Merge pull request #243 from math-comp/measure_minor_cleaning

minor fix, generalizations, cleaning

Author: CohenCyril

CHANGELOG_UNRELEASED.md

@@ -67,6 +67,9 @@
     valid for a `uniformType`
   + moved `continuous_withinNx` from `normedType.v` to `topology.v` and
     generalised it to `uniformType`
+- moved from `measure.v` to `sequences.v`
+  + `ereal_nondecreasing_series`
+  + `ereal_nneg_series_lim_ge` (renamed from `series_nneg`)
 
 ### Renamed
 

theories/measure.v

@@ -6,12 +6,13 @@ Require Import boolp reals ereal.
 Require Import classical_sets posnum topology normedtype sequences.
 
 (******************************************************************************)
-(* This file provides basic elements of a theory of measure illustrated       *)
-(* by a formalization of Boole's inequality.                                  *)
+(* This file provides basic elements of a theory of measure.                  *)
 (*                                                                            *)
 (* {measure set T -> {ereal R}} == type of a measure over sets of elements of *)
 (*                                 type T where R is expected to be a         *)
 (*                                 a realFieldType or a realType              *)
+(*                                                                            *)
+(* Theorems: Boole_inequality, generalized_Boole_inequality                   *)
 (******************************************************************************)
 
 Set Implicit Arguments.
@@ -24,7 +25,7 @@ Local Open Scope ring_scope.
 
 (* TODO: move to classical_sets *)
 Definition triviset T (U : nat -> set T) :=
-  forall j i, (i != j)%nat -> U i `&` U j = set0.
+  forall i j, (i != j)%nat -> U i `&` U j = set0.
 
 Module Measurable.
 
@@ -256,7 +257,7 @@ Definition B_of (A : (set T) ^nat) :=
 Lemma triviset_B_of (A : (set T) ^nat) :
   {homo A : n m / (n <= m)%nat >-> n `<=` m} -> triviset (B_of A).
 Proof.
-move=> ndA j i; wlog : j i / (i < j)%N.
+move=> ndA i j; wlog : i j / (i < j)%N.
   move=> h; rewrite neq_ltn => /orP[|] ?; by
     [rewrite h // ltn_eqF|rewrite setIC h // ltn_eqF].
 move=> ij _; move: j i ij; case => // j [_ /=|n].
@@ -362,25 +363,6 @@ Qed.
 End boole_inequality.
 Notation le_mu_bigsetU := Boole_inequality.
 
-(* NB: see also nondecreasing_series *)
-Lemma ereal_nondecreasing_series (R : realFieldType) (u_ : {ereal R} ^nat) :
-  (forall n, 0%:E <= u_ n)%E ->
-  nondecreasing_seq (fun n => \sum_(i < n) u_ i)%E.
-Proof.
-move=> u_ge0 n m nm; rewrite -(subnKC nm).
-rewrite -[X in (_ <= X)%E](big_mkord xpredT) /index_iota subn0 iota_add.
-rewrite big_cat -[in X in (_ <= X + _)%E](subn0 n) big_mkord lee_addl //=.
-by rewrite sume_ge0.
-Qed.
-
-Lemma series_nneg (R : realType) (u_ : {ereal R} ^nat) k :
-  (forall n, (0%:E <= u_ n)%E) ->
-  (\sum_(i < k.+1) u_ i <= lim (fun n : nat => \sum_(i < n) u_ i))%E.
-Proof.
-move/ereal_nondecreasing_series/nondecreasing_seq_ereal_cvg/cvg_lim => -> //.
-by apply ereal_sup_ub; exists k.+1.
-Qed.
-
 Section generalized_boole_inequality.
 Variables (R : realType) (T : measurableType).
 Variable (mu : {measure set T -> {ereal R}}).
@@ -409,7 +391,7 @@ have -> : lim (mu \o B) = ereal_sup ((mu \o B) @` setT).
   exact: subset_bigsetU.
 have BA : forall m, (mu (B m) <= lim (fun n : nat => \sum_(i < n) mu (A i)))%E.
   move=> m; rewrite (le_trans (le_mu_bigsetU mu mA m.+1)) // -/(B m).
-  by apply: (@series_nneg _ (mu \o A)) => n; exact: measure_ge0.
+  by apply: (@ereal_nneg_series_lim_ge _ (mu \o A)) => n; exact: measure_ge0.
 by apply ub_ereal_sup => /= x [n _ <-{x}]; apply BA.
 Qed.
 

theories/sequences.v

@@ -775,7 +775,7 @@ Qed.
 
 Section sequences_of_extended_real_numbers.
 
-(* TODO: worth keeping in addition to cvgPpinfty? *)
+(* NB: worth keeping in addition to cvgPpinfty? *)
 Lemma cvgPpinfty_lt (R : realFieldType) (u_ : R^o ^nat) :
   u_ --> +oo <-> forall A, A > 0 -> \forall n \near \oo, A < u_ n.
 Proof.
@@ -786,7 +786,7 @@ exists n => // m nm; rewrite (@lt_le_trans _ _ (A *+ 2)) // ?mulr2n ?ltr_addr //
 exact: uoo.
 Qed.
 
-Lemma dvg_ereal_cvg (R : realType) (u_ : (R^o) ^nat) :
+Lemma dvg_ereal_cvg (R : realFieldType) (u_ : (R^o) ^nat) :
   u_ --> +oo -> (fun n => (u_ n)%:E) --> +oo%E.
 Proof.
 move/cvgPpinfty_lt => uoo; apply/cvg_ballP => _/posnumP[e]; rewrite near_map.
@@ -909,4 +909,23 @@ rewrite ltr_subl_addr addrC -ltr_subl_addr (lt_le_trans leum) //.
 by rewrite le_contract nd_u_//; near: n; exists m.
 Grab Existential Variables. all: end_near. Qed.
 
+(* NB: see also nondecreasing_series *)
+Lemma ereal_nondecreasing_series (R : realDomainType) (u_ : {ereal R} ^nat) :
+  (forall n, 0%:E <= u_ n)%E ->
+  nondecreasing_seq (fun n => \sum_(i < n) u_ i)%E.
+Proof.
+move=> u_ge0 n m nm; rewrite -(subnKC nm).
+rewrite -[X in (_ <= X)%E](big_mkord xpredT) /index_iota subn0 iota_add.
+rewrite big_cat -[in X in (_ <= X + _)%E](subn0 n) big_mkord lee_addl //=.
+by rewrite sume_ge0.
+Qed.
+
+Lemma ereal_nneg_series_lim_ge (R : realType) (u_ : {ereal R} ^nat) k :
+  (forall n, (0%:E <= u_ n)%E) ->
+  (\sum_(i < k) u_ i <= lim (fun n : nat => \sum_(i < n) u_ i))%E.
+Proof.
+move/ereal_nondecreasing_series/nondecreasing_seq_ereal_cvg/cvg_lim => -> //.
+by apply ereal_sup_ub; exists k.
+Qed.
+
 End sequences_of_extended_real_numbers.