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 |
|---|---|---|---|---|---|---|---|---|---|---|
ler_peMr x y : 0 <= y -> 1 <= x -> y <= y * x. | Proof. by move=> hy hx; rewrite -{1}[y]mulr1 ler_wpM2l. Qed. | Lemma | ler_peMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_wpM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_neMr x y : y <= 0 -> 1 <= x -> y * x <= y. | Proof. by move=> hy hx; rewrite -{2}[y]mulr1 ler_wnM2l. Qed. | Lemma | ler_neMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_wnM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_piMl x y : 0 <= y -> x <= 1 -> x * y <= y. | Proof. by move=> hy hx; rewrite -{2}[y]mul1r ler_wpM2r. Qed. | Lemma | ler_piMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_wpM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_niMl x y : y <= 0 -> x <= 1 -> y <= x * y. | Proof. by move=> hy hx; rewrite -{1}[y]mul1r ler_wnM2r. Qed. | Lemma | ler_niMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_wnM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_piMr x y : 0 <= y -> x <= 1 -> y * x <= y. | Proof. by move=> hy hx; rewrite -{2}[y]mulr1 ler_wpM2l. Qed. | Lemma | ler_piMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_wpM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_niMr x y : y <= 0 -> x <= 1 -> y <= y * x. | Proof. by move=> hx hy; rewrite -{1}[y]mulr1 ler_wnM2l. Qed. | Lemma | ler_niMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_wnM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_ile1 x y : 0 <= x -> 0 <= y -> x <= 1 -> y <= 1 -> x * y <= 1. | Proof. by move=> *; rewrite (@le_trans _ _ y) ?ler_piMl. Qed. | Lemma | mulr_ile1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"le_trans",
"ler_piMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prodr_ile1 {I : Type} (s : seq I) (P : pred I) (F : I -> R) :
(forall i, P i -> 0 <= F i <= 1) -> \prod_(j <- s | P j) F j <= 1. | Proof.
elim: s => [_ | y s ih xs01]; rewrite ?big_nil// big_cons.
case: ifPn => Py; last by rewrite ih.
have /andP[y0 y1] : 0 <= F y <= 1 by rewrite xs01// mem_head.
rewrite mulr_ile1 ?andbT//; last first.
by rewrite ih// => e xs; rewrite xs01// in_cons xs orbT.
by rewrite prodr_ge0// => x /xs01 /andP[].
Qed. | Lemma | prodr_ile1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"big_cons",
"big_nil",
"in_cons",
"last",
"mem_head",
"mulr_ile1",
"prodr_ge0",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_ilt1 x y : 0 <= x -> 0 <= y -> x < 1 -> y < 1 -> x * y < 1. | Proof. by move=> *; rewrite (@le_lt_trans _ _ y) ?ler_piMl // ltW. Qed. | Lemma | mulr_ilt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"le_lt_trans",
"ler_piMl",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_ilte1 | := (mulr_ile1, mulr_ilt1). | Definition | mulr_ilte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulr_ile1",
"mulr_ilt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_ege1 x y : 1 <= x -> 1 <= y -> 1 <= x * y. | Proof.
by move=> le1x le1y; rewrite (@le_trans _ _ y) ?ler_peMl // (le_trans ler01).
Qed. | Lemma | mulr_ege1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"le1x",
"le_trans",
"ler01",
"ler_peMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_egt1 x y : 1 < x -> 1 < y -> 1 < x * y. | Proof.
by move=> le1x lt1y; rewrite (@lt_trans _ _ y) // ltr_pMl // (lt_trans ltr01).
Qed. | Lemma | mulr_egt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"le1x",
"lt_trans",
"ltr01",
"ltr_pMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_egte1 | := (mulr_ege1, mulr_egt1). | Definition | mulr_egte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulr_ege1",
"mulr_egt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_cp1 | := (mulr_ilte1, mulr_egte1). | Definition | mulr_cp1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulr_egte1",
"mulr_ilte1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_gt0 x : (0 < x^-1) = (0 < x). | Proof.
have [ux | nux] := boolP (x \is a GRing.unit); last by rewrite invr_out.
by apply/idP/idP=> /ltr_pM2r <-; rewrite mul0r (mulrV, mulVr) ?ltr01.
Qed. | Lemma | invr_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"invr_out",
"last",
"ltr01",
"ltr_pM2r",
"mul0r",
"mulVr",
"mulrV",
"unit"
] | ler and ^-1 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
invr_ge0 x : (0 <= x^-1) = (0 <= x). | Proof. by rewrite !le0r invr_gt0 invr_eq0. Qed. | Lemma | invr_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_eq0",
"invr_gt0",
"le0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_lt0 x : (x^-1 < 0) = (x < 0). | Proof. by rewrite -oppr_cp0 -invrN invr_gt0 oppr_cp0. Qed. | Lemma | invr_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invrN",
"invr_gt0",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_le0 x : (x^-1 <= 0) = (x <= 0). | Proof. by rewrite -oppr_cp0 -invrN invr_ge0 oppr_cp0. Qed. | Lemma | invr_le0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invrN",
"invr_ge0",
"oppr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_gte0 | := (invr_ge0, invr_gt0). | Definition | invr_gte0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_ge0",
"invr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_lte0 | := (invr_le0, invr_lt0). | Definition | invr_lte0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_le0",
"invr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divr_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x / y. | Proof. by move=> x_ge0 y_ge0; rewrite mulr_ge0 ?invr_ge0. Qed. | Lemma | divr_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_ge0",
"mulr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divDl_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x / (x + y). | Proof. by move=> *; rewrite divr_ge0// addr_ge0. Qed. | Lemma | divDl_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"addr_ge0",
"divr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divr_gt0 x y : 0 < x -> 0 < y -> 0 < x / y. | Proof. by move=> x_gt0 y_gt0; rewrite pmulr_rgt0 ?invr_gt0. Qed. | Lemma | divr_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_gt0",
"pmulr_rgt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realV : {mono (@GRing.inv R) : x / x \is real}. | Proof. exact: rpredV. Qed. | Lemma | realV | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inv",
"real",
"rpredV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_ge0 n x : 0 <= x -> 0 <= x ^+ n. | Proof. by move=> xge0; rewrite -nnegrE rpredX. Qed. | Lemma | exprn_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"nnegrE",
"rpredX"
] | ler and exprn | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
realX n : {in real, forall x, x ^+ n \is real}. | Proof. exact: rpredX. Qed. | Lemma | realX | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"real",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_gt0 n x : 0 < x -> 0 < x ^+ n. | Proof.
by rewrite !lt0r expf_eq0 => /andP[/negPf-> /exprn_ge0->]; rewrite andbF.
Qed. | Lemma | exprn_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expf_eq0",
"exprn_ge0",
"lt0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_gte0 | := (exprn_ge0, exprn_gt0). | Definition | exprn_gte0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprn_ge0",
"exprn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_ile1 n x : 0 <= x -> x <= 1 -> x ^+ n <= 1. | Proof.
move=> xge0 xle1; elim: n=> [|*]; rewrite ?expr0 // exprS.
by rewrite mulr_ile1 ?exprn_ge0.
Qed. | Lemma | exprn_ile1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr0",
"exprS",
"exprn_ge0",
"mulr_ile1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_ilt1 n x : 0 <= x -> x < 1 -> (x ^+ n < 1) = (n != 0). | Proof.
move=> xge0 xlt1.
case: n; [by rewrite eqxx ltxx | elim=> [|n ihn]; first by rewrite expr1].
by rewrite exprS mulr_ilt1 // exprn_ge0.
Qed. | Lemma | exprn_ilt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqxx",
"expr1",
"exprS",
"exprn_ge0",
"ltxx",
"mulr_ilt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_ilte1 | := (exprn_ile1, exprn_ilt1). | Definition | exprn_ilte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprn_ile1",
"exprn_ilt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_ege1 n x : 1 <= x -> 1 <= x ^+ n. | Proof.
by move=> x_ge1; elim: n=> [|n ihn]; rewrite ?expr0 // exprS mulr_ege1.
Qed. | Lemma | exprn_ege1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr0",
"exprS",
"mulr_ege1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_egt1 n x : 1 < x -> (1 < x ^+ n) = (n != 0). | Proof.
move=> xgt1; case: n; first by rewrite eqxx ltxx.
by elim=> [|n ihn]; rewrite ?expr1// exprS mulr_egt1 // exprn_ge0.
Qed. | Lemma | exprn_egt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqxx",
"expr1",
"exprS",
"exprn_ge0",
"ltxx",
"mulr_egt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_egte1 | := (exprn_ege1, exprn_egt1). | Definition | exprn_egte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprn_ege1",
"exprn_egt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_cp1 | := (exprn_ilte1, exprn_egte1). | Definition | exprn_cp1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprn_egte1",
"exprn_ilte1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_iXnr x n : (0 < n)%N -> 0 <= x -> x <= 1 -> x ^+ n <= x. | Proof. by case: n => n // *; rewrite exprS ler_piMr // exprn_ile1. Qed. | Lemma | ler_iXnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprS",
"exprn_ile1",
"ler_piMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_iXnr x n : 0 < x -> x < 1 -> (x ^+ n < x) = (1 < n)%N. | Proof.
case: n=> [|[|n]] //; first by rewrite expr0 => _ /lt_gtF ->.
by move=> x0 x1; rewrite exprS gtr_pMr // ?exprn_ilt1 // ltW.
Qed. | Lemma | ltr_iXnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr0",
"exprS",
"exprn_ilt1",
"gtr_pMr",
"ltW",
"lt_gtF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_iXnr | := (ler_iXnr, ltr_iXnr). | Definition | lter_iXnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_iXnr",
"ltr_iXnr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_eXnr x n : (0 < n)%N -> 1 <= x -> x <= x ^+ n. | Proof.
case: n => // n _ x_ge1.
by rewrite exprS ler_peMr ?(le_trans _ x_ge1) // exprn_ege1.
Qed. | Lemma | ler_eXnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprS",
"exprn_ege1",
"le_trans",
"ler_peMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_eXnr x n : 1 < x -> (x < x ^+ n) = (1 < n)%N. | Proof.
move=> x_ge1; case: n=> [|[|n]] //; first by rewrite expr0 lt_gtF.
by rewrite exprS ltr_pMr ?(lt_trans _ x_ge1) ?exprn_egt1.
Qed. | Lemma | ltr_eXnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr0",
"exprS",
"exprn_egt1",
"lt_gtF",
"lt_trans",
"ltr_pMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_eXnr | := (ler_eXnr, ltr_eXnr). | Definition | lter_eXnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_eXnr",
"ltr_eXnr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_Xnr | := (lter_iXnr, lter_eXnr). | Definition | lter_Xnr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lter_eXnr",
"lter_iXnr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wiXn2l x :
0 <= x -> x <= 1 -> {homo GRing.exp x : m n / (n <= m)%N >-> m <= n}. | Proof.
move=> xge0 xle1 m n /= hmn.
by rewrite -(subnK hmn) exprD ler_piMl ?(exprn_ge0, exprn_ile1).
Qed. | Lemma | ler_wiXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exp",
"exprD",
"exprn_ge0",
"exprn_ile1",
"ler_piMl",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_weXn2l x : 1 <= x -> {homo GRing.exp x : m n / (m <= n)%N >-> m <= n}. | Proof.
move=> xge1 m n /= hmn; rewrite -(subnK hmn) exprD.
by rewrite ler_peMl ?(exprn_ge0, exprn_ege1) // (le_trans _ xge1) ?ler01.
Qed. | Lemma | ler_weXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exp",
"exprD",
"exprn_ege1",
"exprn_ge0",
"le_trans",
"ler01",
"ler_peMl",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ieexprn_weq1 x n : 0 <= x -> (x ^+ n == 1) = ((n == 0) || (x == 1)). | Proof.
move=> xle0; case: n => [|n]; first by rewrite expr0 eqxx.
case: (@real_ltgtP x 1); do ?by rewrite ?ger0_real.
+ by move=> x_lt1; rewrite 1?lt_eqF // exprn_ilt1.
+ by move=> x_lt1; rewrite 1?gt_eqF // exprn_egt1.
by move->; rewrite expr1n eqxx.
Qed. | Lemma | ieexprn_weq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqxx",
"expr0",
"expr1n",
"exprn_egt1",
"exprn_ilt1",
"ger0_real",
"gt_eqF",
"lt_eqF",
"real_ltgtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ieexprIn x : 0 < x -> x != 1 -> injective (GRing.exp x). | Proof.
move=> x_gt0 x_neq1 m n; without loss /subnK <-: m n / (n <= m)%N.
by move=> IH eq_xmn; case/orP: (leq_total m n) => /IH->.
case: {m}(m - n)%N => // m /eqP/idPn[]; rewrite -[x ^+ n]mul1r exprD.
by rewrite (inj_eq (mulIf _)) ?ieexprn_weq1 ?ltW // expf_neq0 ?gt_eqF.
Qed. | Lemma | ieexprIn | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exp",
"expf_neq0",
"exprD",
"gt_eqF",
"ieexprn_weq1",
"inj_eq",
"leq_total",
"ltW",
"mul1r",
"mulIf",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_iXn2l x :
0 < x -> x < 1 -> {mono GRing.exp x : m n / (n <= m)%N >-> m <= n}. | Proof.
move=> xgt0 xlt1; apply: (le_nmono (inj_nhomo_lt _ _)); last first.
by apply/ler_wiXn2l; exact/ltW.
by apply: ieexprIn; rewrite ?lt_eqF ?ltr_cpable.
Qed. | Lemma | ler_iXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"exp",
"ieexprIn",
"inj_nhomo_lt",
"last",
"le_nmono",
"ler_wiXn2l",
"ltW",
"lt_eqF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_iXn2l x :
0 < x -> x < 1 -> {mono GRing.exp x : m n / (n < m)%N >-> m < n}. | Proof. by move=> xgt0 xlt1; apply: (leW_nmono (ler_iXn2l _ _)). Qed. | Lemma | ltr_iXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"exp",
"leW_nmono",
"ler_iXn2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_iXn2l | := (ler_iXn2l, ltr_iXn2l). | Definition | lter_iXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_iXn2l",
"ltr_iXn2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_eXn2l x :
1 < x -> {mono GRing.exp x : m n / (m <= n)%N >-> m <= n}. | Proof.
move=> xgt1; apply: (le_mono (inj_homo_lt _ _)); last first.
by apply: ler_weXn2l; rewrite ltW.
by apply: ieexprIn; rewrite ?gt_eqF ?gtr_cpable //; apply: lt_trans xgt1.
Qed. | Lemma | ler_eXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"exp",
"gt_eqF",
"ieexprIn",
"inj_homo_lt",
"last",
"le_mono",
"ler_weXn2l",
"ltW",
"lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_eXn2l x :
1 < x -> {mono (GRing.exp x) : m n / (m < n)%N >-> m < n}. | Proof. by move=> xgt1; apply: (leW_mono (ler_eXn2l _)). Qed. | Lemma | ltr_eXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"exp",
"leW_mono",
"ler_eXn2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_eXn2l | := (ler_eXn2l, ltr_eXn2l). | Definition | lter_eXn2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_eXn2l",
"ltr_eXn2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrXn2r n x y : 0 <= x -> x < y -> (x ^+ n < y ^+ n) = (n != 0). | Proof.
move=> xge0 xlty; case: n; first by rewrite ltxx.
elim=> [|n IHn]; rewrite ?[_ ^+ _.+2]exprS //.
rewrite (@le_lt_trans _ _ (x * y ^+ n.+1)) ?ler_wpM2l ?ltr_pM2r ?IHn //.
by rewrite ltW.
by rewrite exprn_gt0 // (le_lt_trans xge0).
Qed. | Lemma | ltrXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprS",
"exprn_gt0",
"le_lt_trans",
"ler_wpM2l",
"ltW",
"ltr_pM2r",
"ltxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerXn2r n : {in nneg & , {homo (@GRing.exp R)^~ n : x y / x <= y}}. | Proof.
move=> x y /= x0 y0 xy; elim: n => [|n IHn]; rewrite !(expr0, exprS) //.
by rewrite (@le_trans _ _ (x * y ^+ n)) ?ler_wpM2l ?ler_wpM2r ?exprn_ge0.
Qed. | Lemma | lerXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exp",
"expr0",
"exprS",
"exprn_ge0",
"le_trans",
"ler_wpM2l",
"ler_wpM2r",
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lterXn2r | := (lerXn2r, ltrXn2r). | Definition | lterXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lerXn2r",
"ltrXn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_wpXn2r n :
(0 < n)%N -> {in nneg & , {homo (@GRing.exp R)^~ n : x y / x < y}}. | Proof. by move=> ngt0 x y /= x0 y0 hxy; rewrite ltrXn2r // -lt0n. Qed. | Lemma | ltr_wpXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exp",
"lt0n",
"ltrXn2r",
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pXn2r n :
(0 < n)%N -> {in nneg & , {mono (@GRing.exp R)^~ n : x y / x <= y}}. | Proof.
case: n => // n _ x y; rewrite !qualifE /= => x_ge0 y_ge0.
have [-> | nzx] := eqVneq x 0; first by rewrite exprS mul0r exprn_ge0.
rewrite -subr_ge0 subrXX pmulr_lge0 ?subr_ge0 //= big_ord_recr /=.
rewrite subnn expr0 mul1r /= ltr_pwDr // ?exprn_gt0 ?lt0r ?nzx //.
by rewrite sumr_ge0 // => i _; rewrite mulr_ge0 ... | Lemma | ler_pXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"big_ord_recr",
"eqVneq",
"exp",
"expr0",
"exprS",
"exprn_ge0",
"exprn_gt0",
"lt0r",
"ltr_pwDr",
"mul0r",
"mul1r",
"mulr_ge0",
"nneg",
"pmulr_lge0",
"subnn",
"subrXX",
"subr_ge0",
"sumr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pXn2r n :
(0 < n)%N -> {in nneg & , {mono (@GRing.exp R)^~ n : x y / x < y}}. | Proof.
by move=> n_gt0 x y x_ge0 y_ge0; rewrite !lt_neqAle !eq_le !ler_pXn2r.
Qed. | Lemma | ltr_pXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eq_le",
"exp",
"ler_pXn2r",
"lt_neqAle",
"n_gt0",
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_pXn2r | := (ler_pXn2r, ltr_pXn2r). | Definition | lter_pXn2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_pXn2r",
"ltr_pXn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pexpIrn n : (0 < n)%N -> {in nneg &, injective ((@GRing.exp R)^~ n)}. | Proof. by move=> n_gt0; apply: inc_inj_in (ler_pXn2r _). Qed. | Lemma | pexpIrn | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"exp",
"inc_inj_in",
"ler_pXn2r",
"n_gt0",
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr_le1 n x : (0 < n)%N -> 0 <= x -> (x ^+ n <= 1) = (x <= 1). | Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ler_pXn2r // [_ \in _]ler01.
Qed. | Lemma | expr_le1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr1n",
"ler01",
"ler_pXn2r"
] | expr and ler/ltr | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
expr_lt1 n x : (0 < n)%N -> 0 <= x -> (x ^+ n < 1) = (x < 1). | Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ltr_pXn2r // [_ \in _]ler01.
Qed. | Lemma | expr_lt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr1n",
"ler01",
"ltr_pXn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr_lte1 | := (expr_le1, expr_lt1). | Definition | expr_lte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr_le1",
"expr_lt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr_ge1 n x : (0 < n)%N -> 0 <= x -> (1 <= x ^+ n) = (1 <= x). | Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ler_pXn2r // [_ \in _]ler01.
Qed. | Lemma | expr_ge1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr1n",
"ler01",
"ler_pXn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr_gt1 n x : (0 < n)%N -> 0 <= x -> (1 < x ^+ n) = (1 < x). | Proof.
by move=> ngt0 xge0; rewrite -{1}[1](expr1n _ n) ltr_pXn2r // [_ \in _]ler01.
Qed. | Lemma | expr_gt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr1n",
"ler01",
"ltr_pXn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expr_gte1 | := (expr_ge1, expr_gt1). | Definition | expr_gte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr_ge1",
"expr_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pexpr_eq1 x n : (0 < n)%N -> 0 <= x -> (x ^+ n == 1) = (x == 1). | Proof. by move=> ngt0 xge0; rewrite !eq_le expr_le1 // expr_ge1. Qed. | Lemma | pexpr_eq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eq_le",
"expr_ge1",
"expr_le1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pexprn_eq1 x n : 0 <= x -> (x ^+ n == 1) = (n == 0) || (x == 1). | Proof. by case: n => [|n] xge0; rewrite ?eqxx // pexpr_eq1 ?gtn_eqF. Qed. | Lemma | pexprn_eq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqxx",
"gtn_eqF",
"pexpr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqrXn2 n x y :
(0 < n)%N -> 0 <= x -> 0 <= y -> (x ^+ n == y ^+ n) = (x == y). | Proof. by move=> ngt0 xge0 yge0; rewrite (inj_in_eq (pexpIrn _)). Qed. | Lemma | eqrXn2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inj_in_eq",
"pexpIrn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrp_eq1 x : 0 <= x -> (x ^+ 2 == 1) = (x == 1). | Proof. by move/pexpr_eq1->. Qed. | Lemma | sqrp_eq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"pexpr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrn_eq1 x : x <= 0 -> (x ^+ 2 == 1) = (x == -1). | Proof. by rewrite -sqrrN -oppr_ge0 -eqr_oppLR => /sqrp_eq1. Qed. | Lemma | sqrn_eq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqr_oppLR",
"oppr_ge0",
"sqrp_eq1",
"sqrrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_sqr : {in nneg &, {mono (fun x => x ^+ 2) : x y / x <= y}}. | Proof. exact: ler_pXn2r. Qed. | Lemma | ler_sqr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_pXn2r",
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_sqr : {in nneg &, {mono (fun x => x ^+ 2) : x y / x < y}}. | Proof. exact: ltr_pXn2r. Qed. | Lemma | ltr_sqr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ltr_pXn2r",
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pV2 :
{in [pred x in GRing.unit | 0 < x] &, {mono (@GRing.inv R) : x y /~ x <= y}}. | Proof.
move=> x y /andP [ux hx] /andP [uy hy] /=.
by rewrite -(ler_pM2l hx) -(ler_pM2r hy) !(divrr, mulrVK) ?unitf_gt0 // mul1r.
Qed. | Lemma | ler_pV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"divrr",
"inv",
"ler_pM2l",
"ler_pM2r",
"mul1r",
"mulrVK",
"unit",
"unitf_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nV2 :
{in [pred x in GRing.unit | x < 0] &, {mono (@GRing.inv R) : x y /~ x <= y}}. | Proof.
move=> x y /andP [ux hx] /andP [uy hy] /=.
by rewrite -(ler_nM2l hx) -(ler_nM2r hy) !(divrr, mulrVK) ?unitf_lt0 // mul1r.
Qed. | Lemma | ler_nV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"divrr",
"inv",
"ler_nM2l",
"ler_nM2r",
"mul1r",
"mulrVK",
"unit",
"unitf_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pV2 :
{in [pred x in GRing.unit | 0 < x] &, {mono (@GRing.inv R) : x y /~ x < y}}. | Proof. exact: leW_nmono_in ler_pV2. Qed. | Lemma | ltr_pV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inv",
"leW_nmono_in",
"ler_pV2",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nV2 :
{in [pred x in GRing.unit | x < 0] &, {mono (@GRing.inv R) : x y /~ x < y}}. | Proof. exact: leW_nmono_in ler_nV2. Qed. | Lemma | ltr_nV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inv",
"leW_nmono_in",
"ler_nV2",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_gt1 x : x \is a GRing.unit -> 0 < x -> (1 < x^-1) = (x < 1). | Proof.
by move=> Ux xgt0; rewrite -{1}[1]invr1 ltr_pV2 ?inE ?unitr1 ?ltr01 ?Ux.
Qed. | Lemma | invr_gt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inE",
"invr1",
"ltr01",
"ltr_pV2",
"unit",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_ge1 x : x \is a GRing.unit -> 0 < x -> (1 <= x^-1) = (x <= 1). | Proof.
by move=> Ux xgt0; rewrite -{1}[1]invr1 ler_pV2 ?inE ?unitr1 ?ltr01 // Ux.
Qed. | Lemma | invr_ge1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inE",
"invr1",
"ler_pV2",
"ltr01",
"unit",
"unitr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_gte1 | := (invr_ge1, invr_gt1). | Definition | invr_gte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_ge1",
"invr_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_le1 x (ux : x \is a GRing.unit) (hx : 0 < x) :
(x^-1 <= 1) = (1 <= x). | Proof. by rewrite -invr_ge1 ?invr_gt0 ?unitrV // invrK. Qed. | Lemma | invr_le1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invrK",
"invr_ge1",
"invr_gt0",
"unit",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_lt1 x (ux : x \is a GRing.unit) (hx : 0 < x) : (x^-1 < 1) = (1 < x). | Proof. by rewrite -invr_gt1 ?invr_gt0 ?unitrV // invrK. Qed. | Lemma | invr_lt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invrK",
"invr_gt0",
"invr_gt1",
"unit",
"unitrV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_lte1 | := (invr_le1, invr_lt1). | Definition | invr_lte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_le1",
"invr_lt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_cp1 | := (invr_gte1, invr_lte1). | Definition | invr_cp1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"invr_gte1",
"invr_lte1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_min (m n : nat) : (Order.min m n)%:R = Order.min m%:R n%:R :> R. | Proof. by rewrite !minElt ltr_nat /Order.lt/= -fun_if. Qed. | Lemma | natr_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lt",
"ltr_nat",
"min",
"minElt",
"nat"
] | max and min | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
natr_max (m n : nat) : (Order.max m n)%:R = Order.max m%:R n%:R :> R. | Proof. by rewrite !maxElt ltr_nat /Order.lt/= -fun_if. Qed. | Lemma | natr_max | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lt",
"ltr_nat",
"max",
"maxElt",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_distr_max_min :
{in real &, forall x y, `|x - y| = max x y - min x y}. | Proof. by move=> x y x_real y_real; case: real_leP. Qed. | Lemma | real_distr_max_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"max",
"min",
"real",
"real_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_pMr x y z : 0 <= x -> x * min y z = min (x * y) (x * z). | Proof.
have [|x_gt0||->]// := comparableP x; last by rewrite !mul0r minxx.
by rewrite !(fun_if, if_arg) lter_pM2l//; case: (y < z).
Qed. | Lemma | minr_pMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"comparableP",
"last",
"lter_pM2l",
"min",
"minxx",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_pMr x y z : 0 <= x -> x * max y z = max (x * y) (x * z). | Proof.
have [|x_gt0||->]// := comparableP x; last by rewrite !mul0r maxxx.
by rewrite !(fun_if, if_arg) lter_pM2l//; case: (y < z).
Qed. | Lemma | maxr_pMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"comparableP",
"last",
"lter_pM2l",
"max",
"maxxx",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_maxr_nMr x y z : x <= 0 -> y \is real -> z \is real ->
x * max y z = min (x * y) (x * z). | Proof.
move=> x0 yr zr; rewrite -[_ * _]opprK -mulrN real_oppr_max// -mulNr.
by rewrite minr_pMr ?oppr_ge0// !(mulNr, mulrN, opprK).
Qed. | Lemma | real_maxr_nMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"max",
"min",
"minr_pMr",
"mulNr",
"mulrN",
"opprK",
"oppr_ge0",
"real",
"real_oppr_max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_minr_nMr x y z : x <= 0 -> y \is real -> z \is real ->
x * min y z = max (x * y) (x * z). | Proof.
move=> x0 yr zr; rewrite -[_ * _]opprK -mulrN real_oppr_min// -mulNr.
by rewrite maxr_pMr ?oppr_ge0// !(mulNr, mulrN, opprK).
Qed. | Lemma | real_minr_nMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"max",
"maxr_pMr",
"min",
"mulNr",
"mulrN",
"opprK",
"oppr_ge0",
"real",
"real_oppr_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_pMl x y z : 0 <= x -> min y z * x = min (y * x) (z * x). | Proof. by move=> *; rewrite mulrC minr_pMr // ![_ * x]mulrC. Qed. | Lemma | minr_pMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"min",
"minr_pMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_pMl x y z : 0 <= x -> max y z * x = max (y * x) (z * x). | Proof. by move=> *; rewrite mulrC maxr_pMr // ![_ * x]mulrC. Qed. | Lemma | maxr_pMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"max",
"maxr_pMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_minr_nMl x y z : x <= 0 -> y \is real -> z \is real ->
min y z * x = max (y * x) (z * x). | Proof. by move=> *; rewrite mulrC real_minr_nMr // ![_ * x]mulrC. Qed. | Lemma | real_minr_nMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"max",
"min",
"mulrC",
"real",
"real_minr_nMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_maxr_nMl x y z : x <= 0 -> y \is real -> z \is real ->
max y z * x = min (y * x) (z * x). | Proof. by move=> *; rewrite mulrC real_maxr_nMr // ![_ * x]mulrC. Qed. | Lemma | real_maxr_nMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"max",
"min",
"mulrC",
"real",
"real_maxr_nMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_maxrN x : x \is real -> max x (- x) = `|x|. | Proof.
move=> x_real; rewrite /max.
by case: real_ge0P => // [/ge0_cp [] | /lt0_cp []];
case: (@real_leP (- x) x); rewrite ?realN.
Qed. | Lemma | real_maxrN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ge0_cp",
"lt0_cp",
"max",
"real",
"realN",
"real_ge0P",
"real_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_maxNr x : x \is real -> max (- x) x = `|x|. | Proof.
by move=> x_real; rewrite comparable_maxC ?real_maxrN ?real_comparable ?realN.
Qed. | Lemma | real_maxNr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"comparable_maxC",
"max",
"real",
"realN",
"real_comparable",
"real_maxrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_minrN x : x \is real -> min x (- x) = - `|x|. | Proof.
by move=> x_real; rewrite -[LHS]opprK real_oppr_min ?opprK ?real_maxNr ?realN.
Qed. | Lemma | real_minrN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"min",
"opprK",
"real",
"realN",
"real_maxNr",
"real_oppr_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_minNr x : x \is real -> min (- x) x = - `|x|. | Proof.
by move=> x_real; rewrite -[LHS]opprK real_oppr_min ?opprK ?real_maxrN ?realN.
Qed. | Lemma | real_minNr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"min",
"opprK",
"real",
"realN",
"real_maxrN",
"real_oppr_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_real : {in P, forall i, F i \is real}. | Hypothesis | F_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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