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 |
|---|---|---|---|---|---|---|---|---|---|---|
lteif_pM2r C x : 0 < x -> {mono *%R^~ x : y z / y < z ?<= if C}. | Proof. by case: C => ? ? ?; rewrite /= lter_pM2r. Qed. | Lemma | lteif_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",... | [
"lter_pM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_nM2l C x : x < 0 -> {mono *%R x : y z /~ y < z ?<= if C}. | Proof. by case: C => ? ? ?; rewrite /= lter_nM2l. Qed. | Lemma | lteif_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",... | [
"lter_nM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_nM2r C x : x < 0 -> {mono *%R^~ x : y z /~ y < z ?<= if C}. | Proof. by case: C => ? ? ?; rewrite /= lter_nM2r. Qed. | Lemma | lteif_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",... | [
"lter_nM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_nnormr C x y : y < 0 ?<= if ~~ C -> (`|x| < y ?<= if C) = false. | Proof. by case: C => ?; rewrite /= lter_nnormr. Qed. | Lemma | lteif_nnormr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lter_nnormr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lteifNE x y C : x \is Num.real -> y \is Num.real ->
(x < y ?<= if ~~ C) = ~~ (y < x ?<= if C). | Proof. by move=> ? ?; rewrite comparable_lteifNE ?real_comparable. Qed. | Lemma | real_lteifNE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"comparable_lteifNE",
"real",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lteif_norml C x y :
x \is Num.real ->
(`|x| < y ?<= if C) = ((- y < x ?<= if C) && (x < y ?<= if C)). | Proof. by case: C => ?; rewrite /= real_lter_norml. Qed. | Lemma | real_lteif_norml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_lter_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lteif_normr C x y :
y \is Num.real ->
(x < `|y| ?<= if C) = ((x < y ?<= if C) || (x < - y ?<= if C)). | Proof. by case: C => ?; rewrite /= real_lter_normr. Qed. | Lemma | real_lteif_normr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_lter_normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lteif_distl C x y e :
x - y \is real ->
(`|x - y| < e ?<= if C) = (y - e < x ?<= if C) && (x < y + e ?<= if C). | Proof. by case: C => /= ?; rewrite real_lter_distl. Qed. | Lemma | real_lteif_distl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_lter_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_leif_mean_square_scaled x y :
x \is real -> y \is real -> x * y *+ 2 <= x ^+ 2 + y ^+ 2 ?= iff (x == y). | Proof.
move=> Rx Ry; rewrite -[_ *+ 2]add0r -leifBRL addrAC -sqrrB -subr_eq0.
by rewrite -sqrf_eq0 eq_sym; apply: leif_eq; rewrite -realEsqr rpredB.
Qed. | Lemma | real_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",... | [
"add0r",
"addrAC",
"apply",
"eq_sym",
"leifBRL",
"leif_eq",
"real",
"realEsqr",
"rpredB",
"sqrf_eq0",
"sqrrB",
"subr_eq0"
] | Mean inequalities. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
real_leif_AGM2_scaled x y :
x \is real -> y \is real -> x * y *+ 4 <= (x + y) ^+ 2 ?= iff (x == y). | Proof.
move=> Rx Ry; rewrite sqrrD addrAC (mulrnDr _ 2) -leifBLR addrK.
exact: real_leif_mean_square_scaled.
Qed. | Lemma | real_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",... | [
"addrAC",
"addrK",
"leifBLR",
"mulrnDr",
"real",
"real_leif_mean_square_scaled",
"sqrrD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_AGM_scaled (I : finType) (A : {pred I}) (E : I -> R) (n := #|A|) :
{in A, forall i, 0 <= E i *+ n} ->
\prod_(i in A) (E i *+ n) <= (\sum_(i in A) E i) ^+ n
?= iff [forall i in A, forall j in A, E i == E j]. | Proof.
have [m leAm] := ubnP #|A|; elim: m => // m IHm in A leAm E n * => Ege0.
apply/leifP; case: ifPn => [/forall_inP-Econstant | Enonconstant].
have [i /= Ai | A0] := pickP [in A]; last by rewrite [n]eq_card0 ?big_pred0.
have /eqfun_inP-E_i := Econstant i Ai; rewrite -(eq_bigr _ E_i) sumr_const.
by rewrite exp... | Lemma | leif_AGM_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",... | [
"A'",
"addrA",
"addrI",
"apply",
"bigD1",
"big_andbC",
"big_pred0",
"cardD1",
"contraTneq",
"eqVneq",
"eq_bigr",
"eq_card0",
"eqfun_inP",
"eqxx",
"exists_inP",
"exists_inPn",
"expn_gt0",
"exprMn_n",
"exprS",
"exprn_gt0",
"forall_inP",
"ger0_real",
"inE",
"last",
"le_l... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_disk_bound p b : {ub | forall x, `|x| <= b -> `|p.[x]| <= ub}. | Proof.
exists (\sum_(j < size p) `|p`_j| * b ^+ j) => x le_x_b.
rewrite horner_coef (le_trans (ler_norm_sum _ _ _)) ?ler_sum // => j _.
rewrite normrM normrX ler_wpM2l ?lerXn2r ?unfold_in //=.
exact: le_trans (normr_ge0 x) le_x_b.
Qed. | Lemma | poly_disk_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",... | [
"horner_coef",
"le_trans",
"lerXn2r",
"ler_norm_sum",
"ler_sum",
"ler_wpM2l",
"normrM",
"normrX",
"normr_ge0",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_BSide_min b x y : x \in Num.real -> y \in Num.real ->
BSide b (Order.min x y) = Order.min (BSide b x) (BSide b y). | Proof. by move=> xr yr; apply/comparable_BSide_min/real_comparable. Qed. | Lemma | real_BSide_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",
"comparable_BSide_min",
"min",
"real",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_BSide_max b x y : x \in Num.real -> y \in Num.real ->
BSide b (Order.max x y) = Order.max (BSide b x) (BSide b y). | Proof. by move=> xr yr; apply/comparable_BSide_max/real_comparable. Qed. | Lemma | real_BSide_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",
"comparable_BSide_max",
"max",
"real",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem0_itvcc_xNx x : (0 \in `[- x, x]) = (0 <= x). | Proof. by rewrite itv_boundlr [in LHS]/<=%O /= oppr_le0 andbb. Qed. | Lemma | mem0_itvcc_xNx | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"itv_boundlr",
"oppr_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem0_itvoo_xNx x : (0 \in `]- x, x[) = (0 < x). | Proof. by rewrite itv_boundlr [in LHS]/<=%O /= oppr_lt0 andbb. Qed. | Lemma | mem0_itvoo_xNx | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"itv_boundlr",
"oppr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_itv ba bb (xa xb x : R) :
(- x \in Interval (BSide ba xa) (BSide bb xb)) =
(x \in Interval (BSide (~~ bb) (- xb)) (BSide (~~ ba) (- xa))). | Proof.
by rewrite !itv_boundlr /<=%O /= !implybF negbK andbC lteifNl lteifNr.
Qed. | Lemma | oppr_itv | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"itv_boundlr",
"lteifNl",
"lteifNr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_itvoo (a b x : R) : (- x \in `]a, b[) = (x \in `]- b, - a[). | Proof. exact: oppr_itv. Qed. | Lemma | oppr_itvoo | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_itvco (a b x : R) : (- x \in `[a, b[) = (x \in `]- b, - a]). | Proof. exact: oppr_itv. Qed. | Lemma | oppr_itvco | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_itvoc (a b x : R) : (- x \in `]a, b]) = (x \in `[- b, - a[). | Proof. exact: oppr_itv. Qed. | Lemma | oppr_itvoc | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppr_itvcc (a b x : R) : (- x \in `[a, b]) = (x \in `[- b, - a]). | Proof. exact: oppr_itv. Qed. | Lemma | oppr_itvcc | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
miditv (R : numDomainType) (i : interval R) : R | :=
match i with
| Interval (BSide _ a) (BSide _ b) => (a + b) / 2%:R
| Interval -oo%O (BSide _ b) => b - 1
| Interval (BSide _ a) +oo%O => a + 1
| Interval -oo%O +oo%O => 0
| _ => 0
end. | Definition | miditv | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"interval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrG_gt0 G : #|G|%:R > 0 :> R. | Proof. by rewrite ltr0n cardG_gt0. Qed. | Lemma | natrG_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",... | [
"cardG_gt0",
"ltr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrG_neq0 G : #|G|%:R != 0 :> R. | Proof. by rewrite gt_eqF // natrG_gt0. Qed. | Lemma | natrG_neq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"gt_eqF",
"natrG_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_indexg_gt0 G B : #|G : B|%g%:R > 0 :> R. | Proof. by rewrite ltr0n indexg_gt0. Qed. | Lemma | natr_indexg_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",... | [
"indexg_gt0",
"ltr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_indexg_neq0 G B : #|G : B|%g%:R != 0 :> R. | Proof. by rewrite gt_eqF // natr_indexg_gt0. Qed. | Lemma | natr_indexg_neq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"gt_eqF",
"natr_indexg_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_real x : x \is real. | Proof. exact: num_real. Qed. | Lemma | num_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerP x y : ler_xor_gt x y (min y x) (min x y) (max y x) (max x y)
`|x - y| `|y - x| (x <= y) (y < x). | Proof. exact: real_leP. Qed. | Lemma | lerP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_xor_gt",
"max",
"min",
"real_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrP x y : ltr_xor_ge x y (min y x) (min x y) (max y x) (max x y)
`|x - y| `|y - x| (y <= x) (x < y). | Proof. exact: real_ltP. Qed. | Lemma | ltrP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_xor_ge",
"max",
"min",
"real_ltP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrgtP x y :
comparer x y (min y x) (min x y) (max y x) (max x y)
`|x - y| `|y - x| (y == x) (x == y)
(x >= y) (x <= y) (x > y) (x < y) . | Proof. exact: real_ltgtP. Qed. | Lemma | ltrgtP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"comparer",
"max",
"min",
"real_ltgtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0P x : ger0_xor_lt0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (x < 0) (0 <= x). | Proof. exact: real_ge0P. Qed. | Lemma | ger0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ger0_xor_lt0",
"max",
"min",
"real_ge0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0P x : ler0_xor_gt0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (0 < x) (x <= 0). | Proof. exact: real_le0P. Qed. | Lemma | ler0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler0_xor_gt0",
"max",
"min",
"real_le0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrgt0P x : comparer0 x (min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (0 == x) (x == 0) (x <= 0) (0 <= x) (x < 0) (x > 0). | Proof. exact: real_ltgt0P. Qed. | Lemma | ltrgt0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"comparer0",
"max",
"min",
"real_ltgt0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_lt0 x y :
(x * y < 0) = [&& x != 0, y != 0 & (x < 0) (+) (y < 0)]. | Proof.
have [x_gt0|x_lt0|->] /= := ltrgt0P x; last by rewrite mul0r.
by rewrite pmulr_rlt0 //; case: ltrgt0P.
by rewrite nmulr_rlt0 //; case: ltrgt0P.
Qed. | Lemma | mulr_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"last",
"ltrgt0P",
"mul0r",
"nmulr_rlt0",
"pmulr_rlt0"
] | sign | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
neq0_mulr_lt0 x y :
x != 0 -> y != 0 -> (x * y < 0) = (x < 0) (+) (y < 0). | Proof. by move=> x_neq0 y_neq0; rewrite mulr_lt0 x_neq0 y_neq0. Qed. | Lemma | neq0_mulr_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_sign_lt0 (b : bool) x :
((-1) ^+ b * x < 0) = (x != 0) && (b (+) (x < 0)%R). | Proof. by rewrite mulr_lt0 signr_lt0 signr_eq0. Qed. | Lemma | mulr_sign_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulr_lt0",
"signr_eq0",
"signr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_sign_norm x : (-1) ^+ (x < 0)%R * `|x| = x. | Proof. by rewrite -realEsign. Qed. | Lemma | mulr_sign_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",... | [
"realEsign"
] | sign & norm | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mulr_Nsign_norm x : (-1) ^+ (0 < x)%R * `|x| = - x. | Proof. by rewrite real_mulr_Nsign_norm. Qed. | Lemma | mulr_Nsign_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",... | [
"real_mulr_Nsign_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numEsign x : x = (-1) ^+ (x < 0)%R * `|x|. | Proof. by rewrite -realEsign. Qed. | Lemma | numEsign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"realEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numNEsign x : -x = (-1) ^+ (0 < x)%R * `|x|. | Proof. by rewrite -realNEsign. Qed. | Lemma | numNEsign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"realNEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrEsign x : `|x| = (-1) ^+ (x < 0)%R * x. | Proof. by rewrite -real_normrEsign. Qed. | Lemma | normrEsign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_normrEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'arg' 'min_' ( i < i0 | P ) F ]" | :=
(Order.arg_min (disp := ring_display) i0 (fun i => P%B) (fun i => F)) :
ring_scope. | Notation | [ 'arg' 'min_' ( i < i0 | P ) F ] | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"arg_min",
"i0",
"ring_display"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'arg' 'min_' ( i < i0 'in' A ) F ]" | :=
[arg min_(i < i0 | i \in A) F] : ring_scope. | Notation | [ 'arg' 'min_' ( i < i0 'in' A ) F ] | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"i0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'arg' 'min_' ( i < i0 ) F ]" | := [arg min_(i < i0 | true) F] :
ring_scope. | Notation | [ 'arg' 'min_' ( i < i0 ) F ] | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"i0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'arg' 'max_' ( i > i0 | P ) F ]" | :=
(Order.arg_max (disp := ring_display) i0 (fun i => P%B) (fun i => F)) :
ring_scope. | Notation | [ 'arg' 'max_' ( i > i0 | P ) F ] | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"arg_max",
"i0",
"ring_display"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'arg' 'max_' ( i > i0 'in' A ) F ]" | :=
[arg max_(i > i0 | i \in A) F] : ring_scope. | Notation | [ 'arg' 'max_' ( i > i0 'in' A ) F ] | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"i0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'arg' 'max_' ( i > i0 ) F ]" | := [arg max_(i > i0 | true) F] :
ring_scope. | Notation | [ 'arg' 'max_' ( i > i0 ) F ] | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"i0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numR_real | := @num_real R. | Let | numR_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"num_real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_cp0 x :
((sg x == 1) = (0 < x)) *
((sg x == -1) = (x < 0)) *
((sg x == 0) = (x == 0)). | Proof.
rewrite -[1]/((-1) ^+ false) -signrN lt0r leNgt sgr_def.
case: (x =P 0) => [-> | _]; first by rewrite !(eq_sym 0) !signr_eq0 ltxx eqxx.
by rewrite !(inj_eq signr_inj) eqb_id eqbF_neg signr_eq0 //.
Qed. | Lemma | sgr_cp0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"eqbF_neg",
"eqb_id",
"eqxx",
"inj_eq",
"leNgt",
"lt0r",
"ltxx",
"sg",
"sgr_def",
"signrN",
"signr_eq0",
"signr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_val x : R -> bool -> bool -> bool -> bool -> bool -> bool
-> bool -> bool -> bool -> bool -> bool -> bool -> R -> Set | :=
| SgrNull of x = 0 : sgr_val x 0 true true true true false false
true false false true false false 0
| SgrPos of x > 0 : sgr_val x x false false true false false true
false false true false false true 1
| SgrNeg of x < 0 : sgr_val x (- x) false true false false true false
false true false false tru... | Variant | sgr_val | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
sgrP x :
sgr_val x `|x| (0 == x) (x <= 0) (0 <= x) (x == 0) (x < 0) (0 < x)
(0 == sg x) (-1 == sg x) (1 == sg x)
(sg x == 0) (sg x == -1) (sg x == 1) (sg x). | Proof.
by rewrite ![_ == sg _]eq_sym !sgr_cp0 /sg; case: ltrgt0P; constructor.
Qed. | Lemma | sgrP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ltrgt0P",
"sg",
"sgr_cp0",
"sgr_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrEsg x : `|x| = sg x * x. | Proof. by case: sgrP; rewrite ?(mul0r, mul1r, mulN1r). Qed. | Lemma | normrEsg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mul0r",
"mul1r",
"mulN1r",
"sg",
"sgrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
numEsg x : x = sg x * `|x|. | Proof. by case: sgrP; rewrite !(mul1r, mul0r, mulrNN). Qed. | Lemma | numEsg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mul0r",
"mul1r",
"mulrNN",
"sg",
"sgrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_sg_norm x : sg x * `|x| = x. | Proof. by rewrite -numEsg. Qed. | Lemma | mulr_sg_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",... | [
"numEsg",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrM x y : sg (x * y) = sg x * sg y. | Proof.
rewrite !sgr_def mulr_lt0 andbA mulrnAr mulrnAl -mulrnA mulnb -negb_or mulf_eq0.
by case: (~~ _) => //; rewrite signr_addb.
Qed. | Lemma | sgrM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"mulnb",
"mulr_lt0",
"mulrnA",
"mulrnAl",
"mulrnAr",
"sg",
"sgr_def",
"signr_addb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrN x : sg (- x) = - sg x. | Proof. by rewrite -mulrN1 sgrM sgrN1 mulrN1. Qed. | Lemma | sgrN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulrN1",
"sg",
"sgrM",
"sgrN1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrX n x : sg (x ^+ n) = (sg x) ^+ n. | Proof. by elim: n => [|n IHn]; rewrite ?sgr1 // !exprS sgrM IHn. Qed. | Lemma | sgrX | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"exprS",
"sg",
"sgr1",
"sgrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_smul x y : sg (sg x * y) = sg x * sg y. | Proof. by rewrite sgrM sgr_id. Qed. | Lemma | sgr_smul | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"sg",
"sgrM",
"sgr_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_gt0 x : (sg x > 0) = (x > 0). | Proof. by rewrite -[LHS]sgr_cp0 sgr_id sgr_cp0. Qed. | Lemma | sgr_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",... | [
"sg",
"sgr_cp0",
"sgr_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_ge0 x : (sgr x >= 0) = (x >= 0). | Proof. by rewrite !leNgt sgr_lt0. Qed. | Lemma | sgr_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",... | [
"leNgt",
"sgr",
"sgr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_norm x : (x <= `|x|). | Proof. exact: real_ler_norm. Qed. | Lemma | ler_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",... | [
"real_ler_norm"
] | norm section | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ler_norml x y : (`|x| <= y) = (- y <= x <= y). | Proof. exact: real_ler_norml. Qed. | Lemma | ler_norml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_normlP x y : reflect ((- x <= y) * (x <= y)) (`|x| <= y). | Proof. exact: real_ler_normlP. Qed. | Lemma | ler_normlP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_normlP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_norml x y : (`|x| == y) = ((x == y) || (x == -y)) && (0 <= y). | Proof. exact: real_eqr_norml. Qed. | Lemma | eqr_norml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_eqr_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_norm2 x y : (`|x| == `|y|) = (x == y) || (x == -y). | Proof. exact: real_eqr_norm2. Qed. | Lemma | eqr_norm2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_eqr_norm2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_norml x y : (`|x| < y) = (- y < x < y). | Proof. exact: real_ltr_norml. Qed. | Lemma | ltr_norml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_norml | := (ler_norml, ltr_norml). | Definition | lter_norml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_norml",
"ltr_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_normlP x y : reflect ((-x < y) * (x < y)) (`|x| < y). | Proof. exact: real_ltr_normlP. Qed. | Lemma | ltr_normlP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_normlP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_normlW x y : `|x| < y -> x < y. | Proof. exact: real_ltr_normlW. Qed. | Lemma | ltr_normlW | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_normlW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltrNnormlW x y : `|x| < y -> - y < x. | Proof. exact: real_ltrNnormlW. Qed. | Lemma | ltrNnormlW | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltrNnormlW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_normlW x y : `|x| <= y -> x <= y. | Proof. exact: real_ler_normlW. Qed. | Lemma | ler_normlW | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_normlW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerNnormlW x y : `|x| <= y -> - y <= x. | Proof. exact: real_lerNnormlW. Qed. | Lemma | lerNnormlW | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_lerNnormlW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_normr x y : (x <= `|y|) = (x <= y) || (x <= - y). | Proof. exact: real_ler_normr. Qed. | Lemma | ler_normr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_normr x y : (x < `|y|) = (x < y) || (x < - y). | Proof. exact: real_ltr_normr. Qed. | Lemma | ltr_normr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_normr | := (ler_normr, ltr_normr). | Definition | lter_normr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_normr",
"ltr_normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distl x y e : (`|x - y| <= e) = (y - e <= x <= y + e). | Proof. exact: real_ler_distl. Qed. | Lemma | ler_distl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_distl x y e : (`|x - y| < e) = (y - e < x < y + e). | Proof. exact: real_ltr_distl. Qed. | Lemma | ltr_distl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_distl | := (ler_distl, ltr_distl). | Definition | lter_distl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_distl",
"ltr_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_distlC x y e : (`|x - y| < e) = (x - e < y < x + e). | Proof. by rewrite distrC ltr_distl. Qed. | Lemma | ltr_distlC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"distrC",
"ltr_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distlC x y e : (`|x - y| <= e) = (x - e <= y <= x + e). | Proof. by rewrite distrC ler_distl. Qed. | Lemma | ler_distlC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"distrC",
"ler_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_distlC | := (ler_distlC, ltr_distlC). | Definition | lter_distlC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_distlC",
"ltr_distlC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_distlDr x y e : `|x - y| < e -> x < y + e. | Proof. exact: real_ltr_distlDr. Qed. | Lemma | ltr_distlDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_distlDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distlDr x y e : `|x - y| <= e -> x <= y + e. | Proof. exact: real_ler_distlDr. Qed. | Lemma | ler_distlDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_distlDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_distlCDr x y e : `|x - y| < e -> y < x + e. | Proof. exact: real_ltr_distlCDr. Qed. | Lemma | ltr_distlCDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_distlCDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distlCDr x y e : `|x - y| <= e -> y <= x + e. | Proof. exact: real_ler_distlCDr. Qed. | Lemma | ler_distlCDr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_distlCDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_distlBl x y e : `|x - y| < e -> x - e < y. | Proof. exact: real_ltr_distlBl. Qed. | Lemma | ltr_distlBl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_distlBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distlBl x y e : `|x - y| <= e -> x - e <= y. | Proof. exact: real_ler_distlBl. Qed. | Lemma | ler_distlBl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_distlBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_distlCBl x y e : `|x - y| < e -> y - e < x. | Proof. exact: real_ltr_distlCBl. Qed. | Lemma | ltr_distlCBl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltr_distlCBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distlCBl x y e : `|x - y| <= e -> y - e <= x. | Proof. exact: real_ler_distlCBl. Qed. | Lemma | ler_distlCBl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ler_distlCBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_even_ge0 n x : ~~ odd n -> 0 <= x ^+ n. | Proof. by move=> even_n; rewrite real_exprn_even_ge0 ?num_real. Qed. | Lemma | exprn_even_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",... | [
"num_real",
"odd",
"real_exprn_even_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_even_gt0 n x : ~~ odd n -> (0 < x ^+ n) = (n == 0)%N || (x != 0). | Proof. by move=> even_n; rewrite real_exprn_even_gt0 ?num_real. Qed. | Lemma | exprn_even_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",... | [
"num_real",
"odd",
"real_exprn_even_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_even_le0 n x : ~~ odd n -> (x ^+ n <= 0) = (n != 0) && (x == 0). | Proof. by move=> even_n; rewrite real_exprn_even_le0 ?num_real. Qed. | Lemma | exprn_even_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",... | [
"num_real",
"odd",
"real_exprn_even_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_even_lt0 n x : ~~ odd n -> (x ^+ n < 0) = false. | Proof. by move=> even_n; rewrite real_exprn_even_lt0 ?num_real. Qed. | Lemma | exprn_even_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"num_real",
"odd",
"real_exprn_even_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_odd_ge0 n x : odd n -> (0 <= x ^+ n) = (0 <= x). | Proof. by move=> even_n; rewrite real_exprn_odd_ge0 ?num_real. Qed. | Lemma | exprn_odd_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",... | [
"num_real",
"odd",
"real_exprn_odd_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_odd_gt0 n x : odd n -> (0 < x ^+ n) = (0 < x). | Proof. by move=> even_n; rewrite real_exprn_odd_gt0 ?num_real. Qed. | Lemma | exprn_odd_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",... | [
"num_real",
"odd",
"real_exprn_odd_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_odd_le0 n x : odd n -> (x ^+ n <= 0) = (x <= 0). | Proof. by move=> even_n; rewrite real_exprn_odd_le0 ?num_real. Qed. | Lemma | exprn_odd_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",... | [
"num_real",
"odd",
"real_exprn_odd_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_odd_lt0 n x : odd n -> (x ^+ n < 0) = (x < 0). | Proof. by move=> even_n; rewrite real_exprn_odd_lt0 ?num_real. Qed. | Lemma | exprn_odd_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"num_real",
"odd",
"real_exprn_odd_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_norml C x y :
(`|x| < y ?<= if C) = (- y < x ?<= if C) && (x < y ?<= if C). | Proof. by case: C; rewrite /= lter_norml. Qed. | Lemma | lteif_norml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lter_norml"
] | lteif | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lteif_normr C x y :
(x < `|y| ?<= if C) = (x < y ?<= if C) || (x < - y ?<= if C). | Proof. by case: C; rewrite /= lter_normr. Qed. | Lemma | lteif_normr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lter_normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_distl C x y e :
(`|x - y| < e ?<= if C) = (y - e < x ?<= if C) && (x < y + e ?<= if C). | Proof. by case: C; rewrite /= lter_distl. Qed. | Lemma | lteif_distl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lter_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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