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 |
|---|---|---|---|---|---|---|---|---|---|---|
sgz_le0 x : (sgz x <= 0) = (x <= 0). | Proof. by case: sgzP. Qed. | Lemma | sgz_le0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"poly",
"orderedzmod",
"numdomain",
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"Order.TTheory",
... | [
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_smul x y : sgz (y *~ (sgz x)) = (sgz x) * (sgz y). | Proof. by rewrite -mulrzl sgzM -sgrEz sgz_sgr. Qed. | Lemma | sgz_smul | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulrzl",
"sgrEz",
"sgz",
"sgzM",
"sgz_sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgrMz m x : sgr (x *~ m) = sgr x *~ sgr m. | Proof. by rewrite -mulrzr sgrM -intr_sg mulrzr. Qed. | Lemma | sgrMz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intr_sg",
"mulrzr",
"sgr",
"sgrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_eq (R R' : realDomainType) (x : R) (y : R') :
(sgz x == sgz y) = ((x == 0) == (y == 0)) && ((0 < x) == (0 < y)). | Proof. by do 2!case: sgzP. Qed. | Lemma | sgz_eq | algebra | algebra/ssrint.v | [
"HB",
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"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"sgz",
"sgzP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_sign (R : pzRingType) s : ((-1) ^+ s)%:~R = (-1) ^+ s :> R. | Proof. exact: rmorph_sign. Qed. | Lemma | intr_sign | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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... | [
"rmorph_sign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_nat (n : nat) : `|n| = n. | Proof. by []. Qed. | Lemma | absz_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszE (m : int) : `|m| = `|m|%R :> int. | Proof. by []. Qed. | Lemma | abszE | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz0 : `|0%R| = 0. | Proof. by []. Qed. | Lemma | absz0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszN m : `|- m| = `|m|. | Proof. by case: (normrN m). Qed. | Lemma | abszN | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"normrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_eq0 m : (`|m| == 0) = (m == 0%R). | Proof. by case: (intP m). Qed. | Lemma | absz_eq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_gt0 m : (`|m| > 0) = (m != 0%R). | Proof. by case: (intP m). Qed. | Lemma | absz_gt0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz1 : `|1%R| = 1. | Proof. by []. Qed. | Lemma | absz1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszN1 : `|-1%R| = 1. | Proof. by []. Qed. | Lemma | abszN1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_id m : `|(`|m|)| = `|m|. | Proof. by []. Qed. | Lemma | absz_id | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszM m1 m2 : `|(m1 * m2)%R| = `|m1| * `|m2|. | Proof. by case: m1 m2 => [[|m1]|m1] [[|m2]|m2] //=; rewrite ?mulnS mulnC. Qed. | Lemma | abszM | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"mulnC",
"mulnS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszX (n : nat) m : `|m ^+ n| = `|m| ^ n. | Proof. by elim: n => // n ihn; rewrite exprS expnS abszM ihn. Qed. | Lemma | abszX | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszM",
"expnS",
"exprS",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_sg m : `|sgr m| = (m != 0%R). | Proof. by case: (intP m). Qed. | Lemma | absz_sg | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intP",
"sgr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gez0_abs m : (0 <= m)%R -> `|m| = m :> int. | Proof. by case: (intP m). Qed. | Lemma | gez0_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gtz0_abs m : (0 < m)%R -> `|m| = m :> int. | Proof. by case: (intP m). Qed. | Lemma | gtz0_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez0_abs m : (m <= 0)%R -> `|m| = - m :> int. | Proof. by case: (intP m). Qed. | Lemma | lez0_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
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"poly",
"orderedzmod",
"numdomain",
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"Order.TTheory",
... | [
"int",
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltz0_abs m : (m < 0)%R -> `|m| = - m :> int. | Proof. by case: (intP m). Qed. | Lemma | ltz0_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lez_abs m : m <= `|m|%N :> int. | Proof. by case: (intP m). Qed. | Lemma | lez_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
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"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int",
"intP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
absz_sign s : `|(-1) ^+ s| = 1. | Proof. by rewrite abszX exp1n. Qed. | Lemma | absz_sign | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
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"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszX",
"exp1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszMsign s m : `|((-1) ^+ s * m)%R| = `|m|. | Proof. by rewrite abszM absz_sign mul1n. Qed. | Lemma | abszMsign | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszM",
"absz_sign",
"mul1n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_sign_abs m : ((-1) ^+ (m < 0)%R * `|m|%:Z)%R = m. | Proof. by rewrite abszE mulr_sign_norm. Qed. | Lemma | mulz_sign_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszE",
"mulr_sign_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulz_Nsign_abs m : ((-1) ^+ (0 < m)%R * `|m|%:Z)%R = - m. | Proof. by rewrite abszE mulr_Nsign_norm. Qed. | Lemma | mulz_Nsign_abs | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszE",
"mulr_Nsign_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intEsign m : m = ((-1) ^+ (m < 0)%R * `|m|%:Z)%R. | Proof. exact: numEsign. Qed. | Lemma | intEsign | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"numEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszEsign m : `|m|%:Z = ((-1) ^+ (m < 0)%R * m)%R. | Proof. exact: normrEsign. Qed. | Lemma | abszEsign | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"normrEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intEsg m : m = (sgz m * `|m|%:Z)%R. | Proof. by rewrite -sgrz -numEsg. Qed. | Lemma | intEsg | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"numEsg",
"sgrz",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abszEsg m : (`|m|%:Z = sgz m * m)%R. | Proof. by rewrite -sgrz -normrEsg. Qed. | Lemma | abszEsg | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
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"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"normrEsg",
"sgrz",
"sgz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulr_absz (x : R) i : x *+ `|i| = x *~ `|i|. | Proof. by rewrite -abszE. Qed. | Lemma | mulr_absz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_absz i : `|i|%:R = `|i|%:~R :> R. | Proof. by rewrite -abszE. Qed. | Lemma | natr_absz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_nmodType : nmodType | := int. | Definition | int_nmodType | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | This notation is supposed to work even if the ssrint library is not Imported.
Since we can't rely on the CS database to contain the zmodule instance on
int we put the instance by hand in the notation. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
int_zmodType : zmodType | := int. | Definition | int_zmodType | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"m - n" | :=
(@GRing.add int_nmodType (m%N : int)
(@GRing.opp int_zmodType (n%N : int))) : distn_scope. | Notation | m - n | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"add",
"int",
"int_nmodType",
"int_zmodType",
"opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnC m1 m2 : `|m1 - m2| = `|m2 - m1|. | Proof. by rewrite -opprB abszN. Qed. | Lemma | distnC | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszN",
"opprB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnDl d n1 n2 : `|d + n1 - (d + n2)| = `|n1 - n2|. | Proof. by rewrite addnC !PoszD addrKA. Qed. | Lemma | distnDl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"addnC",
"addrKA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnDr d n1 n2 : `|n1 + d - (n2 + d)| = `|n1 - n2|. | Proof. by rewrite -!(addnC d) distnDl. Qed. | Lemma | distnDr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addnC",
"distnDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnEr n1 n2 : n1 <= n2 -> `|n1 - n2| = n2 - n1. | Proof. by move/subnK=> {1}<-; rewrite distnC PoszD addrK absz_nat. Qed. | Lemma | distnEr | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"absz_nat",
"addrK",
"distnC",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnEl n1 n2 : n2 <= n1 -> `|n1 - n2| = n1 - n2. | Proof. by move/distnEr <-; rewrite distnC. Qed. | Lemma | distnEl | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"distnC",
"distnEr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distn0 n : `|n - 0| = n. | Proof. by rewrite subr0 absz_nat. Qed. | Lemma | distn0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"absz_nat",
"subr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dist0n n : `|0 - n| = n. | Proof. by rewrite distnC distn0. Qed. | Lemma | dist0n | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"distn0",
"distnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnn m : `|m - m| = 0. | Proof. by rewrite subrr. Qed. | Lemma | distnn | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distn_eq0 n1 n2 : (`|n1 - n2| == 0) = (n1 == n2). | Proof. by rewrite absz_eq0 subr_eq0. Qed. | Lemma | distn_eq0 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"absz_eq0",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distnS n : `|n - n.+1| = 1. | Proof. exact: distnDr n 0 1. Qed. | Lemma | distnS | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"distnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distSn n : `|n.+1 - n| = 1. | Proof. exact: distnDr n 1 0. Qed. | Lemma | distSn | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"distnDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
distn_eq1 n1 n2 :
(`|n1 - n2| == 1) = (if n1 < n2 then n1.+1 == n2 else n1 == n2.+1). | Proof.
case: ltnP => [lt_n12 | le_n21].
by rewrite eq_sym -(eqn_add2r n1) distnEr ?subnK // ltnW.
by rewrite -(eqn_add2r n2) distnEl ?subnK.
Qed. | Lemma | distn_eq1 | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"distnEl",
"distnEr",
"eq_sym",
"eqn_add2r",
"ltnP",
"ltnW",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqD_dist m1 m2 m3 : `|m1 - m3| <= `|m1 - m2| + `|m2 - m3|. | Proof. by rewrite -lez_nat PoszD !abszE ler_distD. Qed. | Lemma | leqD_dist | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"abszE",
"ler_distD",
"lez_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leqifD_distz m1 m2 m3 :
`|m1 - m3| <= `|m1 - m2| + `|m2 - m3|
?= iff (m1 <= m2 <= m3)%R || (m3 <= m2 <= m1)%R. | Proof.
apply/leqifP; rewrite -ltz_nat -eqz_nat PoszD !abszE; apply/leifP.
wlog le_m31 : m1 m3 / (m3 <= m1)%R.
move=> IH; case/orP: (le_total m1 m3) => /IH //.
by rewrite (addrC `|_|)%R orbC !(distrC m1) !(distrC m3).
rewrite ger0_norm ?subr_ge0 // orb_idl => [/andP[le_m12 le_m23]|].
by have /eqP->: m2 == m3; rewr... | Lemma | leqifD_distz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"PoszD",
"abszE",
"addrA",
"addrC",
"apply",
"distrC",
"eq_le",
"eqz_nat",
"ger0_norm",
"le_total",
"le_trans",
"leifD",
"leifP",
"leqifP",
"lexx",
"ltz_nat",
"num_real",
"real_leif_norm",
"subrK",
"subr_ge0"
] | Most of this proof generalizes to all real-ordered rings. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
leqifD_dist n1 n2 n3 :
`|n1 - n3| <= `|n1 - n2| + `|n2 - n3|
?= iff (n1 <= n2 <= n3) || (n3 <= n2 <= n1). | Proof. exact: leqifD_distz. Qed. | Lemma | leqifD_dist | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"leqifD_distz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrn_dist n1 n2 : `|n1 - n2| ^ 2 + 2 * (n1 * n2) = n1 ^ 2 + n2 ^ 2. | Proof.
wlog le_n21: n1 n2 / n2 <= n1.
move=> IH; case/orP: (leq_total n2 n1) => /IH //.
by rewrite (addnC (n2 ^ 2)) (mulnC n2) distnC.
by rewrite distnEl ?sqrnB ?subnK ?nat_Cauchy.
Qed. | Lemma | sqrn_dist | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addnC",
"distnC",
"distnEl",
"leq_total",
"mulnC",
"nat_Cauchy",
"sqrnB",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_norm m : `|m|%:~R = `|m%:~R : R|. | Proof. by rewrite {2}[m]intEsign rmorphMsign normrMsign abszE normr_nat. Qed. | Lemma | intr_norm | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"abszE",
"intEsign",
"normrMsign",
"normr_nat",
"rmorphMsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normrMz m (x : R) : `|x *~ m| = `|x| *~ `|m|. | Proof. by rewrite -mulrzl normrM -intr_norm mulrzl. Qed. | Lemma | normrMz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"intr_norm",
"mulrzl",
"normrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expN1r (i : int) : (-1 : R) ^ i = (-1) ^+ `|i|. | Proof.
case: i => n; first by rewrite exprnP absz_nat.
by rewrite NegzE abszN absz_nat -invr_expz expfV invrN1.
Qed. | Lemma | expN1r | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"NegzE",
"abszN",
"absz_nat",
"expfV",
"exprnP",
"int",
"invrN1",
"invr_expz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coefMrz p n i : (p *~ n)`_i = (p`_i *~ n). | Proof. exact: (raddfMz (coefp i)). Qed. | Lemma | coefMrz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"coefp",
"raddfMz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
polyCMz n : {morph (@polyC R) : c / c *~ n}. | Proof. exact: raddfMz. Qed. | Lemma | polyCMz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"polyC",
"raddfMz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hornerMz n p x : (p *~ n).[x] = p.[x] *~ n. | Proof. exact: (raddfMz (horner_eval _)). Qed. | Lemma | hornerMz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"horner_eval",
"raddfMz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
horner_int n x : (n%:~R : {poly R}).[x] = n%:~R. | Proof. by rewrite hornerMz hornerC. Qed. | Lemma | horner_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"hornerC",
"hornerMz",
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derivMz n p : (p *~ n)^`() = p^`() *~ n. | Proof. exact: raddfMz. Qed. | Lemma | derivMz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
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"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"raddfMz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulpz p n : p *~ n = n%:~R *: p. | Proof. by rewrite scaler_int. Qed. | Lemma | mulpz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"scaler_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredMz (M : zmodType) (S : zmodClosed M) m :
{in S, forall u, u *~ m \in S}. | Proof. by case: m => n u Su; rewrite ?rpredN ?rpredMn. Qed. | Lemma | rpredMz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"rpredMn",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_int (R : pzRingType) (S : subringClosed R) m : m%:~R \in S. | Proof. by rewrite rpredMz ?rpred1. Qed. | Lemma | rpred_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"rpred1",
"rpredMz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredZint (R : pzRingType) (M : lmodType R) (S : zmodClosed M) m :
{in S, forall u, m%:~R *: u \in S}. | Proof. by move=> u Su; rewrite /= scaler_int rpredMz. Qed. | Lemma | rpredZint | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"rpredMz",
"scaler_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredXz (R : unitRingType) (S : divClosed R) m :
{in S, forall x, x ^ m \in S}. | Proof. by case: m => n x Sx; rewrite ?rpredV rpredX. Qed. | Lemma | rpredXz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"rpredV",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredXsign (R : unitRingType) (S : divClosed R) n x :
(x ^ ((-1) ^+ n) \in S) = (x \in S). | Proof. by rewrite -signr_odd; case: (odd n); rewrite ?rpredV. Qed. | Lemma | rpredXsign | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"odd",
"rpredV",
"signr_odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_of_tuple : tuple_of >-> finfun_of. | Coercion | finfun_of_tuple | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"finfun_of",
"tuple_of"
] | Coercion from tuples to finfun: allows writing tensor dimensions as tuples | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of_finfun : finfun_of >-> tuple_of. | Coercion | tuple_of_finfun | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"finfun_of",
"tuple_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
tensor : predArgType | :=
Tensor of 'M[K]_(\prod_(i < k) (u_ i)%:num, \prod_(j < l) (d_ j)%:num)%R. | Variant | tensor | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor_val T | := let: Tensor g := T in g. | Definition | tensor_val | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''T[' R ]_ ( u_ , d_ )" | := (tensor u_ d_ R) (only parsing). | Notation | ''T[' R ]_ ( u_ , d_ ) | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''T[' R ]_ [ u1 , .. , uk ; d1 , .. , dl ]" | := (tensor [tuple of u1%:posnat :: .. [:: uk%:posnat] ..]
[tuple of d1%:posnat :: .. [:: dl%:posnat] ..] R) (only parsing). | Notation | ''T[' R ]_ [ u1 , .. , uk ; d1 , .. , dl ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''T_' ( u_ , d_ )" | := (tensor u_ d_ _). | Notation | ''T_' ( u_ , d_ ) | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''nT[' R ]_ ( u_ )" | := 'T[R]_( u_ , [tuple] ) (only parsing). | Notation | ''nT[' R ]_ ( u_ ) | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''nT[' R ]_ [ u1 , .. , uk ]" | := 'T[R]_( [tuple of u1%:posnat :: .. [:: uk%:posnat] ..] , [tuple] )
(only parsing). | Notation | ''nT[' R ]_ [ u1 , .. , uk ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''oT[' R ]_ ( d_ )" | := 'T[R]_( [tuple] , d_ ) (only parsing). | Notation | ''oT[' R ]_ ( d_ ) | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''oT[' R ]_ [ d1 , .. , dl ]" | := 'T[R]_( [tuple] , [tuple of d1%:posnat :: .. [:: dl%:posnat] ..] )
(only parsing). | Notation | ''oT[' R ]_ [ d1 , .. , dl ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''oT_' ( d_ )" | := (tensor [tuple] d_ _). | Notation | ''oT_' ( d_ ) | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''nT_' ( u_ )" | := (tensor u_ [tuple] _). | Notation | ''nT_' ( u_ ) | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''sT[' R ]" | := (tensor [tuple] [tuple] R) (only parsing). | Notation | ''sT[' R ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''sT'" | := (tensor [tuple] [tuple] _). | Notation | ''sT' | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"tensor",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''T[' R ]" | := 'T[R]_(u_, d_). | Notation | ''T[' R ] | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nmod_closed {m n} (R : nmodType) : @GRing.nmod_closed 'M[R]_(n, m) predT. | Proof. by []. Qed. | Let | nmod_closed | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subsemimod_closed {m n} (R : pzSemiRingType)
: @GRing.subsemimod_closed R 'M[R]_(n, m) predT. | Proof. by []. Qed. | Let | subsemimod_closed | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
const_t {R k l} {u_ : {posnum nat} ^ k} {d_ : {posnum nat} ^ l}
(v : R) : 'T[R]_(u_, d_) | := Tensor (const_mx v). | Definition | const_t | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_mx",
"nat",
"posnum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castt {k l k' l' : nat}
{u_ : {posnum nat} ^ k} {d_ : {posnum nat} ^ l}
{u_' : {posnum nat} ^ k'} {d_' : {posnum nat} ^ l'}
(eq_ud : (\prod_(i < k) (u_ i)%:num = \prod_(i < k') (u_' i)%:num)
* (\prod_(j < l) (d_ j)%:num = \prod_(j < l') (d_' j)%:num))
(t : 'T[R]_(u_, d_)) : 'T[R]_(u_', d_') | :=
Tensor (castmx eq_ud (\val t)). | Definition | castt | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"castmx",
"nat",
"num",
"posnum",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_castt {k l k' l' : nat}
{u_ : {posnum nat} ^ k} {d_ : {posnum nat} ^ l}
{u_' : {posnum nat} ^ k'} {d_' : {posnum nat} ^ l'}
(eq_ud : (\prod_(i < k) (u_ i)%:num = \prod_(i < k') (u_' i)%:num)
* (\prod_(j < l) (d_ j)%:num = \prod_(j < l') (d_' j)%:num))
(t : 'T[R]_(u_, d_)) :
\val (castt ... | Proof. by []. Qed. | Lemma | val_castt | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"castmx",
"castt",
"nat",
"num",
"posnum",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castt_id {k l : nat}
{u_ : {posnum nat} ^ k} {d_ : {posnum nat} ^ l}
erefl_ud (t : 'T[R]_(u_, d_)) :
castt erefl_ud t = t. | Proof. by apply/val_inj; rewrite val_castt castmx_id. Qed. | Lemma | castt_id | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"castmx_id",
"castt",
"nat",
"posnum",
"val_castt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
castt_comp {k1 l1 k2 l2 k3 l3 : nat}
{u1 : {posnum nat} ^ k1} {d1 : {posnum nat} ^ l1}
{u2 : {posnum nat} ^ k2} {d2 : {posnum nat} ^ l2}
{u3 : {posnum nat} ^ k3} {d3 : {posnum nat} ^ l3}
(eq_u1 : \prod_(i < k1) (u1 i)%:num = \prod_(i < k2) (u2 i)%:num)
(eq_d1 : \prod_(j < l1) (d1 j)%:num = \prod_(j ... | Proof. by apply/val_inj; rewrite !val_castt castmx_comp. Qed. | Lemma | castt_comp | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"castmx_comp",
"castt",
"l1",
"l2",
"nat",
"num",
"posnum",
"val_castt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
casttK {k1 l1 k2 l2 : nat}
{u1 : {posnum nat} ^ k1} {d1 : {posnum nat} ^ l1}
{u2 : {posnum nat} ^ k2} {d2 : {posnum nat} ^ l2}
(eq_u : \prod_(i < k1) (u1 i)%:num = \prod_(i < k2) (u2 i)%:num)
(eq_d : \prod_(j < l1) (d1 j)%:num = \prod_(j < l2) (d2 j)%:num) :
cancel (castt (u_' := u2) (d_' := d2) (eq_u... | Proof. by move=> t; apply/val_inj; rewrite !val_castt castmxK. Qed. | Lemma | casttK | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"castmxK",
"castt",
"l1",
"l2",
"nat",
"num",
"posnum",
"val_castt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
casttKV {k1 l1 k2 l2 : nat}
{u1 : {posnum nat} ^ k1} {d1 : {posnum nat} ^ l1}
{u2 : {posnum nat} ^ k2} {d2 : {posnum nat} ^ l2}
(eq_u : \prod_(i < k1) (u1 i)%:num = \prod_(i < k2) (u2 i)%:num)
(eq_d : \prod_(j < l1) (d1 j)%:num = \prod_(j < l2) (d2 j)%:num) :
cancel (castt (u_ := u2) (d_ := d2) (u_' :... | Proof. by move=> t; apply/val_inj; rewrite !val_castt castmxKV. Qed. | Lemma | casttKV | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"apply",
"castmxKV",
"castt",
"l1",
"l2",
"nat",
"num",
"posnum",
"val_castt",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tensor1 | := @const_t _ _ _ u_ d_ (GRing.one R). | Definition | tensor1 | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"const_t",
"one"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmul (t u : 'T[R]_(u_, d_)) | :=
@Tensor _ _ u_ d_ R (map2_mx *%R (\val t) (\val u)). | Definition | hmul | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"map2_mx",
"val"
] | Hadamard product: element-wise multiplication | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
hmulA : associative hmul. | Proof. by move=> x y z; rewrite /hmul map2_mxA. Qed. | Let | hmulA | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_mxA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmul1t : left_id tensor1 hmul. | Proof. by move=> [x]; rewrite /hmul map2_1mx. Qed. | Let | hmul1t | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_1mx",
"tensor1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmul1 : right_id tensor1 hmul. | Proof. by move=> [x]; rewrite /hmul map2_mx1. Qed. | Let | hmul1 | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_mx1",
"tensor1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmulDl : left_distributive hmul +%R. | Proof. by move=> x y z; rewrite /hmul map2_mxDl. Qed. | Let | hmulDl | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_mxDl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmulDr : right_distributive hmul +%R. | Proof. by move=> x y z; rewrite /hmul map2_mxDr. Qed. | Let | hmulDr | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_mxDr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmul0t : left_zero 0%R hmul. | Proof. by move=> x; rewrite /hmul map2_0mx. Qed. | Let | hmul0t | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmul0 : right_zero 0%R hmul. | Proof. by move=> x; rewrite /hmul map2_mx0. Qed. | Let | hmul0 | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_mx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hmulC {R : comPzSemiRingType} : @commutative 'T[R] _ hmul. | Proof. by move=> x y; rewrite /hmul map2_mxC. Qed. | Let | hmulC | algebra | algebra/tensor.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"seq",
"matrix",
"bigop",
"ssrbool",
"eqtype",
"choice",
"fintype",
"ssralg",
"ssrnat",
"ssrfun",
"order",
"finfun",
"tuple",
"finset",
"sesquilinear",
"interval",
"interval_inference",
"numdomain",
"GRing.Theory"
] | [
"hmul",
"map2_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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