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 |
|---|---|---|---|---|---|---|---|---|---|---|
nat_num : qualifier 1 R | := [qualify a x : R | nat_num_subdef x]. | Definition | nat_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_num : qualifier 1 R | := [qualify a x : R | int_num_subdef x]. | Definition | int_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
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"eqtype",
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"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound (x : R) | := (truncn `|x|).+1. | Definition | bound | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc | := truncn (only parsing). | Notation | trunc | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn | := truncn. | Notation | truncn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor | := floor. | Notation | floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"nmodule",
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"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil | := ceil. | Notation | ceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_num | := nat_num. | Notation | nat_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_num | := int_num. | Notation | int_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
archi_bound | := bound. | Notation | archi_bound | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_subproof n :
if n \is real_num then n%:~R <= n < (n + 1)%:~R else n == 0. | Proof. by rewrite num_real !intz ltzD1 lexx. Qed. | Fact | floor_subproof | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"intz",
"lexx",
"ltzD1",
"num_real",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrP n : reflect (exists m, n = m%:~R) true. | Proof. by apply: ReflectT; exists n; rewrite intz. Qed. | Fact | intrP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"intz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrP n : reflect (exists m, n = m%:R) (0 <= n). | Proof.
apply: (iffP idP); last by case=> m ->; rewrite ler0n.
by case: n => // n _; exists n; rewrite natz.
Qed. | Fact | natrP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"last",
"ler0n",
"natz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn | := (@truncn R). | Notation | truncn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor | := (@floor R). | Notation | floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil | := (@ceil R). | Notation | ceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_num | := (@Def.nat_num R). | Notation | nat_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_num | := (@Def.int_num R). | Notation | int_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorP x :
if x \is real_num then (floor x)%:~R <= x < (floor x + 1)%:~R
else floor x == 0. | Proof. exact: floor_subproof. Qed. | Lemma | floorP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"floor_subproof",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorNceil x : floor x = - ceil (- x). | Proof. by rewrite ceil_subproof !opprK. Qed. | Lemma | floorNceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"floor",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilNfloor x : ceil x = - floor (- x). | Proof. exact: ceil_subproof. Qed. | Lemma | ceilNfloor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"floor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncEfloor x : truncn x = if floor x is Posz n then n else 0. | Proof. exact: truncn_subproof. Qed. | Lemma | truncEfloor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Posz",
"floor",
"truncn",
"truncn_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrP x : reflect (exists n, x = n%:R) (x \is a nat_num). | Proof. exact: nat_num_subproof. Qed. | Lemma | natrP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrP x : reflect (exists m, x = m%:~R) (x \is a int_num). | Proof. exact: int_num_subproof. Qed. | Lemma | intrP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_int m : m%:~R \is a int_num. | Proof. by apply/intrP; exists m. Qed. | Lemma | intr_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"int_num",
"intrP"
] | int_num and nat_num | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
natr_nat n : n%:R \is a nat_num. | Proof. by apply/natrP; exists n. Qed. | Lemma | natr_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"nat_num",
"natrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_int_num (S : subringClosed R) x : x \is a int_num -> x \in S. | Proof. by move=> /intrP[n ->]; rewrite rpred_int. Qed. | Lemma | rpred_int_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrP",
"rpred_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpred_nat_num (S : semiringClosed R) x : x \is a nat_num -> x \in S. | Proof. by move=> /natrP[n ->]; apply: rpred_nat. Qed. | Lemma | rpred_nat_num | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"nat_num",
"natrP",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_num0 : 0 \is a int_num. | Proof. exact: (intr_int 0). Qed. | Lemma | int_num0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intr_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_num1 : 1 \is a int_num. | Proof. exact: (intr_int 1). Qed. | Lemma | int_num1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intr_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_num0 : 0 \is a nat_num. | Proof. exact: (natr_nat 0). Qed. | Lemma | nat_num0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"natr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_num1 : 1 \is a nat_num. | Proof. exact: (natr_nat 1). Qed. | Lemma | nat_num1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"natr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_num_subring : subring_closed int_num. | Proof.
by split=> // _ _ /intrP[n ->] /intrP[m ->]; rewrite -(intrB, intrM).
Qed. | Fact | int_num_subring | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrB",
"intrM",
"intrP",
"split",
"subring_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_num_semiring : semiring_closed nat_num. | Proof.
by do 2![split] => //= _ _ /natrP[n ->] /natrP[m ->]; rewrite -(natrD, natrM).
Qed. | Fact | nat_num_semiring | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"natrD",
"natrM",
"natrP",
"semiring_closed",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rreal_nat : {subset nat_num <= real_num}. | Proof. exact: rpred_nat_num. Qed. | Lemma | Rreal_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"real_num",
"rpred_nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_nat : {subset nat_num <= int_num}. | Proof. by move=> _ /natrP[n ->]; rewrite pmulrn intr_int. Qed. | Lemma | intr_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intr_int",
"nat_num",
"natrP",
"pmulrn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Rreal_int : {subset int_num <= real_num}. | Proof. exact: rpred_int_num. Qed. | Lemma | Rreal_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"real_num",
"rpred_int_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrE x : (x \is a int_num) = (x \is a nat_num) || (- x \is a nat_num). | Proof.
apply/idP/orP => [/intrP[[n|n] ->]|[]/intr_nat]; rewrite ?rpredN //.
by left; apply/natrP; exists n.
by rewrite NegzE intrN opprK; right; apply/natrP; exists n.+1.
Qed. | Lemma | intrE | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"NegzE",
"apply",
"int_num",
"intrN",
"intrP",
"intr_nat",
"nat_num",
"natrP",
"opprK",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_int n : n%:R \is a int_num. | Proof. by rewrite intrE natr_nat. Qed. | Lemma | natr_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrE",
"natr_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_normK x : x \is a int_num -> `|x| ^+ 2 = x ^+ 2. | Proof. by move/Rreal_int/real_normK. Qed. | Lemma | intr_normK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Rreal_int",
"int_num",
"real_normK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_normK x : x \is a nat_num -> `|x| ^+ 2 = x ^+ 2. | Proof. by move/Rreal_nat/real_normK. Qed. | Lemma | natr_normK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Rreal_nat",
"nat_num",
"real_normK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_norm_int x : x \is a int_num -> `|x| \is a nat_num. | Proof. by move=> /intrP[m ->]; rewrite -intr_norm rpred_nat_num ?natr_nat. Qed. | Lemma | natr_norm_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrP",
"intr_norm",
"nat_num",
"natr_nat",
"rpred_nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_ge0 x : x \is a nat_num -> 0 <= x. | Proof. by move=> /natrP[n ->]; apply: ler0n. Qed. | Lemma | natr_ge0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"ler0n",
"nat_num",
"natrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_gt0 x : x \is a nat_num -> (0 < x) = (x != 0). | Proof. by move/natr_ge0; case: comparableP. Qed. | Lemma | natr_gt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"comparableP",
"nat_num",
"natr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrEint x : (x \is a nat_num) = (x \is a int_num) && (0 <= x). | Proof.
apply/idP/andP=> [Nx | [Zx x_ge0]]; first by rewrite intr_nat ?natr_ge0.
by rewrite -(ger0_norm x_ge0) natr_norm_int.
Qed. | Lemma | natrEint | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"ger0_norm",
"int_num",
"intr_nat",
"nat_num",
"natr_ge0",
"natr_norm_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrEge0 x : 0 <= x -> (x \is a int_num) = (x \is a nat_num). | Proof. by rewrite natrEint andbC => ->. Qed. | Lemma | intrEge0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"nat_num",
"natrEint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrEsign x : x \is a int_num -> x = (-1) ^+ (x < 0)%R * `|x|. | Proof. by move/Rreal_int/realEsign. Qed. | Lemma | intrEsign | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Rreal_int",
"int_num",
"realEsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_natr x : x \is a nat_num -> `|x| = x. | Proof. by move/natr_ge0/ger0_norm. Qed. | Lemma | norm_natr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ger0_norm",
"nat_num",
"natr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_exp_even x n : ~~ odd n -> x \is a int_num -> x ^+ n \is a nat_num. | Proof.
move=> n_oddF x_intr.
by rewrite natrEint rpredX //= real_exprn_even_ge0 // Rreal_int.
Qed. | Lemma | natr_exp_even | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Rreal_int",
"int_num",
"nat_num",
"natrEint",
"odd",
"real_exprn_even_ge0",
"rpredX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_intr_ge1 x : x \is a int_num -> x != 0 -> 1 <= `|x|. | Proof.
rewrite -normr_eq0 => /natr_norm_int/natrP[n ->].
by rewrite pnatr_eq0 ler1n lt0n.
Qed. | Lemma | norm_intr_ge1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"ler1n",
"lt0n",
"natrP",
"natr_norm_int",
"normr_eq0",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqr_intr_ge1 x : x \is a int_num -> x != 0 -> 1 <= x ^+ 2. | Proof.
by move=> Zx nz_x; rewrite -intr_normK // expr_ge1 ?normr_ge0 ?norm_intr_ge1.
Qed. | Lemma | sqr_intr_ge1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"expr_ge1",
"int_num",
"intr_normK",
"norm_intr_ge1",
"normr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_ler_sqr x : x \is a int_num -> x <= x ^+ 2. | Proof.
move=> Zx; have [-> | nz_x] := eqVneq x 0; first by rewrite expr0n.
apply: le_trans (_ : `|x| <= _); first by rewrite real_ler_norm ?Rreal_int.
by rewrite -intr_normK // ler_eXnr // norm_intr_ge1.
Qed. | Lemma | intr_ler_sqr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Rreal_int",
"apply",
"eqVneq",
"expr0n",
"int_num",
"intr_normK",
"le_trans",
"ler_eXnr",
"norm_intr_ge1",
"real_ler_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_itv x :
x \is real_num -> (floor x)%:~R <= x < (floor x + 1)%:~R. | Proof. by case: ifP (floorP x). Qed. | Lemma | real_floor_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"floorP",
"real_num"
] | floor and int_num | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
real_floor_le x : x \is real_num -> (floor x)%:~R <= x. | Proof. by case/real_floor_itv/andP. Qed. | Lemma | real_floor_le | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floorD1_gt x : x \is real_num -> x < (floor x + 1)%:~R. | Proof. by case/real_floor_itv/andP. Qed. | Lemma | real_floorD1_gt | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_def x m : m%:~R <= x < (m + 1)%:~R -> floor x = m. | Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eq_le -!ltzD1.
move: (ger_real lemx); rewrite realz => /real_floor_itv/andP[lefx ltxf1].
by rewrite -!(ltr_int R) 2?(@le_lt_trans _ _ x).
Qed. | Lemma | floor_def | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eq_le",
"floor",
"ger_real",
"le_lt_trans",
"ltr_int",
"ltzD1",
"real_floor_itv",
"realz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_ge_int x n : x \is real_num -> (n <= floor x) = (n%:~R <= x). | Proof.
move=> /real_floor_itv /andP[lefx ltxf1]; apply/idP/idP => lenx.
by apply: le_trans lefx; rewrite ler_int.
by rewrite -ltzD1 -(ltr_int R); apply: le_lt_trans ltxf1.
Qed. | Lemma | real_floor_ge_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"le_lt_trans",
"le_trans",
"ler_int",
"ltr_int",
"ltzD1",
"real_floor_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_lt_int x n : x \is real_num -> (floor x < n) = (x < n%:~R). | Proof.
by move=> ?; rewrite [RHS]real_ltNge ?realz -?real_floor_ge_int -?ltNge.
Qed. | Lemma | real_floor_lt_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"ltNge",
"real_floor_ge_int",
"real_ltNge",
"real_num",
"realz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_eq x n : x \is real_num ->
(floor x == n) = (n%:~R <= x < (n + 1)%:~R). | Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: real_floor_itv|exact: floor_def].
Qed. | Lemma | real_floor_eq | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"floor_def",
"real_floor_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_floor : {homo floor : x y / x <= y}. | Proof.
move=> x y lexy; move: (floorP x) (floorP y); rewrite (ger_real lexy).
case: ifP => [_ /andP[lefx _] /andP[_] | _ /eqP-> /eqP-> //].
by move=> /(le_lt_trans lexy) /(le_lt_trans lefx); rewrite ltr_int ltzD1.
Qed. | Lemma | le_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"floorP",
"ger_real",
"le_lt_trans",
"ltr_int",
"ltzD1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrKfloor : cancel intr floor. | Proof. by move=> m; apply: floor_def; rewrite lexx rmorphD ltrDl ltr01. Qed. | Lemma | intrKfloor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"floor_def",
"intr",
"lexx",
"ltr01",
"ltrDl",
"rmorphD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrEfloor x : (x \is a int_num) = ((floor x)%:~R == x). | Proof.
by apply/intrP/eqP => [[n ->] | <-]; [rewrite intrKfloor | exists (floor x)].
Qed. | Lemma | intrEfloor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"int_num",
"intrKfloor",
"intrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorK : {in int_num, cancel floor intr}. | Proof. by move=> z; rewrite intrEfloor => /eqP. Qed. | Lemma | floorK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"int_num",
"intr",
"intrEfloor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor0 : floor 0 = 0. | Proof. exact: intrKfloor 0. Qed. | Lemma | floor0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"intrKfloor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor1 : floor 1 = 1. | Proof. exact: intrKfloor 1. Qed. | Lemma | floor1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"intrKfloor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floorDzr : {in int_num & real_num, {morph floor : x y / x + y}}. | Proof.
move=> _ y /intrP[m ->] Ry; apply: floor_def.
by rewrite -addrA 2!rmorphD /= intrKfloor lerD2l ltrD2l real_floor_itv.
Qed. | Lemma | real_floorDzr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"addrA",
"apply",
"floor",
"floor_def",
"int_num",
"intrKfloor",
"intrP",
"lerD2l",
"ltrD2l",
"real_floor_itv",
"real_num",
"rmorphD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floorDrz : {in real_num & int_num, {morph floor : x y / x + y}}. | Proof. by move=> x y xr yz; rewrite addrC real_floorDzr // addrC. Qed. | Lemma | real_floorDrz | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"addrC",
"floor",
"int_num",
"real_floorDzr",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorN : {in int_num, {morph floor : x / - x}}. | Proof. by move=> _ /intrP[m ->]; rewrite -rmorphN !intrKfloor. Qed. | Lemma | floorN | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"int_num",
"intrKfloor",
"intrP",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorM : {in int_num &, {morph floor : x y / x * y}}. | Proof.
by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKfloor.
Qed. | Lemma | floorM | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"int_num",
"intrKfloor",
"intrP",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorX n : {in int_num, {morph floor : x / x ^+ n}}. | Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKfloor. Qed. | Lemma | floorX | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"int_num",
"intrKfloor",
"intrP",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_ge0 x : x \is real_num -> (0 <= floor x) = (0 <= x). | Proof. by move=> ?; rewrite real_floor_ge_int. Qed. | Lemma | real_floor_ge0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_ge_int",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_lt0 x : (floor x < 0) = (x < 0). | Proof.
case: ifP (floorP x) => [xr _ | xr /eqP <-]; first by rewrite real_floor_lt_int.
by rewrite ltxx; apply/esym/(contraFF _ xr)/ltr0_real.
Qed. | Lemma | floor_lt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"floorP",
"ltr0_real",
"ltxx",
"real_floor_lt_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_le0 x : x \is real_num -> (floor x <= 0) = (x < 1). | Proof. by move=> ?; rewrite -ltzD1 add0r real_floor_lt_int. Qed. | Lemma | real_floor_le0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"add0r",
"floor",
"ltzD1",
"real_floor_lt_int",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_gt0 x : (floor x > 0) = (x >= 1). | Proof.
case: ifP (floorP x) => [xr _ | xr /eqP->].
by rewrite gtz0_ge1 real_floor_ge_int.
by rewrite ltxx; apply/esym/(contraFF _ xr)/ger1_real.
Qed. | Lemma | floor_gt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"floorP",
"ger1_real",
"gtz0_ge1",
"ltxx",
"real_floor_ge_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_neq0 x : (floor x != 0) = (x < 0) || (x >= 1). | Proof.
case: ifP (floorP x) => [xr _ | xr /eqP->]; rewrite ?eqxx/=.
by rewrite neq_lt floor_lt0 floor_gt0.
by apply/esym/(contraFF _ xr) => /orP[/ltr0_real|/ger1_real].
Qed. | Lemma | floor_neq0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eqxx",
"floor",
"floorP",
"floor_gt0",
"floor_lt0",
"ger1_real",
"ltr0_real",
"neq_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorpK : {in polyOver int_num, cancel (map_poly floor) (map_poly intr)}. | Proof.
move=> p /(all_nthP 0) Zp; apply/polyP=> i.
rewrite coef_map coef_map_id0 //= -[p]coefK coef_poly.
by case: ifP => [/Zp/floorK // | _]; rewrite floor0.
Qed. | Lemma | floorpK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Zp",
"all_nthP",
"apply",
"coefK",
"coef_map",
"coef_map_id0",
"coef_poly",
"floor",
"floor0",
"floorK",
"int_num",
"intr",
"map_poly",
"polyOver",
"polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorpP (p : {poly R}) :
p \is a polyOver int_num -> {q | p = map_poly intr q}. | Proof. by exists (map_poly floor p); rewrite floorpK. Qed. | Lemma | floorpP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"floorpK",
"int_num",
"intr",
"map_poly",
"poly",
"polyOver"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_itv x : x \is real_num -> (ceil x - 1)%:~R < x <= (ceil x)%:~R. | Proof.
rewrite ceilNfloor -opprD !intrN ltrNl lerNr andbC -realN.
exact: real_floor_itv.
Qed. | Lemma | real_ceil_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"intrN",
"lerNr",
"ltrNl",
"opprD",
"realN",
"real_floor_itv",
"real_num"
] | ceil and int_num | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
real_ceilB1_lt x : x \is real_num -> (ceil x - 1)%:~R < x. | Proof. by case/real_ceil_itv/andP. Qed. | Lemma | real_ceilB1_lt | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_ge x : x \is real_num -> x <= (ceil x)%:~R. | Proof. by case/real_ceil_itv/andP. Qed. | Lemma | real_ceil_ge | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_def x m : (m - 1)%:~R < x <= m%:~R -> ceil x = m. | Proof.
rewrite -ltrN2 -lerN2 andbC -!intrN opprD opprK ceilNfloor.
by move=> /floor_def ->; rewrite opprK.
Qed. | Lemma | ceil_def | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"floor_def",
"intrN",
"lerN2",
"ltrN2",
"opprD",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_le_int x n : x \is real_num -> (ceil x <= n) = (x <= n%:~R). | Proof.
rewrite ceilNfloor lerNl -realN => /real_floor_ge_int ->.
by rewrite intrN lerN2.
Qed. | Lemma | real_ceil_le_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"intrN",
"lerN2",
"lerNl",
"realN",
"real_floor_ge_int",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_gt_int x n : x \is real_num -> (n < ceil x) = (n%:~R < x). | Proof.
by move=> ?; rewrite [RHS]real_ltNge ?realz -?real_ceil_le_int ?ltNge.
Qed. | Lemma | real_ceil_gt_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ltNge",
"real_ceil_le_int",
"real_ltNge",
"real_num",
"realz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_eq x n : x \is real_num ->
(ceil x == n) = ((n - 1)%:~R < x <= n%:~R). | Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: real_ceil_itv|exact: ceil_def].
Qed. | Lemma | real_ceil_eq | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"ceil",
"ceil_def",
"real_ceil_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_ceil : {homo ceil : x y / x <= y}. | Proof. by move=> x y lexy; rewrite !ceilNfloor lerN2 le_floor ?lerN2. Qed. | Lemma | le_ceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"le_floor",
"lerN2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrKceil : cancel intr ceil. | Proof. by move=> m; rewrite ceilNfloor -intrN intrKfloor opprK. Qed. | Lemma | intrKceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"intr",
"intrKfloor",
"intrN",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrEceil x : (x \is a int_num) = ((ceil x)%:~R == x). | Proof. by rewrite -rpredN intrEfloor -eqr_oppLR -intrN -ceilNfloor. Qed. | Lemma | intrEceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"eqr_oppLR",
"int_num",
"intrEfloor",
"intrN",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilK : {in int_num, cancel ceil intr}. | Proof. by move=> z; rewrite intrEceil => /eqP. Qed. | Lemma | ceilK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"int_num",
"intr",
"intrEceil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil0 : ceil 0 = 0. | Proof. exact: intrKceil 0. Qed. | Lemma | ceil0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"intrKceil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil1 : ceil 1 = 1. | Proof. exact: intrKceil 1. Qed. | Lemma | ceil1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"intrKceil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceilDzr : {in int_num & real_num, {morph ceil : x y / x + y}}. | Proof.
move=> x y x_int y_real.
by rewrite ceilNfloor opprD real_floorDzr ?rpredN // opprD -!ceilNfloor.
Qed. | Lemma | real_ceilDzr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"int_num",
"opprD",
"real_floorDzr",
"real_num",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceilDrz : {in real_num & int_num, {morph ceil : x y / x + y}}. | Proof. by move=> x y xr yz; rewrite addrC real_ceilDzr // addrC. Qed. | Lemma | real_ceilDrz | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"addrC",
"ceil",
"int_num",
"real_ceilDzr",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilN : {in int_num, {morph ceil : x / - x}}. | Proof. by move=> ? ?; rewrite !ceilNfloor !opprK floorN. Qed. | Lemma | ceilN | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"floorN",
"int_num",
"opprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilM : {in int_num &, {morph ceil : x y / x * y}}. | Proof.
by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKceil.
Qed. | Lemma | ceilM | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"int_num",
"intrKceil",
"intrP",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilX n : {in int_num, {morph ceil : x / x ^+ n}}. | Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKceil. Qed. | Lemma | ceilX | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"int_num",
"intrKceil",
"intrP",
"rmorphXn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_ge0 x : x \is real_num -> (0 <= ceil x) = (-1 < x). | Proof.
by move=> ?; rewrite ceilNfloor oppr_ge0 real_floor_le0 ?realN 1?ltrNl.
Qed. | Lemma | real_ceil_ge0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"ltrNl",
"oppr_ge0",
"realN",
"real_floor_le0",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_lt0 x : (ceil x < 0) = (x <= -1). | Proof. by rewrite ceilNfloor oppr_lt0 floor_gt0 lerNr. Qed. | Lemma | ceil_lt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"floor_gt0",
"lerNr",
"oppr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_le0 x : x \is real_num -> (ceil x <= 0) = (x <= 0). | Proof. by move=> ?; rewrite real_ceil_le_int. Qed. | Lemma | real_ceil_le0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_le_int",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_gt0 x : (ceil x > 0) = (x > 0). | Proof. by rewrite ceilNfloor oppr_gt0 floor_lt0 oppr_lt0. Qed. | Lemma | ceil_gt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"floor_lt0",
"oppr_gt0",
"oppr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_neq0 x : (ceil x != 0) = (x <= -1) || (x > 0). | Proof. by rewrite ceilNfloor oppr_eq0 floor_neq0 oppr_lt0 lerNr orbC. Qed. | Lemma | ceil_neq0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"ceilNfloor",
"floor_neq0",
"lerNr",
"oppr_eq0",
"oppr_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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