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 |
|---|---|---|---|---|---|---|---|---|---|---|
intq_eq0 n : (n%:~R == 0 :> rat) = (n == 0)%N. | Proof. by rewrite -ratzE /ratz rat_eqE/= /numq /denq/= eqxx andbT. Qed. | Fact | intq_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",
"eqxx",
"numq",
"rat",
"rat_eqE",
"ratz",
"ratzE"
] | Will be subsumed by pnatr_eq0 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
fracqE x : fracq x = x.1%:Q / x.2%:Q. | Proof.
move: x => [m n] /=; apply/val_inj; rewrite val_fracq/=.
case: eqVneq => //= [->|n_neq0]; first by rewrite rat0 invr0 mulr0.
rewrite -[m%:Q]valqK -[n%:Q]valqK.
rewrite [_^-1]invq_frac ?denq_neq0 ?numq_eq0 ?intq_eq0//=.
rewrite [X in valq X]mulq_frac val_fracq /invq_subdef !mulq_subdefE/=.
by rewrite -!/(numq _) ... | Lemma | fracqE | 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",
"denq_int",
"denq_neq0",
"eqVneq",
"fracq",
"intq_eq0",
"invq_frac",
"invq_subdef",
"invr0",
"mul1r",
"mulq_frac",
"mulq_subdefE",
"mulr0",
"mulr1",
"numq",
"numq_eq0",
"numq_int",
"rat0",
"val_fracq",
"val_inj",
"valqK"
] | fracq should never appear, its canonical form is _%:Q / _%:Q | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
divq_num_den x : (numq x)%:Q / (denq x)%:Q = x. | Proof. by rewrite -{3}[x]valqK [valq _]surjective_pairing /= fracqE. Qed. | Lemma | divq_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",
... | [
"denq",
"fracqE",
"numq",
"valqK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divq_spec (n d : int) : int -> int -> rat -> Type | :=
| DivqSpecN of d = 0 : divq_spec n d n 0 0
| DivqSpecP k x of k != 0 : divq_spec n d (k * numq x) (k * denq x) x. | Variant | divq_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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divqP n d : divq_spec n d n d (n%:Q / d%:Q). | Proof.
set x := (n, d); rewrite -[n]/x.1 -[d]/x.2 -fracqE.
by case: fracqP => [_|k fx k_neq0] /=; constructor.
Qed. | Lemma | divqP | 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",
... | [
"divq_spec",
"fracqE",
"fracqP"
] | replaces fracqP | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rat_spec (* (x : rat) *) : rat -> int -> int -> Type | :=
Rat_spec (n : int) (d : nat) & coprime `|n| d.+1
: rat_spec (* x *) (n%:Q / d.+1%:Q) n d.+1. | Variant | rat_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",
... | [
"coprime",
"int",
"nat",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratP x : rat_spec x (numq x) (denq x). | Proof.
rewrite -{1}[x](divq_num_den); case hd: denq => [p|n].
have: 0 < p%:Z by rewrite -hd denq_gt0.
case: p hd=> //= n hd; constructor; rewrite -?hd ?divq_num_den //.
by rewrite -[n.+1]/`|n.+1|%N -hd coprime_num_den.
by move: (denq_gt0 x); rewrite hd.
Qed. | Lemma | ratP | 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_num_den",
"denq",
"denq_gt0",
"divq_num_den",
"numq",
"rat_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimeq_num n d : coprime `|n| `|d| -> numq (n%:~R / d%:~R) = sgr d * n. | Proof.
move=> cnd /=; have <- := fracqE (n, d).
rewrite num_fracq/= (eqP (cnd : _ == 1)) divn1.
have [|d_gt0|d_lt0] := sgrP d;
by rewrite (mul0r, mul1r, mulN1r) //= ?[_ ^ _]signrN ?mulNr mulz_sign_abs.
Qed. | Lemma | coprimeq_num | 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",
"d_gt0",
"divn1",
"fracqE",
"mul0r",
"mul1r",
"mulN1r",
"mulNr",
"mulz_sign_abs",
"num_fracq",
"numq",
"sgr",
"sgrP",
"signrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprimeq_den n d :
coprime `|n| `|d| -> denq (n%:~R / d%:~R) = (if d == 0 then 1 else `|d|). | Proof.
move=> cnd; have <- := fracqE (n, d).
by rewrite den_fracq/= (eqP (cnd : _ == 1)) divn1; case: d {cnd}; case.
Qed. | Lemma | coprimeq_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",
"den_fracq",
"denq",
"divn1",
"fracqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denqVz (i : int) : i != 0 -> denq (i%:~R^-1) = `|i|. | Proof.
move=> h; rewrite -div1r -[1]/(1%:~R).
by rewrite coprimeq_den /= ?coprime1n // (negPf h).
Qed. | Lemma | denqVz | 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",
... | [
"coprime1n",
"coprimeq_den",
"denq",
"div1r",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numqE x : (numq x)%:~R = x * (denq x)%:~R. | Proof. by rewrite -{2}[x]divq_num_den divfK // intq_eq0 denq_eq0. Qed. | Lemma | numqE | 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_eq0",
"divfK",
"divq_num_den",
"intq_eq0",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denqP x : {d | denq x = d.+1}. | Proof. by rewrite /denq; case: x => [[_ [[|d]|]] //= _]; exists d. Qed. | Lemma | denqP | 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 | |
normq '(Rat x _) : rat | := `|x.1|%:~R / (x.2)%:~R. | Definition | normq | 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",
... | [
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_rat '(Rat x _) '(Rat y _) | := x.1 * y.2 <= y.1 * x.2. | Definition | le_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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_rat '(Rat x _) '(Rat y _) | := x.1 * y.2 < y.1 * x.2. | Definition | lt_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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normqE x : normq x = `|numq x|%:~R / (denq x)%:~R. | Proof. by case: x. Qed. | Lemma | normqE | 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",
"normq",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_ratE x y : le_rat x y = (numq x * denq y <= numq y * denq x). | Proof. by case: x; case: y. Qed. | Lemma | le_ratE | 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",
"le_rat",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_ratE x y : lt_rat x y = (numq x * denq y < numq y * denq x). | Proof. by case: x; case: y. Qed. | Lemma | lt_ratE | 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_rat",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt_rat0 x : lt_rat 0 x = (0 < numq x). | Proof. by rewrite lt_ratE mul0r mulr1. Qed. | Lemma | gt_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",
... | [
"lt_rat",
"lt_ratE",
"mul0r",
"mulr1",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_rat0 x : lt_rat x 0 = (numq x < 0). | Proof. by rewrite lt_ratE mul0r mulr1. Qed. | Lemma | lt_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",
... | [
"lt_rat",
"lt_ratE",
"mul0r",
"mulr1",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_rat0 x : le_rat 0 x = (0 <= numq x). | Proof. by rewrite le_ratE mul0r mulr1. Qed. | Lemma | ge_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",
... | [
"le_rat",
"le_ratE",
"mul0r",
"mulr1",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_rat0 x : le_rat x 0 = (numq x <= 0). | Proof. by rewrite le_ratE mul0r mulr1. Qed. | Lemma | le_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",
... | [
"le_rat",
"le_ratE",
"mul0r",
"mulr1",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_rat0D x y : le_rat 0 x -> le_rat 0 y -> le_rat 0 (x + y). | Proof.
rewrite !ge_rat0 => hnx hny.
have hxy: (0 <= numq x * denq y + numq y * denq x).
by rewrite addr_ge0 ?mulr_ge0.
rewrite [_ + _]addq_def /numq /= -!/(denq _) ?mulf_eq0 ?denq_eq0.
rewrite val_fracq/=; case: ifP => //=.
by rewrite ?addq_subdefE !mulr_ge0// !le_gtF ?mulr_ge0 ?denq_ge0//=.
Qed. | Fact | le_rat0D | 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_def",
"addq_subdefE",
"addr_ge0",
"denq",
"denq_eq0",
"denq_ge0",
"ge_rat0",
"le_gtF",
"le_rat",
"mulf_eq0",
"mulr_ge0",
"numq",
"val_fracq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_rat0M x y : le_rat 0 x -> le_rat 0 y -> le_rat 0 (x * y). | Proof.
rewrite !ge_rat0 => hnx hny.
have hxy: (0 <= numq x * denq y + numq y * denq x).
by rewrite addr_ge0 ?mulr_ge0.
rewrite [_ * _]mulq_def /numq /= -!/(denq _) ?mulf_eq0 ?denq_eq0.
rewrite val_fracq/=; case: ifP => //=.
by rewrite ?mulq_subdefE !mulr_ge0// !le_gtF ?mulr_ge0 ?denq_ge0//=.
Qed. | Fact | le_rat0M | 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",
... | [
"addr_ge0",
"denq",
"denq_eq0",
"denq_ge0",
"ge_rat0",
"le_gtF",
"le_rat",
"mulf_eq0",
"mulq_def",
"mulq_subdefE",
"mulr_ge0",
"numq",
"val_fracq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_rat0_anti x : le_rat 0 x -> le_rat x 0 -> x = 0. | Proof.
by move=> hx hy; apply/eqP; rewrite -numq_eq0 eq_le -ge_rat0 -le_rat0 hx hy.
Qed. | Fact | le_rat0_anti | 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",
"eq_le",
"ge_rat0",
"le_rat",
"le_rat0",
"numq_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_numq_div (n d : int) : sgr (numq (n%:Q / d%:Q)) = sgr n * sgr d. | Proof.
set x := (n, d); rewrite -[n]/x.1 -[d]/x.2 -fracqE.
case: fracqP => [|k fx k_neq0] /=; first by rewrite mulr0.
by rewrite !sgrM mulrACA -expr2 sqr_sg k_neq0 sgr_denq mulr1 mul1r.
Qed. | Lemma | sgr_numq_div | 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",
... | [
"expr2",
"fracqE",
"fracqP",
"int",
"mul1r",
"mulr0",
"mulr1",
"mulrACA",
"numq",
"sgr",
"sgrM",
"sgr_denq",
"sqr_sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subq_ge0 x y : le_rat 0 (y - x) = le_rat x y. | Proof.
symmetry; rewrite ge_rat0 !le_ratE -subr_ge0.
case: ratP => nx dx cndx; case: ratP => ny dy cndy.
rewrite -!mulNr addf_div ?intq_eq0 // !mulNr -!rmorphM -rmorphB /=.
symmetry; rewrite !leNgt -sgr_cp0 sgr_numq_div mulrC gtr0_sg //.
by rewrite mul1r sgr_cp0.
Qed. | Fact | subq_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",
... | [
"addf_div",
"ge_rat0",
"gtr0_sg",
"intq_eq0",
"leNgt",
"le_rat",
"le_ratE",
"mul1r",
"mulNr",
"mulrC",
"ratP",
"rmorphB",
"rmorphM",
"sgr_cp0",
"sgr_numq_div",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_rat_total : total le_rat. | Proof. by move=> x y; rewrite !le_ratE; apply: le_total. Qed. | Fact | le_rat_total | 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",
"le_rat",
"le_ratE",
"le_total",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_sign_mul (b : bool) x : numq ((-1) ^+ b * x) = (-1) ^+ b * numq x. | Proof. by case: b; rewrite ?(mul1r, mulN1r) // numqN. Qed. | Fact | numq_sign_mul | 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",
"mulN1r",
"numq",
"numqN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_div_lt0 n d : n != 0 -> d != 0 ->
(numq (n%:~R / d%:~R) < 0)%R = (n < 0)%R (+) (d < 0)%R. | Proof.
move=> n0 d0; rewrite -sgr_cp0 sgr_numq_div !sgr_def n0 d0.
by rewrite !mulr1n -signr_addb; case: (_ (+) _).
Qed. | Fact | numq_div_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",
... | [
"mulr1n",
"numq",
"sgr_cp0",
"sgr_def",
"sgr_numq_div",
"signr_addb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_num_div n d : `|numq (n%:~R / d%:~R)| = numq (`|n|%:~R / `|d|%:~R). | Proof.
rewrite (normrEsg n) (normrEsg d) !rmorphM /= invfM mulrACA !sgr_def.
have [->|n_neq0] := eqVneq; first by rewrite mul0r mulr0.
have [->|d_neq0] := eqVneq; first by rewrite invr0 !mulr0.
rewrite !intr_sign invr_sign -signr_addb numq_sign_mul -numq_div_lt0 //.
by apply: (canRL (signrMK _)); rewrite mulz_sign_abs.... | Lemma | normr_num_div | 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",
"intr_sign",
"invfM",
"invr0",
"invr_sign",
"mul0r",
"mulr0",
"mulrACA",
"mulz_sign_abs",
"normrEsg",
"numq",
"numq_div_lt0",
"numq_sign_mul",
"rmorphM",
"sgr_def",
"signrMK",
"signr_addb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_ratN x : normq (- x) = normq x. | Proof. by rewrite !normqE numqN denqN normrN. Qed. | Fact | norm_ratN | 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",
... | [
"denqN",
"normq",
"normqE",
"normrN",
"numqN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_rat0_norm x : le_rat 0 x -> normq x = x. | Proof.
rewrite ge_rat0; case: ratP=> [] // n d cnd n_ge0.
by rewrite normqE /= normr_num_div ?ger0_norm // divq_num_den.
Qed. | Fact | ge_rat0_norm | 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",
... | [
"divq_num_den",
"ge_rat0",
"ger0_norm",
"le_rat",
"normq",
"normqE",
"normr_num_div",
"ratP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_rat_def x y : (lt_rat x y) = (y != x) && (le_rat x y). | Proof. by rewrite lt_ratE le_ratE lt_def rat_eq. Qed. | Fact | lt_rat_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",
... | [
"le_rat",
"le_ratE",
"lt_def",
"lt_rat",
"lt_ratE",
"rat_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_ge0 x : (0 <= numq x) = (0 <= x). | Proof.
by case: ratP => n d cnd; rewrite ?pmulr_lge0 ?invr_gt0 (ler0z, ltr0z).
Qed. | Lemma | numq_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",
... | [
"invr_gt0",
"ler0z",
"ltr0z",
"numq",
"pmulr_lge0",
"ratP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_le0 x : (numq x <= 0) = (x <= 0). | Proof. by rewrite -oppr_ge0 -numqN numq_ge0 oppr_ge0. Qed. | Lemma | numq_le0 | 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",
"numqN",
"numq_ge0",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_gt0 x : (0 < numq x) = (0 < x). | Proof. by rewrite !ltNge numq_le0. Qed. | Lemma | numq_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",
... | [
"ltNge",
"numq",
"numq_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numq_lt0 x : (numq x < 0) = (x < 0). | Proof. by rewrite !ltNge numq_ge0. Qed. | Lemma | numq_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",
... | [
"ltNge",
"numq",
"numq_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_numq x : sgz (numq x) = sgz x. | Proof.
apply/eqP; case: (sgzP x); rewrite sgz_cp0 ?(numq_gt0, numq_lt0) //.
by move->.
Qed. | Lemma | sgr_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",
... | [
"apply",
"numq",
"numq_gt0",
"numq_lt0",
"sgz",
"sgzP",
"sgz_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_mulr_sign (b : bool) x : denq ((-1) ^+ b * x) = denq x. | Proof. by case: b; rewrite ?(mul1r, mulN1r) // denqN. Qed. | Lemma | denq_mulr_sign | 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",
"denqN",
"mul1r",
"mulN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
denq_norm x : denq `|x| = denq x. | Proof. by rewrite normrEsign denq_mulr_sign. Qed. | Lemma | denq_norm | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"denq_mulr_sign",
"normrEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor x : int | := (numq x %/ denq x)%Z. | Definition | floor | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
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"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"int",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil x : int | := - (- numq x %/ denq x)%Z. | Definition | ceil | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
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"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"int",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn x : nat | :=
if 0 <= x then (`|numq x| %/ `|denq x|)%N else 0%N. | Definition | truncn | algebra | algebra/rat.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
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"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"polydiv",
"zmodp",
"matrix",
... | [
"denq",
"nat",
"numq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_int x | := denq x == 1. | Let | is_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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_nat x | := (0 <= x) && (denq x == 1). | Let | is_nat | 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 | |
floorP x :
if x \is Num.real then (floor x)%:~R <= x < (floor x + 1)%:~R
else floor x == 0. | Proof.
rewrite num_real /floor; case: (ratP x) => n d _ {x}; rewrite ler_pdivlMr//.
by rewrite ltr_pdivrMr// -!intrM ler_int ltr_int lez_floor ?ltz_ceil.
Qed. | Fact | floorP | 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",
... | [
"floor",
"intrM",
"ler_int",
"ler_pdivlMr",
"lez_floor",
"ltr_int",
"ltr_pdivrMr",
"ltz_ceil",
"num_real",
"ratP",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilP x : ceil x = - floor (- x). | Proof. by rewrite /ceil /floor numqN denqN. Qed. | Fact | ceilP | 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",
... | [
"ceil",
"denqN",
"floor",
"numqN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnP x : truncn x = if floor x is Posz n then n else 0. | Proof.
rewrite /truncn /floor; case: (ratP x) => n d _ {x} /=.
by rewrite !ler_pdivlMr// mul0r; case: n => n; rewrite ler0z//= mul1n.
Qed. | Fact | truncnP | 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",
"floor",
"ler0z",
"ler_pdivlMr",
"mul0r",
"mul1n",
"ratP",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrP x : reflect (exists n, x = n%:~R) (is_int x). | Proof.
apply: (iffP idP) => [/eqP d1 | [i ->]]; [|by rewrite /is_int denq_int].
by exists (numq x); case: (ratP x) d1 => n d _ ->; rewrite divr1.
Qed. | Fact | intrP | 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_int",
"divr1",
"is_int",
"numq",
"ratP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrP x : reflect (exists n, x = n%:R) (is_nat x). | Proof.
apply: (iffP idP) => [/andP[]/[swap]/intrP[i ->]|[n ->]].
by rewrite ler0z; case: i => [n _|//]; exists n.
by rewrite /is_nat pmulrn ler0z denq_int.
Qed. | Fact | natrP | 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_int",
"intrP",
"is_nat",
"ler0z",
"pmulrn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorErat (x : rat) : Num.floor x = (numq x %/ denq x)%Z. | Proof. by []. Qed. | Lemma | floorErat | 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",
"floor",
"numq",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilErat (x : rat) : Num.ceil x = - (- numq x %/ denq x)%Z. | Proof. by []. Qed. | Lemma | ceilErat | 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",
... | [
"ceil",
"denq",
"numq",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Qint_def (x : rat) : (x \is a Num.int) = (denq x == 1). | Proof. by []. Qed. | Lemma | Qint_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",
... | [
"denq",
"int",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numqK : {in Num.int, cancel (fun x => numq x) intr}. | Proof. by move=> _ /intrP [x ->]; rewrite numq_int. Qed. | Lemma | numqK | 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",
"intr",
"intrP",
"numq",
"numq_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natq_div m n : (n %| m)%N -> (m %/ n)%:R = m%:R / n%:R :> rat. | Proof. exact/pchar0_natf_div/pchar_num. Qed. | Lemma | natq_div | 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",
... | [
"pchar0_natf_div",
"pchar_num",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr x : R | := (numq x)%:~R / (denq x)%:~R. | Definition | ratr | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_int z : ratr z%:~R = z%:~R. | Proof. by rewrite /ratr numq_int denq_int divr1. Qed. | Lemma | ratr_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_int",
"divr1",
"numq_int",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_nat n : ratr n%:R = n%:R. | Proof. exact: ratr_int n. Qed. | Lemma | ratr_nat | 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",
... | [
"ratr",
"ratr_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_rat (S : divringClosed R) a : ratr a \in S. | Proof. by rewrite rpred_div ?rpred_int. Qed. | Lemma | rpred_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",
... | [
"ratr",
"rpred_div",
"rpred_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_rat (aR : fieldType) rR (f : {rmorphism aR -> rR}) a :
f (ratr _ a) = ratr _ a. | Proof. by rewrite fmorph_div !rmorph_int. Qed. | Lemma | fmorph_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",
... | [
"fmorph_div",
"ratr",
"rmorph_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorph_eq_rat rR (f : {rmorphism rat -> rR}) : f =1 ratr _. | Proof. by move=> a; rewrite -{1}[a]divq_num_den fmorph_div !rmorph_int. Qed. | Lemma | fmorph_eq_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",
... | [
"divq_num_den",
"fmorph_div",
"rat",
"ratr",
"rmorph_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_linear U V (f : U -> V) : zmod_morphism f -> scalable f. | Proof.
move=> fB a u.
pose aM := GRing.isZmodMorphism.Build U V f fB.
pose phi : {additive U -> V} := HB.pack f aM.
rewrite -[f]/(phi : _ -> _) -{2}[a]divq_num_den mulrC -scalerA.
apply: canRL (scalerK _) _; first by rewrite intr_eq0 denq_neq0.
rewrite 2!scaler_int -3!raddfMz /=.
by rewrite -scalerMzr scalerMzl -mulrzr... | Lemma | rat_linear | 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",
... | [
"Build",
"additive",
"apply",
"denq_neq0",
"divq_num_den",
"intr_eq0",
"mulrC",
"mulrzr",
"numqE",
"raddfMz",
"scalable",
"scalerA",
"scalerK",
"scalerMzl",
"scalerMzr",
"scaler_int",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_is_zmod_morphism : zmod_morphism (@ratr F). | Proof.
have injZtoQ: @injective rat int intr by apply: intr_inj.
have nz_den x: (denq x)%:~R != 0 :> F by rewrite intr_eq0 denq_eq0.
move=> x y.
apply: (canLR (mulfK (nz_den _))); apply: (mulIf (nz_den x)).
rewrite mulrAC mulrBl divfK ?nz_den // mulrAC -!rmorphM.
apply: (mulIf (nz_den y)); rewrite mulrAC mulrBl divfK ?... | Fact | ratr_is_zmod_morphism | 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",
"denq_eq0",
"divfK",
"int",
"intr",
"intr_eq0",
"intr_inj",
"mulIf",
"mulfK",
"mulrA",
"mulrAC",
"mulrBl",
"numqE",
"rat",
"ratr",
"rmorphB",
"rmorphM",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_is_additive | := ratr_is_zmod_morphism. | Definition | ratr_is_additive | 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",
... | [
"ratr_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_is_monoid_morphism : monoid_morphism (@ratr F). | Proof.
have injZtoQ: @injective rat int intr by apply: intr_inj.
have nz_den x: (denq x)%:~R != 0 :> F by rewrite intr_eq0 denq_eq0.
split=> [|x y]; first by rewrite /ratr divr1.
rewrite /ratr mulrC mulrAC; apply: canLR (mulKf (nz_den _)) _; rewrite !mulrA.
do 2!apply: canRL (mulfK (nz_den _)) _; rewrite -!rmorphM; con... | Fact | ratr_is_monoid_morphism | 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",
"denq_eq0",
"divr1",
"int",
"intr",
"intr_eq0",
"intr_inj",
"monoid_morphism",
"mulKf",
"mulfK",
"mulrA",
"mulrAC",
"mulrC",
"mulrCA",
"numqE",
"rat",
"ratr",
"rmorphM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_is_multiplicative | :=
(fun g => (g.2,g.1)) ratr_is_monoid_morphism. | Definition | ratr_is_multiplicative | 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",
... | [
"ratr_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_rat : {mono (@ratr F) : x y / x <= y}. | Proof.
move=> x y /=; case: (ratP x) => nx dx cndx; case: (ratP y) => ny dy cndy.
rewrite !fmorph_div /= !ratr_int !ler_pdivlMr ?ltr0z //.
by rewrite ![_ / _ * _]mulrAC !ler_pdivrMr ?ltr0z // -!rmorphM /= !ler_int.
Qed. | Lemma | ler_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",
... | [
"fmorph_div",
"ler_int",
"ler_pdivlMr",
"ler_pdivrMr",
"ltr0z",
"mulrAC",
"ratP",
"ratr",
"ratr_int",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_rat : {mono (@ratr F) : x y / x < y}. | Proof. exact: leW_mono ler_rat. Qed. | Lemma | ltr_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",
... | [
"leW_mono",
"ler_rat",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0q x : (0 <= ratr F x) = (0 <= x). | Proof. by rewrite (_ : 0 = ratr F 0) ?ler_rat ?rmorph0. Qed. | Lemma | ler0q | 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",
... | [
"ler_rat",
"ratr",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerq0 x : (ratr F x <= 0) = (x <= 0). | Proof. by rewrite (_ : 0 = ratr F 0) ?ler_rat ?rmorph0. Qed. | Lemma | lerq0 | 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",
... | [
"ler_rat",
"ratr",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0q x : (0 < ratr F x) = (0 < x). | Proof. by rewrite (_ : 0 = ratr F 0) ?ltr_rat ?rmorph0. Qed. | Lemma | ltr0q | 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",
... | [
"ltr_rat",
"ratr",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrq0 x : (ratr F x < 0) = (x < 0). | Proof. by rewrite (_ : 0 = ratr F 0) ?ltr_rat ?rmorph0. Qed. | Lemma | ltrq0 | 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",
... | [
"ltr_rat",
"ratr",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_sg x : ratr F (sgr x) = sgr (ratr F x). | Proof. by rewrite !sgr_def fmorph_eq0 ltrq0 rmorphMn /= rmorph_sign. Qed. | Lemma | ratr_sg | 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",
... | [
"fmorph_eq0",
"ltrq0",
"ratr",
"rmorphMn",
"rmorph_sign",
"sgr",
"sgr_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ratr_norm x : ratr F `|x| = `|ratr F x|. | Proof.
by rewrite {2}[x]numEsign rmorphMsign normrMsign [`|ratr F _|]ger0_norm ?ler0q.
Qed. | Lemma | ratr_norm | 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",
... | [
"ger0_norm",
"ler0q",
"normrMsign",
"numEsign",
"ratr",
"rmorphMsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_rat : {morph ratr F : x y / Num.min x y}. | Proof. by move=> x y; rewrite !minEle ler_rat; case: leP. Qed. | Lemma | minr_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",
... | [
"leP",
"ler_rat",
"min",
"minEle",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_rat : {morph ratr F : x y / Num.max x y}. | Proof. by move=> x y; rewrite !maxEle ler_rat; case: leP. Qed. | Lemma | maxr_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",
... | [
"leP",
"ler_rat",
"max",
"maxEle",
"ratr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_rat : {mono (@ratr F) : x / Num.floor x}. | Proof.
move=> x; apply: floor_def; apply/andP; split.
- by rewrite -ratr_int ler_rat floor_le.
- by rewrite -ratr_int ltr_rat floorD1_gt.
Qed. | Lemma | floor_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",
... | [
"apply",
"floor",
"floorD1_gt",
"floor_def",
"floor_le",
"ler_rat",
"ltr_rat",
"ratr",
"ratr_int",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_rat : {mono (@ratr F) : x / Num.ceil x}. | Proof. by move=> x; rewrite !ceilNfloor -rmorphN floor_rat. Qed. | Lemma | ceil_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",
... | [
"ceil",
"ceilNfloor",
"floor_rat",
"ratr",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Qint_dvdz (m d : int) : (d %| m)%Z -> (m%:~R / d%:~R : rat) \is a Num.int. | Proof.
case/dvdzP=> z ->; rewrite rmorphM /=; have [->|dn0] := eqVneq d 0.
by rewrite mulr0 mul0r.
by rewrite mulfK ?intr_eq0.
Qed. | Lemma | Qint_dvdz | 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",
... | [
"dvdzP",
"eqVneq",
"int",
"intr_eq0",
"mul0r",
"mulfK",
"mulr0",
"rat",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Qnat_dvd (m d : nat) : (d %| m)%N -> (m%:R / d%:R : rat) \is a Num.nat. | Proof. by move=> h; rewrite natrEint divr_ge0 ?ler0n // !pmulrn Qint_dvdz. Qed. | Lemma | Qnat_dvd | 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",
... | [
"Qint_dvdz",
"divr_ge0",
"ler0n",
"nat",
"natrEint",
"pmulrn",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pZtoQ | := (map_poly (intr : int -> rat)). | Notation | pZtoQ | 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",
"intr",
"map_poly",
"rat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_rat_int_poly p : size (pZtoQ p) = size p. | Proof. by apply: size_map_inj_poly; first apply: intr_inj. Qed. | Lemma | size_rat_int_poly | 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",
"intr_inj",
"pZtoQ",
"size",
"size_map_inj_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_poly_scale (p : {poly rat}) :
{q : {poly int} & {a | a != 0 & p = a%:~R^-1 *: pZtoQ q}}. | Proof.
pose a := \prod_(i < size p) denq p`_i.
have nz_a: a != 0 by apply/prodf_neq0=> i _; apply: denq_neq0.
exists (map_poly numq (a%:~R *: p)), a => //.
apply: canRL (scalerK _) _; rewrite ?intr_eq0 //.
apply/polyP=> i; rewrite !(coefZ, coef_map_id0) // numqK // Qint_def mulrC.
have [ltip | /(nth_default 0)->] := lt... | Lemma | rat_poly_scale | 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",
... | [
"Qint_def",
"apply",
"bigD1",
"coefZ",
"coef_map_id0",
"denq",
"denq_int",
"denq_neq0",
"int",
"intr_eq0",
"last",
"ltnP",
"map_poly",
"mul0r",
"mulrA",
"mulrC",
"nth_default",
"numq",
"numqE",
"numqK",
"pZtoQ",
"poly",
"polyP",
"prodf_neq0",
"rat",
"rmorphM",
"sc... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdp_rat_int p q : (pZtoQ p %| pZtoQ q) = (p %| q). | Proof.
apply/dvdpP/Pdiv.Idomain.dvdpP=> [[/= r1 Dq] | [[/= a r] nz_a Dq]]; last first.
exists (a%:~R^-1 *: pZtoQ r).
by rewrite -scalerAl -rmorphM -Dq /= linearZ/= scalerK ?intr_eq0.
have [r [a nz_a Dr1]] := rat_poly_scale r1; exists (a, r) => //=.
apply: (map_inj_poly _ _ : injective pZtoQ) => //; first exact: int... | Lemma | dvdp_rat_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",
... | [
"apply",
"dvdpP",
"intr_eq0",
"intr_inj",
"last",
"linearZ",
"map_inj_poly",
"pZtoQ",
"r1",
"rat_poly_scale",
"rmorphM",
"scalerAl",
"scalerK",
"scalerKV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpP_rat_int p q :
p %| pZtoQ q ->
{p1 : {poly int} & {a | a != 0 & p = a *: pZtoQ p1} & {r | q = p1 * r}}. | Proof.
have{p} [p [a nz_a ->]] := rat_poly_scale p.
rewrite dvdpZl ?invr_eq0 ?intr_eq0 // dvdp_rat_int => dv_p_q.
exists (zprimitive p); last exact: dvdpP_int.
have [-> | nz_p] := eqVneq p 0.
by exists 1; rewrite ?oner_eq0 // zprimitive0 map_poly0 !scaler0.
exists ((zcontents p)%:~R / a%:~R).
by rewrite mulf_neq0 ?... | Lemma | dvdpP_rat_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",
... | [
"dvdpP_int",
"dvdpZl",
"dvdp_rat_int",
"eqVneq",
"int",
"intr_eq0",
"invr_eq0",
"last",
"map_poly0",
"map_polyZ",
"mulf_neq0",
"mulrC",
"nz_p",
"oner_eq0",
"pZtoQ",
"poly",
"rat_poly_scale",
"scaler0",
"scalerA",
"zcontents",
"zcontents_eq0",
"zpolyEprim",
"zprimitive",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irreducible_rat_int p :
irreducible_poly (pZtoQ p) <-> irreducible_poly p. | Proof.
rewrite /irreducible_poly size_rat_int_poly; split=> -[] p1 p_irr; split=> //.
move=> q q1; rewrite /eqp -!dvdp_rat_int => rq.
by apply/p_irr => //; rewrite size_rat_int_poly.
move=> q + /dvdpP_rat_int [] r [] c c0 qE [] s sE.
rewrite qE size_scale// size_rat_int_poly => r1.
apply/(eqp_trans (eqp_scale _ c0)... | Lemma | irreducible_rat_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",
... | [
"apply",
"c0",
"dvdpP_rat_int",
"dvdp_mulIl",
"dvdp_rat_int",
"eqp",
"eqp_scale",
"eqp_trans",
"irreducible_poly",
"pZtoQ",
"r1",
"size_rat_int_poly",
"size_scale",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inIntSpan (V : zmodType) m (s : m.-tuple V) v | :=
exists a : int ^ m, v = \sum_(i < m) s`_i *~ a i. | Definition | inIntSpan | 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",
"tuple"
] | Integral spans. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
solve_Qint_span (vT : vectType rat) m (s : m.-tuple vT) v :
{b : int ^ m &
{p : seq (int ^ m) &
forall a : int ^ m,
v = \sum_(i < m) s`_i *~ a i <->
exists c : seq int, a = b + \sum_(i < size p) p`_i *~ c`_i}} +
(~ inIntSpan s v). | Proof.
have s_s (i : 'I_m): s`_i \in <<s>>%VS by rewrite memv_span ?memt_nth.
have s_Zs a: \sum_(i < m) s`_i *~ a i \in <<s>>%VS.
by apply/rpred_sum => i _; apply/rpredMz.
case s_v: (v \in <<s>>%VS); last by right=> [[a Dv]]; rewrite Dv s_Zs in s_v.
move SE : (\matrix_(i < m, j < _) coord (vbasis <<s>>) j s`_i) => S.... | Lemma | solve_Qint_span | 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",
"addKr",
"addNKr",
"addr0",
"addrI",
"apply",
"basis_free",
"big1",
"bigD1",
"big_split",
"block_mxKul",
"card_ord",
"castmx",
"const_mx",
"coord",
"coord_free",
"coord_span",
"coord_vbasis",
"denq",
"denq_int",
"denq_neq0",
"dsubmx",
"eq_bigr",
"eq_row_base",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dec_Qint_span (vT : vectType rat) m (s : m.-tuple vT) v :
decidable (inIntSpan s v). | Proof.
have [[b [p aP]]|] := solve_Qint_span s v; last by right.
left; exists b; apply/(aP b); exists [::]; rewrite big1 ?addr0 // => i _.
by rewrite nth_nil mulr0z.
Qed. | Lemma | dec_Qint_span | 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",
... | [
"aP",
"addr0",
"apply",
"big1",
"decidable",
"inIntSpan",
"last",
"mulr0z",
"nth_nil",
"rat",
"solve_Qint_span",
"tuple",
"vT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eisenstein_crit (p : nat) (q : {poly int}) : prime p -> (size q != 1)%N ->
~~ (p %| lead_coef q)%Z -> ~~ (p ^+ 2 %| q`_0)%Z ->
(forall i, (i < (size q).-1)%N -> p %| q`_i)%Z ->
irreducible_poly q. | Proof.
move=> p_prime qN1 Ndvd_pql Ndvd_pq0 dvd_pq.
apply/irreducible_rat_int.
have qN0 : q != 0 by rewrite -lead_coef_eq0; apply: contraNneq Ndvd_pql => ->.
split.
rewrite size_map_poly_id0 ?intr_eq0 ?lead_coef_eq0//.
by rewrite ltn_neqAle eq_sym qN1 size_poly_gt0.
move=> f' +/dvdpP_rat_int[f [d dN0 feq]]; rewrite... | Lemma | eisenstein_crit | 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",
"big_ord1",
"coefM",
"coefXn",
"coefZ",
"coef_map",
"coef_poly",
"contraNneq",
"contra_neq",
"dvdpP_rat_int",
"dvdp_exp_XsubCP",
"dvdp_mull",
"dvdp_mulr",
"dvdz_mul",
"dvdz_pcharf",
"eqVneq",
"eq_from_nth",
"eq_sym",
"eqpP",
"eqp_dvdr",
"eqp_scale",
"eqp_sym",
"e... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rat_to_ring | :=
rewrite -?[0%Q]/(0 : rat)%R -?[1%Q]/(1 : rat)%R
-?[(_ - _)%Q]/(_ - _ : rat)%R -?[(_ / _)%Q]/(_ / _ : rat)%R
-?[(_ + _)%Q]/(_ + _ : rat)%R -?[(_ * _)%Q]/(_ * _ : rat)%R
-?[(- _)%Q]/(- _ : rat)%R -?[(_ ^-1)%Q]/(_ ^-1 : rat)%R /=. | Ltac | rat_to_ring | 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",
... | [
"rat"
] | Connecting rationals to the ring and field tactics | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ring_to_rat | :=
rewrite -?[0%R]/0%Q -?[1%R]/1%Q
-?[(_ - _)%R]/(_ - _)%Q -?[(_ / _)%R]/(_ / _)%Q
-?[(_ + _)%R]/(_ + _)%Q -?[(_ * _)%R]/(_ * _)%Q
-?[(- _)%R]/(- _)%Q -?[(_ ^-1)%R]/(_ ^-1)%Q /=. | Ltac | ring_to_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",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'rat' x // y ]" | := (@Rat (x, y) _) (only printing) : ring_scope. | Notation | [ 'rat' x // y ] | 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",
... | [] | Pretty printing or normal element of rat. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"[ 'rat' x // y ]" | :=
(@Rat (x : int, y : int) (fracq_subproof (x : int, y : int))) : ring_scope. | Notation | [ 'rat' x // y ] | 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_subproof",
"int"
] | For debugging purposes we provide the parsable version | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rat_vm_compute n (x : rat) : vm_compute_eq n%:Q x -> n%:Q = x. | Proof. exact. Qed. | Lemma | rat_vm_compute | 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",
... | [
"rat"
] | A specialization of vm_compute rewrite rule for pattern _%:Q | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pi_zero_quot_morph zqT | := PiMorph (@pi_zeror _ _ _ _ _ zqT). | Canonical | pi_zero_quot_morph | algebra | algebra/ring_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"Quotient.Exports"
] | [
"PiMorph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_opp_quot_morph zqT | := PiMorph1 (@pi_oppr _ _ _ _ _ zqT). | Canonical | pi_opp_quot_morph | algebra | algebra/ring_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"Quotient.Exports"
] | [
"PiMorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_add_quot_morph zqT | := PiMorph2 (@pi_addr _ _ _ _ _ zqT). | Canonical | pi_add_quot_morph | algebra | algebra/ring_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"Quotient.Exports"
] | [
"PiMorph2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_is_zmod_morphism : zmod_morphism \pi_Q. | Proof. by move=> x y /=; rewrite !piE. Qed. | Lemma | pi_is_zmod_morphism | algebra | algebra/ring_quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"generic_quotient",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"GRing.Theory",
"Quotient.Exports"
] | [
"piE",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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