phanerozoic/Coq-Analysis · Datasets at Hugging Face

fact stringlengths 9 12.1k	type stringclasses 25 values	library stringclasses 6 values	imports listlengths 0 15	filename stringclasses 105 values	symbolic_name stringlengths 1 50
continuity_pt_nbhs (f : R -> R) x : continuity_pt f x <-> forall eps : {posnum R}, nbhs x (fun u => `\|f u - f x\| < eps%:num). Proof. split=> [fcont e\|fcont _/RltP/posnumP[e]]; last first. have [_/posnumP[d] xd_fxe] := fcont e. exists d%:num; split; first by apply/RltP; have := [gt0 of d%:num]. by move=> y [_ /RltP yxd]; apply/RltP/xd_fxe; rewrite /= distrC. have /RltP egt0 := [gt0 of e%:num]. have [_ [/RltP/posnumP[d] dx_fxe]] := fcont e%:num egt0. exists d%:num => //= y xyd; case: (eqVneq x y) => [->\|xney]. by rewrite subrr normr0. apply/RltP/dx_fxe; split; first by split=> //; apply/eqP. by have /RltP := xyd; rewrite distrC. Qed.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	continuity_pt_nbhs
continuity_pt_cvg (f : R -> R) (x : R) : continuity_pt f x <-> {for x, continuous f}. Proof. eapply iff_trans; first exact: continuity_pt_nbhs. apply iff_sym. have FF : Filter (f @ x)%classic. by typeclasses eauto. (by apply fmap_filter; apply: @filter_filter' (locally_filter _).) case: (@fcvg_ballP _ _ (f @ x)%classic FF (f x)) => {FF}H1 H2. (* TODO: in need for lemmas and/or refactoring of already existing lemmas (ball vs. Rabs) *) split => [{H2} - /H1 {}H1 eps\|{H1} H]. - have {H1} [//\|_/posnumP[x0] Hx0] := H1 eps%:num. exists x0%:num => //= Hx0' /Hx0 /=. by rewrite /= distrC; apply. - apply H2 => _ /posnumP[eps]; move: (H eps) => {H} [_ /posnumP[x0] Hx0]. exists x0%:num => //= y /Hx0 /= {}Hx0. by rewrite /ball /= distrC. Qed.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	continuity_pt_cvg
continuity_ptE (f : R -> R) (x : R) : continuity_pt f x <-> {for x, continuous f}. Proof. exact: continuity_pt_cvg. Qed. Local Open Scope classical_set_scope.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	continuity_ptE
continuity_pt_cvg' f x : continuity_pt f x <-> f @ x^' --> f x. Proof. by rewrite continuity_ptE continuous_withinNx. Qed.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	continuity_pt_cvg'
continuity_pt_dnbhs f x : continuity_pt f x <-> forall eps, 0 < eps -> x^' (fun u => `\|f x - f u\| < eps). Proof. by rewrite continuity_pt_cvg' -filter_fromP cvg_ballP -filter_fromP. Qed.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	continuity_pt_dnbhs
nbhs_pt_comp (P : R -> Prop) (f : R -> R) (x : R) : nbhs (f x) P -> continuity_pt f x -> \near x, P (f x). Proof. by move=> Lf /continuity_pt_cvg; apply. Qed.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	nbhs_pt_comp
RexpE (x : R) : Rtrigo_def.exp x = expR x. Proof. apply/esym; rewrite /exp /exist_exp; case: Alembert_C3 => y. rewrite /Pser /infinite_sum /= => exp_ub. rewrite /expR /exp_coeff /series/=; apply: (@cvg_lim R^o) => //. rewrite -cvg_shiftS /=; apply/cvgrPdist_lt => /= e /RltP /exp_ub[N Nexp_ub]. near=> n. have nN : (n >= N)%coq_nat by apply/ssrnat.leP; near: n; exact: nbhs_infty_ge. move: Nexp_ub => /(_ _ nN) /[!RdistE] /RltP /=. rewrite distrC sum_f_R0E; congr (`\| _ - _ \| < e). by apply: eq_bigr=> k _; rewrite RinvE RpowE mulrC factE INRE. Unshelve. all: by end_near. Qed.	Lemma	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	RexpE
RexpE := RexpE.RexpE.	Definition	analysis_stdlib	[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]	analysis_stdlib/Rstruct_topology.v	RexpE
functional_extensionality_dep : forall (A : Type) (B : A -> Type) (f g : forall x : A, B x), (forall x : A, f x = g x) -> f = g.	Axiom	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	functional_extensionality_dep
propositional_extensionality : forall P Q : Prop, P <-> Q -> P = Q.	Axiom	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	propositional_extensionality
constructive_indefinite_description : forall (A : Type) (P : A -> Prop), (exists x : A, P x) -> {x : A \| P x}.	Axiom	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	constructive_indefinite_description
cid := constructive_indefinite_description.	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	cid
cid2 (A : Type) (P Q : A -> Prop) : (exists2 x : A, P x & Q x) -> {x : A \| P x & Q x}. Proof. move=> PQA; suff: {x \| P x /\ Q x} by move=> [a [*]]; exists a. by apply: cid; case: PQA => x; exists x. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	cid2
existT_inj1 (T : Type) (P : T -> Type) (x y : T) (Px : P x) (Py : P y) : existT P x Px = existT P y Py -> x = y. Proof. by case. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	existT_inj1
existT_inj2 (T : eqType) (P : T -> Type) (x : T) (Px1 Px2 : P x) : existT P x Px1 = existT P x Px2 -> Px1 = Px2. Proof. apply: internal_Eqdep_dec.inj_pair2_eq_dec => y z. by have [\|/eqP] := eqVneq y z; [left\|right]. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	existT_inj2
surjective_existT (T : Type) (P : T -> Type) (p : {x : T & P x}): existT [eta P] (projT1 p) (projT2 p) = p. Proof. by case: p. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	surjective_existT
mextensionality := { _ : forall (P Q : Prop), (P <-> Q) -> (P = Q); _ : forall {T U : Type} (f g : T -> U), (forall x, f x = g x) -> f = g; }.	Record	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	mextensionality
extensionality : mextensionality. Proof. split. - exact: propositional_extensionality. - by move=> T U f g; apply: functional_extensionality_dep. Qed.	Fact	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	extensionality
propext (P Q : Prop) : (P <-> Q) -> (P = Q). Proof. by have [propext _] := extensionality; apply: propext. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	propext
eqProp := apply: propext; split.	Ltac	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eqProp
funext {T U : Type} (f g : T -> U) : (f =1 g) -> f = g. Proof. by case: extensionality=> _; apply. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funext
propeqE (P Q : Prop) : (P = Q) = (P <-> Q). Proof. by apply: propext; split=> [->\|/propext]. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	propeqE
propeqP (P Q : Prop) : (P = Q) <-> (P <-> Q). Proof. by rewrite propeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	propeqP
funeqE {T U : Type} (f g : T -> U) : (f = g) = (f =1 g). Proof. by rewrite propeqE; split=> [->//\|/funext]. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funeqE
funeq2E {T U V : Type} (f g : T -> U -> V) : (f = g) = (f =2 g). Proof. by rewrite propeqE; split=> [->//\|?]; rewrite funeqE=> x; rewrite funeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funeq2E
funeq3E {T U V W : Type} (f g : T -> U -> V -> W) : (f = g) = (forall x y z, f x y z = g x y z). Proof. by rewrite propeqE; split=> [->//\|?]; rewrite funeq2E=> x y; rewrite funeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funeq3E
funeqP {T U : Type} (f g : T -> U) : (f = g) <-> (f =1 g). Proof. by rewrite funeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funeqP
funeq2P {T U V : Type} (f g : T -> U -> V) : (f = g) <-> (f =2 g). Proof. by rewrite funeq2E. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funeq2P
funeq3P {T U V W : Type} (f g : T -> U -> V -> W) : (f = g) <-> (forall x y z, f x y z = g x y z). Proof. by rewrite funeq3E. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	funeq3P
predeqE {T} (P Q : T -> Prop) : (P = Q) = (forall x, P x <-> Q x). Proof. by rewrite propeqE; split=> [->//\|?]; rewrite funeqE=> x; rewrite propeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	predeqE
predeq2E {T U} (P Q : T -> U -> Prop) : (P = Q) = (forall x y, P x y <-> Q x y). Proof. by rewrite propeqE; split=> [->//\|?]; rewrite funeq2E=> ??; rewrite propeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	predeq2E
predeq3E {T U V} (P Q : T -> U -> V -> Prop) : (P = Q) = (forall x y z, P x y z <-> Q x y z). Proof. by rewrite propeqE; split=> [->//\|?]; rewrite funeq3E=> ???; rewrite propeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	predeq3E
predeqP {T} (A B : T -> Prop) : (A = B) <-> (forall x, A x <-> B x). Proof. by rewrite predeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	predeqP
predeq2P {T U} (P Q : T -> U -> Prop) : (P = Q) <-> (forall x y, P x y <-> Q x y). Proof. by rewrite predeq2E. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	predeq2P
predeq3P {T U V} (P Q : T -> U -> V -> Prop) : (P = Q) <-> (forall x y z, P x y z <-> Q x y z). Proof. by rewrite predeq3E. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	predeq3P
propT {P : Prop} : P -> P = True. Proof. by move=> p; rewrite propeqE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	propT
Prop_irrelevance (P : Prop) (x y : P) : x = y. Proof. by move: x (x) y => /propT-> [] []. Qed. #[global] Hint Resolve Prop_irrelevance : core.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	Prop_irrelevance
mclassic := { _ : forall (P : Prop), {P} + {~P}; _ : forall T, hasChoice T }.	Record	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	mclassic
choice X Y (P : X -> Y -> Prop) : (forall x, exists y, P x y) -> {f & forall x, P x (f x)}. Proof. by move=> /(_ _)/constructive_indefinite_description -/all_tag. Qed. (* Diaconescu Theorem *)	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	choice
EM P : P \/ ~ P. Proof. pose U val := fun Q : bool => Q = val \/ P. have Uex val : exists b, U val b by exists val; left. pose f val := projT1 (cid (Uex val)). pose Uf val : U val (f val) := projT2 (cid (Uex val)). have : f true != f false \/ P. have [] := (Uf true, Uf false); rewrite /U. by move=> [->\|?] [->\|?] ; do ?[by right]; left. move=> [/eqP fTFN\|]; [right=> p\|by left]; apply: fTFN. have UTF : U true = U false by rewrite predeqE /U => b; split=> _; right. rewrite /f; move: (Uex true) (Uex false); rewrite UTF => p1 p2. by congr (projT1 (cid _)). Qed.	Theorem	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	EM
pselect (P : Prop): {P} + {~P}. Proof. have : exists b, if b then P else ~ P. by case: (EM P); [exists true\|exists false]. by move=> /cid [[]]; [left\|right]. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	pselect
pselectT T : (T -> False) + T. Proof. have [/cid[]//\|NT] := pselect (exists t : T, True); first by right. by left=> t; case: NT; exists t. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	pselectT
classic : mclassic. Proof. split=> [\|T]; first exact: pselect. exists (fun (P : pred T) (n : nat) => if pselect (exists x, P x) isn't left ex then None else Some (projT1 (cid ex))) => [P n x\|P [x Px]\|P Q /funext -> //]. by case: pselect => // ex [<- ]; case: cid. by exists 0; case: pselect => // -[]; exists x. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	classic
gen_choiceMixin (T : Type) : hasChoice T. Proof. by case: classic. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	gen_choiceMixin
lem (P : Prop): P \/ ~P. Proof. by case: (pselect P); tauto. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	lem
trueE : true = True :> Prop. Proof. by rewrite propeqE; split. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	trueE
falseE : false = False :> Prop. Proof. by rewrite propeqE; split. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	falseE
propF (P : Prop) : ~ P -> P = False. Proof. by move=> p; rewrite propeqE; tauto. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	propF
eq_fun T rT (U V : T -> rT) : (forall x : T, U x = V x) -> (fun x => U x) = (fun x => V x). Proof. by move=> /funext->. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_fun
eq2_fun T1 T2 rT (U V : T1 -> T2 -> rT) : (forall x y, U x y = V x y) -> (fun x y => U x y) = (fun x y => V x y). Proof. by move=> UV; rewrite funeq2E => x y; rewrite UV. Qed. #[deprecated(since="mathcomp-analysis 1.10.0", note="renamed to `eq2_fun`.")]	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq2_fun
eq_fun2 := eq2_fun (only parsing).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_fun2
eq3_fun T1 T2 T3 rT (U V : T1 -> T2 -> T3 -> rT) : (forall x y z, U x y z = V x y z) -> (fun x y z => U x y z) = (fun x y z => V x y z). Proof. by move=> UV; rewrite funeq3E => x y z; rewrite UV. Qed. #[deprecated(since="mathcomp-analysis 1.10.0", note="renamed to `eq3_fun`.")]	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq3_fun
eq_fun3 := eq3_fun (only parsing).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_fun3
eq_forall T (U V : T -> Prop) : (forall x : T, U x = V x) -> (forall x, U x) = (forall x, V x). Proof. by move=> e; rewrite propeqE; split=> ??; rewrite (e,=^~e). Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_forall
eq2_forall T S (U V : forall x : T, S x -> Prop) : (forall x y, U x y = V x y) -> (forall x y, U x y) = (forall x y, V x y). Proof. by move=> UV; apply/eq_forall => x; exact/eq_forall. Qed. #[deprecated(since="mathcomp-analysis 1.10.0", note="renamed to `eq2_forall`.")]	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq2_forall
eq_forall2 := eq2_forall (only parsing).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_forall2
eq3_forall T S R (U V : forall (x : T) (y : S x), R x y -> Prop) : (forall x y z, U x y z = V x y z) -> (forall x y z, U x y z) = (forall x y z, V x y z). Proof. by move=> UV; apply/eq2_forall => x y; exact/eq_forall. Qed. #[deprecated(since="mathcomp-analysis 1.10.0", note="renamed to `eq3_forall`.")]	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq3_forall
eq_forall3 := eq3_forall (only parsing).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_forall3
eq_exists T (U V : T -> Prop) : (forall x : T, U x = V x) -> (exists x, U x) = (exists x, V x). Proof. by move=> e; rewrite propeqE; split=> - [] x ?; exists x; rewrite (e,=^~e). Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_exists
eq2_exists T S (U V : forall x : T, S x -> Prop) : (forall x y, U x y = V x y) -> (exists x y, U x y) = (exists x y, V x y). Proof. by move=> UV; apply/eq_exists => x; exact/eq_exists. Qed. #[deprecated(since="mathcomp-analysis 1.10.0", note="renamed to `eq2_exists`.")]	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq2_exists
eq_exists2 := eq2_exists (only parsing).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_exists2
eq3_exists T S R (U V : forall (x : T) (y : S x), R x y -> Prop) : (forall x y z, U x y z = V x y z) -> (exists x y z, U x y z) = (exists x y z, V x y z). Proof. by move=> UV; apply/eq2_exists => x y; exact/eq_exists. Qed. #[deprecated(since="mathcomp-analysis 1.10.0", note="renamed to `eq3_exists`.")]	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq3_exists
eq_exists3 := eq3_exists (only parsing).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_exists3
eq_exist T (P : T -> Prop) (s t : T) (p : P s) (q : P t) : s = t -> exist P s p = exist P t q. Proof. by move=> st; case: _ / st in q *; apply/congr1. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_exist
forall_swap T S (U : forall (x : T) (y : S), Prop) : (forall x y, U x y) = (forall y x, U x y). Proof. by rewrite propeqE; split. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	forall_swap
exists_swap T S (U : forall (x : T) (y : S), Prop) : (exists x y, U x y) = (exists y x, U x y). Proof. by rewrite propeqE; split => -[x [y]]; exists y, x. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	exists_swap
reflect_eq (P : Prop) (b : bool) : reflect P b -> P = b. Proof. by rewrite propeqE; exact: rwP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	reflect_eq
asbool (P : Prop) := if pselect P then true else false.	Definition	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asbool
asboolE (P : Prop) : `[<P>] = P :> Prop. Proof. by rewrite propeqE /asbool; case: pselect; split. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolE
asboolP (P : Prop) : reflect P `[<P>]. Proof. by apply: (equivP idP); rewrite asboolE. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolP
asboolb (b : bool) : `[< b >] = b. Proof. by apply/asboolP/idP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolb
asboolPn (P : Prop) : reflect (~ P) (~~ `[<P>]). Proof. by rewrite /asbool; case: pselect=> h; constructor. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolPn
asboolW (P : Prop) : `[<P>] -> P. Proof. by case: asboolP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolW
orW A B : A \/ B -> A + B. Proof. have [\|NA] := asboolP A; first by left. have [\|NB] := asboolP B; first by right. by move=> AB; exfalso; case: AB. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	orW
or3W A B C : [\/ A, B \| C] -> A + B + C. Proof. have [\|NA] := asboolP A; first by left; left. have [\|NB] := asboolP B; first by left; right. have [\|NC] := asboolP C; first by right. by move=> ABC; exfalso; case: ABC. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	or3W
or4W A B C D : [\/ A, B, C \| D] -> A + B + C + D. Proof. have [\|NA] := asboolP A; first by left; left; left. have [\|NB] := asboolP B; first by left; left; right. have [\|NC] := asboolP C; first by left; right. have [\|ND] := asboolP D; first by right. by move=> ABCD; exfalso; case: ABCD. Qed. (* Shall this be a coercion ?*)	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	or4W
asboolT (P : Prop) : P -> `[<P>]. Proof. by case: asboolP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolT
asboolF (P : Prop) : ~ P -> `[<P>] = false. Proof. by apply/introF/asboolP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asboolF
eq_opE (T : eqType) (x y : T) : (x == y : Prop) = (x = y). Proof. by apply/propext; split=> /eqP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eq_opE
is_true_inj : injective is_true. Proof. by move=> [] []; rewrite ?(trueE, falseE) ?propeqE; tauto. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	is_true_inj
gen_eq (T : Type) (u v : T) := `[<u = v>].	Definition	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	gen_eq
gen_eqP (T : Type) : Equality.axiom (@gen_eq T). Proof. by move=> x y; apply: (iffP (asboolP _)). Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	gen_eqP
gen_eqMixin (T : Type) : hasDecEq T := hasDecEq.Build T (@gen_eqP T). HB.instance Definition _ (T : Type) (T' : T -> eqType) := gen_eqMixin (forall t : T, T' t). HB.instance Definition _ (T : Type) (T' : T -> choiceType) := gen_choiceMixin (forall t : T, T' t). HB.instance Definition _ := gen_eqMixin Prop. HB.instance Definition _ := gen_choiceMixin Prop.	Definition	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	gen_eqMixin
classicType := T. HB.instance Definition _ := gen_eqMixin classicType. HB.instance Definition _ := gen_choiceMixin classicType.	Definition	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	classicType
eclassicType : Type := T. HB.instance Definition _ := Equality.copy eclassicType T. HB.instance Definition _ := gen_choiceMixin eclassicType.	Definition	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eclassicType
canonical_of T U (sort : U -> T) := forall (G : T -> Type), (forall x', G (sort x')) -> forall x, G x.	Definition	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	canonical_of
canonical_ sort := (@canonical_of _ _ sort).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	canonical_
canonical T E := (@canonical_of T E id).	Notation	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	canonical
canon T U (sort : U -> T) : (forall x, exists y, sort y = x) -> canonical_ sort. Proof. by move=> + G Gs x => /(_ x)/cid[x' <-]. Qed. Arguments canon {T U sort} x.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	canon
Peq : canonical Type eqType. Proof. by apply: canon => T; exists {classic T}. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	Peq
Pchoice : canonical Type choiceType. Proof. by apply: canon => T; exists {classic T}. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	Pchoice
eqPchoice : canonical eqType choiceType. Proof. by apply: canon => T; exists {eclassic T}; case: T => //= T [?]//. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	eqPchoice
not_True : (~ True) = False. Proof. exact/propext. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	not_True
not_False : (~ False) = True. Proof. by apply/propext; split=> _. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	not_False
asbool_equiv_eq {P Q : Prop} : (P <-> Q) -> `[<P>] = `[<Q>]. Proof. by rewrite -propeqE => ->. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asbool_equiv_eq
asbool_equiv_eqP {P Q : Prop} b : reflect Q b -> (P <-> Q) -> `[<P>] = b. Proof. by move=> Q_b [PQ QP]; apply/asboolP/Q_b. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asbool_equiv_eqP
asbool_equiv {P Q : Prop} : (P <-> Q) -> (`[<P>] <-> `[<Q>]). Proof. by move/asbool_equiv_eq->. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asbool_equiv
asbool_eq_equiv {P Q : Prop} : `[<P>] = `[<Q>] -> (P <-> Q). Proof. by move=> eq; split=> /asboolP; rewrite (eq, =^~ eq) => /asboolP. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	asbool_eq_equiv
and_asboolP (P Q : Prop) : reflect (P /\ Q) (`[< P >] && `[< Q >]). Proof. apply: (iffP idP); first by case/andP => /asboolP p /asboolP q. by case=> /asboolP-> /asboolP->. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	and_asboolP
and3_asboolP (P Q R : Prop) : reflect [/\ P, Q & R] [&& `[< P >], `[< Q >] & `[< R >]]. Proof. apply: (iffP idP); first by case/and3P => /asboolP p /asboolP q /asboolP r. by case => /asboolP -> /asboolP -> /asboolP ->. Qed.	Lemma	classical	[ "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect.", "From mathcomp Require Import mathcomp_extra.", "From mathcomp Require internal_Eqdep_dec." ]	classical/boolp.v	and3_asboolP

continuity_pt_nbhs (f : R -> R) x : continuity_pt f x <-> forall eps : {posnum R}, nbhs x (fun u => `|f u - f x| < eps%:num). Proof. split=> [fcont e|fcont _/RltP/posnumP[e]]; last first. have [_/posnumP[d] xd_fxe] := fcont e. exists d%:num; split; first by apply/RltP; have := [gt0 of d%:num]. by move=> y [_ /RltP yxd]; apply/RltP/xd_fxe; rewrite /= distrC. have /RltP egt0 := [gt0 of e%:num]. have [_ [/RltP/posnumP[d] dx_fxe]] := fcont e%:num egt0. exists d%:num => //= y xyd; case: (eqVneq x y) => [->|xney]. by rewrite subrr normr0. apply/RltP/dx_fxe; split; first by split=> //; apply/eqP. by have /RltP := xyd; rewrite distrC. Qed.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

continuity_pt_nbhs

continuity_pt_cvg (f : R -> R) (x : R) : continuity_pt f x <-> {for x, continuous f}. Proof. eapply iff_trans; first exact: continuity_pt_nbhs. apply iff_sym. have FF : Filter (f @ x)%classic. by typeclasses eauto. (*by apply fmap_filter; apply: @filter_filter' (locally_filter _).*) case: (@fcvg_ballP _ _ (f @ x)%classic FF (f x)) => {FF}H1 H2. (* TODO: in need for lemmas and/or refactoring of already existing lemmas (ball vs. Rabs) *) split => [{H2} - /H1 {}H1 eps|{H1} H]. - have {H1} [//|_/posnumP[x0] Hx0] := H1 eps%:num. exists x0%:num => //= Hx0' /Hx0 /=. by rewrite /= distrC; apply. - apply H2 => _ /posnumP[eps]; move: (H eps) => {H} [_ /posnumP[x0] Hx0]. exists x0%:num => //= y /Hx0 /= {}Hx0. by rewrite /ball /= distrC. Qed.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

continuity_pt_cvg

continuity_ptE (f : R -> R) (x : R) : continuity_pt f x <-> {for x, continuous f}. Proof. exact: continuity_pt_cvg. Qed. Local Open Scope classical_set_scope.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

continuity_ptE

continuity_pt_cvg' f x : continuity_pt f x <-> f @ x^' --> f x. Proof. by rewrite continuity_ptE continuous_withinNx. Qed.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

continuity_pt_cvg'

continuity_pt_dnbhs f x : continuity_pt f x <-> forall eps, 0 < eps -> x^' (fun u => `|f x - f u| < eps). Proof. by rewrite continuity_pt_cvg' -filter_fromP cvg_ballP -filter_fromP. Qed.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

continuity_pt_dnbhs

nbhs_pt_comp (P : R -> Prop) (f : R -> R) (x : R) : nbhs (f x) P -> continuity_pt f x -> \near x, P (f x). Proof. by move=> Lf /continuity_pt_cvg; apply. Qed.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

nbhs_pt_comp

RexpE (x : R) : Rtrigo_def.exp x = expR x. Proof. apply/esym; rewrite /exp /exist_exp; case: Alembert_C3 => y. rewrite /Pser /infinite_sum /= => exp_ub. rewrite /expR /exp_coeff /series/=; apply: (@cvg_lim R^o) => //. rewrite -cvg_shiftS /=; apply/cvgrPdist_lt => /= e /RltP /exp_ub[N Nexp_ub]. near=> n. have nN : (n >= N)%coq_nat by apply/ssrnat.leP; near: n; exact: nbhs_infty_ge. move: Nexp_ub => /(_ _ nN) /[!RdistE] /RltP /=. rewrite distrC sum_f_R0E; congr (`| _ - _ | < e). by apply: eq_bigr=> k _; rewrite RinvE RpowE mulrC factE INRE. Unshelve. all: by end_near. Qed.

Lemma

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

RexpE

RexpE := RexpE.RexpE.

Definition

analysis_stdlib

[ "Require Import Rdefinitions Raxioms RIneq Rbasic_fun Zwf.", "Require Import Epsilon FunctionalExtensionality Ranalysis1 Rsqrt_def.", "Require Import Rtrigo1 Reals.", "From HB Require Import structures.", "From mathcomp Require Import all_ssreflect ssralg ssrnum archimedean.", "From mathcomp Require Import boolp classical_sets reals interval_inference.", "From mathcomp Require Export Rstruct.", "From mathcomp Require Import topology.", "From mathcomp Require normedtype sequences." ]

analysis_stdlib/Rstruct_topology.v

RexpE

functional_extensionality_dep : forall (A : Type) (B : A -> Type) (f g : forall x : A, B x), (forall x : A, f x = g x) -> f = g.

Axiom

classical