Sorgenfrey line and properties

_CoqProject

@@ -30,7 +30,6 @@ reals/constructive_ereal.v
 reals/reals.v
 reals/real_interval.v
 reals/signed.v
-reals/interval_inference.v
 reals/prodnormedzmodule.v
 reals/all_reals.v
 experimental_reals/xfinmap.v
@@ -120,5 +119,6 @@ theories/kernel.v
 theories/pi_irrational.v
 theories/gauss_integral.v
 theories/showcase/summability.v
+theories/showcase/sorgenfreyline.v
 analysis_stdlib/Rstruct_topology.v
 analysis_stdlib/showcase/uniform_bigO.v

experimental_reals/discrete.v

@@ -4,7 +4,7 @@
 (* Copyright (c) - 2016--2018 - Polytechnique                           *)
 
 (* -------------------------------------------------------------------- *)
-From Corelib Require Setoid.
+From Coq Require Setoid.
 From HB Require Import structures.
 From mathcomp Require Import all_ssreflect all_algebra.
 From mathcomp.classical Require Import boolp.

reals/all_reals.v

@@ -1,4 +1,3 @@
-From mathcomp Require Export interval_inference.
 From mathcomp Require Export constructive_ereal.
 From mathcomp Require Export reals.
 From mathcomp Require Export real_interval.

reals/reals.v

@@ -43,7 +43,7 @@
 (*                                                                            *)
 (******************************************************************************)
 
-From Corelib Require Import Setoid.
+From Coq Require Import Setoid.
 From HB Require Import structures.
 From mathcomp Require Import all_ssreflect all_algebra archimedean.
 From mathcomp Require Import boolp classical_sets set_interval.

theories/Make

@@ -86,3 +86,4 @@ pi_irrational.v
 gauss_integral.v
 all_analysis.v
 showcase/summability.v
+showcase/sorgenfreyline.v

theories/borel_hierarchy.v

@@ -19,10 +19,13 @@ Set Implicit Arguments.
 Unset Strict Implicit.
 Unset Printing Implicit Defensive.
 
-Import Order.TTheory GRing.Theory Num.Theory.
+Import Order.TTheory GRing.Theory Num.Theory Num.Def.
 Import numFieldNormedType.Exports.
+Import numFieldTopology.Exports.
+Import exp.
 
 Local Open Scope classical_set_scope.
+Local Open Scope ring_scope.
 
 Section Gdelta_Fsigma.
 Context {T : topologicalType}.
@@ -94,3 +97,288 @@ have /Baire : forall n, open (C n) /\ dense (C n).
   - by apply: denseI => //; apply oB.
 by rewrite -C0; exact: dense0.
 Qed.
+
+Section perfectlynormalspace.
+Context (R : realType) (T : topologicalType).
+
+Definition perfectly_normal_space (x : R) :=
+  forall E : set T, closed E -> 
+    exists f : T -> R, continuous f /\ E = f @^-1` [set x].
+
+Lemma perfectly_normal_spaceP x y : perfectly_normal_space x -> perfectly_normal_space y.
+Proof.
+move=>px E cE.
+case:(px E cE) => f [] cf ->.
+pose f' := f + cst (y - x). 
+exists f'.
+split.
+  rewrite /f'.
+  move=> z.
+  apply: continuousD.
+    exact:cf.
+  exact:cst_continuous.
+apply/seteqP.
+rewrite /f' /cst /=.
+split => z /=.
+  rewrite addrfctE => ->.
+  by rewrite subrKC.
+rewrite addrfctE.
+move/eqP.
+by rewrite eq_sym -subr_eq opprB subrKC eq_sym => /eqP.
+Qed.
+
+Definition perfectly_normal_space' (x : R) :=
+  forall E : set T, open E -> 
+    exists f : T -> R, continuous f /\ E = f @^-1` ~`[set x].
+
+Definition perfectly_normal_space01 :=
+  forall E F : set T, closed E -> closed F -> [disjoint E & F] ->
+    exists f : T -> R, continuous f /\ E = f @^-1` [set 0] /\ F = f @^-1` [set 1] 
+      /\ f @` [set: T] = `[0, 1]%classic.
+
+Definition perfectly_normal_space_Gdelta :=
+  normal_space T /\ forall E : set T, closed E -> Gdelta E.
+
+Lemma perfectly_normal_space01_normal :
+  perfectly_normal_space01 -> normal_space T.
+Proof.
+move=> pns01 A cA B /set_nbhsP[C] [oC AC CB].
+case: (pns01 A (~` C) cA).
+- by rewrite closedC.
+- exact/disj_setPCl.
+move=> f [/continuousP /= cf] [f0] [f1] f01.
+exists (f @^-1` `]-oo, 1/2]).
+  apply/set_nbhsP.
+  exists (f @^-1` `]-oo, 1/2[).
+  split => //.
+  - exact: cf.
+  - by rewrite f0 => x /= ->; rewrite in_itv /=.
+  - by apply: preimage_subset => x /=; rewrite !in_itv /=; apply: ltW.
+apply: subset_trans CB.
+have<-:= proj1 (closure_id _).
+  have<-:= (setCK C).
+  rewrite f1 preimage_setC.
+  apply: preimage_subset => x /=; rewrite in_itv /=.
+  apply: contraTnot => ->.
+  by rewrite -ltNge ltr_pdivrMr // mul1r ltr1n.
+have/continuousP /continuous_closedP:= cf.
+apply.
+exact: lray_closed.
+Qed.
+
+Lemma EFin_series (f : R^nat) : EFin \o series f = eseries (EFin \o f).
+Proof.
+apply/boolp.funext => n.
+rewrite /series /eseries /=.
+elim: n => [|n IH]; first by rewrite !big_geq.
+by rewrite !big_nat_recr //= EFinD IH.
+Qed.
+
+Let perfectly_normal_space_12 : perfectly_normal_space_Gdelta -> perfectly_normal_space 0.
+Proof.
+move=> pnsGd E cE.
+case: (pnsGd) => nT cEGdE.
+have[U oU HE]:= cEGdE E cE.
+have/boolp.choice[f_n Hn]: forall n, exists f : T -> R, 
+  [/\ continuous f, range f `<=` `[0, 1], f @` E `<=` [set 0] & f @` (~` U n) `<=` [set 1]].
+  move=> n.
+  apply/uniform_separatorP.
+  apply: normal_uniform_separator => //.
+  - by rewrite closedC.
+  - rewrite HE -subsets_disjoint.
+    exact: bigcap_inf.
+pose f_sum := fun n => \sum_(0 <= k < n) (f_n k \* cst (2^-k.+1)).
+have cf_sum n : continuous (f_sum n).
+  rewrite /f_sum => x; elim: n => [|n IH].
+    rewrite big_geq //; exact: cst_continuous.
+  rewrite big_nat_recr //=; apply: continuousD => //.
+  apply/continuousM/cst_continuous.
+  by case: (Hn n) => /(_ x).
+have f_sumE x : f_sum ^~ x = [series f_n k x * 2^-k.+1]_k.
+  apply/boolp.funext => n.
+  rewrite /f_sum /series /mk_sequence.
+  elim: n => [|n IH]; first by rewrite !big_geq.
+  by rewrite !big_nat_recr //= [X in X _ _ x]/GRing.add /= IH.
+pose f := fun x => limn (f_sum^~ x).
+rewrite /= in f.
+exists f.
+have ndf_sum y : {homo f_sum^~ y : a b / (a <= b)%N >-> a <= b}.
+  move=> a b ab.
+  rewrite /f_sum.
+  rewrite (big_cat_nat _ ab) //= lerDl.
+  elim/big_rec: _ => //= i f0 _.
+  rewrite /GRing.add /= => /le_trans; apply.
+  rewrite lerDr mulr_ge0 //.
+    case: (Hn i) => _ Hr _ _.
+    have /Hr /= := imageT (f_n i) y.
+    by rewrite in_itv /= => /andP[].
+  by rewrite invr_ge0 exprn_ge0.
+(*have Hub y m k : (\sum_(m <= i < k) f_n i \* cst (2 ^- i.+1)) y <= 2^-m.
+  apply: (@le_trans _ _ (2^-m - 2^-k)); last first.
+  by rewrite gerBl invr_ge0 exprn_ge0.*)
+have Hcvg y : cvgn (f_sum^~ y).
+  apply: nondecreasing_is_cvgn => //.
+  exists 1 => z /= [m] _ <-.
+  rewrite /f_sum.
+  apply: (@le_trans _ _ (1 - 2^-m)); last by rewrite gerBl invr_ge0 exprn_ge0.
+  elim: m => [|m IH]; first by rewrite big_geq // expr0 invr1 subrr.
+  rewrite big_nat_recr //=.
+  have Hm : f_n m y * 2 ^- m.+1 <= 2 ^- m.+1.
+    case: (Hn m) => _ Hr _ _.
+    have /Hr /= := imageT (f_n m) y.
+    rewrite in_itv /= => /andP[f0 f1].
+    by rewrite -[leRHS]mul1r ler_pM.
+  apply: (le_trans (lerD IH Hm)).
+  rewrite -addrA.
+  have -> : 2^- m = 2 * 2^-m.+1 :> R.
+    by rewrite exprS invfM mulrA mulfV // mul1r.
+  rewrite -{1}add1n {1}mulrnDr mulrDl !mul1r opprD !addrA -addrA.
+  by rewrite [_ + 2^- _]addrC subrr addr0.
+split.
+  move=> x Nfx.
+  rewrite -filter_from_ballE.
+  case => eps /= eps0 HB.
+  pose n := (2 + truncn (- ln eps / ln 2))%N.
+  have eps0' : eps / 2 > 0 by exact: divr_gt0.
+  move/continuousP/(_ _ (ball_open (f_sum n x) eps0')) : (cf_sum n) => /= ofs.
+  rewrite nbhs_filterE.
+  rewrite fmapE.
+  rewrite nbhsE /=.
+  set B := _ @^-1` _ in ofs.
+  exists B.
+    split => //.
+    exact: ballxx.
+  apply: subset_trans (preimage_subset (f:=f) HB).
+  rewrite /B /preimage /ball => t /=.
+  have Hf y :
+    f y = f_sum n y + limn (fun n' => \sum_(n <= k < n') f_n k y * 2^- k.+1).
+    have /= := nondecreasing_telescope_sumey n _ (ndf_sum y).
+    rewrite EFin_lim // fin_numE /= => /(_ isT).
+    rewrite (eq_eseriesr (g:=fun i => ((f_n i \* cst (2 ^- i.+1)) y)%:E));
+      last first.
+      move => i _.
+      rewrite /f_sum -EFinD big_nat_recr //=.
+      by rewrite [X in X _ _ y]/GRing.add /= addrAC subrr add0r.
+    move/(f_equal (fun x => (f_sum n y)%:E + x)).
+    rewrite addrA addrAC -EFinB subrr add0r => H.
+    apply: EFin_inj.
+    rewrite -H EFinD; congr (_ + _).
+    rewrite -EFin_lim.
+      apply/congr_lim/boolp.funext => k /=.
+      elim: k => [|k IH]; first by rewrite !big_geq.
+      case: (leP n k) => nk.
+        by rewrite !big_nat_recr //= IH EFinD.
+      by rewrite !big_geq.
+    move: (Hcvg y).
+    by rewrite is_cvg_series_restrict f_sumE.
+  move=> feps.
+  rewrite !Hf opprD addrA (addrAC (f_sum n x)) -(addrA (_ - _)).
+  apply: (le_lt_trans (ler_normD _ _)).
+  rewrite (splitr eps).
+  apply: ltr_leD => //.
+  admit.
+apply/seteqP; split => x /= Hx.
+  apply: EFin_inj.
+  rewrite -EFin_lim // f_sumE EFin_series /= eseries0 // => i _ _ /=.
+  case: (Hn i) => _ _ /(_ (f_n i x)) /= => ->.
+    by rewrite mul0r.
+  by exists x.
+apply: contraPP Hx.
+rewrite (_ : (~ E x) = setC E x) // HE setC_bigcap /= => -[] i _ HU.
+case: (Hn i) => _ _ _ /(_ (f_n i x)) /= => Hfix.
+rewrite /f => Hf.
+have := (nondecreasing_cvgn_le (ndf_sum x) (Hcvg x) i.+1).
+have := ndf_sum x _ _ (leq0n i.+1).
+rewrite Hf {1}/f_sum big_geq // => /[swap] fi_le0.
+rewrite le0r ltNge fi_le0 orbF.
+rewrite /f_sum big_nat_recr //= [X in X _ _ x]/GRing.add /= -/(f_sum i).
+rewrite Hfix; last by exists x.
+rewrite mul1r /cst => /eqP fi0.
+have := ndf_sum x _ _ (leq0n i).
+rewrite {1}/f_sum big_geq // -[0 x]fi0 gerDl.
+by rewrite leNgt invr_gt0 exprn_gt0.
+Admitted.
+
+Let perfectly_normal_space_13 : perfectly_normal_space_Gdelta -> perfectly_normal_space' 0.
+move=> pnsGd E oE.
+case: (pnsGd) => nT cEGdE.
+have clcpE: closed (~`E).
+  by rewrite closedC.
+have[U oU hE]:= cEGdE (~` E) clcpE.
+have/boolp.choice[f_n Hn]: forall n, exists f : T -> R, 
+  [/\ continuous f, range f `<=` `[0, 1], f @` (~` E) `<=` [set 0] & f @` (~` U n) `<=` [set 1]].
+  move=> n.
+  apply/uniform_separatorP.
+  apply: normal_uniform_separator => //.
+  - by rewrite closedC.
+  - rewrite hE.
+    rewrite -subsets_disjoint.
+    exact: bigcap_inf.
+Admitted.
+
+Let perfectly_normal_space_23 : perfectly_normal_space 0 -> perfectly_normal_space' 0.
+Proof.
+Admitted.
+
+Let perfectly_normal_space_32 : perfectly_normal_space' 0 -> perfectly_normal_space 0.
+Proof.
+move=> pns' E cE; case: (pns' (~`E)).
+  by rewrite openC.
+move=> f [cf f0]; exists f.
+split.
+  by [].
+by rewrite -[RHS]setCK preimage_setC -f0 setCK.
+Qed.
+
+Let perfectly_normal_space_24 : perfectly_normal_space 0 -> perfectly_normal_space01.
+Proof.
+Admitted.
+
+Let perfectly_normal_space_34 : perfectly_normal_space' 0 -> perfectly_normal_space01.
+Proof.
+Admitted.
+
+Local Lemma perfectly_normal_space_42 :                                 
+perfectly_normal_space01  -> perfectly_normal_space 0.                  
+Proof.                                                                  
+move=> + E cE => /(_ E set0 cE closed0).                                
+rewrite disj_set2E setI0 eqxx => /(_ erefl).                            
+case=> f [] cf [] Ef [] _ _.                                            
+by exists f; split.                                                     
+Qed.                                                                    
+
+Let perfectly_normal_space_41 :
+  perfectly_normal_space01 -> perfectly_normal_space_Gdelta.
+Proof.            
+move=> pns01; split; first exact: perfectly_normal_space01_normal.
+move=> E cE; case:(perfectly_normal_space_42 pns01 cE) => // f [] cf Ef.
+exists (fun n => f @^-1` `]-n.+1%:R^-1,n.+1%:R^-1[%classic).
+  by move=> n; move/continuousP: cf; apply; exact: itv_open.
+rewrite -preimage_bigcap (_ : bigcap _ _ = [set 0])//.
+rewrite eqEsubset; split; last first.
+  by move=> x -> n _ /=; rewrite in_itv//= oppr_lt0 andbb invr_gt0.
+apply: subsetC2; rewrite setC_bigcap => x /= /eqP x0.
+exists (trunc `|x^-1|) => //=; rewrite in_itv/= -ltr_norml ltNge.
+apply/negP; rewrite negbK.
+by rewrite invf_ple ?posrE// ?normr_gt0// -normfV ltW// truncnS_gt.
+Qed.
+
+Theorem Vedenissoff_closed : perfectly_normal_space_Gdelta <-> perfectly_normal_space 0.
+Proof.
+move: perfectly_normal_space_12 perfectly_normal_space_23 perfectly_normal_space_34 perfectly_normal_space_41.
+tauto.
+Qed.
+
+Theorem Vedenissoff_open : perfectly_normal_space_Gdelta <-> perfectly_normal_space' 0.
+Proof.
+move: perfectly_normal_space_12 perfectly_normal_space_23 perfectly_normal_space_34 perfectly_normal_space_41.
+tauto.
+Qed.
+
+Theorem Vedenissoff01 : perfectly_normal_space_Gdelta <-> perfectly_normal_space01.
+Proof.
+move: perfectly_normal_space_12 perfectly_normal_space_23 perfectly_normal_space_34 perfectly_normal_space_41.
+tauto.
+Qed.
+
+End perfectlynormalspace.

theories/normedtype_theory/pseudometric_normed_Zmodule.v

@@ -95,6 +95,22 @@ Proof. by rewrite funeqE => x /=; rewrite addr0. Qed.
 Lemma center0 : center 0 = id.
 Proof. by rewrite oppr0 shift0. Qed.
 
+Lemma centerN (x y : R) : center x (- y) = - shift x y.
+Proof. by rewrite /shift opprD. Qed.
+
+Lemma shiftN (x y : R) : shift x (- y) = - center x y.
+Proof. by rewrite /shift opprD opprK. Qed.
+
+Lemma image_centerN (E : set R) (x : R) :
+  [set center x (- y) | y in E] =[set - (shift x y) | y in E].
+Proof.
+by apply/seteqP; split => y [] u Eu <- /=; exists u => //; rewrite opprD.
+Qed.
+
+Lemma image_shiftN (E : set R) (x : R) :
+  [set shift x (- y) | y in E] = [set - (center x y) | y in E].
+Proof. by rewrite -(opprK x) image_centerN opprK. Qed.
+
 End Shift.
 Arguments shift {R} x / y.
 Notation center c := (shift (- c)).