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 |
|---|---|---|---|---|---|---|---|---|---|---|
mul_conjC_gt0 x : (0 < x * x^* ) = (x != 0). | Proof.
have [->|x_neq0] := eqVneq; first by rewrite rmorph0 mulr0.
by rewrite -normCK exprn_gt0 ?normr_gt0.
Qed. | Lemma | mul_conjC_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... | [
"eqVneq",
"exprn_gt0",
"mulr0",
"normCK",
"normr_gt0",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_conjC_eq0 x : (x * x^* == 0) = (x == 0). | Proof. by rewrite -normCK expf_eq0 normr_eq0. Qed. | Lemma | mul_conjC_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... | [
"expf_eq0",
"normCK",
"normr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_ge0 x : (0 <= x^* ) = (0 <= x). | Proof.
wlog suffices: x / 0 <= x -> 0 <= x^*.
by move=> IH; apply/idP/idP=> /IH; rewrite ?conjCK.
rewrite [in X in X -> _]le0r => /predU1P[-> | x_gt0]; first by rewrite rmorph0.
by rewrite -(pmulr_rge0 _ x_gt0) mul_conjC_ge0.
Qed. | Lemma | 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... | [
"apply",
"conjCK",
"le0r",
"mul_conjC_ge0",
"pmulr_rge0",
"predU1P",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_nat n : (n%:R)^* = n%:R :> C. | Proof. exact: rmorph_nat. Qed. | Lemma | conjC_nat | 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... | [
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC0 : 0^* = 0 :> C. | Proof. exact: rmorph0. Qed. | Lemma | conjC0 | 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... | [
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC1 : 1^* = 1 :> C. | Proof. exact: rmorph1. Qed. | Lemma | conjC1 | 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... | [
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjCN1 : (- 1)^* = - 1 :> C. | Proof. exact: rmorphN1. Qed. | Lemma | conjCN1 | 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... | [
"rmorphN1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_eq0 x : (x^* == 0) = (x == 0). | Proof. exact: fmorph_eq0. Qed. | Lemma | conjC_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... | [
"fmorph_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invC_norm x : x^-1 = `|x| ^- 2 * x^*. | Proof.
have [-> | nx_x] := eqVneq x 0; first by rewrite conjC0 mulr0 invr0.
by rewrite normCK invfM divfK ?conjC_eq0.
Qed. | Lemma | invC_norm | 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... | [
"conjC0",
"conjC_eq0",
"divfK",
"eqVneq",
"invfM",
"invr0",
"mulr0",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CrealE x : (x \is real) = (x^* == x). | Proof.
rewrite realEsqr ger0_def normrX normCK.
by have [-> | /mulfI/inj_eq-> //] := eqVneq x 0; rewrite rmorph0 !eqxx.
Qed. | Lemma | CrealE | 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... | [
"eqVneq",
"eqxx",
"ger0_def",
"inj_eq",
"mulfI",
"normCK",
"normrX",
"real",
"realEsqr",
"rmorph0"
] | Real number subset. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
CrealP {x} : reflect (x^* = x) (x \is real). | Proof. by rewrite CrealE; apply: eqP. Qed. | Lemma | CrealP | 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... | [
"CrealE",
"apply",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_Creal x : x \is real -> x^* = x. | Proof. by move/CrealP. Qed. | Lemma | conj_Creal | 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... | [
"CrealP",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_normC z : `|z|^* = `|z|. | Proof. by rewrite conj_Creal ?normr_real. Qed. | Lemma | conj_normC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conj_Creal",
"normr_real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CrealJ : {mono (@conj C) : x / x \is Num.real}. | Proof. by apply: (homo_mono1 conjCK) => x xreal; rewrite conj_Creal. Qed. | Lemma | CrealJ | 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",
"conjCK",
"conj_Creal",
"homo_mono1",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geC0_conj x : 0 <= x -> x^* = x. | Proof. by move=> /ger0_real/CrealP. Qed. | Lemma | geC0_conj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"CrealP",
"ger0_real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
geC0_unit_exp x n : 0 <= x -> (x ^+ n.+1 == 1) = (x == 1). | Proof. by move=> x_ge0; rewrite pexpr_eq1. Qed. | Lemma | geC0_unit_exp | 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... | [
"pexpr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
case_rootC | := rewrite /nthroot; case: (rootC_subproof _ _). | Ltac | case_rootC | 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",
"rootC_subproof"
] | Elementary properties of roots. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
root0C x : 0.-root x = 0. | Proof. by case_rootC. Qed. | Lemma | root0C | 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... | [
"case_rootC",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootCK n : (n > 0)%N -> cancel n.-root (fun x => x ^+ n). | Proof. by case: n => //= n _ x; case_rootC. Qed. | Lemma | rootCK | 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... | [
"case_rootC",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root1C x : 1.-root x = x. | Proof. exact: (@rootCK 1). Qed. | Lemma | root1C | 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",
"rootCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC0 n : n.-root 0 = 0. | Proof.
have [-> | n_gt0] := posnP n; first by rewrite root0C.
by have /eqP := rootCK n_gt0 0; rewrite expf_eq0 n_gt0 /= => /eqP.
Qed. | Lemma | rootC0 | 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... | [
"expf_eq0",
"n_gt0",
"posnP",
"root",
"root0C",
"rootCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_inj n : (n > 0)%N -> injective n.-root. | Proof. by move/rootCK/can_inj. Qed. | Lemma | rootC_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... | [
"root",
"rootCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_rootC n : (n > 0)%N -> {mono n.-root : x y / x == y}. | Proof. by move/rootC_inj/inj_eq. Qed. | Lemma | eqr_rootC | 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... | [
"inj_eq",
"root",
"rootC_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_eq0 n x : (n > 0)%N -> (n.-root x == 0) = (x == 0). | Proof. by move=> n_gt0; rewrite -{1}(rootC0 n) eqr_rootC. Qed. | Lemma | rootC_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... | [
"eqr_rootC",
"n_gt0",
"root",
"rootC0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonRealCi : ('i : C) \isn't real. | Proof. by rewrite realEsqr sqrCi oppr_ge0 lt_geF ?ltr01. Qed. | Lemma | nonRealCi | 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_geF",
"ltr01",
"oppr_ge0",
"real",
"realEsqr",
"sqrCi"
] | Rectangular coordinates. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
neq0Ci : 'i != 0 :> C. | Proof. by apply: contraNneq nonRealCi => ->. Qed. | Lemma | neq0Ci | 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",
"contraNneq",
"nonRealCi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normCi : `|'i| = 1 :> C. | Proof. by apply/eqP; rewrite -(@pexpr_eq1 _ _ 2) // -normrX sqrCi normrN1. Qed. | Lemma | normCi | 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",
"normrN1",
"normrX",
"pexpr_eq1",
"sqrCi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invCi : 'i^-1 = - 'i :> C. | Proof. by rewrite -div1r -[1]opprK -sqrCi mulNr mulfK ?neq0Ci. Qed. | Lemma | invCi | 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... | [
"div1r",
"mulNr",
"mulfK",
"neq0Ci",
"opprK",
"sqrCi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjCi : 'i^* = - 'i :> C. | Proof. by rewrite -invCi invC_norm normCi expr1n invr1 mul1r. Qed. | Lemma | conjCi | 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",
"invC_norm",
"invCi",
"invr1",
"mul1r",
"normCi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Crect x : x = 'Re x + 'i * 'Im x. | Proof.
rewrite !(ReE, ImE) 2!mulrA mulCii mulN1r opprB -mulrDl.
by rewrite addrCA addrK mulrDl -splitr.
Qed. | Lemma | Crect | 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",
"ImE",
"Re",
"ReE",
"addrCA",
"addrK",
"mulCii",
"mulN1r",
"mulrA",
"mulrDl",
"opprB",
"splitr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqCP x y : x = y <-> ('Re x = 'Re y) /\ ('Im x = 'Im y). | Proof. by split=> [->//|[eqRe eqIm]]; rewrite [x]Crect [y]Crect eqRe eqIm. Qed. | Lemma | eqCP | 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... | [
"Crect",
"Im",
"Re",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqC x y : (x == y) = ('Re x == 'Re y) && ('Im x == 'Im y). | Proof. by apply/eqP/(andPP eqP eqP) => /eqCP. Qed. | Lemma | eqC | 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",
"Re",
"apply",
"eqCP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Creal_Re x : 'Re x \is real. | Proof. by rewrite ReE CrealE fmorph_div rmorph_nat rmorphD /= conjCK addrC. Qed. | Lemma | Creal_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... | [
"CrealE",
"Re",
"ReE",
"addrC",
"conjCK",
"fmorph_div",
"real",
"rmorphD",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Creal_Im x : 'Im x \is real. | Proof.
rewrite ImE CrealE fmorph_div rmorph_nat rmorphM/= rmorphB/= conjCK.
by rewrite conjCi -opprB mulrNN.
Qed. | Lemma | Creal_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... | [
"CrealE",
"Im",
"ImE",
"conjCK",
"conjCi",
"fmorph_div",
"mulrNN",
"opprB",
"real",
"rmorphB",
"rmorphM",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_is_zmod_morphism : zmod_morphism Re. | Proof. by move=> x y; rewrite !ReE rmorphB addrACA -opprD mulrBl. Qed. | Fact | Re_is_zmod_morphism | 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",
"ReE",
"addrACA",
"mulrBl",
"opprD",
"rmorphB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_is_additive | := Re_is_zmod_morphism. | Definition | Re_is_additive | 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_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_is_zmod_morphism : zmod_morphism Im. | Proof.
by move=> x y; rewrite !ImE rmorphB opprD addrACA -opprD mulrBr mulrBl.
Qed. | Fact | Im_is_zmod_morphism | 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",
"ImE",
"addrACA",
"mulrBl",
"mulrBr",
"opprD",
"rmorphB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_is_additive | := Im_is_zmod_morphism. | Definition | Im_is_additive | 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_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Creal_ImP z : reflect ('Im z = 0) (z \is real). | Proof.
rewrite ImE CrealE -subr_eq0 -(can_eq (mulKf neq0Ci)) mulr0.
by rewrite -(can_eq (divfK nz2)) mul0r; apply: eqP.
Qed. | Lemma | Creal_ImP | 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... | [
"CrealE",
"Im",
"ImE",
"apply",
"can_eq",
"divfK",
"mul0r",
"mulKf",
"mulr0",
"neq0Ci",
"nz2",
"real",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Creal_ReP z : reflect ('Re z = z) (z \in real). | Proof.
rewrite (sameP (Creal_ImP z) eqP) -(can_eq (mulKf neq0Ci)) mulr0.
by rewrite -(inj_eq (addrI ('Re z))) addr0 -Crect eq_sym; apply: eqP.
Qed. | Lemma | Creal_ReP | 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... | [
"Creal_ImP",
"Crect",
"Re",
"addr0",
"addrI",
"apply",
"can_eq",
"eq_sym",
"inj_eq",
"mulKf",
"mulr0",
"neq0Ci",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReMl : {in real, forall x, {morph Re : z / x * z}}. | Proof.
by move=> x Rx z /=; rewrite !ReE rmorphM /= (conj_Creal Rx) -mulrDr -mulrA.
Qed. | Lemma | ReMl | 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",
"ReE",
"conj_Creal",
"mulrA",
"mulrDr",
"real",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReMr : {in real, forall x, {morph Re : z / z * x}}. | Proof. by move=> x Rx z /=; rewrite mulrC ReMl // mulrC. Qed. | Lemma | ReMr | 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",
"ReMl",
"mulrC",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImMl : {in real, forall x, {morph Im : z / x * z}}. | Proof.
by move=> x Rx z; rewrite !ImE rmorphM /= (conj_Creal Rx) -mulrBr mulrCA !mulrA.
Qed. | Lemma | ImMl | 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",
"ImE",
"conj_Creal",
"mulrA",
"mulrBr",
"mulrCA",
"real",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImMr : {in real, forall x, {morph Im : z / z * x}}. | Proof. by move=> x Rx z /=; rewrite mulrC ImMl // mulrC. Qed. | Lemma | ImMr | 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",
"ImMl",
"mulrC",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_i : 'Re 'i = 0. | Proof. by rewrite ReE conjCi subrr mul0r. Qed. | Lemma | Re_i | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re",
"ReE",
"conjCi",
"mul0r",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_i : 'Im 'i = 1. | Proof.
rewrite ImE conjCi -opprD mulrN -mulr2n mulrnAr mulCii.
by rewrite mulNrn opprK divff.
Qed. | Lemma | Im_i | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im",
"ImE",
"conjCi",
"divff",
"mulCii",
"mulNrn",
"mulr2n",
"mulrN",
"mulrnAr",
"opprD",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_conj z : 'Re z^* = 'Re z. | Proof. by rewrite !ReE addrC conjCK. Qed. | Lemma | Re_conj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Re",
"ReE",
"addrC",
"conjCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_conj z : 'Im z^* = - 'Im z. | Proof. by rewrite !ImE -mulNr -mulrN opprB conjCK. Qed. | Lemma | Im_conj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im",
"ImE",
"conjCK",
"mulNr",
"mulrN",
"opprB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_rect : {in real &, forall x y, 'Re (x + 'i * y) = x}. | Proof.
move=> x y Rx Ry; rewrite /= raddfD /= (Creal_ReP x Rx).
by rewrite ReMr // Re_i mul0r addr0.
Qed. | Lemma | Re_rect | 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... | [
"Creal_ReP",
"Re",
"ReMr",
"Re_i",
"addr0",
"mul0r",
"raddfD",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_rect : {in real &, forall x y, 'Im (x + 'i * y) = y}. | Proof.
move=> x y Rx Ry; rewrite /= raddfD /= (Creal_ImP x Rx) add0r.
by rewrite ImMr // Im_i mul1r.
Qed. | Lemma | Im_rect | 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... | [
"Creal_ImP",
"Im",
"ImMr",
"Im_i",
"add0r",
"mul1r",
"raddfD",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_rect : {in real &, forall x y, (x + 'i * y)^* = x - 'i * y}. | Proof.
by move=> x y Rx Ry; rewrite /= rmorphD rmorphM /= conjCi mulNr !conj_Creal.
Qed. | Lemma | conjC_rect | 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... | [
"conjCi",
"conj_Creal",
"mulNr",
"real",
"rmorphD",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addC_rect x1 y1 x2 y2 :
(x1 + 'i * y1) + (x2 + 'i * y2) = x1 + x2 + 'i * (y1 + y2). | Proof. by rewrite addrACA -mulrDr. Qed. | Lemma | addC_rect | 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... | [
"addrACA",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppC_rect x y : - (x + 'i * y) = - x + 'i * (- y). | Proof. by rewrite mulrN -opprD. Qed. | Lemma | oppC_rect | 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... | [
"mulrN",
"opprD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subC_rect x1 y1 x2 y2 :
(x1 + 'i * y1) - (x2 + 'i * y2) = x1 - x2 + 'i * (y1 - y2). | Proof. by rewrite oppC_rect addC_rect. Qed. | Lemma | subC_rect | 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... | [
"addC_rect",
"oppC_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulC_rect x1 y1 x2 y2 : (x1 + 'i * y1) * (x2 + 'i * y2) =
x1 * x2 - y1 * y2 + 'i * (x1 * y2 + x2 * y1). | Proof.
rewrite mulrDl !mulrDr mulrACA -expr2 sqrCi mulN1r.
by rewrite [_ - _]addrC addrACA mulrCA -mulrA [_ * y1]mulrC.
Qed. | Lemma | mulC_rect | 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... | [
"addrACA",
"addrC",
"expr2",
"mulN1r",
"mulrA",
"mulrACA",
"mulrC",
"mulrCA",
"mulrDl",
"mulrDr",
"sqrCi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImM x y : 'Im (x * y) = 'Re x * 'Im y + 'Re y * 'Im x. | Proof.
rewrite [x in LHS]Crect [y in LHS]Crect mulC_rect.
by rewrite !(Im_rect, rpredB, rpredD, rpredM).
Qed. | Lemma | ImM | 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... | [
"Crect",
"Im",
"Im_rect",
"Re",
"mulC_rect",
"rpredB",
"rpredD",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImMil x : 'Im ('i * x) = 'Re x. | Proof. by rewrite ImM Re_i Im_i mul0r mulr1 add0r. Qed. | Lemma | ImMil | 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",
"ImM",
"Im_i",
"Re",
"Re_i",
"add0r",
"mul0r",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReMil x : 'Re ('i * x) = - 'Im x. | Proof. by rewrite -ImMil mulrA mulCii mulN1r raddfN. Qed. | Lemma | ReMil | 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",
"ImMil",
"Re",
"mulCii",
"mulN1r",
"mulrA",
"raddfN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReMir x : 'Re (x * 'i) = - 'Im x. | Proof. by rewrite mulrC ReMil. Qed. | Lemma | ReMir | 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",
"Re",
"ReMil",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImMir x : 'Im (x * 'i) = 'Re x. | Proof. by rewrite mulrC ImMil. Qed. | Lemma | ImMir | 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",
"ImMil",
"Re",
"mulrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReM x y : 'Re (x * y) = 'Re x * 'Re y - 'Im x * 'Im y. | Proof. by rewrite -ImMil mulrCA ImM ImMil ReMil mulNr ['Im _ * _]mulrC. Qed. | Lemma | ReM | 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",
"ImM",
"ImMil",
"Re",
"ReMil",
"mulNr",
"mulrC",
"mulrCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC2_rect :
{in real &, forall x y, `|x + 'i * y| ^+ 2 = x ^+ 2 + y ^+ 2}. | Proof.
move=> x y Rx Ry; rewrite /= normCK rmorphD rmorphM /= conjCi !conj_Creal //.
by rewrite mulrC mulNr -subr_sqr exprMn sqrCi mulN1r opprK.
Qed. | Lemma | normC2_rect | 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... | [
"conjCi",
"conj_Creal",
"exprMn",
"mulN1r",
"mulNr",
"mulrC",
"normCK",
"opprK",
"real",
"rmorphD",
"rmorphM",
"sqrCi",
"subr_sqr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC2_Re_Im z : `|z| ^+ 2 = 'Re z ^+ 2 + 'Im z ^+ 2. | Proof. by rewrite -normC2_rect -?Crect. Qed. | Lemma | normC2_Re_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... | [
"Crect",
"Im",
"Re",
"normC2_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invC_Crect x y : (x + 'i * y)^-1 = (x^* - 'i * y^*) / `|x + 'i * y| ^+ 2. | Proof. by rewrite /= invC_norm mulrC !rmorphE rmorphM /= conjCi mulNr. Qed. | Lemma | invC_Crect | 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... | [
"conjCi",
"invC_norm",
"mulNr",
"mulrC",
"rmorphE",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invC_rect :
{in real &, forall x y, (x + 'i * y)^-1 = (x - 'i * y) / (x ^+ 2 + y ^+ 2)}. | Proof. by move=> x y Rx Ry; rewrite invC_Crect normC2_rect ?conj_Creal. Qed. | Lemma | invC_rect | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conj_Creal",
"invC_Crect",
"normC2_rect",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ImV x : 'Im x^-1 = - 'Im x / `|x| ^+ 2. | Proof.
rewrite [x in LHS]Crect invC_rect// ImMr ?(rpredV, rpredD, rpredX)//.
by rewrite -mulrN Im_rect ?rpredN// -normC2_rect// -Crect.
Qed. | Lemma | ImV | 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... | [
"Crect",
"Im",
"ImMr",
"Im_rect",
"invC_rect",
"mulrN",
"normC2_rect",
"rpredD",
"rpredN",
"rpredV",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ReV x : 'Re x^-1 = 'Re x / `|x| ^+ 2. | Proof.
rewrite [x in LHS]Crect invC_rect// ReMr ?(rpredV, rpredD, rpredX)//.
by rewrite -mulrN Re_rect ?rpredN// -normC2_rect// -Crect.
Qed. | Lemma | ReV | 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... | [
"Crect",
"Re",
"ReMr",
"Re_rect",
"invC_rect",
"mulrN",
"normC2_rect",
"rpredD",
"rpredN",
"rpredV",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rectC_mulr x y z : (x + 'i * y) * z = x * z + 'i * (y * z). | Proof. by rewrite mulrDl mulrA. Qed. | Lemma | rectC_mulr | 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... | [
"mulrA",
"mulrDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rectC_mull x y z : z * (x + 'i * y) = z * x + 'i * (z * y). | Proof. by rewrite mulrDr mulrCA. Qed. | Lemma | rectC_mull | 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... | [
"mulrCA",
"mulrDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divC_Crect x1 y1 x2 y2 :
(x1 + 'i * y1) / (x2 + 'i * y2) =
(x1 * x2^* + y1 * y2^* + 'i * (x2^* * y1 - x1 * y2^*)) /
`|x2 + 'i * y2| ^+ 2. | Proof.
rewrite invC_Crect// -mulrN [_ / _]rectC_mulr mulC_rect !mulrA -mulrBl.
rewrite [_ * _ * y1]mulrAC -mulrDl mulrA -mulrDl !(mulrN, mulNr) opprK.
by rewrite [- _ + _]addrC.
Qed. | Lemma | divC_Crect | 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",
"invC_Crect",
"mulC_rect",
"mulNr",
"mulrA",
"mulrAC",
"mulrBl",
"mulrDl",
"mulrN",
"opprK",
"rectC_mulr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divC_rect x1 y1 x2 y2 :
x1 \is real -> y1 \is real -> x2 \is real -> y2 \is real ->
(x1 + 'i * y1) / (x2 + 'i * y2) =
(x1 * x2 + y1 * y2 + 'i * (x2 * y1 - x1 * y2)) /
(x2 ^+ 2 + y2 ^+ 2). | Proof. by move=> *; rewrite divC_Crect normC2_rect ?conj_Creal. Qed. | Lemma | divC_rect | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conj_Creal",
"divC_Crect",
"normC2_rect",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_div x y : 'Im (x / y) = ('Re y * 'Im x - 'Re x * 'Im y) / `|y| ^+ 2. | Proof. by rewrite ImM ImV ReV mulrA [X in _ + X]mulrAC -mulrDl mulrN addrC. Qed. | Lemma | Im_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... | [
"Im",
"ImM",
"ImV",
"Re",
"ReV",
"addrC",
"mulrA",
"mulrAC",
"mulrDl",
"mulrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Re_div x y : 'Re (x / y) = ('Re x * 'Re y + 'Im x * 'Im y) / `|y| ^+ 2. | Proof. by rewrite ReM ImV ReV !mulrA -mulrBl mulrN opprK. Qed. | Lemma | Re_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... | [
"Im",
"ImV",
"Re",
"ReM",
"ReV",
"mulrA",
"mulrBl",
"mulrN",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_normC_Re_Creal z : `|'Re z| <= `|z| ?= iff (z \is real). | Proof.
rewrite -(mono_in_leif ler_sqr); try by rewrite qualifE /=.
rewrite [`|'Re _| ^+ 2]normCK conj_Creal // normC2_Re_Im -expr2.
rewrite addrC -leifBLR subrr (sameP (Creal_ImP _) eqP) -sqrf_eq0 eq_sym.
by apply: leif_eq; rewrite -realEsqr.
Qed. | Lemma | leif_normC_Re_Creal | 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... | [
"Creal_ImP",
"Re",
"addrC",
"apply",
"conj_Creal",
"eq_sym",
"expr2",
"leifBLR",
"leif_eq",
"ler_sqr",
"mono_in_leif",
"normC2_Re_Im",
"normCK",
"real",
"realEsqr",
"sqrf_eq0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_Re_Creal z : 'Re z <= `|z| ?= iff (0 <= z). | Proof.
have ubRe: 'Re z <= `|'Re z| ?= iff (0 <= 'Re z).
by rewrite ger0_def eq_sym; apply/leif_eq/real_ler_norm.
congr (_ <= _ ?= iff _): (leif_trans ubRe (leif_normC_Re_Creal z)).
apply/andP/idP=> [[zRge0 /Creal_ReP <- //] | z_ge0].
by have Rz := ger0_real z_ge0; rewrite (Creal_ReP _ _).
Qed. | Lemma | leif_Re_Creal | 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... | [
"Creal_ReP",
"Re",
"apply",
"eq_sym",
"ger0_def",
"ger0_real",
"leif_eq",
"leif_normC_Re_Creal",
"leif_trans",
"real_ler_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqC_semipolar x y :
`|x| = `|y| -> 'Re x = 'Re y -> 0 <= 'Im x * 'Im y -> x = y. | Proof.
move=> eq_norm eq_Re sign_Im.
rewrite [x]Crect [y]Crect eq_Re; congr (_ + 'i * _).
have /eqP := congr1 (fun z => z ^+ 2) eq_norm.
rewrite !normC2_Re_Im eq_Re (can_eq (addKr _)) eqf_sqr => /pred2P[] // eq_Im.
rewrite eq_Im mulNr -expr2 oppr_ge0 real_exprn_even_le0 //= in sign_Im.
by rewrite eq_Im (eqP sign_Im) op... | Lemma | eqC_semipolar | 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... | [
"Crect",
"Im",
"Re",
"addKr",
"can_eq",
"eqf_sqr",
"expr2",
"mulNr",
"normC2_Re_Im",
"oppr0",
"oppr_ge0",
"pred2P",
"real_exprn_even_le0"
] | Equality from polar coordinates, for the upper plane. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
argCleP y z :
reflect (0 <= 'Im z -> 0 <= 'Im y /\ 'Re z <= 'Re y) (argCle y z). | Proof.
suffices dIm x: nnegIm x = (0 <= 'Im x).
rewrite /argCle !dIm !(ImE, ReE) ler_pM2r ?invr_gt0 ?ltr0n //.
by apply: (iffP implyP) => geZyz /geZyz/andP.
by rewrite (ImE x) pmulr_lge0 ?invr_gt0 ?ltr0n //; congr (0 <= _ * _).
Qed. | Let | argCleP | 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",
"ImE",
"Re",
"ReE",
"apply",
"argCle",
"invr_gt0",
"ler_pM2r",
"ltr0n",
"nnegIm",
"pmulr_lge0"
] | Nth roots. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rootC_Re_max n x y :
(n > 0)%N -> y ^+ n = x -> 0 <= 'Im y -> 'Re y <= 'Re (n.-root x). | Proof.
by move=> n_gt0 yn_x leI0y; case_rootC=> z /= _ /(_ y n_gt0 yn_x)/argCleP[].
Qed. | Lemma | rootC_Re_max | 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",
"Re",
"argCleP",
"case_rootC",
"n_gt0",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neg_unity_root n : (n > 1)%N -> exists2 w : C, w ^+ n = 1 & 'Re w < 0. | Proof.
move=> n_gt1; have [|w /eqP pw_0] := closed_rootP (\poly_(i < n) (1 : C)) _.
by rewrite size_poly_eq ?oner_eq0 // -(subnKC n_gt1).
rewrite horner_poly (eq_bigr _ (fun _ _ => mul1r _)) in pw_0.
have wn1: w ^+ n = 1 by apply/eqP; rewrite -subr_eq0 subrX1 pw_0 mulr0.
suffices /existsP[i ltRwi0]: [exists i : 'I_n,... | Let | neg_unity_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... | [
"Creal_ReP",
"Re",
"apply",
"bigD1",
"closed_rootP",
"contra_eqT",
"eq_bigr",
"existsP",
"existsPn",
"expr1n",
"exprAC",
"gt_eqF",
"horner_poly",
"ltnW",
"ltr01",
"ltr_wpDr",
"mul1r",
"mulr0",
"oner_eq0",
"raddf0",
"raddf_sum",
"real_leNgt",
"rpred0",
"rpred1",
"size_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Im_rootC_ge0 n x : (n > 1)%N -> 0 <= 'Im (n.-root x). | Proof.
set y := n.-root x => n_gt1; have n_gt0 := ltnW n_gt1.
apply: wlog_neg; rewrite -real_ltNge ?rpred0 // => ltIy0.
suffices [z zn_x leI0z]: exists2 z, z ^+ n = x & 'Im z >= 0.
by rewrite /y; case_rootC => /= y1 _ /(_ z n_gt0 zn_x)/argCleP[].
have [w wn1 ltRw0] := neg_unity_root n_gt1.
wlog leRI0yw: w wn1 ltRw0 /... | Lemma | Im_rootC_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... | [
"Crect",
"Im",
"Im_conj",
"Im_rect",
"Re",
"Re_conj",
"addr_ge0",
"apply",
"argCleP",
"case_rootC",
"exprMn",
"ltW",
"ltnW",
"mul1r",
"mulC_rect",
"mulrN",
"n_gt0",
"neg_unity_root",
"nmulr_rgt0",
"oppr_ge0",
"real",
"real_ge0P",
"real_ltNge",
"rmorph1",
"rmorphXn",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_lt0 n x : (1 < n)%N -> (n.-root x < 0) = false. | Proof.
set y := n.-root x => n_gt1; have n_gt0 := ltnW n_gt1.
apply: negbTE; apply: wlog_neg => /negbNE lt0y; rewrite le_gtF //.
have Rx: x \is real by rewrite -[x](rootCK n_gt0) rpredX // ltr0_real.
have Re_y: 'Re y = y by apply/Creal_ReP; rewrite ltr0_real.
have [z zn_x leR0z]: exists2 z, z ^+ n = x & 'Re z >= 0.
h... | Lemma | rootC_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Creal_ReP",
"Im",
"Im_conj",
"Re",
"ReMr",
"Re_conj",
"apply",
"conj_Creal",
"exprMn",
"le_gtF",
"le_trans",
"ltW",
"ltnW",
"ltr0_real",
"mul1r",
"n_gt0",
"neg_unity_root",
"nmulr_lgt0",
"oppr_ge0",
"real",
"real_ge0P",
"rmorphXn",
"root",
"rootCK",
"rootC_Re_max",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_ge0 n x : (n > 0)%N -> (0 <= n.-root x) = (0 <= x). | Proof.
set y := n.-root x => n_gt0.
apply/idP/idP=> [/(exprn_ge0 n) | x_ge0]; first by rewrite rootCK.
rewrite -(ge_leif (leif_Re_Creal y)).
have Ray: `|y| \is real by apply: normr_real.
rewrite -(Creal_ReP _ Ray) rootC_Re_max ?(Creal_ImP _ Ray) //.
by rewrite -normrX rootCK // ger0_norm.
Qed. | Lemma | rootC_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... | [
"Creal_ImP",
"Creal_ReP",
"apply",
"exprn_ge0",
"ge_leif",
"ger0_norm",
"leif_Re_Creal",
"n_gt0",
"normrX",
"normr_real",
"real",
"root",
"rootCK",
"rootC_Re_max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_gt0 n x : (n > 0)%N -> (n.-root x > 0) = (x > 0). | Proof. by move=> n_gt0; rewrite !lt0r rootC_ge0 ?rootC_eq0. Qed. | Lemma | rootC_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",
"n_gt0",
"root",
"rootC_eq0",
"rootC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_le0 n x : (1 < n)%N -> (n.-root x <= 0) = (x == 0). | Proof.
by move=> n_gt1; rewrite le_eqVlt rootC_lt0 // orbF rootC_eq0 1?ltnW.
Qed. | Lemma | rootC_le0 | 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... | [
"le_eqVlt",
"ltnW",
"root",
"rootC_eq0",
"rootC_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_rootCl n : (n > 0)%N -> {in Num.nneg, {mono n.-root : x y / x <= y}}. | Proof.
move=> n_gt0 x x_ge0 y; have [y_ge0 | not_y_ge0] := boolP (0 <= y).
by rewrite -(ler_pXn2r n_gt0) ?qualifE /= ?rootC_ge0 ?rootCK.
rewrite (contraNF (@le_trans _ _ _ 0 _ _)) ?rootC_ge0 //.
by rewrite (contraNF (le_trans x_ge0)).
Qed. | Lemma | ler_rootCl | 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... | [
"le_trans",
"ler_pXn2r",
"n_gt0",
"nneg",
"root",
"rootCK",
"rootC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_rootC n : (n > 0)%N -> {in Num.nneg &, {mono n.-root : x y / x <= y}}. | Proof. by move=> n_gt0 x y x_ge0 _; apply: ler_rootCl. Qed. | Lemma | ler_rootC | 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_rootCl",
"n_gt0",
"nneg",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_rootCl n : (n > 0)%N -> {in Num.nneg, {mono n.-root : x y / x < y}}. | Proof. by move=> n_gt0 x x_ge0 y; rewrite !lt_def ler_rootCl ?eqr_rootC. Qed. | Lemma | ltr_rootCl | 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... | [
"eqr_rootC",
"ler_rootCl",
"lt_def",
"n_gt0",
"nneg",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_rootC n : (n > 0)%N -> {in Num.nneg &, {mono n.-root : x y / x < y}}. | Proof. by move/ler_rootC/leW_mono_in. Qed. | Lemma | ltr_rootC | 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... | [
"leW_mono_in",
"ler_rootC",
"nneg",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprCK n x : (0 < n)%N -> 0 <= x -> n.-root (x ^+ n) = x. | Proof.
move=> n_gt0 x_ge0; apply/eqP.
by rewrite -(eqrXn2 n_gt0) ?rootC_ge0 ?exprn_ge0 ?rootCK.
Qed. | Lemma | exprCK | 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",
"eqrXn2",
"exprn_ge0",
"n_gt0",
"root",
"rootCK",
"rootC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_rootC n x : `|n.-root x| = n.-root `|x|. | Proof.
have [-> | n_gt0] := posnP n; first by rewrite !root0C normr0.
by apply/eqP; rewrite -(eqrXn2 n_gt0) ?rootC_ge0 // -normrX !rootCK.
Qed. | Lemma | norm_rootC | 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",
"eqrXn2",
"n_gt0",
"normr0",
"normrX",
"posnP",
"root",
"root0C",
"rootCK",
"rootC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootCX n x k : (n > 0)%N -> 0 <= x -> n.-root (x ^+ k) = n.-root x ^+ k. | Proof.
move=> n_gt0 x_ge0; apply/eqP.
by rewrite -(eqrXn2 n_gt0) ?(exprn_ge0, rootC_ge0) // 1?exprAC !rootCK.
Qed. | Lemma | rootCX | 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",
"eqrXn2",
"exprAC",
"exprn_ge0",
"n_gt0",
"root",
"rootCK",
"rootC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC1 n : (n > 0)%N -> n.-root 1 = 1. | Proof. by move/(rootCX 0)/(_ ler01). Qed. | Lemma | rootC1 | 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... | [
"ler01",
"root",
"rootCX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootCpX n x k : (k > 0)%N -> 0 <= x -> n.-root (x ^+ k) = n.-root x ^+ k. | Proof.
by case: n => [|n] k_gt0; [rewrite !root0C expr0n gtn_eqF | apply: rootCX].
Qed. | Lemma | rootCpX | 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",
"expr0n",
"gtn_eqF",
"root",
"root0C",
"rootCX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootCV n x : 0 <= x -> n.-root x^-1 = (n.-root x)^-1. | Proof.
move=> x_ge0; have [->|n_gt0] := posnP n; first by rewrite !root0C invr0.
apply/eqP.
by rewrite -(eqrXn2 n_gt0) ?(invr_ge0, rootC_ge0) // !exprVn !rootCK.
Qed. | Lemma | rootCV | 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",
"eqrXn2",
"exprVn",
"invr0",
"invr_ge0",
"n_gt0",
"posnP",
"root",
"root0C",
"rootCK",
"rootC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_eq1 n x : (n > 0)%N -> (n.-root x == 1) = (x == 1). | Proof. by move=> n_gt0; rewrite -{1}(rootC1 n_gt0) eqr_rootC. Qed. | Lemma | rootC_eq1 | 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... | [
"eqr_rootC",
"n_gt0",
"root",
"rootC1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_ge1 n x : (n > 0)%N -> (n.-root x >= 1) = (x >= 1). | Proof.
by move=> n_gt0; rewrite -{1}(rootC1 n_gt0) ler_rootCl // qualifE /= ler01.
Qed. | Lemma | rootC_ge1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler01",
"ler_rootCl",
"n_gt0",
"root",
"rootC1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_gt1 n x : (n > 0)%N -> (n.-root x > 1) = (x > 1). | Proof. by move=> n_gt0; rewrite !lt_def rootC_eq1 ?rootC_ge1. Qed. | Lemma | rootC_gt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lt_def",
"n_gt0",
"root",
"rootC_eq1",
"rootC_ge1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_le1 n x : (n > 0)%N -> 0 <= x -> (n.-root x <= 1) = (x <= 1). | Proof. by move=> n_gt0 x_ge0; rewrite -{1}(rootC1 n_gt0) ler_rootCl. Qed. | Lemma | rootC_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... | [
"ler_rootCl",
"n_gt0",
"root",
"rootC1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootC_lt1 n x : (n > 0)%N -> 0 <= x -> (n.-root x < 1) = (x < 1). | Proof. by move=> n_gt0 x_ge0; rewrite !lt_neqAle rootC_eq1 ?rootC_le1. Qed. | Lemma | rootC_lt1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"lt_neqAle",
"n_gt0",
"root",
"rootC_eq1",
"rootC_le1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootCMl n x z : 0 <= x -> n.-root (x * z) = n.-root x * n.-root z. | Proof.
rewrite le0r => /predU1P[-> | x_gt0]; first by rewrite !(mul0r, rootC0).
have [| n_gt1 | ->] := ltngtP n 1; last by rewrite !root1C.
by case: n => //; rewrite !root0C mul0r.
have [x_ge0 n_gt0] := (ltW x_gt0, ltnW n_gt1).
have nx_gt0: 0 < n.-root x by rewrite rootC_gt0.
have Rnx: n.-root x \is real by rewrite g... | Lemma | rootCMl | 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... | [
"ImMl",
"Im_rootC_ge0",
"ReMl",
"apply",
"eqC_semipolar",
"eq_le",
"eqrXn2",
"exprMn",
"exprVn",
"ger0_real",
"gt_eqF",
"invr_ge0",
"last",
"le0r",
"ler_pM2l",
"ltW",
"ltnW",
"ltngtP",
"mul0r",
"mulKf",
"mulVKf",
"mulr_ge0",
"n_gt0",
"normrX",
"predU1P",
"real",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.