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 |
|---|---|---|---|---|---|---|---|---|---|---|
qpoly_inv_out (p : {poly %/ h}) : ~~ coprimep hQ p -> qpoly_inv p = p. | Proof. by rewrite /qpoly_inv => /negPf->. Qed. | Lemma | qpoly_inv_out | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"coprimep",
"hQ",
"poly",
"qpoly_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irreducible_poly_coprime (A : idomainType) (p q : {poly A}) :
irreducible_poly p -> coprimep p q = ~~(p %| q)%R. | Proof.
case => H1 H2; apply/coprimepP/negP.
move=> sPq H.
by have := sPq p (dvdpp _) H; rewrite -size_poly_eq1; case: size H1 => [|[]].
move=> pNDq d dDp dPq.
rewrite -size_poly_eq1; case: eqP => // /eqP /(H2 _) => /(_ dDp) dEp.
by case: pNDq; rewrite -(eqp_dvdl _ dEp).
Qed. | Lemma | irreducible_poly_coprime | algebra | algebra/qpoly.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"poly",
"polydiv",
"matrix",
"mxalgebra",
"... | [
"apply",
"coprimep",
"coprimepP",
"dvdpp",
"eqp_dvdl",
"irreducible_poly",
"poly",
"size",
"size_poly_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr | := Num.sg. | Notation | sgr | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat : Set | := Rat {
valq : (int * int);
_ : (0 < valq.2) && coprime `|valq.1| `|valq.2|
}. | Record | rat | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"coprime",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratz (n : int) | := @Rat (n, 1) (coprimen1 _). | Definition | ratz | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"coprimen1",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_isSub | := Eval hnf in [isSub for valq]. | Definition | rat_isSub | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | Coercion ratz (n : int) := @Rat (n, 1) (coprimen1 _). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
numq x | := (valq x).1. | Definition | numq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq x | := (valq x).2. | Definition | denq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_gt0 x : 0 < denq x. | Proof. by rewrite /denq; case: x=> [[a b] /= /andP []]. Qed. | Lemma | denq_gt0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_ge0 x | := ltW (denq_gt0 x). | Definition | denq_ge0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq_gt0",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_lt0 x : (denq x < 0) = false. | Proof. by rewrite lt_gtF. Qed. | Lemma | denq_lt0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"lt_gtF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_neq0 x : denq x != 0. | Proof. by rewrite /denq gt_eqF ?denq_gt0. Qed. | Lemma | denq_neq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"denq_gt0",
"gt_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_eq0 x : (denq x == 0) = false. | Proof. exact: negPf (denq_neq0 _). Qed. | Lemma | denq_eq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"denq_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_num_den x : coprime `|numq x| `|denq x|. | Proof. by rewrite /numq /denq; case: x=> [[a b] /= /andP []]. Qed. | Lemma | coprime_num_den | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"coprime",
"denq",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
RatK x P : @Rat (numq x, denq x) P = x. | Proof. by move: x P => [[a b] P'] P; apply: val_inj. Qed. | Fact | RatK | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"apply",
"denq",
"numq",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_subdef x | :=
if x.2 != 0 then let g := gcdn `|x.1| `|x.2| in
((-1) ^ ((x.2 < 0) (+) (x.1 < 0)) * (`|x.1| %/ g)%:Z, (`|x.2| %/ g)%:Z)
else (0, 1). | Definition | fracq_subdef | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"gcdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_opt_subdef (x : int * int) | :=
if (0 < x.2) && coprime `|x.1| `|x.2| then x else fracq_subdef x. | Definition | fracq_opt_subdef | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"coprime",
"fracq_subdef",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_opt_subdefE x : fracq_opt_subdef x = fracq_subdef x. | Proof.
rewrite /fracq_opt_subdef; case: ifP => //; case: x => n d /= /andP[d_gt0 cnd].
rewrite /fracq_subdef gt_eqF//= lt_gtF//= (eqP cnd) !divn1 abszEsg gtz0_abs//.
rewrite mulrA sgz_def mulrnAr -signr_addb addbb expr0.
by have [->|] := eqVneq n 0; rewrite (mulr0, mul1r).
Qed. | Lemma | fracq_opt_subdefE | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszEsg",
"d_gt0",
"divn1",
"eqVneq",
"expr0",
"fracq_opt_subdef",
"fracq_subdef",
"gt_eqF",
"gtz0_abs",
"lt_gtF",
"mul1r",
"mulr0",
"mulrA",
"mulrnAr",
"sgz_def",
"signr_addb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_subproof x (y := fracq_opt_subdef x) :
(0 < y.2) && (coprime `|y.1| `|y.2|). | Proof.
rewrite {}/y fracq_opt_subdefE /=; have [] //= := eqVneq x.2 0.
case: x => [/= n d]; rewrite -absz_gt0 => dN0.
have ggt0 : (0 < gcdn `|n| `|d|)%N by rewrite gcdn_gt0 dN0 orbT.
rewrite ltz_nat divn_gt0// dvdn_leq ?dvdn_gcdr//=.
rewrite abszM abszX abszN1 exp1n mul1n absz_nat.
rewrite /coprime -(@eqn_pmul2r (gcdn ... | Fact | fracq_subproof | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszM",
"abszN1",
"abszX",
"absz_gt0",
"absz_nat",
"coprime",
"divnK",
"divn_gt0",
"dvdn_gcdl",
"dvdn_gcdr",
"dvdn_leq",
"eqVneq",
"eqn_pmul2r",
"exp1n",
"fracq_opt_subdef",
"fracq_opt_subdefE",
"gcdn",
"gcdn_gt0",
"ltz_nat",
"mul1n",
"muln_gcdl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_opt_subdef_id x :
fracq_opt_subdef (fracq_opt_subdef x) = fracq_subdef x. | Proof.
rewrite [fracq_opt_subdef (_ x)]/fracq_opt_subdef.
by rewrite fracq_subproof fracq_opt_subdefE.
Qed. | Lemma | fracq_opt_subdef_id | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq_opt_subdef",
"fracq_opt_subdefE",
"fracq_subdef",
"fracq_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq '((n', d')) : rat | :=
match d', n' with
| Posz 0 as d, _ as n => Rat (fracq_subproof (1, 0))
| _ as d, Posz _ as n | _ as d, _ as n =>
Rat (fracq_subproof (fracq_opt_subdef (n, d)))
end. | Definition | fracq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Posz",
"fracq_opt_subdef",
"fracq_subproof",
"n'",
"rat"
] | As a consequence val (fracq x) = fracq_opt_subdef (fracq_opt_subdef x)) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Irat_prf | := Ifracq_subproof : (int * int) -> Irat_prf. | Variant | Irat_prf | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"int"
] | inductive type. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Irat | := IRat : (int * int) -> Irat_prf -> Irat. | Variant | Irat | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Irat_prf",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
parse (x : Number.number) : option Irat | :=
let parse_pos i f :=
let nf := Decimal.nb_digits f in
let d := (10 ^ nf)%nat in
let n := (Nat.of_uint i * d + Nat.of_uint f)%nat in
valq (fracq (Posz n, Posz d)) in
let parse i f :=
match i with
| Decimal.Pos i => parse_pos i f
| Decimal.Neg i => let (n, d) := parse_pos i f in ((- n)%... | Definition | parse | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Irat",
"Posz",
"fracq",
"nat",
"number",
"of_uint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
print (r : Irat) : option Number.number | :=
let print_pos n d :=
if d == 1%nat then Some (Nat.to_uint n, Decimal.Nil) else
let d2d5 :=
match prime_decomp d with
| [:: (2, d2); (5, d5)] => Some (d2, d5)
| [:: (2, d2)] => Some (d2, O)
| [:: (5, d5)] => Some (O, d5)
| _ => None
end in
match d2d5 w... | Definition | print | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Irat",
"Nil",
"Posz",
"edivn",
"nat",
"number",
"prime_decomp",
"to_uint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_fracq x : val (fracq x) = fracq_subdef x. | Proof. by case: x => [[n|n] [[|[|d]]|d]]//=; rewrite !fracq_opt_subdef_id. Qed. | Lemma | val_fracq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"fracq_opt_subdef_id",
"fracq_subdef",
"val"
] | Check 0.2%Q. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
num_fracq x : numq (fracq x) = if x.2 != 0 then
(-1) ^ ((x.2 < 0) (+) (x.1 < 0)) * (`|x.1| %/ gcdn `|x.1| `|x.2|)%:Z else 0. | Proof. by rewrite /numq val_fracq/=; case: ifP. Qed. | Lemma | num_fracq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"gcdn",
"numq",
"val_fracq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
den_fracq x : denq (fracq x) =
if x.2 != 0 then (`|x.2| %/ gcdn `|x.1| `|x.2|)%:Z else 1. | Proof. by rewrite /denq val_fracq/=; case: ifP. Qed. | Lemma | den_fracq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"fracq",
"gcdn",
"val_fracq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratz_frac n : ratz n = fracq (n, 1). | Proof.
by apply: val_inj; rewrite val_fracq/= gcdn1 !divn1 abszE mulr_sign_norm.
Qed. | Fact | ratz_frac | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszE",
"apply",
"divn1",
"fracq",
"gcdn1",
"mulr_sign_norm",
"ratz",
"val_fracq",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valqK x : fracq (valq x) = x. | Proof.
move: x => [[n d] /= Pnd]; apply: val_inj; rewrite ?val_fracq/=.
move: Pnd; rewrite /coprime /fracq /= => /andP[] hd -/eqP hnd.
by rewrite lt_gtF ?gt_eqF //= hnd !divn1 mulz_sign_abs abszE gtr0_norm.
Qed. | Fact | valqK | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszE",
"apply",
"coprime",
"divn1",
"fracq",
"gt_eqF",
"gtr0_norm",
"lt_gtF",
"mulz_sign_abs",
"val_fracq",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalq '(n, d) | := sgr d * (gcdn `|n| `|d|)%:Z. | Definition | scalq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"gcdn",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalq_def x : scalq x = sgr x.2 * (gcdn `|x.1| `|x.2|)%:Z. | Proof. by case: x. Qed. | Lemma | scalq_def | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"gcdn",
"scalq",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalq_eq0 x : (scalq x == 0) = (x.2 == 0). | Proof.
case: x => n d; rewrite scalq_def /= mulf_eq0 sgr_eq0 /= eqz_nat.
rewrite -[gcdn _ _ == 0]negbK -lt0n gcdn_gt0 ?absz_gt0 [X in ~~ X]orbC.
by case: sgrP.
Qed. | Fact | scalq_eq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"absz_gt0",
"eqz_nat",
"gcdn",
"gcdn_gt0",
"lt0n",
"mulf_eq0",
"scalq",
"scalq_def",
"sgrP",
"sgr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_scalq x : sgr (scalq x) = sgr x.2. | Proof.
rewrite scalq_def sgrM sgr_id -[(gcdn _ _)%:Z]intz sgr_nat.
by rewrite -lt0n gcdn_gt0 ?absz_gt0 orbC; case: sgrP; rewrite // mul0r.
Qed. | Lemma | sgr_scalq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"absz_gt0",
"gcdn",
"gcdn_gt0",
"intz",
"lt0n",
"mul0r",
"scalq",
"scalq_def",
"sgr",
"sgrM",
"sgrP",
"sgr_id",
"sgr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signr_scalq x : (scalq x < 0) = (x.2 < 0). | Proof. by rewrite -!sgr_cp0 sgr_scalq. Qed. | Lemma | signr_scalq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"scalq",
"sgr_cp0",
"sgr_scalq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalqE x :
x.2 != 0 -> scalq x = (-1) ^+ (x.2 < 0)%R * (gcdn `|x.1| `|x.2|)%:Z. | Proof. by rewrite scalq_def; case: sgrP. Qed. | Lemma | scalqE | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"gcdn",
"scalq",
"scalq_def",
"sgrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valq_frac x :
x.2 != 0 -> x = (scalq x * numq (fracq x), scalq x * denq (fracq x)). | Proof.
move=> x2_neq0; rewrite scalqE//; move: x2_neq0.
case: x => [n d] /= d_neq0; rewrite num_fracq den_fracq/= ?d_neq0.
rewrite mulr_signM -mulrA -!PoszM addKb.
do 2!rewrite muln_divCA ?(dvdn_gcdl, dvdn_gcdr) // divnn.
by rewrite gcdn_gt0 !absz_gt0 d_neq0 orbT !muln1 !mulz_sign_abs.
Qed. | Fact | valq_frac | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"PoszM",
"absz_gt0",
"den_fracq",
"denq",
"divnn",
"dvdn_gcdl",
"dvdn_gcdr",
"fracq",
"gcdn_gt0",
"muln1",
"muln_divCA",
"mulrA",
"mulr_signM",
"mulz_sign_abs",
"num_fracq",
"numq",
"scalq",
"scalqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zeroq | := 0%Q. | Definition | zeroq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oneq | := 1%Q. | Definition | oneq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
frac0q x : fracq (0, x) = zeroq. | Proof.
apply: val_inj; rewrite //= val_fracq/= div0n !gcd0n !mulr0 !divnn.
by have [//|x_neq0] := eqVneq; rewrite absz_gt0 x_neq0.
Qed. | Fact | frac0q | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"absz_gt0",
"apply",
"div0n",
"divnn",
"eqVneq",
"fracq",
"gcd0n",
"mulr0",
"val_fracq",
"val_inj",
"zeroq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq0 x : fracq (x, 0) = zeroq. | Proof. exact/eqP. Qed. | Fact | fracq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"zeroq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_spec (x : int * int) : int * int -> rat -> Type | :=
| FracqSpecN of x.2 = 0 : fracq_spec x (x.1, 0) zeroq
| FracqSpecP k fx of k != 0 : fracq_spec x (k * numq fx, k * denq fx) fx. | Variant | fracq_spec | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"int",
"numq",
"rat",
"zeroq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracqP x : fracq_spec x x (fracq x). | Proof.
case: x => n d /=; have [d_eq0 | d_neq0] := eqVneq d 0.
by rewrite d_eq0 fracq0; constructor.
by rewrite {2}[(_, _)]valq_frac //; constructor; rewrite scalq_eq0.
Qed. | Fact | fracqP | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"eqVneq",
"fracq",
"fracq0",
"fracq_spec",
"scalq_eq0",
"valq_frac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_eqE x y : (x == y) = (numq x == numq y) && (denq x == denq y). | Proof.
rewrite -val_eqE [val x]surjective_pairing [val y]surjective_pairing /=.
by rewrite xpair_eqE.
Qed. | Lemma | rat_eqE | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"numq",
"val",
"val_eqE",
"xpair_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_denq x : sgr (denq x) = 1. | Proof. by apply/eqP; rewrite sgr_cp0. Qed. | Lemma | sgr_denq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"apply",
"denq",
"sgr",
"sgr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_denq x : `|denq x| = denq x. | Proof. by rewrite gtr0_norm. Qed. | Lemma | normr_denq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"gtr0_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_denq x : `|denq x|%N = denq x :> int. | Proof. by rewrite abszE normr_denq. Qed. | Lemma | absz_denq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszE",
"denq",
"int",
"normr_denq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_eq x y : (x == y) = (numq x * denq y == numq y * denq x). | Proof.
symmetry; rewrite rat_eqE andbC.
have [->|] /= := eqVneq (denq _); first by rewrite (inj_eq (mulIf _)).
apply: contraNF => /eqP hxy; rewrite -absz_denq -[eqbRHS]absz_denq.
rewrite eqz_nat /= eqn_dvd.
rewrite -(@Gauss_dvdr _ `|numq x|) 1?coprime_sym ?coprime_num_den // andbC.
rewrite -(@Gauss_dvdr _ `|numq y|) 1?... | Lemma | rat_eq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Gauss_dvdr",
"abszM",
"absz_denq",
"apply",
"coprime_num_den",
"coprime_sym",
"denq",
"dvdn_mull",
"dvdnn",
"eqVneq",
"eqbRHS",
"eqn_dvd",
"eqz_nat",
"inj_eq",
"mulIf",
"numq",
"rat_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_eq x y : x.2 != 0 -> y.2 != 0 ->
(fracq x == fracq y) = (x.1 * y.2 == y.1 * x.2). | Proof.
case: fracqP=> //= u fx u_neq0 _; case: fracqP=> //= v fy v_neq0 _; symmetry.
rewrite [eqbRHS]mulrC mulrACA [eqbRHS]mulrACA.
by rewrite [denq _ * _]mulrC (inj_eq (mulfI _)) ?mulf_neq0 // rat_eq.
Qed. | Fact | fracq_eq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"eqbRHS",
"fracq",
"fracqP",
"inj_eq",
"mulfI",
"mulf_neq0",
"mulrACA",
"mulrC",
"rat_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracq_eq0 x : (fracq x == zeroq) = (x.1 == 0) || (x.2 == 0). | Proof.
move: x=> [n d] /=; have [->|d0] := eqVneq d 0.
by rewrite fracq0 eqxx orbT.
by rewrite -[zeroq]valqK orbF fracq_eq ?d0 //= mulr1 mul0r.
Qed. | Fact | fracq_eq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"eqVneq",
"eqxx",
"fracq",
"fracq0",
"fracq_eq",
"mul0r",
"mulr1",
"valqK",
"zeroq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fracqMM x n d : x != 0 -> fracq (x * n, x * d) = fracq (n, d). | Proof.
move=> x_neq0; apply/eqP.
have [->|d_neq0] := eqVneq d 0; first by rewrite mulr0 !fracq0.
by rewrite fracq_eq ?mulf_neq0 //= mulrCA mulrA.
Qed. | Fact | fracqMM | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"apply",
"eqVneq",
"fracq",
"fracq0",
"fracq_eq",
"mulf_neq0",
"mulr0",
"mulrA",
"mulrCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addq_subdef (x y : int * int) | :=
let: (x1, x2) := x in
let: (y1, y2) := y in
match x2, y2 with
| Posz 1, Posz 1 =>
match x1, y1 with
| Posz 0, _ => (y1, 1)
| _, Posz 0 => (x1, 1)
| Posz n, Posz 1 => (Posz n.+1, 1)
| Posz 1, Posz n => (Posz n.+1, 1)
| _, _ => (x1 + y1, 1)
end
| ... | Definition | addq_subdef | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Posz",
"int"
] | of terms of the form N%:Q when N is a concrete natural number. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
addq '(Rat x xP) '(Rat y yP) | := fracq (addq_subdef x y). | Definition | addq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq_subdef",
"fracq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addq_def x y : addq x y = fracq (addq_subdef (valq x) (valq y)). | Proof. by case: x; case: y. Qed. | Lemma | addq_def | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"addq_subdef",
"fracq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addq_subdefE x y : addq_subdef x y = (x.1 * y.2 + y.1 * x.2, x.2 * y.2). | Proof.
case: x y => [x1 [[|[|x2]]|x2]] [y1 [[|[|y2]]|y2]]/=; rewrite ?Monoid.simpm//.
by case: x1 y1 => [[|[|m]]|m] [[|[|n]]|n]; rewrite ?Monoid.simpm// -PoszD addn1.
Qed. | Lemma | addq_subdefE | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"PoszD",
"addn1",
"addq_subdef",
"simpm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppq_subdef (x : int * int) | := (- x.1, x.2). | Definition | oppq_subdef | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppq '(Rat x xP) | := fracq (oppq_subdef x). | Definition | oppq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"oppq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppq_def x : oppq x = fracq (oppq_subdef (valq x)). | Proof. by case: x. Qed. | Definition | oppq_def | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"oppq",
"oppq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addq_subdefC : commutative addq_subdef. | Proof. by move=> x y; rewrite !addq_subdefE addrC [x.2 * _]mulrC. Qed. | Fact | addq_subdefC | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq_subdef",
"addq_subdefE",
"addrC",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addq_subdefA : associative addq_subdef. | Proof.
move=> x y z; rewrite !addq_subdefE/=.
by rewrite !mulrDl addrA !mulrA 2![_ * _ * x.2]mulrAC.
Qed. | Fact | addq_subdefA | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq_subdef",
"addq_subdefE",
"addrA",
"mulrA",
"mulrAC",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addq_frac x y : x.2 != 0 -> y.2 != 0 ->
(addq (fracq x) (fracq y)) = fracq (addq_subdef x y). | Proof.
case: fracqP => // u fx u_neq0 _; case: fracqP => // v fy v_neq0 _.
rewrite addq_def !addq_subdefE /=.
rewrite ![(_ * numq _) * _]mulrACA [(_ * denq _) * _]mulrACA.
by rewrite [v * _]mulrC -mulrDr fracqMM ?mulf_neq0.
Qed. | Fact | addq_frac | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"addq_def",
"addq_subdef",
"addq_subdefE",
"denq",
"fracq",
"fracqMM",
"fracqP",
"mulf_neq0",
"mulrACA",
"mulrC",
"mulrDr",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratzD : {morph ratz : x y / x + y >-> addq x y}. | Proof. by move=> x y; rewrite !ratz_frac addq_frac// addq_subdefE/= !mulr1. Qed. | Fact | ratzD | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"addq_frac",
"addq_subdefE",
"mulr1",
"ratz",
"ratz_frac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppq_frac x : oppq (fracq x) = fracq (oppq_subdef x). | Proof.
rewrite /oppq_subdef; case: fracqP => /= [|u fx u_neq0].
by rewrite fracq0.
by rewrite oppq_def -mulrN fracqMM.
Qed. | Fact | oppq_frac | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"fracq0",
"fracqMM",
"fracqP",
"mulrN",
"oppq",
"oppq_def",
"oppq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratzN : {morph ratz : x / - x >-> oppq x}. | Proof. by move=> x /=; rewrite !ratz_frac // /add /= !mulr1. Qed. | Fact | ratzN | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"add",
"mulr1",
"oppq",
"ratz",
"ratz_frac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addqC : commutative addq. | Proof. by move=> x y; rewrite !addq_def /= addq_subdefC. Qed. | Fact | addqC | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"addq_def",
"addq_subdefC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addqA : associative addq. | Proof.
move=> x y z; rewrite -[x]valqK -[y]valqK -[z]valqK.
by rewrite ?addq_frac ?addq_subdefA// ?addq_subdefE ?mulf_neq0 ?denq_neq0.
Qed. | Fact | addqA | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"addq_frac",
"addq_subdefA",
"addq_subdefE",
"denq_neq0",
"mulf_neq0",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add0q : left_id zeroq addq. | Proof.
move=> x; rewrite -[x]valqK -[zeroq]valqK addq_frac ?denq_neq0 // !addq_subdefE.
by rewrite mul0r add0r mulr1 mul1r -surjective_pairing.
Qed. | Fact | add0q | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"add0r",
"addq",
"addq_frac",
"addq_subdefE",
"denq_neq0",
"mul0r",
"mul1r",
"mulr1",
"valqK",
"zeroq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addNq : left_inverse (fracq (0, 1)) oppq addq. | Proof.
move=> x; rewrite -[x]valqK !(addq_frac, oppq_frac) ?denq_neq0 //.
rewrite !addq_subdefE /oppq_subdef //= mulNr addNr; apply/eqP.
by rewrite fracq_eq ?mulf_neq0 ?denq_neq0 //= !mul0r.
Qed. | Fact | addNq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addNr",
"addq",
"addq_frac",
"addq_subdefE",
"apply",
"denq_neq0",
"fracq",
"fracq_eq",
"mul0r",
"mulNr",
"mulf_neq0",
"oppq",
"oppq_frac",
"oppq_subdef",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq_subdef (x y : int * int) | :=
let: (x1, x2) := x in
let: (y1, y2) := y in
match x2, y2 with
| Posz 1, Posz 1 => (x1 * y1, 1)
| Posz 1, _ => (x1 * y1, y2)
| _, Posz 1 => (x1 * y1, x2)
| _, _ => (x1 * y1, x2 * y2)
end. | Definition | mulq_subdef | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"Posz",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq '(Rat x xP) '(Rat y yP) | := fracq (mulq_subdef x y). | Definition | mulq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"mulq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq_def x y : mulq x y = fracq (mulq_subdef (valq x) (valq y)). | Proof. by case: x; case: y. Qed. | Lemma | mulq_def | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"mulq",
"mulq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq_subdefE x y : mulq_subdef x y = (x.1 * y.1, x.2 * y.2). | Proof.
by case: x y => [x1 [[|[|x2]]|x2]] [y1 [[|[|y2]]|y2]]/=; rewrite ?Monoid.simpm.
Qed. | Lemma | mulq_subdefE | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mulq_subdef",
"simpm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq_subdefC : commutative mulq_subdef. | Proof. by move=> x y; rewrite !mulq_subdefE mulrC [_ * x.2]mulrC. Qed. | Fact | mulq_subdefC | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mulq_subdef",
"mulq_subdefE",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_subdefA : associative mulq_subdef. | Proof. by move=> x y z; rewrite !mulq_subdefE !mulrA. Qed. | Fact | mul_subdefA | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mulq_subdef",
"mulq_subdefE",
"mulrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invq_subdef (x : int * int) | := (x.2, x.1). | Definition | invq_subdef | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invq '(Rat x xP) | := fracq (invq_subdef x). | Definition | invq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"invq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invq_def x : invq x = fracq (invq_subdef (valq x)). | Proof. by case: x. Qed. | Lemma | invq_def | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"invq",
"invq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq_frac x y : (mulq (fracq x) (fracq y)) = fracq (mulq_subdef x y). | Proof.
rewrite mulq_def !mulq_subdefE; case: (fracqP x) => /= [|u fx u_neq0].
by rewrite !mul0r !mul1r fracq0 frac0q.
case: (fracqP y) => /= [|v fy v_neq0].
by rewrite !mulr0 !mulr1 fracq0 frac0q.
by rewrite ![_ * (v * _)]mulrACA [RHS]fracqMM ?mulf_neq0.
Qed. | Fact | mulq_frac | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"frac0q",
"fracq",
"fracq0",
"fracqMM",
"fracqP",
"mul0r",
"mul1r",
"mulf_neq0",
"mulq",
"mulq_def",
"mulq_subdef",
"mulq_subdefE",
"mulr0",
"mulr1",
"mulrACA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratzM : {morph ratz : x y / x * y >-> mulq x y}. | Proof. by move=> x y /=; rewrite !ratz_frac //= !mulr1. Qed. | Fact | ratzM | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mulq",
"mulr1",
"ratz",
"ratz_frac"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invq_frac x :
x.1 != 0 -> x.2 != 0 -> invq (fracq x) = fracq (invq_subdef x). | Proof. by rewrite invq_def; case: (fracqP x) => // k ? k0; rewrite fracqMM. Qed. | Fact | invq_frac | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"fracq",
"fracqMM",
"fracqP",
"invq",
"invq_def",
"invq_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulqC : commutative mulq. | Proof. by move=> x y; rewrite !mulq_def mulq_subdefC. Qed. | Fact | mulqC | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mulq",
"mulq_def",
"mulq_subdefC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulqA : associative mulq. | Proof.
by move=> x y z; rewrite -[x]valqK -[y]valqK -[z]valqK !mulq_frac mul_subdefA.
Qed. | Fact | mulqA | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mul_subdefA",
"mulq",
"mulq_frac",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul1q : left_id oneq mulq. | Proof.
move=> x; rewrite -[x]valqK -[oneq]valqK; rewrite mulq_frac !mulq_subdefE.
by rewrite !mul1r -surjective_pairing.
Qed. | Fact | mul1q | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"mul1r",
"mulq",
"mulq_frac",
"mulq_subdefE",
"oneq",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulq_addl : left_distributive mulq addq. | Proof.
move=> x y z; rewrite -[x]valqK -[y]valqK -[z]valqK /=.
rewrite !(mulq_frac, addq_frac, mulq_subdefE, addq_subdefE) ?mulf_neq0 ?denq_neq0 //=.
apply/eqP; rewrite fracq_eq ?mulf_neq0 ?denq_neq0 //= !mulrDl; apply/eqP.
by rewrite !mulrA ![_ * (valq z).1]mulrC !mulrA ![_ * (valq x).2]mulrC !mulrA.
Qed. | Fact | mulq_addl | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"addq_frac",
"addq_subdefE",
"apply",
"denq_neq0",
"fracq_eq",
"mulf_neq0",
"mulq",
"mulq_frac",
"mulq_subdefE",
"mulrA",
"mulrC",
"mulrDl",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonzero1q : oneq != zeroq. | Proof. by []. Qed. | Fact | nonzero1q | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"oneq",
"zeroq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulVq x : x != 0 -> mulq (invq x) x = 1. | Proof.
rewrite -[x]valqK -[0]valqK fracq_eq ?denq_neq0 //= mulr1 mul0r=> nx0.
rewrite !(mulq_frac, invq_frac, mulq_subdefE) ?denq_neq0 // -[1]valqK.
by apply/eqP; rewrite fracq_eq ?mulf_neq0 ?denq_neq0 //= mulr1 mul1r mulrC.
Qed. | Fact | mulVq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"apply",
"denq_neq0",
"fracq_eq",
"invq",
"invq_frac",
"mul0r",
"mul1r",
"mulf_neq0",
"mulq",
"mulq_frac",
"mulq_subdefE",
"mulr1",
"mulrC",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invq0 : invq 0 = 0. | Proof. exact/eqP. Qed. | Fact | invq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"invq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_eq0 x : (numq x == 0) = (x == 0). | Proof.
rewrite -[x]valqK fracq_eq0; case: fracqP=> /= [|k {}x k0].
by rewrite eqxx orbT.
by rewrite !mulf_eq0 (negPf k0) /= denq_eq0 orbF.
Qed. | Lemma | numq_eq0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq_eq0",
"eqxx",
"fracqP",
"fracq_eq0",
"mulf_eq0",
"numq",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n %:Q" | := ((n : int)%:~R : rat) : ring_scope. | Notation | n %:Q | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"int",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subq (x y : rat) : rat | := (addq x (oppq y)). | Definition | subq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"addq",
"oppq",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divq (x y : rat) : rat | := (mulq x (invq y)). | Definition | divq | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"invq",
"mulq",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"- x" | := (oppq x) : rat_scope. | Notation | - x | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"oppq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ^-1" | := (invq x) : rat_scope. | Notation | x ^-1 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"invq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratzE n : ratz n = n%:Q. | Proof.
elim: n=> [|n ihn|n ihn]; first by rewrite mulr0z ratz_frac.
by rewrite intS mulrzDr ratzD ihn.
by rewrite intS opprD mulrzDr ratzD ihn.
Qed. | Lemma | ratzE | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"intS",
"mulr0z",
"mulrzDr",
"opprD",
"ratz",
"ratzD",
"ratz_frac"
] | ratz should not be used, %:Q should be used instead | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
numq_int n : numq n%:Q = n. | Proof. by rewrite -ratzE. Qed. | Lemma | numq_int | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"numq",
"ratzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_int n : denq n%:Q = 1. | Proof. by rewrite -ratzE. Qed. | Lemma | denq_int | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"ratzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat0 : 0%:Q = 0. | Proof. by []. Qed. | Lemma | rat0 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat1 : 1%:Q = 1. | Proof. by []. Qed. | Lemma | rat1 | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numqN x : numq (- x) = - numq x. | Proof.
rewrite [- _]oppq_def/= num_fracq.
case: x => -[a b]; rewrite /numq/= => /andP[b_gt0].
rewrite /coprime => /eqP cab.
by rewrite lt_gtF ?gt_eqF // {2}abszN cab divn1 mulz_sign_abs.
Qed. | Lemma | numqN | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszN",
"coprime",
"divn1",
"gt_eqF",
"lt_gtF",
"mulz_sign_abs",
"num_fracq",
"numq",
"oppq_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denqN x : denq (- x) = denq x. | Proof.
rewrite [- _]oppq_def den_fracq.
case: x => -[a b]; rewrite /denq/= => /andP[b_gt0].
by rewrite /coprime=> /eqP cab; rewrite gt_eqF // abszN cab divn1 gtz0_abs.
Qed. | Lemma | denqN | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"abszN",
"coprime",
"den_fracq",
"denq",
"divn1",
"gt_eqF",
"gtz0_abs",
"oppq_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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