",
"dvdp_gcdr",
"dvdp_mod",
"dvdp_trans",
"eqVneq",
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"gtn_min",
"last",
"leqP",
"leq_trans",
"ltnS",
"ltn_modp",
"minn",
"size",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdpC p q : gcdp p q %= gcdp q p. | Proof. by rewrite /eqp !dvdp_gcd !dvdp_gcdl !dvdp_gcdr. Qed. | Lemma | gcdpC | 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_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"eqp",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcd1p p : gcdp 1 p %= 1. | Proof.
rewrite -size_poly_eq1 gcdpE size_poly1; case: ltnP.
by rewrite modp1 gcd0p size_poly1 eqxx.
move/size1_polyC=> e; rewrite e.
have [->|p00] := eqVneq p`_0 0; first by rewrite modp0 gcdp0 size_poly1.
by rewrite modpC // gcd0p size_polyC p00.
Qed. | Lemma | gcd1p | 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",
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"ltnP",
"modp0",
"modp1",
"modpC",
"size1_polyC",
"size_poly1",
"size_polyC",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp1 p : gcdp p 1 %= 1. | Proof. by rewrite (eqp_ltrans (gcdpC _ _)) gcd1p. Qed. | Lemma | gcdp1 | 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_ltrans",
"gcd1p",
"gcdp",
"gcdpC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_addl_mul p q r: gcdp r (p * r + q) %= gcdp r q. | Proof.
suff h m n d : gcdp d n %| gcdp d (m * d + n).
apply/andP; split => //.
by rewrite {2}(_: q = (-p) * r + (p * r + q)) ?H // mulNr addKr.
by rewrite dvdp_gcd dvdp_gcdl /= dvdp_addr ?dvdp_gcdr ?dvdp_mull ?dvdp_gcdl.
Qed. | Lemma | gcdp_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... | [
"addKr",
"apply",
"dvdp_addr",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_mull",
"gcdp",
"mulNr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_addl m n : gcdp m (m + n) %= gcdp m n. | Proof. by rewrite -[m in m + _]mul1r gcdp_addl_mul. Qed. | Lemma | gcdp_addl | 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... | [
"gcdp",
"gcdp_addl_mul",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_addr m n : gcdp m (n + m) %= gcdp m n. | Proof. by rewrite addrC gcdp_addl. Qed. | Lemma | gcdp_addr | 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... | [
"addrC",
"gcdp",
"gcdp_addl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_mull m n : gcdp n (m * n) %= n. | Proof.
have [-> | nn0] := eqVneq n 0; first by rewrite gcd0p mulr0 eqpxx.
have [-> | mn0] := eqVneq m 0; first by rewrite mul0r gcdp0 eqpxx.
rewrite gcdpE modp_mull gcd0p size_mul //; case: leqP; last by rewrite eqpxx.
rewrite (polySpred mn0) addSn /= -[leqRHS]add0n leq_add2r -ltnS.
rewrite -polySpred //= leq_eqVlt ltn... | Lemma | gcdp_mull | 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... | [
"add0n",
"addSn",
"apply",
"dvdpZl",
"eqVneq",
"eqp_scale",
"eqpxx",
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"last",
"leqP",
"leqRHS",
"leq_add2r",
"leq_eqVlt",
"ltnS",
"modp_eq0P",
"modp_mull",
"mul0r",
"mul_polyC",
"mulr0",
"polySpred",
"size_mul",
"size_poly1P",
"size... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_mulr m n : gcdp n (n * m) %= n. | Proof. by rewrite mulrC gcdp_mull. Qed. | Lemma | gcdp_mulr | 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... | [
"gcdp",
"gcdp_mull",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_scalel c m n : c != 0 -> gcdp (c *: m) n %= gcdp m n. | Proof.
move=> cn0; rewrite /eqp dvdp_gcd [gcdp m n %| _]dvdp_gcd !dvdp_gcdr !andbT.
apply/andP; split; last first.
by apply: dvdp_trans (dvdp_gcdl _ _) _; rewrite dvdpZr.
by apply: dvdp_trans (dvdp_gcdl _ _) _; rewrite dvdpZl.
Qed. | Lemma | gcdp_scalel | 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",
"dvdpZl",
"dvdpZr",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"dvdp_trans",
"eqp",
"gcdp",
"last",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_scaler c m n : c != 0 -> gcdp m (c *: n) %= gcdp m n. | Proof.
move=> cn0; apply: eqp_trans (gcdpC _ _) _.
by apply: eqp_trans (gcdp_scalel _ _ _) _ => //; apply: gcdpC.
Qed. | Lemma | gcdp_scaler | 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",
"eqp_trans",
"gcdp",
"gcdpC",
"gcdp_scalel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gcd_idl m n : m %| n -> gcdp m n %= m. | Proof.
have [-> | mn0] := eqVneq m 0.
by rewrite dvd0p => /eqP ->; rewrite gcdp0 eqpxx.
rewrite dvdp_eq; move/eqP/(f_equal (gcdp m)) => h.
apply: eqp_trans (gcdp_mull (n %/ m) _).
by rewrite -h eqp_sym gcdp_scaler // expf_neq0 // lead_coef_eq0.
Qed. | Lemma | dvdp_gcd_idl | 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",
"dvdp_eq",
"eqVneq",
"eqp_sym",
"eqp_trans",
"eqpxx",
"expf_neq0",
"gcdp",
"gcdp0",
"gcdp_mull",
"gcdp_scaler",
"lead_coef_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_gcd_idr m n : n %| m -> gcdp m n %= n. | Proof. by move/dvdp_gcd_idl; exact/eqp_trans/gcdpC. Qed. | Lemma | dvdp_gcd_idr | 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_gcd_idl",
"eqp_trans",
"gcdp",
"gcdpC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_exp p k l : gcdp (p ^+ k) (p ^+ l) %= p ^+ minn k l. | Proof.
case: leqP => [|/ltnW] /subnK <-; rewrite exprD; first exact: gcdp_mull.
exact/(eqp_trans (gcdpC _ _))/gcdp_mull.
Qed. | Lemma | gcdp_exp | 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_trans",
"exprD",
"gcdp",
"gcdpC",
"gcdp_mull",
"leqP",
"ltnW",
"minn",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_eq0 p q : (gcdp p q == 0) = (p == 0) && (q == 0). | Proof.
apply/idP/idP; last by case/andP => /eqP -> /eqP ->; rewrite gcdp0.
have h m n: gcdp m n == 0 -> (m == 0).
by rewrite -(dvd0p m); move/eqP<-; rewrite dvdp_gcdl.
by move=> ?; rewrite (h _ q) // (h _ p) // -eqp0 (eqp_ltrans (gcdpC _ _)) eqp0.
Qed. | Lemma | gcdp_eq0 | 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",
"dvdp_gcdl",
"eqp0",
"eqp_ltrans",
"gcdp",
"gcdp0",
"gcdpC",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_gcdr p q r : q %= r -> gcdp p q %= gcdp p r. | Proof.
move=> eqr; rewrite /eqp !(dvdp_gcd, dvdp_gcdl, andbT) /=.
by rewrite -(eqp_dvdr _ eqr) dvdp_gcdr (eqp_dvdr _ eqr) dvdp_gcdr.
Qed. | Lemma | eqp_gcdr | 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_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"eqp",
"eqp_dvdr",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_gcdl r p q : p %= q -> gcdp p r %= gcdp q r. | Proof.
move=> eqr; rewrite /eqp !(dvdp_gcd, dvdp_gcdr, andbT) /=.
by rewrite -(eqp_dvdr _ eqr) dvdp_gcdl (eqp_dvdr _ eqr) dvdp_gcdl.
Qed. | Lemma | eqp_gcdl | 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_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"eqp",
"eqp_dvdr",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_gcd p1 p2 q1 q2 : p1 %= p2 -> q1 %= q2 -> gcdp p1 q1 %= gcdp p2 q2. | Proof. move=> e1 e2; exact: eqp_trans (eqp_gcdr _ e2) (eqp_gcdl _ e1). Qed. | Lemma | eqp_gcd | 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_gcdl",
"eqp_gcdr",
"eqp_trans",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqp_rgcd_gcd p q : rgcdp p q %= gcdp p q. | Proof.
move: {2}(minn (size p) (size q)) (leqnn (minn (size p) (size q))) => n.
elim: n p q => [p q|n ihn p q hs].
rewrite leqn0; case: ltnP => _; rewrite size_poly_eq0; move/eqP->.
by rewrite gcd0p rgcd0p eqpxx.
by rewrite gcdp0 rgcdp0 eqpxx.
have [-> | pn0] := eqVneq p 0; first by rewrite gcd0p rgcd0p eqpxx.
... | Lemma | eqp_rgcd_gcd | 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",
"eqp_gcdl",
"eqp_rmod_mod",
"eqp_size",
"eqp_trans",
"eqpxx",
"gcd0p",
"gcdp",
"gcdp0",
"gcdpE",
"geq_min",
"leq_trans",
"leqn0",
"leqnn",
"ltnP",
"ltnS",
"ltn_modp",
"minn",
"rgcd0p",
"rgcdp",
"rgcdp0",
"rgcdpE",
"size",
"size_poly_eq0",
"sp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_modl m n : gcdp (m %% n) n %= gcdp m n. | Proof.
have [/modp_small -> // | lenm] := ltnP (size m) (size n).
by rewrite (gcdpE m n) ltnNge lenm.
Qed. | Lemma | gcdp_modl | 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... | [
"gcdp",
"gcdpE",
"ltnNge",
"ltnP",
"modp_small",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_modr m n : gcdp m (n %% m) %= gcdp m n. | Proof.
apply: eqp_trans (gcdpC _ _); apply: eqp_trans (gcdp_modl _ _); exact: gcdpC.
Qed. | Lemma | gcdp_modr | 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",
"eqp_trans",
"gcdp",
"gcdpC",
"gcdp_modl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gcdp_def d m n :
d %| m -> d %| n -> (forall d', d' %| m -> d' %| n -> d' %| d) ->
gcdp m n %= d. | Proof.
move=> dm dn h; rewrite /eqp dvdp_gcd dm dn !andbT.
by apply: h; rewrite (dvdp_gcdl, dvdp_gcdr).
Qed. | Lemma | gcdp_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... | [
"apply",
"dvdp_gcd",
"dvdp_gcdl",
"dvdp_gcdr",
"eqp",
"gcdp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep p q | := size (gcdp p q) == 1%N. | Definition | coprimep | 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... | [
"gcdp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_size_gcd p q : coprimep p q -> size (gcdp p q) = 1. | Proof. by rewrite /coprimep=> /eqP. Qed. | Lemma | coprimep_size_gcd | 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... | [
"coprimep",
"gcdp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimep_def p q : coprimep p q = (size (gcdp p q) == 1). | Proof. done. Qed. | Lemma | coprimep_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... | [
"coprimep",
"gcdp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepZl c m n : c != 0 -> coprimep (c *: m) n = coprimep m n. | Proof. by move=> ?; rewrite !coprimep_def (eqp_size (gcdp_scalel _ _ _)). Qed. | Lemma | coprimepZl | 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... | [
"coprimep",
"coprimep_def",
"eqp_size",
"gcdp_scalel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepZr c m n: c != 0 -> coprimep m (c *: n) = coprimep m n. | Proof. by move=> ?; rewrite !coprimep_def (eqp_size (gcdp_scaler _ _ _)). Qed. | Lemma | coprimepZr | 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... | [
"coprimep",
"coprimep_def",
"eqp_size",
"gcdp_scaler"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimepp p : coprimep p p = (size p == 1). | Proof. by rewrite coprimep_def gcdpp. Qed. | Lemma | 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... | [
"coprimep",
"coprimep_def",
"gcdpp",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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