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 |
|---|---|---|---|---|---|---|---|---|---|---|
lter_pM2r | := (ler_pM2r, ltr_pM2r). | Definition | lter_pM2r | 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_pM2r",
"ltr_pM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nM2l x : x < 0 -> {mono *%R x : x y /~ x <= y}. | Proof. by move=> x_lt0 y z /=; rewrite -lerN2 -!mulNr ler_pM2l ?oppr_gt0. Qed. | Lemma | ler_nM2l | 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",... | [
"lerN2",
"ler_pM2l",
"mulNr",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nM2l x : x < 0 -> {mono *%R x : x y /~ x < y}. | Proof. by move=> x_lt0; apply: leW_nmono (ler_nM2l _). Qed. | Lemma | ltr_nM2l | 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",
"leW_nmono",
"ler_nM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_nM2l | := (ler_nM2l, ltr_nM2l). | Definition | lter_nM2l | 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_nM2l",
"ltr_nM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nM2r x : x < 0 -> {mono *%R^~ x : x y /~ x <= y}. | Proof. by move=> x_lt0 y z /=; rewrite ![_ * x]mulrC ler_nM2l. Qed. | Lemma | ler_nM2r | 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_nM2l",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nM2r x : x < 0 -> {mono *%R^~ x : x y /~ x < y}. | Proof. by move=> x_lt0; apply: leW_nmono (ler_nM2r _). Qed. | Lemma | ltr_nM2r | 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",
"leW_nmono",
"ler_nM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_nM2r | := (ler_nM2r, ltr_nM2r). | Definition | lter_nM2r | 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_nM2r",
"ltr_nM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wpM2l x : 0 <= x -> {homo *%R x : y z / y <= z}. | Proof.
by rewrite le0r => /orP[/eqP-> y z | /ler_pM2l/mono2W//]; rewrite !mul0r.
Qed. | Lemma | ler_wpM2l | 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",... | [
"le0r",
"ler_pM2l",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wpM2r x : 0 <= x -> {homo *%R^~ x : y z / y <= z}. | Proof. by move=> x_ge0 y z leyz; rewrite ![_ * x]mulrC ler_wpM2l. Qed. | Lemma | ler_wpM2r | 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",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wnM2l x : x <= 0 -> {homo *%R x : y z /~ y <= z}. | by move=> x_le0 y z leyz; rewrite -![x * _]mulrNN ler_wpM2l ?lterNE. Qed. | Lemma | ler_wnM2l | 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",
"lterNE",
"mulrNN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wnM2r x : x <= 0 -> {homo *%R^~ x : y z /~ y <= z}. | Proof. by move=> x_le0 y z leyz; rewrite -![_ * x]mulrNN ler_wpM2r ?lterNE. Qed. | Lemma | ler_wnM2r | 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",
"lterNE",
"mulrNN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pM x1 y1 x2 y2 :
0 <= x1 -> 0 <= x2 -> x1 <= y1 -> x2 <= y2 -> x1 * x2 <= y1 * y2. | Proof.
move=> x1ge0 x2ge0 le_xy1 le_xy2; have y1ge0 := le_trans x1ge0 le_xy1.
exact: le_trans (ler_wpM2r x2ge0 le_xy1) (ler_wpM2l y1ge0 le_xy2).
Qed. | Lemma | ler_pM | 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_wpM2l",
"ler_wpM2r"
] | Binary forms, for backchaining. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltr_pM x1 y1 x2 y2 :
0 <= x1 -> 0 <= x2 -> x1 < y1 -> x2 < y2 -> x1 * x2 < y1 * y2. | Proof.
move=> x1ge0 x2ge0 lt_xy1 lt_xy2; have y1gt0 := le_lt_trans x1ge0 lt_xy1.
by rewrite (le_lt_trans (ler_wpM2r x2ge0 (ltW lt_xy1))) ?ltr_pM2l.
Qed. | Lemma | ltr_pM | 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_wpM2r",
"ltW",
"ltr_pM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pMn2r n : (0 < n)%N -> {mono (@GRing.natmul R)^~ n : x y / x <= y}. | Proof.
by case: n => // n _ x y /=; rewrite -mulr_natl -[y *+ _]mulr_natl ler_pM2l.
Qed. | Lemma | ler_pMn2r | 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_pM2l",
"mulr_natl",
"natmul"
] | complement for x *+ n and <= or < | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltr_pMn2r n : (0 < n)%N -> {mono (@GRing.natmul R)^~ n : x y / x < y}. | Proof. by move/ler_pMn2r/leW_mono. Qed. | Lemma | ltr_pMn2r | 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",... | [
"leW_mono",
"ler_pMn2r",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrnI n : (0 < n)%N -> injective ((@GRing.natmul R)^~ n). | Proof. by move/ler_pMn2r/inc_inj. Qed. | Lemma | pmulrnI | 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",... | [
"inc_inj",
"ler_pMn2r",
"natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_pMn2r n : (0 < n)%N -> {mono (@GRing.natmul R)^~ n : x y / x == y}. | Proof. by move/pmulrnI/inj_eq. Qed. | Lemma | eqr_pMn2r | 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_eq",
"natmul",
"pmulrnI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_lgt0 x n : (0 < n)%N -> (0 < x *+ n) = (0 < x). | Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ltr_pMn2r // mul0rn. Qed. | Lemma | pmulrn_lgt0 | 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_pMn2r",
"mul0rn",
"n_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_llt0 x n : (0 < n)%N -> (x *+ n < 0) = (x < 0). | Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ltr_pMn2r // mul0rn. Qed. | Lemma | pmulrn_llt0 | 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_pMn2r",
"mul0rn",
"n_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_lge0 x n : (0 < n)%N -> (0 <= x *+ n) = (0 <= x). | Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ler_pMn2r // mul0rn. Qed. | Lemma | pmulrn_lge0 | 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_pMn2r",
"mul0rn",
"n_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_lle0 x n : (0 < n)%N -> (x *+ n <= 0) = (x <= 0). | Proof. by move=> n_gt0; rewrite -(mul0rn _ n) ler_pMn2r // mul0rn. Qed. | Lemma | pmulrn_lle0 | 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_pMn2r",
"mul0rn",
"n_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_wMn2r x y n : x < y -> (x *+ n < y *+ n) = (0 < n)%N. | Proof. by move=> ltxy; case: n=> // n; rewrite ltr_pMn2r. Qed. | Lemma | ltr_wMn2r | 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_pMn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerMn2r n x y : (x *+ n <= y *+ n) = ((n == 0) || (x <= y)). | Proof. by case: n => [|n]; rewrite ?lexx ?eqxx // ler_pMn2r. Qed. | Lemma | lerMn2r | 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",
"ler_pMn2r",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrMn2r n x y : (x *+ n < y *+ n) = ((0 < n)%N && (x < y)). | Proof. by case: n => [|n]; rewrite ?lexx ?eqxx // ltr_pMn2r. Qed. | Lemma | ltrMn2r | 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",
"lexx",
"ltr_pMn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqrMn2r n x y : (x *+ n == y *+ n) = (n == 0)%N || (x == y). | Proof. by rewrite !(@eq_le _ R) !lerMn2r -orb_andr. Qed. | Lemma | eqrMn2r | 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",
"lerMn2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrn_eq0 x n : (x *+ n == 0) = ((n == 0)%N || (x == 0)). | Proof. by rewrite -mulr_natl mulf_eq0 pnatr_eq0. Qed. | Lemma | mulrn_eq0 | 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",... | [
"mulf_eq0",
"mulr_natl",
"pnatr_eq0"
] | More characteristic zero properties. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eqNr x : (- x == x) = (x == 0). | Proof. by rewrite eq_sym -addr_eq0 -mulr2n mulrn_eq0. Qed. | Lemma | eqNr | 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_eq0",
"eq_sym",
"mulr2n",
"mulrn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrIn x : x != 0 -> injective (GRing.natmul x). | Proof.
move=> x_neq0 m n; without loss /subnK <-: m n / (n <= m)%N.
by move=> IH eq_xmn; case/orP: (leq_total m n) => /IH->.
by move/eqP; rewrite mulrnDr -subr_eq0 addrK mulrn_eq0 => /predU1P[-> | /idPn].
Qed. | Lemma | mulrIn | 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",... | [
"addrK",
"leq_total",
"mulrnDr",
"mulrn_eq0",
"natmul",
"predU1P",
"subnK",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrn_wgt0 x n : 0 < x -> (0 < x *+ n) = (0 < n)%N. | Proof. by case: n => // n hx; rewrite pmulrn_lgt0. Qed. | Lemma | mulrn_wgt0 | 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",... | [
"pmulrn_lgt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulrn_wlt0 x n : x < 0 -> (x *+ n < 0) = (0 < n)%N. | Proof. by case: n => // n hx; rewrite pmulrn_llt0. Qed. | Lemma | mulrn_wlt0 | 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",... | [
"pmulrn_llt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nat m n : (m%:R <= n%:R :> R) = (m <= n)%N. | Proof. by rewrite ler_pMn2l. Qed. | Lemma | ler_nat | 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_pMn2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nat m n : (m%:R < n%:R :> R) = (m < n)%N. | Proof. by rewrite ltr_pMn2l. Qed. | Lemma | ltr_nat | 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_pMn2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_nat m n : (m%:R == n%:R :> R) = (m == n)%N. | Proof. by rewrite (inj_eq (mulrIn _)) ?oner_eq0. Qed. | Lemma | eqr_nat | 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_eq",
"mulrIn",
"oner_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pnatr_eq1 n : (n%:R == 1 :> R) = (n == 1)%N. | Proof. exact: eqr_nat 1. Qed. | Lemma | pnatr_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_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lern0 n : (n%:R <= 0 :> R) = (n == 0). | Proof. by rewrite -[0]/0%:R ler_nat leqn0. Qed. | Lemma | lern0 | 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",... | [
"leqn0",
"ler_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrn0 n : (n%:R < 0 :> R) = false. | Proof. by rewrite -[0]/0%:R ltr_nat ltn0. Qed. | Lemma | ltrn0 | 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",... | [
"ltn0",
"ltr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler1n n : (1 <= n%:R :> R) = (1 <= n)%N. | Proof. by rewrite -ler_nat. Qed. | Lemma | ler1n | 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_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr1n n : (1 < n%:R :> R) = (1 < n)%N. | Proof. by rewrite -ltr_nat. Qed. | Lemma | ltr1n | 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_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lern1 n : (n%:R <= 1 :> R) = (n <= 1)%N. | Proof. by rewrite -ler_nat. Qed. | Lemma | lern1 | 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_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrn1 n : (n%:R < 1 :> R) = (n < 1)%N. | Proof. by rewrite -ltr_nat. Qed. | Lemma | ltrn1 | 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_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrN10 : -1 < 0 :> R. | Proof. by rewrite oppr_lt0. Qed. | Lemma | ltrN10 | 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",... | [
"oppr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerN10 : -1 <= 0 :> R. | Proof. by rewrite oppr_le0. Qed. | Lemma | lerN10 | 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",... | [
"oppr_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr10 : (1 < 0 :> R) = false. | Proof. by rewrite le_gtF. Qed. | Lemma | ltr10 | 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_gtF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler10 : (1 <= 0 :> R) = false. | Proof. by rewrite lt_geF. Qed. | Lemma | ler10 | 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_geF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0N1 : (0 < -1 :> R) = false. | Proof. by rewrite le_gtF // lerN10. Qed. | Lemma | ltr0N1 | 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_gtF",
"lerN10"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0N1 : (0 <= -1 :> R) = false. | Proof. by rewrite lt_geF // ltrN10. Qed. | Lemma | ler0N1 | 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_geF",
"ltrN10"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_rgt0 x n : 0 < x -> (0 < x *+ n) = (0 < n)%N. | Proof. exact: mulrn_wgt0. Qed. | Lemma | pmulrn_rgt0 | 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",... | [
"mulrn_wgt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_rlt0 x n : 0 < x -> (x *+ n < 0) = false. | Proof. by move=> x_gt0; rewrite -(mulr0n x) ltr_pMn2l. Qed. | Lemma | pmulrn_rlt0 | 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_pMn2l",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_rge0 x n : 0 < x -> 0 <= x *+ n. | Proof. by move=> x_gt0; rewrite -(mulr0n x) ler_pMn2l. Qed. | Lemma | pmulrn_rge0 | 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_pMn2l",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulrn_rle0 x n : 0 < x -> (x *+ n <= 0) = (n == 0)%N. | Proof. by move=> x_gt0; rewrite -(mulr0n x) ler_pMn2l ?leqn0. Qed. | Lemma | pmulrn_rle0 | 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",... | [
"leqn0",
"ler_pMn2l",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulrn_rgt0 x n : x < 0 -> (0 < x *+ n) = false. | Proof. by move=> x_lt0; rewrite -(mulr0n x) ltr_nMn2l. Qed. | Lemma | nmulrn_rgt0 | 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_nMn2l",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulrn_rge0 x n : x < 0 -> (0 <= x *+ n) = (n == 0)%N. | Proof. by move=> x_lt0; rewrite -(mulr0n x) ler_nMn2l ?leqn0. Qed. | Lemma | nmulrn_rge0 | 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",... | [
"leqn0",
"ler_nMn2l",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulrn_rle0 x n : x < 0 -> x *+ n <= 0. | Proof. by move=> x_lt0; rewrite -(mulr0n x) ler_nMn2l. Qed. | Lemma | nmulrn_rle0 | 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_nMn2l",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulr_rlt0 x y : 0 < x -> (x * y < 0) = (y < 0). | Proof.
by move=> x_gt0; rewrite -[LHS]oppr_gt0 -mulrN pmulr_rgt0 // oppr_gt0.
Qed. | Lemma | pmulr_rlt0 | 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",... | [
"mulrN",
"oppr_gt0",
"pmulr_rgt0"
] | x positive and y right | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pmulr_rle0 x y : 0 < x -> (x * y <= 0) = (y <= 0). | Proof.
by move=> x_gt0; rewrite -[LHS]oppr_ge0 -mulrN pmulr_rge0 // oppr_ge0.
Qed. | Lemma | pmulr_rle0 | 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",... | [
"mulrN",
"oppr_ge0",
"pmulr_rge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulr_lgt0 x y : 0 < x -> (0 < y * x) = (0 < y). | Proof. by move=> x_gt0; rewrite mulrC pmulr_rgt0. Qed. | Lemma | pmulr_lgt0 | 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",... | [
"mulrC",
"pmulr_rgt0"
] | x positive and y left | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pmulr_lge0 x y : 0 < x -> (0 <= y * x) = (0 <= y). | Proof. by move=> x_gt0; rewrite mulrC pmulr_rge0. Qed. | Lemma | pmulr_lge0 | 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",... | [
"mulrC",
"pmulr_rge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulr_llt0 x y : 0 < x -> (y * x < 0) = (y < 0). | Proof. by move=> x_gt0; rewrite mulrC pmulr_rlt0. Qed. | Lemma | pmulr_llt0 | 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",... | [
"mulrC",
"pmulr_rlt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulr_lle0 x y : 0 < x -> (y * x <= 0) = (y <= 0). | Proof. by move=> x_gt0; rewrite mulrC pmulr_rle0. Qed. | Lemma | pmulr_lle0 | 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",... | [
"mulrC",
"pmulr_rle0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulr_rgt0 x y : x < 0 -> (0 < x * y) = (y < 0). | Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rgt0 lterNE. Qed. | Lemma | nmulr_rgt0 | 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",... | [
"lterNE",
"mulrNN",
"pmulr_rgt0"
] | x negative and y right | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nmulr_rge0 x y : x < 0 -> (0 <= x * y) = (y <= 0). | Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rge0 lterNE. Qed. | Lemma | nmulr_rge0 | 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",... | [
"lterNE",
"mulrNN",
"pmulr_rge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulr_rlt0 x y : x < 0 -> (x * y < 0) = (0 < y). | Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rlt0 lterNE. Qed. | Lemma | nmulr_rlt0 | 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",... | [
"lterNE",
"mulrNN",
"pmulr_rlt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulr_rle0 x y : x < 0 -> (x * y <= 0) = (0 <= y). | Proof. by move=> x_lt0; rewrite -mulrNN pmulr_rle0 lterNE. Qed. | Lemma | nmulr_rle0 | 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",... | [
"lterNE",
"mulrNN",
"pmulr_rle0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulr_lgt0 x y : x < 0 -> (0 < y * x) = (y < 0). | Proof. by move=> x_lt0; rewrite mulrC nmulr_rgt0. Qed. | Lemma | nmulr_lgt0 | 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",... | [
"mulrC",
"nmulr_rgt0"
] | x negative and y left | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nmulr_lge0 x y : x < 0 -> (0 <= y * x) = (y <= 0). | Proof. by move=> x_lt0; rewrite mulrC nmulr_rge0. Qed. | Lemma | nmulr_lge0 | 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",... | [
"mulrC",
"nmulr_rge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulr_llt0 x y : x < 0 -> (y * x < 0) = (0 < y). | Proof. by move=> x_lt0; rewrite mulrC nmulr_rlt0. Qed. | Lemma | nmulr_llt0 | 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",... | [
"mulrC",
"nmulr_rlt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmulr_lle0 x y : x < 0 -> (y * x <= 0) = (0 <= y). | Proof. by move=> x_lt0; rewrite mulrC nmulr_rle0. Qed. | Lemma | nmulr_lle0 | 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",... | [
"mulrC",
"nmulr_rle0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x * y. | Proof. by move=> x_ge0 y_ge0; rewrite -(mulr0 x) ler_wpM2l. Qed. | Lemma | mulr_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",... | [
"ler_wpM2l",
"mulr0"
] | weak and symmetric lemmas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulr_le0 x y : x <= 0 -> y <= 0 -> 0 <= x * y. | Proof. by move=> x_le0 y_le0; rewrite -(mulr0 x) ler_wnM2l. Qed. | Lemma | mulr_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",... | [
"ler_wnM2l",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_ge0_le0 x y : 0 <= x -> y <= 0 -> x * y <= 0. | Proof. by move=> x_le0 y_le0; rewrite -(mulr0 x) ler_wpM2l. Qed. | Lemma | mulr_ge0_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",... | [
"ler_wpM2l",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_le0_ge0 x y : x <= 0 -> 0 <= y -> x * y <= 0. | Proof. by move=> x_le0 y_le0; rewrite -(mulr0 x) ler_wnM2l. Qed. | Lemma | mulr_le0_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",... | [
"ler_wnM2l",
"mulr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_gt0 x y : 0 < x -> 0 < y -> 0 < x * y. | Proof. by move=> x_gt0 y_gt0; rewrite pmulr_rgt0. Qed. | Lemma | mulr_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",... | [
"pmulr_rgt0"
] | mulr_gt0 with only one case | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulr_ge0_gt0 x y : 0 <= x -> 0 <= y -> (0 < x * y) = (0 < x) && (0 < y). | Proof.
rewrite le_eqVlt => /predU1P[<-|x0]; first by rewrite mul0r ltxx.
rewrite le_eqVlt => /predU1P[<-|y0]; first by rewrite mulr0 ltxx andbC.
by apply/idP/andP=> [|_]; rewrite pmulr_rgt0.
Qed. | Lemma | mulr_ge0_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",
"le_eqVlt",
"ltxx",
"mul0r",
"mulr0",
"pmulr_rgt0",
"predU1P"
] | and reverse direction | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
prodr_ge0 I r (P : pred I) (E : I -> R) :
(forall i, P i -> 0 <= E i) -> 0 <= \prod_(i <- r | P i) E i. | Proof. by move=> Ege0; rewrite -nnegrE rpred_prod. Qed. | Lemma | prodr_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",
"rpred_prod"
] | Iterated products | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
prodr_gt0 I r (P : pred I) (E : I -> R) :
(forall i, P i -> 0 < E i) -> 0 < \prod_(i <- r | P i) E i. | Proof. by move=> Ege0; rewrite -posrE rpred_prod. Qed. | Lemma | prodr_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",... | [
"posrE",
"rpred_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_prod I r (P : pred I) (E1 E2 : I -> R) :
(forall i, P i -> 0 <= E1 i <= E2 i) ->
\prod_(i <- r | P i) E1 i <= \prod_(i <- r | P i) E2 i. | Proof.
move=> leE12; elim/(big_load (fun x => 0 <= x)): _.
elim/big_rec2: _ => // i x2 x1 /leE12/andP[le0Ei leEi12] [x1ge0 le_x12].
by rewrite mulr_ge0 // ler_pM.
Qed. | Lemma | ler_prod | 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_load",
"big_rec2",
"ler_pM",
"mulr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_prod I r (P : pred I) (E1 E2 : I -> R) :
has P r -> (forall i, P i -> 0 <= E1 i < E2 i) ->
\prod_(i <- r | P i) E1 i < \prod_(i <- r | P i) E2 i. | Proof.
elim: r => //= i r IHr; rewrite !big_cons; case: ifP => {IHr}// Pi _ ltE12.
have /andP[le0E1i ltE12i] := ltE12 i Pi; set E2r := \prod_(j <- r | P j) E2 j.
apply: le_lt_trans (_ : E1 i * E2r < E2 i * E2r).
by rewrite ler_wpM2l ?ler_prod // => j /ltE12/andP[-> /ltW].
by rewrite ltr_pM2r ?prodr_gt0 // => j /ltE12... | Lemma | ltr_prod | 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",
"big_cons",
"has",
"le_lt_trans",
"ler_prod",
"ler_wpM2l",
"ltW",
"ltr_pM2r",
"prodr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_prod_nat (E1 E2 : nat -> R) (n m : nat) :
(m < n)%N -> (forall i, (m <= i < n)%N -> 0 <= E1 i < E2 i) ->
\prod_(m <= i < n) E1 i < \prod_(m <= i < n) E2 i. | Proof.
move=> lt_mn ltE12; rewrite !big_nat ltr_prod {ltE12}//.
by apply/hasP; exists m; rewrite ?mem_index_iota leqnn.
Qed. | Lemma | ltr_prod_nat | 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",
"big_nat",
"hasP",
"leqnn",
"ltr_prod",
"mem_index_iota",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realMr x y : x != 0 -> x \is real -> (x * y \is real) = (y \is real). | Proof.
move=> x_neq0 xR; case: real_ltgtP x_neq0 => // hx _; rewrite !realE.
by rewrite nmulr_rge0 // nmulr_rle0 // orbC.
by rewrite pmulr_rge0 // pmulr_rle0 // orbC.
Qed. | Lemma | realMr | 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",... | [
"nmulr_rge0",
"nmulr_rle0",
"pmulr_rge0",
"pmulr_rle0",
"real",
"realE",
"real_ltgtP"
] | real of mul | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
realrM x y : y != 0 -> y \is real -> (x * y \is real) = (x \is real). | Proof. by move=> y_neq0 yR; rewrite mulrC realMr. Qed. | Lemma | realrM | 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",... | [
"mulrC",
"real",
"realMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realM : {in real &, forall x y, x * y \is real}. | Proof. exact: rpredM. Qed. | Lemma | realM | 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",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realrMn x n : (n != 0)%N -> (x *+ n \is real) = (x \is real). | Proof. by move=> n_neq0; rewrite -mulr_natl realMr ?realn ?pnatr_eq0. Qed. | Lemma | realrMn | 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_natl",
"pnatr_eq0",
"real",
"realMr",
"realn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_pMl x y : 0 < y -> (x * y <= y) = (x <= 1). | Proof. by move=> hy; rewrite -{2}[y]mul1r ler_pM2r. Qed. | Lemma | ger_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",... | [
"ler_pM2r",
"mul1r"
] | ler/ltr and multiplication between a positive/negative | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gtr_pMl x y : 0 < y -> (x * y < y) = (x < 1). | Proof. by move=> hy; rewrite -{2}[y]mul1r ltr_pM2r. Qed. | Lemma | gtr_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",... | [
"ltr_pM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_pMr x y : 0 < y -> (y * x <= y) = (x <= 1). | Proof. by move=> hy; rewrite -{2}[y]mulr1 ler_pM2l. Qed. | Lemma | ger_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",... | [
"ler_pM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr_pMr x y : 0 < y -> (y * x < y) = (x < 1). | Proof. by move=> hy; rewrite -{2}[y]mulr1 ltr_pM2l. Qed. | Lemma | gtr_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",... | [
"ltr_pM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pMl x y : 0 < y -> (y <= x * y) = (1 <= x). | Proof. by move=> hy; rewrite -{1}[y]mul1r ler_pM2r. Qed. | Lemma | ler_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",... | [
"ler_pM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pMl x y : 0 < y -> (y < x * y) = (1 < x). | Proof. by move=> hy; rewrite -{1}[y]mul1r ltr_pM2r. Qed. | Lemma | ltr_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",... | [
"ltr_pM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pMr x y : 0 < y -> (y <= y * x) = (1 <= x). | Proof. by move=> hy; rewrite -{1}[y]mulr1 ler_pM2l. Qed. | Lemma | ler_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",... | [
"ler_pM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pMr x y : 0 < y -> (y < y * x) = (1 < x). | Proof. by move=> hy; rewrite -{1}[y]mulr1 ltr_pM2l. Qed. | Lemma | ltr_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",... | [
"ltr_pM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_nMl x y : y < 0 -> (x * y <= y) = (1 <= x). | Proof. by move=> hy; rewrite -{2}[y]mul1r ler_nM2r. Qed. | Lemma | ger_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",... | [
"ler_nM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr_nMl x y : y < 0 -> (x * y < y) = (1 < x). | Proof. by move=> hy; rewrite -{2}[y]mul1r ltr_nM2r. Qed. | Lemma | gtr_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",... | [
"ltr_nM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_nMr x y : y < 0 -> (y * x <= y) = (1 <= x). | Proof. by move=> hy; rewrite -{2}[y]mulr1 ler_nM2l. Qed. | Lemma | ger_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",... | [
"ler_nM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr_nMr x y : y < 0 -> (y * x < y) = (1 < x). | Proof. by move=> hy; rewrite -{2}[y]mulr1 ltr_nM2l. Qed. | Lemma | gtr_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",... | [
"ltr_nM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nMl x y : y < 0 -> (y <= x * y) = (x <= 1). | Proof. by move=> hy; rewrite -{1}[y]mul1r ler_nM2r. Qed. | Lemma | ler_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",... | [
"ler_nM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nMl x y : y < 0 -> (y < x * y) = (x < 1). | Proof. by move=> hy; rewrite -{1}[y]mul1r ltr_nM2r. Qed. | Lemma | ltr_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",... | [
"ltr_nM2r",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nMr x y : y < 0 -> (y <= y * x) = (x <= 1). | Proof. by move=> hy; rewrite -{1}[y]mulr1 ler_nM2l. Qed. | Lemma | ler_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",... | [
"ler_nM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nMr x y : y < 0 -> (y < y * x) = (x < 1). | Proof. by move=> hy; rewrite -{1}[y]mulr1 ltr_nM2l. Qed. | Lemma | ltr_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",... | [
"ltr_nM2l",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_peMl x y : 0 <= y -> 1 <= x -> y <= x * y. | Proof. by move=> hy hx; rewrite -{1}[y]mul1r ler_wpM2r. Qed. | Lemma | ler_peMl | 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"
] | ler/ltr and multiplication between a positive/negative
and a exterior (1 <= _) or interior (0 <= _ <= 1) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ler_neMl x y : y <= 0 -> 1 <= x -> x * y <= y. | Proof. by move=> hy hx; rewrite -{2}[y]mul1r ler_wnM2r. Qed. | Lemma | ler_neMl | 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 |
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