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 |
|---|---|---|---|---|---|---|---|---|---|---|
real_arg_minP : extremum_spec <=%R P F [arg min_(i < i0 | P i) F i]%O. | Proof.
by apply: comparable_arg_minP => // i j iP jP; rewrite real_comparable ?F_real.
Qed. | Lemma | real_arg_minP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"F_real",
"apply",
"comparable_arg_minP",
"extremum_spec",
"i0",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_arg_maxP : extremum_spec >=%R P F [arg max_(i > i0 | P i) F i]%O. | Proof.
by apply: comparable_arg_maxP => // i j iP jP; rewrite real_comparable ?F_real.
Qed. | Lemma | real_arg_maxP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"F_real",
"apply",
"comparable_arg_maxP",
"extremum_spec",
"i0",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_norm x : x \is real -> x <= `|x|. | Proof.
by case/real_ge0P=> hx //; rewrite (le_trans (ltW hx)) // oppr_ge0 ltW.
Qed. | Lemma | real_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",... | [
"le_trans",
"ltW",
"oppr_ge0",
"real",
"real_ge0P"
] | norm | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
normr_real v : `|v| \is real. | Proof. by apply/ger0_real. Qed. | Lemma | normr_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",
"ger0_real",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_norm_sum I r (G : I -> V) (P : pred I):
`|\sum_(i <- r | P i) G i| <= \sum_(i <- r | P i) `|G i|. | Proof.
elim/big_rec2: _ => [|i y x _]; first by rewrite normr0.
by rewrite -(lerD2l `|G i|); apply: le_trans; apply: ler_normD.
Qed. | Lemma | ler_norm_sum | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_rec2",
"le_trans",
"lerD2l",
"ler_normD",
"normr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_normB v w : `|v - w| <= `|v| + `|w|. | Proof. by rewrite (le_trans (ler_normD _ _)) ?normrN. Qed. | Lemma | ler_normB | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"le_trans",
"ler_normD",
"normrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_distD u v w : `|v - w| <= `|v - u| + `|u - w|. | Proof. by rewrite (le_trans _ (ler_normD _ _)) // addrA addrNK. Qed. | Lemma | ler_distD | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"le_trans",
"ler_normD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerB_normD v w : `|v| - `|w| <= `|v + w|. | Proof.
by rewrite -{1}[v](addrK w) lterBDl (le_trans (ler_normD _ _))// addrC normrN.
Qed. | Lemma | lerB_normD | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"addrC",
"addrK",
"le_trans",
"ler_normD",
"lterBDl",
"normrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lerB_dist v w : `|v| - `|w| <= `|v - w|. | Proof. by rewrite -[`|w|]normrN lerB_normD. Qed. | Lemma | lerB_dist | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lerB_normD",
"normrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_dist_dist v w : `| `|v| - `|w| | <= `|v - w|. | Proof.
have [||_|_] // := @real_leP `|v| `|w|; last by rewrite lerB_dist.
by rewrite distrC lerB_dist.
Qed. | Lemma | ler_dist_dist | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"last",
"lerB_dist",
"real_leP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_dist_normD v w : `| `|v| - `|w| | <= `|v + w|. | Proof. by rewrite -[w]opprK normrN ler_dist_dist. Qed. | Lemma | ler_dist_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",... | [
"ler_dist_dist",
"normrN",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_nnorml v x : x < 0 -> (`|v| <= x) = false. | Proof. by move=> h; rewrite lt_geF //; apply/(lt_le_trans h). Qed. | Lemma | ler_nnorml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"lt_geF",
"lt_le_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_nnorml v x : x <= 0 -> (`|v| < x) = false. | Proof. by move=> h; rewrite le_gtF //; apply/(le_trans h). Qed. | Lemma | ltr_nnorml | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"le_gtF",
"le_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_nnormr | := (ler_nnorml, ltr_nnorml). | Definition | lter_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",... | [
"ler_nnorml",
"ltr_nnorml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_norml x y : x \is real -> (`|x| <= y) = (- y <= x <= y). | Proof.
move=> xR; wlog x_ge0 : x xR / 0 <= x => [hwlog|].
move: (xR) => /(@real_leVge _ 0) /orP [|/hwlog->|hx] //.
by rewrite -[x]opprK normrN lerN2 andbC lerNl hwlog ?realN ?oppr_ge0.
rewrite ger0_norm //; have [le_xy|] := boolP (x <= y); last by rewrite andbF.
by rewrite (le_trans _ x_ge0) // oppr_le0 (le_trans x... | Lemma | real_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",... | [
"ger0_norm",
"last",
"le_trans",
"lerN2",
"lerNl",
"normrN",
"opprK",
"oppr_ge0",
"oppr_le0",
"real",
"realN",
"real_leVge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_normlP x y :
x \is real -> reflect ((-x <= y) * (x <= y)) (`|x| <= y). | Proof.
by move=> Rx; rewrite real_ler_norml // lerNl; apply: (iffP andP) => [] [].
Qed. | Lemma | real_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",... | [
"apply",
"lerNl",
"real",
"real_ler_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_eqr_norml x y :
x \is real -> (`|x| == y) = ((x == y) || (x == -y)) && (0 <= y). | Proof.
move=> Rx.
apply/idP/idP=> [|/andP[/pred2P[]-> /ger0_norm/eqP]]; rewrite ?normrE //.
case: real_le0P => // hx; rewrite 1?eqr_oppLR => /eqP exy.
by move: hx; rewrite exy ?oppr_le0 eqxx orbT //.
by move: hx=> /ltW; rewrite exy eqxx.
Qed. | Lemma | real_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",... | [
"apply",
"eqr_oppLR",
"eqxx",
"ger0_norm",
"ltW",
"normrE",
"oppr_le0",
"pred2P",
"real",
"real_le0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_eqr_norm2 x y :
x \is real -> y \is real -> (`|x| == `|y|) = (x == y) || (x == -y). | Proof.
move=> Rx Ry; rewrite real_eqr_norml // normrE andbT.
by case: real_le0P; rewrite // opprK orbC.
Qed. | Lemma | real_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",... | [
"normrE",
"opprK",
"real",
"real_eqr_norml",
"real_le0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_norml x y : x \is real -> (`|x| < y) = (- y < x < y). | Proof.
move=> Rx; wlog x_ge0 : x Rx / 0 <= x => [hwlog|].
move: (Rx) => /(@real_leVge _ 0) /orP [|/hwlog->|hx] //.
by rewrite -[x]opprK normrN ltrN2 andbC ltrNl hwlog ?realN ?oppr_ge0.
rewrite ger0_norm //; have [le_xy|] := boolP (x < y); last by rewrite andbF.
by rewrite (lt_le_trans _ x_ge0) // oppr_lt0 (le_lt_tr... | Lemma | real_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",... | [
"ger0_norm",
"last",
"le_lt_trans",
"lt_le_trans",
"ltrN2",
"ltrNl",
"normrN",
"opprK",
"oppr_ge0",
"oppr_lt0",
"real",
"realN",
"real_leVge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lter_norml | := (real_ler_norml, real_ltr_norml). | Definition | real_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",... | [
"real_ler_norml",
"real_ltr_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_normlP x y :
x \is real -> reflect ((-x < y) * (x < y)) (`|x| < y). | Proof.
by move=> Rx; rewrite real_ltr_norml // ltrNl; apply: (iffP (@andP _ _)); case.
Qed. | Lemma | real_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",... | [
"apply",
"ltrNl",
"real",
"real_ltr_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_normr x y : y \is real -> (x <= `|y|) = (x <= y) || (x <= - y). | Proof.
move=> Ry.
have [xR|xNR] := boolP (x \is real); last by rewrite ?Nreal_leF ?realN.
rewrite real_leNgt ?real_ltr_norml // negb_and -?real_leNgt ?realN //.
by rewrite orbC lerNr.
Qed. | Lemma | real_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",... | [
"Nreal_leF",
"last",
"lerNr",
"real",
"realN",
"real_leNgt",
"real_ltr_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_normr x y : y \is real -> (x < `|y|) = (x < y) || (x < - y). | Proof.
move=> Ry.
have [xR|xNR] := boolP (x \is real); last by rewrite ?Nreal_ltF ?realN.
rewrite real_ltNge ?real_ler_norml // negb_and -?real_ltNge ?realN //.
by rewrite orbC ltrNr.
Qed. | Lemma | real_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",... | [
"Nreal_ltF",
"last",
"ltrNr",
"real",
"realN",
"real_ler_norml",
"real_ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lter_normr | := (real_ler_normr, real_ltr_normr). | Definition | real_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",... | [
"real_ler_normr",
"real_ltr_normr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_normlW x y : x \is real -> `|x| < y -> x < y. | Proof. by move=> ?; case/real_ltr_normlP. Qed. | Lemma | real_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",
"real_ltr_normlP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltrNnormlW x y : x \is real -> `|x| < y -> - y < x. | Proof. by move=> ?; case/real_ltr_normlP => //; rewrite ltrNl. Qed. | Lemma | real_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",... | [
"ltrNl",
"real",
"real_ltr_normlP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_normlW x y : x \is real -> `|x| <= y -> x <= y. | Proof. by move=> ?; case/real_ler_normlP. Qed. | Lemma | real_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",
"real_ler_normlP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lerNnormlW x y : x \is real -> `|x| <= y -> - y <= x. | Proof. by move=> ?; case/real_ler_normlP => //; rewrite lerNl. Qed. | Lemma | real_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",... | [
"lerNl",
"real",
"real_ler_normlP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_distl x y e :
x - y \is real -> (`|x - y| <= e) = (y - e <= x <= y + e). | Proof. by move=> Rxy; rewrite real_lter_norml // !lterBDl. Qed. | Lemma | real_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",... | [
"lterBDl",
"real",
"real_lter_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_distl x y e :
x - y \is real -> (`|x - y| < e) = (y - e < x < y + e). | Proof. by move=> Rxy; rewrite real_lter_norml // !lterBDl. Qed. | Lemma | real_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",... | [
"lterBDl",
"real",
"real_lter_norml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lter_distl | := (real_ler_distl, real_ltr_distl). | Definition | real_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",... | [
"real_ler_distl",
"real_ltr_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_distlDr x y e : x - y \is real -> `|x - y| < e -> x < y + e. | Proof. by move=> ?; rewrite real_ltr_distl // => /andP[]. Qed. | Lemma | real_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",
"real_ltr_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_distlDr x y e : x - y \is real -> `|x - y| <= e -> x <= y + e. | Proof. by move=> ?; rewrite real_ler_distl // => /andP[]. Qed. | Lemma | real_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",
"real_ler_distl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_distlCDr x y e : x - y \is real -> `|x - y| < e -> y < x + e. | Proof. by rewrite realBC (distrC x) => ? /real_ltr_distlDr; apply. Qed. | Lemma | real_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",... | [
"apply",
"distrC",
"real",
"realBC",
"real_ltr_distlDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_distlCDr x y e : x - y \is real -> `|x - y| <= e -> y <= x + e. | Proof. by rewrite realBC distrC => ? /real_ler_distlDr; apply. Qed. | Lemma | real_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",... | [
"apply",
"distrC",
"real",
"realBC",
"real_ler_distlDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_distlBl x y e : x - y \is real -> `|x - y| < e -> x - e < y. | Proof. by move/real_ltr_distlDr; rewrite ltrBlDr; apply. Qed. | Lemma | real_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",... | [
"apply",
"ltrBlDr",
"real",
"real_ltr_distlDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_distlBl x y e : x - y \is real -> `|x - y| <= e -> x - e <= y. | Proof. by move/real_ler_distlDr; rewrite lerBlDr; apply. Qed. | Lemma | real_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",... | [
"apply",
"lerBlDr",
"real",
"real_ler_distlDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ltr_distlCBl x y e : x - y \is real -> `|x - y| < e -> y - e < x. | Proof. by rewrite realBC distrC => ? /real_ltr_distlBl; apply. Qed. | Lemma | real_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",... | [
"apply",
"distrC",
"real",
"realBC",
"real_ltr_distlBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ler_distlCBl x y e : x - y \is real -> `|x - y| <= e -> y - e <= x. | Proof. by rewrite realBC distrC => ? /real_ler_distlBl; apply. Qed. | Lemma | real_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",... | [
"apply",
"distrC",
"real",
"realBC",
"real_ler_distlBl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_norm_id x : (`|x| == x) = (0 <= x). | Proof. by rewrite ger0_def. Qed. | Lemma | eqr_norm_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",... | [
"ger0_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_normN x : (`|x| == - x) = (x <= 0). | Proof. by rewrite ler0_def. Qed. | Lemma | eqr_normN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_norm_idVN | := =^~ (ger0_def, ler0_def). | Definition | eqr_norm_idVN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"ler0_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_even_ge0 n x : x \is real -> ~~ odd n -> 0 <= x ^+ n. | Proof.
move=> xR even_n; have [/exprn_ge0 -> //|x_lt0] := real_ge0P xR.
rewrite -[x]opprK -mulN1r exprMn -signr_odd (negPf even_n) expr0 mul1r.
by rewrite exprn_ge0 ?oppr_ge0 ?ltW.
Qed. | Lemma | real_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",... | [
"expr0",
"exprMn",
"exprn_ge0",
"ltW",
"mul1r",
"mulN1r",
"odd",
"opprK",
"oppr_ge0",
"real",
"real_ge0P",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_even_gt0 n x :
x \is real -> ~~ odd n -> (0 < x ^+ n) = (n == 0)%N || (x != 0). | Proof.
move=> xR n_even; rewrite lt0r real_exprn_even_ge0 ?expf_eq0 //.
by rewrite andbT negb_and lt0n negbK.
Qed. | Lemma | real_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",... | [
"expf_eq0",
"lt0n",
"lt0r",
"odd",
"real",
"real_exprn_even_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_even_le0 n x :
x \is real -> ~~ odd n -> (x ^+ n <= 0) = (n != 0) && (x == 0). | Proof.
move=> xR n_even; rewrite !real_leNgt ?rpred0 ?rpredX //.
by rewrite real_exprn_even_gt0 // negb_or negbK.
Qed. | Lemma | real_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",... | [
"odd",
"real",
"real_exprn_even_gt0",
"real_leNgt",
"rpred0",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_even_lt0 n x :
x \is real -> ~~ odd n -> (x ^+ n < 0) = false. | Proof. by move=> xR n_even; rewrite le_gtF // real_exprn_even_ge0. Qed. | Lemma | real_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",... | [
"le_gtF",
"odd",
"real",
"real_exprn_even_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_odd_ge0 n x :
x \is real -> odd n -> (0 <= x ^+ n) = (0 <= x). | Proof.
case/real_ge0P => [x_ge0|x_lt0] n_odd; first by rewrite exprn_ge0.
apply: negbTE; rewrite lt_geF //.
case: n n_odd => // n /= n_even; rewrite exprS pmulr_llt0 //.
by rewrite real_exprn_even_gt0 ?ler0_real ?ltW // (lt_eqF x_lt0) ?orbT.
Qed. | Lemma | real_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",... | [
"apply",
"exprS",
"exprn_ge0",
"ler0_real",
"ltW",
"lt_eqF",
"lt_geF",
"odd",
"pmulr_llt0",
"real",
"real_exprn_even_gt0",
"real_ge0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_odd_gt0 n x : x \is real -> odd n -> (0 < x ^+ n) = (0 < x). | Proof.
by move=> xR n_odd; rewrite !lt0r expf_eq0 real_exprn_odd_ge0; case: n n_odd.
Qed. | Lemma | real_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",... | [
"expf_eq0",
"lt0r",
"odd",
"real",
"real_exprn_odd_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_odd_le0 n x : x \is real -> odd n -> (x ^+ n <= 0) = (x <= 0). | Proof.
by move=> xR n_odd; rewrite !real_leNgt ?rpred0 ?rpredX // real_exprn_odd_gt0.
Qed. | Lemma | real_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",... | [
"odd",
"real",
"real_exprn_odd_gt0",
"real_leNgt",
"rpred0",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_exprn_odd_lt0 n x : x \is real -> odd n -> (x ^+ n < 0) = (x < 0). | Proof.
by move=> xR n_odd; rewrite !real_ltNge ?rpred0 ?rpredX // real_exprn_odd_ge0.
Qed. | Lemma | real_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",... | [
"odd",
"real",
"real_exprn_odd_ge0",
"real_ltNge",
"rpred0",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realEsqr x : (x \is real) = (0 <= x ^+ 2). | Proof. by rewrite ger0_def normrX eqf_sqr -ger0_def -ler0_def. Qed. | Lemma | realEsqr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqf_sqr",
"ger0_def",
"ler0_def",
"normrX",
"real"
] | GG: Could this be a better definition of "real" ? | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
real_normK x : x \is real -> `|x| ^+ 2 = x ^+ 2. | Proof. by move=> Rx; rewrite -normrX ger0_norm -?realEsqr. Qed. | Lemma | real_normK | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"normrX",
"real",
"realEsqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_sign s : `|(-1) ^+ s : R| = 1. | Proof. by rewrite normrX normrN1 expr1n. Qed. | Lemma | normr_sign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr1n",
"normrN1",
"normrX"
] | Binary sign ((-1) ^+ s). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
normrMsign s x : `|(-1) ^+ s * x| = `|x|. | Proof. by rewrite normrM normr_sign mul1r. Qed. | Lemma | normrMsign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mul1r",
"normrM",
"normr_sign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signr_gt0 (b : bool) : (0 < (-1) ^+ b :> R) = ~~ b. | Proof. by case: b; rewrite (ltr01, ltr0N1). Qed. | Lemma | signr_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",... | [
"ltr01",
"ltr0N1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signr_lt0 (b : bool) : ((-1) ^+ b < 0 :> R) = b. | Proof. by case: b; rewrite // ?(ltrN10, ltr10). Qed. | Lemma | signr_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",... | [
"ltr10",
"ltrN10"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signr_ge0 (b : bool) : (0 <= (-1) ^+ b :> R) = ~~ b. | Proof. by rewrite le0r signr_eq0 signr_gt0. Qed. | Lemma | signr_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",... | [
"le0r",
"signr_eq0",
"signr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signr_le0 (b : bool) : ((-1) ^+ b <= 0 :> R) = b. | Proof. by rewrite le_eqVlt signr_eq0 signr_lt0. Qed. | Lemma | signr_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",... | [
"le_eqVlt",
"signr_eq0",
"signr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signr_inj : injective (fun b : bool => (-1) ^+ b : R). | Proof. exact: can_inj (fun x => 0 >= x) signr_le0. Qed. | Lemma | signr_inj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"signr_le0"
] | This actually holds for char R != 2. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sgr_def x : sg x = (-1) ^+ (x < 0)%R *+ (x != 0). | Proof. by rewrite /sg; do 2!case: ifP => //. Qed. | Lemma | sgr_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",... | [
"sg"
] | Ternary sign (sg). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
neqr0_sign x : x != 0 -> (-1) ^+ (x < 0)%R = sgr x. | Proof. by rewrite sgr_def => ->. Qed. | Lemma | neqr0_sign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"sgr_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtr0_sg x : 0 < x -> sg x = 1. | Proof. by move=> x_gt0; rewrite /sg gt_eqF // lt_gtF. Qed. | Lemma | gtr0_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",... | [
"gt_eqF",
"lt_gtF",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0_sg x : x < 0 -> sg x = -1. | Proof. by move=> x_lt0; rewrite /sg x_lt0 lt_eqF. Qed. | Lemma | ltr0_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",... | [
"lt_eqF",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr0 : sg 0 = 0 :> R. | Proof. by rewrite /sgr eqxx. Qed. | Lemma | sgr0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"sg",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr1 : sg 1 = 1 :> R. | Proof. by rewrite gtr0_sg // ltr01. Qed. | Lemma | sgr1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_sg",
"ltr01",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrN1 : sg (-1) = -1 :> R. | Proof. by rewrite ltr0_sg // ltrN10. Qed. | Lemma | sgrN1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"ltr0_sg",
"ltrN10",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrE | := (sgr0, sgr1, sgrN1). | Definition | sgrE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"sgr0",
"sgr1",
"sgrN1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqr_sg x : sg x ^+ 2 = (x != 0)%:R. | Proof. by rewrite sgr_def exprMn_n sqrr_sign -mulnn mulnb andbb. Qed. | Lemma | sqr_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",... | [
"exprMn_n",
"mulnb",
"mulnn",
"sg",
"sgr_def",
"sqrr_sign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_sg_eq1 x y : (sg x * y == 1) = (x != 0) && (sg x == y). | Proof.
rewrite /sg eq_sym; case: ifP => _; first by rewrite mul0r oner_eq0.
by case: ifP => _; rewrite ?mul1r // mulN1r eqr_oppLR.
Qed. | Lemma | mulr_sg_eq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eq_sym",
"eqr_oppLR",
"mul0r",
"mul1r",
"mulN1r",
"oner_eq0",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_sg_eqN1 x y : (sg x * sg y == -1) = (x != 0) && (sg x == - sg y). | Proof.
move/sg: y => y; rewrite /sg eq_sym eqr_oppLR.
case: ifP => _; first by rewrite mul0r oppr0 oner_eq0.
by case: ifP => _; rewrite ?mul1r // mulN1r eqr_oppLR.
Qed. | Lemma | mulr_sg_eqN1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"eqr_oppLR",
"mul0r",
"mul1r",
"mulN1r",
"oner_eq0",
"oppr0",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_eq0 x : (sg x == 0) = (x == 0). | Proof. by rewrite -sqrf_eq0 sqr_sg pnatr_eq0; case: (x == 0). Qed. | Lemma | sgr_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",... | [
"pnatr_eq0",
"sg",
"sqr_sg",
"sqrf_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_odd n x : x != 0 -> (sg x) ^+ n = (sg x) ^+ (odd n). | Proof. by rewrite /sg; do 2!case: ifP => // _; rewrite ?expr1n ?signr_odd. Qed. | Lemma | sgr_odd | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"expr1n",
"odd",
"sg",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrMn x n : sg (x *+ n) = (n != 0)%:R * sg x. | Proof.
case: n => [|n]; first by rewrite mulr0n sgr0 mul0r.
by rewrite !sgr_def mulrn_eq0 mul1r pmulrn_llt0.
Qed. | Lemma | sgrMn | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"mulr0n",
"mulrn_eq0",
"pmulrn_llt0",
"sg",
"sgr0",
"sgr_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_nat n : sg n%:R = (n != 0)%:R :> R. | Proof. by rewrite sgrMn sgr1 mulr1. Qed. | Lemma | sgr_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",... | [
"mulr1",
"sg",
"sgr1",
"sgrMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_id x : sg (sg x) = sg x. | Proof. by rewrite !(fun_if sg) !sgrE. Qed. | Lemma | sgr_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",... | [
"sg",
"sgrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_lt0 x : (sg x < 0) = (x < 0). | Proof.
rewrite /sg; case: eqP => [-> // | _].
by case: ifP => _; rewrite ?ltrN10 // lt_gtF.
Qed. | Lemma | sgr_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",... | [
"lt_gtF",
"ltrN10",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_le0 x : (sgr x <= 0) = (x <= 0). | Proof. by rewrite !le_eqVlt sgr_eq0 sgr_lt0. Qed. | Lemma | sgr_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",... | [
"le_eqVlt",
"sgr",
"sgr_eq0",
"sgr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realEsign x : x \is real -> x = (-1) ^+ (x < 0)%R * `|x|. | Proof. by case/real_ge0P; rewrite (mul1r, mulN1r) ?opprK. Qed. | Lemma | realEsign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"mul1r",
"mulN1r",
"opprK",
"real",
"real_ge0P"
] | sign and norm | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
realNEsign x : x \is real -> - x = (-1) ^+ (0 < x)%R * `|x|. | Proof. by move=> Rx; rewrite -normrN -oppr_lt0 -realEsign ?rpredN. Qed. | Lemma | realNEsign | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"oppr_lt0",
"real",
"realEsign",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_normrEsign (x : R) (xR : x \is real) : `|x| = (-1) ^+ (x < 0)%R * x. | Proof. by rewrite {3}[x]realEsign // signrMK. Qed. | Lemma | real_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",
"realEsign",
"signrMK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_mulr_sign_norm x : x \is real -> (-1) ^+ (x < 0)%R * `|x| = x. | Proof. by move/realEsign. Qed. | Lemma | real_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",... | [
"real",
"realEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_mulr_Nsign_norm x : x \is real -> (-1) ^+ (0 < x)%R * `|x| = - x. | Proof. by move/realNEsign. Qed. | Lemma | real_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",
"realNEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realEsg x : x \is real -> x = sgr x * `|x|. | Proof.
move=> xR; have [-> | ] := eqVneq x 0; first by rewrite normr0 mulr0.
by move=> /neqr0_sign <-; rewrite -realEsign.
Qed. | Lemma | realEsg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqVneq",
"mulr0",
"neqr0_sign",
"normr0",
"real",
"realEsign",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normr_sg x : `|sg x| = (x != 0)%:R. | Proof. by rewrite sgr_def -mulr_natr normrMsign normr_nat. Qed. | Lemma | normr_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",... | [
"mulr_natr",
"normrMsign",
"normr_nat",
"sg",
"sgr_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgr_norm x : sg `|x| = (x != 0)%:R. | Proof. by rewrite /sg le_gtF // normr_eq0 mulrb if_neg. Qed. | Lemma | sgr_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",... | [
"le_gtF",
"mulrb",
"normr_eq0",
"sg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_nat_r m n C : (m%:R <= n%:R ?= iff C :> R) = (m <= n ?= iff C)%N. | Proof. by rewrite /leif !ler_nat eqr_nat. Qed. | Lemma | leif_nat_r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"eqr_nat",
"leif",
"ler_nat"
] | leif | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
leifBLR x y z C : (x - y <= z ?= iff C) = (x <= z + y ?= iff C). | Proof. by rewrite /leif !eq_le lerBlDr lerBrDr. Qed. | Lemma | leifBLR | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"leif",
"lerBlDr",
"lerBrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leifBRL x y z C : (x <= y - z ?= iff C) = (x + z <= y ?= iff C). | Proof. by rewrite -leifBLR opprK. Qed. | Lemma | leifBRL | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"leifBLR",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leifD x1 y1 C1 x2 y2 C2 :
x1 <= y1 ?= iff C1 -> x2 <= y2 ?= iff C2 ->
x1 + x2 <= y1 + y2 ?= iff C1 && C2. | Proof.
rewrite -(mono_leif (C := C1) (lerD2r x2)).
rewrite -(mono_leif (C := C2) (lerD2l y1)).
exact: leif_trans.
Qed. | Lemma | leifD | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"leif_trans",
"lerD2l",
"lerD2r",
"mono_leif"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_sum (I : finType) (P C : pred I) (E1 E2 : I -> R) :
(forall i, P i -> E1 i <= E2 i ?= iff C i) ->
\sum_(i | P i) E1 i <= \sum_(i | P i) E2 i ?= iff [forall (i | P i), C i]. | Proof.
move=> leE12; rewrite -big_andE.
elim/big_rec3: _ => [|i Ci m2 m1 /leE12]; first by rewrite /leif lexx eqxx.
exact: leifD.
Qed. | Lemma | leif_sum | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_andE",
"big_rec3",
"eqxx",
"leif",
"leifD",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_0_sum (I : finType) (P C : pred I) (E : I -> R) :
(forall i, P i -> 0 <= E i ?= iff C i) ->
0 <= \sum_(i | P i) E i ?= iff [forall (i | P i), C i]. | Proof. by move/leif_sum; rewrite big1_eq. Qed. | Lemma | leif_0_sum | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"big1_eq",
"leif_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_leif_norm x : x \is real -> x <= `|x| ?= iff (0 <= x). | Proof.
by move=> xR; rewrite ger0_def eq_sym; apply: leif_eq; rewrite real_ler_norm.
Qed. | Lemma | real_leif_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",... | [
"apply",
"eq_sym",
"ger0_def",
"leif_eq",
"real",
"real_ler_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_pM x1 x2 y1 y2 C1 C2 :
0 <= x1 -> 0 <= x2 -> x1 <= y1 ?= iff C1 -> x2 <= y2 ?= iff C2 ->
x1 * x2 <= y1 * y2 ?= iff (y1 * y2 == 0) || C1 && C2. | Proof.
move=> x1_ge0 x2_ge0 le_xy1 le_xy2; have [y_0 | ] := eqVneq _ 0.
apply/leifP; rewrite y_0 /= mulf_eq0 !eq_le x1_ge0 x2_ge0 !andbT.
move/eqP: y_0; rewrite mulf_eq0.
by case/pred2P=> <-; rewrite (le_xy1, le_xy2) ?orbT.
rewrite /= mulf_eq0 => /norP[y1nz y2nz].
have y1_gt0: 0 < y1 by rewrite lt_def y1nz (le_tr... | Lemma | leif_pM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"apply",
"eqVneq",
"eq_le",
"eq_sym",
"le_trans",
"leifP",
"leif_trans",
"ler_pM2l",
"ler_pM2r",
"lt_def",
"mono_leif",
"mulf_eq0",
"mulr0",
"mulr_gt0",
"pred2P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_nM x1 x2 y1 y2 C1 C2 :
y1 <= 0 -> y2 <= 0 -> x1 <= y1 ?= iff C1 -> x2 <= y2 ?= iff C2 ->
y1 * y2 <= x1 * x2 ?= iff (x1 * x2 == 0) || C1 && C2. | Proof.
rewrite -!oppr_ge0 -mulrNN -[x1 * x2]mulrNN => y1le0 y2le0 le_xy1 le_xy2.
by apply: leif_pM => //; rewrite (nmono_leif lerN2).
Qed. | Lemma | leif_nM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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",
"leif_pM",
"lerN2",
"mulrNN",
"nmono_leif",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_pprod (I : finType) (P C : pred I) (E1 E2 : I -> R) :
(forall i, P i -> 0 <= E1 i) ->
(forall i, P i -> E1 i <= E2 i ?= iff C i) ->
let pi E := \prod_(i | P i) E i in
pi E1 <= pi E2 ?= iff (pi E2 == 0) || [forall (i | P i), C i]. | Proof.
move=> E1_ge0 leE12 /=; rewrite -big_andE; elim/(big_load (fun x => 0 <= x)): _.
elim/big_rec3: _ => [|i Ci m2 m1 Pi [m1ge0 le_m12]].
by split=> //; apply/leifP; rewrite orbT.
have Ei_ge0 := E1_ge0 i Pi; split; first by rewrite mulr_ge0.
congr (leif _ _ _): (leif_pM Ei_ge0 m1ge0 (leE12 i Pi) le_m12).
by rewrit... | Lemma | leif_pprod | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"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_andE",
"big_load",
"big_rec3",
"leif",
"leifP",
"leif_pM",
"mulf_eq0",
"mulr_ge0",
"pi",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_lteifr0 C x y : (y - x < 0 ?<= if C) = (y < x ?<= if C). | Proof. by case: C => /=; rewrite subr_lte0. Qed. | Lemma | subr_lteifr0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"subr_lte0"
] | lteif | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
subr_lteif0r C x y : (0 < y - x ?<= if C) = (x < y ?<= if C). | Proof. by case: C => /=; rewrite subr_gte0. Qed. | Lemma | subr_lteif0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"subr_gte0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subr_lteif0 | := (subr_lteifr0, subr_lteif0r). | Definition | subr_lteif0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"subr_lteif0r",
"subr_lteifr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif01 C : 0 < 1 ?<= if C :> R. | Proof. by case: C; rewrite /= lter01. Qed. | Lemma | lteif01 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numdomain.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"poly",
"orderedzmod",
"Order.TTheory",
"GRing.Theory",... | [
"lter01"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_pM2l C x : 0 < x -> {mono *%R x : y z / y < z ?<= if C}. | Proof. by case: C => ? ? ?; rewrite /= lter_pM2l. Qed. | Lemma | lteif_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",... | [
"lter_pM2l"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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