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 |
|---|---|---|---|---|---|---|---|---|---|---|
comSemiAlgType | := (comNzSemiAlgType) (only parsing). | Notation | comSemiAlgType | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comAlgType | := (comNzAlgType) (only parsing). | Notation | comAlgType | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subLSemiAlgType | := (subNzLSemiAlgType) (only parsing). | Notation | subLSemiAlgType | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subLalgType | := (subNzLalgType) (only parsing). | Notation | subLalgType | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subSemiAlgType | := (subNzSemiAlgType) (only parsing). | Notation | subSemiAlgType | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subAlgType | := (subNzAlgType) (only parsing). | Notation | subAlgType | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'char' R ]" | := (GRing.pchar R) : ring_scope. | Notation | [ 'char' R ] | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_char0 R | := (GRing.pchar R =i pred0). | Notation | has_char0 | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Frobenius_aut chRp | := (pFrobenius_aut chRp). | Notation | Frobenius_aut | algebra.algebraic_hierarchy | algebra/algebraic_hierarchy/ssralg.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"prime",
"binomial",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"decfield",
"GRing",
"GRing.Theory",
"AllExports"
] | [
"pFrobenius_aut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wD2l (x : R) : {homo +%R x : y z / y <= z}. | Proof. by move=> y z; rewrite !ler_def ![_ + z]addrC addrKA. Qed. | Fact | ler_wD2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"addrKA",
"ler_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_nmod_closed : nmod_closed (@Num.real R). | Proof.
split=> [|x y Rx Ry]; first by rewrite realE lexx.
without loss{Rx} x_ge0: x y Ry / 0 <= x.
case/orP: Rx => [? | x_le0]; first exact.
by rewrite -rpredN opprD; apply; rewrite ?rpredN ?oppr_ge0.
case/orP: Ry => [y_ge0 | y_le0]; first by rewrite realE -nnegrE rpredD.
by rewrite realE -[y]opprK orbC -oppr_ge0 o... | Fact | real_nmod_closed | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ger_leVge",
"lexx",
"nmod_closed",
"nnegrE",
"opprB",
"opprD",
"opprK",
"oppr_ge0",
"real",
"realE",
"rpredD",
"rpredN",
"split",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparabler_trans : transitive (Num.comparable : rel R). | Proof.
move=> y x z; rewrite !comparablerE => xBy_real yBz_real.
by have := rpredD xBy_real yBz_real; rewrite addrA addrNK.
Qed. | Fact | comparabler_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",
"addrNK",
"comparable",
"comparablerE",
"rel",
"rpredD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr | := norm. | Notation | 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",... | [
"norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`| x |" | := (norm x) : ring_scope. | Notation | `| x | | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr (x : R) : R | := if x == 0 then 0 else if x < 0 then -1 else 1. | Definition | sgr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
sg | := sgr. | Notation | sg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_axiom : Prop | := forall x : R, x \is real. | Definition | real_axiom | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
archimedean_axiom : Prop | := forall x : R, exists ub, `|x| < ub%:R. | Definition | archimedean_axiom | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
real_closed_axiom : Prop | :=
forall (p : {poly R}) (a b : R),
a <= b -> p.[a] <= 0 <= p.[b] -> exists2 x, a <= x <= b & root p x. | Definition | real_closed_axiom | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"poly",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_def x : (0 <= x) = (`|x| == x). | Proof. by rewrite ler_def subr0. Qed. | Lemma | ger0_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",... | [
"ler_def",
"subr0"
] | Basic consequences (just enough to get predicate closure properties). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ler01 : 0 <= 1 :> R. | Proof.
have n1_nz: `|1 : R| != 0 by apply: contraNneq (@oner_neq0 R) => /normr0_eq0->.
by rewrite ger0_def -(inj_eq (mulfI n1_nz)) -normrM !mulr1.
Qed. | Lemma | ler01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ger0_def",
"inj_eq",
"mulfI",
"mulr1",
"normr0_eq0",
"normrM",
"oner_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr01 : 0 < 1 :> R. | Proof. by rewrite lt_def oner_neq0 ler01. Qed. | Lemma | ltr01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler01",
"lt_def",
"oner_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter01 | := (ler01, ltr01). | Definition | lter01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler01",
"ltr01"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulr_rgt0 x y : 0 < x -> (0 < x * y) = (0 < y). | Proof.
rewrite !lt_def !ger0_def normrM mulf_eq0 negb_or => /andP[x_neq0 /eqP->].
by rewrite x_neq0 (inj_eq (mulfI x_neq0)).
Qed. | Lemma | pmulr_rgt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ger0_def",
"inj_eq",
"lt_def",
"mulfI",
"mulf_eq0",
"normrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pos_divr_closed : divr_closed (@pos R). | Proof.
split=> [|x y x_gt0 y_gt0]; rewrite posrE ?ltr01 //.
have [Uy|/invr_out->] := boolP (y \is a GRing.unit); last by rewrite pmulr_rgt0.
by rewrite -(pmulr_rgt0 _ y_gt0) mulrC divrK.
Qed. | Fact | pos_divr_closed | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"divrK",
"divr_closed",
"invr_out",
"last",
"ltr01",
"mulrC",
"pmulr_rgt0",
"pos",
"posrE",
"split",
"unit"
] | Closure properties of the real predicates. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nneg_divr_closed : divr_closed (@nneg R). | Proof.
split=> [|x y]; rewrite !nnegrE ?ler01 ?le0r // -!posrE.
case/predU1P=> [-> _ | x_gt0]; first by rewrite mul0r eqxx.
by case/predU1P=> [-> | y_gt0]; rewrite ?invr0 ?mulr0 ?eqxx // orbC rpred_div.
Qed. | Fact | nneg_divr_closed | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"divr_closed",
"eqxx",
"invr0",
"le0r",
"ler01",
"mul0r",
"mulr0",
"nneg",
"nnegrE",
"posrE",
"predU1P",
"rpred_div",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_divr_closed : divr_closed (@real R). | Proof.
split=> [|x y Rx Ry]; first by rewrite realE ler01.
without loss{Rx} x_ge0: x / 0 <= x.
case/orP: Rx => [? | x_le0]; first exact.
by rewrite -rpredN -mulNr; apply; rewrite ?oppr_ge0.
without loss{Ry} y_ge0: y / 0 <= y; last by rewrite realE -nnegrE rpred_div.
case/orP: Ry => [? | y_le0]; first exact.
by rewr... | Fact | real_divr_closed | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"divr_closed",
"invrN",
"last",
"ler01",
"mulNr",
"mulrN",
"nnegrE",
"oppr_ge0",
"real",
"realE",
"rpredN",
"rpred_div",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_real (R : realDomainType) (x : R) : x \is real. | Proof. exact: le_total. 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",... | [
"le_total",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_normD V (x y : V) : `|x + y| <= `|x| + `|y| | :=
ler_normD x y. | Definition | ler_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",... | [] | Lemmas from the signature (reexported). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
addr_gt0 x y : 0 < x -> 0 < y -> 0 < x + y | := @addr_gt0 R x y. | Definition | addr_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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr0_eq0 W (x : W) : `|x| = 0 -> x = 0 | := @normr0_eq0 R W x. | Definition | normr0_eq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger_leVge x y : 0 <= x -> 0 <= y -> (x <= y) || (y <= x) | :=
@ger_leVge R x y. | Definition | ger_leVge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
normrM : {morph norm : x y / (x : R) * y} | := @normrM R. | Definition | normrM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_def x y : (x <= y) = (`|y - x| == y - x) | := ler_def x y. | Definition | ler_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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrMn V (x : V) n : `|x *+ n| = `|x| *+ n | := normrMn x n. | Definition | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrN V (x : V) : `|- x| = `|x| | := normrN x. | Definition | 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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmulr_rgt0 x y : 0 < x -> (0 < x * y) = (0 < y). | Proof. exact: pmulr_rgt0. Qed. | Lemma | pmulr_rgt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [] | General properties of <= and < | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pmulr_rge0 x y : 0 < x -> (0 <= x * y) = (0 <= y). | Proof. by move=> x_gt0; rewrite !le0r mulf_eq0 pmulr_rgt0 // gt_eqF. Qed. | Lemma | pmulr_rge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"gt_eqF",
"le0r",
"mulf_eq0",
"pmulr_rgt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler01 : 0 <= 1 :> R. | Proof. exact: ler01. Qed. | Lemma | ler01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [] | Integer comparisons and characteristic 0. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltr01 : 0 < 1 :> R. | Proof. exact: ltr01. Qed. | Lemma | ltr01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
lter01 | := lter01. | Definition | lter01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
ler0n n : 0 <= n%:R :> R. | Proof. by rewrite -nnegrE rpred_nat. Qed. | Lemma | ler0n | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"nnegrE",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0Sn n : 0 < n.+1%:R :> R. | Proof. by elim: n => // n; apply: addr_gt0. Qed. | Lemma | ltr0Sn | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0n n : (0 < n%:R :> R) = (0 < n)%N. | Proof. by case: n => //= n; apply: ltr0Sn. Qed. | Lemma | ltr0n | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ltr0Sn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pnatr_eq0 n : (n%:R == 0 :> R) = (n == 0)%N. | Proof. by case: n => [|n]; rewrite ?mulr0n ?eqxx // gt_eqF. Qed. | Lemma | pnatr_eq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqxx",
"gt_eqF",
"mulr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pchar_num : [pchar R] =i pred0. | Proof. by case=> // p /=; rewrite !inE pnatr_eq0 andbF. Qed. | Lemma | pchar_num | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"inE",
"pchar",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_def x : (0 <= x) = (`|x| == x). | Proof. exact: ger0_def. Qed. | Lemma | ger0_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",... | [] | Properties of the norm. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
normr_idP {x} : reflect (`|x| = x) (0 <= x). | Proof. by rewrite ger0_def; apply: eqP. Qed. | Lemma | normr_idP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ger0_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_norm x : 0 <= x -> `|x| = x. | Proof. exact: normr_idP. Qed. | Lemma | ger0_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",... | [
"normr_idP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr1 : `|1 : R| = 1. | Proof. exact: ger0_norm. Qed. | Lemma | normr1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_nat n : `|n%:R : R| = n%:R. | Proof. exact: ger0_norm. Qed. | Lemma | normr_nat | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ger0_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_prod I r (P : pred I) (F : I -> R) :
`|\prod_(i <- r | P i) F i| = \prod_(i <- r | P i) `|F i|. | Proof. exact: (big_morph norm normrM normr1). Qed. | Lemma | normr_prod | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"big_morph",
"norm",
"normr1",
"normrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrX n x : `|x ^+ n| = `|x| ^+ n. | Proof. by rewrite -(card_ord n) -!prodr_const normr_prod. Qed. | Lemma | normrX | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"card_ord",
"normr_prod",
"prodr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_unit : {homo (@norm _ (* R *) R) : x / x \is a GRing.unit}. | Proof.
move=> x /= /unitrP [y [yx xy]]; apply/unitrP; exists `|y|.
by rewrite -!normrM xy yx normr1.
Qed. | Lemma | normr_unit | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"norm",
"normr1",
"normrM",
"unit",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrV : {in GRing.unit, {morph (@norm _ (* R *) R) : x / x ^-1}}. | Proof.
move=> x ux; apply: (mulrI (normr_unit ux)).
by rewrite -normrM !divrr ?normr1 ?normr_unit.
Qed. | Lemma | normrV | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"divrr",
"mulrI",
"norm",
"normr1",
"normrM",
"normr_unit",
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrN1 : `|-1 : R| = 1. | Proof.
have: `|-1 : R| ^+ 2 == 1 by rewrite -normrX -signr_odd normr1.
rewrite sqrf_eq1 => /orP[/eqP //|]; rewrite -ger0_def le0r oppr_eq0 oner_eq0.
by move/(addr_gt0 ltr01); rewrite subrr ltxx.
Qed. | 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",
"ger0_def",
"le0r",
"ltr01",
"ltxx",
"normr1",
"normrX",
"oner_eq0",
"oppr_eq0",
"signr_odd",
"sqrf_eq1",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_real I (P : pred I) (F : I -> R) (s : seq I) :
{in P, forall i, F i \is real} -> \prod_(i <- s | P i) F i \is real. | Proof. by apply/big_real; [apply: rpredM | apply: rpred1]. Qed. | Lemma | prod_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",... | [
"apply",
"big_real",
"real",
"rpred1",
"rpredM",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr0 : `|0 : V| = 0. | Proof. by rewrite -(mulr0n 0) normrMn mulr0n. Qed. | Lemma | normr0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulr0n",
"normrMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distrC v w : `|v - w| = `|w - v|. | Proof. by rewrite -opprB normrN. Qed. | Lemma | distrC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"normrN",
"opprB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_id v : `| `|v| | = `|v|. | Proof.
have nz2: 2 != 0 :> R by rewrite pnatr_eq0.
apply: (mulfI nz2); rewrite -{1}normr_nat -normrM mulr_natl mulr2n ger0_norm //.
by rewrite -{2}normrN -normr0 -(subrr v) ler_normD.
Qed. | Lemma | normr_id | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ger0_norm",
"ler_normD",
"mulfI",
"mulr2n",
"mulr_natl",
"normr0",
"normrM",
"normrN",
"normr_nat",
"nz2",
"pnatr_eq0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_ge0 v : 0 <= `|v|. | Proof. by rewrite ger0_def normr_id. Qed. | Lemma | normr_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",... | [
"ger0_def",
"normr_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_lt0 v : (`|v| < 0) = false. | Proof. by rewrite le_gtF// normr_ge0. Qed. | Lemma | normr_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",... | [
"le_gtF",
"normr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_norm_neq0 v : `|v| > 0 -> (v != 0). | Proof. by apply: contra_ltN => /eqP->; rewrite normr0. Qed. | Lemma | gtr0_norm_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",... | [
"apply",
"contra_ltN",
"normr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_norm_eq0F v : `|v| > 0 -> (v == 0) = false. | Proof. by move=> /gtr0_norm_neq0/negPf->. Qed. | Lemma | gtr0_norm_eq0F | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"gtr0_norm_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr0P v : reflect (`|v| = 0) (v == 0). | Proof. by apply: (iffP eqP)=> [->|/normr0_eq0 //]; apply: normr0. Qed. | Lemma | normr0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"normr0",
"normr0_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_eq0 v | := sameP (`|v| =P 0) (normr0P v). | Definition | normr_eq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"normr0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_le0 v : (`|v| <= 0) = (v == 0). | Proof. by rewrite -normr_eq0 eq_le normr_ge0 andbT. Qed. | Lemma | normr_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",... | [
"eq_le",
"normr_eq0",
"normr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_gt0 v : (`|v| > 0) = (v != 0). | Proof. by rewrite lt_def normr_eq0 normr_ge0 andbT. Qed. | Lemma | normr_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",
"normr_eq0",
"normr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrE | := (normr_id, normr0, normr1, normrN1, normr_ge0, normr_eq0,
normr_lt0, normr_le0, normr_gt0, normrN). | Definition | normrE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"normr0",
"normr1",
"normrN",
"normrN1",
"normr_eq0",
"normr_ge0",
"normr_gt0",
"normr_id",
"normr_le0",
"normr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0_def x : (x <= 0) = (`|x| == - x). | Proof. by rewrite ler_def sub0r normrN. Qed. | Lemma | ler0_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",... | [
"ler_def",
"normrN",
"sub0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0_norm x : x <= 0 -> `|x| = - x. | Proof. by move=> x_le0; rewrite -[r in _ = r]ger0_norm ?normrN ?oppr_ge0. Qed. | Lemma | ler0_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",... | [
"ger0_norm",
"normrN",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_norm x (hx : 0 < x) | := ger0_norm (ltW hx). | Definition | gtr0_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",... | [
"ger0_norm",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0_norm x (hx : x < 0) | := ler0_norm (ltW hx). | Definition | ltr0_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",... | [
"ler0_norm",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_le_norm :
{in nneg &, {mono (@normr _ R) : x y / x <= y}}. | Proof. by move=> x y; rewrite !nnegrE => x0 y0; rewrite !ger0_norm. Qed. | Lemma | ger0_le_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",... | [
"ger0_norm",
"nneg",
"nnegrE",
"normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_le_norm :
{in pos &, {mono (@normr _ R) : x y / x <= y}}. | Proof. by move=> x y; rewrite !posrE => /ltW x0 /ltW y0; exact: ger0_le_norm. Qed. | Lemma | gtr0_le_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",... | [
"ger0_le_norm",
"ltW",
"normr",
"pos",
"posrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0_ge_norm :
{in npos &, {mono (@normr _ R) : x y / x <= y >-> x >= y}}. | Proof.
move=> x y; rewrite !nposrE => x0 y0.
by rewrite !ler0_norm// -[LHS]subr_ge0 opprK addrC subr_ge0.
Qed. | Lemma | ler0_ge_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",... | [
"addrC",
"ler0_norm",
"normr",
"npos",
"nposrE",
"opprK",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0_ge_norm :
{in neg &, {mono (@normr _ R) : x y / x <= y >-> x >= y}}. | Proof. by move=> x y; rewrite !negrE => /ltW x0 /ltW y0; exact: ler0_ge_norm. Qed. | Lemma | ltr0_ge_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",... | [
"ler0_ge_norm",
"ltW",
"neg",
"negrE",
"normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_num | := pchar_num (only parsing). | Notation | char_num | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"pchar_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_nneg (R : numDomainType) (x : R) : `|x| \is Num.nneg. | Proof. by rewrite qualifE /=. Qed. | Lemma | normr_nneg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"nneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real1 : 1 \is @real R. | Proof. exact: rpred1. Qed. | Lemma | real1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"rpred1"
] | Properties of the real subset. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
realn n : n%:R \is @real R. | Proof. exact: rpred_nat. Qed. | Lemma | realn | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_xor_gt (x y : R) : R -> R -> R -> R -> R -> R ->
bool -> bool -> Set | :=
| LerNotGt of x <= y : ler_xor_gt x y x x y y (y - x) (y - x) true false
| GtrNotLe of y < x : ler_xor_gt x y y y x x (x - y) (x - y) false true. | Variant | ler_xor_gt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [] | dichotomy and trichotomy | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltr_xor_ge (x y : R) : R -> R -> R -> R -> R -> R ->
bool -> bool -> Set | :=
| LtrNotGe of x < y : ltr_xor_ge x y x x y y (y - x) (y - x) false true
| GerNotLt of y <= x : ltr_xor_ge x y y y x x (x - y) (x - y) true false. | Variant | ltr_xor_ge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
comparer x y : R -> R -> R -> R -> R -> R ->
bool -> bool -> bool -> bool -> bool -> bool -> Set | :=
| ComparerLt of x < y : comparer x y x x y y (y - x) (y - x)
false false false true false true
| ComparerGt of x > y : comparer x y y y x x (x - y) (x - y)
false false true false true false
| ComparerEq of x = y : comparer x y x x x x 0 0
true true true true false false. | Variant | comparer | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
real_leP x y : x \is real -> y \is real ->
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.
move=> xR yR; case: (comparable_leP (real_leVge xR yR)) => xy.
- by rewrite [`|x - y|]distrC !ger0_norm ?subr_cp0 //; constructor.
- by rewrite [`|y - x|]distrC !gtr0_norm ?subr_cp0 //; constructor.
Qed. | Lemma | real_leP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_leP",
"distrC",
"ger0_norm",
"gtr0_norm",
"ler_xor_gt",
"max",
"min",
"real",
"real_leVge",
"subr_cp0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltP x y : x \is real -> y \is real ->
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. by move=> xR yR; case: real_leP=> //; constructor. Qed. | Lemma | real_ltP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltgtP x y : x \is real -> y \is real ->
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.
move=> xR yR; case: (comparable_ltgtP (real_leVge yR xR)) => [?|?|->].
- by rewrite [`|y - x|]distrC !gtr0_norm ?subr_gt0//; constructor.
- by rewrite [`|x - y|]distrC !gtr0_norm ?subr_gt0//; constructor.
- by rewrite subrr normr0; constructor.
Qed. | Lemma | real_ltgtP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_ltgtP",
"comparer",
"distrC",
"gtr0_norm",
"max",
"min",
"normr0",
"real",
"real_leVge",
"subr_gt0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger0_xor_lt0 (x : R) : R -> R -> R -> R -> R ->
bool -> bool -> Set | :=
| Ger0NotLt0 of 0 <= x : ger0_xor_lt0 x 0 0 x x x false true
| Ltr0NotGe0 of x < 0 : ger0_xor_lt0 x x x 0 0 (- x) true false. | Variant | ger0_xor_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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0_xor_gt0 (x : R) : R -> R -> R -> R -> R ->
bool -> bool -> Set | :=
| Ler0NotLe0 of x <= 0 : ler0_xor_gt0 x x x 0 0 (- x) false true
| Gtr0NotGt0 of 0 < x : ler0_xor_gt0 x 0 0 x x x true false. | Variant | ler0_xor_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",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparer0 x : R -> R -> R -> R -> R ->
bool -> bool -> bool -> bool -> bool -> bool -> Set | :=
| ComparerGt0 of 0 < x : comparer0 x 0 0 x x x false false false true false true
| ComparerLt0 of x < 0 : comparer0 x x x 0 0 (- x) false false true false true false
| ComparerEq0 of x = 0 : comparer0 x 0 0 0 0 0 true true true true false false. | Variant | comparer0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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 | |
real_ge0P x : x \is real -> ger0_xor_lt0 x
(min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (x < 0) (0 <= x). | Proof.
move=> hx; rewrite -[X in `|X|]subr0; case: real_leP;
by rewrite ?subr0 ?sub0r //; constructor.
Qed. | Lemma | real_ge0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_leP",
"sub0r",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_le0P x : x \is real -> ler0_xor_gt0 x
(min 0 x) (min x 0) (max 0 x) (max x 0)
`|x| (0 < x) (x <= 0). | Proof.
move=> hx; rewrite -[X in `|X|]subr0; case: real_ltP;
by rewrite ?subr0 ?sub0r //; constructor.
Qed. | Lemma | real_le0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_ltP",
"sub0r",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltgt0P x : x \is real ->
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.
move=> hx; rewrite -[X in `|X|]subr0; case: (@real_ltgtP 0 x);
by rewrite ?subr0 ?sub0r //; constructor.
Qed. | Lemma | real_ltgt0P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"real_ltgtP",
"sub0r",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ger1_real x : 1 <= x -> x \is real. | Proof. by move=> /ger_real->. Qed. | Lemma | ger1_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",... | [
"ger_real",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler1_real x : x <= 1 -> x \is real. | Proof. by move=> /ler_real->. Qed. | Lemma | ler1_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",... | [
"ler_real",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pM2l x : 0 < x -> {mono *%R x : x y / x <= y}. | Proof.
by move=> x_gt0 y z /=; rewrite -subr_ge0 -mulrBr pmulr_rge0 // subr_ge0.
Qed. | Lemma | ler_pM2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mulrBr",
"pmulr_rge0",
"subr_ge0"
] | mulr and ler/ltr | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
ltr_pM2l x : 0 < x -> {mono *%R x : x y / x < y}. | Proof. by move=> x_gt0; apply: leW_mono (ler_pM2l _). Qed. | Lemma | ltr_pM2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"leW_mono",
"ler_pM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_pM2l | := (ler_pM2l, ltr_pM2l). | Definition | lter_pM2l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_pM2l",
"ltr_pM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pM2r x : 0 < x -> {mono *%R^~ x : x y / x <= y}. | Proof. by move=> x_gt0 y z /=; rewrite ![_ * x]mulrC ler_pM2l. Qed. | Lemma | ler_pM2r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ler_pM2l",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pM2r x : 0 < x -> {mono *%R^~ x : x y / x < y}. | Proof. by move=> x_gt0; apply: leW_mono (ler_pM2r _). Qed. | Lemma | ltr_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",... | [
"apply",
"leW_mono",
"ler_pM2r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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