diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index dd5e81e1d..dac6ec183 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -110,6 +110,59 @@ + structures `POrderedNbhs`, `POrderedTopological`, `POrderedUniform`, `POrderedPseudoMetric`, `POrderedPointedTopological` +- in `measurable_function.v`: + + lemma `preimage_set_system_compS` + +- in `numfun.v`: + + defintions `funrpos`, `funrneg` with notations `^\+` and `^\-` + + lemmas `funrpos_ge0`, `funrneg_ge0`, `funrposN`, `funrnegN`, `ge0_funrposE`, + `ge0_funrnegE`, `le0_funrposE`, `le0_funrnegE`, `ge0_funrposM`, `ge0_funrnegM`, + `le0_funrposM`, `le0_funrnegM`, `funrposDneg`, `funrposBneg`, + `funrD_posD`, `funrpos_le`, `funrneg_le` + + lemmas `funerpos`, `funerneg` + +- in `measurable_structure.v`: + + definitions `preimage_display`, `g_sigma_algebra_preimageType`, + `g_sigma_algebra_preimage` + + notations `.-preimage`, `.-preimage.-measurable` + +- in `measurable_realfun.v`: + + lemmas `measurable_funrpos`, `measurable_funrneg` + +- new file `independence.v`: + + definition `independent_events` + + definition `mutual_independence` + + lemma `eq_mutual_independence` + + definition `independence2`, `independence2P` + + lemmas `setI_closed_setT`, `setI_closed_set0` + + lemma `g_sigma_algebra_finite_measure_unique` + + lemma `mutual_independence_fset` + + lemma `mutual_independence_finiteS` + + theorem `mutual_independence_finite_g_sigma` + + lemma `mutual_dependence_bigcup` + + lemmas `g_sigma_algebra_preimage_comp`, `g_sigma_algebra_preimage_funrpos`, + `g_sigma_algebra_preimage_funrneg` + + definition `independent_RVs` + + lemma `independent_RVsD1` + + theorem `independent_generators` + + definition `independent_RVs2` + + lemmas `independent_RVs2_comp`, `independent_RVs2_funrposneg`, + `independent_RVs2_funrnegpos`, `independent_RVs2_funrnegneg`, + `independent_RVs2_funrpospos` + + definition `pairRV`, lemma `measurable_pairRV` + + lemmas `independent_RVs2_product_measure1` + + lemmas `independent_RVs2_setI_preimage`, + `independent_Lfun1_expectation_product_measure_lty` + + lemmas `expectationM_nnsfun`, `expectationM_ge0`, + `ge0_independent_expectationM`, `independent_Lfun1_expectationM_lty`, + `independent_Lfun1M`, `independent_expectationM` + +- in `functions.v`: + + lemma `addBrfctE` + +- in `ereal.v`: + + lemma `ge0_addBefctE` + ### Changed - in `charge.v`: @@ -163,6 +216,11 @@ `bounded_variationN`, `bounded_variationl`, `bounded_variationr`, `variations_opp`, `nondecreasing_bounded_variation` +- in `numfun.v`: + + `fune_abse` renamed to `funeposDneg` and direction of the equality changed + + `funeposneg` renamed to `funeposBneg` and direction of the equality changed + + `funeD_posD` renamed to `funeDB` and direction of the equality changed + ### Renamed - in `probability.v`: diff --git a/_CoqProject b/_CoqProject index aaec25f6d..32999d79a 100644 --- a/_CoqProject +++ b/_CoqProject @@ -124,6 +124,7 @@ theories/lebesgue_integral_theory/giry.v theories/ftc.v theories/hoelder.v theories/probability.v +theories/independence.v theories/convex.v theories/charge.v theories/kernel.v diff --git a/classical/functions.v b/classical/functions.v index 9f5b894c3..823d4c8a8 100644 --- a/classical/functions.v +++ b/classical/functions.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat. From HB Require Import structures. From mathcomp Require Import mathcomp_extra unstable boolp classical_sets. @@ -2700,6 +2700,10 @@ Proof. by elim: n => [//|n h]; rewrite funeqE=> ?; rewrite !mulrSr h. Qed. Lemma opprfctE (T : Type) (K : zmodType) (f : T -> K) : - f = (fun x => - f x). Proof. by []. Qed. +Lemma addBrfctE (U : Type) (K : zmodType) : + @interchange (U -> K) (fun a b => a - b) (fun a b => a + b). +Proof. by move=> a b c d; apply/funext => x; rewrite addrACA -opprD. Qed. + Lemma mulrfctE (T : Type) (K : pzRingType) (f g : T -> K) : f * g = (fun x => f x * g x). Proof. by []. Qed. diff --git a/classical/unstable.v b/classical/unstable.v index b12853ecc..b452fd14a 100644 --- a/classical/unstable.v +++ b/classical/unstable.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2022 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint. From mathcomp Require Import archimedean interval mathcomp_extra. diff --git a/reals/constructive_ereal.v b/reals/constructive_ereal.v index d9cd247e1..830b08c36 100644 --- a/reals/constructive_ereal.v +++ b/reals/constructive_ereal.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) (* -------------------------------------------------------------------- *) (* Copyright (c) - 2015--2016 - IMDEA Software Institute *) (* Copyright (c) - 2015--2018 - Inria *) diff --git a/theories/Make b/theories/Make index d67f6b544..727b5ddff 100644 --- a/theories/Make +++ b/theories/Make @@ -91,6 +91,7 @@ lebesgue_integral_theory/giry.v ftc.v hoelder.v probability.v +independence.v convex.v charge.v kernel.v diff --git a/theories/charge.v b/theories/charge.v index 2b4c9a09c..83bc873e4 100644 --- a/theories/charge.v +++ b/theories/charge.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval. From mathcomp Require Import finmap fingroup perm rat. @@ -2174,7 +2174,7 @@ Lemma Radon_Nikodym_change_of_variables f E : measurable E -> \int[mu]_(x in E) (f x * ('d (charge_of_finite_measure nu) '/d mu) x) = \int[nu]_(x in E) f x. Proof. -move=> mE mf; rewrite [in RHS](funeposneg f) integralB //; last 2 first. +move=> mE mf; rewrite -[in RHS](funeposBneg f) integralB //; last 2 first. - exact: integrable_funepos. - exact: integrable_funeneg. transitivity (\int[mu]_(x in E) (f x * Radon_Nikodym_SigmaFinite.f nu mu x)). @@ -2186,7 +2186,7 @@ transitivity (\int[mu]_(x in E) (f x * Radon_Nikodym_SigmaFinite.f nu mu x)). exact: measurable_int (Radon_Nikodym_SigmaFinite.f_integrable _). - apply: ae_eqe_mul2l. exact/ae_eq_sym/ae_eq_Radon_Nikodym_SigmaFinite. -rewrite [in LHS](funeposneg f). +rewrite -[in LHS](funeposBneg f). under [in LHS]eq_integral => x xE. rewrite muleBl; last 2 first. - exact: Radon_Nikodym_SigmaFinite.f_fin_num. - exact: add_def_funeposneg. diff --git a/theories/ereal.v b/theories/ereal.v index 8c0eb51a6..d7d4dc4c3 100644 --- a/theories/ereal.v +++ b/theories/ereal.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) (* -------------------------------------------------------------------- *) (* Copyright (c) - 2015--2016 - IMDEA Software Institute *) (* Copyright (c) - 2015--2018 - Inria *) @@ -58,11 +58,16 @@ Import numFieldTopology.Exports. From mathcomp Require Import mathcomp_extra unstable. Local Open Scope ring_scope. - Local Open Scope ereal_scope. - Local Open Scope classical_set_scope. +Lemma ge0_addBefctE (T : Type) (R : realDomainType) (a b c d : T -> \bar R) : + (forall x, 0 <= c x) -> (forall x, 0 <= d x) -> + a + b \- (c + d) = a \- c + (b \- d). +Proof. +by move=> ? ?; apply/funext=> x; rewrite !fctE addeACA oppeD ?ge0_adde_def ?inE. +Qed. + Lemma EFin_bigcup T (F : nat -> set T) : EFin @` (\bigcup_i F i) = \bigcup_i (EFin @` F i). Proof. diff --git a/theories/esum.v b/theories/esum.v index a25e061a8..9d4e14a75 100644 --- a/theories/esum.v +++ b/theories/esum.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From mathcomp Require Import all_ssreflect ssralg ssrnum finmap. From mathcomp Require Import mathcomp_extra boolp classical_sets functions. From mathcomp Require Import cardinality fsbigop reals ereal interval_inference. @@ -507,14 +507,14 @@ Proof. by move=> Df; rewrite summableN; exact: summableD. Qed. Lemma summable_funepos D f : summable D f -> summable D f^\+. Proof. -apply: le_lt_trans; apply le_esum => t Dt. -by rewrite -/((abse \o f) t) fune_abse gee0_abs// leeDl. +apply: le_lt_trans; apply: le_esum => t Dt. +by rewrite -/((abse \o f) t) -funeposDneg gee0_abs// leeDl. Qed. Lemma summable_funeneg D f : summable D f -> summable D f^\-. Proof. -apply: le_lt_trans; apply le_esum => t Dt. -by rewrite -/((abse \o f) t) fune_abse gee0_abs// leeDr. +apply: le_lt_trans; apply: le_esum => t Dt. +by rewrite -/((abse \o f) t) -funeposDneg gee0_abs// leeDr. Qed. End summable_lemmas. @@ -596,7 +596,7 @@ have -> : (C_ = A_ \- B_)%R. apply/funext => k. rewrite /= /A_ /C_ /B_ -sumrN -big_split/= -summable_fine_sum//. apply eq_bigr => i Pi; rewrite -fineB//. - - by rewrite [in LHS](funeposneg f). + - by rewrite -[in LHS](funeposBneg f). - by rewrite fin_num_abs (@summable_pinfty _ _ P) //; exact/summable_funepos. - by rewrite fin_num_abs (@summable_pinfty _ _ P) //; exact/summable_funeneg. by rewrite distrC; apply: hN; near: n; exists N. @@ -653,14 +653,14 @@ rewrite [X in _ == X -> _]addeC -sube_eq; last 2 first. - rewrite fin_num_adde_defr// ge0_esum_posneg//. rewrite (@eq_esum _ _ _ _ (abse \o f))// -?summableE// => i Di. by rewrite /= gee0_abs// f0. -rewrite -addeA addeCA eq_sym [X in _ == X -> _]addeC -sube_eq; last 2 first. - - rewrite ge0_esum_posneg//. - rewrite (@eq_esum _ _ _ _ (abse \o f))// -?summableE// => i Di. - by rewrite /= gee0_abs// f0. - - rewrite fin_num_adde_defl// ge0_esum_posneg//. - rewrite (@eq_esum _ _ _ _ (abse \o g))// -?summableE// => i Di. - by rewrite /= gee0_abs// g0. -by rewrite ge0_esum_posneg// ge0_esum_posneg// => /eqP ->. +rewrite -addeA addeCA eq_sym [X in _ == X -> _]addeC -sube_eq. +- by rewrite ge0_esum_posneg// ge0_esum_posneg// => /eqP ->. +- rewrite ge0_esum_posneg//. + rewrite (@eq_esum _ _ _ _ (abse \o f))// -?summableE// => i Di. + by rewrite /= gee0_abs// f0. +- rewrite fin_num_adde_defl// ge0_esum_posneg//. + rewrite (@eq_esum _ _ _ _ (abse \o g))// -?summableE// => i Di. + by rewrite /= gee0_abs// g0. Qed. End esumB. diff --git a/theories/independence.v b/theories/independence.v new file mode 100644 index 000000000..4bb172f63 --- /dev/null +++ b/theories/independence.v @@ -0,0 +1,1009 @@ +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) +From mathcomp Require Import all_ssreflect interval_inference. +From mathcomp Require Import ssralg poly ssrnum ssrint interval finmap. +From mathcomp Require Import mathcomp_extra boolp classical_sets functions. +From mathcomp Require Import cardinality fsbigop interval_inference. +From HB Require Import structures. +From mathcomp Require Import exp numfun lebesgue_measure lebesgue_integral. +From mathcomp Require Import reals ereal topology normedtype sequences. +From mathcomp Require Import esum measure exp numfun lebesgue_measure. +From mathcomp Require Import lebesgue_integral kernel probability. +From mathcomp Require Import hoelder. + +(**md**************************************************************************) +(* # Independence *) +(* *) +(* The status of this file is experimental. *) +(* *) +(* ``` *) +(* independent_events I F == the events F indexed by I are independent *) +(* mutual_independence I F == the set systems F indexed by I are independent *) +(* independent_RVs I X == the random variables X indexed by I are *) +(* independent *) +(* independent_RVs2 X Y == the random variables X and Y are independent *) +(* ``` *) +(* *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Import Order.TTheory GRing.Theory Num.Def Num.Theory. +Import numFieldTopology.Exports. + +Local Open Scope classical_set_scope. +Local Open Scope ring_scope. + +Section independent_events. +Context {R : realType} d {T : measurableType d} (P : probability T R) + {I0 : choiceType}. +Local Open Scope ereal_scope. + +Definition independent_events (I : set I0) (E : I0 -> set T) := + (forall i, I i -> measurable (E i)) /\ + forall J : {fset I0}, [set` J] `<=` I -> + P (\bigcap_(i in [set` J]) E i) = \prod_(i <- J) P (E i). + +End independent_events. + +Section mutual_independence. +Context {R : realType} d {T : measurableType d} (P : probability T R) + {I0 : choiceType}. +Local Open Scope ereal_scope. + +Definition mutual_independence (I : set I0) (F : I0 -> set_system T) := + (forall i, I i -> F i `<=` measurable) /\ + forall J : {fset I0}, [set` J] `<=` I -> + forall E, (forall i, i \in J -> E i \in F i) -> + P (\big[setI/setT]_(j <- J) E j) = \prod_(j <- J) P (E j). + +Lemma eq_mutual_independence (I : set I0) (F F' : I0 -> set_system T) : + (forall i, I i -> F i = F' i) -> + mutual_independence I F -> mutual_independence I F'. +Proof. +move=> FF' IF; split => [i Ii|J JI E EF']. + by rewrite -FF'//; apply IF. +apply IF => // i iJ. +by rewrite FF' ?EF'//; exact: JI. +Qed. + +End mutual_independence. + +Section independence. +Context {R : realType} d {T : measurableType d} (P : probability T R). +Local Open Scope ereal_scope. + +Definition independence2 (F G : set_system T) := + [/\ F `<=` measurable , G `<=` measurable & + forall A B, A \in F -> B \in G -> P (A `&` B) = P A * P B]. + +Lemma independence2P (F G : set_system T) : independence2 F G <-> + mutual_independence P [set: bool] (fun b => if b then F else G). +Proof. +split=> [[mF mG FG]|/= indeF]. + split=> [/= [|]//|/= J Jbool E EF]. + have [/eqP Jtrue|/eqP Jfalse| |] := set_bool [set` J]. + - rewrite -bigcap_fset Jtrue bigcap_set1. + by rewrite fsbig_seq ?Jtrue ?fsbig_set1. + - rewrite -bigcap_fset Jfalse bigcap_set1. + by rewrite fsbig_seq// ?Jfalse fsbig_set1. + - rewrite set_seq_eq0 => /eqP ->. + by rewrite !big_nil probability_setT. + - rewrite setT_bool => /eqP {}Jbool. + rewrite -bigcap_fset Jbool bigcap_setU1 bigcap_set1. + rewrite FG//. + + rewrite -(set_fsetK J) Jbool. + rewrite fset_setU1//= fset_set1. + rewrite big_fsetU1 ?inE//=. + by rewrite big_seq_fset1. + + rewrite EF//. + rewrite -(set_fsetK J) Jbool. + by rewrite in_fset_set// !inE/=; left. + + rewrite EF//. + rewrite -(set_fsetK J) Jbool. + by rewrite in_fset_set// !inE/=; right. +split. +- by case: indeF => /= [+ _] => /(_ true); exact. +- by case: indeF => /= [+ _] => /(_ false); exact. +- move=> A B AF BG. + case: indeF => _ /= /(_ [fset true; false]%fset _ (fun b => if b then A else B)). + do 2 rewrite big_fsetU1/= ?inE//= big_seq_fset1. + by apply => // -[]. +Qed. + +End independence. + +Lemma setI_closed_setT T (F : set_system T) : + setI_closed F -> setI_closed (F `|` [set setT]). +Proof. +move=> IF=> C D [FC|/= ->{C}]. +- move=> [FD|/= ->{D}]. + by left; exact: IF. + by rewrite setIT; left. +- move=> [FD|->{D}]. + by rewrite setTI; left. + by rewrite !setTI; right. +Qed. + +Lemma setI_closed_set0 T (F : set_system T) : + setI_closed F -> setI_closed (F `|` [set set0]). +Proof. +move=> IF=> C D [FC|/= ->{C}]. +- move=> [FD|/= ->{D}]. + by left; exact: IF. + by rewrite setI0; right. +- move=> [FD|->{D}]. + by rewrite set0I; right. + by rewrite !set0I; right. +Qed. + +Lemma g_sigma_algebra_finite_measure_unique {d} {T : measurableType d} + {R : realType} (G : set_system T) : + G `<=` d.-measurable -> + setI_closed G -> + forall m1 m2 : {finite_measure set T -> \bar R}, + m1 [set: T] = m2 [set: T] -> + (forall A : set T, G A -> m1 A = m2 A) -> + forall E : set T, <> E -> m1 E = m2 E. +Proof. +move=> Gm IG m1 m2 m1m2T m1m2 E sGE. +apply: (@g_sigma_algebra_measure_unique _ _ _ + (G `|` [set setT]) _ (fun=> setT)) => //. +- by move=> A [/Gm//|/= ->//]. +- by right. +- by rewrite bigcup_const. +- exact: setI_closed_setT. +- by move=> B [/m1m2 //|/= ->]. +- by move=> n; apply: fin_num_fun_lty; exact: fin_num_measure. +- move: E sGE; apply: smallest_sub => // C GC. + by apply: sub_gen_smallest; left. +Qed. + +Section mutual_independence_properties. +Context {R : realType} d {T : measurableType d} (P : probability T R). +Local Open Scope ereal_scope. + +(**md see Achim Klenke's Probability Theory, Ch.2, sec.2.1, thm.2.13(i) *) +Lemma mutual_independence_fset {I0 : choiceType} (I : {fset I0}) + (F : I0 -> set_system T) : + (forall i, i \in I -> F i `<=` measurable /\ (F i) [set: T]) -> + mutual_independence P [set` I] F <-> + forall E, (forall i, i \in I -> E i \in F i) -> + P (\big[setI/setT]_(j <- I) E j) = \prod_(j <- I) P (E j). +Proof. +move=> mF; split=> [indeF E EF|indeF]; first by apply indeF. +split=> [i /mF[]//|J JI E EF]. +pose E' i := if i \in J then E i else [set: T]. +have /indeF : forall i, i \in I -> E' i \in F i. + move=> i iI; rewrite /E'; case: ifPn => [|iJ]; first exact: EF. + by rewrite inE; apply mF. +move/fsubsetP : (JI) => /(big_fset_incl _) <- /=; last first. + by move=> j jI jJ; rewrite /E' (negbTE jJ). +move/fsubsetP : (JI) => /(big_fset_incl _) <- /=; last first. + by move=> j jI jJ; rewrite /E' (negbTE jJ); rewrite probability_setT. +rewrite big_seq [in X in X = _ -> _](eq_bigr E); last first. + by move=> i iJ; rewrite /E' iJ. +rewrite -big_seq => ->. +by rewrite !big_seq; apply: eq_bigr => i iJ; rewrite /E' iJ. +Qed. + +(**md see Achim Klenke's Probability Theory, Ch.2, sec.2.1, thm.2.13(ii) *) +Lemma mutual_independence_finiteS {I0 : choiceType} (I : set I0) + (F : I0 -> set_system T) : + mutual_independence P I F <-> + (forall J : {fset I0}%fset, [set` J] `<=` I -> + mutual_independence P [set` J] F). +Proof. +split=> [indeF J JI|indeF]. + split=> [i /JI Ii|K KJ E EF]. + by apply indeF. + by apply indeF => // i /KJ /JI. +split=> [i Ii|J JI E EF]. + have : [set` [fset i]%fset] `<=` I. + by move=> j; rewrite /= inE => /eqP ->. + by move/indeF => [+ _]; apply; rewrite /= inE. +by have [_] := indeF _ JI; exact. +Qed. + +(**md see Achim Klenke's Probability Theory, Ch.2, sec.2.1, thm.2.13(iii) *) +Theorem mutual_independence_finite_g_sigma {I0 : choiceType} (I : set I0) + (F : I0 -> set_system T) : + (forall i, i \in I -> setI_closed (F i `|` [set set0])) -> + mutual_independence P I F <-> mutual_independence P I (fun i => <>). +Proof. +split=> indeF; last first. + split=> [i Ii|J JI E EF]. + case: indeF => /(_ _ Ii) + _. + by apply: subset_trans; exact: sub_gen_smallest. + apply indeF => // i iJ; rewrite inE. + by apply: sub_gen_smallest; exact/set_mem/EF. +split=> [i Ii|K KI E EF]. + case: indeF => + _ => /(_ _ Ii). + by apply: smallest_sub; exact: sigma_algebra_measurable. +suff: forall J J' : {fset I0}%fset, (J `<=` J')%fset -> [set` J'] `<=` I -> + forall E, (forall i, i \in J -> E i \in <>) -> + (forall i, i \in [set` J'] `\` [set` J] -> E i \in F i) -> + P (\big[setI/setT]_(j <- J') E j) = \prod_(j <- J') P (E j). + move=> /(_ K K (@fsubset_refl _ _) KI E); apply. + - by move=> i iK; exact: EF. + - by move=> i; rewrite setDv inE. +move=> {E EF K KI}. +apply: finSet_rect => K ih J' KJ' J'I E EsF EF. +have [K0|/fset0Pn[j jK]] := eqVneq K fset0. + apply indeF => // i iJ'. + by apply: EF; rewrite !inE; split => //=; rewrite K0 inE. +pose J := (K `\ j)%fset. +have jI : j \in I by apply/mem_set/J'I => /=; move/fsubsetP : KJ'; exact. +have JK : (J `<` K)%fset by rewrite fproperD1. +have JjJ' : (j |` J `<=` J')%fset by apply: fsubset_trans KJ'; rewrite fsetD1K. +have JJ' : (J `<=` J')%fset by apply: fsubset_trans JjJ'; exact: fsubsetU1. +have J'mE i : i \in J' -> d.-measurable (E i). + move=> iJ'. + case: indeF => + _ => /(_ i (J'I _ iJ')) Fim. + suff: <> (E i). + by apply: smallest_sub => //; exact: sigma_algebra_measurable. + apply/set_mem. + have [iK|iK] := boolP (i \in K); first by rewrite EsF. + have /set_mem : E i \in F i. + by rewrite EF// !inE/=; split => //; exact/negP. + by move/sub_sigma_algebra => /(_ setT)/mem_set. +have mmu : measurable (\big[setI/[set: T]]_(j0 <- (J' `\ j)%fset) (E j0)). + rewrite big_seq; apply: bigsetI_measurable => i /[!inE] /andP[_ iJ']. + exact: J'mE. +pose mu0 A := P (A `&` \big[setI/[set: T]]_(j0 <- (J' `\ j)%fset) (E j0)). +pose mu := [the {finite_measure set _ -> \bar _} of mrestr P mmu]. +pose nu0 A := P A * \prod_(j0 <- (J' `\ j)%fset) P (E j0). +have nuk : (0 <= fine (\prod_(j0 <- (J' `\ j)%fset) P (E j0)))%R. + by rewrite fine_ge0// prode_ge0. +pose nu := [the measure _ _ of mscale (NngNum nuk) P]. +have nuEj A : nu A = P A * \prod_(j0 <- (J' `\ j)%fset) P (E j0). + rewrite /nu/= /mscale muleC/= fineK// big_seq. + rewrite prode_fin_num// => i /[!inE]/= /andP[ij iJ']. + by rewrite fin_num_measure//; exact: J'mE. +have JJ'j : (J `<=` J' `\ j)%fset by exact: fsetSD. +have J'jI : [set` (J' `\ j)%fset] `<=` I. + by apply: subset_trans J'I; apply/fsubsetP; exact: fsubsetDl. +have jJ' : j \in J' by move/fsubsetP : JjJ'; apply; rewrite !inE eqxx. +have Fjmunu A : (F j `|` [set set0; setT]) A -> mu A = nu A. + move=> [FjA|[->|->]]. + - rewrite nuEj. + pose E' i := if i == j then A else E i. + transitivity (P (\big[setI/[set: T]]_(j <- J') E' j)). + rewrite [in RHS](big_fsetD1 j)//= {1}/E' eqxx//; congr (P (_ `&` _)). + rewrite !big_seq; apply: eq_bigr => i /[!inE] /andP[ij _]. + by rewrite /E' (negbTE ij). + transitivity (\prod_(j <- J') P (E' j)); last first. + rewrite [in LHS](big_fsetD1 j)//= {1}/E' eqxx//; congr (P _ * _). + rewrite !big_seq; apply: eq_bigr => i /[!inE] /andP[ij _]. + by rewrite /E' (negbTE ij). + apply: (ih _ JK _ JJ' J'I). + + move=> i iJ. + rewrite /E'; case: ifPn => [/eqP ->|ij]. + by rewrite inE; exact: sub_sigma_algebra. + rewrite EsF//. + by move/fproper_sub : JK => /fsubsetP; apply. + + move=> i. + rewrite ![in X in X -> _]inE /= ![in X in X -> _]inE => -[] iJ' /negP. + rewrite negb_and negbK => /predU1P[ij|iK]. + by rewrite /E' ij eqxx inE. + rewrite /E' ifF//; last first. + apply/negP => /eqP ij. + by rewrite ij jK in iK. + by rewrite EF// !inE/=; split => //; exact/negP. + - by rewrite !measure0. + - rewrite /mu /= /mrestr /= nuEj setTI probability_setT mul1e. + apply: (ih _ JK _ JJ'j J'jI). + + move=> i iJ. + rewrite EsF//. + by move/fproper_sub : JK => /fsubsetP; apply. + + move=> i. + rewrite ![in X in X -> _]inE /= ![in X in X -> _]inE => -[]. + move=> /andP[-> iJ'] iK. + by rewrite EF// !inE. +have sFjmunu A : <> A -> mu A = nu A. + move=> sFjA. + apply: (@g_sigma_algebra_finite_measure_unique _ _ _ (F j `|` [set set0])). + - move=> B /= [|->//]. + move: B. + case: indeF => + _. + by apply; exact/set_mem. + - exact: H. + - rewrite /mu/= /mrestr/= nuEj setTI probability_setT mul1e. + apply: (ih _ JK _ JJ'j J'jI). + + move=> i iJ. + rewrite EsF//. + by move/fproper_sub : JK => /fsubsetP; apply. + + move=> i. + rewrite ![in X in X -> _]inE /= ![in X in X -> _]inE => -[]. + move=> /andP[ij iJ']. + rewrite ij/= => iK. + by rewrite EF// !inE. + - move=> B FjB; apply: Fjmunu. + case: FjB => [|->//]; first by left. + by right; left. + - by move: sFjA; exact: sub_smallest2r. +rewrite [in LHS](big_fsetD1 j)//= [in RHS](big_fsetD1 j)//=. +have /sFjmunu : <> (E j) by apply/set_mem; rewrite EsF. +by rewrite /mu/= /mrestr/= nuEj. +Qed. + +(* pick an i from I_ k s.t. E k \in F (f k) *) +Local Definition f (K0 : choiceType) (K : {fset K0}) + (I0 : pointedType) (F : I0 -> set_system T) E I_ + (EF : forall k, k \in K -> E k \in \bigcup_(i in I_ k) F i) k := + if pselect (k \in K) is left kK then + sval (cid2 (set_mem (EF _ kK))) + else + point. + +Local Lemma f_prop (K0 : choiceType) (K : {fset K0}) + (I0 : pointedType) (F : I0 -> set_system T) E I_ + (EF : forall k, k \in K -> E k \in \bigcup_(i in I_ k) F i) k : + k \in K -> E k \in F (f EF k). +Proof. +move=> kK; rewrite /f; case: pselect => [kK_/=|//]. +by case: cid2 => // i/= ? /mem_set. +Qed. + +Local Lemma f_inj (K0 : choiceType) (K K' : {fset K0}) + (K'K : [set` K'] `<=` [set` K]) (I0 : pointedType) (F : I0 -> set_system T) + E I_ (EF : forall k, k \in K' -> E k \in \bigcup_(i in I_ k) F i) + (KI : trivIset [set` K] I_) k1 k2 : + k1 \in K' -> k2 \in K'-> k1 != k2 -> + ([fset f EF k1] `&` [fset f EF k2] = fset0)%fset. +Proof. +move=> k1K' k2K' k1k2; apply/eqP; rewrite -fsubset0. +apply/fsubsetP => i /[!inE] /andP[]. +rewrite /f; case: pselect => // k1K'_. +case: cid2 => // i' i'k1 Fi' /eqP ->{i}. +case: pselect => // k2K'_. +case: cid2 => // j ik2 FjEk2 /eqP/= i'j. +rewrite -{j}i'j in ik2 FjEk2 *. +move/trivIsetP : KI => /(_ _ _ (K'K _ k1K') (K'K _ k2K') k1k2). +by move/seteqP => [+ _] => /(_ i')/=; rewrite -falseE; exact. +Qed. + +Local Definition g (K0 : pointedType) (K' : {fset K0}) + (I0 : Type) (I_ : K0 -> set I0) (i : I0) : K0 := + if pselect (exists2 k, k \in K' & i \in I_ k) is left e then + sval (cid2 e) + else + point. + +Local Lemma gf (K0 I0 : pointedType) (F : I0 -> set_system T) + (K K' : {fset K0}) (K'K : [set` K'] `<=` [set` K]) + E I_ (EF : forall k, k \in K' -> E k \in \bigcup_(i in I_ k) F i) + (KI : trivIset [set` K] (fun i => I_ i)) i : i \in K' -> g K I_ (f EF i) = i. +Proof. +move=> iK'; rewrite /f; case: pselect => // iK'_. +case: cid2 => // j jIi FjEi. +rewrite /g; case: pselect => // k'; last first. + exfalso; apply: k'. + by exists i => //; [exact: K'K|exact/mem_set]. +case: cid2 => //= k'' k''K jIk'. +apply/eqP/negP => /negP k'i. +move/trivIsetP : KI => /(_ k'' i) /= /(_ k''K (K'K _ iK') k'i)/= /eqP. +apply/negP/set0P; exists j; split => //. +exact/set_mem. +Qed. + +(**md see Achim Klenke's Probability Theory, Ch.2, sec.2.1, thm.2.13(iv) *) +Lemma mutual_independence_bigcup (K0 I0 : pointedType) (K : {fset K0}) + (I_ : K0 -> set I0) (I : set I0) (F : I0 -> set_system T) : + trivIset [set` K] (fun i => I_ i) -> + (forall k, k \in K -> I_ k `<=` I) -> + mutual_independence P I F -> + mutual_independence P [set` K] (fun k => \bigcup_(i in I_ k) F i). +Proof. +move=> KI I_I PIF. +split=> [i Ki A [j ij FjA]|K' K'K E EF]. + by case: PIF => + _ => /(_ j); apply => //; exact: (I_I _ Ki). +case: PIF => Fm PIF. +pose f' := f EF. +pose g' := g K I_. +pose J' := (\bigcup_(k <- K') [fset f' k])%fset. +pose E' := E \o g'. +suff: P (\big[setI/[set: T]]_(j <- J') E' j) = \prod_(j <- J') P (E' j). + move=> suf. + transitivity (P (\big[setI/[set: T]]_(j <- J') E' j)). + congr (P _). + apply/seteqP; split=> [t|]. + rewrite -!bigcap_fset => L j => /bigfcupP[k] /[!andbT] kK'. + rewrite !inE => /eqP ->{j}. + apply: L => //=. + by rewrite /g' /f' gf. + move=> t. + rewrite -!bigcap_fset => L j/= jK'. + have /= := L (f' j). + rewrite /E' /g' /f'/= gf//. + by apply; apply/bigfcupP; exists j;[rewrite jK'|rewrite inE]. + rewrite suf partition_disjoint_bigfcup//=. + rewrite [LHS]big_seq [RHS]big_seq; apply: eq_big => // k kK'. + by rewrite big_seq_fset1 /E' /g' /f' /= gf. + move=> k1 k2 k1K' k2K' k1k2 /=. + by rewrite -fsetI_eq0 -fsubset0 (f_inj K'K). +apply: PIF. +- move=> i/= /bigfcupP[k] /[!andbT] kK' /[!inE] /eqP ->. + apply: (I_I k) => /=; first exact: K'K. + by rewrite /f' /f; case: pselect => // kK'_; case: cid2. +- move=> i /bigfcupP[k'] /[!andbT] k'K /[!inE] /eqP ->{i}. + by rewrite /E' /g' /f'/= gf//; exact/set_mem/f_prop. +Qed. + +End mutual_independence_properties. + +Section g_sigma_algebra_preimage_lemmas. +Context d {T : measurableType d} {R : realType}. + +Lemma g_sigma_algebra_preimage_comp (X : {mfun T >-> R}) (f : R -> R) : + measurable_fun setT f -> + g_sigma_algebra_preimage (f \o X)%R `<=` g_sigma_algebra_preimage X. +Proof. exact: preimage_set_system_compS. Qed. + +Lemma g_sigma_algebra_preimage_funrpos (X : {mfun T >-> R}) : + g_sigma_algebra_preimage X^\+%R `<=` d.-measurable. +Proof. +by move=> A/= -[B mB] <-; have := measurable_funrpos (measurable_funP X); exact. +Qed. + +Lemma g_sigma_algebra_preimage_funrneg (X : {mfun T >-> R}) : + g_sigma_algebra_preimage X^\-%R `<=` d.-measurable. +Proof. +by move=> A/= -[B mB] <-; have := measurable_funrneg (measurable_funP X); exact. +Qed. + +End g_sigma_algebra_preimage_lemmas. +Arguments g_sigma_algebra_preimage_comp {d T R X} f. + +Section independent_RVs. +Context {R : realType} d (T : measurableType d). +Context {I0 : choiceType}. +Context {d' : I0 -> _} (T' : forall i : I0, measurableType (d' i)). +Variable P : probability T R. + +Definition independent_RVs (I : set I0) + (X : forall i : I0, {mfun T >-> T' i}) : Prop := + mutual_independence P I (fun i => g_sigma_algebra_preimage (X i)). + +End independent_RVs. + +Section independent_RVs_properties. +Context {R : realType} d d' (T : measurableType d) (T' : measurableType d'). +Variable P : probability T R. + +Lemma independent_RVsD1 (I : {fset nat}) i0 (X : {RV P >-> T'}^nat) : + independent_RVs P [set` I] X -> independent_RVs P [set` (I `\ i0)%fset] X. +Proof. +move=> PIX; split => [/= i|/= J JIi0 E EK]. + by rewrite !inE => /andP[ii0 iI]; exact: PIX.1. +by apply: PIX.2 => //= x /JIi0 /=; rewrite !inE => /andP[]. +Qed. + +End independent_RVs_properties. + +Section independent_generators. +Context {R : realType} d (T : measurableType d). +Context {I0 : choiceType}. +Context {d' : I0 -> _} (T' : forall i : I0, measurableType (d' i)). +Variable P : probability T R. + +(**md see Achim Klenke's Probability Theory, Ch.2, sec.2.1, thm.2.16 *) +Theorem independent_generators (I : set I0) (F : forall i : I0, set_system (T' i)) + (X : forall i, {RV P >-> T' i}) : + (forall i, i \in I -> setI_closed (F i)) -> + (forall i, i \in I -> F i `<=` @measurable _ (T' i)) -> + (forall i, i \in I -> @measurable _ (T' i) = <>) -> + mutual_independence P I (fun i => preimage_set_system setT (X i) (F i)) -> + independent_RVs P I X. +Proof. +move=> IF FA AsF indeX1. +have closed_preimage i : I i -> setI_closed (preimage_set_system setT (X i) (F i)). + move=> Ii A B [A' FiA']; rewrite setTI => <-{A}. + move=> [B' FiB']; rewrite setTI => <-{B}. + rewrite /preimage_set_system/=; exists (A' `&` B'). + apply: IF => //. + - exact/mem_set. + - by rewrite setTI. +have gen_preimage i : I i -> + <> = + g_sigma_algebra_preimage (X i). + move=> Ii. + rewrite /g_sigma_algebra_preimage AsF; last exact/mem_set. + by rewrite -g_sigma_preimageE. +rewrite /independent_RVs. +suff: mutual_independence P I + (fun i => <>). + exact: eq_mutual_independence. +apply: (mutual_independence_finite_g_sigma _ _).1. +- by move=> i Ii; apply: setI_closed_set0; exact/closed_preimage/set_mem. +- split => [i Ii A [A' mA'] <-{A}|J JI E EF]. + by apply/measurable_funP => //; apply: FA => //; exact/mem_set. + by apply indeX1. +Qed. + +End independent_generators. + +Section independent_RVs2. +Context {R : realType} d d' (T : measurableType d) (T' : measurableType d'). +Variable P : probability T R. + +Definition independent_RVs2 (X Y : {mfun T >-> T'}) := + independent_RVs P [set: bool] (fun b => if b then Y else X). + +End independent_RVs2. + +Section independent_RVs2_properties. +Context {R : realType} d d' (T : measurableType d) (T' : measurableType d'). +Variable P : probability T R. +Local Open Scope ring_scope. + +Lemma independent_RVs2_comp (X Y : {RV P >-> R}) (f g : {mfun R >-> R}) : + independent_RVs2 P X Y -> independent_RVs2 P (f \o X) (g \o Y). +Proof. +move=> indeXY; split => /=. +- move=> [] _ /= A. + + by rewrite /g_sigma_algebra_preimage/= /preimage_set_system/= => -[B mB <-]; + exact/measurableT_comp. + + by rewrite /g_sigma_algebra_preimage/= /preimage_set_system/= => -[B mB <-]; + exact/measurableT_comp. +- move=> J _ E JE. + apply indeXY => //= i iJ; have := JE _ iJ. + by move: i {iJ} =>[|]//=; rewrite !inE => Eg; + exact: g_sigma_algebra_preimage_comp Eg. +Qed. + +Lemma independent_RVs2_funrposneg (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> independent_RVs2 P X^\+ Y^\-. +Proof. +move=> indeXY; split=> [[|]/= _|J J2 E JE]. +- exact: g_sigma_algebra_preimage_funrneg. +- exact: g_sigma_algebra_preimage_funrpos. +- apply indeXY => //= i iJ; have := JE _ iJ. + move/J2 : iJ; move: i => [|]// _; rewrite !inE. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr (- x) 0)%R). + exact: measurable_funrneg. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr x 0)%R) => //. + exact: measurable_funrpos. +Qed. + +Lemma independent_RVs2_funrnegpos (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> independent_RVs2 P X^\- Y^\+. +Proof. +move=> indeXY; split=> [/= [|]// _ |J J2 E JE]. +- exact: g_sigma_algebra_preimage_funrpos. +- exact: g_sigma_algebra_preimage_funrneg. +- apply indeXY => //= i iJ; have := JE _ iJ. + move/J2 : iJ; move: i => [|]// _; rewrite !inE. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr x 0)%R). + exact: measurable_funrpos. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr (- x) 0)%R). + exact: measurable_funrneg. +Qed. + +Lemma independent_RVs2_funrnegneg (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> independent_RVs2 P X^\- Y^\-. +Proof. +move=> indeXY; split=> [/= [|]// _ |J J2 E JE]. +- exact: g_sigma_algebra_preimage_funrneg. +- exact: g_sigma_algebra_preimage_funrneg. +- apply indeXY => //= i iJ; have := JE _ iJ. + move/J2 : iJ; move: i => [|]// _; rewrite !inE. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr (- x) 0)%R). + exact: measurable_funrneg. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr (- x) 0)%R). + exact: measurable_funrneg. +Qed. + +Lemma independent_RVs2_funrpospos (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> independent_RVs2 P X^\+ Y^\+. +Proof. +move=> indeXY; split=> [/= [|]//= _ |J J2 E JE]. +- exact: g_sigma_algebra_preimage_funrpos. +- exact: g_sigma_algebra_preimage_funrpos. +- apply indeXY => //= i iJ; have := JE _ iJ. + move/J2 : iJ; move: i => [|]// _; rewrite !inE. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr x 0)%R). + exact: measurable_funrpos. + + apply: (g_sigma_algebra_preimage_comp (fun x => maxr x 0)%R). + exact: measurable_funrpos. +Qed. + +End independent_RVs2_properties. + +Section pairRV. +Context d d' {T : measurableType d} {T' : measurableType d'} {R : realType} + (P : probability T R). + +Definition pairRV (X Y : {RV P >-> T'}) : T * T -> T' * T' := + (fun x => (X x.1, Y x.2)). + +Lemma measurable_pairM (X Y : {RV P >-> T'}) : measurable_fun setT (pairRV X Y). +Proof. +rewrite /pairRV. +apply/measurable_fun_pairP; split => //=. +- rewrite [X in measurable_fun _ X](_ : _ = X \o fst)//. + exact/measurableT_comp. +- rewrite [X in measurable_fun _ X](_ : _ = Y \o snd)//. + exact/measurableT_comp. +Qed. + +HB.instance Definition _ (X Y : {RV P >-> T'}) := + @isMeasurableFun.Build _ _ _ _ (pairRV X Y) (measurable_pairM X Y). + +End pairRV. + +Section independent_RVs2_properties_realType. +Context {R : realType} d (T : measurableType d). +Variable P : probability T R. +Local Open Scope ereal_scope. + +Lemma independent_RVs2_setI_preimage (X Y : {mfun T >-> R}) (A1 A2 : set R) : + measurable A1 -> measurable A2 -> + independent_RVs2 P X Y -> + P (X @^-1` A1 `&` Y @^-1` A2) = P (X @^-1` A1) * P (Y @^-1` A2). +Proof. +move=> mA1 mA2. +rewrite /independent_RVs2 /independent_RVs /mutual_independence /= => -[_]. +move/(_ [fset false; true]%fset (@subsetT _ _) + (fun b => if b then Y @^-1` A2 else X @^-1` A1)). +rewrite !big_fsetU1 ?inE//= !big_seq_fset1/=. +apply => -[|] /= _; rewrite !inE; rewrite /g_sigma_algebra_preimage. +by exists A2 => //; rewrite setTI. +by exists A1 => //; rewrite setTI. +Qed. + +Lemma independent_RVs2_product_measure1 (X Y : {RV P >-> R}) (A1 A2 : set R) : + measurable A1 -> measurable A2 -> + independent_RVs2 P X Y -> + (P \x P) (pairRV X Y @^-1` (A1 `*` A2)) = P (X @^-1` A1 `&` Y @^-1` A2). +Proof. +move=> mA1 mA2 iPXY. +rewrite (_ : (pairRV X Y @^-1` (A1 `*` A2)) = + (X @^-1` A1) `*` (Y @^-1` A2)); last first. + by apply/seteqP; split => [[x1 x2]|[x1 x2]]. +rewrite product_measure1E. +- by rewrite independent_RVs2_setI_preimage. +- by rewrite -[_ @^-1` _]setTI; exact: (measurable_funP X). +- by rewrite -[_ @^-1` _]setTI; exact: (measurable_funP Y). +Qed. + +End independent_RVs2_properties_realType. + +Section product_expectation_over_product_measure. +Context {R : realType} d (T : measurableType d). +Variable P : probability T R. +Local Open Scope ereal_scope. + +Lemma independent_Lfun1_expectation_product_measure_lty (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> + (X : _ -> _) \in Lfun P 1 -> (Y : _ -> _) \in Lfun P 1 -> + 'E_(P \x P) [(fun x => `|X x.1 * Y x.2|)%R] < +oo. +Proof. +move=> indeXY iX iY. +rewrite unlock. +rewrite [ltLHS](_ : _ = + \int[distribution (P \x P) (pairRV X Y)%R]_x `|x.1 * x.2|%:E); last first. + rewrite ge0_integral_distribution//=; last first. + apply/measurable_EFinP => //=. + by apply/measurableT_comp => //=; exact/measurable_funM. +rewrite [ltLHS](_ : _ = + \int[distribution P X \x distribution P Y]_x `|x.1 * x.2|%:E); last first. + apply: eq_measure_integral => // A mA _. + apply/esym. (* NB: don't simpl here! *) + apply: product_measure_unique => //= A1 A2 mA1 mA2. + by rewrite independent_RVs2_product_measure1// independent_RVs2_setI_preimage. +rewrite fubini_tonelli1//=; last first. + apply/measurable_EFinP => /=; apply/measurableT_comp => //=. + exact/measurable_funM. +rewrite /fubini_F/=. +rewrite [ltLHS](_ : _ = \int[distribution P X]_x `|x|%:E * + \int[distribution P Y]_y `|y|%:E); last first. + rewrite -ge0_integralZr//=; last 2 first. + exact/measurable_EFinP. + exact: integral_ge0. + apply: eq_integral => x _. + rewrite -ge0_integralZl//=. + by under eq_integral do rewrite normrM. + exact/measurable_EFinP. +rewrite ge0_integral_distribution//=; last exact/measurable_EFinP. +rewrite ge0_integral_distribution//=; last exact/measurable_EFinP. +rewrite lte_mul_pinfty//. +- exact: integral_ge0. +- apply: integrable_fin_num => //=. + by move/Lfun1_integrable : iX => /integrable_abse. +- apply: integrable_lty => //. + by move/Lfun1_integrable : iY => /integrable_abse. +Qed. + +End product_expectation_over_product_measure. + +Section product_expectation. +Context {R : realType} d (T : measurableType d). +Variable P : probability T R. +Local Open Scope ereal_scope. + +Import HBNNSimple. + +Lemma expectationM_nnsfun (f g : {nnsfun T >-> R}) : + (forall y y', y \in range f -> y' \in range g -> + P (f @^-1` [set y] `&` g @^-1` [set y']) = + P (f @^-1` [set y]) * P (g @^-1` [set y'])) -> + 'E_P [f \* g] = 'E_P [f] * 'E_P [g]. +Proof. +move=> fg; transitivity + ('E_P [(fun x => (\sum_(y \in range f) y * \1_(f @^-1` [set y]) x)%R) + \* (fun x => (\sum_(y \in range g) y * \1_(g @^-1` [set y]) x)%R)]). + by congr ('E_P [_]); apply/funext => t/=; rewrite (fimfunE f) (fimfunE g). +transitivity ('E_P [(fun x => (\sum_(y \in range f) \sum_(y' \in range g) y * y' + * \1_(f @^-1` [set y] `&` g @^-1` [set y']) x)%R)]). + congr ('E_P [_]); apply/funext => t/=. + rewrite mulrC; rewrite fsbig_distrr//=. (* TODO: lemma fsbig_distrl *) + apply: eq_fsbigr => y yf; rewrite mulrC; rewrite fsbig_distrr//=. + by apply: eq_fsbigr => y' y'g; rewrite indicI mulrCA !mulrA (mulrC y'). +rewrite unlock. +under eq_integral do rewrite -fsumEFin//. +transitivity (\sum_(y \in range f) (\sum_(y' \in range g) + ((y * y')%:E * \int[P]_w (\1_(f @^-1` [set y] `&` g @^-1` [set y']) w)%:E))). + rewrite ge0_integral_fsum//=; last 2 first. + - move=> r; under eq_fun do rewrite -fsumEFin//. + apply: emeasurable_fsum => // s. + apply/measurable_EFinP/measurable_funM => //. + exact/measurable_indic/measurableI. + - move=> r t _; rewrite lee_fin sumr_ge0 // => s _; rewrite -lee_fin. + by rewrite indicI/= indicE -mulrACA EFinM mule_ge0// nnfun_muleindic_ge0. + apply: eq_fsbigr => y yf. + under eq_integral do rewrite -fsumEFin//. + rewrite ge0_integral_fsum//=; last 2 first. + - move=> r; apply/measurable_EFinP; apply: measurable_funM => //. + exact/measurable_indic/measurableI. + - move=> r t _. + by rewrite indicI/= indicE -mulrACA EFinM mule_ge0// nnfun_muleindic_ge0. + apply: eq_fsbigr => y' y'g. + under eq_integral do rewrite EFinM. + by rewrite integralZl//; exact/integrable_indic/measurableI. +transitivity (\sum_(y \in range f) (\sum_(y' \in range g) + ((y * y')%:E * (\int[P]_w (\1_(f @^-1` [set y]) w)%:E * + \int[P]_w (\1_(g @^-1` [set y']) w)%:E)))). + apply: eq_fsbigr => y fy; apply: eq_fsbigr => y' gy'; congr *%E. + transitivity ('E_P[\1_(f @^-1` [set y] `&` g @^-1` [set y'])]). + by rewrite unlock. + transitivity ('E_P[\1_(f @^-1` [set y])] * 'E_P[\1_(g @^-1` [set y'])]); + last by rewrite unlock. + rewrite expectation_indic//; last exact: measurableI. + by rewrite !expectation_indic// fg. +transitivity ( + (\sum_(y \in range f) (y%:E * (\int[P]_w (\1_(f @^-1` [set y]) w)%:E))) * + (\sum_(y' \in range g) (y'%:E * \int[P]_w (\1_(g @^-1` [set y']) w)%:E))). + transitivity (\sum_(y \in range f) (\sum_(y' \in range g) + (y'%:E * \int[P]_w (\1_(g @^-1` [set y']) w)%:E)%E)%R * + (y%:E * \int[P]_w (\1_(f @^-1` [set y]) w)%:E)%E%R); last first. + rewrite !fsbig_finite//= ge0_sume_distrl//; last first. + move=> r _; rewrite -integralZl//; last exact: integrable_indic. + by apply: integral_ge0 => t _; rewrite nnfun_muleindic_ge0. + by apply: eq_bigr => r _; rewrite muleC. + apply: eq_fsbigr => y fy. + rewrite !fsbig_finite//= ge0_sume_distrl//; last first. + move=> r _; rewrite -integralZl//; last exact: integrable_indic. + by apply: integral_ge0 => t _; rewrite nnfun_muleindic_ge0. + apply: eq_bigr => r _; rewrite (mulrC y) EFinM. + by rewrite [X in _ * X]muleC muleACA. +suff: forall h : {nnsfun T >-> R}, + (\sum_(y \in range h) (y%:E * \int[P]_w (\1_(h @^-1` [set y]) w)%:E)%E)%R + = \int[P]_w (h w)%:E. + by move=> suf; congr *%E; rewrite suf. +move=> h. +apply/esym. +under eq_integral do rewrite (fimfunE h). +under eq_integral do rewrite -fsumEFin//. +rewrite ge0_integral_fsum//; last 2 first. +- by move=> r; exact/measurable_EFinP/measurable_funM. +- by move=> r t _; rewrite lee_fin -lee_fin nnfun_muleindic_ge0. +by apply: eq_fsbigr => y fy; rewrite -integralZl//; exact/integrable_indic. +Qed. + +Lemma ge0_independent_expectationM (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> + 'E_P[X] *? 'E_P[Y] -> + (forall t, 0 <= X t)%R -> (forall t, 0 <= Y t)%R -> + 'E_P [X * Y] = 'E_P [X] * 'E_P [Y]. +Proof. +move=> indeXY defXY X0 Y0. +have mX : measurable_fun setT (EFin \o X) by exact/measurable_EFinP. +have mY : measurable_fun setT (EFin \o Y) by exact/measurable_EFinP. +pose X_ := nnsfun_approx measurableT mX. +pose Y_ := nnsfun_approx measurableT mY. +have EXY : 'E_P[X_ n \* Y_ n] @[n --> \oo] --> 'E_P [X * Y]. + rewrite unlock; have -> : \int[P]_w ((X * Y) w)%:E = + \int[P]_x limn (fun n => (EFin \o (X_ n \* Y_ n)%R) x). + apply: eq_integral => t _; apply/esym/cvg_lim => //=. + rewrite fctE EFinM; under eq_fun do rewrite EFinM. + by apply: cvgeM; [rewrite mule_def_fin//| + apply: cvg_nnsfun_approx => //= x _; rewrite lee_fin..]. + apply: cvg_monotone_convergence => //. + - by move=> n; apply/measurable_EFinP; exact: measurable_funM. + - by move=> n t _; rewrite lee_fin. + - move=> t _ m n mn. + by rewrite lee_fin/= ler_pM//; exact/lefP/nd_nnsfun_approx. +have EX : 'E_P[X_ n] @[n --> \oo] --> 'E_P [X]. + rewrite unlock. + have -> : \int[P]_w (X w)%:E = \int[P]_x limn (fun n => (EFin \o X_ n) x). + by apply: eq_integral => t _; apply/esym/cvg_lim => //=; + apply: cvg_nnsfun_approx => // x _; rewrite lee_fin. + apply: cvg_monotone_convergence => //. + - by move=> n; exact/measurable_EFinP. + - by move=> n t _; rewrite lee_fin. + - by move=> t _ m n mn; rewrite lee_fin/=; exact/lefP/nd_nnsfun_approx. +have EY : 'E_P[Y_ n] @[n --> \oo] --> 'E_P [Y]. + rewrite unlock. + have -> : \int[P]_w (Y w)%:E = \int[P]_x limn (fun n => (EFin \o Y_ n) x). + by apply: eq_integral => t _; apply/esym/cvg_lim => //=; + apply: cvg_nnsfun_approx => // x _; rewrite lee_fin. + apply: cvg_monotone_convergence => //. + - by move=> n; exact/measurable_EFinP. + - by move=> n t _; rewrite lee_fin. + - by move=> t _ m n mn; rewrite lee_fin/=; exact/lefP/nd_nnsfun_approx. +have {EX EY}EXY' : 'E_P[X_ n] * 'E_P[Y_ n] @[n --> \oo] --> 'E_P[X] * 'E_P[Y]. + apply: cvgeM => //. +suff : forall n, 'E_P[X_ n \* Y_ n] = 'E_P[X_ n] * 'E_P[Y_ n]. + by move=> suf; apply: (cvg_unique _ EXY) => //=; under eq_fun do rewrite suf. +move=> n; apply: expectationM_nnsfun => x y xX_ yY_. +suff : P (\big[setI/setT]_(j <- [fset false; true]%fset) + [eta fun=> set0 with 0%N |-> X_ n @^-1` [set x], + 1%N |-> Y_ n @^-1` [set y]] j) = + \prod_(j <- [fset false; true]%fset) + P ([eta fun=> set0 with 0%N |-> X_ n @^-1` [set x], + 1%N |-> Y_ n @^-1` [set y]] j). + by rewrite !big_fsetU1/= ?inE//= !big_seq_fset1/=. +move: indeXY => [/= _]; apply => // i. +pose AX := dyadic_approx setT (EFin \o X). +pose AY := dyadic_approx setT (EFin \o Y). +pose BX := integer_approx setT (EFin \o X). +pose BY := integer_approx setT (EFin \o Y). +have mA (Z : {RV P >-> R}) m k : (k < m * 2 ^ m)%N -> + g_sigma_algebra_preimage Z (dyadic_approx setT (EFin \o Z) m k). + move=> mk; rewrite /g_sigma_algebra_preimage /dyadic_approx mk setTI. + rewrite /preimage_set_system/=; exists [set` dyadic_itv R m k] => //. + rewrite setTI/=; apply/seteqP; split => z/=. + by rewrite inE/= => Zz; exists (Z z). + by rewrite inE/= => -[r rmk] [<-]. +have mB (Z : {RV P >-> R}) k : + g_sigma_algebra_preimage Z (integer_approx setT (EFin \o Z) k). + rewrite /g_sigma_algebra_preimage /integer_approx setTI /preimage_set_system/=. + by exists `[k%:R, +oo[%classic => //; rewrite setTI preimage_itvcy. +have m1A (Z : {RV P >-> R}) : forall k, (k < n * 2 ^ n)%N -> + measurable_fun setT + (\1_(dyadic_approx setT (EFin \o Z) n k) : g_sigma_algebra_preimageType Z -> R). + move=> k kn. + exact/(@measurable_indicP _ (g_sigma_algebra_preimageType Z))/mA. +rewrite !inE => /orP[|]/eqP->{i} //=. +- have : @measurable_fun _ _ (g_sigma_algebra_preimageType X) _ setT (X_ n). + rewrite nnsfun_approxE//. + apply: measurable_funD => //=. + apply: measurable_sum => //= k'; apply: measurable_funM => //. + by apply: measurable_indic; exact: mA. + apply: measurable_funM => //. + by apply: measurable_indic; exact: mB. + rewrite /measurable_fun => /(_ measurableT _ (measurable_set1 x)). + by rewrite setTI. +- have : @measurable_fun _ _ (g_sigma_algebra_preimageType Y) _ setT (Y_ n). + rewrite nnsfun_approxE//. + apply: measurable_funD => //=. + apply: measurable_sum => //= k'; apply: measurable_funM => //. + by apply: measurable_indic; exact: mA. + apply: measurable_funM => //. + by apply: measurable_indic; exact: mB. + move=> /(_ measurableT [set y] (measurable_set1 y)). + by rewrite setTI. +Qed. + +Lemma independent_Lfun1_expectationM_lty (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> + (X : _ -> _) \in Lfun P 1 -> (Y : _ -> _) \in Lfun P 1 -> + `|'E_P [X * Y]| < +oo. +Proof. +move=> indeXY iX iY. +apply: (@le_lt_trans _ _ 'E_P[(@normr _ _ \o X) * (@normr _ _ \o Y)]). + rewrite unlock/= (le_trans (le_abse_integral _ _ _))//. + apply/measurable_EFinP/measurable_funM. + by move/Lfun1_integrable : iX => /measurable_int/measurable_EFinP. + by move/Lfun1_integrable : iY => /measurable_int/measurable_EFinP. + apply: ge0_le_integral => //=. + - by apply/measurable_EFinP; exact/measurableT_comp. + - by apply/measurable_EFinP; apply/measurable_funM; exact/measurableT_comp. + - by move=> t _; rewrite lee_fin/= normrM. +rewrite ge0_independent_expectationM//=. +- rewrite lte_mul_pinfty//. + + by rewrite expectation_ge0/=. + + rewrite expectation_fin_num//=. + by move: iX => /Lfun_norm. + + move : iY => /Lfun1_integrable/integrableP[_]. + by rewrite unlock. +- exact: independent_RVs2_comp. +- apply: mule_def_fin; rewrite unlock integrable_fin_num//. + + by move/Lfun_norm : iX => /Lfun1_integrable. + + by move/Lfun_norm : iY => /Lfun1_integrable. +Qed. + +Lemma independent_Lfun1M (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> + (X : _ -> _) \in Lfun P 1 -> (Y : _ -> _) \in Lfun P 1 -> + (X \* Y)%R \in Lfun P 1. +Proof. +move=> indeXY iX iY. +apply/Lfun1_integrable. +apply/integrableP; split => //. + move/Lfun1_integrable : iX => /integrableP[iX _]. + move/Lfun1_integrable : iY => /integrableP[iY _]. + exact/measurable_EFinP/measurable_funM/measurable_EFinP. +have := independent_Lfun1_expectationM_lty indeXY iX iY. +rewrite unlock => /abse_integralP; apply => //. +exact/measurable_EFinP/measurable_funM. +Qed. + +Lemma independent_expectationM (X Y : {RV P >-> R}) : + independent_RVs2 P X Y -> + (X : _ -> _) \in Lfun P 1 -> (Y : _ -> _) \in Lfun P 1 -> + 'E_P [X * Y] = 'E_P [X] * 'E_P [Y]. +Proof. +move=> XY iX iY. +transitivity ('E_P[(X^\+ - X^\-) * (Y^\+ - Y^\-)]). + by rewrite !funrposBneg. +have ? : X^\-%R \in Lfun P 1. + apply/Lfun1_integrable; rewrite -funerneg; apply/integrable_funeneg => //. + exact/Lfun1_integrable. +have ? : X^\-%R \in Lfun P 1. + apply/Lfun1_integrable; rewrite -funerneg; apply/integrable_funeneg => //. + exact/Lfun1_integrable. +have ? : X^\+%R \in Lfun P 1. + apply/Lfun1_integrable; rewrite -funerpos; apply/integrable_funepos => //. + exact/Lfun1_integrable. +have ? : Y^\+%R \in Lfun P 1. + apply/Lfun1_integrable; rewrite -funerpos; apply/integrable_funepos => //. + exact/Lfun1_integrable. +have ? : Y^\-%R \in Lfun P 1. + apply/Lfun1_integrable; rewrite -funerneg; apply/integrable_funeneg => //. + exact/Lfun1_integrable. +have ? : (X^\+ \* Y^\+)%R \in Lfun P 1. + by apply: independent_Lfun1M => //=; exact: independent_RVs2_funrpospos. +have ? : (X^\- \* Y^\+)%R \in Lfun P 1. + by apply: independent_Lfun1M => //=; exact: independent_RVs2_funrnegpos. +have ? : (X^\+ \* Y^\-)%R \in Lfun P 1. + by apply: independent_Lfun1M => //=; exact: independent_RVs2_funrposneg. +have ? : (X^\- \* Y^\-)%R \in Lfun P 1. + by apply: independent_Lfun1M => //=; exact: independent_RVs2_funrnegneg. +transitivity ('E_P[X^\+ * Y^\+] - 'E_P[X^\- * Y^\+] + - 'E_P[X^\+ * Y^\-] + 'E_P[X^\- * Y^\-]). + rewrite mulrDr !mulrDl -!expectationB//=; last by rewrite rpredB. + rewrite -expectationD//=; last by rewrite !rpredB. + congr ('E_P[_]); apply/funext => t/=. + by rewrite !fctE !(mulNr,mulrN,opprK,addrA). +rewrite [in LHS]ge0_independent_expectationM//=; last 2 first. + exact: independent_RVs2_funrpospos. + by rewrite mule_def_fin// expectation_fin_num. +rewrite [in LHS]ge0_independent_expectationM//=; last 2 first. + exact: independent_RVs2_funrnegpos. + by rewrite mule_def_fin// expectation_fin_num. +rewrite [in LHS]ge0_independent_expectationM//=; last 2 first. + exact: independent_RVs2_funrposneg. + by rewrite mule_def_fin// expectation_fin_num. +rewrite [in LHS]ge0_independent_expectationM//=; last 2 first. + exact: independent_RVs2_funrnegneg. + by rewrite mule_def_fin// expectation_fin_num. +transitivity ('E_P[X^\+ - X^\-] * 'E_P[Y^\+ - Y^\-]). + rewrite -addeA -addeACA -muleBr; last 2 first. + by rewrite expectation_fin_num. + by rewrite fin_num_adde_defr// expectation_fin_num. + rewrite -oppeB; last first. + by rewrite fin_num_adde_defr// fin_numM// expectation_fin_num. + rewrite -muleBr; last 2 first. + by rewrite expectation_fin_num. + by rewrite fin_num_adde_defr// expectation_fin_num. + rewrite -muleBl. + - by rewrite -!expectationB. + - by rewrite fin_numB// !expectation_fin_num. + - by rewrite fin_num_adde_defr// expectation_fin_num. +by rewrite !funrposBneg. +Qed. + +End product_expectation. diff --git a/theories/lebesgue_integral_theory/lebesgue_integrable.v b/theories/lebesgue_integral_theory/lebesgue_integrable.v index fec5bd46e..7704f2478 100644 --- a/theories/lebesgue_integral_theory/lebesgue_integrable.v +++ b/theories/lebesgue_integral_theory/lebesgue_integrable.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. From mathcomp Require Import archimedean. @@ -207,10 +207,9 @@ Proof. by move=> fi gi; exact/(integrableD fi)/integrableN. Qed. Lemma integrable_add_def f : mu_int f -> \int[mu]_(x in D) f^\+ x +? - (\int[mu]_(x in D) f^\- x). Proof. -move=> /integrableP[mf]; rewrite -[fun x => _]/(abse \o f) fune_abse => foo. -rewrite ge0_integralD // in foo; last 2 first. -- exact: measurable_funepos. -- exact: measurable_funeneg. +move=> /integrableP[mf]; rewrite -[fun x => _]/(abse \o f) -funeposDneg => foo. +rewrite ge0_integralD // in foo; [|exact: measurable_funepos + |exact: measurable_funeneg]. apply: ltpinfty_adde_def. - by apply: le_lt_trans foo; rewrite leeDl// integral_ge0. - by rewrite inE (@le_lt_trans _ _ 0)// leeNl oppe0 integral_ge0. @@ -223,7 +222,7 @@ move=> /integrableP[Df foo]; apply/integrableP; split. apply: le_lt_trans foo; apply: ge0_le_integral => //. - by apply/measurableT_comp => //; exact: measurable_funepos. - exact/measurableT_comp. -- by move=> t Dt; rewrite -/((abse \o f) t) fune_abse gee0_abs// leeDl. +- by move=> t Dt; rewrite -/((abse \o f) t) -funeposDneg gee0_abs// leeDl. Qed. Lemma integrable_funeneg f : mu_int f -> mu_int f^\-. @@ -233,7 +232,7 @@ move=> /integrableP[Df foo]; apply/integrableP; split. apply: le_lt_trans foo; apply: ge0_le_integral => //. - by apply/measurableT_comp => //; exact: measurable_funeneg. - exact/measurableT_comp. -- by move=> t Dt; rewrite -/((abse \o f) t) fune_abse gee0_abs// leeDr. +- by move=> t Dt; rewrite -/((abse \o f) t) -funeposDneg gee0_abs// leeDr. Qed. Lemma integral_funeneg_lt_pinfty f : mu_int f -> \int[mu]_(x in D) f^\- x < +oo. @@ -241,9 +240,8 @@ Proof. move=> /integrableP[mf]; apply: le_lt_trans; apply: ge0_le_integral => //. - exact: measurable_funeneg. - exact: measurableT_comp. -- move=> x Dx; have [fx0|/ltW fx0] := leP (f x) 0. - rewrite lee0_abs// funenegE. - by move: fx0; rewrite -{1}oppe0 -leeNr => /max_idPl ->. +- move=> x Dx; have /orP[fx0|fx0] := le_total (f x) 0. + by rewrite lee0_abs// funenegE ge_max lexx leeNr oppe0 fx0. rewrite gee0_abs// funenegE. by move: (fx0); rewrite -{1}oppe0 -leeNl => /max_idPr ->. Qed. @@ -268,7 +266,7 @@ rewrite fin_numElt; apply/andP; split. case: fi => mf; apply: le_lt_trans; apply: ge0_le_integral => //. - exact/measurable_funeneg. - exact/measurableT_comp. -- by move=> x Dx; rewrite -/((abse \o f) x) (fune_abse f) leeDr. +- by move=> x Dx; rewrite -/((abse \o f) x) -funeposDneg leeDr. Qed. Lemma integrable_pos_fin_num f : @@ -280,7 +278,7 @@ rewrite fin_numElt; apply/andP; split. case: fi => mf; apply: le_lt_trans; apply: ge0_le_integral => //. - exact/measurable_funepos. - exact/measurableT_comp. -- by move=> x Dx; rewrite -/((abse \o f) x) (fune_abse f) leeDl. +- by move=> x Dx; rewrite -/((abse \o f) x) -funeposDneg leeDl. Qed. Lemma integrableMr (h : T -> R) g : @@ -593,8 +591,8 @@ have : (g1 \+ g2)^\+ \+ g1^\- \+ g2^\- = (g1 \+ g2)^\- \+ g1^\+ \+ g2^\+. by rewrite !funeposE -!fine_max. by rewrite funeposE !funenegE -!fine_max. apply/eqP. - rewrite -[LHS]/((g1^\+ \+ g2^\+ \- (g1^\- \+ g2^\-)) x) -funeD_posD. - by rewrite -[RHS]/((_ \- _) x) -funeD_Dpos. + rewrite -[LHS]/((g1^\+ \+ g2^\+ \- (g1^\- \+ g2^\-)) x) funeDB. + by rewrite -[RHS]/((_ \- _) x) funeposBneg. move/(congr1 (fun y => \int[mu]_(x in D) (y x) )). rewrite (ge0_integralD mu mD); last 4 first. - by move=> x _; rewrite adde_ge0. @@ -741,7 +739,7 @@ Local Open Scope ereal_scope. Lemma integrable_lty (f : T -> \bar R) : mu.-integrable D f -> \int[mu]_(x in D) f x < +oo. Proof. -move=> intf; rewrite (funeposneg f) integralB//; +move=> intf; rewrite -(funeposBneg f) integralB//; [|exact: integrable_funepos|exact: integrable_funeneg]. rewrite lte_add_pinfty ?integral_funepos_lt_pinfty// lteNl ltNye_eq. by rewrite integrable_neg_fin_num. @@ -799,7 +797,7 @@ rewrite -[X in _ = _ - X]ge0_integral_pushforward//; last first. rewrite -integralB//=; last first. - by apply: integrable_funeneg => //=; exact: integrable_pushforward. - by apply: integrable_funepos => //=; exact: integrable_pushforward. -- by apply/eq_integral=> // x _; rewrite /= [in LHS](funeposneg f). +- by apply/eq_integral=> // x _; rewrite -[in LHS](funeposBneg f). Qed. End transfer. @@ -813,7 +811,7 @@ Lemma negligible_integral (D N : set T) (f : T -> \bar R) : measurable N -> measurable D -> mu.-integrable D f -> mu N = 0 -> \int[mu]_(x in D) f x = \int[mu]_(x in D `\` N) f x. Proof. -move=> mN mD mf muN0; rewrite [f]funeposneg ?integralB //; first last. +move=> mN mD mf muN0; rewrite -[f]funeposBneg ?integralB//; first last. - exact: integrable_funeneg. - exact: integrable_funepos. - apply: (integrableS mD) => //; first exact: measurableD. @@ -859,7 +857,7 @@ Lemma integral_measure_add : \int[measure_add m1 m2]_(x in D) f x = Proof. transitivity (\int[m1]_(x in D) (f^\+ \- f^\-) x + \int[m2]_(x in D) (f^\+ \- f^\-) x); last first. - by congr +%E; apply: eq_integral => x _; rewrite [in RHS](funeposneg f). + by congr +%E; apply: eq_integral => x _; rewrite -[in RHS](funeposBneg f). rewrite integralB//; [|exact: integrable_funepos|exact: integrable_funeneg]. rewrite integralB//; [|exact: integrable_funepos|exact: integrable_funeneg]. rewrite addeACA -ge0_integral_measure_add//; last first. @@ -908,7 +906,7 @@ have ? : \int[mu]_(x in \bigcup_i F i) g x \is a fin_num. transitivity (\int[mu]_(x in \bigcup_i F i) g^\+ x - \int[mu]_(x in \bigcup_i F i) g^\- x)%E. rewrite -integralB. - - by apply: eq_integral => t Ft; rewrite [in LHS](funeposneg g). + - by apply: eq_integral => t Ft; rewrite -[in LHS](funeposBneg g). - exact: bigcupT_measurable. - by apply: integrable_funepos => //; exact: bigcupT_measurable. - by apply: integrable_funeneg => //; exact: bigcupT_measurable. diff --git a/theories/lebesgue_integral_theory/lebesgue_integral_dominated_convergence.v b/theories/lebesgue_integral_theory/lebesgue_integral_dominated_convergence.v index 3622ce6a1..1c5525ebf 100644 --- a/theories/lebesgue_integral_theory/lebesgue_integral_dominated_convergence.v +++ b/theories/lebesgue_integral_theory/lebesgue_integral_dominated_convergence.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. From mathcomp Require Import archimedean. @@ -297,11 +297,11 @@ exists (fun n => p_ n - n_ n)%R; split. - move=> n; rewrite /comp; under eq_fun => ? do rewrite sfunB /= EFinB. by apply: integrableB => //; [exact: intp | exact: intn]. - move=> ? ?; rewrite /comp; under eq_fun => ? do rewrite sfunB /= EFinB. - rewrite [f]funeposneg; apply: cvgeB => //;[|exact: pf|exact:nf]. + rewrite -[f]funeposBneg; apply: cvgeB => //;[|exact: pf|exact:nf]. exact: add_def_funeposneg. have fpn z n : f z - ((p_ n - n_ n) z)%:E = (f^\+ z - (p_ n z)%:E) - (f^\- z - (n_ n z)%:E). - rewrite sfunB EFinB fin_num_oppeB // {1}[f]funeposneg -addeACA. + rewrite sfunB EFinB fin_num_oppeB // -{1}[f]funeposBneg -addeACA. by congr (_ _); rewrite fin_num_oppeB. case/integrableP: (intf) => mf _. have mfpn n : mu.-integrable E (fun z => f z - ((p_ n - n_ n) z)%:E). diff --git a/theories/lebesgue_integral_theory/lebesgue_integral_fubini.v b/theories/lebesgue_integral_theory/lebesgue_integral_fubini.v index 91897623a..12808fd21 100644 --- a/theories/lebesgue_integral_theory/lebesgue_integral_fubini.v +++ b/theories/lebesgue_integral_theory/lebesgue_integral_fubini.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. From mathcomp Require Import archimedean. @@ -922,7 +922,7 @@ apply: ge0_le_integral; [by []|by []|..]. + apply: measurableT_comp => //. by apply: measurableT_comp => //; exact: measurable_funepos. + by apply: measurableT_comp => //; exact/measurableT_comp. - + by move=> y _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse leeDl. + + by move=> y _; rewrite gee0_abs// -/((abse \o f) _) -funeposDneg leeDl. Qed. Let integrable_Fminus : m1.-integrable setT Fminus. @@ -939,7 +939,7 @@ apply: ge0_le_integral; [by []|by []|..]. + apply: measurableT_comp => //; apply: measurableT_comp => //. exact: measurable_funeneg. + by apply: measurableT_comp => //; exact: measurableT_comp. - + by move=> y _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse leeDr. + + by move=> y _; rewrite gee0_abs// -/((abse \o f) _) -funeposDneg leeDr. Qed. Lemma integrable_fubini_F : m1.-integrable setT F. @@ -982,7 +982,7 @@ apply: ge0_le_integral; [by []|by []|exact: measurableT_comp|..]. + apply: measurableT_comp => //. by apply: measurableT_comp => //; exact: measurable_funepos. + by apply: measurableT_comp => //; exact: measurableT_comp. - + by move=> x _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse leeDl. + + by move=> x _; rewrite gee0_abs// -/((abse \o f) _) -funeposDneg leeDl. Qed. Let integrable_Gminus : m2.-integrable setT Gminus. @@ -997,7 +997,7 @@ apply: ge0_le_integral; [by []|by []|exact: measurableT_comp|..]. + apply: measurableT_comp => //. by apply: measurableT_comp => //; exact: measurable_funeneg. + by apply: measurableT_comp => //; exact: measurableT_comp. - + by move=> x _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse leeDr. + + by move=> x _; rewrite gee0_abs// -/((abse \o f) _) -funeposDneg leeDr. Qed. Lemma integral12_prod_meas1 : \int[m1]_x F x = \int[m1 \x m2]_z f z. diff --git a/theories/lebesgue_integral_theory/lebesgue_integral_nonneg.v b/theories/lebesgue_integral_theory/lebesgue_integral_nonneg.v index 087a36bd8..91592f830 100644 --- a/theories/lebesgue_integral_theory/lebesgue_integral_nonneg.v +++ b/theories/lebesgue_integral_theory/lebesgue_integral_nonneg.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. From mathcomp Require Import archimedean. @@ -605,7 +605,7 @@ Proof. have [/[!inE] aD|aD] := boolP (a \in D). rewrite integralE ge0_integral_dirac//; last exact/measurable_funepos. rewrite ge0_integral_dirac//; last exact/measurable_funeneg. - by rewrite [in RHS](funeposneg f) diracE mem_set// mul1e. + by rewrite -[in RHS](funeposBneg f) diracE mem_set// mul1e. rewrite diracE (negbTE aD) mul0e -(integral_measure_zero D f)//. apply: eq_measure_integral => //= S mS DS; rewrite /dirac indicE memNset//. by move=> /DS/mem_set; exact/negP. @@ -613,8 +613,8 @@ Qed. End integral_dirac. -Lemma summable_integral_dirac {R : realType} (a : nat -> \bar R) : summable setT a -> - (\sum_(n (\sum_(n sa. apply: (@le_lt_trans _ _ (\sum_(i \int[measure_add (msum m_ N) (m_ N)]_(x in D) (f_ n x)%:E)). rewrite -monotone_convergence//; last first. by move=> t Dt a b ab; rewrite lee_fin; exact/lefP/nd_nnsfun_approx. - by apply: eq_integral => t /[!inE] Dt; apply/esym/cvg_lim => //; exact: cvg_nnsfun_approx. + apply: eq_integral => t /[!inE] Dt; apply/esym/cvg_lim => //. + exact: cvg_nnsfun_approx. transitivity (limn (fun n => \int[msum m_ N]_(x in D) (f_ n x)%:E + \int[m_ N]_(x in D) (f_ n x)%:E)). by congr (limn _); apply/funext => n; by rewrite integral_measure_add_nnsfun. @@ -1111,14 +1112,14 @@ End ge0_cvgn_integral. Lemma le_abse_integral d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}) (D : set T) (f : T -> \bar R) (mD : measurable D) : measurable_fun D f -> - (`| \int[mu]_(x in D) (f x) | <= \int[mu]_(x in D) `|f x|)%E. + (`| \int[mu]_(x in D) f x | <= \int[mu]_(x in D) `|f x|)%E. Proof. move=> mf. rewrite integralE (le_trans (lee_abs_sub _ _))// gee0_abs; last first. exact: integral_ge0. rewrite gee0_abs; last exact: integral_ge0. -by rewrite -ge0_integralD // -?fune_abse//; - [exact: measurable_funepos | exact: measurable_funeneg]. +rewrite -ge0_integralD//; [|exact: measurable_funepos|exact: measurable_funeneg]. +by under [in leRHS]eq_integral do rewrite -/((abse \o f) _) -funeposDneg. Qed. Lemma abse_integralP d (T : measurableType d) (R : realType) @@ -1128,7 +1129,7 @@ Lemma abse_integralP d (T : measurableType d) (R : realType) Proof. move=> mD mf; split => [|] foo; last first. exact: (le_lt_trans (le_abse_integral mu mD mf) foo). -under eq_integral do rewrite -/((abse \o f) _) fune_abse. +under eq_integral do rewrite -/((abse \o f) _) -funeposDneg. rewrite ge0_integralD//;[|exact/measurable_funepos|exact/measurable_funeneg]. move: foo; rewrite integralE/= -fin_num_abs fin_numB => /andP[fpoo fnoo]. by rewrite lte_add_pinfty// ltey_eq ?fpoo ?fnoo. diff --git a/theories/lebesgue_integral_theory/measurable_fun_approximation.v b/theories/lebesgue_integral_theory/measurable_fun_approximation.v index 88af112a6..e67b078f4 100644 --- a/theories/lebesgue_integral_theory/measurable_fun_approximation.v +++ b/theories/lebesgue_integral_theory/measurable_fun_approximation.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. From mathcomp Require Import archimedean. @@ -404,7 +404,7 @@ exists (fun n => fp_ n \+ cst (-1) \* fn_ n) => x /=. rewrite [X in X @ \oo --> _](_ : _ = EFin \o fp_^~ x \+ (-%E \o EFin \o fn_^~ x))%E; last first. by apply/funext => n/=; rewrite EFinD mulN1r. -by move=> Dx; rewrite (funeposneg f); apply: cvgeD; +by move=> Dx; rewrite -(funeposBneg f); apply: cvgeD; [exact: add_def_funeposneg|apply: cvg_nnsfun_approx|apply:cvgeN; apply: cvg_nnsfun_approx]. Qed. diff --git a/theories/measurable_realfun.v b/theories/measurable_realfun.v index 0e7947c12..997feedb7 100644 --- a/theories/measurable_realfun.v +++ b/theories/measurable_realfun.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint interval. From mathcomp Require Import archimedean rat. @@ -991,6 +991,12 @@ by move=> mf mg mD; move: (mD); apply: measurable_fun_if => //; [exact: measurable_fun_ltr|exact: measurable_funS mg|exact: measurable_funS mf]. Qed. +Lemma measurable_funrpos D f : measurable_fun D f -> measurable_fun D f^\+. +Proof. by move=> mf; exact: measurable_maxr. Qed. + +Lemma measurable_funrneg D f : measurable_fun D f -> measurable_fun D f^\-. +Proof. by move=> mf; apply: measurable_maxr => //; exact: measurableT_comp. Qed. + Lemma measurable_minr D f g : measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \min g). Proof. @@ -1083,6 +1089,19 @@ Qed. End measurable_fun_realType. +Section funrposneg_measurable. +Context {d} {aT : measurableType d} {rT : realType}. + +HB.instance Definition _ (f : {mfun aT >-> rT}) := + @isMeasurableFun.Build d _ _ _ f^\+ + (measurable_funrpos (@measurable_funPT _ _ _ _ f)). + +HB.instance Definition _ (f : {mfun aT >-> rT}) := + @isMeasurableFun.Build d _ _ _ f^\- + (measurable_funrneg (@measurable_funPT _ _ _ _ f)). + +End funrposneg_measurable. + Section mono_measurable. Context {R : realType}. diff --git a/theories/measure_theory/measurable_function.v b/theories/measure_theory/measurable_function.v index ce3f705fd..300cce8b9 100644 --- a/theories/measure_theory/measurable_function.v +++ b/theories/measure_theory/measurable_function.v @@ -331,6 +331,17 @@ HB.instance Definition _ := isMeasurableFun.Build _ _ _ _ (f \o g) End mfun_measurableType. +Lemma preimage_set_system_compS (aT : Type) + d (rT : measurableType d) d' (T : sigmaRingType d') + (g : rT -> T) (f : aT -> rT) (D : set aT) : + measurable_fun setT g -> + preimage_set_system D (g \o f) measurable `<=` + preimage_set_system D f measurable. +Proof. +move=> mg A; rewrite /preimage_set_system => -[B GB]; exists (g @^-1` B) => //. +by rewrite -[X in measurable X]setTI; exact: mg. +Qed. + Section measurability. (* f is measurable on the sigma-algebra generated by itself *) diff --git a/theories/measure_theory/measurable_structure.v b/theories/measure_theory/measurable_structure.v index d9196e229..faeed5a70 100644 --- a/theories/measure_theory/measurable_structure.v +++ b/theories/measure_theory/measurable_structure.v @@ -90,11 +90,20 @@ From mathcomp Require Import ereal topology normedtype sequences. (* ## Other measure-theoretic definitions *) (* *) (* ``` *) -(* preimage_set_system D f G == set system of the preimages by f of sets in G *) -(* image_set_system D f G == set system of the sets with a preimage by f *) -(* in G *) -(* subset_sigma_subadditive mu == alternative predicate defining *) -(* sigma-subadditivity *) +(* preimage_set_system D f G == set system of the preimages by f of sets *) +(* in G *) +(* g_sigma_algebra_preimage f == sigma-algebra generated by the *) +(* function f *) +(* g_sigma_algebra_preimageType f == the measurableType corresponding to *) +(* g_sigma_algebra_preimage f *) +(* This is an HB alias. *) +(* f.-preimage.-measurable A == A is measurable for *) +(* g_sigma_algebra_preimage f *) +(* image_set_system D f G == set system of the sets with a preimage *) +(* by f in G *) +(* *) +(* subset_sigma_subadditive mu == alternative predicate defining *) +(* sigma-subadditivity *) (* ``` *) (* *) (* ## Product of measurable spaces *) @@ -1343,6 +1352,58 @@ case=> h0 hC hU; split; first by exists set0 => //; rewrite preimage_set0 setI0. exact: (mF' i).2. Qed. +Definition preimage_display {T T'} : (T -> T') -> measure_display. +Proof. exact. Qed. + +Definition g_sigma_algebra_preimageType d' (T : pointedType) + (T' : measurableType d') (f : T -> T') : Type := T. + +Definition g_sigma_algebra_preimage d' (T : pointedType) + (T' : measurableType d') (f : T -> T') := + preimage_set_system setT f (@measurable _ T'). + +Section preimage_generated_sigma_algebra. +Context {d'} (T : pointedType) (T' : measurableType d'). +Variable f : T -> T'. + +Let preimage_set0 : g_sigma_algebra_preimage f set0. +Proof. +rewrite /g_sigma_algebra_preimage /preimage_set_system/=. +by exists set0 => //; rewrite preimage_set0 setI0. +Qed. + +Let preimage_setC A : + g_sigma_algebra_preimage f A -> g_sigma_algebra_preimage f (~` A). +Proof. +rewrite /g_sigma_algebra_preimage /preimage_set_system/= => -[B mB] <-{A}. +by exists (~` B); [exact: measurableC|rewrite !setTI preimage_setC]. +Qed. + +Let preimage_bigcup (F : (set T)^nat) : + (forall i, g_sigma_algebra_preimage f (F i)) -> + g_sigma_algebra_preimage f (\bigcup_i (F i)). +Proof. +move=> mF; rewrite /g_sigma_algebra_preimage /preimage_set_system/=. +pose g := fun i => sval (cid2 (mF i)). +pose mg := fun i => svalP (cid2 (mF i)). +exists (\bigcup_i g i). + by apply: bigcup_measurable => k; case: (mg k). +rewrite setTI /g preimage_bigcup; apply: eq_bigcupr => k _. +by case: (mg k) => _; rewrite setTI. +Qed. + +HB.instance Definition _ := Pointed.on (g_sigma_algebra_preimageType f). + +HB.instance Definition _ := @isMeasurable.Build (preimage_display f) + (g_sigma_algebra_preimageType f) (g_sigma_algebra_preimage f) + preimage_set0 preimage_setC preimage_bigcup. + +End preimage_generated_sigma_algebra. + +Notation "f .-preimage" := (preimage_display f) : measure_display_scope. +Notation "f .-preimage.-measurable" := + (measurable : set (set (g_sigma_algebra_preimageType f))) : classical_set_scope. + Definition image_set_system (aT rT : Type) (D : set aT) (f : aT -> rT) (G : set (set aT)) : set (set rT) := [set B : set rT | G (D `&` f @^-1` B)]. diff --git a/theories/measure_theory/measure_negligible.v b/theories/measure_theory/measure_negligible.v index 05c563edc..e9efb02f8 100644 --- a/theories/measure_theory/measure_negligible.v +++ b/theories/measure_theory/measure_negligible.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect all_algebra. From mathcomp Require Import boolp classical_sets functions cardinality reals. @@ -273,7 +273,8 @@ Lemma ae_eq_funeposneg (f g : T -> \bar R) : Proof. split=> [fg|[pfg nfg]]. by split; near=> x => Dx; rewrite !(funeposE,funenegE) (near fg). -by near=> x => Dx; rewrite (funeposneg f) (funeposneg g) ?(near pfg, near nfg). +near=> x => Dx. +by rewrite -(funeposBneg f) -(funeposBneg g) ?(near pfg, near nfg). Unshelve. all: by end_near. Qed. Local Close Scope ereal_scope. @@ -283,7 +284,8 @@ Proof. by symmetry. Qed. Lemma ae_eq_trans U (f g h : T -> U) : ae_eq f g -> ae_eq g h -> ae_eq f h. Proof. by apply transitivity. Qed. -Lemma ae_eq_sub W (f g h i : T -> W) : ae_eq f g -> ae_eq h i -> ae_eq (f \- h) (g \- i). +Lemma ae_eq_sub W (f g h i : T -> W) : ae_eq f g -> ae_eq h i -> + ae_eq (f \- h) (g \- i). Proof. by apply: filterS2 => x + + Dx => /= /(_ Dx) -> /(_ Dx) ->. Qed. Lemma ae_eq_mul2r W (f g h : T -> W) : ae_eq f g -> ae_eq (f \* h) (g \* h). @@ -295,7 +297,8 @@ Proof. by move=>/(ae_eq_comp2 (fun x y => h x * y)). Qed. Lemma ae_eq_mul1l W (f g : T -> W) : ae_eq f (cst 1) -> ae_eq g (g \* f). Proof. by apply: filterS => x /= /[apply] ->; rewrite mulr1. Qed. -Lemma ae_eq_abse (f g : T -> \bar R) : ae_eq f g -> ae_eq (abse \o f) (abse \o g). +Lemma ae_eq_abse (f g : T -> \bar R) : ae_eq f g -> + ae_eq (abse \o f) (abse \o g). Proof. by apply: filterS => x /[apply] /= ->. Qed. Lemma ae_foralln (P : nat -> T -> Prop) : @@ -306,8 +309,8 @@ have seqDUAmeas := seqDU_measurable Ameas. exists (\bigcup_n A n); split => //. - exact/bigcup_measurable. - rewrite seqDU_bigcup_eq measure_bigcup// eseries0// => i _ _. - by rewrite (@subset_measure0 _ _ _ _ _ (A i))//=; apply: subset_seqDU. -- by move=> x /=; rewrite -existsNP => -[n NPnx]; exists n => //; apply: NPA. + by rewrite (@subset_measure0 _ _ _ _ _ (A i))//=; exact: subset_seqDU. +- by move=> x /=; rewrite -existsNP => -[n NPnx]; exists n => //; exact: NPA. Qed. End ae_eq. diff --git a/theories/numfun.v b/theories/numfun.v index 0ce6da7b8..c8b549d6e 100644 --- a/theories/numfun.v +++ b/theories/numfun.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. From mathcomp Require Import mathcomp_extra boolp classical_sets fsbigop. @@ -25,12 +25,15 @@ From mathcomp Require Import sequences function_spaces. (* bounded_variation a b f == all variations of f are bounded *) (* {nnfun T >-> R} == type of non-negative functions *) (* f ^\+ == the function formed by the non-negative outputs *) -(* of f (from a type to the type of extended real *) -(* numbers) and 0 otherwise *) -(* rendered as f ⁺ with company-coq (U+207A) *) +(* of f and 0 otherwise *) +(* The codomain of f is the real numbers in scope *) +(* ring_scope and the extended real numbers in *) +(* scope ereal_scope. *) +(* Rendered as f ⁺ with company-coq (U+207A). *) (* f ^\- == the function formed by the non-positive outputs *) (* of f and 0 o.w. *) -(* rendered as f ⁻ with company-coq (U+207B) *) +(* Similar to ^\+. *) +(* Rendered as f ⁻ with company-coq (U+207B). *) (* \1_ A == indicator function 1_A *) (* ``` *) (* *) @@ -679,6 +682,133 @@ Proof. by apply/funext=> x; rewrite /patch/=; case: ifP; rewrite ?mule0. Qed. End erestrict_lemmas. +Section funrposneg. +Local Open Scope ring_scope. + +Definition funrpos T (R : realDomainType) (f : T -> R) := + fun x => Num.max (f x) 0. +Definition funrneg T (R : realDomainType) (f : T -> R) := + fun x => Num.max (- f x) 0. + +End funrposneg. + +Notation "f ^\+" := (funrpos f) : ring_scope. +Notation "f ^\-" := (funrneg f) : ring_scope. + +Section funrposneg_lemmas. +Local Open Scope ring_scope. +Variables (T : Type) (R : realDomainType) (D : set T). +Implicit Types (f g : T -> R) (r : R). + +Lemma funrpos_ge0 f x : 0 <= f^\+ x. +Proof. by rewrite /funrpos /= le_max lexx orbT. Qed. + +Lemma funrneg_ge0 f x : 0 <= f^\- x. +Proof. by rewrite /funrneg le_max lexx orbT. Qed. + +Lemma funrposN f : (\- f)^\+ = f^\-. Proof. exact/funext. Qed. + +Lemma funrnegN f : (\- f)^\- = f^\+. +Proof. by apply/funext => x; rewrite /funrneg opprK. Qed. + +(* TODO: the following lemmas require a pointed type and realDomainType does +not seem to be at this point + +Lemma funrpos_restrict f : (f \_ D)^\+ = (f^\+) \_ D. +Proof. +by apply/funext => x; rewrite /patch/_^\+; case: ifP; rewrite //= maxxx. +Qed. + +Lemma funrneg_restrict f : (f \_ D)^\- = (f^\-) \_ D. +Proof. +by apply/funext => x; rewrite /patch/_^\-; case: ifP; rewrite //= oppr0 maxxx. +Qed.*) + +Lemma ge0_funrposE f : (forall x, D x -> 0 <= f x) -> {in D, f^\+ =1 f}. +Proof. by move=> f0 x; rewrite inE => Dx; apply/max_idPl/f0. Qed. + +Lemma ge0_funrnegE f : (forall x, D x -> 0 <= f x) -> {in D, f^\- =1 cst 0}. +Proof. +by move=> f0 x; rewrite inE => Dx; apply/max_idPr; rewrite lerNl oppr0 f0. +Qed. + +Lemma le0_funrposE f : (forall x, D x -> f x <= 0) -> {in D, f^\+ =1 cst 0}. +Proof. by move=> f0 x; rewrite inE => Dx; exact/max_idPr/f0. Qed. + +Lemma le0_funrnegE f : (forall x, D x -> f x <= 0) -> {in D, f^\- =1 \- f}. +Proof. +by move=> f0 x; rewrite inE => Dx; apply/max_idPl; rewrite lerNr oppr0 f0. +Qed. + +Lemma ge0_funrposM r f : (0 <= r)%R -> + (fun x => r * f x)^\+ = (fun x => r * (f^\+ x)). +Proof. by move=> ?; rewrite funeqE => x; rewrite /funrpos maxr_pMr// mulr0. Qed. + +Lemma ge0_funrnegM r f : (0 <= r)%R -> + (fun x => r * f x)^\- = (fun x => r * (f^\- x)). +Proof. +by move=> r0; rewrite funeqE => x; rewrite /funrneg -mulrN maxr_pMr// mulr0. +Qed. + +Lemma le0_funrposM r f : (r <= 0)%R -> + (fun x => r * f x)^\+ = (fun x => - r * (f^\- x)). +Proof. +move=> r0; rewrite -[in LHS](opprK r); under eq_fun do rewrite mulNr. +by rewrite funrposN ge0_funrnegM ?oppr_ge0. +Qed. + +Lemma le0_funrnegM r f : (r <= 0)%R -> + (fun x => r * f x)^\- = (fun x => - r * (f^\+ x)). +Proof. +move=> r0; rewrite -[in LHS](opprK r); under eq_fun do rewrite mulNr. +by rewrite funrnegN ge0_funrposM ?oppr_ge0. +Qed. + +Lemma funrposDneg f : f^\+ + f^\- = Num.norm \o f. +Proof. +rewrite funeqE => x /=; rewrite !fctE/=; have [fx0|/ltW fx0] := leP (f x) 0. +- rewrite ler0_norm// /funrpos /funrneg. + move/max_idPr : (fx0) => ->; rewrite add0r. + by move: fx0; rewrite -{1}oppr0 lerNr => /max_idPl ->. +- rewrite ger0_norm// /funrpos /funrneg; move/max_idPl : (fx0) => ->. + by move: fx0; rewrite -{1}oppr0 lerNl => /max_idPr ->; rewrite addr0. +Qed. + +Lemma funrposBneg f : f^\+ - f^\- = f. +Proof. +apply/funext => x. +rewrite /funrpos /funrneg/= !fctE; have [|/ltW] := leP (f x) 0. + by rewrite -{1}oppr0 -lerNr => /max_idPl ->; rewrite opprK add0r. +by rewrite -{1}oppr0 -lerNl => /max_idPr ->; rewrite subr0. +Qed. + +Lemma funrDB f g : (f^\+ + g^\+) - (f^\- + g^\-) = f + g. +Proof. by rewrite addBrfctE !funrposBneg. Qed. + +Lemma funrpos_le f g : + {in D, forall x, f x <= g x} -> {in D, forall x, f^\+ x <= g^\+ x}. +Proof. +move=> fg x Dx; rewrite /funrpos /Num.max; case: ifPn => fx; case: ifPn => gx//. +- by rewrite leNgt. +- by move: fx; rewrite -leNgt => /(lt_le_trans gx); rewrite ltNge fg. +- exact: fg. +Qed. + +Lemma funrneg_le f g : + {in D, forall x, f x <= g x} -> {in D, forall x, g^\- x <= f^\- x}. +Proof. +move=> fg x Dx; rewrite /funrneg /Num.max; case: ifPn => gx; case: ifPn => fx//. +- by rewrite leNgt. +- by move: gx; rewrite -leNgt => /(lt_le_trans fx); rewrite ltrN2 ltNge fg. +- by rewrite lerN2; exact: fg. +Qed. + +End funrposneg_lemmas. +#[global] +Hint Extern 0 (is_true (0%R <= _ ^\+ _)%R) => solve [apply: funrpos_ge0] : core. +#[global] +Hint Extern 0 (is_true (0%R <= _ ^\- _)%R) => solve [apply: funrneg_ge0] : core. + HB.lock Definition funepos T (R : realDomainType) (f : T -> \bar R) := fun x => maxe (f x) 0. @@ -774,9 +904,9 @@ move=> r0; rewrite -[in LHS](opprK r); under eq_fun do rewrite EFinN mulNe. by rewrite funenegN ge0_funeposM ?oppr_ge0. Qed. -Lemma fune_abse f : abse \o f = f^\+ \+ f^\-. +Lemma funeposDneg f : f^\+ + f^\- = abse \o f. Proof. -rewrite funeqE => x /=; have [fx0|/ltW fx0] := leP (f x) 0. +rewrite funeqE => x /=; rewrite !fctE/=; have /orP[fx0|fx0] := le_total (f x) 0. - rewrite lee0_abs// funeposE funenegE. move/max_idPr : (fx0) => ->; rewrite add0e. by move: fx0; rewrite -{1}oppe0 leeNr => /max_idPl ->. @@ -784,7 +914,7 @@ rewrite funeqE => x /=; have [fx0|/ltW fx0] := leP (f x) 0. by move: fx0; rewrite -{1}oppe0 leeNl => /max_idPr ->; rewrite adde0. Qed. -Lemma funeposneg f : f = (fun x => f^\+ x - f^\- x). +Lemma funeposBneg f : f^\+ \- f^\- = f. Proof. rewrite funeqE => x; rewrite funeposE funenegE; have [|/ltW] := leP (f x) 0. by rewrite -{1}oppe0 -leeNr => /max_idPl ->; rewrite oppeK add0e. @@ -797,27 +927,13 @@ by rewrite funenegE funeposE; case: (f x) => [r| |]; [rewrite -fine_max/=|rewrite /maxe /= ltNyr|rewrite /maxe /= ltNyr]. Qed. -Lemma funeD_Dpos f g : f \+ g = (f \+ g)^\+ \- (f \+ g)^\-. -Proof. -apply/funext => x; rewrite funeposE funenegE; have [|/ltW] := leP 0 (f x + g x). -- by rewrite -{1}oppe0 -leeNl => /max_idPr ->; rewrite sube0. -- by rewrite -{1}oppe0 -leeNr => /max_idPl ->; rewrite oppeK add0e. -Qed. +#[deprecated(since="mathcomp-analysis 1.15.0", note="use `funeposBneg` instead")] +Lemma funeD_Dpos f g : (f + g)^\+ \- (f + g)^\- = f + g. +Proof. by rewrite funeposBneg. Qed. -Lemma funeD_posD f g : f \+ g = (f^\+ \+ g^\+) \- (f^\- \+ g^\-). +Lemma funeDB f g : (f^\+ + g^\+) \- (f^\- + g^\-) = f + g. Proof. -apply/funext => x; rewrite !funeposE !funenegE. -have [|fx0] := leP 0 (f x); last rewrite add0e. -- rewrite -{1}oppe0 leeNl => /max_idPr ->; have [|/ltW] := leP 0 (g x). - by rewrite -{1}oppe0 leeNl => /max_idPr ->; rewrite adde0 sube0. - by rewrite -{1}oppe0 -leeNr => /max_idPl ->; rewrite adde0 sub0e oppeK. -- move/ltW : (fx0); rewrite -{1}oppe0 leeNr => /max_idPl ->. - have [|] := leP 0 (g x); last rewrite add0e. - by rewrite -{1}oppe0 leeNl => /max_idPr ->; rewrite adde0 oppeK addeC. - move gg' : (g x) => g'; move: g' gg' => [g' gg' g'0|//|goo _]. - + move/ltW : (g'0); rewrite -{1}oppe0 -leeNr => /max_idPl => ->. - by rewrite fin_num_oppeD// 2!oppeK. - + by rewrite /maxe /=; case: (f x) fx0. +by rewrite ge0_addBefctE ?funeposBneg//; by move=> x; rewrite funeneg_ge0. Qed. Lemma funepos_le f g : @@ -839,12 +955,27 @@ move=> fg x Dx; rewrite !funenegE /maxe; case: ifPn => gx; case: ifPn => fx //. Qed. End funposneg_lemmas. +#[deprecated(since="mathcomp-analysis 1.15.0", note="use `-funeDB` instead")] +Notation funeD_posD := funeDB (only parsing). + #[global] Hint Extern 0 (is_true (0%R <= _ ^\+ _)%E) => solve [apply: funepos_ge0] : core. #[global] Hint Extern 0 (is_true (0%R <= _ ^\- _)%E) => solve [apply: funeneg_ge0] : core. +Section funrpos_funepos_lemmas. +Context {T : Type} {R : realDomainType}. + +Lemma funerpos (f : T -> R) : (EFin \o f)^\+%E = (EFin \o f^\+). +Proof. by apply/funext => x; rewrite funeposE /funrpos/= EFin_max. Qed. + +Lemma funerneg (f : T -> R) : (EFin \o f)^\-%E = (EFin \o f^\-). +Proof. by apply/funext => x; rewrite funenegE /funrneg/= EFin_max. Qed. + +End funrpos_funepos_lemmas. + Definition indic {T} {R : pzRingType} (A : set T) (x : T) : R := (x \in A)%:R. + Reserved Notation "'\1_' A" (at level 8, A at level 2, format "'\1_' A") . Notation "'\1_' A" := (indic A) : ring_scope. diff --git a/theories/probability.v b/theories/probability.v index 0c8762181..060dbce25 100644 --- a/theories/probability.v +++ b/theories/probability.v @@ -1,4 +1,4 @@ -(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *) +(* mathcomp analysis (c) 2026 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg. From mathcomp Require Import poly ssrnum ssrint interval archimedean finmap. @@ -1646,7 +1646,7 @@ pose f_ := nnsfun_approx measurableT mf. transitivity (lim (\int[uniform_prob ab]_x (f_ n x)%:E @[n --> \oo])%E). rewrite -monotone_convergence//=. - apply: eq_integral => ? /[!inE] xD; apply/esym/cvg_lim => //=. - exact: cvg_nnsfun_approx. + exact/cvg_nnsfun_approx. - by move=> n; exact/measurable_EFinP/measurable_funTS. - by move=> n ? _; rewrite lee_fin. - by move=> ? _ ? ? mn; rewrite lee_fin; exact/lefP/nd_nnsfun_approx.