Module mathcomp.analysis.ereal

From HB Require Import structures.
From mathcomp Require Import boot order algebra interval_inference finmap.
#[warning="-warn-library-file-internal-analysis"]
From mathcomp Require Import unstable.
From mathcomp Require Import boolp classical_sets functions fsbigop cardinality
set_interval.
From mathcomp Require Import reals.
From mathcomp Require Export constructive_ereal.
From mathcomp Require Import topology.

# Extended real numbers, classical part ($\overline{\mathbb{R}}$) This is an addition to the file constructive_ereal.v with classical logic elements. ``` (\sum_(i \in A) f i)%E == finitely supported sum, see fsbigop.v ereal_sup E == supremum of E ereal_inf E == infimum of E ereal_supremums_neq0 S == S has a supremum ``` ## Topology of extended real numbers ``` ereal_topologicalType R == topology for extended real numbers over R, a realFieldType ereal_pseudoMetricType R == pseudometric space for extended reals over R where is a realFieldType; the distance between x and y is defined by `|contract x - contract y| ``` ## Filters ``` ereal_dnbhs x == filter on extended real numbers that corresponds to the deleted neighborhood x^' if x is a real number and to predicates that are eventually true if x is +oo/-oo. ereal_nbhs x == same as ereal_dnbhs where dnbhs is replaced with nbhs. ereal_loc_seq x == sequence that converges to x in the set of extended real numbers. ```

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Import Order.TTheory GRing.Theory Num.Theory.
Import numFieldTopology.Exports.

Local Open Scope ring_scope.
Local Open Scope ereal_scope.
Local Open Scope classical_set_scope.

Lemma ( : Type) ( : realDomainType) ( : T -> \bar R) :
(forall , 0 <= c x) -> (forall , 0 <= d x) ->
a \+ b \- (c \+ d) = a \- c \+ (b \- d).

Proof.

by move=> ? ?; apply/funext=> x; rewrite addeACA oppeD ?ge0_adde_def ?inE.
Qed.

Lemma ( : nat -> set T) :
EFin @` (\bigcup_ F i) = \bigcup_ (EFin @` F i).

Proof.

rewrite eqEsubset; split => [_ [r [n _ Fnr <-]]|]; first by exists n => //; exists r.
by move=> x [n _ [r Fnr <- /=]]; exists r => //; exists n.
Qed.

Lemma ( : set T) :
EFin @` (~` A) = (~` (EFin @` A)) `\` [set -oo; +oo].

Proof.

rewrite eqEsubset; split => [_ [r Ar <-]|[r | |]].
by split => [|[]//]; apply: contra_not Ar => -[? ? [] <-].
- move=> [Ar _]; apply/not_exists2P; apply: contra_not Ar => h.
by exists r => //; have [|//] := h r; apply: contrapT.
- by move=> -[_] /not_orP[_ /=].
- by move=> -[_] /not_orP[/=].
Qed.

Local Close Scope classical_set_scope.

Notation := (\big[+%dE/0%dE]_( \in A) F%dE) :
ereal_dual_scope.
Notation := (\big[+%E/0%E]_( \in A) F%E) :
ereal_scope.

Section ERealArith.
Context { : numDomainType}.
Implicit Types x y z : \bar R.

Local Open Scope classical_set_scope.

Lemma : @abse R @^-1` [set +oo] = [set -oo; +oo].

Proof.

by rewrite predeqE => y; split ; move: y => [y//| |]//=; [right | left | case].
Qed.

Lemma : (@abse R @^-1` [set -oo])%classic = set0.

Proof.

rewrite predeqE => t; split => //=; apply/eqP.
by rewrite gt_eqF// (lt_le_trans _ (abse_ge0 t)).
Qed.

Lemma ( : R -> \bar R) ( : T -> R) :
{morph h : / (x + y)%R >-> (x + y)%E} ->
h \o (f \+ g)%R = ((h \o f) \+ (h \o g))%E.

Proof.

by move=> mh; apply/funext => t /=; rewrite mh. Qed.

Lemma ( : R -> \bar R) ( : T -> R) :
{morph h : / (- x)%R >-> (- x)%E} ->
h \o (\- f)%R = \- (h \o f)%E.

Proof.

by move=> mh; apply/funext => t /=; rewrite mh. Qed.

Lemma ( : R -> \bar R) ( : T -> R) :
{morph h : / (x - y)%R >-> (x - y)%E} ->
h \o (f \- g)%R = ((h \o f) \- (h \o g))%E.

Proof.

by move=> mh; apply/funext => t /=; rewrite mh. Qed.

Lemma ( : R -> \bar R) ( : T -> R) :
{morph h : / (x * y)%R >-> (x * y)%E} ->
h \o (k \o* f) = (fun => h k * h (f t))%E.

Proof.

by move=> mf; apply/funext => t /=; rewrite mf; rewrite muleC. Qed.

Local Close Scope classical_set_scope.

End ERealArith.

Section ERealArithTh_numDomainType.
Context { : numDomainType}.
Implicit Types (x y z : \bar R) (r : R).

Lemma : range -%E = [set: \bar R]%classic.

Proof.

by apply/seteqP; split => [//|x] _; exists (- x); rewrite ?oppeK. Qed.

Lemma ( : set (\bar R)) :
((A `<=` B) <-> (-%E @` A `<=` -%E @` B))%classic.

Proof.

split=> [AB _ [] x ? <-|AB x Ax]; first by exists x => //; exact: AB.
have /AB[y By] : ((-%E @` A) (- x))%classic by exists x.
by rewrite eqe_oppP => <-.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
(forall , P i -> 0 <= F i) -> 0 <= \sum_( \in P) F i.

Proof.

move=> PF; case: finite_supportP; rewrite ?big_nil// => X XP F0 _.
by rewrite big_seq_cond big_mkcondr sume_ge0// => i /XP/PF.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
(forall , P t -> F t <= 0) -> \sum_( \in P) F i <= 0.

Proof.

move=> PF; case: finite_supportP; rewrite ?big_nil// => X XP F0 _.
by rewrite big_seq_cond big_mkcondr sume_le0// => i /XP/PF.
Qed.

Lemma ( : choiceType) ( : I -> R) : finite_set A ->
\sum_( \in A) (F i)%:E = (\sum_( \in A) F i)%:E.

Proof.

by move=> fs; rewrite fsbig_finite//= sumEFin -fsbig_finite. Qed.

End ERealArithTh_numDomainType.

Section ERealArithTh_realDomainType.
Context { : realDomainType}.
Implicit Types (x y z u a b : \bar R) (r : R).

Lemma ( : set R) ( : R -> R) :
{in A &, {homo f : / (x <= y)%O}} ->
{in (EFin @` A)%classic &, {homo er_map f : / (x <= y)%E}}.

Proof.

move=> h x y; rewrite !inE/= => -[r Ar <-{x}] [s As <-{y}].
by rewrite !lee_fin/= => /h; apply; rewrite inE.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
0 < \sum_( \in P) F i -> exists2 , P i & 0 < F i.

Proof.

apply: contraPP => /forall2NP xNPF; rewrite le_gtF// fsume_le0// => i Pi.
by case: (xNPF i) => // /negP; case: ltP.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
\sum_( \in P) F i < 0 -> exists2 , P i & F i < 0.

Proof.

apply: contraPP => /forall2NP xNPF; rewrite le_gtF// fsume_ge0// => i Pi.
by case: (xNPF i) => // /negP; case: ltP.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
  finite_set P ->
  (forall , P i -> 0 <= F i) ->
  \sum_( \in P) F i = 0 -> forall , P i -> F i = 0.

Proof.

move=> Pfin F0 /eqP; apply: contraTP => /existsPNP[i Pi /eqP Fi0].
rewrite (fsbigD1 i)//= padde_eq0 ?F0 ?negb_and ?Fi0//.
by rewrite fsume_ge0// => j [/F0->].
Qed.

Lemma [ : choiceType] [ : set T] [ : T -> \bar R] :
  finite_set A -> finite_set B ->
  {subset A <= B} -> {in [predD B & A], forall : T, 0 <= f t}%E ->
  (\sum_( \in A) f t <= \sum_( \in B) f t)%E.

Proof.

move=> finA finB AB f0; rewrite !fsbig_finite//=; apply: lee_sum_nneg_subfset.
by apply/fsubsetP; rewrite -fset_set_sub//; apply/subsetP.
by move=> t; rewrite !inE !in_fset_set// => /f0.
Qed.

Lemma [ : choiceType] ( : set T) ( : T -> \bar R) :
finite_set I ->
(forall , I i -> a i <= b i)%E -> (\sum_( \in I) a i <= \sum_( \in I) b i)%E.

Proof.

move=> finI ab.
rewrite !fsbig_finite// big_seq [in leRHS]big_seq lee_sum //.
by move=> i; rewrite in_fset_set// inE; exact: ab.
Qed.

Lemma ( : choiceType) ( : T -> \bar R) ( : set T) :
(forall , P i -> 0 <= F i) ->
x * (\sum_( \in P) F i) = \sum_( \in P) x * F i.

Proof.

move=> F0; have [->{x}|x0] := eqVneq x 0%E.
  by rewrite mul0e big1// => ? _; rewrite mul0e.
rewrite big_seq ge0_sume_distrr.
  by move=> t; case: finite_supportP => // X XP _ _ /XP/F0.
rewrite -big_seq; apply: eq_fbigl => y.
rewrite !unlock; congr (_ \in fset_set _).
apply/seteqP; rewrite /preimage; split=> [|] z/= [Pz Fz0];
  split=> //; apply: contra_not Fz0.
by move=> /eqP; rewrite mule_eq0 (negbTE x0)/= => /eqP.
by move=> ->; rewrite mule0.
Qed.

Lemma ( : choiceType) ( : T -> \bar R) ( : set T) :
(forall , P i -> 0 <= F i) ->
(\sum_( \in P) F i) * x = \sum_( \in P) F i * x.

Proof.

move=> F0; rewrite muleC ge0_mule_fsumr//.
by apply: eq_fsbigr => i; rewrite muleC.
Qed.

End ERealArithTh_realDomainType.
Arguments lee_fsum [R T I a b].

Module .

Import DualAddTheory.

Section DualERealArithTh_numDomainType.

Local Open Scope ereal_dual_scope.

Context { : numDomainType}.

Implicit Types x y z : \bar R.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
finite_support 0%E P (\- F)%E = finite_support 0%E P F.

Proof.

rewrite /finite_support !unlock; congr fset_set; congr setI.
by rewrite seteqP; split=> x /= /eqP + /ltac:(apply/eqP); rewrite oppe_eq0.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
(\sum_( \in P) F i)%dE = (- (\sum_( \in P) (- F i)))%E.

Proof.

rewrite finite_supportNe.
apply: (big_ind2 (fun => x = - y)%E) => [|_ x _ y -> ->|i _].
- by rewrite oppe0.
- by rewrite dual_addeE !oppeK.
- by rewrite oppeK.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar^d R) :
(forall , P i -> 0 <= F i) -> 0 <= \sum_( \in P) F i.

Proof.

move=> PF; case: finite_supportP; rewrite ?big_nil// => X XP F0 _.
by rewrite big_seq_cond big_mkcondr dsume_ge0// => i /XP/PF.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar R) :
(forall , P t -> F t <= 0) -> \sum_( \in P) F i <= 0.

Proof.

move=> PF; case: finite_supportP; rewrite ?big_nil// => X XP F0 _.
by rewrite big_seq_cond big_mkcondr dsume_le0// => i /XP/PF.
Qed.

End DualERealArithTh_numDomainType.

Section DualERealArithTh_realDomainType.

Import DualAddTheory.

Local Open Scope ereal_dual_scope.

Context { : realDomainType}.
Implicit Types x y z a b : \bar^d R.

Lemma ( : choiceType) ( : set I) ( : I -> \bar^d R) :
0 < \sum_( \in P) F i -> exists2 , P i & 0 < F i.

Proof.

rewrite dual_fsumeE oppe_gt0 => /fsume_lt0[i Pi].
by rewrite oppe_lt0 => ?; exists i.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar^d R) :
\sum_( \in P) F i < 0 -> exists2 , P i & F i < 0.

Proof.

rewrite dual_fsumeE oppe_lt0 => /fsume_gt0[i Pi].
by rewrite oppe_gt0 => ?; exists i.
Qed.

Lemma ( : choiceType) ( : set I) ( : I -> \bar^d R) :
  finite_set P ->
  (forall , P i -> 0 <= F i) ->
  \sum_( \in P) F i = 0 -> forall , P i -> F i = 0.

Proof.

move=> Pfin F0 /eqP; apply: contraTP => /existsPNP[i Pi /eqP Fi0].
rewrite (fsbigD1 i)//= pdadde_eq0 ?F0 ?negb_and ?Fi0//.
by rewrite dfsume_ge0// => j [/F0->].
Qed.

Lemma ( : choiceType) ( : T -> \bar^d R) ( : set T) :
(forall : T, F i <= 0) -> x * (\sum_( \in P) F i) = \sum_( \in P) x * F i.

Proof.

move=> Fge0.
rewrite !dual_fsumeE muleN ge0_mule_fsumr; first by move=> ?; rewrite oppe_ge0.
rewrite (eq_bigr _ (fun _ _ => muleN _ _)).
by rewrite (eq_finite_support _ (fun _ => muleN _ _)).
Qed.

Lemma ( : choiceType) ( : T -> \bar^d R) ( : set T) :
(forall : T, F i <= 0) -> (\sum_( \in P) F i) * x = \sum_( \in P) F i * x.

Proof.

move=> F0; rewrite muleC le0_mule_dfsumr//.
by apply: eq_fsbigr => i; rewrite muleC.
Qed.

End DualERealArithTh_realDomainType.

End DualAddTheoryNumDomain.

Module .
Export ConstructiveDualAddTheory.
Export DualAddTheoryNumDomain.
End DualAddTheory.

. () := isPointed.Build (\bar R) 0%E.

Lemma { : Type} ( : set aT) { : numDomainType}
( : aT -> \bar R) : f = (f \_ (~` D)) \+ (f \_ D).

Proof.

by apply/funext => x; rewrite !patchE in_setC; case: ifPn => [xD|/negPn ->];
rewrite ?(negbTE xD) ?(addr0,add0r).
Qed.

Section ereal_supremum.
Variable : realFieldType.
Local Open Scope classical_set_scope.
Implicit Types (S : set (\bar R)) (x y : \bar R).

Lemma : ubound [set: \bar R] = [set +oo].

Proof.

apply/seteqP; split => /= [x Tx|x -> ?]; last by rewrite leey.
by apply/eqP; rewrite eq_le leey /= Tx.
Qed.

Lemma : ubound S +oo.

Proof.

by apply/ubP=> x _; rewrite leey. Qed.

Lemma : ubound S -oo -> S = set0 \/ S = [set -oo].

Proof.

have [->|/set0P[x Sx] Snoo] := eqVneq S set0; first by left.
right; rewrite predeqE => y; split => [/Snoo|->{y}].
by rewrite leeNy_eq => /eqP ->.
by have := Snoo _ Sx; rewrite leeNy_eq => /eqP <-.
Qed.

Lemma : supremums [set: \bar R] = [set +oo].

Proof.

rewrite /supremums uboundT.
by apply/seteqP; split=> [x []//|x -> /=]; split => // y ->.
Qed.

Lemma : supremums (@set0 (\bar R)) -oo.

Proof.

by split; [exact/ubP | apply/lbP=> y _; rewrite leNye]. Qed.

Lemma : supremum -oo [set: \bar R] = +oo.

Proof.

rewrite /supremum (negbTE setT0) supremumsT.
by case: xgetP => // /(_ +oo)/= /eqP; rewrite eqxx.
Qed.

Lemma : S +oo -> supremum x0 S = +oo.

Proof.

move=> Spoo; rewrite /supremum ifF; first by apply/eqP => S0; rewrite S0 in Spoo.
have sSoo : supremums S +oo.
  split; first exact: ereal_ub_pinfty.
  by move=> /= y /(_ _ Spoo); rewrite leye_eq => /eqP ->.
case: xgetP.
  by move=> _ -> sSxget; move: (is_subset1_supremums sSoo sSxget).
by move=> /(_ +oo); exact: contra_notP.
Qed.

Definition := supremum -oo S.

Definition := - ereal_sup (-%E @` S).

Lemma : ereal_sup set0 = -oo

Proof.

exact: supremum0. Qed.

Lemma : ereal_sup [set: \bar R] = +oo.

Proof.

by rewrite /ereal_sup/= supremumT. Qed.

Lemma : ereal_sup [set x] = x

Proof.

exact: supremum1. Qed.

Lemma : ereal_inf set0 = +oo.

Proof.

by rewrite /ereal_inf image_set0 ereal_sup0. Qed.

Lemma : ereal_inf [set: \bar R] = -oo.

Proof.

by rewrite /ereal_inf range_oppe/= ereal_supT. Qed.

Lemma : ereal_inf [set x] = x.

Proof.

by rewrite /ereal_inf image_set1 ereal_sup1 oppeK. Qed.

Lemma : ubound S M -> ereal_sup S <= M.

Proof.

rewrite /ereal_sup /supremum; case: ifPn => [/eqP ->|]; first by rewrite leNye.
- by move=> _ SM; case: xgetP => [_ -> [_]| _] /=; [exact |rewrite leNye].
Qed.

Lemma : lbound S M -> M <= ereal_inf S.

Proof.

move=> SM; rewrite /ereal_inf leeNr; apply: ge_ereal_sup => x [y Sy <-{x}].
by rewrite leeNl oppeK; exact: SM.
Qed.

Lemma ( : R) : (0 < e)%R ->
ereal_sup S \is a fin_num -> exists2 , S x & (ereal_sup S - e%:E < x).

Proof.

move=> e0 Sr; have : ~ ubound S (ereal_sup S - e%:E).
by move/ge_ereal_sup; apply/negP; rewrite -ltNge lteBlDr// lteDl// lte_fin.
move/asboolP; rewrite asbool_neg; case/existsp_asboolPn => /= x.
by rewrite not_implyE => -[? ?]; exists x => //; rewrite ltNge; apply/negP.
Qed.

Lemma ( : R) : (0 < e)%R ->
ereal_inf S \is a fin_num -> exists2 , S x & (x < ereal_inf S + e%:E).

Proof.

move=> e0; rewrite fin_numN => /(ub_ereal_sup_adherent e0)[x []].
move=> y Sy <-; rewrite -lteNr => /lt_le_trans ex; exists y => //.
by apply: ex; rewrite fin_num_oppeD// oppeK.
Qed.

Lemma : x < ereal_sup S -> exists2 , S y & x < y.

Proof.

rewrite not_exists2P => + g; apply/negP; rewrite -leNgt.
by apply: ge_ereal_sup => y Sy; move: (g y) => [//|/negP]; rewrite leNgt.
Qed.

Lemma : ereal_inf S < x -> exists2 , S y & y < x.

Proof.

rewrite lteNl => /ereal_sup_gt[_ [y Sy <-]].
by rewrite lteNl oppeK => xlty; exists y.
Qed.

Lemma : ereal_inf S = - ereal_sup (-%E @` S).

Proof.

by []. Qed.

Lemma : ereal_sup (-%E @` S) = - ereal_inf S.

Proof.

by rewrite oppeK. Qed.

Lemma : ereal_inf (-%E @` S) = - ereal_sup S.

Proof.

rewrite /ereal_inf; congr (- ereal_sup _) => /=.
by rewrite image_comp -[RHS]image_id; apply: eq_imagel => /= ?; rewrite oppeK.
Qed.

Lemma : ereal_sup S = - ereal_inf (-%E @` S).

Proof.

by rewrite ereal_infN oppeK. Qed.

End ereal_supremum.
#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `ge_ereal_sup`.")]
Notation := ge_ereal_sup (only parsing).

Section ereal_supremum_realType.
Variable : realType.
Local Open Scope classical_set_scope.
Implicit Types S : set (\bar R).
Implicit Types x : \bar R.

Let : R := if x is r%:E then r else r0.

Lemma : supremums S !=set0.

Proof.

have [->|Snoo] := eqVneq S [set -oo].
  by exists -oo; split; [rewrite ub_set1 |exact: lb_ub_refl].
have [->|S0] := eqVneq S set0.
  by exists -oo; exact: ereal_supremums_set0_ninfty.
have [Spoo|Spoo] := pselect (S +oo).
  by exists +oo; split; [apply/ereal_ub_pinfty | apply/lbP => /= y /ubP; apply].
have [r Sr] : exists , S r%:E.
  move: S0 => /set0P[] [r Sr| // |Snoo1]; first by exists r.
  apply/not_existsP => nS; move/negP : Snoo; apply.
  by apply/eqP; rewrite predeqE => -[] // r; split => // /nS.
set U := fine_def r @` S.
have [|] := eqVneq (ubound U) set0.
  rewrite -subset0 => U0; exists +oo.
  split; [exact/ereal_ub_pinfty | apply/lbP => /= -[r0 /ubP Sr0|//|]].
  - suff : ubound U r0 by move/U0.
    by apply/ubP=> _ -[] [r1 Sr1 <-|//| /= _ <-]; rewrite -lee_fin; exact: Sr0.
  - by move/ereal_ub_ninfty => [|]; by [move/eqP : S0|move/eqP : Snoo].
set u : R := sup U.
exists u%:E; split; last first.
  apply/lbP=> -[r0 /ubP Sr0| |].
  - rewrite lee_fin; apply: ge_sup; first by exists r, r%:E.
    by apply/ubP => _ -[[r2 ? <-| // | /= _ <-]]; rewrite -lee_fin; exact: Sr0.
  - by rewrite leey.
  - by move/ereal_ub_ninfty=> [|/eqP //]; [move/eqP : S0|rewrite (negbTE Snoo)].
apply/ubP => -[r0 Sr0|//|_]; last by rewrite leNye.
rewrite lee_fin.
suff : has_sup U by move/sup_upper_bound/ubP; apply; exists r0%:E.
split; first by exists r0, r0%:E.
exists u; apply/ubP => y; move=> [] y' Sy' <-{y}.
have : has_sup U by split; [exists r, r%:E | exact/set0P].
move/sup_upper_bound/ubP; apply.
by case: y' Sy' => [r1 /= Sr1 | // | /= _]; [exists r1%:E | exists r%:E].
Qed.

Lemma : ubound S (ereal_sup S).

Proof.

move=> y Sy; rewrite /ereal_sup /supremum ifF.
by apply/eqP; rewrite predeqE => /(_ y)[+ _]; exact.
case: xgetP => /=; first by move=> _ -> -[] /ubP geS _; apply: geS.
by case: (ereal_supremums_neq0 S) => /= x0 Sx0 /(_ x0).
Qed.

Lemma : S +oo -> ereal_sup S = +oo.

Proof.

by move=> Soo; apply/eqP; rewrite eq_le leey/=; exact: ereal_sup_ubound.
Qed.

Lemma : (exists2 , S y & x <= y) -> x <= ereal_sup S.

Proof.

by move=> [y Sy] /le_trans; apply; exact: ereal_sup_ubound. Qed.

Lemma : ereal_sup S = -oo <-> S `<=` [set -oo].

Proof.

split.
by move=> supS [r /ereal_sup_ubound|/ereal_sup_ubound|//]; rewrite supS.
by move=> /(@subset_set1 _ S) [] ->; [exact: ereal_sup0|exact: ereal_sup1].
Qed.

Lemma : lbound S (ereal_inf S).

Proof.

by move=> x Sx; rewrite /ereal_inf leeNl; apply: ereal_sup_ubound; exists x.
Qed.

Lemma : (exists2 , S y & y <= x) -> ereal_inf S <= x.

Proof.

by move=> [y Sy]; apply: le_trans; exact: ereal_inf_lbound. Qed.

Lemma : ereal_inf S = +oo <-> S `<=` [set +oo].

Proof.

rewrite eqe_oppLRP oppe_subset image_set1; exact: ereal_sup_ninfty. Qed.

Lemma : {homo @ereal_sup R : / A `<=` B >-> A <= B}.

Proof.

by move=> A B AB; apply: ge_ereal_sup => x Ax; exact/ereal_sup_ubound/AB.
Qed.

Lemma : {homo @ereal_inf R : / A `<=` B >-> B <= A}.

Proof.

by move=> A B AB; apply: le_ereal_inf_tmp => x Bx; exact/ereal_inf_lbound/AB.
Qed.

Lemma ( : set R) : ~ has_ubound A ->
A !=set0 -> ereal_sup (EFin @` A) = +oo%E.

Proof.

move=> + A0; apply: contra_notP => /eqP; rewrite -ltey => Aoo.
exists (fine (ereal_sup (EFin @` A))) => x Ax.
rewrite -lee_fin -(@fineK _ x%:E)// lee_fin fine_le//; last first.
by apply: ereal_sup_ubound => /=; exists x.
rewrite fin_numE// -ltey Aoo andbT.
apply/eqP => /ereal_sup_ninfty/(_ x%:E).
by have /[swap] /[apply]: (EFin @` A) x%:E by exists x.
Qed.

Lemma ( : set R) :
has_ubound A -> A !=set0 -> ereal_sup (EFin @` A) = (sup A)%:E.

Proof.

move=> has_ubA A0; apply/eqP; rewrite eq_le; apply/andP; split.
  by apply: ge_ereal_sup => /= y [r Ar <-{y}]; rewrite lee_fin ub_le_sup.
set esup := ereal_sup _; have := leey esup.
rewrite [X in _ X]le_eqVlt => /predU1P[->|esupoo]; first by rewrite leey.
have := leNye esup; rewrite [in X in X -> _]le_eqVlt => /predU1P[/esym|ooesup].
  case: A0 => i Ai.
  by move=> /ereal_sup_ninfty /(_ i%:E) /(_ (ex_intro2 A _ i Ai erefl)).
have esup_fin_num : esup \is a fin_num.
  by rewrite fin_numE -leeNy_eq -ltNge ooesup /= -leye_eq -ltNge esupoo.
rewrite -(@fineK _ esup) // lee_fin leNgt.
apply/negP => /(sup_gt A0)[r Ar]; apply/negP; rewrite -leNgt.
by rewrite -lee_fin fineK//; apply: ereal_sup_ubound; exists r.
Qed.

Lemma ( : set R) : has_lbound A -> A !=set0 ->
ereal_inf (EFin @` A) = (inf A)%:E.

Proof.

move=> has_lbA A0; rewrite /ereal_inf /inf EFinN; congr (- _)%E.
rewrite -ereal_sup_EFin; [exact/has_lb_ubN|exact/nonemptyN|].
by rewrite !image_comp.
Qed.

Lemma :
reflect (forall : \bar R, S y -> y <= x) (ereal_sup S <= x).

Proof.

apply/(iffP idP) => [+ y Sy|].
by move=> /(le_trans _)->//; rewrite le_ereal_sup_tmp//; exists y.
apply: contraPP => /negP; rewrite -ltNge -existsPNP.
by move=> /ereal_sup_gt[y Sy ltyx]; exists y => //; rewrite lt_geF.
Qed.

Lemma :
reflect (forall : \bar R, S y -> x <= y) (x <= ereal_inf S).

Proof.

rewrite leeNr; apply/(equivP (ereal_supP _ _)); setoid_rewrite leeNr.
split=> [ge_x y Sy|ge_x _ [y Sy <-]]; rewrite ?oppeK// ?ge_x//.
by rewrite -[y]oppeK ge_x//; exists y.
Qed.

Lemma { : Type} ( : X -> Y -> \bar R)
    ( : set X) ( : set Y) :
  ereal_sup [set ereal_sup [set f x y | in B] | in A] =
  ereal_sup [set ereal_sup [set f x y | in A] | in B].

Proof.

suff suf : forall ( : Type) ( : U -> V -> \bar R) ( : set U) ( : set V),
    ereal_sup [set ereal_sup [set g x y | in D] | in C] <=
    ereal_sup [set ereal_sup [set g x y | in C] | in D].
  by apply/le_anti/andP; split; exact: suf.
move=> U V g C D.
apply/ereal_supP => _ [x Cx <-]; apply/ereal_supP => _ [y Dy <-].
apply: le_ereal_sup_tmp; exists (ereal_sup [set g x y | in C]).
- by exists y.
- by apply: le_ereal_sup_tmp; exists (g x y) => //; exists x.
Qed.

Lemma :
reflect (exists2 : \bar R, S y & x < y) (x < ereal_sup S).

Proof.

rewrite ltNge; apply/(equivP negP); rewrite -(rwP (ereal_supP _ _)) -existsPNP.
by apply/eq_exists2r => y; rewrite (rwP2 negP idP) -ltNge.
Qed.

Lemma :
reflect (exists2 : \bar R, S y & y < x) (ereal_inf S < x).

Proof.

rewrite ltNge; apply/(equivP negP); rewrite -(rwP (ereal_infP _ _)) -existsPNP.
by apply/eq_exists2r => y; rewrite (rwP2 negP idP) -ltNge.
Qed.

Lemma : S (ereal_inf S) ->
reflect (exists2 : \bar R, S y & y <= x) (ereal_inf S <= x).

Proof.

move=> Sinf; apply: (iffP idP); last exact: ge_ereal_inf.
by move=> Sx; exists (ereal_inf S).
Qed.

Lemma : S (ereal_sup S) ->
reflect (exists2 : \bar R, S y & x <= y) (x <= ereal_sup S).

Proof.

move=> Ssup; apply: (iffP idP); last exact: le_ereal_sup_tmp.
by move=> Sx; exists (ereal_sup S).
Qed.

Lemma :
ereal_inf S = -oo -> exists2 : \bar R, S x & x < e%:E.

Proof.

by move=> infNy; apply/ereal_inf_ltP; rewrite infNy ltNyr. Qed.

Lemma : @ereal_sup R (range EFin) = +oo.

Proof.

rewrite hasNub_ereal_sup//; last by exists 0%R.
by apply/has_ubPn => x; exists (x+1)%R => //; rewrite ltrDl.
Qed.

Lemma : @ereal_inf R (range EFin) = -oo.

Proof.

rewrite /ereal_inf [X in ereal_sup X](_ : _ = range EFin); last first.
by rewrite ereal_sup_real.
apply/seteqP; split => x/=[y].
by move=> [z] _ <- <-; exists (-z)%R.
by move=> _ <-; exists (-y%:E); first (by exists (-y)%R); rewrite oppeK.
Qed.

End ereal_supremum_realType.
#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `ge_ereal_inf`.")]
Notation := ge_ereal_inf.
#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `ereal_sup_le`.")]
Notation := ereal_sup_le.
#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `ereal_inf_le_tmp`.")]
Notation := ereal_inf_le_tmp.
#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `le_ereal_sup_tmp`.")]
Notation := le_ereal_sup_tmp.
Arguments ereal_supP {R S x}.
Arguments ereal_infP {R S x}.
Arguments ereal_sup_gtP {R S x}.
Arguments ereal_inf_ltP {R S x}.
Arguments ereal_sup_geP {R S x}.
Arguments ereal_inf_leP {R S x}.

Section ereal_sup_cst.
Context { : realFieldType}.
Implicit Types (x : \bar R) (X : set (\bar R)).
Local Open Scope ereal_scope.

Lemma ( : set T) : A != set0 -> ereal_sup (cst x @` A) = x.

Proof.

by move=> AN0; rewrite set_cst ifN// ereal_sup1. Qed.

Lemma ( : set T) : A != set0 -> ereal_inf (cst x @` A) = x.

Proof.

by move=> AN0; rewrite set_cst ifN// ereal_inf1. Qed.

End ereal_sup_cst.

Section ereal_supZ.
Context { : realType}.
Implicit Types (r s : R) (X : set (\bar R)).
Local Open Scope ereal_scope.

Lemma : (0 < r)%R ->
ereal_sup [set r%:E * x | in X] = r%:E * ereal_sup X.

Proof.

move=> /[dup] r_gt0; rewrite lt0r => /andP[r_neq0 r_ge0].
gen have gen : r r_gt0 {r_ge0 r_neq0} X /
    ereal_sup [set r%:E * x | in X] <= r%:E * ereal_sup X.
  apply/ereal_supP => y/= [x Ax <-]; rewrite lee_pmul2l//=.
  by apply/ereal_supP => //=; exists x.
apply/eqP; rewrite eq_le gen//= -lee_pdivlMl//.
rewrite (le_trans _ (gen _ _ _)) ?invr_gt0 ?image_comp//=.
by under eq_imagel do rewrite /= muleA -EFinM mulVf ?mul1e//=; rewrite image_id.
Qed.

Lemma : X != set0 -> (0 <= r)%R ->
ereal_sup [set r%:E * x | in X] = r%:E * ereal_sup X.

Proof.

move=> AN0; have [r_gt0|//|<-] := ltgtP => _; first by rewrite ereal_sup_pZl.
by rewrite mul0e; under eq_imagel do rewrite mul0e/=; rewrite ereal_sup_cst.
Qed.

Lemma : (0 < r)%R ->
ereal_inf [set r%:E * x | in X] = r%:E * ereal_inf X.

Proof.

move=> r_gt0; rewrite !ereal_infEN muleN image_comp/=; congr (- _).
by under eq_imagel do rewrite /= -muleN; rewrite -image_comp ereal_sup_pZl.
Qed.

Lemma : X != set0 -> (0 < r)%R ->
ereal_sup [set r%:E * x | in X] = r%:E * ereal_sup X.

Proof.

move=> XN0 r_gt0; rewrite !ereal_supEN muleN image_comp/=; congr (- _).
by under eq_imagel do rewrite /= -muleN; rewrite -image_comp ereal_inf_pZl.
Qed.

Lemma ( : \bar R) : 0 <= c -> X != set0 ->
(forall , X x -> 0 <= x) ->
ereal_sup [set c * x | in X] = c * ereal_sup X.

Proof.

move=> c0 /[dup] Xneq0 /set0P[x Xx] X_ge0.
case: c c0 => [r|_|//].
- rewrite lee_fin le_eqVlt => /predU1P[<-|r0].
  + rewrite mul0e.
    under eq_imagel do rewrite mul0e.
    by rewrite ereal_sup_cst.
  + exact/(ereal_supZl Xneq0)/ltW.
- have [Xall0|] := pselect (forall , X a -> a = 0).
  + rewrite [X in ereal_sup X = _](_ : _ = [set 0]%classic).
      apply/seteqP; split.
      * by move=> _ [z Xz <-]; rewrite (Xall0 _ Xz) mule0.
      * by move=> y /= ->; exists x => //; rewrite (Xall0 _ Xx) mule0.
    have -> : X = [set 0]%classic.
      apply/seteqP; split.
      * by move=> y /Xall0 ->.
      * by move=> y /= ->; rewrite -(Xall0 _ Xx).
    by rewrite ereal_sup1 mule0.
  + rewrite -existsNE => -[y /not_implyP[Xy /eqP]].
    rewrite neq_lt ltNge X_ge0//= => y0.
    rewrite gt0_mulye//; first  by rewrite (lt_le_trans y0)// ereal_sup_ubound.
    by rewrite ereal_supy//=; exists y => //; exact: gt0_mulye.
Qed.

Section ge0_ereal_supZl_range.
Context { : choiceType} ( : T -> nat -> \bar R).
Hypothesis : forall , 0 <= f t n.

Lemma ( : \bar R) ( : T) : 0 <= c ->
c * ereal_sup (range (f x)) = ereal_sup (range (fun => c * f x n)).

Proof.

move=> c0.
rewrite [X in _ = ereal_sup X](_ : _ = [set c * y | in range (f x)]%classic).
  apply/seteqP; split.
  - by move=> _ [n _ <-]; exists (f x n) => //; exists n.
  - by move=> _ [_ [n _ <-] <-]; exists n.
rewrite ge0_ereal_supZl//.
- by apply/set0P; exists (f x 0%N), 0%N.
- by move=> _ [n _ <-]; exact: f_ge0.
Qed.

End ge0_ereal_supZl_range.

End ereal_supZ.

Lemma ( : numDomainType) ( : T -> \bar R) ( : set T) :
(abse \o f) \_ D = abse \o (f \_ D).

Proof.

by apply/funext=> t; rewrite /restrict/=; case: ifPn => // _; rewrite abse0.
Qed.

Lemma ( : numFieldType) ( : T -> R) ( : set T) :
(EFin \o f) \_ D = EFin \o (f \_ D).

Proof.

by apply/funext => x; rewrite /= !patchE; case: ifPn. Qed.

Section SignedRealFieldStability.
Context { : realFieldType}.

Lemma ( : Itv.def (@ext_num_sem R) i -> Prop)
( := Itv.real1 IntItv.keep_nonpos i) :
Itv.spec (@ext_num_sem R) r (ereal_sup [set x%:num | in S]).

Proof.

rewrite {}/r; case: i S => [//| [l u]] S /=.
apply/and3P; split.
- rewrite real_fine -real_leey.
  by rewrite ge_ereal_sup// => _ [[[x||] /=/and3P[? ? ?]] _ <-].
- by case: ereal_sup.
- case: u S => [[] [[| u] | u] S |//]; rewrite /= bnd_simp//;
     apply: ge_ereal_sup => _ [[x /=/and3P[_ _ /= +]] _ <-]; rewrite bnd_simp//.
  + by move/ltW.
  + by move=> /ltW /le_trans; apply; rewrite lee_fin lerz0.
  + by move=> /le_trans; apply; rewrite lee_fin lerz0.
Qed.

Canonical ( : Itv.def (@ext_num_sem R) i -> Prop) :=
  Itv.mk (ext_num_spec_ereal_sup S).

Lemma ( : Itv.def (@ext_num_sem R) i -> Prop)
    ( := Itv.real1 IntItv.keep_nonneg i) :
  Itv.spec (@ext_num_sem R) r (ereal_inf [set x%:num | in S]).

Proof.

rewrite {}/r; case: i S => [//| [l u]] S /=.
apply/and3P; split.
- by rewrite real_fine -real_leNye leNye.
- case: l S => [[] [l | l] S |//]; rewrite /= bnd_simp//;
     apply: le_ereal_inf_tmp => _ [[x /=/and3P[_ /= + _]] _ <-]; rewrite bnd_simp.
  + by apply: le_trans; rewrite lee_fin ler0z.
  + by move=> /ltW; apply: le_trans; rewrite lee_fin ler0z.
- by case: ereal_inf.
Qed.

Canonical ( : Itv.def (@ext_num_sem R) i -> Prop) :=
  Itv.mk (ext_num_spec_ereal_inf S).

End SignedRealFieldStability.

Section ereal_nbhs.
Context { : numFieldType}.
Local Open Scope ereal_scope.
Local Open Scope classical_set_scope.

Definition ( : \bar R) : set_system (\bar R) :=
  [set | match x with
    | r%:E => r^' (fun r => P r%:E)
    | +oo => exists , M \is Num.real /\ forall , M%:E < y -> P y
    | -oo => exists , M \is Num.real /\ forall , y < M%:E -> P y
  end].
Definition ( : \bar R) : set_system (\bar R) :=
  [set | match x with
    | x%:E => nbhs x (fun => P r%:E)
    | +oo => exists , M \is Num.real /\ forall , M%:E < y -> P y
    | -oo => exists , M \is Num.real /\ forall , y < M%:E -> P y
  end].
..Build (\bar R) ereal_nbhs.
End ereal_nbhs.

Section ereal_nbhs_instances.
Context { : numFieldType}.

Global Instance :
  forall : \bar R, ProperFilter (ereal_dnbhs x).

Proof.

case=> [x| |].
- case: (Proper_dnbhs_numFieldType x) => x0 [//= xT xI xS].
    by apply: Build_ProperFilter => //=; exact: Build_Filter.
- apply: Build_ProperFilter_ex.
    move=> P [x [xr xP]] //; exists (x + 1)%:E; apply: xP => /=.
    by rewrite lte_fin ltrDl.
  split=> /= [|P Q [MP [MPr gtMP]] [MQ [MQr gtMQ]] |P Q sPQ [M [Mr gtM]]].
  + by exists 0%R.
  + have [MP0|MP0] := eqVneq MP 0%R.
      have [MQ0|MQ0] := eqVneq MQ 0%R.
        by exists 0%R; split => // x x0; split;
          [apply/gtMP; rewrite MP0 | apply/gtMQ; rewrite MQ0].
      exists `|MQ|%R; rewrite realE normr_ge0; split => // x MQx; split.
        by apply: gtMP; rewrite (le_lt_trans _ MQx) // MP0 lee_fin.
      apply: gtMQ.
      by rewrite (le_lt_trans _ MQx)// lee_fin real_ler_normr ?lexx.
    have [MQ0|MQ0] := eqVneq MQ 0%R.
      exists `|MP|%R; rewrite realE normr_ge0; split => // x MPx; split.
        apply: gtMP.
        by rewrite (le_lt_trans _ MPx)// lee_fin real_ler_normr ?lexx.
      by apply: gtMQ; rewrite (le_lt_trans _ MPx) // lee_fin MQ0.
    have {}MP0 : (0 < `|MP|)%R by rewrite normr_gt0.
    have {}MQ0 : (0 < `|MQ|)%R by rewrite normr_gt0.
    exists (Num.max (PosNum MP0) (PosNum MQ0))%:num.
    rewrite realE /= ge0 /=; split => //.
    case=> [r| |//].
    * rewrite lte_fin/= num_max num_gt_max /= => /andP[MPx MQx]; split.
      by apply/gtMP; rewrite lte_fin (le_lt_trans _ MPx)// real_ler_normr ?lexx.
      by apply/gtMQ; rewrite lte_fin (le_lt_trans _ MQx)// real_ler_normr ?lexx.
    * by move=> _; split; [apply/gtMP | apply/gtMQ].
  + by exists M; split => // ? /gtM /sPQ.
- apply: Build_ProperFilter_ex.
  + move=> P [M [Mr ltMP]]; exists (M - 1)%:E.
    by apply: ltMP; rewrite lte_fin gtrDl oppr_lt0.
  + split=> /= [|P Q [MP [MPr ltMP]] [MQ [MQr ltMQ]] |P Q sPQ [M [Mr ltM]]].
    * by exists 0%R.
    * have [MP0|MP0] := eqVneq MP 0%R.
        have [MQ0|MQ0] := eqVneq MQ 0%R.
          by exists 0%R; split => // x x0; split;
          [apply/ltMP; rewrite MP0 | apply/ltMQ; rewrite MQ0].
        exists (- `|MQ|)%R; rewrite realN realE normr_ge0; split => // x xMQ.
        split.
          by apply: ltMP; rewrite (lt_le_trans xMQ)// lee_fin MP0 lerNl oppr0.
       apply: ltMQ; rewrite (lt_le_trans xMQ) // lee_fin lerNl -normrN.
       by rewrite real_ler_normr ?realN // lexx.
    * have [MQ0|MQ0] := eqVneq MQ 0%R.
        exists (- `|MP|)%R; rewrite realN realE normr_ge0; split => // x MPx.
        split.
          apply: ltMP; rewrite (lt_le_trans MPx) // lee_fin lerNl -normrN.
          by rewrite real_ler_normr ?realN // lexx.
        by apply: ltMQ; rewrite (lt_le_trans MPx) // lee_fin MQ0 lerNl oppr0.
      have {}MP0 : (0 < `|MP|)%R by rewrite normr_gt0.
      have {}MQ0 : (0 < `|MQ|)%R by rewrite normr_gt0.
      exists (- (Num.max (PosNum MP0) (PosNum MQ0))%:num)%R.
      rewrite realN realE /= ge0 /=; split => //.
      case=> [r|//|].
      - rewrite lte_fin ltrNr num_max num_gt_max => /andP[].
        rewrite ltrNr => MPx; rewrite ltrNr => MQx; split.
          apply/ltMP; rewrite lte_fin (lt_le_trans MPx) //= lerNl -normrN.
          by rewrite real_ler_normr ?realN // lexx.
        apply/ltMQ; rewrite lte_fin (lt_le_trans MQx) //= lerNl -normrN.
        by rewrite real_ler_normr ?realN // lexx.
      - by move=> _; split; [apply/ltMP | apply/ltMQ].
    * by exists M; split => // x /ltM /sPQ.
Qed.

Typeclasses Opaque ereal_dnbhs.

Global Instance : forall , ProperFilter (@ereal_nbhs R x).

Proof.

case=> [r| |].
- case: (ereal_dnbhs_filter r%:E) => r0 [//= nrT rI rS].
apply: Build_ProperFilter_ex => P /nbhs_ballP[r2 r20 rr2].
by exists r%:E; exact/rr2/ballxx.
- exact: (ereal_dnbhs_filter +oo).
- exact: (ereal_dnbhs_filter -oo).
Qed.

Typeclasses Opaque ereal_nbhs.

End ereal_nbhs_instances.

Section ereal_nbhs_infty.
Context ( : numFieldType).
Implicit Type r : R.

Lemma : r \is Num.real -> \forall \near +oo, r%:E < x.

Proof.

by exists r. Qed.

Lemma : r \is Num.real -> \forall \near +oo, r%:E <= x.

Proof.

by exists r; split => //; apply: ltW. Qed.

Lemma : r \is Num.real -> \forall \near -oo, r%:E > x.

Proof.

by exists r. Qed.

Lemma : r \is Num.real -> \forall \near -oo, r%:E >= x.

Proof.

by exists r; split => // ?; apply: ltW. Qed.

Lemma : \forall \near +oo, fine x \is @Num.real R.

Proof.

apply: filterS (ereal_nbhs_pinfty_gt (@real0 _)) => x.
by case: x => //= x; apply: gtr0_real.
Qed.

Lemma : \forall \near -oo, fine x \is @Num.real R.

Proof.

apply: filterS (ereal_nbhs_ninfty_lt (@real0 _)) => x.
by case: x => //= x; apply: ltr0_real.
Qed.

End ereal_nbhs_infty.

Section ereal_topologicalType.
Variable : realFieldType.

Lemma ( : \bar R) ( : set (\bar R)) :
ereal_nbhs p A -> A p.

Proof.

move: p => -[p | [M [Mreal MA]] | [M [Mreal MA]]] /=; [|exact: MA | exact: MA].
move=> /nbhs_ballP[_/posnumP[e]]; apply; exact/ballxx.
Qed.

Lemma ( : \bar R) ( : set (\bar R)) :
ereal_nbhs p A -> ereal_nbhs p (ereal_nbhs^~ A).

Proof.

move: p => -[p| [M [Mreal MA]] | [M [Mreal MA]]] //=.
- move=> /nbhs_ballP[_/posnumP[e]] ballA.
  apply/nbhs_ballP; exists (e%:num / 2)%R => //= r per.
  apply/nbhs_ballP; exists (e%:num / 2)%R => //= x rex.
  apply/ballA/(@ball_splitl _ _ r) => //; exact/ball_sym.
- exists (M + 1)%R; split; first by rewrite realD.
  move=> -[x| _ |_] //=; last by exists M.
  rewrite lte_fin => M'x /=.
  apply/nbhs_ballP; exists 1%R => //= y x1y.
  apply: MA; rewrite lte_fin.
  rewrite addrC -ltrBrDl in M'x.
  rewrite (lt_le_trans M'x) // lerBlDl addrC -lerBlDl.
  rewrite (le_trans _ (ltW x1y)) // real_ler_norm // realB //.
    rewrite ltrBrDr in M'x.
    by rewrite -comparabler0 (@comparabler_trans _ (M + 1)%R).
  by rewrite num_real. (* where we really use realFieldType *)
- exists (M - 1)%R; split; first by rewrite realB.
  move=> -[x| _ |_] //=; last by exists M.
  rewrite lte_fin => M'x /=.
  apply/nbhs_ballP; exists 1%R => //= y x1y.
  apply: MA; rewrite lte_fin.
  rewrite ltrBrDl in M'x.
  rewrite (le_lt_trans _ M'x) // addrC -lerBlDl.
  rewrite (le_trans _ (ltW x1y)) // distrC real_ler_norm // realB //.
    by rewrite num_real. (* where we really use realFieldType *)
  rewrite addrC -ltrBrDr in M'x.
  by rewrite -comparabler0 (@comparabler_trans _ (M - 1)%R).
Qed.

End ereal_topologicalType.

Local Open Scope classical_set_scope.

Lemma ( : realFieldType) ( : \bar R) :
nbhs (- x) = [set (-%E @` A) | in nbhs x].

Proof.

case: x => [r /=| |].
- rewrite /nbhs /= /ereal_nbhs -nbhs_ballE.
  rewrite predeqE => S; split => [[_/posnumP[e] reS]|[S' [_ /posnumP[e] reS' <-]]].
    exists (-%E @` S).
      exists e%:num => //= r1 rer1; exists (- r1%:E); last by rewrite oppeK.
      by apply: reS; rewrite /ball /= opprK -normrN opprD opprK.
    rewrite predeqE => s; split => [[y [z Sz] <- <-]|Ss].
      by rewrite oppeK.
    by exists (- s); [exists s | rewrite oppeK].
  exists e%:num => //= r1 rer1; exists (- r1%:E); last by rewrite oppeK.
  by apply: reS'; rewrite /ball /= opprK -normrN opprD.
- rewrite predeqE => S; split=> [[M [Mreal MS]]|[x [M [Mreal Mx]] <-]].
    exists (-%E @` S).
      exists (- M)%R; rewrite realN Mreal; split => // x Mx.
      by exists (- x); [apply: MS; rewrite lteNl | rewrite oppeK].
    rewrite predeqE => x; split=> [[y [z Sz <- <-]]|Sx]; first by rewrite oppeK.
    by exists (- x); [exists x | rewrite oppeK].
  exists (- M)%R; rewrite realN; split => // y yM.
  exists (- y); by [apply: Mx; rewrite lteNr|rewrite oppeK].
- rewrite predeqE => S; split=> [[M [Mreal MS]]|[x [M [Mreal Mx]] <-]].
    exists (-%E @` S).
      exists (- M)%R; rewrite realN Mreal; split => // x Mx.
      by exists (- x); [apply: MS; rewrite lteNr | rewrite oppeK].
    rewrite predeqE => x; split=> [[y [z Sz <- <-]]|Sx]; first by rewrite oppeK.
    by exists (- x); [exists x | rewrite oppeK].
  exists (- M)%R; rewrite realN; split => // y yM.
  exists (- y); by [apply: Mx; rewrite lteNl|rewrite oppeK].
Qed.

Lemma ( : realFieldType) ( : \bar R) ( : set (\bar R)) :
nbhs (- z) (-%E @` A) -> nbhs z A.

Proof.

rewrite nbhsNe => -[S zS] SA; rewrite -(oppeK z) nbhsNe.
exists (-%E @` S); first by rewrite nbhsNe; exists S.
rewrite predeqE => x; split => [[y [u Su <-{y} <-]]|Ax].
  rewrite oppeK.
  move: SA; rewrite predeqE => /(_ (- u)) [h _].
  have : (exists2 , S y & - y = - u) by exists u.
  by move/h => -[y Ay] /eqP; rewrite eqe_opp => /eqP <-.
exists (- x); last by rewrite oppeK.
exists x => //.
move: SA; rewrite predeqE => /(_ (- x)) [_ h].
have : (-%E @` A) (- x) by exists x.
by move/h => [y Sy] /eqP; rewrite eqe_opp => /eqP <-.
Qed.

Lemma ( : realFieldType) :
continuous (-%E : \bar R -> \bar R).

Proof.

move=> x S /= xS; apply: nbhsNKe; rewrite image_preimage //.
by rewrite predeqE => y; split => // _; exists (- y) => //; rewrite oppeK.
Qed.

Section contract_expand.
Variable : realFieldType.
Implicit Types (x : \bar R) (r : R).
Local Open Scope ereal_scope.

Lemma ( : set (\bar R)) :
(@contract R) @` (-%E @` S) = -%R @` ((@contract R) @` S).

Proof.

rewrite predeqE => r; split => [[y [z Sz <-{y} <-{r}]]|[s [y Sy <-{s} <-{r}]]].
by exists (contract z); [exists z | rewrite contractN].
by exists (- y); [exists y | rewrite contractN].
Qed.

Lemma : cancel (@contract R) (@expand R).

Proof.

apply: (onS_can [pred | `|r| <= 1]%R (@contract_le1 R)).
exact: inj_can_sym_on (@expandK R) (on2W (@contract_inj R)).
Qed.

Lemma : {on [pred | `|r| <= 1]%R, bijective (@contract R)}.

Proof.

exists (@expand R); [exact: in1W contractK | exact: (@expandK R)]. Qed.

Definition := monoLR_in
  (in_onW_can _ predT contractK) (fun _ => contract_le1 x) (@le_expand_in R).
Definition := monoLR_in
  (in_onW_can _ predT contractK) (fun _ => contract_le1 x) (@lt_expand R).
Definition := monoRL_in
  (in_onW_can _ predT contractK) (fun _ => contract_le1 x) (@le_expand_in R).
Definition := monoRL_in
  (in_onW_can _ predT contractK) (fun _ => contract_le1 x) (@lt_expand R).

Lemma : (contract x == 0%R) = (x == 0).

Proof.

by rewrite -(can_eq contractK) contract0. Qed.

Lemma : (contract x == (- 1)%R) = (x == -oo).

Proof.

by rewrite -(can_eq contractK). Qed.

Lemma : (contract x == 1%R) = (x == +oo).

Proof.

by rewrite -(can_eq contractK). Qed.

End contract_expand.

Section contract_expand_realType.
Variable : realType.

Let := @contract R.

Lemma : S !=set0 -> (`|sup (contract @` S)| <= 1)%R.

Proof.

case=> x Sx; rewrite ler_norml; apply/andP; split; last first.
  apply: ge_sup; first by exists (contract x), x.
  by move=> r [y Sy] <-; case/ler_normlP : (contract_le1 y).
rewrite (@le_trans _ _ (contract x)) //.
  by case/ler_normlP : (contract_le1 x); rewrite lerNl.
apply: ub_le_sup; last by exists x.
by exists 1%R => r [y Sy <-]; case/ler_normlP : (contract_le1 y).
Qed.

Lemma : S !=set0 -> contract (ereal_sup S) = sup (contract @` S).

Proof.

move=> S0; apply/eqP; rewrite eq_le; apply/andP; split; last first.
  apply: ge_sup.
    by case: S0 => x Sx; exists (contract x), x.
  by move=> x [y Sy] <-{x}; rewrite le_contract; exact/ereal_sup_ubound.
rewrite leNgt; apply/negP.
set supc := sup _; set csup := contract _; move=> ltsup.
suff [y [ysupS ?]] : exists , y < ereal_sup S /\ ubound S y.
  have : ereal_sup S <= y by exact: ge_ereal_sup.
  by move/(lt_le_trans ysupS); rewrite ltxx.
suff [x [? [ubSx x1]]] : exists , (x < csup)%R /\ ubound (contract @` S) x /\
    (`|x| <= 1)%R.
  exists (expand x); split => [|y Sy].
    by rewrite -(contractK (ereal_sup S)) lt_expand // inE // contract_le1.
  by rewrite -(contractK y) le_expand //; apply: ubSx; exists y.
exists ((supc + csup) / 2)%R; split; first by rewrite midf_lt.
split => [r [y Sy <-{r}]|].
  rewrite (@le_trans _ _ supc) ?midf_le //; last by rewrite ltW.
  apply: ub_le_sup; last by exists y.
  by exists 1%R => r [z Sz <-]; case/ler_normlP : (contract_le1 z).
rewrite ler_norml; apply/andP; split; last first.
  rewrite ler_pdivrMr // mul1r (_ : 2 = 1 + 1)%R // lerD //.
  by case/ler_normlP : (sup_contract_le1 S0).
  by case/ler_normlP : (contract_le1 (ereal_sup S)).
rewrite ler_pdivlMr // (_ : 2 = 1 + 1)%R // mulN1r opprD lerD //.
by case/ler_normlP : (sup_contract_le1 S0); rewrite lerNl.
by case/ler_normlP : (contract_le1 (ereal_sup S)); rewrite lerNl.
Qed.

Lemma : S !=set0 -> contract (ereal_inf S) = inf (contract @` S).

Proof.

move=> -[x Sx]; rewrite /ereal_inf /contract (contractN (ereal_sup (-%E @` S))).
by rewrite -/contract contract_sup /inf;
[exists (- x), x|rewrite contract_imageN].
Qed.

End contract_expand_realType.

Section ereal_PseudoMetric.
Variable : realFieldType.
Implicit Types (x y : \bar R) (r : R).

Lemma : {homo ereal_ball x : / (e <= e')%R >-> e `<=` e'}.

Proof.

by move=> e e' ee' y; rewrite /ereal_ball => /lt_le_trans; exact. Qed.

Lemma { : {posnum R}} : (e%:num <= 1)%R ->
expand (1 - e%:num)%R < r%:E -> ereal_ball +oo e%:num r%:E.

Proof.

move=> e1 er; rewrite /ereal_ball gtr0_norm ?subr_gt0.
by case/ltr_normlP : (contract_lt1 r).
rewrite ltrBlDl addrC -ltrBlDl -[ltLHS]expandK ?lt_contract//.
by rewrite inE ger0_norm ?lerBlDl ?lerDr // subr_ge0.
Qed.

Lemma ( : {posnum R}) : (r <= r')%R ->
(1 <= contract r%:E + e%:num)%R -> ereal_ball r%:E e%:num r'%:E.

Proof.

rewrite le_eqVlt => /predU1P[<-{r'} _|rr' re1]; first exact: ereal_ball_center.
rewrite /ereal_ball ltr0_norm; first by rewrite subr_lt0 lt_contract lte_fin.
rewrite opprB ltrBlDl (lt_le_trans _ re1) //.
by case/ltr_normlP : (contract_lt1 r').
Qed.

Lemma ( : {posnum R}) : (r' < r)%R ->
(contract r%:E - e%:num <= -1)%R -> ereal_ball r%:E e%:num r'%:E.

Proof.

move=> r'r reN1; rewrite /ereal_ball.
rewrite gtr0_norm ?subr_gt0 ?lt_contract ?lte_fin//.
rewrite ltrBlDl addrC -ltrBlDl (le_lt_trans reN1) //.
by move: (contract_lt1 r'); rewrite ltr_norml => /andP[].
Qed.

Lemma ( : {posnum R}) : (r' < r)%R ->
expand (contract r%:E - e%:num)%R < r'%:E ->
(`|contract r%:E - e%:num| < 1)%R -> ereal_ball r%:E e%:num r'%:E.

Proof.

move=> r'r ? r'e'r.
rewrite /ereal_ball gtr0_norm ?subr_gt0 ?lt_contract ?lte_fin//.
by rewrite ltrBlDl addrC -ltrBlDl -lt_expandLR ?inE ?ltW.
Qed.

Lemma ( : {posnum R}) :
  let := (r - fine (expand (contract r%:E - e%:num)))%R in
  ball r e' r' -> (r' < r)%R ->
  (`|contract r%:E - e%:num| < 1)%R ->
  ereal_ball r%:E e%:num r'%:E.

Proof.

move=> e' re'r' rr' X; rewrite /ereal_ball.
rewrite gtr0_norm ?subr_gt0// ?lt_contract ?lte_fin//.
move: re'r'; rewrite /ball /= gtr0_norm ?subr_gt0// /e'.
rewrite ltrD2l ltrN2 -lte_fin fine_expand// lt_expandLR ?inE ?ltW//.
by rewrite ltrBlDl addrC -ltrBlDl.
Qed.

Lemma ( : {posnum R}) :
  let : R := (fine (expand (contract r%:E + e%:num)) - r)%R in
  ball r e' r' -> (r <= r')%R ->
  (`| contract r%:E + e%:num | < 1)%R ->
  ereal_ball r%:E e%:num r'%:E.

Proof.

move=> e' r'e'r rr' re1; rewrite /ereal_ball.
move: rr'; rewrite le_eqVlt => /predU1P[->|rr']; first by rewrite subrr normr0.
rewrite /ball /= ltr0_norm ?subr_lt0// opprB in r'e'r.
rewrite ltr0_norm ?subr_lt0 ?lt_contract ?lte_fin//.
rewrite opprB; move: r'e'r.
rewrite /e' -ltrBlDr opprK subrK -lte_fin fine_expand //.
by rewrite lt_expandRL ?inE ?ltW// ltrBlDl.
Qed.

Lemma ( : set (\bar R)) ( : {posnum R}) : (e%:num <= 1)%R ->
ereal_ball +oo e%:num `<=` A -> nbhs +oo A.

Proof.

move=> e1 ooeA.
exists (fine (expand (1 - e%:num)%R)); rewrite num_real; split => //.
case => [r | | //].
- rewrite fine_expand.
by rewrite ger0_norm ?ltrBlDl ?ltrDr // subr_ge0.
by move=> ?; exact/ooeA/expand_ereal_ball_pinfty.
- by move=> _; exact/ooeA/ereal_ball_center.
Qed.

Lemma ( : set (\bar R)) ( : {posnum R}) : (e%:num <= 1)%R ->
ereal_ball -oo e%:num `<=` A -> nbhs -oo A.

Proof.

move=> e1 reA; suff h : nbhs +oo (-%E @` A).
  rewrite (_ : -oo = - +oo) // nbhsNe; exists (-%E @` A) => //.
  rewrite predeqE => x; split=> [[y [z Az <- <-]]|Ax]; rewrite ?oppeK //.
  by exists (- x); [exists x | rewrite oppeK].
apply/ (@nbhs_oo_up_e1 _ e) => // x x1e; exists (- x); last by rewrite oppeK.
by apply/reA/ereal_ballN; rewrite oppeK.
Qed.

Lemma ( : set (\bar R)) ( : {posnum R}) : (1 < e%:num)%R ->
ereal_ball +oo e%:num `<=` A -> nbhs +oo A.

Proof.

move=> e1 reA; have [e2{e1}|e2] := ltrP 2 e%:num.
  suff -> : A = setT by exists 0%R.
  rewrite predeqE => x; split => // _; apply: reA.
  exact/ereal_ballN/ereal_ball_ninfty_oversize.
have /andP[e10 e11] : (0 < e%:num - 1 <= 1)%R.
  by rewrite subr_gt0 e1 /= lerBlDl.
apply: nbhsNKe.
have : ((PosNum e10)%:num <= 1)%R by [].
move/(@nbhs_oo_down_e1 (-%E @` A) (PosNum e10)); apply=> y ye.
exists (- y); last by rewrite oppeK.
apply/reA/ereal_ballN; rewrite oppeK /=.
by apply: le_ereal_ball ye => /=; rewrite lerBlDl lerDr.
Qed.

Lemma ( : set (\bar R)) ( : {posnum R}) : (1 < e%:num)%R ->
ereal_ball -oo e%:num `<=` A -> nbhs -oo A.

Proof.

move=> e1 reA; have [e2{e1}|e2] := ltrP 2 e%:num.
  suff -> : A = setT by exists 0%R.
  by rewrite predeqE => x; split => // _; exact/reA/ereal_ball_ninfty_oversize.
have /andP[e10 e11] : (0 < e%:num - 1 <= 1)%R.
  by rewrite subr_gt0 e1 /= lerBlDl.
apply: nbhsNKe.
have : ((PosNum e10)%:num <= 1)%R by [].
move/(@nbhs_oo_up_e1 (-%E @` A) (PosNum e10)); apply.
move=> y ye; exists (- y); last by rewrite oppeK.
apply/reA/ereal_ballN; rewrite /= oppeK.
by apply: le_ereal_ball ye => /=; rewrite lerBlDl lerDr.
Qed.

Lemma ( : {posnum R}) ( : set (\bar R)) :
  ereal_ball r%:E e%:num `<=` A ->
  (- 1 < contract r%:E - e%:num)%R ->
  (1 <= contract r%:E + e%:num)%R ->
  nbhs r%:E A.

Proof.

move=> reA reN1 re1.
have er1 : (`|contract r%:E - e%:num| < 1)%R.
  rewrite ltr_norml reN1 andTb ltrBlDl ltr_pwDl //.
  by move: (contract_le1 r%:E); rewrite ler_norml => /andP[].
pose e' := (r - fine (expand (contract r%:E - e%:num)))%R.
have e'0 : (0 < e')%R.
  rewrite subr_gt0 -lte_fin -[ltRHS](contractK r%:E).
  rewrite fine_expand // lt_expand ?inE ?contract_le1// ?ltW//.
  by rewrite ltrBlDl ltrDr.
apply/nbhs_ballP; exists e' => // r' re'r'; apply: reA.
by have [?|?] := lerP r r';
  [exact: contract_ereal_ball_fin_le | exact: ball_ereal_ball_fin_lt].
Qed.

Lemma ( : {posnum R}) ( : set (\bar R)) :
  ereal_ball r%:E e%:num `<=` A ->
  (contract r%:E - e%:num <= - 1)%R ->
  (contract r%:E + e%:num < 1)%R ->
  nbhs r%:E A.

Proof.

move=> reA reN1 re1.
have ? : (`|contract r%:E + e%:num| < 1)%R.
  rewrite ltr_norml re1 andbT (@lt_le_trans _ _ (contract r%:E)) // ?lerDl //.
  by move: (contract_lt1 r); rewrite ltr_norml => /andP[].
pose e' : R := (fine (expand (contract r%:E + e%:num)) - r)%R.
have e'0 : (0 < e')%R.
  rewrite /e' subr_gt0 -lte_fin -[in ltLHS](contractK r%:E).
  by rewrite fine_expand // lt_expand ?inE ?contract_le1 ?ltrDl ?ltW.
apply/nbhs_ballP; exists e' => // r' r'e'r; apply: reA.
by have [?|?] := lerP r r';
  [exact: ball_ereal_ball_fin_le | exact: contract_ereal_ball_fin_lt].
Qed.

Lemma ( : {posnum R}) ( : set (\bar R)) :
  ereal_ball r%:E e%:num `<=` A ->
  (contract r%:E - e%:num < -1)%R ->
  (1 < contract r%:E + e%:num)%R ->
  nbhs r%:E A.

Proof.

move=> reA reN1 re1; suff : A = setT by move->; apply: filterT.
rewrite predeqE => x; split => // _; apply: reA.
case: x => [r'| |] //.
- have [?|?] := lerP r r'.
  + by apply: contract_ereal_ball_fin_le => //; exact/ltW.
  + by apply: contract_ereal_ball_fin_lt => //; exact/ltW.
- exact/contract_ereal_ball_pinfty.
- apply/ereal_ballN/contract_ereal_ball_pinfty.
  by rewrite EFinN contractN -(opprK 1%R) ltrNl opprD opprK.
Qed.

Lemma ( : {posnum R}) ( : set (\bar R)) :
ereal_ball r%:E e%:num `<=` A -> nbhs r%:E A.

Proof.

move=> reA.
have [/eqP|reN1] := eqVneq (contract r%:E - e%:num)%R (-1)%R.
  rewrite subr_eq addrC => /eqP reN1.
  have [re1|] := eqVneq (contract r%:E + e%:num)%R 1%R.
    move/eqP: reN1; rewrite -re1 -subr_eq0 opprB addrK.
    rewrite -mulr2n mulrn_eq0 orFb contract_eq0 => /eqP[r0].
    move: re1; rewrite r0 contract0 add0r => e1.
    apply/nbhs_ballP; exists 1%R => //= r'; rewrite /ball /= sub0r normrN => r'1.
    apply: reA.
    by rewrite /ereal_ball r0 contract0 sub0r normrN e1 contract_lt1.
  rewrite neq_lt => /orP[re1|re1].
    by apply: (@nbhs_fin_out_below _ e) => //; rewrite reN1 addrAC subrr sub0r.
  have e1 : (1 < e%:num)%R.
    move: re1; rewrite reN1 addrAC ltrBrDl -!mulr2n -(mulr_natl e%:num).
    by rewrite -{1}(mulr1 2%:R) => ?; rewrite -(@ltr_pM2l _ 2).
  have Aoo : setT `\ -oo `<=` A.
    move=> x [_]; rewrite /set1 /= => xnoo; apply: reA.
    case: x xnoo => [r' _ | _ |//].
      have [rr'|r'r] := lerP (contract r%:E) (contract r'%:E).
        apply: contract_ereal_ball_fin_le; last exact/ltW.
        by rewrite -lee_fin -(contractK r%:E) -(contractK r'%:E) le_expand.
      apply: contract_ereal_ball_fin_lt; last by rewrite reN1 lerBlDl.
      rewrite -lte_fin -(contractK r%:E) -(contractK r'%:E).
      by rewrite lt_expand // inE contract_le1.
    exact: contract_ereal_ball_pinfty.
  have : nbhs r%:E (setT `\ -oo) by apply/nbhs_ballP; exists 1%R => /=.
  move=> /nbhs_ballP[_/posnumP[e']] /=; rewrite /ball /= => h.
  by apply/nbhs_ballP; exists e'%:num => //= y /h; apply: Aoo.
move: reN1; rewrite eq_sym neq_lt => /orP[reN1|reN1].
  have [re1|re1] := eqVneq (contract r%:E + e%:num)%R 1%R.
    by apply: (@nbhs_fin_out_above _ e) => //; rewrite re1.
  move: re1; rewrite neq_lt => /orP[re1|re1].
    have ? : (`|contract r%:E - e%:num| < 1)%R.
      rewrite ltr_norml reN1/= -/(contract _%:E) ltrBlDl.
      rewrite (@lt_le_trans _ _ 1%R) // ?lerDr//.
      by rewrite (le_lt_trans (ler_norm _))// contract_lt1.
    have ? : (`|contract r%:E + e%:num| < 1)%R.
      rewrite ltr_norml re1 andbT -(addr0 (- 1)%R) ler_ltD //.
      by move: (contract_le1 r%:E); rewrite ler_norml => /andP[].
    pose e' : R := Num.min
      (r - fine (expand (contract r%:E - e%:num)))%R
      (fine (expand (contract r%:E + e%:num)) - r)%R.
    have e'0 : (0 < e')%R.
      rewrite /e' lt_min; apply/andP; split.
        rewrite subr_gt0 -lte_fin -[in ltRHS](contractK r%:E).
        rewrite fine_expand // lt_expand// ?inE ?contract_le1 ?ltW//.
        by rewrite ltrBlDl ltrDr.
      rewrite subr_gt0 -lte_fin -[in ltLHS](contractK r%:E).
      by rewrite fine_expand// lt_expand ?inE ?contract_le1 ?ltrDl ?ltW.
    apply/nbhs_ballP; exists e' => // r' re'r'; apply: reA.
    have [|r'r] := lerP r r'.
      move=> rr'; apply: ball_ereal_ball_fin_le => //.
      by apply: le_ball re'r'; rewrite ge_min lexx orbT.
    move: re'r'; rewrite /ball /= lt_min => /andP[].
    rewrite gtr0_norm ?subr_gt0// ltrD2l ltrN2 -lte_fin fine_expand// => re'r _.
    exact: expand_ereal_ball_fin_lt.
  by apply: (@nbhs_fin_out_above _ e) => //; rewrite ltW.
have [re1|re1] := ltrP 1 (contract r%:E + e%:num).
  exact: (@nbhs_fin_out_above_below _ e).
move: re1; rewrite le_eqVlt => /orP[re1|re1].
  have {}re1 : contract r%:E = (1 - e%:num)%R.
    by move: re1; rewrite eq_sym -subr_eq => /eqP <-.
  have e1 : (1 < e%:num)%R.
    move: reN1.
    rewrite re1 -addrA -opprD ltrBlDl ltrBrDl -!mulr2n.
    rewrite -(mulr_natl e%:num) -{1}(mulr1 2%:R) => ?.
    by rewrite -(@ltr_pM2l _ 2).
  have Aoo : (setT `\ +oo `<=` A).
    move=> x [_]; rewrite /set1 /= => xpoo; apply: reA.
    case: x xpoo => [r' _ | // |_].
      rewrite /ereal_ball.
      have [rr'|r'r] := lerP (contract r%:E) (contract r'%:E).
        rewrite re1 opprB addrCA -[ltRHS]addr0 ltrD2 subr_lt0.
        by case/ltr_normlP : (contract_lt1 r').
      rewrite /ereal_ball.
      rewrite re1 addrAC ltrBlDl ltrD // (lt_trans _ e1) // ltrNl.
      by move: (contract_lt1 r'); rewrite ltr_norml => /andP[].
    rewrite /ereal_ball.
    rewrite [contract -oo]/= opprK gtr0_norm ?subr_gt0.
      rewrite -ltrBlDl add0r ltrNl.
      by move: (contract_lt1 r); rewrite ltr_norml => /andP[].
    by rewrite re1 addrAC ltrBlDl ltrD.
   have : nbhs r%:E (setT `\ +oo) by exists 1%R => /=.
   case => _/posnumP[x] /=; rewrite /ball_ => h.
   by exists x%:num => //= y /h; exact: Aoo.
by apply: (@nbhs_fin_out_below _ e) => //; rewrite ltW.
Qed.

Lemma : nbhs = nbhs_ (entourage_ (@ereal_ball R)).

Proof.

set diag := fun => [set | ereal_ball xy.1 r xy.2].
rewrite predeq2E => x A; split.
- rewrite {1}/nbhs /= /ereal_nbhs.
  case: x => [/= r [_/posnumP[e] reA]| [M [/= Mreal MA]]| [M [/= Mreal MA]]].
  + pose e' : R := Num.min (contract r%:E - contract (r%:E - e%:num%:E))%R
                           (contract (r%:E + e%:num%:E) - contract r%:E)%R.
    exists (diag e'); rewrite /diag.
      exists e' => //.
      rewrite /= /e' lt_min; apply/andP; split.
        by rewrite subr_gt0 lt_contract lte_fin ltrBlDr ltrDl.
      by rewrite subr_gt0 lt_contract lte_fin ltrDl.
    case=> [r' /= /xsectionP/= re'r'| |]/=.
    * rewrite /ereal_ball in re'r'.
      have [r'r|rr'] := lerP (contract r'%:E) (contract r%:E).
        apply: reA; rewrite /ball /= ltr_norml//.
        rewrite ger0_norm ?subr_ge0// in re'r'.
        have : (contract (r%:E - e%:num%:E) < contract r'%:E)%R.
          move: re'r'; rewrite /e' lt_min => /andP[+ _].
          rewrite /e' ltrBrDl addrC -ltrBrDl => /lt_le_trans.
          by apply; rewrite opprB addrC subrK.
        rewrite -lt_expandRL ?inE ?contract_le1 // !contractK lte_fin.
        rewrite ltrBlDr addrC -ltrBlDr => ->; rewrite andbT.
        by rewrite (@lt_le_trans _ _ 0%R)// subr_ge0 -lee_fin -le_contract.
      apply: reA; rewrite /ball /= ltr_norml//.
      rewrite ltr0_norm ?subr_lt0// opprB in re'r'.
      apply/andP; split; last first.
        by rewrite (@lt_trans _ _ 0%R) // subr_lt0 -lte_fin -lt_contract.
      rewrite ltrNl opprB.
      rewrite /e' in re'r'.
      have r're : (contract r'%:E < contract (r%:E + e%:num%:E))%R.
        move: re'r'; rewrite lt_min => /andP[_].
        by rewrite ltrBlDr subrK.
      rewrite ltrBlDr -lte_fin -(contractK (_ + r)%:E)%R.
      by rewrite addrC -(contractK r'%:E) // lt_expand ?inE ?contract_le1.
    * move=> /xsectionP/=; rewrite /ereal_ball [contract +oo]/=.
      rewrite lt_min => /andP[re'1 re'2].
      have [cr0|cr0] := lerP 0 (contract r%:E).
        move: re'2; rewrite ler0_norm.
          by rewrite subr_le0; case/ler_normlP : (contract_le1 r%:E).
        rewrite opprB ltrBrDl addrC subrK ltNge; apply: contraNP => _.
        by rewrite (le_trans (ler_norm _))// contract_le1.
      move: re'2; rewrite ler0_norm.
        by rewrite subr_le0; case/ler_normlP : (contract_le1 r%:E).
      rewrite opprB ltrBrDl addrC subrK ltNge; apply: contraNP => _.
      by rewrite (le_trans (ler_norm _))// contract_le1.
    * move=> /xsectionP/=; rewrite /ereal_ball [contract -oo]/= opprK.
      rewrite lt_min => /andP[re'1 _].
      move: re'1.
      rewrite ger0_norm.
        rewrite addrC -lerBlDl add0r.
        by move: (contract_le1 r%:E); rewrite ler_norml => /andP[].
      rewrite ltrD2l ltNge; apply: contraNP => _.
      by rewrite lerNl lerNnormlW// contract_le1.
  + exists (diag (1 - contract M%:E))%R; rewrite /diag.
      exists (1 - contract M%:E)%R => //=.
      by rewrite subr_gt0 (le_lt_trans _ (contract_lt1 M)) // ler_norm.
    case=> [r| |]/= /xsectionP/=.
    * rewrite /ereal_ball [_ +oo]/= ger0_norm.
        by rewrite subr_ge0 // (le_trans _ (contract_le1 r%:E)) // ler_norm.
      by rewrite ltrD2l ltrN2 => rM1; apply/MA; rewrite -lt_contract.
    * by rewrite /ereal_ball /= subrr normr0 => h; exact: MA.
    * rewrite /ereal_ball /= opprK ltNge; apply: contraNP => _.
      rewrite [in leRHS]ger0_norm // lerD2l.
      by rewrite -/(contract M%:E) lerNl lerNnormlW// contract_le1.
  + exists (diag (1 + contract M%:E)%R); rewrite /diag.
      exists (1 + contract M%:E)%R => //=.
      by rewrite -/(contract M%:E) -ltrBlDl sub0r ltrNnormlW// contract_lt1.
    case=> [r| |] /xsectionP/=.
    * rewrite /ereal_ball => /= rM1.
      apply: MA; rewrite lte_fin.
      rewrite ler0_norm in rM1.
        by rewrite subr_le0 -/(contract r%:E) lerNnormlW// contract_le1.
      move: rM1; rewrite opprB opprK -ltrBlDl addrK.
      by rewrite -!/(contract _%:E) lt_contract lte_fin.
    * rewrite /ereal_ball/= -opprD normrN ger0_norm// ltrD2l.
      by rewrite -/(contract M%:E) ltNge (le_trans (ler_norm _))// contract_le1.
    * by rewrite /ereal_ball /= => _; exact: MA.
- case: x => [r [E [_/posnumP[e] reA] sEA] | [E [_/posnumP[e] reA] sEA] |
               [E [_/posnumP[e] reA] sEA]] //=.
  + by apply: (@nbhs_fin_inbound _ e) => ? ?; exact/sEA/xsectionP/reA.
  + have [|] := lerP e%:num 1%R.
    * by move/nbhs_oo_up_e1; apply => x ex; exact/sEA/xsectionP/reA.
    * by move/nbhs_oo_up_1e; apply => x ex; exact/sEA/xsectionP/reA.
  + have [|] := lerP e%:num 1%R.
    * by move/nbhs_oo_down_e1; apply => x ex; exact/sEA/xsectionP/reA.
    * by move/nbhs_oo_down_1e; apply => x ex; exact/sEA/xsectionP/reA.
Qed.

..Build R (\bar R)
  ereal_nbhsE ereal_ball_center ereal_ball_sym ereal_ball_triangle erefl.

End ereal_PseudoMetric.

Lemma ( : realFieldType) ( : R -> Prop) ( : R) ( : \bar R) :
  a < r%:E -> r%:E < b ->
  (forall , a < y%:E -> y%:E < b -> P y) ->
  nbhs r P.

Proof.

move => ar rb abP; case: (lt_ereal_nbhs ar rb) => d rd.
exists d%:num => //= y; rewrite /= distrC.
by move=> /rd /andP[? ?]; apply: abP.
Qed.

Lemma ( : numFieldType) ( : \bar R) :
ereal_dnbhs x --> ereal_nbhs x.

Proof.

move: x => [r P [_/posnumP[e] reP] |r P|r P] //=.
by exists e%:num => //= ? ? ?; apply: reP.
Qed.

Lemma ( : numFieldType) ( : R) :
ereal_dnbhs r%:E --> nbhs r%:E.

Proof.

by move=> P [_/posnumP[e] reP] //=; exists e%:num => //= ? ? ?; exact: reP.
Qed.

Definition ( : numDomainType) ( : \bar R) ( : nat) :=
  match x with
    | x%:E => (x + (n%:R + 1)^-1)%:E
    | +oo => n%:R%:E
    | -oo => - n%:R%:E
  end.

Lemma ( : realType) ( : \bar R) :
  ereal_loc_seq x  @ \oo --> ereal_dnbhs x.

Proof.

move=> P; rewrite /ereal_loc_seq.
case: x => /= [x [_/posnumP[d] dP] |[d [dreal dP]] |[d [dreal dP]]]; last 2 first.
- exists (Num.truncn (d + 1)) => // n /=.
  by rewrite truncn_le_nat -natr1 ltrD2r => ltdn; apply: dP; rewrite lte_fin.
- exists (Num.truncn (- d + 1)) => // n /=.
  rewrite truncn_le_nat -natr1 ltrD2r => ltNdn.
  by apply: dP; rewrite lte_fin ltrNl.
exists (Num.truncn d%:num^-1) => // n /=.
rewrite truncn_le_nat invf_plt ?posrE// => ltnd; apply: dP; last first.
  by rewrite addrC -subr_eq0 addrK invr_eq0 lt0r_neq0.
by rewrite /= opprD addNKr normrN normfV natr1 gtr0_norm.
Qed.

Section nbhs_ereal.
Context { : numFieldType} ( : \bar R -> Prop).

Lemma ( : R) : (\forall \near x%:E, P y) <-> \near , P x%:E.

Proof.

done. Qed.

End nbhs_ereal.

Section cvg_fin.
Context { : numFieldType} { : set_system \bar R} { : Filter F}.

Lemma : F --> a%:E -> \near , F \is a fin_num.

Proof.

by apply; apply/nbhs_EFin; near=> x. Unshelve. all: by end_near. Qed.

End cvg_fin.

Section filter.
Context { : numFieldType} { : set_system \bar R} { : Filter F}.

Lemma : F --> a%:E -> fine @ F --> a.

Proof.

move=> /(_ _)/= Fa.
apply/(@metricType_numDomainType.cvgrPdist_lt _ R^o) => // _/posnumP[e].
rewrite near_simpl.
apply: Fa; apply/nbhs_EFin.
exact: (@metricType_numDomainType.cvgr_dist_lt _ _ _ (nbhs a)).
Qed.

End filter.

Section limit.
Context { : numFieldType} { : Type} { : set_system I} { : Filter F}
( : I -> \bar R).

Lemma : (\near , f F \is a fin_num) -> fine \o f @ F --> a ->
f @ F --> a%:E.

Proof.

move=> Ffin Fa P/= /nbhs_EFin /Fa; rewrite !near_simpl.
by apply: filterS2 Ffin => x /fineK->.
Qed.

Lemma : f @ F --> a%:E -> fine \o f @ F --> a.

Proof.

exact: fine_fcvg. Qed.

Lemma : f @ F --> a%:E -> \near , f F \is a fin_num.

Proof.

exact: fcvg_is_fine. Qed.

Lemma :
f @ F --> a%:E <-> (\near , f F \is a fin_num) /\ fine \o f @ F --> a.

Proof.

by split;[split;[exact: (@cvg_is_fine a)|exact: fine_cvg]|case; apply: cvg_EFin].
Qed.

End limit.

Files

Module mathcomp.analysis.ereal