statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
lter_pdivlMl | := (ler_pdivlMl, ltr_pdivlMl). | Definition | lter_pdivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_pdivlMl",
"ltr_pdivlMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_pdivrMl z x y : 0 < z -> (z^-1 * y <= x) = (y <= z * x). | Proof. by move=> z_gt0; rewrite mulrC ler_pdivrMr ?[z * _]mulrC. Qed. | Lemma | ler_pdivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_pdivrMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_pdivrMl z x y : 0 < z -> (z^-1 * y < x) = (y < z * x). | Proof. by move=> z_gt0; rewrite mulrC ltr_pdivrMr ?[z * _]mulrC. Qed. | Lemma | ltr_pdivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr_pdivrMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_pdivrMl | := (ler_pdivrMl, ltr_pdivrMl). | Definition | lter_pdivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_pdivrMl",
"ltr_pdivrMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_ndivlMr z x y : z < 0 -> (x <= y / z) = (y <= x * z). | Proof. by move=> z_lt0; rewrite -(@ler_nM2r _ z) ?mulfVK ?lt_eqF. Qed. | Lemma | ler_ndivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_nM2r",
"lt_eqF",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_ndivlMr z x y : z < 0 -> (x < y / z) = (y < x * z). | Proof. by move=> z_lt0; rewrite -(@ltr_nM2r _ z) ?mulfVK ?lt_eqF. Qed. | Lemma | ltr_ndivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lt_eqF",
"ltr_nM2r",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_ndivlMr | := (ler_ndivlMr, ltr_ndivlMr). | Definition | lter_ndivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_ndivlMr",
"ltr_ndivlMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_ndivrMr z x y : z < 0 -> (y / z <= x) = (x * z <= y). | Proof. by move=> z_lt0; rewrite -(@ler_nM2r _ z) ?mulfVK ?lt_eqF. Qed. | Lemma | ler_ndivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_nM2r",
"lt_eqF",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_ndivrMr z x y : z < 0 -> (y / z < x) = (x * z < y). | Proof. by move=> z_lt0; rewrite -(@ltr_nM2r _ z) ?mulfVK ?lt_eqF. Qed. | Lemma | ltr_ndivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lt_eqF",
"ltr_nM2r",
"mulfVK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_ndivrMr | := (ler_ndivrMr, ltr_ndivrMr). | Definition | lter_ndivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_ndivrMr",
"ltr_ndivrMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_ndivlMl z x y : z < 0 -> (x <= z^-1 * y) = (y <= z * x). | Proof. by move=> z_lt0; rewrite mulrC ler_ndivlMr ?[z * _]mulrC. Qed. | Lemma | ler_ndivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_ndivlMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_ndivlMl z x y : z < 0 -> (x < z^-1 * y) = (y < z * x). | Proof. by move=> z_lt0; rewrite mulrC ltr_ndivlMr ?[z * _]mulrC. Qed. | Lemma | ltr_ndivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr_ndivlMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_ndivlMl | := (ler_ndivlMl, ltr_ndivlMl). | Definition | lter_ndivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_ndivlMl",
"ltr_ndivlMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_ndivrMl z x y : z < 0 -> (z^-1 * y <= x) = (z * x <= y). | Proof. by move=> z_lt0; rewrite mulrC ler_ndivrMr ?[z * _]mulrC. Qed. | Lemma | ler_ndivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_ndivrMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_ndivrMl z x y : z < 0 -> (z^-1 * y < x) = (z * x < y). | Proof. by move=> z_lt0; rewrite mulrC ltr_ndivrMr ?[z * _]mulrC. Qed. | Lemma | ltr_ndivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr_ndivrMr",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lter_ndivrMl | := (ler_ndivrMl, ltr_ndivrMl). | Definition | lter_ndivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_ndivrMl",
"ltr_ndivrMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divDl_le1 x y : 0 <= x -> 0 <= y -> x / (x + y) <= 1. | Proof.
rewrite le0r => /predU1P[-> _|*]; first by rewrite mul0r.
by rewrite ler_pdivrMr ?ltr_wpDr// mul1r lerDl.
Qed. | Lemma | divDl_le1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"le0r",
"lerDl",
"ler_pdivrMr",
"ltr_wpDr",
"mul0r",
"mul1r",
"predU1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natf_div m d : (d %| m)%N -> (m %/ d)%:R = m%:R / d%:R :> F. | Proof. by apply: pchar0_natf_div; apply: (@pchar_num F). Qed. | Lemma | natf_div | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"pchar0_natf_div",
"pchar_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normfV : {morph (norm : F -> F) : x / x ^-1}. | Proof.
move=> x /=; have [/normrV //|Nux] := boolP (x \is a GRing.unit).
by rewrite !invr_out // unitfE normr_eq0 -unitfE.
Qed. | Lemma | normfV | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invr_out",
"norm",
"normrV",
"normr_eq0",
"unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normf_div : {morph (norm : F -> F) : x y / x / y}. | Proof. by move=> x y /=; rewrite normrM normfV. Qed. | Lemma | normf_div | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"norm",
"normfV",
"normrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_sg x : (sg x)^-1 = sgr x. | Proof. by rewrite !(fun_if GRing.inv) !(invr0, invrN, invr1). Qed. | Lemma | invr_sg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"inv",
"invr0",
"invr1",
"invrN",
"sg",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrV x : sgr x^-1 = sgr x. | Proof. by rewrite /sgr invr_eq0 invr_lt0. Qed. | Lemma | sgrV | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"invr_eq0",
"invr_lt0",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitr x : x = x / 2%:R + x / 2%:R. | Proof. by rewrite -mulr2n -[RHS]mulr_natr mulfVK //= pnatr_eq0. Qed. | Lemma | splitr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"mulfVK",
"mulr2n",
"mulr_natr",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_pdivlMr C z x y :
0 < z -> (x < y / z ?<= if C) = (x * z < y ?<= if C). | Proof. by case: C => ? /=; rewrite lter_pdivlMr. Qed. | Lemma | lteif_pdivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_pdivlMr"
] | lteif | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lteif_pdivrMr C z x y :
0 < z -> (y / z < x ?<= if C) = (y < x * z ?<= if C). | Proof. by case: C => ? /=; rewrite lter_pdivrMr. Qed. | Lemma | lteif_pdivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_pdivrMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_pdivlMl C z x y :
0 < z -> (x < z^-1 * y ?<= if C) = (z * x < y ?<= if C). | Proof. by case: C => ? /=; rewrite lter_pdivlMl. Qed. | Lemma | lteif_pdivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_pdivlMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_pdivrMl C z x y :
0 < z -> (z^-1 * y < x ?<= if C) = (y < z * x ?<= if C). | Proof. by case: C => ? /=; rewrite lter_pdivrMl. Qed. | Lemma | lteif_pdivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_pdivrMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_ndivlMr C z x y :
z < 0 -> (x < y / z ?<= if C) = (y < x * z ?<= if C). | Proof. by case: C => ? /=; rewrite lter_ndivlMr. Qed. | Lemma | lteif_ndivlMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_ndivlMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_ndivrMr C z x y :
z < 0 -> (y / z < x ?<= if C) = (x * z < y ?<= if C). | Proof. by case: C => ? /=; rewrite lter_ndivrMr. Qed. | Lemma | lteif_ndivrMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_ndivrMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_ndivlMl C z x y :
z < 0 -> (x < z^-1 * y ?<= if C) = (y < z * x ?<= if C). | Proof. by case: C => ? /=; rewrite lter_ndivlMl. Qed. | Lemma | lteif_ndivlMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_ndivlMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_ndivrMl C z x y :
z < 0 -> (z^-1 * y < x ?<= if C) = (z * x < y ?<= if C). | Proof. by case: C => ? /=; rewrite lter_ndivrMl. Qed. | Lemma | lteif_ndivrMl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lter_ndivrMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mid x y | := ((x + y) / 2). | Notation | mid | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | Interval midpoint. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
midf_le x y : x <= y -> (x <= mid x y) * (mid x y <= y). | Proof.
move=> lexy; rewrite ler_pdivlMr ?ler_pdivrMr ?ltr0Sn //.
by rewrite !mulrDr !mulr1 !lerD2.
Qed. | Lemma | midf_le | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lerD2",
"ler_pdivlMr",
"ler_pdivrMr",
"ltr0Sn",
"mid",
"mulr1",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
midf_lt x y : x < y -> (x < mid x y) * (mid x y < y). | Proof.
move=> ltxy; rewrite ltr_pdivlMr ?ltr_pdivrMr ?ltr0Sn //.
by rewrite !mulrDr !mulr1 !ltrD2.
Qed. | Lemma | midf_lt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr0Sn",
"ltrD2",
"ltr_pdivlMr",
"ltr_pdivrMr",
"mid",
"mulr1",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
midf_lte | := (midf_le, midf_lt). | Definition | midf_lte | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"midf_le",
"midf_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_addgt0Pr x y : reflect (forall e, e > 0 -> x <= y + e) (x <= y). | Proof.
apply/(iffP idP)=> [lexy e e_gt0 | lexye]; first by rewrite ler_wpDr// ltW.
have [||ltyx]// := comparable_leP.
rewrite (@comparabler_trans _ (y + 1))// /Order.comparable ?lexye ?ltr01//.
by rewrite lerDl ler01 orbT.
have /midf_lt [_] := ltyx; rewrite le_gtF//.
by rewrite addrC -(subrKA y) addrC 2!mulrDl -spl... | Lemma | ler_addgt0Pr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"apply",
"comparable",
"comparable_leP",
"comparabler_trans",
"divr_gt0",
"le_gtF",
"ler01",
"lerDl",
"ler_wpDr",
"ltW",
"ltr01",
"midf_lt",
"mulrDl",
"splitr",
"subrKA",
"subr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_addgt0Pl x y : reflect (forall e, e > 0 -> x <= e + y) (x <= y). | Proof.
by apply/(equivP (ler_addgt0Pr x y)); split=> lexy e /lexy; rewrite addrC.
Qed. | Lemma | ler_addgt0Pl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"apply",
"ler_addgt0Pr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_le a b : (forall x, x < a -> x < b) -> a <= b. | Proof.
move=> ab; apply/ler_addgt0Pr => e e_gt0; rewrite -lerBDr ltW//.
by rewrite ab// ltrBlDr ltrDl.
Qed. | Lemma | lt_le | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"lerBDr",
"ler_addgt0Pr",
"ltW",
"ltrBlDr",
"ltrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt_ge a b : (forall x, b < x -> a < x) -> a <= b. | Proof.
by move=> ab; apply/ler_addgt0Pr => e e_gt0; rewrite ltW// ab// ltrDl.
Qed. | Lemma | gt_ge | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"ler_addgt0Pr",
"ltW",
"ltrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_leif_mean_square x y :
x \is real -> y \is real -> x * y <= mid (x ^+ 2) (y ^+ 2) ?= iff (x == y). | Proof.
move=> Rx Ry; rewrite -(mono_leif (ler_pM2r (ltr_nat F 0 2))).
by rewrite divfK ?pnatr_eq0 // mulr_natr; apply: real_leif_mean_square_scaled.
Qed. | Lemma | real_leif_mean_square | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"divfK",
"ler_pM2r",
"ltr_nat",
"mid",
"mono_leif",
"mulr_natr",
"pnatr_eq0",
"real",
"real_leif_mean_square_scaled"
] | The AGM, unscaled but without the nth root. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
real_leif_AGM2 x y :
x \is real -> y \is real -> x * y <= mid x y ^+ 2 ?= iff (x == y). | Proof.
move=> Rx Ry; rewrite -(mono_leif (ler_pM2r (ltr_nat F 0 4))).
rewrite mulr_natr (natrX F 2 2) -exprMn divfK ?pnatr_eq0 //.
exact: real_leif_AGM2_scaled.
Qed. | Lemma | real_leif_AGM2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"divfK",
"exprMn",
"ler_pM2r",
"ltr_nat",
"mid",
"mono_leif",
"mulr_natr",
"natrX",
"pnatr_eq0",
"real",
"real_leif_AGM2_scaled"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_AGM (I : finType) (A : {pred I}) (E : I -> F) :
let n := #|A| in let mu := (\sum_(i in A) E i) / n%:R in
{in A, forall i, 0 <= E i} ->
\prod_(i in A) E i <= mu ^+ n
?= iff [forall i in A, forall j in A, E i == E j]. | Proof.
move=> n mu Ege0; have [n0 | n_gt0] := posnP n.
by rewrite n0 -big_andE !(big_pred0 _ _ _ _ (card0_eq n0)); apply/leifP.
pose E' i := E i / n%:R.
have defE' i: E' i *+ n = E i by rewrite -mulr_natr divfK ?pnatr_eq0 -?lt0n.
have /leif_AGM_scaled (i): i \in A -> 0 <= E' i *+ n by rewrite defE' => /Ege0.
rewrite ... | Lemma | leif_AGM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"big_andE",
"big_pred0",
"card0_eq",
"divfK",
"eq_bigr",
"eq_forallb_in",
"eqr_pMn2r",
"leifP",
"leif_AGM_scaled",
"lt0n",
"mulr_natr",
"mulr_suml",
"n_gt0",
"pnatr_eq0",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cauchy_root_bound p : p != 0 -> {b | forall x, root p x -> `|x| <= b}. | Proof.
move=> nz_p; set a := lead_coef p; set n := (size p).-1.
have [q Dp]: {q | forall x, x != 0 -> p.[x] = (a - q.[x^-1] / x) * x ^+ n}.
exists (- \poly_(i < n) p`_(n - i.+1)) => x nz_x.
rewrite hornerN mulNr opprK horner_poly mulrDl !mulr_suml addrC.
rewrite horner_coef polySpred // big_ord_recr (reindex_inj ... | Lemma | Cauchy_root_bound | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"apply",
"big_ord_recr",
"contraTneq",
"divfK",
"divr_ge0",
"eq_bigr",
"eq_sym",
"expf_neq0",
"exprB",
"exprSr",
"exprVn",
"ger0_real",
"hornerN",
"horner_coef",
"horner_poly",
"invf_le1",
"le_gtF",
"le_trans",
"lead_coef",
"lead_coefE",
"lead_coef_eq0",
"ler_wpD... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natf_indexg (gT : finGroupType) (G H : {group gT}) :
H \subset G -> #|G : H|%g%:R = (#|G|%:R / #|H|%:R)%R :> F. | Proof. by move=> sHG; rewrite -divgS // natf_div ?cardSg. Qed. | Lemma | natf_indexg | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"cardSg",
"divgS",
"gT",
"group",
"natf_div",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mid_in_itv : forall ba bb (xa xb : R), xa < xb ?<= if ba && ~~ bb ->
mid xa xb \in Interval (BSide ba xa) (BSide bb xb). | Proof.
by move=> [] [] xa xb /= ?; apply/itv_dec; rewrite /= ?midf_lte // ?ltW.
Qed. | Lemma | mid_in_itv | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"itv_dec",
"ltW",
"mid",
"midf_lte"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mid_in_itvoo : forall (xa xb : R), xa < xb -> mid xa xb \in `]xa, xb[. | Proof. by move=> xa xb ?; apply: mid_in_itv. Qed. | Lemma | mid_in_itvoo | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"mid",
"mid_in_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mid_in_itvcc : forall (xa xb : R), xa <= xb -> mid xa xb \in `[xa, xb]. | Proof. by move=> xa xb ?; apply: mid_in_itv. Qed. | Lemma | mid_in_itvcc | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"mid",
"mid_in_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_miditv i : (i.1 < i.2)%O -> miditv i \in i. | Proof.
move: i => [[ba a|[]] [bb b|[]]] //= ab; first exact: mid_in_itv.
by rewrite !in_itv -lteifBlDl subrr lteif01.
by rewrite !in_itv lteifBlDr -lteifBlDl subrr lteif01.
Qed. | Lemma | mem_miditv | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"in_itv",
"lteif01",
"lteifBlDl",
"lteifBlDr",
"mid_in_itv",
"miditv",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
miditv_le_left i b : (i.1 < i.2)%O -> (BSide b (miditv i) <= i.2)%O. | Proof.
case: i => [x y] lti; have := mem_miditv lti; rewrite inE => /andP[_ ].
by apply: le_trans; rewrite !bnd_simp.
Qed. | Lemma | miditv_le_left | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"bnd_simp",
"inE",
"le_trans",
"mem_miditv",
"miditv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
miditv_ge_right i b : (i.1 < i.2)%O -> (i.1 <= BSide b (miditv i))%O. | Proof.
case: i => [x y] lti; have := mem_miditv lti; rewrite inE => /andP[+ _].
by move=> /le_trans; apply; rewrite !bnd_simp.
Qed. | Lemma | miditv_ge_right | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"bnd_simp",
"inE",
"le_trans",
"mem_miditv",
"miditv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_segmentDgt0Pr x y z :
reflect (forall e, e > 0 -> y \in `[x - e, z + e]) (y \in `[x, z]). | Proof.
apply/(iffP idP)=> [xyz e /[dup] e_gt0 /ltW e_ge0 | xyz_e].
by rewrite in_itv /= lerBDr !ler_wpDr// (itvP xyz).
by rewrite in_itv /= ; apply/andP; split; apply/ler_addgt0Pr => ? /xyz_e;
rewrite in_itv /= lerBDr => /andP [].
Qed. | Lemma | in_segmentDgt0Pr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"in_itv",
"itvP",
"lerBDr",
"ler_addgt0Pr",
"ler_wpDr",
"ltW",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_segmentDgt0Pl x y z :
reflect (forall e, e > 0 -> y \in `[- e + x, e + z]) (y \in `[x, z]). | Proof.
apply/(equivP (in_segmentDgt0Pr x y z)).
by split=> zxy e /zxy; rewrite [z + _]addrC [_ + x]addrC.
Qed. | Lemma | in_segmentDgt0Pl | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"apply",
"in_segmentDgt0Pr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_mean_square x y : x * y <= (x ^+ 2 + y ^+ 2) / 2 ?= iff (x == y). | Proof. exact: real_leif_mean_square. Qed. | Lemma | leif_mean_square | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"real_leif_mean_square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_AGM2 x y : x * y <= ((x + y) / 2)^+ 2 ?= iff (x == y). | Proof. exact: real_leif_AGM2. Qed. | Lemma | leif_AGM2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"real_leif_AGM2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxr_absE x y : max x y = (x + y + `|x - y|) / 2. | Proof.
by rewrite addrC -[x + y]addr_min_max distr_max_min subrKA mulrDl -splitr.
Qed. | Lemma | maxr_absE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"addr_min_max",
"distr_max_min",
"max",
"mulrDl",
"splitr",
"subrKA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minr_absE x y : min x y = (x + y - `|x - y|) / 2. | Proof.
by rewrite addrC -[x + y]addr_max_min distr_max_min opprB subrKA mulrDl -splitr.
Qed. | Lemma | minr_absE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"addr_max_min",
"distr_max_min",
"min",
"mulrDl",
"opprB",
"splitr",
"subrKA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
poly_ivt : real_closed_axiom R. | Proof. exact: poly_ivt. Qed. | Lemma | poly_ivt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"real_closed_axiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_ge0 a : 0 <= sqrt a. | Proof. by rewrite /sqrt; case: (sig2W _). Qed. | Lemma | sqrtr_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"sig2W",
"sqrt"
] | Square Root theory | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sqr_sqrtr a : 0 <= a -> sqrt a ^+ 2 = a. | Proof.
by rewrite /sqrt => a_ge0; case: (sig2W _) => /= x _; rewrite a_ge0 => /eqP.
Qed. | Lemma | sqr_sqrtr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"sig2W",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler0_sqrtr a : a <= 0 -> sqrt a = 0. | Proof.
rewrite /sqrtr; case: (sig2W _) => x /= _.
by have [//|_ /eqP//|->] := ltrgt0P a; rewrite mulf_eq0 orbb => /eqP.
Qed. | Lemma | ler0_sqrtr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltrgt0P",
"mulf_eq0",
"sig2W",
"sqrt",
"sqrtr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr0_sqrtr a : a < 0 -> sqrt a = 0. | Proof. by move=> /ltW; apply: ler0_sqrtr. Qed. | Lemma | ltr0_sqrtr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"ler0_sqrtr",
"ltW",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_spec a : R -> bool -> bool -> R -> Type | :=
| IsNoSqrtr of a < 0 : sqrtr_spec a a false true 0
| IsSqrtr b of 0 <= b : sqrtr_spec a (b ^+ 2) true false b. | Variant | sqrtr_spec | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtrP a : sqrtr_spec a a (0 <= a) (a < 0) (sqrt a). | Proof.
have [a_ge0|a_lt0] := ger0P a.
by rewrite -{1 2}[a]sqr_sqrtr //; constructor.
by rewrite ltr0_sqrtr //; constructor.
Qed. | Lemma | sqrtrP | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ger0P",
"ltr0_sqrtr",
"sqr_sqrtr",
"sqrt",
"sqrtr_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_sqr a : sqrt (a ^+ 2) = `|a|. | Proof.
have /eqP : sqrt (a ^+ 2) ^+ 2 = `|a| ^+ 2.
by rewrite -normrX ger0_norm ?sqr_sqrtr ?sqr_ge0.
rewrite eqf_sqr => /predU1P[-> //|ha].
have := sqrtr_ge0 (a ^+ 2); rewrite (eqP ha) oppr_ge0 normr_le0 => /eqP ->.
by rewrite normr0 oppr0.
Qed. | Lemma | sqrtr_sqr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"eqf_sqr",
"ger0_norm",
"normr0",
"normrX",
"normr_le0",
"oppr0",
"oppr_ge0",
"predU1P",
"sqr_ge0",
"sqr_sqrtr",
"sqrt",
"sqrtr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqr_inj : {in Num.nneg &, injective (fun x : R => x ^+ 2)}. | Proof. by move=> ? ? ? ? /(congr1 sqrt); rewrite !sqrtr_sqr !ger0_norm. Qed. | Lemma | sqr_inj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ger0_norm",
"nneg",
"sqrt",
"sqrtr_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_inj : {in Num.nneg &, injective (fun x : R => sqrt x)}. | Proof. by move=> x y xge0 yge0 sxy; rewrite -[x]sqr_sqrtr// sxy sqr_sqrtr. Qed. | Lemma | sqrtr_inj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"nneg",
"sqr_sqrtr",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtrM a b : 0 <= a -> sqrt (a * b) = sqrt a * sqrt b. | Proof.
case: (sqrtrP a) => // {}a a_ge0 _; case: (sqrtrP b) => [b_lt0 | {}b b_ge0].
by rewrite mulr0 ler0_sqrtr // nmulr_lle0 ?mulr_ge0.
by rewrite mulrACA sqrtr_sqr ger0_norm ?mulr_ge0.
Qed. | Lemma | sqrtrM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ger0_norm",
"ler0_sqrtr",
"mulr0",
"mulrACA",
"mulr_ge0",
"nmulr_lle0",
"sqrt",
"sqrtrP",
"sqrtr_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr0 : sqrt 0 = 0 :> R. | Proof. by move: (sqrtr_sqr 0); rewrite exprS mul0r => ->; rewrite normr0. Qed. | Lemma | sqrtr0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"exprS",
"mul0r",
"normr0",
"sqrt",
"sqrtr_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr1 : sqrt 1 = 1 :> R. | Proof. by move: (sqrtr_sqr 1); rewrite expr1n => ->; rewrite normr1. Qed. | Lemma | sqrtr1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"expr1n",
"normr1",
"sqrt",
"sqrtr_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_eq0 a : (sqrt a == 0) = (a <= 0). | Proof.
case: sqrtrP => [/ltW ->|b]; first by rewrite eqxx.
case: ltrgt0P => [b_gt0|//|->]; last by rewrite exprS mul0r lexx.
by rewrite lt_geF ?pmulr_rgt0.
Qed. | Lemma | sqrtr_eq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"eqxx",
"exprS",
"last",
"lexx",
"ltW",
"lt_geF",
"ltrgt0P",
"mul0r",
"pmulr_rgt0",
"sqrt",
"sqrtrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtr_gt0 a : (0 < sqrt a) = (0 < a). | Proof. by rewrite lt0r sqrtr_ge0 sqrtr_eq0 -ltNge andbT. Qed. | Lemma | sqrtr_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lt0r",
"ltNge",
"sqrt",
"sqrtr_eq0",
"sqrtr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_sqrt a b : 0 <= a -> 0 <= b -> (sqrt a == sqrt b) = (a == b). | Proof.
move=> a_ge0 b_ge0; apply/eqP/eqP=> [HS|->] //.
by move: (sqr_sqrtr a_ge0); rewrite HS (sqr_sqrtr b_ge0).
Qed. | Lemma | eqr_sqrt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"sqr_sqrtr",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_wsqrtr : {homo @sqrt R : a b / a <= b}. | Proof.
move=> a b /= le_ab; case: (boolP (0 <= a))=> [pa|]; last first.
by rewrite -ltNge; move/ltW; rewrite -sqrtr_eq0; move/eqP->.
rewrite -(@ler_pXn2r R 2) ?nnegrE ?sqrtr_ge0 //.
by rewrite !sqr_sqrtr // (le_trans pa).
Qed. | Lemma | ler_wsqrtr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"last",
"le_trans",
"ler_pXn2r",
"ltNge",
"ltW",
"nnegrE",
"sqr_sqrtr",
"sqrt",
"sqrtr_eq0",
"sqrtr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_psqrt : {in @nneg R &, {mono sqrt : a b / a <= b}}. | Proof.
apply: le_mono_in => x y x_gt0 y_gt0.
rewrite !lt_neqAle => /andP[neq_xy le_xy].
by rewrite ler_wsqrtr // eqr_sqrt // neq_xy.
Qed. | Lemma | ler_psqrt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"eqr_sqrt",
"le_mono_in",
"ler_wsqrtr",
"lt_neqAle",
"nneg",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_sqrt a b : 0 <= b -> (sqrt a <= sqrt b) = (a <= b). | Proof.
move=> b_ge0; have [a_le0|a_gt0] := ler0P a; last first.
by rewrite ler_psqrt // nnegrE ltW.
by rewrite ler0_sqrtr // sqrtr_ge0 (le_trans a_le0).
Qed. | Lemma | ler_sqrt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"last",
"le_trans",
"ler0P",
"ler0_sqrtr",
"ler_psqrt",
"ltW",
"nnegrE",
"sqrt",
"sqrtr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_sqrt a b : 0 < b -> (sqrt a < sqrt b) = (a < b). | Proof.
move=> b_gt0; have [a_le0|a_gt0] := ler0P a; last first.
by rewrite (leW_mono_in ler_psqrt)//; apply: ltW.
by rewrite ler0_sqrtr // sqrtr_gt0 b_gt0 (le_lt_trans a_le0).
Qed. | Lemma | ltr_sqrt | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"last",
"leW_mono_in",
"le_lt_trans",
"ler0P",
"ler0_sqrtr",
"ler_psqrt",
"ltW",
"sqrt",
"sqrtr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtrV x : 0 <= x -> sqrt (x^-1) = (sqrt x)^-1. | Proof.
case: ltrgt0P => // [x_gt0 _|->]; last by rewrite !(invr0, sqrtr0).
have sx_neq0 : sqrt x != 0 by rewrite sqrtr_eq0 -ltNge.
apply: (mulfI sx_neq0).
by rewrite -sqrtrM !(divff, ltW, sqrtr1) // lt0r_neq0.
Qed. | Lemma | sqrtrV | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"divff",
"invr0",
"last",
"lt0r_neq0",
"ltNge",
"ltW",
"ltrgt0P",
"mulfI",
"sqrt",
"sqrtr0",
"sqrtr1",
"sqrtrM",
"sqrtr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normCK : forall x, `|x| ^+ 2 = x * x^* | := normCK_subdef. | Definition | normCK | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrCi : 'i ^+ 2 = -1 :> C | := sqrCi. | Definition | sqrCi | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulCii : 'i * 'i = -1 :> C. | Proof. exact: sqrCi. Qed. | Lemma | mulCii | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"sqrCi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjCK : involutive (@conj C). | Proof.
have JE x : x^* = `|x|^+2 / x.
have [->|x_neq0] := eqVneq x 0; first by rewrite rmorph0 invr0 mulr0.
by apply: (canRL (mulfK _)) => //; rewrite mulrC -normCK.
move=> x; have [->|x_neq0] := eqVneq x 0; first by rewrite !rmorph0.
rewrite !JE normrM normfV exprMn normrX normr_id.
by rewrite exprVn -mulrA -invfM... | Lemma | conjCK | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"conj",
"divKf",
"eqVneq",
"expr2",
"exprMn",
"exprVn",
"invfM",
"invr0",
"mulfK",
"mulr0",
"mulrA",
"mulrC",
"normCK",
"normfV",
"normrM",
"normrX",
"normr_eq0",
"normr_id",
"rmorph0",
"sqrf_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re2 z | := z + z^*. | Let | Re2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nnegIm z | := (0 <= 'i * (z^* - z)). | Definition | nnegIm | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
argCle y z | := nnegIm z ==> nnegIm y && (Re2 z <= Re2 y). | Definition | argCle | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re2",
"nnegIm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_spec n (x : C) : Type | :=
RootCspec (y : C) of if (n > 0)%N then y ^+ n = x else y = 0
& forall z, (n > 0)%N -> z ^+ n = x -> argCle y z. | Variant | rootC_spec | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"argCle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_subproof n x : rootC_spec n x. | Proof.
have realRe2 u : Re2 u \is Num.real by
rewrite realEsqr expr2 {2}/Re2 -{2}[u]conjCK addrC -rmorphD -normCK exprn_ge0.
have argCle_total : total argCle.
move=> u v; rewrite /total /argCle.
by do 2!case: (nnegIm _) => //; rewrite ?orbT //= real_leVge.
have argCle_trans : transitive argCle.
move=> u v w /im... | Fact | rootC_subproof | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re2",
"addrC",
"allP",
"apply",
"argCle",
"closed_field_poly_normal",
"coefB",
"coefC",
"coefXn",
"conjCK",
"eqxx",
"expr2",
"exprn_ge0",
"hornerE",
"inE",
"le_trans",
"lead_coefE",
"lexx",
"lt0n",
"ltnS",
"mem_nth",
"mulnb",
"mulrb",
"mulrnA",
"mulrn_eq0",
"n_gt0"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nthroot n x | := let: RootCspec y _ _ := rootC_subproof n x in y. | Definition | nthroot | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .-root" | := (nthroot n) : ring_scope. | Notation | n .-root | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"nthroot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC | := 2.-root. | Notation | sqrtC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_lock : unit. | Proof. exact: tt. Qed. | Fact | Re_lock | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_lock : unit. | Proof. exact: tt. Qed. | Fact | Im_lock | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re z | := locked_with Re_lock ((z + z^*) / 2%:R). | Definition | Re | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re_lock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im z | := locked_with Im_lock ('i * (z^* - z) / 2%:R). | Definition | Im | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im_lock"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Re z" | := (Re z) : ring_scope. | Notation | 'Re z | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Im z" | := (Im z) : ring_scope. | Notation | 'Im z | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReE z : 'Re z = (z + z^*) / 2%:R. | Proof. by rewrite ['Re _]unlock. Qed. | Lemma | ReE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImE z : 'Im z = 'i * (z^* - z) / 2%:R. | Proof. by rewrite ['Im _]unlock. Qed. | Lemma | ImE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz2 : 2 != 0 :> C. | Proof. by rewrite pnatr_eq0. Qed. | Let | nz2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normCKC x : `|x| ^+ 2 = x^* * x. | Proof. by rewrite normCK mulrC. Qed. | Lemma | normCKC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"mulrC",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_conjC_ge0 x : 0 <= x * x^*. | Proof. by rewrite -normCK exprn_ge0. Qed. | Lemma | mul_conjC_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"exprn_ge0",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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