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 |
|---|---|---|---|---|---|---|---|---|---|---|
ltrz1 n : (n%:~R < 1 :> R) = (n < 1). | Proof. by rewrite -[1]/(1%:~R) ltr_int. Qed. | Lemma | ltrz1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"ltr_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_eq0 n : (n%:~R == 0 :> R) = (n == 0). | Proof. by rewrite -(mulr0z 1) (inj_eq (mulrIz _)) // oner_eq0. Qed. | Lemma | intr_eq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"inj_eq",
"mulr0z",
"mulrIz",
"oner_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrz_eq0 x n : (x *~ n == 0) = ((n == 0) || (x == 0)). | Proof. by rewrite -mulrzl mulf_eq0 intr_eq0. Qed. | Lemma | mulrz_eq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intr_eq0",
"mulf_eq0",
"mulrzl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrz_neq0 x n : (x *~ n != 0) = ((n != 0) && (x != 0)). | Proof. by rewrite mulrz_eq0 negb_or. Qed. | Lemma | mulrz_neq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulrz_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realz n : (n%:~R : R) \in Num.real. | Proof. by rewrite -topredE /Num.real /= ler0z lerz0 le_total. Qed. | Lemma | realz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"le_total",
"ler0z",
"lerz0",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_inj | := @mulrIz 1 (oner_neq0 R). | Definition | intr_inj | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulrIz",
"oner_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz (R : unitRingType) (x : R) (n : int) | :=
match n with
| Posz n => x ^+ n
| Negz n => x ^- (n.+1)
end. | Definition | exprz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x ^ n" | := (exprz x n) : ring_scope. | Notation | x ^ n | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprnP x (n : nat) : x ^+ n = x ^ n. | Proof. by []. Qed. | Lemma | exprnP | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprnN x (n : nat) : x ^- n = x ^ (-n%:Z). | Proof. by case: n=> //; rewrite oppr0 expr0 invr1. Qed. | Lemma | exprnN | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"expr0",
"invr1",
"nat",
"oppr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr0z x : x ^ 0 = 1. | Proof. by []. Qed. | Lemma | expr0z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr1z x : x ^ 1 = x. | Proof. by []. Qed. | Lemma | expr1z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprN1 x : x ^ (-1) = x^-1. | Proof. by []. Qed. | Lemma | exprN1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_expz x n : (x ^ n)^-1 = x ^ (- n). | Proof. by case: (intP n)=> // [|m]; rewrite ?opprK ?expr0z ?invr1 // invrK. Qed. | Lemma | invr_expz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"expr0z",
"intP",
"invr1",
"invrK",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_inv x n : (x^-1) ^ n = x ^ (- n). | Proof. by case: (intP n)=> // m; rewrite -[_ ^ (- _)]exprVn ?opprK ?invrK. Qed. | Lemma | exprz_inv | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprVn",
"intP",
"invrK",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp1rz n : 1 ^ n = 1 :> R. | Proof. by case: (intP n)=> // m; rewrite -?exprz_inv ?invr1; apply: expr1n. Qed. | Lemma | exp1rz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"expr1n",
"exprz_inv",
"intP",
"invr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprSz x (n : nat) : x ^ n.+1 = x * x ^ n. | Proof. exact: exprS. Qed. | Lemma | exprSz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprS",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprSzr x (n : nat) : x ^ n.+1 = x ^ n * x. | Proof. exact: exprSr. Qed. | Lemma | exprSzr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprSr",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzD_nat x (m n : nat) : x ^ (m%:Z + n) = x ^ m * x ^ n. | Proof. exact: exprD. Qed. | Fact | exprzD_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprD",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzD_Nnat x (m n : nat) : x ^ (-m%:Z + -n%:Z) = x ^ (-m%:Z) * x ^ (-n%:Z). | Proof. by rewrite -opprD -!exprz_inv exprzD_nat. Qed. | Fact | exprzD_Nnat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprzD_nat",
"exprz_inv",
"nat",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzD_ss x m n : (0 <= m) && (0 <= n) || (m <= 0) && (n <= 0)
-> x ^ (m + n) = x ^ m * x ^ n. | Proof.
case: (intP m)=> {m} [|m|m]; case: (intP n)=> {n} [|n|n] //= _;
by rewrite ?expr0z ?mul1r ?exprzD_nat ?exprzD_Nnat ?sub0r ?addr0 ?mulr1.
Qed. | Lemma | exprzD_ss | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addr0",
"expr0z",
"exprzD_Nnat",
"exprzD_nat",
"intP",
"mul1r",
"mulr1",
"sub0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp0rz n : 0 ^ n = (n == 0)%:~R :> R. | Proof. by case: (intP n)=> // m; rewrite -?exprz_inv ?invr0 exprSz mul0r. Qed. | Lemma | exp0rz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprSz",
"exprz_inv",
"intP",
"invr0",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commrXz x y n : GRing.comm x y -> GRing.comm x (y ^ n). | Proof.
rewrite /GRing.comm; elim: n x y=> [|n ihn|n ihn] x y com_xy //=.
* by rewrite expr0z mul1r mulr1.
* by rewrite -exprnP commrX //.
rewrite -exprz_inv -exprnP commrX //.
case: (boolP (y \is a GRing.unit))=> uy; last by rewrite invr_out.
by apply/eqP; rewrite (can2_eq (mulrVK _) (mulrK _)) // -mulrA com_xy mulKr.
... | Lemma | commrXz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"can2_eq",
"comm",
"commrX",
"expr0z",
"exprnP",
"exprz_inv",
"invr_out",
"last",
"mul1r",
"mulKr",
"mulr1",
"mulrA",
"mulrK",
"mulrVK",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprMz_comm x y n : x \is a GRing.unit -> y \is a GRing.unit ->
GRing.comm x y -> (x * y) ^ n = x ^ n * y ^ n. | Proof.
move=> ux uy com_xy; elim: n => [|n _|n _]; first by rewrite expr0z mulr1.
by rewrite -!exprnP exprMn_comm.
rewrite -!exprnN -!exprVn com_xy -exprMn_comm ?invrM//.
exact/commrV/commr_sym/commrV.
Qed. | Lemma | exprMz_comm | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"comm",
"commrV",
"commr_sym",
"expr0z",
"exprMn_comm",
"exprVn",
"exprnN",
"exprnP",
"invrM",
"mulr1",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commrXz_wmulls x y n :
0 <= n -> GRing.comm x y -> (x * y) ^ n = x ^ n * y ^ n. | Proof.
move=> n0 com_xy; elim: n n0 => [|n _|n _] //; first by rewrite expr0z mulr1.
by rewrite -!exprnP exprMn_comm.
Qed. | Lemma | commrXz_wmulls | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"comm",
"expr0z",
"exprMn_comm",
"exprnP",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitrXz x n (ux : x \is a GRing.unit) : x ^ n \is a GRing.unit. | Proof.
case: (intP n)=> {n} [|n|n]; rewrite ?expr0z ?unitr1 ?unitrX //.
by rewrite -invr_expz unitrV unitrX.
Qed. | Lemma | unitrXz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"expr0z",
"intP",
"invr_expz",
"unit",
"unitr1",
"unitrV",
"unitrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzDr x (ux : x \is a GRing.unit) m n : x ^ (m + n) = x ^ m * x ^ n. | Proof.
move: n m; apply: wlog_le=> n m hnm.
by rewrite addrC hnm commrXz //; exact/commr_sym/commrXz.
case: (intP m) hnm=> {m} [|m|m]; rewrite ?mul1r ?add0r //;
case: (intP n)=> {n} [|n|n _]; rewrite ?mulr1 ?addr0 //;
do ?by rewrite exprzD_ss.
rewrite -invr_expz subzSS !exprSzr invrM ?unitrX // -mulrA mulVKr //.
... | Lemma | exprzDr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"add0r",
"addr0",
"addrC",
"apply",
"commrXz",
"commr_sym",
"exprSzr",
"exprzD_nat",
"exprzD_ss",
"intP",
"invrM",
"invr_expz",
"leqP",
"ltnW",
"mul1r",
"mulVKr",
"mulr1",
"mulrA",
"mulrK",
"opprB",
"subnK",
"subzSS",
"subzn",
"unit",
"unitrX",
"unitrXz",
"wlog_le... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_exp x m n : (x ^ m) ^ n = (x ^ (m * n)). | Proof.
wlog: n / 0 <= n.
by case: n=> [n -> //|n]; rewrite ?NegzE mulrN -?invr_expz=> -> /=.
elim: n x m=> [|n ihn|n ihn] x m // _; first by rewrite mulr0 !expr0z.
rewrite exprSz ihn // intS mulrDr mulr1 exprzD_ss //.
by case: (intP m)=> // m'; rewrite ?oppr_le0 //.
Qed. | Lemma | exprz_exp | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"expr0z",
"exprSz",
"exprzD_ss",
"intP",
"intS",
"invr_expz",
"mulr0",
"mulr1",
"mulrDr",
"mulrN",
"oppr_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzAC x m n : (x ^ m) ^ n = (x ^ n) ^ m. | Proof. by rewrite !exprz_exp mulrC. Qed. | Lemma | exprzAC | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz_exp",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_out x n (nux : x \isn't a GRing.unit) (hn : 0 <= n) :
x ^ (- n) = x ^ n. | Proof. by case: (intP n) hn=> //= m; rewrite -exprnN -exprVn invr_out. Qed. | Lemma | exprz_out | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprVn",
"exprnN",
"intP",
"invr_out",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_pMzl x m n : 0 <= n -> (x *~ m) ^ n = x ^ n *~ (m ^ n). | Proof.
by elim: n=> [|n ihn|n _] // _; rewrite !exprSz ihn // mulrzAr mulrzAl -mulrzA.
Qed. | Lemma | exprz_pMzl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprSz",
"mulrzA",
"mulrzAl",
"mulrzAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_pintl m n (hn : 0 <= n) : m%:~R ^ n = (m ^ n)%:~R :> R. | Proof. by rewrite exprz_pMzl // exp1rz. Qed. | Lemma | exprz_pintl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exp1rz",
"exprz_pMzl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzMzl x m n (ux : x \is a GRing.unit) (um : m%:~R \is a @GRing.unit R):
(x *~ m) ^ n = (m%:~R ^ n) * x ^ n :> R. | Proof. rewrite -[x *~ _]mulrzl exprMz_comm //; exact/commr_sym/commr_int. Qed. | Lemma | exprzMzl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"commr_int",
"commr_sym",
"exprMz_comm",
"mulrzl",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expNrz x n : (- x) ^ n = (-1) ^ n * x ^ n :> R. | Proof.
case: n=> [] n; rewrite ?NegzE; first exact: exprNn.
by rewrite -!exprz_inv !invrN invr1; apply: exprNn.
Qed. | Lemma | expNrz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"apply",
"exprNn",
"exprz_inv",
"invr1",
"invrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitr_n0expz x n :
n != 0 -> (x ^ n \is a GRing.unit) = (x \is a GRing.unit). | Proof.
by case: n => *; rewrite ?NegzE -?exprz_inv ?unitrX_pos ?unitrV ?lt0n.
Qed. | Lemma | unitr_n0expz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"exprz_inv",
"lt0n",
"unit",
"unitrV",
"unitrX_pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrV (n : int) :
n \in [:: 0; 1; -1] -> n%:~R ^-1 = n%:~R :> R. | Proof.
by case: (intP n)=> // [|[]|[]] //; rewrite ?rmorphN ?invrN (invr0, invr1).
Qed. | Lemma | intrV | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP",
"invr0",
"invr1",
"invrN",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rmorphXz (R' : unitRingType) (f : {rmorphism R -> R'}) n :
{in GRing.unit, {morph f : x / x ^ n}}. | Proof. by case: n => n x Ux; rewrite ?rmorphV ?rpredX ?rmorphXn. Qed. | Lemma | rmorphXz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"rmorphV",
"rmorphXn",
"rpredX",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expfz_eq0 x n : (x ^ n == 0) = (n != 0) && (x == 0). | Proof.
by case: n=> n; rewrite ?NegzE -?exprz_inv ?expf_eq0 ?lt0n ?invr_eq0.
Qed. | Lemma | expfz_eq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"expf_eq0",
"exprz_inv",
"invr_eq0",
"lt0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expfz_neq0 x n : x != 0 -> x ^ n != 0. | Proof. by move=> x_nz; rewrite expfz_eq0; apply/nandP; right. Qed. | Lemma | expfz_neq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"expfz_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprzMl x y n (ux : x \is a GRing.unit) (uy : y \is a GRing.unit) :
(x * y) ^ n = x ^ n * y ^ n. | Proof. by rewrite exprMz_comm //; apply: mulrC. Qed. | Lemma | exprzMl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprMz_comm",
"mulrC",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expfV (x : R) (i : int) : (x ^ i) ^-1 = (x ^-1) ^ i. | Proof. by rewrite invr_expz exprz_inv. Qed. | Lemma | expfV | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz_inv",
"int",
"invr_expz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expfzDr x m n : x != 0 -> x ^ (m + n) = x ^ m * x ^ n. | Proof. by move=> hx; rewrite exprzDr ?unitfE. Qed. | Lemma | expfzDr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprzDr",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expfz_n0addr x m n : m + n != 0 -> x ^ (m + n) = x ^ m * x ^ n. | Proof.
have [-> hmn|nx0 _] := eqVneq x 0; last exact: expfzDr.
rewrite !exp0rz (negPf hmn).
case: (eqVneq m 0) hmn => [->|]; rewrite (mul0r, mul1r) //.
by rewrite add0r=> /negPf->.
Qed. | Lemma | expfz_n0addr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"add0r",
"eqVneq",
"exp0rz",
"expfzDr",
"last",
"mul0r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expfzMl x y n : (x * y) ^ n = x ^ n * y ^ n. | Proof.
have [->|/negPf n0] := eqVneq n 0; first by rewrite !expr0z mulr1.
case: (boolP ((x * y) == 0)); rewrite ?mulf_eq0.
by case/pred2P=> ->; rewrite ?(mul0r, mulr0, exp0rz, n0).
by case/norP=> x0 y0; rewrite exprzMl ?unitfE.
Qed. | Lemma | expfzMl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eqVneq",
"exp0rz",
"expr0z",
"exprzMl",
"mul0r",
"mulf_eq0",
"mulr0",
"mulr1",
"pred2P",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmorphXz (R : unitRingType) (f : {rmorphism F -> R}) n :
{morph f : x / x ^ n}. | Proof. by case: n => n x; rewrite ?fmorphV rmorphXn. Qed. | Lemma | fmorphXz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"fmorphV",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_ge0 n x (hx : 0 <= x) : (0 <= x ^ n). | Proof. by case: n => n; rewrite ?invr_ge0 ?exprn_ge0. Qed. | Lemma | exprz_ge0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprn_ge0",
"invr_ge0"
] | ler and exprz | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
exprz_gt0 n x (hx : 0 < x) : (0 < x ^ n). | Proof. by case: n => n; rewrite ?invr_gt0 ?exprn_gt0. Qed. | Lemma | exprz_gt0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprn_gt0",
"invr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_gte0 | := (exprz_ge0, exprz_gt0). | Definition | exprz_gte0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz_ge0",
"exprz_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wpiXz2l x (x0 : 0 <= x) (x1 : x <= 1) :
{in >= 0 &, {homo exprz x : x y /~ x <= y}}. | Proof.
move=> [] m [] n; rewrite -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
by rewrite lez_nat -?exprnP => /ler_wiXn2l; apply.
Qed. | Lemma | ler_wpiXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprnP",
"exprz",
"ler_wiXn2l",
"lez_nat",
"ltz_nat",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wpeXz2l x (x1 : 1 <= x) : {in >= 0 &, {homo exprz x : x y / x <= y}}. | Proof.
move=> [] m [] n; rewrite -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
by rewrite lez_nat -?exprnP=> /ler_weXn2l; apply.
Qed. | Fact | ler_wpeXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprnP",
"exprz",
"ler_weXn2l",
"lez_nat",
"ltz_nat",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pexprz_eq1 x n (x0 : 0 <= x) : (x ^ n == 1) = ((n == 0) || (x == 1)). | Proof.
case: n=> n; rewrite ?NegzE -?exprz_inv ?oppr_eq0 pexprn_eq1 // ?invr_eq1 //.
by rewrite invr_ge0.
Qed. | Lemma | pexprz_eq1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"exprz_inv",
"invr_eq1",
"invr_ge0",
"oppr_eq0",
"pexprn_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wpXz2r n (hn : 0 <= n) :
{in >= 0 & , {homo (@exprz R)^~ n : x y / x <= y}}. | Proof. by case: n hn=> // n _; exact: lerXn2r. Qed. | Lemma | ler_wpXz2r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"lerXn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wniXz2l x (x0 : 0 <= x) (x1 : x <= 1) :
{in < 0 &, {homo exprz x : x y /~ x <= y}}. | Proof.
move=> [] m [] n; rewrite ?NegzE -!topredE /= ?oppr_cp0 ?ltz_nat // => _ _.
rewrite lerN2 lez_nat -?invr_expz=> hmn; have := x0.
rewrite le0r=> /predU1P [->|lx0]; first by rewrite !exp0rz invr0.
by rewrite lef_pV2 -?topredE /= ?exprz_gt0 // ler_wiXn2l.
Qed. | Lemma | ler_wniXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"exp0rz",
"exprz",
"exprz_gt0",
"invr0",
"invr_expz",
"le0r",
"lef_pV2",
"lerN2",
"ler_wiXn2l",
"lez_nat",
"ltz_nat",
"oppr_cp0",
"predU1P"
] | ler and exprz | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ler_wneXz2l x (x1 : 1 <= x) : {in <= 0 &, {homo exprz x : x y / x <= y}}. | Proof.
move=> m n hm hn /= hmn.
rewrite -lef_pV2 -?topredE /= ?exprz_gt0 ?(lt_le_trans ltr01) //.
by rewrite !invr_expz ler_wpeXz2l ?lerN2 -?topredE //= oppr_cp0.
Qed. | Fact | ler_wneXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"exprz_gt0",
"invr_expz",
"lef_pV2",
"lerN2",
"ler_wpeXz2l",
"lt_le_trans",
"ltr01",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_weXz2l x (x1 : 1 <= x) : {homo exprz x : x y / x <= y}. | Proof.
move=> m n /= hmn; case: (lerP 0 m)=> [|/ltW] hm.
by rewrite ler_wpeXz2l // [_ \in _](le_trans hm).
case: (lerP n 0)=> [|/ltW] hn.
by rewrite ler_wneXz2l // [_ \in _](le_trans hmn).
apply: (@le_trans _ _ (x ^ 0)); first by rewrite ler_wneXz2l.
by rewrite ler_wpeXz2l.
Qed. | Lemma | ler_weXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprz",
"le_trans",
"lerP",
"ler_wneXz2l",
"ler_wpeXz2l",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ieexprIz x (x0 : 0 < x) (nx1 : x != 1) : injective (exprz x). | Proof.
apply: wlog_lt=> // m n hmn; first by move=> hmn'; rewrite hmn.
move=> /(f_equal ( *%R^~ (x ^ (- n)))).
rewrite -!expfzDr ?gt_eqF // subrr expr0z=> /eqP.
by rewrite pexprz_eq1 ?(ltW x0) // (negPf nx1) subr_eq0 orbF=> /eqP.
Qed. | Lemma | ieexprIz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"expfzDr",
"expr0z",
"exprz",
"gt_eqF",
"ltW",
"pexprz_eq1",
"subr_eq0",
"subrr",
"wlog_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_piXz2l x (x0 : 0 < x) (x1 : x < 1) :
{in >= 0 &, {mono exprz x : x y /~ x <= y}}. | Proof.
apply: (le_nmono_in (inj_nhomo_lt_in _ _)).
by move=> n m hn hm /=; apply: ieexprIz; rewrite // lt_eqF.
by apply: ler_wpiXz2l; rewrite ?ltW.
Qed. | Lemma | ler_piXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprz",
"ieexprIz",
"inj_nhomo_lt_in",
"le_nmono_in",
"ler_wpiXz2l",
"ltW",
"lt_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_piXz2l x (x0 : 0 < x) (x1 : x < 1) :
{in >= 0 &, {mono exprz x : x y /~ x < y}}. | Proof. exact: (leW_nmono_in (ler_piXz2l _ _)). Qed. | Lemma | ltr_piXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"leW_nmono_in",
"ler_piXz2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_niXz2l x (x0 : 0 < x) (x1 : x < 1) :
{in < 0 &, {mono exprz x : x y /~ x <= y}}. | Proof.
apply: (le_nmono_in (inj_nhomo_lt_in _ _)).
by move=> n m hn hm /=; apply: ieexprIz; rewrite // lt_eqF.
by apply: ler_wniXz2l; rewrite ?ltW.
Qed. | Lemma | ler_niXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprz",
"ieexprIz",
"inj_nhomo_lt_in",
"le_nmono_in",
"ler_wniXz2l",
"ltW",
"lt_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_niXz2l x (x0 : 0 < x) (x1 : x < 1) :
{in < 0 &, {mono (exprz x) : x y /~ x < y}}. | Proof. exact: (leW_nmono_in (ler_niXz2l _ _)). Qed. | Lemma | ltr_niXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"leW_nmono_in",
"ler_niXz2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_eXz2l x (x1 : 1 < x) : {mono exprz x : x y / x <= y}. | Proof.
apply: (le_mono (inj_homo_lt _ _)).
by apply: ieexprIz; rewrite ?(lt_trans ltr01) // gt_eqF.
by apply: ler_weXz2l; rewrite ?ltW.
Qed. | Lemma | ler_eXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprz",
"gt_eqF",
"ieexprIz",
"inj_homo_lt",
"le_mono",
"ler_weXz2l",
"ltW",
"lt_trans",
"ltr01"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_eXz2l x (x1 : 1 < x) : {mono exprz x : x y / x < y}. | Proof. exact: (leW_mono (ler_eXz2l _)). Qed. | Lemma | ltr_eXz2l | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"leW_mono",
"ler_eXz2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wnXz2r n (hn : n <= 0) :
{in > 0 & , {homo (@exprz R)^~ n : x y /~ x <= y}}. | Proof.
move=> x y /= hx hy hxy; rewrite -lef_pV2 ?[_ \in _]exprz_gt0 //.
by rewrite !invr_expz ler_wpXz2r ?[_ \in _]ltW // oppr_cp0.
Qed. | Lemma | ler_wnXz2r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"exprz_gt0",
"invr_expz",
"lef_pV2",
"ler_wpXz2r",
"ltW",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pexpIrz n (n0 : n != 0) : {in >= 0 &, injective ((@exprz R)^~ n)}. | Proof.
move=> x y; rewrite ![_ \in _]le0r=> /predU1P [-> _ /eqP|hx].
by rewrite exp0rz ?(negPf n0) eq_sym expfz_eq0=> /andP [_ /eqP->].
case/predU1P=> [-> /eqP|hy].
by rewrite exp0rz ?(negPf n0) expfz_eq0=> /andP [_ /eqP].
move=> /(f_equal ( *%R^~ (y ^ (- n)))) /eqP.
rewrite -expfzDr ?(gt_eqF hy) // subrr expr0z -e... | Lemma | pexpIrz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"can2_eq",
"eq_sym",
"exp0rz",
"expfzDr",
"expfzMl",
"expfz_eq0",
"expr0z",
"exprz",
"exprz_inv",
"gt_eqF",
"invr_ge0",
"le0r",
"ltW",
"mul1r",
"mulrK",
"mulrVK",
"mulr_ge0",
"pexprz_eq1",
"predU1P",
"subrr",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nexpIrz n (n0 : n != 0) : {in <= 0 &, injective ((@exprz R)^~ n)}. | Proof.
move=> x y; rewrite ![_ \in _]le_eqVlt => /predU1P [-> _ /eqP|hx].
by rewrite exp0rz ?(negPf n0) eq_sym expfz_eq0=> /andP [_ /eqP->].
case/predU1P=> [-> /eqP|hy].
by rewrite exp0rz ?(negPf n0) expfz_eq0=> /andP [_ /eqP].
move=> /(f_equal ( *%R^~ (y ^ (- n)))) /eqP.
rewrite -expfzDr ?(lt_eqF hy) // subrr expr... | Lemma | nexpIrz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"can2_eq",
"eq_sym",
"exp0rz",
"expfzDr",
"expfzMl",
"expfz_eq0",
"expr0z",
"exprz",
"exprz_inv",
"invr_le0",
"le_eqVlt",
"ltW",
"lt_eqF",
"mul1r",
"mulrK",
"mulrVK",
"mulr_le0",
"pexprz_eq1",
"predU1P",
"subrr",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pXz2r n (hn : 0 < n) :
{in >= 0 & , {mono ((@exprz R)^~ n) : x y / x <= y}}. | Proof.
apply: le_mono_in (inj_homo_lt_in _ _).
by move=> x y hx hy /=; apply: pexpIrz; rewrite // gt_eqF.
by apply: ler_wpXz2r; rewrite ltW.
Qed. | Lemma | ler_pXz2r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprz",
"gt_eqF",
"inj_homo_lt_in",
"le_mono_in",
"ler_wpXz2r",
"ltW",
"pexpIrz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pXz2r n (hn : 0 < n) :
{in >= 0 & , {mono ((@exprz R)^~ n) : x y / x < y}}. | Proof. exact: leW_mono_in (ler_pXz2r _). Qed. | Lemma | ltr_pXz2r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"leW_mono_in",
"ler_pXz2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nXz2r n (hn : n < 0) :
{in > 0 & , {mono ((@exprz R)^~ n) : x y /~ x <= y}}. | Proof.
apply: le_nmono_in (inj_nhomo_lt_in _ _); last first.
by apply: ler_wnXz2r; rewrite ltW.
by move=> x y hx hy /=; apply: pexpIrz; rewrite ?[_ \in _]ltW ?lt_eqF.
Qed. | Lemma | ler_nXz2r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"apply",
"exprz",
"inj_nhomo_lt_in",
"last",
"le_nmono_in",
"ler_wnXz2r",
"ltW",
"lt_eqF",
"pexpIrz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nXz2r n (hn : n < 0) :
{in > 0 & , {mono ((@exprz R)^~ n) : x y /~ x < y}}. | Proof. exact: leW_nmono_in (ler_nXz2r _). Qed. | Lemma | ltr_nXz2r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprz",
"leW_nmono_in",
"ler_nXz2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqrXz2 n x y : n != 0 -> 0 <= x -> 0 <= y -> (x ^ n == y ^ n) = (x == y). | Proof. by move=> *; rewrite (inj_in_eq (pexpIrz _)). Qed. | Lemma | eqrXz2 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"inj_in_eq",
"pexpIrz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz x : int | := if x == 0 then 0 else if x < 0 then -1 else 1. | Definition | sgz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_def x : sgz x = (-1) ^+ (x < 0)%R *+ (x != 0). | Proof. by rewrite /sgz; case: (_ == _); case: (_ < _). Qed. | Lemma | sgz_def | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrEz x : sgr x = (sgz x)%:~R. | Proof. by rewrite !(fun_if intr). Qed. | Lemma | sgrEz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intr",
"sgr",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_sgz x : 0 < x -> sgz x = 1. | Proof. by move=> x_gt0; rewrite /sgz lt_neqAle andbC eq_le lt_geF. Qed. | Lemma | gtr0_sgz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eq_le",
"lt_geF",
"lt_neqAle",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0_sgz x : x < 0 -> sgz x = -1. | Proof. by move=> x_lt0; rewrite /sgz eq_sym eq_le x_lt0 lt_geF. Qed. | Lemma | ltr0_sgz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eq_le",
"eq_sym",
"lt_geF",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz0 : sgz (0 : R) = 0. | Proof. by rewrite /sgz eqxx. Qed. | Lemma | sgz0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eqxx",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz1 : sgz (1 : R) = 1. | Proof. by rewrite gtr0_sgz // ltr01. Qed. | Lemma | sgz1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"gtr0_sgz",
"ltr01",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzN1 : sgz (-1 : R) = -1. | Proof. by rewrite ltr0_sgz // ltrN10. Qed. | Lemma | sgzN1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"ltr0_sgz",
"ltrN10",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzE | := (sgz0, sgz1, sgzN1). | Definition | sgzE | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz0",
"sgz1",
"sgzN1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_sgr x : sgz (sgr x) = sgz x. | Proof. by rewrite !(fun_if sgz) !sgzE. Qed. | Lemma | sgz_sgr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgr",
"sgz",
"sgzE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_sgz x : `|sgz x| = (x != 0). | Proof. by rewrite sgz_def -mulr_natr normrMsign normr_nat natz. Qed. | Lemma | normr_sgz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulr_natr",
"natz",
"normrMsign",
"normr_nat",
"sgz",
"sgz_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_sg x : `|sgr x| = (x != 0)%:~R. | Proof. by rewrite sgr_def -mulr_natr normrMsign normr_nat. Qed. | Lemma | normr_sg | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulr_natr",
"normrMsign",
"normr_nat",
"sgr",
"sgr_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_int m : sgz (m%:~R : R) = sgz m. | Proof. by rewrite /sgz intr_eq0 ltrz0. Qed. | Lemma | sgz_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intr_eq0",
"ltrz0",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrz (n : int) : sgr n = sgz n. | Proof. by rewrite sgrEz intz. Qed. | Lemma | sgrz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intz",
"sgr",
"sgrEz",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_sg m : (sgr m)%:~R = sgr (m%:~R) :> R. | Proof. by rewrite sgrz -sgz_int -sgrEz. Qed. | Lemma | intr_sg | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgr",
"sgrEz",
"sgrz",
"sgz_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_id (x : R) : sgz (sgz x) = sgz x. | Proof. by rewrite !(fun_if (@sgz _)). Qed. | Lemma | sgz_id | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_cp0 x :
((sgz x == 1) = (0 < x)) *
((sgz x == -1) = (x < 0)) *
((sgz x == 0) = (x == 0)). | Proof. by rewrite /sgz; case: ltrgtP. Qed. | Lemma | sgz_cp0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"ltrgtP",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_val x : bool -> bool -> bool -> bool -> bool -> bool
-> bool -> bool -> bool -> bool -> bool -> bool
-> bool -> bool -> bool -> bool -> bool -> bool
-> R -> R -> int -> Set | :=
| SgzNull of x = 0 : sgz_val x true true true true false false
true false false true false false true false false true false false 0 0 0
| SgzPos of x > 0 : sgz_val x false false true false false true
false false true false false true false false true false false true x 1 1
| SgzNeg of x < 0 : sgz_val ... | Variant | sgz_val | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzP x :
sgz_val x (0 == x) (x <= 0) (0 <= x) (x == 0) (x < 0) (0 < x)
(0 == sgr x) (-1 == sgr x) (1 == sgr x)
(sgr x == 0) (sgr x == -1) (sgr x == 1)
(0 == sgz x) (-1 == sgz x) (1 == sgz x)
(sgz x == 0) (sgz x == -1) (sgz x == 1) `|x| (sgr x) (sgz x). | Proof.
rewrite ![_ == sgz _]eq_sym ![_ == sgr _]eq_sym !sgr_cp0 !sgz_cp0.
by rewrite /sgz; case: sgrP; constructor.
Qed. | Lemma | sgzP | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eq_sym",
"sgr",
"sgrP",
"sgr_cp0",
"sgz",
"sgz_cp0",
"sgz_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzN x : sgz (- x) = - sgz x. | Proof. by rewrite /sgz oppr_eq0 oppr_lt0; case: ltrgtP. Qed. | Lemma | sgzN | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"ltrgtP",
"oppr_eq0",
"oppr_lt0",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_sg x : sgz x * sgz x = (x != 0)%:~R. | Proof. by case: sgzP; rewrite ?(mulr0, mulr1, mulrNN). Qed. | Lemma | mulz_sg | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulr0",
"mulr1",
"mulrNN",
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_sg_eq1 x y : (sgz x * sgz y == 1) = (x != 0) && (sgz x == sgz y). | Proof.
do 2?case: sgzP=> _; rewrite ?(mulr0, mulr1, mulrN1, opprK, oppr0, eqxx);
by rewrite ?[0 == 1]eq_sym ?oner_eq0 //= eqr_oppLR oppr0 oner_eq0.
Qed. | Lemma | mulz_sg_eq1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eq_sym",
"eqr_oppLR",
"eqxx",
"mulr0",
"mulr1",
"mulrN1",
"oner_eq0",
"oppr0",
"opprK",
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_sg_eqN1 x y : (sgz x * sgz y == -1) = (x != 0) && (sgz x == - sgz y). | Proof. by rewrite -eqr_oppLR -mulrN -sgzN mulz_sg_eq1. Qed. | Lemma | mulz_sg_eqN1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eqr_oppLR",
"mulrN",
"mulz_sg_eq1",
"sgz",
"sgzN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzM x y : sgz (x * y) = sgz x * sgz y. | Proof.
rewrite -sgz_sgr -(sgz_sgr x) -(sgz_sgr y) sgrM.
by case: sgrP; case: sgrP; rewrite /sgz ?(mulNr, mul0r, mul1r);
rewrite ?(oppr_eq0, oppr_cp0, eqxx, ltxx, ltr01, ltr10, oner_eq0).
Qed. | Lemma | sgzM | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"eqxx",
"ltr01",
"ltr10",
"ltxx",
"mul0r",
"mul1r",
"mulNr",
"oner_eq0",
"oppr_cp0",
"oppr_eq0",
"sgrM",
"sgrP",
"sgz",
"sgz_sgr"
] | Proof. by do 3!case: sgrP=> _. Qed. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sgzX (n : nat) x : sgz (x ^+ n) = (sgz x) ^+ n. | Proof. by elim: n => [|n IHn]; rewrite ?sgz1 // !exprS sgzM IHn. Qed. | Lemma | sgzX | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"exprS",
"nat",
"sgz",
"sgz1",
"sgzM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_eq0 x : (sgz x == 0) = (x == 0). | Proof. by rewrite sgz_cp0. Qed. | Lemma | sgz_eq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz",
"sgz_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_odd (n : nat) x : x != 0 -> (sgz x) ^+ n = (sgz x) ^+ (odd n). | Proof. by case: sgzP => //=; rewrite ?expr1n // signr_odd. Qed. | Lemma | sgz_odd | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"expr1n",
"nat",
"odd",
"sgz",
"sgzP",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_gt0 x : (sgz x > 0) = (x > 0). | Proof. by case: sgzP. Qed. | Lemma | sgz_gt0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_lt0 x : (sgz x < 0) = (x < 0). | Proof. by case: sgzP. Qed. | Lemma | sgz_lt0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_ge0 x : (sgz x >= 0) = (x >= 0). | Proof. by case: sgzP. Qed. | Lemma | sgz_ge0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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