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Coq-MathComp

fact stringlengths 8 1.54k	type stringclasses 19 values	library stringclasses 8 values	imports listlengths 1 10	filename stringclasses 98 values	symbolic_name stringlengths 1 42
RecordNumDomain_hasFloorCeilTruncn R of Num.NumDomain R := { floor : R -> int; ceil : R -> int; truncn : R -> nat; int_num_subdef : pred R; nat_num_subdef : pred R; floor_subproof : forall x, if x \is real_num then (floor x)%:~R <= x < (floor x + 1)%:~R else floor x == 0; ceil_subproof : forall x, ceil x = - floor (- x); truncn_subproof : forall x, truncn x = if floor x is Posz n then n else 0; int_num_subproof : forall x, reflect (exists n, x = n%:~R) (int_num_subdef x); nat_num_subproof : forall x, reflect (exists n, x = n%:R) (nat_num_subdef x); }. #[short(type="archiNumDomainType")] HB.structure Definition ArchiNumDomain := { R of NumDomain_hasFloorCeilTruncn R & Num.NumDomain R }.	HB.mixin	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Record
DefinitionArchiNumField := { R of NumDomain_hasFloorCeilTruncn R & Num.NumField R }.	HB.structure	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Definition
DefinitionArchiClosedField := { R of NumDomain_hasFloorCeilTruncn R & Num.ClosedField R }.	HB.structure	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Definition
DefinitionArchiRealDomain := { R of NumDomain_hasFloorCeilTruncn R & Num.RealDomain R }.	HB.structure	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Definition
DefinitionArchiRealField := { R of NumDomain_hasFloorCeilTruncn R & Num.RealField R }.	HB.structure	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Definition
DefinitionArchiRealClosedField := { R of NumDomain_hasFloorCeilTruncn R & Num.RealClosedField R }.	HB.structure	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Definition
nat_num: qualifier 1 R := [qualify a x : R \| nat_num_subdef x].	Definition	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	nat_num
int_num: qualifier 1 R := [qualify a x : R \| int_num_subdef x].	Definition	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	int_num
bound(x : R) := (truncn `\|x\|).+1.	Definition	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	bound
trunc:= truncn.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	trunc
truncn:= truncn. #[deprecated(since="mathcomp 2.4.0", note="Renamed to truncn.")]	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	truncn
trunc:= truncn.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	trunc
floor:= floor.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor
ceil:= ceil.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil
nat_num:= nat_num.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	nat_num
int_num:= int_num.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	int_num
archi_bound:= bound.	Notation	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	archi_bound
floorPn : if n \is real_num then n%:~R <= n < (n + 1)%:~R else n == 0. Proof. by rewrite num_real !intz ltzD1 lexx. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorP
intrPn : reflect (exists m, n = m%:~R) true. Proof. by apply: ReflectT; exists n; rewrite intz. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrP
natrPn : 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.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natrP
Definition_ := @NumDomain_hasFloorCeilTruncn.Build int id id _ xpredT nneg_num_pred intArchimedean.floorP (fun=> esym (opprK _)) (fun=> erefl) intArchimedean.intrP intArchimedean.natrP.	HB.instance	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Definition
floorNceilx : floor x = - ceil (- x). Proof. by rewrite ceil_subproof !opprK. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorNceil
ceilNfloorx : ceil x = - floor (- x). Proof. exact: ceil_subproof. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceilNfloor
truncEfloorx : truncn x = if floor x is Posz n then n else 0. Proof. exact: truncn_subproof. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	truncEfloor
natrPx : reflect (exists n, x = n%:R) (x \is a nat_num). Proof. exact: nat_num_subproof. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natrP
intrPx : reflect (exists m, x = m%:~R) (x \is a int_num). Proof. exact: int_num_subproof. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrP
intr_intm : m%:~R \is a int_num. Proof. by apply/intrP; exists m. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intr_int
natr_natn : n%:R \is a nat_num. Proof. by apply/natrP; exists n. Qed. #[local] Hint Resolve intr_int natr_nat : core.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_nat
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	rpred_int_num
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	rpred_nat_num
int_num0: 0 \is a int_num. Proof. exact: (intr_int 0). Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	int_num0
int_num1: 1 \is a int_num. Proof. exact: (intr_int 1). Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	int_num1
nat_num0: 0 \is a nat_num. Proof. exact: (natr_nat 0). Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	nat_num0
nat_num1: 1 \is a nat_num. Proof. exact: (natr_nat 1). Qed. #[local] Hint Resolve int_num0 int_num1 nat_num0 nat_num1 : core. Fact int_num_subring : subring_closed int_num. Proof. by split=> // _ _ /intrP[n ->] /intrP[m ->]; rewrite -(intrB, intrM). Qed. #[export] HB.instance Definition _ := GRing.isSubringClosed.Build R int_num_subdef int_num_subring. Fact nat_num_semiring : semiring_closed nat_num. Proof. by do 2![split] => //= _ _ /natrP[n ->] /natrP[m ->]; rewrite -(natrD, natrM). Qed. #[export] HB.instance Definition _ := GRing.isSemiringClosed.Build R nat_num_subdef nat_num_semiring.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	nat_num1
Rreal_nat: {subset nat_num <= real_num}. Proof. exact: rpred_nat_num. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Rreal_nat
intr_nat: {subset nat_num <= int_num}. Proof. by move=> _ /natrP[n ->]; rewrite pmulrn intr_int. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intr_nat
Rreal_int: {subset int_num <= real_num}. Proof. exact: rpred_int_num. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	Rreal_int
intrEx : (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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrE
intr_normKx : x \is a int_num -> `\|x\| ^+ 2 = x ^+ 2. Proof. by move/Rreal_int/real_normK. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intr_normK
natr_normKx : x \is a nat_num -> `\|x\| ^+ 2 = x ^+ 2. Proof. by move/Rreal_nat/real_normK. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_normK
natr_norm_intx : 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_norm_int
natr_ge0x : x \is a nat_num -> 0 <= x. Proof. by move=> /natrP[n ->]; apply: ler0n. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_ge0
natr_gt0x : x \is a nat_num -> (0 < x) = (x != 0). Proof. by move/natr_ge0; case: comparableP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_gt0
natrEintx : (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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natrEint
intrEge0x : 0 <= x -> (x \is a int_num) = (x \is a nat_num). Proof. by rewrite natrEint andbC => ->. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrEge0
intrEsignx : x \is a int_num -> x = (-1) ^+ (x < 0)%R * `\|x\|. Proof. by move/Rreal_int/realEsign. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrEsign
norm_natrx : x \is a nat_num -> `\|x\| = x. Proof. by move/natr_ge0/ger0_norm. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	norm_natr
natr_exp_evenx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_exp_even
norm_intr_ge1x : 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	norm_intr_ge1
sqr_intr_ge1x : 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	sqr_intr_ge1
intr_ler_sqrx : 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intr_ler_sqr
real_floor_itvx : x \is real_num -> (floor x)%:~R <= x < (floor x + 1)%:~R. Proof. by case: ifP (floorP x). Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_itv
real_floor_lex : x \is real_num -> (floor x)%:~R <= x. Proof. by case/real_floor_itv/andP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_le
real_floorD1_gtx : x \is real_num -> x < (floor x + 1)%:~R. Proof. by case/real_floor_itv/andP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floorD1_gt
floor_defx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor_def
real_floor_ge_intx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_ge_int
real_floor_lt_intx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_lt_int
real_floor_eqx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_eq
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	le_floor
intrKfloor: cancel intr floor. Proof. by move=> m; apply: floor_def; rewrite lexx rmorphD ltrDl ltr01. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrKfloor
natr_intn : n%:R \is a int_num. Proof. by rewrite intrE natr_nat. Qed. #[local] Hint Resolve natr_int : core.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	natr_int
intrEfloorx : x \is a int_num = ((floor x)%:~R == x). Proof. by apply/intrP/eqP => [[n ->] \| <-]; [rewrite intrKfloor \| exists (floor x)]. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrEfloor
floorK: {in int_num, cancel floor intr}. Proof. by move=> z; rewrite intrEfloor => /eqP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorK
floor0: floor 0 = 0. Proof. exact: intrKfloor 0. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor0
floor1: floor 1 = 1. Proof. exact: intrKfloor 1. Qed. #[local] Hint Resolve floor0 floor1 : core.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor1
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floorDzr
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floorDrz
floorN: {in int_num, {morph floor : x / - x}}. Proof. by move=> _ /intrP[m ->]; rewrite -rmorphN !intrKfloor. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorN
floorM: {in int_num &, {morph floor : x y / x * y}}. Proof. by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKfloor. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorM
floorXn : {in int_num, {morph floor : x / x ^+ n}}. Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKfloor. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorX
real_floor_ge0x : x \is real_num -> (0 <= floor x) = (0 <= x). Proof. by move=> ?; rewrite real_floor_ge_int. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_ge0
floor_lt0x : (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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor_lt0
real_floor_le0x : x \is real_num -> (floor x <= 0) = (x < 1). Proof. by move=> ?; rewrite -ltzD1 add0r real_floor_lt_int. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_floor_le0
floor_gt0x : (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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor_gt0
floor_neq0x : (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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floor_neq0
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorpK
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	floorpP
real_ceil_itvx : 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_itv
real_ceilB1_ltx : x \is real_num -> (ceil x - 1)%:~R < x. Proof. by case/real_ceil_itv/andP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceilB1_lt
real_ceil_gex : x \is real_num -> x <= (ceil x)%:~R. Proof. by case/real_ceil_itv/andP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_ge
ceil_defx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil_def
real_ceil_le_intx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_le_int
real_ceil_gt_intx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_gt_int
real_ceil_eqx 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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_eq
le_ceil: {homo ceil : x y / x <= y}. Proof. by move=> x y lexy; rewrite !ceilNfloor lerN2 le_floor ?lerN2. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	le_ceil
intrKceil: cancel intr ceil. Proof. by move=> m; rewrite ceilNfloor -intrN intrKfloor opprK. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrKceil
intrEceilx : x \is a int_num = ((ceil x)%:~R == x). Proof. by rewrite -rpredN intrEfloor -eqr_oppLR -intrN -ceilNfloor. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	intrEceil
ceilK: {in int_num, cancel ceil intr}. Proof. by move=> z; rewrite intrEceil => /eqP. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceilK
ceil0: ceil 0 = 0. Proof. exact: intrKceil 0. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil0
ceil1: ceil 1 = 1. Proof. exact: intrKceil 1. Qed. #[local] Hint Resolve ceil0 ceil1 : core.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil1
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceilDzr
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	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceilDrz
ceilN: {in int_num, {morph ceil : x / - x}}. Proof. by move=> ? ?; rewrite !ceilNfloor !opprK floorN. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceilN
ceilM: {in int_num &, {morph ceil : x y / x * y}}. Proof. by move=> _ _ /intrP[m1 ->] /intrP[m2 ->]; rewrite -rmorphM !intrKceil. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceilM
ceilXn : {in int_num, {morph ceil : x / x ^+ n}}. Proof. by move=> _ /intrP[m ->]; rewrite -rmorphXn !intrKceil. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceilX
real_ceil_ge0x : x \is real_num -> (0 <= ceil x) = (-1 < x). Proof. by move=> ?; rewrite ceilNfloor oppr_ge0 real_floor_le0 ?realN 1?ltrNl. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_ge0
ceil_lt0x : (ceil x < 0) = (x <= -1). Proof. by rewrite ceilNfloor oppr_lt0 floor_gt0 lerNr. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil_lt0
real_ceil_le0x : x \is real_num -> (ceil x <= 0) = (x <= 0). Proof. by move=> ?; rewrite real_ceil_le_int. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	real_ceil_le0
ceil_gt0x : (ceil x > 0) = (x > 0). Proof. by rewrite ceilNfloor oppr_gt0 floor_lt0 oppr_lt0. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil_gt0
ceil_neq0x : (ceil x != 0) = (x <= -1) \|\| (x > 0). Proof. by rewrite ceilNfloor oppr_eq0 floor_neq0 oppr_lt0 lerNr orbC. Qed.	Lemma	algebra	[ "From HB Require Import structures", "From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice", "From mathcomp Require Import fintype bigop order ssralg poly ssrnum ssrint" ]	algebra/archimedean.v	ceil_neq0

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Coq-MathComp

Structured dataset from the Mathematical Components library (MathComp) for Coq.

Schema

Column	Type	Description
fact	string	Declaration body
type	string	Lemma, Definition, HB.structure, HB.mixin, Canonical, etc.
library	string	Module (algebra, boot, fingroup, character, etc.)
imports	list	Require/Import statements
filename	string	Source file path
symbolic_name	string	Declaration identifier

Statistics

By Type

Type	Count
Lemma	14,917
Definition	2,850
Notation	808
Canonical	425
HB.instance	247
Fixpoint	138
Coercion	116
Variant	104
Theorem	63
HB.structure	51
HB.mixin	50

By Library

Library	Count
algebra	7,455
boot	4,591
fingroup	1,995
character	1,864
solvable	1,747
order	1,197
field	1,048

About MathComp

Mathematical Components is a library of formalized mathematics for Coq, including algebra, number theory, and finite group theory. It uses the SSReflect proof language and Hierarchy Builder (HB) for structure definitions.