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 |
|---|---|---|---|---|---|---|---|---|---|---|
sqr_ge0 x : 0 <= x ^+ 2. | Proof. by rewrite exprn_even_ge0. Qed. | Lemma | sqr_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",... | [
"exprn_even_ge0"
] | Special lemmas for squares. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sqr_norm_eq1 x : (x ^+ 2 == 1) = (`|x| == 1). | Proof. by rewrite sqrf_eq1 eqr_norml ler01 andbT. Qed. | Lemma | sqr_norm_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_norml",
"ler01",
"sqrf_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_mean_square_scaled x y :
x * y *+ 2 <= x ^+ 2 + y ^+ 2 ?= iff (x == y). | Proof. exact: real_leif_mean_square_scaled. Qed. | Lemma | leif_mean_square_scaled | 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_leif_mean_square_scaled"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_AGM2_scaled x y : x * y *+ 4 <= (x + y) ^+ 2 ?= iff (x == y). | Proof. exact: real_leif_AGM2_scaled. Qed. | Lemma | leif_AGM2_scaled | 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_leif_AGM2_scaled"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distr_max_min x y : `|x - y| = max x y - min x y. | Proof. exact: real_distr_max_min. Qed. | Lemma | 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_distr_max_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_max : {morph -%R : x y / max x y >-> min x y : R}. | Proof. by move=> x y; apply: real_oppr_max. Qed. | Lemma | oppr_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",... | [
"apply",
"max",
"min",
"real_oppr_max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_min : {morph -%R : x y / min x y >-> max x y : R}. | Proof. by move=> x y; apply: real_oppr_min. Qed. | Lemma | oppr_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",... | [
"apply",
"max",
"min",
"real_oppr_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_minl : @left_distributive R R +%R min. | Proof. by move=> x y z; apply: real_addr_minl. Qed. | Lemma | addr_minl | 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",
"min",
"real_addr_minl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_minr : @right_distributive R R +%R min. | Proof. by move=> x y z; apply: real_addr_minr. Qed. | Lemma | addr_minr | 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",
"min",
"real_addr_minr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_maxl : @left_distributive R R +%R max. | Proof. by move=> x y z; apply: real_addr_maxl. Qed. | Lemma | addr_maxl | 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",
"max",
"real_addr_maxl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addr_maxr : @right_distributive R R +%R max. | Proof. by move=> x y z; apply: real_addr_maxr. Qed. | Lemma | addr_maxr | 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",
"max",
"real_addr_maxr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_nMr x y z : x <= 0 -> x * min y z = max (x * y) (x * z). | Proof. by move=> x_le0; apply: real_minr_nMr. Qed. | Lemma | 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",... | [
"apply",
"max",
"min",
"real_minr_nMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_nMr x y z : x <= 0 -> x * max y z = min (x * y) (x * z). | Proof. by move=> x_le0; apply: real_maxr_nMr. Qed. | Lemma | 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",... | [
"apply",
"max",
"min",
"real_maxr_nMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_nMl x y z : x <= 0 -> min y z * x = max (y * x) (z * x). | Proof. by move=> x_le0; apply: real_minr_nMl. Qed. | Lemma | 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",... | [
"apply",
"max",
"min",
"real_minr_nMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_nMl x y z : x <= 0 -> max y z * x = min (y * x) (z * x). | Proof. by move=> x_le0; apply: real_maxr_nMl. Qed. | Lemma | 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",... | [
"apply",
"max",
"min",
"real_maxr_nMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxrN x : max x (- x) = `|x|. | Proof. exact: real_maxrN. Qed. | Lemma | 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",... | [
"max",
"real_maxrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxNr x : max (- x) x = `|x|. | Proof. exact: real_maxNr. Qed. | Lemma | 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",... | [
"max",
"real_maxNr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minrN x : min x (- x) = - `|x|. | Proof. exact: real_minrN. Qed. | Lemma | 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",
"real_minrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minNr x : min (- x) x = - `|x|. | Proof. exact: real_minNr. Qed. | Lemma | 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",
"real_minNr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_itv_bound a b : {ub | forall x, a <= x <= b -> `|p.[x]| <= ub}. | Proof.
have [ub le_p_ub] := poly_disk_bound p (Num.max `|a| `|b|).
exists ub => x /andP[le_a_x le_x_b]; rewrite le_p_ub // le_max !ler_normr.
by have [_|_] := ler0P x; rewrite ?lerN2 ?le_a_x ?le_x_b orbT.
Qed. | Lemma | poly_itv_bound | 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_max",
"ler0P",
"lerN2",
"ler_normr",
"max",
"poly_disk_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
monic_Cauchy_bound : p \is monic -> {b | forall x, x >= b -> p.[x] > 0}. | Proof.
move/monicP=> mon_p; pose n := (size p - 2)%N.
have [p_le1 | p_gt1] := leqP (size p) 1.
exists 0 => x _; rewrite (size1_polyC p_le1) hornerC.
by rewrite -[p`_0]lead_coefC -size1_polyC // mon_p ltr01.
pose lb := \sum_(j < n.+1) `|p`_j|; exists (lb + 1) => x le_ub_x.
have x_ge1: 1 <= x; last have x_gt0 := lt_l... | Lemma | monic_Cauchy_bound | 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",... | [
"addn1",
"addnS",
"apply",
"big_ord_recr",
"exprS",
"exprn_gt0",
"ger0_norm",
"hornerC",
"horner_coef",
"last",
"le_lt_trans",
"le_trans",
"lead_coefC",
"lead_coefE",
"leqP",
"leq_ord",
"lerD2l",
"ler_norm",
"ler_sum",
"ler_weXn2l",
"ler_wpDl",
"ler_wpM2l",
"ltW",
"lt_l... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x <= y" | := (Rle x y) : ring_scope. | Notation | x <= y | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x < y" | := (Rlt x y) : ring_scope. | Notation | x < y | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrr x : (x < x) = false. | Proof. by rewrite lt_def eqxx. Qed. | Lemma | ltrr | 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",
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge0_def x : (0 <= x) = (`|x| == x). | Proof. by rewrite le_def subr0. Qed. | Lemma | ge0_def | 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_def",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_ge0 x y : (0 <= x - y) = (y <= x). | Proof. by rewrite ge0_def -le_def. Qed. | Lemma | subr_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",... | [
"ge0_def",
"le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_gt0 x y : (0 < y - x) = (x < y). | Proof. by rewrite !lt_def subr_eq0 subr_ge0. Qed. | Lemma | subr_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",... | [
"lt_def",
"subr_eq0",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_trans : transitive Rlt. | Proof.
move=> y x z le_xy le_yz.
by rewrite -subr_gt0 -(subrK y z) -addrA addr_gt0 // subr_gt0.
Qed. | Lemma | lt_trans | 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",... | [
"addrA",
"addr_gt0",
"subrK",
"subr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le01 : 0 <= 1. | Proof.
have n1_nz: `|1| != 0 :> R by apply: contraNneq (@oner_neq0 R) => /norm_eq0->.
by rewrite ge0_def -(inj_eq (mulfI n1_nz)) -normM !mulr1.
Qed. | Lemma | le01 | 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",
"contraNneq",
"ge0_def",
"inj_eq",
"mulfI",
"mulr1",
"normM",
"norm_eq0",
"oner_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt01 : 0 < 1. | Proof. by rewrite lt_def oner_neq0 le01. Qed. | Lemma | lt01 | 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",... | [
"le01",
"lt_def",
"oner_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltW x y : x < y -> x <= y. | Proof. by rewrite lt_def => /andP[]. Qed. | Lemma | ltW | 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_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerr x : x <= x. | Proof.
have n2: `|2| == 2 :> R by rewrite -ge0_def ltW ?addr_gt0 ?lt01.
rewrite le_def subrr -(inj_eq (addrI `|0|)) addr0 -mulr2n -mulr_natr.
by rewrite -(eqP n2) -normM mul0r.
Qed. | Lemma | lerr | 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",... | [
"addr0",
"addrI",
"addr_gt0",
"ge0_def",
"inj_eq",
"le_def",
"lt01",
"ltW",
"mul0r",
"mulr2n",
"mulr_natr",
"normM",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_def' x y : (x <= y) = (x == y) || (x < y). | Proof. by rewrite lt_def; case: eqVneq => //= ->; rewrite lerr. Qed. | Lemma | le_def' | 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",... | [
"eqVneq",
"lerr",
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_trans : transitive Rle. | by move=> y x z; rewrite !le_def' => /predU1P [->|hxy] // /predU1P [<-|hyz];
rewrite ?hxy ?(lt_trans hxy hyz) orbT.
Qed. | Lemma | le_trans | 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_def'",
"lt_trans",
"predU1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrMn x n : `|x *+ n| = `|x| *+ n. | Proof.
rewrite -mulr_natr -[RHS]mulr_natr normM.
congr (_ * _); apply/eqP; rewrite -ge0_def.
elim: n => [|n ih]; [exact: lerr | apply: (le_trans ih)].
by rewrite le_def -natrB // subSnn -[_%:R]subr0 -le_def mulr1n le01.
Qed. | Lemma | normrMn | 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",
"ge0_def",
"le01",
"le_def",
"le_trans",
"lerr",
"mulr1n",
"mulr_natr",
"natrB",
"normM",
"subSnn",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrN1 : `|-1| = 1 :> R. | Proof.
have: `|-1| ^+ 2 == 1 :> R
by rewrite expr2 /= -normM mulrNN mul1r -[1]subr0 -le_def le01.
rewrite sqrf_eq1 => /predU1P [] //; rewrite -[-1]subr0 -le_def.
have ->: (0 <= -1) = (-1 == 0 :> R) || (0 < -1)
by rewrite lt_def; case: eqP => // ->; rewrite lerr.
by rewrite oppr_eq0 oner_eq0 => /(addr_gt... | Lemma | normrN1 | 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_gt0",
"expr2",
"le01",
"le_def",
"lerr",
"lt01",
"lt_def",
"ltrr",
"mul1r",
"mulrNN",
"normM",
"oner_eq0",
"oppr_eq0",
"predU1P",
"sqrf_eq1",
"subr0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrN x : `|- x| = `|x|. | Proof. by rewrite -mulN1r normM -[RHS]mul1r normrN1. Qed. | Lemma | normrN | 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",... | [
"mul1r",
"mulN1r",
"normM",
"normrN1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_total : Order.POrder_isTotal ring_display R. | Proof.
constructor=> x y; move: (real (x - y)).
by rewrite unfold_in /= !ler_def subr0 add0r opprB orbC.
Qed. | Lemma | le_total | 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",... | [
"add0r",
"ler_def",
"opprB",
"real",
"ring_display",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le | := Rle. | Notation | le | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt | := Rlt. | Notation | lt | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x <= y" | := (le x y) : ring_scope. | Notation | x <= y | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x < y" | := (lt x y) : ring_scope. | Notation | x < y | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le0N x : (0 <= - x) = (x <= 0). | Proof. by rewrite -sub0r sub_ge0. Qed. | Let | le0N | 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",... | [
"sub0r",
"sub_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leN_total x : 0 <= x \/ 0 <= - x. | Proof. by apply/orP; rewrite le0N le0_total. Qed. | Let | leN_total | 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",
"le0N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le00 : 0 <= 0. | Proof. by have:= le0_total 0; rewrite orbb. Qed. | Let | le00 | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt0_add x y : 0 < x -> 0 < y -> 0 < x + y. | Proof.
rewrite !lt_def => /andP [x_neq0 l0x] /andP [y_neq0 l0y]; rewrite le0_add //.
rewrite andbT addr_eq0; apply: contraNneq x_neq0 => hxy.
by rewrite [x](@le0_anti) // hxy -le0N opprK.
Qed. | Fact | lt0_add | 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",
"apply",
"contraNneq",
"le0N",
"le0_add",
"lt_def",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq0_norm x : `|x| = 0 -> x = 0. | Proof.
case: (leN_total x) => /ge0_norm => [-> // | Dnx nx0].
by rewrite -[x]opprK -Dnx normN nx0 oppr0.
Qed. | Fact | eq0_norm | 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",... | [
"leN_total",
"normN",
"oppr0",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_def x y : (x <= y) = (`|y - x| == y - x). | Proof.
wlog ->: x y / x = 0 by move/(_ 0 (y - x)); rewrite subr0 sub_ge0 => ->.
rewrite {x}subr0; apply/idP/eqP=> [/ge0_norm// | Dy].
by have [//| ny_ge0] := leN_total y; rewrite -Dy -normN ge0_norm.
Qed. | Fact | le_def | 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",
"leN_total",
"normN",
"sub_ge0",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normM : {morph norm : x y / x * y}. | Proof.
move=> x y /=; wlog x_ge0 : x / 0 <= x.
by move=> IHx; case: (leN_total x) => /IHx//; rewrite mulNr !normN.
wlog y_ge0 : y / 0 <= y; last by rewrite ?ge0_norm ?le0_mul.
by move=> IHy; case: (leN_total y) => /IHy//; rewrite mulrN !normN.
Qed. | Fact | normM | 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",... | [
"last",
"le0_mul",
"leN_total",
"mulNr",
"mulrN",
"norm",
"normN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_normD x y : `|x + y| <= `|x| + `|y|. | Proof.
wlog x_ge0 : x y / 0 <= x.
by move=> IH; case: (leN_total x) => /IH// /(_ (- y)); rewrite -opprD !normN.
rewrite -sub_ge0 ge0_norm //; have [y_ge0 | ny_ge0] := leN_total y.
by rewrite !ge0_norm ?subrr ?le0_add.
rewrite -normN ge0_norm //; have [hxy|hxy] := leN_total (x + y).
by rewrite ge0_norm... | Fact | le_normD | 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",... | [
"addrC",
"addrCA",
"addrKA",
"addrNK",
"le0_add",
"leN_total",
"normN",
"opprD",
"opprK",
"sub_ge0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_total : total le. | Proof. by move=> x y; rewrite -sub_ge0 -opprB le0N orbC -sub_ge0 le0_total. Qed. | Fact | le_total | 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",
"le0N",
"opprB",
"sub_ge0",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt0N x : (- x < 0) = (0 < x). | Proof. by rewrite -sub_gt0 add0r opprK. Qed. | Fact | lt0N | 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",... | [
"add0r",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leN_total x : 0 <= x \/ 0 <= - x. | Proof.
rewrite !le_def [_ == - x]eq_sym oppr_eq0 -[0 < - x]lt0N opprK.
apply/orP; case: (eqVneq x) => //=; exact: lt0_total.
Qed. | Let | leN_total | 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",
"eqVneq",
"eq_sym",
"le_def",
"lt0N",
"opprK",
"oppr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le00 : (0 <= 0). | Proof. by rewrite le_def eqxx. Qed. | Let | le00 | 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",
"le_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_ge0 x y : (0 <= y - x) = (x <= y). | Proof. by rewrite !le_def eq_sym subr_eq0 eq_sym sub_gt0. Qed. | Fact | sub_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",... | [
"eq_sym",
"le_def",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le0_add x y : 0 <= x -> 0 <= y -> 0 <= x + y. | Proof.
rewrite !le_def => /predU1P [<-|x_gt0]; first by rewrite add0r.
by case/predU1P=> [<-|y_gt0]; rewrite ?addr0 ?x_gt0 ?lt0_add // orbT.
Qed. | Fact | le0_add | 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",... | [
"add0r",
"addr0",
"le_def",
"lt0_add",
"predU1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le0_mul x y : 0 <= x -> 0 <= y -> 0 <= x * y. | Proof.
rewrite !le_def => /predU1P [<-|x_gt0]; first by rewrite mul0r eqxx.
by case/predU1P=> [<-|y_gt0]; rewrite ?mulr0 ?eqxx ?lt0_mul // orbT.
Qed. | Fact | le0_mul | 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",
"le_def",
"mul0r",
"mulr0",
"predU1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_def' x y : (x <= y) = (`|y - x| == y - x). | Proof.
wlog ->: x y / x = 0 by move/(_ 0 (y - x)); rewrite subr0 sub_ge0 => ->.
rewrite {x}subr0; apply/idP/eqP=> [/ge0_norm// | Dy].
by have [//| ny_ge0] := leN_total y; rewrite -Dy -normN ge0_norm.
Qed. | Fact | le_def' | 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",
"leN_total",
"normN",
"sub_ge0",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_def x y : (x < y) = (y != x) && (x <= y). | Proof.
rewrite le_def; case: eqVneq => //= ->; rewrite -sub_gt0 subrr.
by apply/idP=> lt00; case/negP: (lt0_ngt0 lt00).
Qed. | Fact | lt_def | 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",
"eqVneq",
"le_def",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_total : total le. | Proof.
move=> x y; rewrite !le_def; have [->|] //= := eqVneq; rewrite -subr_eq0.
by move/lt0_total; rewrite -(sub_gt0 (x - y)) sub0r opprB !sub_gt0 orbC.
Qed. | Fact | le_total | 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",... | [
"eqVneq",
"le",
"le_def",
"opprB",
"sub0r",
"subr_eq0",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj {C : numClosedFieldType} : C -> C | := @conj_subdef C. | Definition | conj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_op | := conj (only parsing). | Notation | conj_op | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_ivt : real_closed_axiom R. | Proof. exact: poly_ivt_subproof. Qed. | Lemma | poly_ivt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"real_closed_axiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_subproof (x : R) :
exists2 y, 0 <= y & (if 0 <= x then y ^+ 2 == x else y == 0) : bool. | Proof.
case x_ge0: (0 <= x); last by exists 0.
have le0x1: 0 <= x + 1 by rewrite -nnegrE rpredD ?rpred1.
have [|y /andP[y_ge0 _]] := @poly_ivt ('X^2 - x%:P) _ _ le0x1.
rewrite !hornerE expr0n/= sub0r oppr_le0 x_ge0/= subr_ge0.
by rewrite -[leLHS]mul1r ler_pM// (lerDl, lerDr).
by rewrite rootE !hornerE subr_eq0; exi... | Fact | sqrtr_subproof | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"expr0n",
"hornerE",
"last",
"leLHS",
"lerDl",
"lerDr",
"ler_pM",
"mul1r",
"nnegrE",
"oppr_le0",
"poly_ivt",
"rootE",
"rpred1",
"rpredD",
"sub0r",
"subr_eq0",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC | := conj. | Notation | conjC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr {R} x | := s2val (sig2W (@sqrtr_subproof R x)). | Definition | sqrtr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"sig2W",
"sqrtr_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt | := sqrtr. | Notation | sqrt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"sqrtr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"z ^*" | := (conj z) : ring_scope. | Notation | z ^* | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'i" | := imaginary : ring_scope. | Notation | 'i | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitf_gt0 x : 0 < x -> x \is a GRing.unit. | Proof. by move=> hx; rewrite unitfE eq_sym lt_eqF. Qed. | Lemma | unitf_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"eq_sym",
"lt_eqF",
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitf_lt0 x : x < 0 -> x \is a GRing.unit. | Proof. by move=> hx; rewrite unitfE lt_eqF. Qed. | Lemma | unitf_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lt_eqF",
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lef_pV2 : {in pos &, {mono (@GRing.inv F) : x y /~ x <= y}}. | Proof. by move=> x y hx hy /=; rewrite ler_pV2 ?inE ?unitf_gt0. Qed. | Lemma | lef_pV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"inE",
"inv",
"ler_pV2",
"pos",
"unitf_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lef_nV2 : {in neg &, {mono (@GRing.inv F) : x y /~ x <= y}}. | Proof. by move=> x y hx hy /=; rewrite ler_nV2 ?inE ?unitf_lt0. Qed. | Lemma | lef_nV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"inE",
"inv",
"ler_nV2",
"neg",
"unitf_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltf_pV2 : {in pos &, {mono (@GRing.inv F) : x y /~ x < y}}. | Proof. exact: leW_nmono_in lef_pV2. Qed. | Lemma | ltf_pV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"inv",
"leW_nmono_in",
"lef_pV2",
"pos"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltf_nV2 : {in neg &, {mono (@GRing.inv F) : x y /~ x < y}}. | Proof. exact: leW_nmono_in lef_nV2. Qed. | Lemma | ltf_nV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"inv",
"leW_nmono_in",
"lef_nV2",
"neg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltef_pV2 | := (lef_pV2, ltf_pV2). | Definition | ltef_pV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lef_pV2",
"ltf_pV2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltef_nV2 | := (lef_nV2, ltf_nV2). | Definition | ltef_nV2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lef_nV2",
"ltf_nV2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_pgt : {in pos &, forall x y, (x < y^-1) = (y < x^-1)}. | Proof. by move=> x y *; rewrite -[x in LHS]invrK ltf_pV2// posrE invr_gt0. Qed. | Lemma | invf_pgt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_gt0",
"ltf_pV2",
"pos",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_pge : {in pos &, forall x y, (x <= y^-1) = (y <= x^-1)}. | Proof. by move=> x y *; rewrite -[x in LHS]invrK lef_pV2// posrE invr_gt0. Qed. | Lemma | invf_pge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_gt0",
"lef_pV2",
"pos",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_ngt : {in neg &, forall x y, (x < y^-1) = (y < x^-1)}. | Proof. by move=> x y *; rewrite -[x in LHS]invrK ltf_nV2// negrE invr_lt0. Qed. | Lemma | invf_ngt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_lt0",
"ltf_nV2",
"neg",
"negrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_nge : {in neg &, forall x y, (x <= y^-1) = (y <= x^-1)}. | Proof. by move=> x y *; rewrite -[x in LHS]invrK lef_nV2// negrE invr_lt0. Qed. | Lemma | invf_nge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_lt0",
"lef_nV2",
"neg",
"negrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_gt1 x : 0 < x -> (1 < x^-1) = (x < 1). | Proof. by move=> x0; rewrite invf_pgt ?invr1 ?posrE. Qed. | Lemma | invf_gt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_pgt",
"invr1",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_ge1 x : 0 < x -> (1 <= x^-1) = (x <= 1). | Proof. by move=> x0; rewrite invf_pge ?invr1 ?posrE. Qed. | Lemma | invf_ge1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_pge",
"invr1",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_gte1 | := (invf_ge1, invf_gt1). | Definition | invf_gte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_ge1",
"invf_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_plt : {in pos &, forall x y, (x^-1 < y) = (y^-1 < x)}. | Proof. by move=> x y *; rewrite -[y in LHS]invrK ltf_pV2// posrE invr_gt0. Qed. | Lemma | invf_plt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_gt0",
"ltf_pV2",
"pos",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_ple : {in pos &, forall x y, (x^-1 <= y) = (y^-1 <= x)}. | Proof. by move=> x y *; rewrite -[y in LHS]invrK lef_pV2// posrE invr_gt0. Qed. | Lemma | invf_ple | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_gt0",
"lef_pV2",
"pos",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_nlt : {in neg &, forall x y, (x^-1 < y) = (y^-1 < x)}. | Proof. by move=> x y *; rewrite -[y in LHS]invrK ltf_nV2// negrE invr_lt0. Qed. | Lemma | invf_nlt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_lt0",
"ltf_nV2",
"neg",
"negrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_nle : {in neg &, forall x y, (x^-1 <= y) = (y^-1 <= x)}. | Proof. by move=> x y *; rewrite -[y in LHS]invrK lef_nV2// negrE invr_lt0. Qed. | Lemma | invf_nle | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invrK",
"invr_lt0",
"lef_nV2",
"neg",
"negrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_le1 x : 0 < x -> (x^-1 <= 1) = (1 <= x). | Proof. by move=> x0; rewrite -invf_ple ?invr1 ?posrE. Qed. | Lemma | invf_le1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_ple",
"invr1",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_lt1 x : 0 < x -> (x^-1 < 1) = (1 < x). | Proof. by move=> x0; rewrite invf_plt ?invr1 ?posrE. Qed. | Lemma | invf_lt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_plt",
"invr1",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_lte1 | := (invf_le1, invf_lt1). | Definition | invf_lte1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_le1",
"invf_lt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invf_cp1 | := (invf_gte1, invf_lte1). | Definition | invf_cp1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invf_gte1",
"invf_lte1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pdivlMr z x y : 0 < z -> (x <= y / z) = (x * z <= y). | Proof. by move=> z_gt0; rewrite -(@ler_pM2r _ z _ x) ?mulfVK ?gt_eqF. Qed. | Lemma | ler_pdivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"gt_eqF",
"ler_pM2r",
"mulfVK"
] | These lemma are all combinations of mono(LR|RL) with ler_[pn]mul2[rl]. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltr_pdivlMr z x y : 0 < z -> (x < y / z) = (x * z < y). | Proof. by move=> z_gt0; rewrite -(@ltr_pM2r _ z _ x) ?mulfVK ?gt_eqF. Qed. | Lemma | ltr_pdivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"gt_eqF",
"ltr_pM2r",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_pdivlMr | := (ler_pdivlMr, ltr_pdivlMr). | Definition | lter_pdivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_pdivlMr",
"ltr_pdivlMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pdivrMr z x y : 0 < z -> (y / z <= x) = (y <= x * z). | Proof. by move=> z_gt0; rewrite -(@ler_pM2r _ z) ?mulfVK ?gt_eqF. Qed. | Lemma | ler_pdivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"gt_eqF",
"ler_pM2r",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pdivrMr z x y : 0 < z -> (y / z < x) = (y < x * z). | Proof. by move=> z_gt0; rewrite -(@ltr_pM2r _ z) ?mulfVK ?gt_eqF. Qed. | Lemma | ltr_pdivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"gt_eqF",
"ltr_pM2r",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_pdivrMr | := (ler_pdivrMr, ltr_pdivrMr). | Definition | lter_pdivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_pdivrMr",
"ltr_pdivrMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pdivlMl z x y : 0 < z -> (x <= z^-1 * y) = (z * x <= y). | Proof. by move=> z_gt0; rewrite mulrC ler_pdivlMr ?[z * _]mulrC. Qed. | Lemma | ler_pdivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_pdivlMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pdivlMl z x y : 0 < z -> (x < z^-1 * y) = (z * x < y). | Proof. by move=> z_gt0; rewrite mulrC ltr_pdivlMr ?[z * _]mulrC. Qed. | Lemma | ltr_pdivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr_pdivlMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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