From eb858aa4f0fafdea997e8a509ff2e3f61450321b Mon Sep 17 00:00:00 2001 From: zstone1 Date: Mon, 29 Aug 2022 09:57:45 -0400 Subject: [PATCH 01/42] Pasting Lemma (#735) * trying to write path stuff * pasting lemma * lint * updating changelog * reverting change * Update theories/topology.v Co-authored-by: Cyril Cohen Co-authored-by: Cyril Cohen --- CHANGELOG_UNRELEASED.md | 2 ++ theories/topology.v | 26 ++++++++++++++++++++++++++ 2 files changed, 28 insertions(+) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 99aa513be0..ea987f2e55 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -13,6 +13,8 @@ - in `topology.v`: + lemma `near_inftyS` + + lemma `continuous_closedP`, `closedU`, `pasting` + - in `sequences.v`: + lemmas `contraction_dist`, `contraction_cvg`, `contraction_cvg_fixed`, `banach_fixed_point`, diff --git a/theories/topology.v b/theories/topology.v index 1aec5e6756..eba7e2d981 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -2507,6 +2507,18 @@ by move=> /(filterI FAu_) => /filter_ex[t [Au_t u_Bt]]; exists (u_ t). Qed. Arguments closed_cvg {T V F FF u_} _ _ _ _ _. +Lemma continuous_closedP (S T : topologicalType) (f : S -> T) : + continuous f <-> forall A, closed A -> closed (f @^-1` A). +Proof. +rewrite continuousP; split=> ctsf ? ?. + by rewrite -openC preimage_setC; apply ctsf; rewrite openC. +by rewrite -closedC preimage_setC; apply ctsf; rewrite closedC. +Qed. + +Lemma closedU (T : topologicalType) (D E : set T) : + closed D -> closed E -> closed (D `|` E). +Proof. by rewrite -?openC setCU; exact: openI. Qed. + Section closure_lemmas. Variable T : topologicalType. Implicit Types E A B U : set T. @@ -5900,6 +5912,20 @@ by split => + x /[dup] Ax /oA Aox; rewrite /filter_of /= => /(_ _ Ax); rewrite -(nbhs_subspace_interior Aox). Qed. +Lemma pasting {U} A B (f : T -> U) : closed A -> closed B -> + {within A, continuous f} -> {within B, continuous f} -> + {within A `|` B, continuous f}. +Proof. +move=> ? ? ctsA ctsB; apply/continuous_closedP => W oW. +case/continuous_closedP/(_ _ oW)/closed_subspaceP: ctsA => V1 [? V1W]. +case/continuous_closedP/(_ _ oW)/closed_subspaceP: ctsB => V2 [? V2W]. +apply/closed_subspaceP; exists ((V1 `&` A) `|` (V2 `&` B)); split. + by apply: closedU; exact: closedI. +rewrite [RHS]setIUr -V2W -V1W eqEsubset; split=> ?. + by case=> [[][]] ? ? [] ?; [left | left | right | right]; split. +by case=> [][] ? ?; split=> []; [left; split | left | right; split | right]. +Qed. + End SubspaceRelative. Lemma continuous_compact {T U : topologicalType} (f : T -> U) A : From 8de211e8912328604742129eab7d5cfbfd8946ad Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 24 Aug 2022 22:25:10 +0900 Subject: [PATCH 02/42] fine is a measurable_fun --- CHANGELOG_UNRELEASED.md | 2 ++ theories/lebesgue_measure.v | 16 ++++++++++++++++ 2 files changed, 18 insertions(+) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index ea987f2e55..e874929fa6 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -25,6 +25,8 @@ `onemX_ge0`, `onemX_lt1`, `onemD`, `onemMr`, `onemM` - in `signed.v`: + lemmas `onem_PosNum`, `onemX_NngNum` +- in `lebesgue_measure.v`: + + lemma `measurable_fun_fine` ### Changed diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 3870ffd810..c41e3cc836 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -834,6 +834,22 @@ End salgebra_R_ssets. Hint Extern 0 (measurable [set _]) => solve [apply: measurable_set1| apply: emeasurable_set1] : core. +Lemma measurable_fun_fine (R : realType) (D : set (\bar R)) : measurable D -> + measurable_fun D fine. +Proof. +move=> mD _ /= B mB; rewrite [X in measurable X](_ : _ `&` _ = if 0%R \in B then + D `&` ((EFin @` B) `|` [set -oo; +oo]%E) else D `&` EFin @` B); last first. + apply/seteqP; split=> [[r [Dr Br]|[Doo B0]|[Doo B0]]|[r| |]]. + - by case: ifPn => _; split => //; left; exists r. + - by rewrite mem_set//; split => //; right; right. + - by rewrite mem_set//; split => //; right; left. + - by case: ifPn => [_ [Dr [[s + [sr]]|[]//]]|_ [Dr [s + [sr]]]]; rewrite sr. + - by case: ifPn => [/[!inE] B0 [Doo [[]//|]] [//|_]|B0 [Doo//] []]. + - by case: ifPn => [/[!inE] B0 [Doo [[]//|]] [//|_]|B0 [Doo//] []]. +case: ifPn => B0; apply/measurableI => //; last exact: measurable_EFin. +by apply: measurableU; [exact: measurable_EFin|exact: measurableU]. +Qed. + Section lebesgue_measure_itv. Variable R : realType. From 06421649a1aab2a2b58d3ff70ab8005f2d64b05f Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 22 Jul 2022 22:17:48 +0900 Subject: [PATCH 03/42] ge0_integral_mscale --- CHANGELOG_UNRELEASED.md | 16 ++++ theories/lebesgue_integral.v | 162 +++++++++++++++++++++++++---------- 2 files changed, 133 insertions(+), 45 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index e874929fa6..99ab541eb7 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -27,6 +27,22 @@ + lemmas `onem_PosNum`, `onemX_NngNum` - in `lebesgue_measure.v`: + lemma `measurable_fun_fine` + + lemma `big_const_idem` + + lemma `big_id_idem` + + lemma `big_rem_AC` + + lemma `bigD1_AC` + + lemma `big_mkcond_idem` + + lemma `big_split_idem` + + lemma `big_id_idem_AC` + + lemma `bigID_idem` +- in `mathcomp_extra.v`: + + lemmas `bigmax_le` and `bigmax_lt` + + lemma `bigmin_idr` + + lemma `bigmax_idr` +- in `classical_sets.v`: + + lemma `subset_refl` +- in lemma `lebesgue_integral.v`: + + lemma `ge0_integral_mscale` ### Changed diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index 36183f2cdb..a0581c2787 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -2146,6 +2146,123 @@ Unshelve. all: by end_near. Qed. End ge0_integralM. +Section integral_indic. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType) + (mu : {measure set T -> \bar R}) (D : set T) (mD : measurable D). + +Lemma integral_indic (E : set T) : measurable E -> + \int[mu]_(x in D) (\1_E x)%:E = mu (E `&` D). +Proof. +move=> mE; rewrite (_ : \1_E = indic_nnsfun R mE)// integral_nnsfun//=. +by rewrite restrict_indic sintegral_indic//; exact: measurableI. +Qed. + +End integral_indic. + +Section integralM_indic. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). + +Lemma integralM_indic (f : R -> set T) (k : R) : + ((k < 0)%R -> f k = set0) -> measurable (f k) -> + \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. +Proof. +move=> fk0 mfk; have [k0|k0] := ltP k 0%R. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. + by move=> x _; rewrite fk0// indic0 mulr0. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. + by rewrite fk0// indic0. +under eq_integral do rewrite EFinM. +rewrite ge0_integralM//. +- apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. + by rewrite (_ : \1_(f k) = mindic R mfk). +- by move=> y _; rewrite lee_fin. +Qed. + +Lemma integralM_indic_nnsfun (f : {nnsfun T >-> R}) (k : R) : + \int[m]_(x in D) (k * \1_(f @^-1` [set k]) x)%:E = + k%:E * \int[m]_(x in D) (\1_(f @^-1` [set k]) x)%:E. +Proof. +rewrite (@integralM_indic (fun k => f @^-1` [set k]))// => k0. +by rewrite preimage_nnfun0. +Qed. + +End integralM_indic. + +Section integral_mscale. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). +Variables (k : {nonneg R}) (f : T -> \bar R). + +Let integral_mscale_indic E : measurable E -> + \int[mscale k m]_(x in D) (\1_E x)%:E = + k%:num%:E * \int[m]_(x in D) (\1_E x)%:E. +Proof. by move=> ?; rewrite !integral_indic. Qed. + +Let integral_mscale_nnsfun (h : {nnsfun T >-> R}) : + \int[mscale k m]_(x in D) (h x)%:E = k%:num%:E * \int[m]_(x in D) (h x)%:E. +Proof. +rewrite -ge0_integralM//; last 2 first. + by apply/EFin_measurable_fun; exact: measurable_funS (@measurable_funP _ _ _ h). + by move=> x _; rewrite lee_fin. +under [LHS]eq_integral do rewrite fimfunE -sumEFin. +rewrite ge0_integral_sum//; last 2 first. + move=> r; apply/EFin_measurable_fun/measurable_funrM. + apply: (@measurable_funS _ _ _ _ setT) => //. + by rewrite (_ : \1__ = mindic R (@measurable_sfunP _ _ _ h r)). + by move=> n x Dx; rewrite EFinM muleindic_ge0. +under [RHS]eq_integral. + move=> x xD; rewrite fimfunE -sumEFin ge0_sume_distrr//; last first. + by move=> r _; rewrite EFinM muleindic_ge0. + over. +rewrite ge0_integral_sum//; last 2 first. + move=> r; apply/EFin_measurable_fun/measurable_funrM/measurable_funrM. + apply: (@measurable_funS _ _ _ _ setT) => //. + by rewrite (_ : \1__ = mindic R (@measurable_sfunP _ _ _ h r)). + by move=> n x Dx; rewrite EFinM mule_ge0// muleindic_ge0. +apply eq_bigr => r _; rewrite ge0_integralM//. +- by rewrite !integralM_indic_nnsfun//= integral_mscale_indic// muleCA. + apply/EFin_measurable_fun/measurable_funrM. + apply: (@measurable_funS _ _ _ _ setT) => //. + by rewrite (_ : \1__ = mindic R (@measurable_sfunP _ _ _ h r)). +- by move=> t Dt; rewrite muleindic_ge0. +Qed. + +Lemma ge0_integral_mscale (mf : measurable_fun D f) : + (forall x, D x -> 0 <= f x) -> + \int[mscale k m]_(x in D) f x = k%:num%:E * \int[m]_(x in D) f x. +Proof. +move=> f0; have [f_ [ndf_ f_f]] := approximation mD mf f0. +transitivity (lim (fun n => \int[mscale k m]_(x in D) (f_ n x)%:E)). + rewrite -monotone_convergence//=. + - by apply eq_integral => x /[!inE] xD; apply/esym/cvg_lim => //=; exact: f_f. + - move=> n; apply/EFin_measurable_fun. + exact: (@measurable_funS _ _ _ _ setT). + - by move=> n x Dx; rewrite lee_fin. + - by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. +rewrite (_ : \int[m]_(x in D) _ = + lim (fun n => \int[m]_(x in D) (f_ n x)%:E)); last first. + rewrite -monotone_convergence//. + - by apply: eq_integral => x /[!inE] xD; apply/esym/cvg_lim => //; exact: f_f. + - move=> n; apply/EFin_measurable_fun. + exact: (@measurable_funS _ _ _ _ setT). + - by move=> n x Dx; rewrite lee_fin. + - by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. +rewrite -ereal_limrM//. + by congr (lim _); apply/funext => n /=; rewrite integral_mscale_nnsfun. +apply/ereal_nondecreasing_is_cvg => a b ab; apply ge0_le_integral => //. +- by move=> x Dx; rewrite lee_fin. +- exact/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT). +- by move=> x Dx; rewrite lee_fin. +- exact/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT). + by move=> x Dx; rewrite lee_fin; move/ndf_ : ab => /lefP. +Qed. + +End integral_mscale. + Section fatou. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). @@ -2272,51 +2389,6 @@ Qed. End integral_cst. -Section integral_ind. -Local Open Scope ereal_scope. -Variables (d : measure_display) (T : measurableType d) (R : realType). -Variables (mu : {measure set T -> \bar R}) (D : set T) (mD : measurable D). - -Lemma integral_indic (E : set T) : measurable E -> - \int[mu]_(x in D) (\1_E x)%:E = mu (E `&` D). -Proof. -move=> mE; rewrite (_ : \1_E = indic_nnsfun R mE)// integral_nnsfun//=. -by rewrite restrict_indic sintegral_indic//; exact: measurableI. -Qed. - -End integral_ind. - -Section integralM_indic. -Local Open Scope ereal_scope. -Variables (d : measure_display) (T : measurableType d) (R : realType). -Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). - -Lemma integralM_indic (f : R -> set T) (k : R) : - ((k < 0)%R -> f k = set0) -> measurable (f k) -> - \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. -Proof. -move=> fk0 mfk; have [k0|k0] := ltP k 0%R. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. - by move=> x _; rewrite fk0// indic0 mulr0. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. - by rewrite fk0// indic0. -under eq_integral do rewrite EFinM. -rewrite ge0_integralM//. -- apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1_(f k) = mindic R mfk). -- by move=> y _; rewrite lee_fin. -Qed. - -Lemma integralM_indic_nnsfun (f : {nnsfun T >-> R}) (k : R) : - \int[m]_(x in D) (k * \1_(f @^-1` [set k]) x)%:E = - k%:E * \int[m]_(x in D) (\1_(f @^-1` [set k]) x)%:E. -Proof. -rewrite (@integralM_indic (fun k => f @^-1` [set k]))// => k0. -by rewrite preimage_nnfun0. -Qed. - -End integralM_indic. - Section integral_dirac. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (a : T) (R : realType). From 3b2d205f9b16b7ccbacb02c7783c24bcdcda3a98 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Tue, 2 Aug 2022 19:55:19 +0900 Subject: [PATCH 04/42] add and use specialized lemmas to shorten - renaming --- CHANGELOG_UNRELEASED.md | 24 ++-- theories/lebesgue_integral.v | 244 +++++++++++++++-------------------- theories/measure.v | 6 +- 3 files changed, 121 insertions(+), 153 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 99ab541eb7..b92599fa25 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -27,22 +27,12 @@ + lemmas `onem_PosNum`, `onemX_NngNum` - in `lebesgue_measure.v`: + lemma `measurable_fun_fine` - + lemma `big_const_idem` - + lemma `big_id_idem` - + lemma `big_rem_AC` - + lemma `bigD1_AC` - + lemma `big_mkcond_idem` - + lemma `big_split_idem` - + lemma `big_id_idem_AC` - + lemma `bigID_idem` -- in `mathcomp_extra.v`: - + lemmas `bigmax_le` and `bigmax_lt` - + lemma `bigmin_idr` - + lemma `bigmax_idr` -- in `classical_sets.v`: - + lemma `subset_refl` -- in lemma `lebesgue_integral.v`: +- in `lebesgue_integral.v`: + lemma `ge0_integral_mscale` +- in `measure.v`: + + lemma `measurable_funTS` +- in `lebesgue_measure.v`: + + lemma `measurable_fun_indic` ### Changed @@ -66,6 +56,10 @@ + `lee_ninfty_eq` -> `leeNy_eq` - in `measure.v`: + `cvg_mu_inc` -> `nondecreasing_cvg_mu` +- in `lebesgue_integral.v`: + + `muleindic_ge0` -> `nnfun_muleindic_ge0` + + `mulem_ge0` -> `mulemu_ge0` + + `nnfun_mulem_ge0` -> `nnsfun_mulemu_ge0` ### Removed diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index a0581c2787..b27a2b902f 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -330,6 +330,13 @@ Definition max_mfun f g := [the {mfun aT >-> _} of f \max g]. End ring. Arguments indic_mfun {d aT rT} _. +Lemma measurable_fun_indic (d : measure_display) (T : measurableType d) + (R : realType) (D A : set T) : measurable A -> + measurable_fun D (\1_A : T -> R). +Proof. +by move=> mA; apply/measurable_funTS; rewrite (_ : \1__ = mindic R mA). +Qed. + Section sfun_pred. Context {d} {aT : measurableType d} {rT : realType}. Definition sfun : {pred _ -> _} := [predI @mfun _ aT rT & fimfun]. @@ -626,33 +633,33 @@ Section mulem_ge0. Local Open Scope ereal_scope. Let mulef_ge0 (R : realDomainType) x (f : R -> \bar R) : - (forall x, 0 <= f x) -> ((x < 0)%R -> f x = 0) -> 0 <= x%:E * f x. + 0 <= f x -> ((x < 0)%R -> f x = 0) -> 0 <= x%:E * f x. Proof. move=> A0 xA /=; have [x0|x0] := ltP x 0%R; first by rewrite (xA x0) mule0. by rewrite mule_ge0. Qed. -Lemma muleindic_ge0 d (T : measurableType d) (R : realDomainType) +Lemma nnfun_muleindic_ge0 d (T : measurableType d) (R : realDomainType) (f : {nnfun T >-> R}) r z : 0 <= r%:E * (\1_(f @^-1` [set r]) z)%:E. Proof. apply: (@mulef_ge0 _ _ (fun r => (\1_(f @^-1` [set r]) z)%:E)). - by move=> x; rewrite lee_fin /indic. + by rewrite lee_fin// indicE. by move=> r0; rewrite preimage_nnfun0// indic0. Qed. -Lemma mulem_ge0 d (T : measurableType d) (R : realType) +Lemma mulemu_ge0 d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}) x (A : R -> set T) : ((x < 0)%R -> A x = set0) -> 0 <= x%:E * mu (A x). Proof. by move=> xA; rewrite (@mulef_ge0 _ _ (mu \o _))//= => /xA ->; rewrite measure0. Qed. -Arguments mulem_ge0 {d T R mu x} A. +Global Arguments mulemu_ge0 {d T R mu x} A. -Lemma nnfun_mulem_ge0 d (T : measurableType d) (R : realType) - (mu : {measure set T -> \bar R})(f : {nnsfun T >-> R}) x : +Lemma nnsfun_mulemu_ge0 d (T : measurableType d) (R : realType) + (mu : {measure set T -> \bar R}) (f : {nnsfun T >-> R}) x : 0 <= x%:E * mu (f @^-1` [set x]). Proof. -by apply: (mulem_ge0 (fun x => f @^-1` [set x])); exact: preimage_nnfun0. +by apply: (mulemu_ge0 (fun x => f @^-1` [set x])); exact: preimage_nnfun0. Qed. End mulem_ge0. @@ -694,7 +701,7 @@ by case: ifPn => [/[!inE] <-|]; rewrite ?mul0e// measure0 mule0. Qed. Lemma sintegral_ge0 (f : {nnsfun T >-> R}) : 0 <= sintegral mu f. -Proof. by rewrite sintegralE fsume_ge0// => r _; exact: nnfun_mulem_ge0. Qed. +Proof. by rewrite sintegralE fsume_ge0// => r _; exact: nnsfun_mulemu_ge0. Qed. Lemma sintegral_indic (A : set T) : sintegral mu \1_A = mu A. Proof. @@ -742,7 +749,7 @@ transitivity (\sum_(x \in [set: R]) r%:E * (x%:E * m (f @^-1` [set x]))). rewrite (reindex_fsbigT (fun x => r * x)%R)//; last first. by exists ( *%R r ^-1)%R; [exact: mulKf|exact: mulVKf]. by apply: eq_fsbigr => x; rewrite mulrAC divrr ?unitfE// mul1r muleA EFinM. -by rewrite ge0_mule_fsumr// => x; exact: nnfun_mulem_ge0. +by rewrite ge0_mule_fsumr// => x; exact: nnsfun_mulemu_ge0. Qed. End sintegralrM. @@ -844,7 +851,7 @@ rewrite /fleg [X in _ X](_ : _ = \big[setU/set0]_(y <- fset_set (range f)) \big[setU/set0]_(x <- fset_set (range (g n)) | c * y <= x) (f @^-1` [set y] `&` (g n @^-1` [set x]))). apply: bigsetU_measurable => r _; apply: bigsetU_measurable => r' crr'. - by apply: measurableI; apply/measurable_sfunP. + exact/measurableI/measurable_sfunP. rewrite predeqE => t; split => [/= cfgn|]. - rewrite -bigcup_set; exists (f t); first by rewrite /= in_fset_set//= mem_set. rewrite -bigcup_set_cond; exists (g n t) => //=. @@ -924,7 +931,7 @@ rewrite [X in X --> _](_ : _ = fun n => \sum_(x <- fset_set (range f)) by rewrite mulr0 => /esym/eqP; rewrite (negbTE r0). by rewrite /preimage/= => -[fxr cnx]; rewrite mindicE mem_set// mulr1. rewrite sintegralE fsbig_finite//=; apply: ereal_lim_sum => [r n _|r _]. - apply: (@mulem_ge0 _ _ _ _ _ (fun x => f @^-1` [set x] `&` fleg c n)) => r0. + apply: (mulemu_ge0 (fun x => f @^-1` [set x] `&` fleg c n)) => r0. by rewrite preimage_nnfun0// set0I. apply: ereal_cvgrM => //; rewrite [X in _ --> X](_ : _ = mu (\bigcup_n (f @^-1` [set r] `&` fleg c n))); last first. @@ -1768,8 +1775,7 @@ wlog fg : D mD mf mg mfg / forall x, D x -> f x +? g x => [hwlogD|]; last first. have [f_ f_cvg] := approximation_sfun mD mf. have [g_ g_cvg] := approximation_sfun mD mg. apply: (emeasurable_fun_cvg (fun n x => (f_ n x + g_ n x)%:E)) => //. - move=> n; apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT) => //; exact: measurable_funD. + by move=> n; apply/EFin_measurable_fun/measurable_funTS/measurable_funD. move=> x Dx; under eq_fun do rewrite EFinD. by apply: ereal_cvgD; [exact: fg|exact: f_cvg|exact: g_cvg]. move=> A mA; wlog NAnoo: A mD mf mg mA / ~ (A -oo) => [hwlogA|]. @@ -1847,7 +1853,7 @@ wlog fg : D mD mf mg mfg / forall x, D x -> f x *? g x => [hwlogM|]; last first. have [g_ g_cvg] := approximation_sfun mD mg. apply: (emeasurable_fun_cvg (fun n x => (f_ n x * g_ n x)%:E)) => //. move=> n; apply/EFin_measurable_fun. - by apply: measurable_funM => //; exact: (@measurable_funS _ _ _ _ setT). + by apply: measurable_funM => //; exact: measurable_funTS. move=> x Dx; under eq_fun do rewrite EFinM. by apply: ereal_cvgM; [exact: fg|exact: f_cvg|exact: g_cvg]. move=> A mA; wlog NA0: A mD mf mg mA / ~ (A 0) => [hwlogA|]. @@ -2166,8 +2172,9 @@ Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). Lemma integralM_indic (f : R -> set T) (k : R) : - ((k < 0)%R -> f k = set0) -> measurable (f k) -> - \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. + ((k < 0)%R -> f k = set0) -> measurable (f k) -> + \int[m]_(x in D) (k * \1_(f k) x)%:E = + k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. Proof. move=> fk0 mfk; have [k0|k0] := ltP k 0%R. rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. @@ -2176,8 +2183,7 @@ move=> fk0 mfk; have [k0|k0] := ltP k 0%R. by rewrite fk0// indic0. under eq_integral do rewrite EFinM. rewrite ge0_integralM//. -- apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1_(f k) = mindic R mfk). +- exact/EFin_measurable_fun/measurable_fun_indic. - by move=> y _; rewrite lee_fin. Qed. @@ -2205,30 +2211,26 @@ Proof. by move=> ?; rewrite !integral_indic. Qed. Let integral_mscale_nnsfun (h : {nnsfun T >-> R}) : \int[mscale k m]_(x in D) (h x)%:E = k%:num%:E * \int[m]_(x in D) (h x)%:E. Proof. -rewrite -ge0_integralM//; last 2 first. - by apply/EFin_measurable_fun; exact: measurable_funS (@measurable_funP _ _ _ h). - by move=> x _; rewrite lee_fin. under [LHS]eq_integral do rewrite fimfunE -sumEFin. -rewrite ge0_integral_sum//; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM. - apply: (@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1__ = mindic R (@measurable_sfunP _ _ _ h r)). - by move=> n x Dx; rewrite EFinM muleindic_ge0. +rewrite [LHS]ge0_integral_sum//; last 2 first. + - move=> r. + exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. + - by move=> n x _; rewrite EFinM nnfun_muleindic_ge0. +rewrite -[RHS]ge0_integralM//; last 2 first. + - exact/EFin_measurable_fun/measurable_funTS. + - by move=> x _; rewrite lee_fin. under [RHS]eq_integral. move=> x xD; rewrite fimfunE -sumEFin ge0_sume_distrr//; last first. - by move=> r _; rewrite EFinM muleindic_ge0. + by move=> r _; rewrite EFinM nnfun_muleindic_ge0. over. -rewrite ge0_integral_sum//; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM/measurable_funrM. - apply: (@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1__ = mindic R (@measurable_sfunP _ _ _ h r)). - by move=> n x Dx; rewrite EFinM mule_ge0// muleindic_ge0. +rewrite [RHS]ge0_integral_sum//; last 2 first. + - move=> r; apply/EFin_measurable_fun/measurable_funrM/measurable_funrM. + exact/measurable_fun_indic. + - by move=> n x _; rewrite EFinM mule_ge0// nnfun_muleindic_ge0. apply eq_bigr => r _; rewrite ge0_integralM//. - by rewrite !integralM_indic_nnsfun//= integral_mscale_indic// muleCA. - apply/EFin_measurable_fun/measurable_funrM. - apply: (@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1__ = mindic R (@measurable_sfunP _ _ _ h r)). -- by move=> t Dt; rewrite muleindic_ge0. +- exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. +- by move=> t _; rewrite nnfun_muleindic_ge0. Qed. Lemma ge0_integral_mscale (mf : measurable_fun D f) : @@ -2239,26 +2241,24 @@ move=> f0; have [f_ [ndf_ f_f]] := approximation mD mf f0. transitivity (lim (fun n => \int[mscale k m]_(x in D) (f_ n x)%:E)). rewrite -monotone_convergence//=. - by apply eq_integral => x /[!inE] xD; apply/esym/cvg_lim => //=; exact: f_f. - - move=> n; apply/EFin_measurable_fun. - exact: (@measurable_funS _ _ _ _ setT). - - by move=> n x Dx; rewrite lee_fin. - - by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. + - by move=> n; exact/EFin_measurable_fun/measurable_funTS. + - by move=> n x _; rewrite lee_fin. + - by move=> x _ a b /ndf_ /lefP; rewrite lee_fin. rewrite (_ : \int[m]_(x in D) _ = - lim (fun n => \int[m]_(x in D) (f_ n x)%:E)); last first. - rewrite -monotone_convergence//. + lim (fun n => \int[m]_(x in D) (f_ n x)%:E)); last first. + rewrite -monotone_convergence//=. - by apply: eq_integral => x /[!inE] xD; apply/esym/cvg_lim => //; exact: f_f. - - move=> n; apply/EFin_measurable_fun. - exact: (@measurable_funS _ _ _ _ setT). - - by move=> n x Dx; rewrite lee_fin. - - by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. + - by move=> n; exact/EFin_measurable_fun/measurable_funTS. + - by move=> n x _; rewrite lee_fin. + - by move=> x _ a b /ndf_ /lefP; rewrite lee_fin. rewrite -ereal_limrM//. by congr (lim _); apply/funext => n /=; rewrite integral_mscale_nnsfun. apply/ereal_nondecreasing_is_cvg => a b ab; apply ge0_le_integral => //. -- by move=> x Dx; rewrite lee_fin. -- exact/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT). -- by move=> x Dx; rewrite lee_fin. -- exact/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT). - by move=> x Dx; rewrite lee_fin; move/ndf_ : ab => /lefP. +- by move=> x _; rewrite lee_fin. +- exact/EFin_measurable_fun/measurable_funTS. +- by move=> x _; rewrite lee_fin. +- exact/EFin_measurable_fun/measurable_funTS. + by move=> x _; rewrite lee_fin; move/ndf_ : ab => /lefP. Qed. End integral_mscale. @@ -2403,7 +2403,7 @@ transitivity (lim (fun n => \int[\d_ a]_(x in D) (f_ n x)%:E)). rewrite -monotone_convergence//. - apply: eq_integral => x Dx; apply/esym/cvg_lim => //; apply: f_f. by rewrite inE in Dx. - - by move=> n; apply/EFin_measurable_fun; exact/(@measurable_funS _ _ _ _ setT). + - by move=> n; apply/EFin_measurable_fun; exact/measurable_funTS. - by move=> *; rewrite lee_fin. - by move=> x _ m n mn; rewrite lee_fin; exact/lefP/ndf_. rewrite (_ : (fun _ => _) = (fun n => (f_ n a)%:E)). @@ -2417,11 +2417,8 @@ rewrite ge0_integral_sum//. rewrite big1_fset ?adde0// => r; rewrite !inE => /andP[rfna _] _. rewrite integral_indic//= diracE memNset ?mule0//. by apply/not_andP; left; exact/nesym/eqP. -- move=> r; apply/EFin_measurable_fun. - apply: measurable_funM => //; first exact: measurable_fun_cst. - apply: (@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (f_ n) r)). -- by move=> r x _; rewrite muleindic_ge0. +- by move=> r; exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. +- by move=> r x _; rewrite nnfun_muleindic_ge0. Qed. Lemma integral_dirac (f : T -> \bar R) (mf : measurable_fun D f) : @@ -2458,10 +2455,8 @@ Proof. under eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//; last 2 first. - move=> r /=; apply: measurable_fun_comp => //. - apply: measurable_funM => //. - exact: measurable_fun_cst. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f r)). - - by move=> r t _; rewrite EFinM muleindic_ge0. + exact/measurable_funrM/measurable_fun_indic. + - by move=> r t _; rewrite EFinM nnfun_muleindic_ge0. transitivity (\sum_(i <- fset_set (range f)) (\sum_(n < N) i%:E * \int[m_ n]_x (\1_(f @^-1` [set i]) x)%:E)). apply eq_bigr => r _. @@ -2515,7 +2510,7 @@ rewrite big_ord_recr/= -ih. rewrite (_ : _ m_ N.+1 = measure_add [the measure _ _ of msum m_ N] (m_ N)); last first. by apply/funext => A; rewrite measure_addE /msum/= big_ord_recr. have mf_ n : measurable_fun D (fun x => (f_ n x)%:E). - by apply: (@measurable_funS _ _ _ _ setT) => //; exact/EFin_measurable_fun. + exact/measurable_funTS/EFin_measurable_fun. have f_ge0 n x : D x -> 0 <= (f_ n x)%:E by move=> Dx; rewrite lee_fin. have cvg_f_ (m : {measure set T -> \bar R}) : cvg (fun x => \int[m]_(x0 in D) (f_ x x0)%:E). apply: ereal_nondecreasing_is_cvg => a b ab. @@ -2570,11 +2565,9 @@ Lemma integral_measure_series_nnsfun (D : set T) (mD : measurable D) Proof. under eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//; last 2 first. - - move=> r /=. - apply: measurable_fun_comp => //. - apply: measurable_funM => //; first exact: measurable_fun_cst. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f r)). - - by move=> r t _; rewrite EFinM muleindic_ge0. + - move=> r /=; apply: measurable_fun_comp => //. + exact/measurable_funrM/measurable_fun_indic. + - by move=> r t _; rewrite EFinM nnfun_muleindic_ge0. transitivity (\sum_(i <- fset_set (range f)) (\sum_(n r _. @@ -3049,17 +3042,13 @@ have [M M0 muM] : exists2 M, (0 <= M)%R & exists (fine (\int[mu]_(x in D) `|f x|)); first exact/fine_ge0/integral_ge0. move=> n. rewrite -integral_indic// -ge0_integralM//; last 2 first. - - apply: measurable_fun_comp=> //; apply: (@measurable_funS _ _ _ _ setT)=>//. - by rewrite (_ : \1_ _ = indic_nnsfun R mE). + - by apply: measurable_fun_comp=> //; exact/measurable_fun_indic. - by move=> *; rewrite lee_fin. rewrite fineK//; last first. by case: fint => _ foo; rewrite ge0_fin_numE//; exact: integral_ge0. apply: ge0_le_integral => //. - by move=> *; rewrite lee_fin /indic. - - apply/EFin_measurable_fun; apply: measurable_funM=>//. - + exact: measurable_fun_cst. - + apply: (@measurable_funS _ _ _ _ setT)=>//. - by rewrite (_ : \1_ _ = indic_nnsfun R mE)//. + - exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. - by apply: measurable_fun_comp => //; case: fint. - move=> x Dx; rewrite /= indicE. have [|xE] := boolP (x \in E); last by rewrite mule0. @@ -3361,16 +3350,13 @@ have le_f_M t : D t -> `|f t| <= M%:E * (f' t)%:E. have : 0 <= \int[mu]_(x in D) `|f x| <= `|M|%:E * mu Df_neq0. rewrite integral_ge0//= /Df_neq0 -{2}(setIid D) setIAC -integral_indic//. rewrite -/Df_neq0 -ge0_integralM//; last 2 first. - - apply: measurable_fun_comp=> //; apply: (@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1_ _ = mindic R mDf_neq0). - - by move=> x Dx; rewrite lee_fin. + - by apply: measurable_fun_comp=> //; exact: measurable_fun_indic. + - by move=> x ?; rewrite lee_fin. apply: ge0_le_integral => //. - exact: measurable_fun_comp. - by move=> x Dx; rewrite mule_ge0// lee_fin. - apply: emeasurable_funM; first exact: measurable_fun_cst. - apply: measurable_fun_comp => //. - apply: (@measurable_funS _ _ _ _ setT)=> //. - by rewrite (_ : \1_ _ = mindic R mDf_neq0)//. + by apply: measurable_fun_comp => //; exact: measurable_fun_indic. - move=> x Dx. rewrite (le_trans (le_f_M _ Dx))// lee_fin /f' indicE. by case: (_ \in _) => //; rewrite ?mulr1 ?mulr0// ler_norm. @@ -3481,9 +3467,8 @@ move=> mN mD ND mf muN0; rewrite integralEindic//. rewrite (eq_integral (fun x => `|f x * (\1_N x)%:E|)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_N t)%:E)// lee_fin. apply/ae_eq_integral_abs => //. - apply: emeasurable_funM => //; first exact: (@measurable_funS _ _ _ _ D). - apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1_N = mindic R mN). + apply: emeasurable_funM => //; first exact: (measurable_funS mD). + exact/EFin_measurable_fun/measurable_fun_indic. exists N; split => // t /= /not_implyP[_]; rewrite indicE. by have [|] := boolP (t \in N); rewrite ?inE ?mule0. Qed. @@ -3511,7 +3496,7 @@ pose oneN : {nnsfun T >-> R} := [the {nnsfun T >-> R} of mindic R mN]. have intone : mu.-integrable D (fun x => f x * (oneN x)%:E). split. apply: emeasurable_funM=> //; apply/EFin_measurable_fun. - exact: (@measurable_funS _ _ _ _ setT). + exact: measurable_funTS. rewrite (eq_integral (fun x => `|f x| * (\1_N x)%:E)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_N t)%:E) // lee_fin. rewrite -integral_setI_indic// (@integral_abs_eq0 D)//. @@ -3521,18 +3506,17 @@ have h1 : mu.-integrable D f <-> mu.-integrable D (fun x => f x * (oneCN x)%:E). split=> [intf|intCf]. split. apply: emeasurable_funM=> //; apply/EFin_measurable_fun => //. - exact: (@measurable_funS _ _ _ _ setT). + exact: measurable_funTS. rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E) // lee_fin. rewrite -integral_setI_indic//; case: intf => _; apply: le_lt_trans. by apply: subset_integral => //; [exact:measurableI|exact:measurable_fun_comp]. split => //; rewrite (funID mN f) -/oneCN -/oneN. have ? : measurable_fun D (fun x : T => f x * (oneCN x)%:E). - apply: emeasurable_funM=> //. - by apply/EFin_measurable_fun; exact: (@measurable_funS _ _ _ _ setT). + by apply: emeasurable_funM=> //; exact/EFin_measurable_fun/measurable_funTS. have ? : measurable_fun D (fun x : T => f x * (oneN x)%:E). apply: emeasurable_funM => //. - by apply/EFin_measurable_fun; apply: (@measurable_funS _ _ _ _ setT). + exact/EFin_measurable_fun/measurable_funTS. apply: (@le_lt_trans _ _ (\int[mu]_(x in D) (`|f x * (oneCN x)%:E| + `|f x * (oneN x)%:E|))). apply: ge0_le_integral => //. @@ -3547,7 +3531,7 @@ have h2 : mu.-integrable (D `\` N) f <-> split=> [intCf|intCf]. split. apply: emeasurable_funM=> //; apply/EFin_measurable_fun => //. - exact: (@measurable_funS _ _ _ _ setT). + exact: measurable_funTS. rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E)// lee_fin. rewrite -integral_setI_indic //; case: intCf => _; apply: le_lt_trans. @@ -3571,10 +3555,10 @@ move=> mN mD mf f0 muN0. rewrite {1}(funID mN f) ge0_integralD//; last 4 first. - by move=> x Dx; apply: mule_ge0 => //; [exact: f0|rewrite lee_fin]. - apply: emeasurable_funM=> //; apply/EFin_measurable_fun=> //. - exact: (@measurable_funS _ _ _ _ setT). + exact: measurable_funTS. - by move=> x Dx; apply: mule_ge0 => //; [exact: f0|rewrite lee_fin]. - apply: emeasurable_funM=> //; apply/EFin_measurable_fun=> //. - exact: (@measurable_funS _ _ _ _ setT). + exact: measurable_funTS. rewrite -integral_setI_indic//; last exact: measurableC. rewrite -integral_setI_indic// [X in _ + X = _](_ : _ = 0) ?adde0//. rewrite (eq_integral (abse \o f)); last first. @@ -3592,12 +3576,12 @@ Proof. move=> mD mf mg f0 g0 [N [mN N0 subN]]. rewrite integralEindic// [RHS]integralEindic//. rewrite (negligible_integral mN)//; last 2 first. - - apply: emeasurable_funM => //; apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT) => //; rewrite (_ : \1_D = mindic R mD). + - apply: emeasurable_funM => //. + exact/EFin_measurable_fun/measurable_fun_indic. - by move=> x Dx; apply: mule_ge0 => //; [exact: f0|rewrite lee_fin]. rewrite [RHS](negligible_integral mN)//; last 2 first. - - apply: emeasurable_funM => //; apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT) => //; rewrite (_ : \1_D = mindic R mD). + - apply: emeasurable_funM => //. + exact/EFin_measurable_fun/measurable_fun_indic. - by move=> x Dx; apply: mule_ge0 => //; [exact: g0|rewrite lee_fin]. - apply: eq_integral => x;rewrite in_setD => /andP[_ xN]. apply: contrapT; rewrite indicE; have [|?] := boolP (x \in D). @@ -4045,9 +4029,6 @@ have f_g' n x : D x -> `|f_' n x| <= g' x. have [/=|/= xN] := boolP (x \in N); first by rewrite normr0. apply: contrapT => fg; move: xN; apply/negP; rewrite negbK inE; left; right. by apply: subN2 => /= /(_ n Dx). -have ? : measurable_fun D (\1_(D `\` N) : T -> R). - apply: (@measurable_funS _ _ _ _ setT) => //. - by rewrite (_ : \1_ _ = mindic R (measurableD mD mN)). have mu_ n : measurable_fun D (f_' n). apply/(measurable_restrict (f_ n) (measurableD mD mN) _ _).1 => //. by apply: measurable_funS (mf_ _) => // x []. @@ -4073,7 +4054,7 @@ split. apply/funext => n; apply: ae_eq_integral => //. + apply: measurable_fun_comp => //; apply: emeasurable_funB => //. apply/(measurable_restrict _ (measurableD _ _) _ _).1 => //. - by apply: (@measurable_funS _ _ _ _ D) => // x []. + by apply: (measurable_funS mD) => // x []. + by rewrite /g_; apply: measurable_fun_comp => //; exact: emeasurable_funB. + exists N; split => //; rewrite -(setCK N); apply: subsetC => x /= Nx Dx. by rewrite /f_' /f' /restrict mem_set. @@ -4085,7 +4066,7 @@ split. set Y := (X in _ -> _ --> X); rewrite [X in _ --> X -> _](_ : _ = Y) //. apply: ae_eq_integral => //. apply/(measurable_restrict _ (measurableD _ _) _ _).1 => //. - by apply: (@measurable_funS _ _ _ _ D) => // x []. + by apply: (measurable_funS mD) => // x []. exists N; split => //; rewrite -(setCK N); apply: subsetC => x /= Nx Dx. by rewrite /f' /restrict mem_set. Qed. @@ -4223,8 +4204,7 @@ have CB : C `<=` B. rewrite funeqE => x; rewrite indicE /phi /m2/= /mrestr. have [xX1|xX1] := boolP (x \in X1); first by rewrite mule1 in_xsectionM. by rewrite mule0 notin_xsectionM// set0I measure0. - apply/measurable_funeM/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R mX1). + exact/measurable_funeM/EFin_measurable_fun/measurable_fun_indic. suff monoB : monotone_class setT B by exact: monotone_class_subset. split => //; [exact: CB| |exact: xsection_ndseq_closed]. move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. @@ -4265,8 +4245,7 @@ have CB : C `<=` B. rewrite funeqE => y; rewrite indicE /psi /m1/= /mrestr. have [yX2|yX2] := boolP (y \in X2); first by rewrite mule1 in_ysectionM. by rewrite mule0 notin_ysectionM// set0I measure0. - apply/measurable_funeM/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R mX2). + exact/measurable_funeM/EFin_measurable_fun/measurable_fun_indic. suff monoB : monotone_class setT B by exact: monotone_class_subset. split => //; [exact: CB| |exact: ysection_ndseq_closed]. move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. @@ -4433,7 +4412,7 @@ rewrite (_ : (fun _ => _) = fun x => m2 A2 * (\1_A1 x)%:E); last first. [rewrite in_xsectionM// mule1|rewrite mule0 notin_xsectionM]. rewrite ge0_integralM//. - by rewrite muleC integral_indic// setIT. -- by apply: measurable_fun_comp => //; rewrite (_ : \1_ _ = mindic R mA1). +- by apply: measurable_fun_comp => //; exact/measurable_fun_indic. - by move=> x _; rewrite lee_fin. Qed. @@ -4547,7 +4526,7 @@ have mA1A2 : measurable (A1 `*` A2) by apply: measurableM. transitivity (\int[m2]_y (m1 \o ysection (A1 `*` A2)) y) => //. rewrite (_ : _ \o _ = fun y => m1 A1 * (\1_A2 y)%:E). rewrite ge0_integralM//; last 2 first. - - by apply: measurable_fun_comp => //; rewrite (_ : \1_ _ = mindic R mA2). + - by apply: measurable_fun_comp => //; exact/measurable_fun_indic. - by move=> y _; rewrite lee_fin. by rewrite integral_indic ?setIT ?mul1e. rewrite funeqE => y; rewrite indicE. @@ -4662,16 +4641,13 @@ rewrite funeqE => x; rewrite /F /fubini_F [in LHS]/=. under eq_fun do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun => //; apply: measurable_funrM => //. - apply/measurable_fun_prod1 => //. - (*NB: we shouldn't need the following rewriting*) - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - - by move=> r y _; rewrite EFinM; exact: muleindic_ge0. + exact/measurable_fun_prod1/measurable_fun_indic. + - by move=> r y _; rewrite EFinM nnfun_muleindic_ge0. apply: eq_fbigr => i; rewrite in_fset_set// inE => -[/= t _ <-{i} _]. under eq_fun do rewrite EFinM. rewrite ge0_integralM//; last by rewrite lee_fin. - by rewrite -/((m2 \o xsection _) x) -indic_fubini_tonelli_FE. -- apply/EFin_measurable_fun/measurable_fun_prod1. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f t))). +- exact/EFin_measurable_fun/measurable_fun_prod1/measurable_fun_indic. - by move=> y _; rewrite lee_fin. Qed. @@ -4688,15 +4664,13 @@ rewrite funeqE => y; rewrite /G /fubini_G [in LHS]/=. under eq_fun do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun => //; apply: measurable_funrM => //. - apply/measurable_fun_prod2 => //. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - - by move=> r x _; rewrite EFinM; exact: muleindic_ge0. + exact/measurable_fun_prod2/measurable_fun_indic. + - by move=> r x _; rewrite EFinM nnfun_muleindic_ge0. apply: eq_fbigr => i; rewrite in_fset_set// inE => -[/= t _ <-{i} _]. under eq_fun do rewrite EFinM. rewrite ge0_integralM//; last by rewrite lee_fin. - by rewrite -/((m1 \o ysection _) y) -indic_fubini_tonelli_GE. -- apply/EFin_measurable_fun/measurable_fun_prod2. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f t))). +- exact/EFin_measurable_fun/measurable_fun_prod2/measurable_fun_indic. - by move=> x _; rewrite lee_fin. Qed. @@ -4714,15 +4688,14 @@ Lemma sfun_fubini_tonelli1 : \int[m]_z (f z)%:E = \int[m1]_x F x. Proof. under [LHS]eq_integral do rewrite EFinf; rewrite ge0_integral_sum //; last 2 first. - - move=> r; apply/EFin_measurable_fun/measurable_funrM => //. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f r)). - - by move=> r /= z _; exact: muleindic_ge0. + - move=> r. + exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. + - by move=> r /= z _; exact: nnfun_muleindic_ge0. transitivity (\sum_(k <- fset_set (range f)) \int[m1]_x (k%:E * (fubini_F m2 (EFin \o \1_(f @^-1` [set k])) x))). apply: eq_fbigr => i; rewrite in_fset_set// inE => -[z _ <-{i} _]. rewrite ge0_integralM//; last 3 first. - - apply/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f z)))//. + - exact/EFin_measurable_fun/measurable_fun_indic. - by move=> /= x _; rewrite lee_fin. - by rewrite lee_fin. rewrite indic_fubini_tonelli1// -ge0_integralM//; last by rewrite lee_fin. @@ -4743,15 +4716,14 @@ Lemma sfun_fubini_tonelli2 : \int[m']_z (f z)%:E = \int[m2]_y G y. Proof. under [LHS]eq_integral do rewrite EFinf; rewrite ge0_integral_sum //; last 2 first. - - move=> i; apply/EFin_measurable_fun/measurable_funrM => //. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - - by move=> r /= z _; exact: muleindic_ge0. + - move=> i. + exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. + - by move=> r /= z _; exact: nnfun_muleindic_ge0. transitivity (\sum_(k <- fset_set (range f)) \int[m2]_x (k%:E * (fubini_G m1 (EFin \o \1_(f @^-1` [set k])) x))). apply: eq_fbigr => i; rewrite in_fset_set// inE => -[z _ <-{i} _]. rewrite ge0_integralM//; last 3 first. - - apply/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f z))). + - exact/EFin_measurable_fun/measurable_fun_indic. - by move=> /= x _; rewrite lee_fin. - by rewrite lee_fin. rewrite indic_fubini_tonelli2// -ge0_integralM//; last by rewrite lee_fin. @@ -4773,27 +4745,25 @@ Proof. under eq_integral do rewrite EFinf. under [RHS]eq_integral do rewrite EFinf. rewrite ge0_integral_sum //; last 2 first. - - move=> i; apply/EFin_measurable_fun/measurable_funrM => //. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - - by move=> r z _; exact: muleindic_ge0. + - move=> i. + exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. + - by move=> r z _; exact: nnfun_muleindic_ge0. transitivity (\sum_(k <- fset_set (range f)) k%:E * \int[m']_z ((EFin \o \1_(f @^-1` [set k])) z)). apply: eq_fbigr => i; rewrite in_fset_set// inE => -[t _ <- _]. rewrite ge0_integralM//; last 3 first. - - apply/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f t))). + - exact/EFin_measurable_fun/measurable_fun_indic. - by move=> /= x _; rewrite lee_fin. - by rewrite lee_fin. rewrite indic_fubini_tonelli1// indic_fubini_tonelli//. by rewrite -indic_fubini_tonelli2. apply/esym; rewrite ge0_integral_sum //; last 2 first. - - move=> i; apply/EFin_measurable_fun/measurable_funrM => //. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - - by move=> r z _; exact: muleindic_ge0. + - move=> i. + exact/EFin_measurable_fun/measurable_funrM/measurable_fun_indic. + - by move=> r z _; exact: nnfun_muleindic_ge0. apply: eq_fbigr => i; rewrite in_fset_set// inE => -[x _ <- _]. rewrite ge0_integralM//; last by rewrite lee_fin. -- apply/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f x))). +- exact/EFin_measurable_fun/measurable_fun_indic. - by move=> /= y _; rewrite lee_fin. Qed. diff --git a/theories/measure.v b/theories/measure.v index d96c859b97..5803e37a6e 100644 --- a/theories/measure.v +++ b/theories/measure.v @@ -959,7 +959,7 @@ split. Qed. Lemma measurable_funS (E D : set T1) (f : T1 -> T2) : - measurable E -> D `<=` E -> measurable_fun E f -> + measurable E -> D `<=` E -> measurable_fun E f -> measurable_fun D f. Proof. move=> mE DE mf mD; have mC : measurable (E `\` D) by exact: measurableD. @@ -969,6 +969,10 @@ suff -> : D `|` (E `\` D) = E by move=> [[]] //. by rewrite setDUK. Qed. +Lemma measurable_funTS (D : set T1) (f : T1 -> T2) : + measurable_fun setT f -> measurable_fun D f. +Proof. exact: measurable_funS. Qed. + Lemma measurable_fun_ext (D : set T1) (f g : T1 -> T2) : {in D, f =1 g} -> measurable_fun D f -> measurable_fun D g. Proof. From 79b829f45788b3b0f2609e4ff84ef70068dc0983 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Tue, 26 Jul 2022 18:54:30 +0900 Subject: [PATCH 05/42] use set instead of fset in esum --- CHANGELOG_UNRELEASED.md | 45 ++++++++ theories/esum.v | 199 +++++++++++++++++++----------------- theories/lebesgue_measure.v | 2 +- theories/measure.v | 9 +- 4 files changed, 154 insertions(+), 101 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index b92599fa25..5c8a7e1ecd 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -38,6 +38,43 @@ - in `measure.v`: + generalize `measurable_uncurry` + + generalize `pushforward` +- in `lebesgue_integral.v` + + change `Arguments` of `eq_integrable` +- in `lebesgue_integral.v`: + + fix pretty-printing of `{mfun _ >-> _}`, `{sfun _ >-> _}`, `{nnfun _ >-> _}` +- in `lebesgue_integral.v` + + minor generalization of `eq_measure_integral` +- from `topology.v` to `mathcomp_extra.v`: + + generalize `ltr_bigminr` to `porderType` and rename to `bigmin_lt` + + generalize `bigminr_ler` to `orderType` and rename to `bigmin_le` +- moved out of module `Bigminr` in `normedtype.v` to `mathcomp_extra.v` and generalized to `orderType`: + + lemma `bigminr_ler_cond`, renamed to `bigmin_le_cond` +- moved out of module `Bigminr` in `normedtype.v` to `mathcomp_extra.v`: + + lemma `bigminr_maxr` +- moved from from module `Bigminr` in `normedtype.v` + + to `mathcomp_extra.v` and generalized to `orderType` + * `bigminr_mkcond` -> `bigmin_mkcond` + * `bigminr_split` -> `bigmin_split` + * `bigminr_idl` -> `bigmin_idl` + * `bigminrID` -> `bigminID` + * `bigminrD1` -> `bigminD1` + * `bigminr_inf` -> `bigmin_inf` + * `bigminr_gerP` -> `bigmin_geP` + * `bigminr_gtrP` -> ``bigmin_gtP`` + * `bigminr_eq_arg` -> `bigmin_eq_arg` + * `eq_bigminr` -> `eq_bigmin` + + to `topology.v` and generalized to `orderType` + * `bigminr_lerP` -> `bigmin_leP` + * `bigminr_ltrP` -> `bigmin_ltP` +- moved from `topology.v` to `mathcomp_extra.v`: + + `bigmax_lerP` -> `bigmax_leP` + + `bigmax_ltrP` -> `bigmax_ltP` + + `ler_bigmax_cond` -> `le_bigmax_cond` + + `ler_bigmax` -> `le_bigmax` + + `le_bigmax` -> `homo_le_bigmax` +- in `esum.v`: + + definition `esum` ### Renamed @@ -63,6 +100,14 @@ ### Removed +- in `normedtype.v` (module `Bigminr`) + + `bigminr_ler_cond`, `bigminr_ler`. + + `bigminr_seq1`, `bigminr_pred1_eq`, `bigminr_pred1` +- in `topology.v`: + + `bigmax_seq1`, `bigmax_pred1_eq`, `bigmax_pred1` +- in `esum.v`: + + lemma `fsetsP`, `sum_fset_set` + ### Infrastructure ### Misc diff --git a/theories/esum.v b/theories/esum.v index 3e01f55ff4..b137796212 100644 --- a/theories/esum.v +++ b/theories/esum.v @@ -35,20 +35,17 @@ Section set_of_fset_in_a_set. Variable (T : choiceType). Implicit Type S : set T. -Definition fsets S : set {fset T} := [set F : {fset T} | [set` F] `<=` S]. +Definition fsets S : set (set T) := [set F | finite_set F /\ F `<=` S]. -Lemma fsets_set0 S : fsets S fset0. Proof. by []. Qed. +Lemma fsets_set0 S : fsets S set0. Proof. by split. Qed. -Lemma fsets_self (F : {fset T}) : fsets [set x | x \in F] F. -Proof. by []. Qed. +Lemma fsets_self (F : set T) : finite_set F -> fsets F F. +Proof. by move=> finF; split. Qed. -Lemma fsetsP S (F : {fset T}) : [set` F] `<=` S <-> fsets S F. -Proof. by []. Qed. - -Lemma fsets0 : fsets set0 = [set fset0]. +Lemma fsets0 : fsets set0 = [set set0]. Proof. rewrite predeqE => A; split => [|->]; last exact: fsets_set0. -by rewrite /fsets /= subset0 => /eqP; rewrite set_fset_eq0 => /eqP. +by rewrite /fsets/= subset0 => -[]. Qed. End set_of_fset_in_a_set. @@ -57,15 +54,15 @@ Section esum. Variables (R : realFieldType) (T : choiceType). Implicit Types (S : set T) (a : T -> \bar R). -Definition esum S a := ereal_sup [set \sum_(x <- A) a x | A in fsets S]. +Definition esum S a := ereal_sup [set \sum_(x <- fset_set A) a x | A in fsets S]. Local Notation "\esum_ ( i 'in' P ) A" := (esum P (fun i => A)). Lemma esum_set0 a : \esum_(i in set0) a i = 0. Proof. rewrite /esum fsets0 [X in ereal_sup X](_ : _ = [set 0%E]) ?ereal_sup1//. -rewrite predeqE => x; split; first by move=> [_ /= ->]; rewrite big_seq_fset0. -by move=> -> /=; exists fset0 => //; rewrite big_seq_fset0. +apply/seteqP; split=> [x [_ /= ->]|x]; first by rewrite fset_set0 big_seq_fset0. +by move=> -> /=; exists set0 => //; rewrite fset_set0 big_seq_fset0. Qed. End esum. @@ -76,42 +73,37 @@ Section esum_realType. Variables (R : realType) (T : choiceType). Implicit Types (a : T -> \bar R). -Lemma esum_ge0 (S : set T) a : (forall x, S x -> 0 <= a x) -> 0 <= \esum_(i in S) a i. +Lemma esum_ge0 (S : set T) a : + (forall x, S x -> 0 <= a x) -> 0 <= \esum_(i in S) a i. Proof. -move=> a0. -by apply: ereal_sup_ub; exists fset0; [exact: fsets_set0|rewrite big_nil]. +move=> a0; apply: ereal_sup_ub. +by exists set0; [exact: fsets_set0|rewrite fset_set0 big_nil]. Qed. -Lemma esum_fset (F : {fset T}) a : (forall i, i \in F -> 0 <= a i) -> - \esum_(i in [set` F]) a i = \sum_(i <- F) a i. +Lemma esum_fset (F : set T) a : finite_set F -> + (forall i, i \in F -> 0 <= a i) -> + \esum_(i in F) a i = \sum_(i <- fset_set F) a i. Proof. -move=> f0; apply/eqP; rewrite eq_le; apply/andP; split; last first. +move=> finF f0; apply/eqP; rewrite eq_le; apply/andP; split; last first. by apply ereal_sup_ub; exists F => //; exact: fsets_self. -apply ub_ereal_sup => /= ? -[F' F'F <-]; apply/lee_sum_nneg_subfset. - exact/fsetsP. -by move=> t; rewrite inE => /andP[_ /f0]. +apply ub_ereal_sup => /= ? -[F' [finF' F'F] <-]; apply/lee_sum_nneg_subfset. + by apply/fsubsetP; rewrite -fset_set_sub. +by move=> t; rewrite inE/= !in_fset_set// => /andP[_] /f0. Qed. Lemma esum_set1 t a : 0 <= a t -> \esum_(i in [set t]) a i = a t. Proof. -by move=> ?; rewrite -set_fset1 esum_fset ?big_seq_fset1// => t' /[!inE] /eqP->. -Qed. - -Lemma sum_fset_set (A : set T) a : finite_set A -> - (forall i, A i -> 0 <= a i) -> - \sum_(i <- fset_set A) a i = \esum_(i in A) a i. -Proof. -move=> Afin a0; rewrite -esum_fset => [|i]; rewrite ?fset_setK//. -by rewrite in_fset_set ?inE//; apply: a0. +by move=> ?; rewrite esum_fset// ?fset_set1// ?big_seq_fset1// => t' /[!inE] ->. Qed. Lemma fsbig_esum (A : set T) a : finite_set A -> (forall x, 0 <= a x) -> \sum_(x \in A) (a x) = \esum_(x in A) a x. -Proof. by move=> *; rewrite fsbig_finite//= sum_fset_set. Qed. +Proof. by move=> *; rewrite fsbig_finite//= -esum_fset. Qed. + End esum_realType. Lemma esum_ge [R : realType] [T : choiceType] (I : set T) (a : T -> \bar R) x : - (exists2 X : {fset T}, fsets I X & x <= \sum_(i <- X) a i) -> + (exists2 X : set T, fsets I X & x <= \sum_(i <- fset_set X) a i) -> x <= \esum_(i in I) a i. Proof. by move=> [X IX /le_trans->//]; apply: ereal_sup_ub => /=; exists X. Qed. @@ -119,18 +111,18 @@ Lemma esum0 [R : realFieldType] [I : choiceType] (D : set I) (a : I -> \bar R) : (forall i, D i -> a i = 0) -> \esum_(i in D) a i = 0. Proof. move=> a0; rewrite /esum (_ : [set _ | _ in _] = [set 0]) ?ereal_sup1//. -apply/seteqP; split=> x //= => [[X XI] <-|->]. - by rewrite big_seq_cond big1// => i /andP[Xi _]; rewrite a0//; apply: XI. -by exists fset0; rewrite ?big_seq_fset0. +apply/seteqP; split=> x //= => [[X [finX XI]] <-|->]. + by rewrite big_seq big1// => i; rewrite in_fset_set// inE=> /XI/a0. +by exists set0; rewrite ?fset_set0 ?big_seq_fset0//; exact: fsets_set0. Qed. Lemma le_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : (forall i, I i -> a i <= b i) -> \esum_(i in I) a i <= \esum_(i in I) b i. Proof. -move=> le_ab; rewrite ub_ereal_sup => //= _ [X XI] <-; rewrite esum_ge//. -exists X => //; rewrite big_seq_cond [x in _ <= x]big_seq_cond lee_sum => // i. -by rewrite andbT => /XI /le_ab. +move=> le_ab; rewrite ub_ereal_sup => //= _ [X [finX XI]] <-; rewrite esum_ge//. +exists X => //; rewrite big_seq [x in _ <= x]big_seq lee_sum => // i. +by rewrite in_fset_set// inE => /XI /le_ab. Qed. Lemma eq_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : @@ -150,39 +142,42 @@ wlog : a b ag0 bg0 / \esum_(i in I) a i \isn't a fin_num => [saoo|]; last first. rewrite (@le_trans _ _ +oo)//; first by rewrite /adde/=; case: esum. rewrite leye_eq; apply/eqP/eq_infty => y; rewrite esum_ge//. have : y%:E < \esum_(i in I) a i by rewrite aoo// ltey. - move=> /ereal_sup_gt[_ [X XI] <-] /ltW yle; exists X => //=. - rewrite (le_trans yle)// big_split lee_addl// big_seq_cond sume_ge0 => // i. - by rewrite andbT => /XI; apply: bg0. + move=> /ereal_sup_gt[_ [X [finX XI]] <-] /ltW yle; exists X => //=. + rewrite (le_trans yle)// big_split lee_addl// big_seq sume_ge0 => // i. + by rewrite in_fset_set// inE => /XI; exact: bg0. case: (boolP (\esum_(i in I) a i \is a fin_num)) => sa; last exact: saoo. case: (boolP (\esum_(i in I) b i \is a fin_num)) => sb; last first. by rewrite addeC (eq_esum (fun _ _ => addeC _ _)) saoo. -rewrite -lee_subr_addr// ub_ereal_sup//= => _ [X XI] <-. -have saX : \sum_(i <- X) a i \is a fin_num. +rewrite -lee_subr_addr// ub_ereal_sup//= => _ [X [finX XI]] <-. +have saX : \sum_(i <- fset_set X) a i \is a fin_num. apply: contraTT sa => /fin_numPn[] sa. - suff : \sum_(i <- X) a i >= 0 by rewrite sa. - by rewrite big_seq_cond sume_ge0 => // i; rewrite ?andbT => /XI/ag0. + suff : \sum_(i <- fset_set X) a i >= 0 by rewrite sa. + by rewrite big_seq sume_ge0// => t; rewrite in_fset_set// inE => /XI/ag0. apply/fin_numPn; right; apply/eqP; rewrite -leye_eq esum_ge//. by exists X; rewrite // sa. -rewrite lee_subr_addr// addeC -lee_subr_addr// ub_ereal_sup//= => _ [Y YI] <-. -rewrite lee_subr_addr// addeC esum_ge//; exists (X `|` Y)%fset. - by move=> i/=; rewrite inE => /orP[/XI|/YI]. -rewrite big_split/= lee_add//=. - rewrite lee_sum_nneg_subfset//=; first exact/fsubsetP/fsubsetUl. - by move=> x; rewrite !inE/= => /andP[/negPf->]/= => /YI/ag0. -rewrite lee_sum_nneg_subfset//=; first exact/fsubsetP/fsubsetUr. -by move=> x; rewrite !inE/= => /andP[/negPf->]/orP[]// => /XI/bg0. -Qed. - -Lemma esum_mkcond [R : realType] [T : choiceType] (I : set T) (a : T -> \bar R) : +rewrite lee_subr_addr// addeC -lee_subr_addr// ub_ereal_sup//= => _ [Y [finY YI]] <-. +rewrite lee_subr_addr// addeC esum_ge//; exists (X `|` Y). + by split; [rewrite finite_setU|rewrite subUset]. +rewrite big_split/= lee_add//= lee_sum_nneg_subfset//=. +- by apply/fsubsetP; rewrite -fset_set_sub// finite_setU. +- move=> x; rewrite !inE fset_setU// in_fsetU !in_fset_set// andb_orr andNb/=. + by move=> /andP[_] /[!inE] /YI/ag0. +- by apply/fsubsetP; rewrite -fset_set_sub// finite_setU. +- move=> x; rewrite !inE fset_setU// in_fsetU !in_fset_set// andb_orr andNb/=. + by rewrite orbF => /andP[_] /[!inE] /XI/bg0. +Qed. + +Lemma esum_mkcond [R : realType] [T : choiceType] (I : set T) + (a : T -> \bar R) : \esum_(i in I) a i = \esum_(i in [set: T]) if i \in I then a i else 0. Proof. -apply/eqP; rewrite eq_le !ub_ereal_sup//= => _ [X XI] <-; rewrite -?big_mkcond//=. - rewrite big_fset_condE/=; set Y := [fset _ | _ in X & _]%fset. - rewrite ereal_sup_ub//; exists Y => //= i /=. - by rewrite 2!inE/= => /andP[_]; rewrite inE. +apply/eqP; rewrite eq_le !ub_ereal_sup//= => _ [X [finX XI]] <-. + rewrite -big_mkcond/= big_fset_condE/=; set Y := [fset _ | _ in _ & _]%fset. + rewrite ereal_sup_ub//=; exists [set` Y]; last by rewrite set_fsetK. + by split => // i/=; rewrite !inE/= => /andP[_]; rewrite inE. rewrite ereal_sup_ub//; exists X => //; rewrite -big_mkcond/=. -rewrite big_seq_cond [RHS]big_seq_cond; apply: eq_bigl => i. -by case: (boolP (i \in X)) => //= /XI Ii; apply/mem_set. +rewrite big_seq_cond [RHS]big_seq; apply: eq_bigl => i. +by rewrite in_fset_set// andb_idr// 2!inE => /XI. Qed. Lemma esum_mkcondr [R : realType] [T : choiceType] (I J : set T) (a : T -> \bar R) : @@ -231,39 +226,46 @@ Lemma esum_esum [R : realType] [T1 T2 : choiceType] \esum_(i in I) \esum_(j in J i) a i j = \esum_(k in I `*`` J) a k.1 k.2. Proof. move=> a_ge0; apply/eqP; rewrite eq_le; apply/andP; split. - apply: ub_ereal_sup => /= _ [X IX] <-. + apply: ub_ereal_sup => /= _ [X [finX IX]] <-. under eq_bigr do rewrite esum_mkcond. rewrite -esum_sum; last by move=> i j _ _; case: ifP. under eq_esum do rewrite -big_mkcond/=. - apply: ub_ereal_sup => /= _ [Y _ <-]; apply: ereal_sup_ub => /=. - exists [fset z | z in X `*` Y & z.2 \in J z.1]%fset => //=. - move=> z/=; rewrite !inE/= -andbA => /and3P[Xz1 Yz2 zJ]. - by split; [exact: IX | rewrite inE in zJ]. + apply: ub_ereal_sup => /= _ [Y [finY _] <-]; apply: ereal_sup_ub => /=. + have ? : finite_set [set z | z \in X `*` Y /\ z.2 \in J z.1]. + apply: sub_finite_set (finite_setM finX finY) => z/=. + by rewrite in_setM => -[/andP[] /[!inE]]. + exists [set z | z \in X `*` Y /\ z.2 \in J z.1] => /=. + by split => //= z/=; rewrite !inE/= => -[[/IX]]. rewrite (exchange_big_dep xpredT)//= pair_big_dep_cond/=. - apply: eq_fbigl => -[/= k1 k2]; rewrite !inE -andbA. - apply/idP/imfset2P => /= [/and3P[kX kY kJ]|]. - exists k1; rewrite ?(andbT, inE)//=. - by exists k2; rewrite ?(andbT, inE)//= kY kJ. - by move=> [{}k1 + [{}k2 + [-> ->]]]; rewrite !inE andbT => -> /andP[-> ->]. -apply: ub_ereal_sup => _ /= [X/= XIJ] <-; apply: esum_ge. -pose X1 := [fset x.1 | x in X]%fset. -pose X2 := [fset x.2 | x in X]%fset. -exists X1; first by move=> x/= /imfsetP[z /= zX ->]; have [] := XIJ z. -apply: (@le_trans _ _ (\sum_(i <- X1) \sum_(j <- X2 | j \in J i) a i j)). + apply: eq_fbigl => -[/= k1 k2]; rewrite in_fset_set//; apply/idP/imfset2P. + rewrite !inE/= !inE/= -andA => -[kX [kY kJ]]. + exists k1; first by rewrite !inE/= andbT/= in_fset_set// inE. + by exists k2 => //; rewrite !inE in_fset_set//; apply/andP; split=> /[!inE]. + move=> [t1]; rewrite !inE andbT/= !inE/= in_fset_set// inE => Xt1. + by move=> [t2]; rewrite !inE in_fset_set// =>/andP[/[!inE] Yt1 Jt1t2] [-> ->]. +apply: ub_ereal_sup => _ /= [X/= [finX XIJ]] <-; apply: esum_ge. +exists X.`1; first by split=> [|x [y /XIJ[]//]]; exact: finite_set_fst. +apply: (@le_trans _ _ + (\sum_(i <- fset_set X.`1) \sum_(j <- fset_set X.`2 | j \in J i) a i j)). rewrite pair_big_dep_cond//=; set Y := Imfset.imfset2 _ _ _ _. rewrite [leRHS](big_fsetID _ (mem X))/=. - rewrite (_ : [fset x | x in Y & x \in X] = Y `&` X)%fset; last first. - by apply/fsetP => x; rewrite 2!inE. + rewrite (_ : [fset x | x in Y & x \in X] = Y `&` fset_set X)%fset; last first. + by apply/fsetP => x; rewrite 2!inE/= in_fset_set. rewrite (fsetIidPr _); first by rewrite lee_addl// sume_ge0. - apply/fsubsetP => -[i j] Xij; apply/imfset2P. - exists i => //=; rewrite ?inE ?andbT//=. - by apply/imfsetP; exists (i, j). + apply/fsubsetP => -[i j]; rewrite in_fset_set// inE => Xij; apply/imfset2P. + exists i => /=. + rewrite !inE/= in_fset_set//; last exact: finite_set_fst. + by rewrite andbT inE; exists j. exists j => //; rewrite !inE/=; have /XIJ[/= _ Jij] := Xij. - by apply/andP; split; rewrite ?inE//; apply/imfsetP; exists (i, j). + rewrite in_fset_set; last exact: finite_set_snd. + by apply/andP; split; rewrite ?inE//; exists i. rewrite big_mkcond [leRHS]big_mkcond. apply: lee_sum => i Xi; rewrite ereal_sup_ub => //=. -exists [fset j in X2 | j \in J i]%fset; last by rewrite -big_fset_condE. -by move=> j/=; rewrite !inE => /andP[_]; rewrite inE. +have ? : finite_set (X.`2 `&` J i). + by apply: finite_setI; left; apply: finite_set_snd. +exists (X.`2 `&` J i) => //. +rewrite [in RHS]big_fset_condE/=; apply eq_fbigl => j. +by rewrite in_fset_set// !inE/= in_setI in_fset_set//; exact: finite_set_snd. Qed. Lemma lee_sum_fset_nat (R : realDomainType) @@ -301,12 +303,14 @@ Lemma nneseries_esum (R : realType) (a : nat -> \bar R) (P : pred nat) : Proof. move=> a0; apply/eqP; rewrite eq_le; apply/andP; split. apply: (ereal_lim_le (is_cvg_nneseries_cond a0)); apply: nearW => n. - apply: ereal_sup_ub => /=; exists [fset val i | i in 'I_n & P i]%fset. + apply: ereal_sup_ub => /=; exists [set` [fset val i | i in 'I_n & P i]%fset]. + split; first exact: finite_fset. by move=> /= k /imfsetP[/= i]; rewrite inE => + ->. - rewrite big_imfset/=; last by move=> ? ? ? ? /val_inj. + rewrite set_fsetK big_imfset/=; last by move=> ? ? ? ? /val_inj. by rewrite big_filter big_enum_cond/= big_mkord. -apply: ub_ereal_sup => _ [/= F /fsetsP PF <-]. -rewrite -(big_rmcond_in P)/=; last by move=> i /PF ->. +apply: ub_ereal_sup => _ [/= F [finF PF] <-]. +rewrite -(big_rmcond_in P)/=; last first. + by move=> k; rewrite in_fset_set// inE => /PF ->. by apply: lee_sum_fset_lim. Qed. @@ -325,16 +329,19 @@ gen have le_esum : T T' a P Q e / apply/eqP; rewrite eq_le le_esum//=. rewrite [leRHS](_ : _ = \esum_(j in Q) a (e (e^-1%FUN j))); last first. by apply: eq_esum => i Qi; rewrite invK ?inE. - by rewrite le_esum => //= i Qi; rewrite a_ge0//; apply: funS. -rewrite ub_ereal_sup => //= _ [X XQ <-]; rewrite ereal_sup_ub => //=. -exists (e^-1 @` X)%fset; first by move=> _ /imfsetP[t' /= /XQ Qt' ->]; apply: funS. -rewrite big_imfset => //=; last first. - by move=> x y /XQ Qx /XQ Qy /(congr1 e); rewrite !invK ?inE. -by apply: eq_big_seq => i /XQ Qi; rewrite invK ?inE. + by rewrite le_esum => //= i Qi; rewrite a_ge0//; exact: funS. +rewrite ub_ereal_sup => //= _ [X [finX XQ] <-]; rewrite ereal_sup_ub => //=. +exists [set` (e^-1 @` (fset_set X))%fset]. + split=> [|t /= /imfsetP[t'/=]]; first exact: finite_fset. + by rewrite in_fset_set// inE => /XQ Qt' ->; exact: funS. +rewrite set_fsetK big_imfset => //=; last first. + move=> x y/=; rewrite !in_fset_set// !inE => /XQ Qx /XQ Qy /(congr1 e). + by rewrite !invK ?inE. +apply: eq_big_seq => i; rewrite in_fset_set// inE => /XQ Qi. +by rewrite invK ?inE. Qed. Arguments reindex_esum {R T T'} P Q e a. - Section nneseries_interchange. Local Open Scope ereal_scope. diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index c41e3cc836..5de4b68350 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -370,7 +370,7 @@ move=> /(_ _ _ _)/Box[]//=; apply: le_le_trans. rewrite hlength_itv ?lte_fin -?EFinD/= -addrA -opprD. by case: ltP => //; rewrite lee_fin subr_le0. rewrite nneseries_esum//; last by move=> *; rewrite adde_ge0//= ?lee_fin. -rewrite esum_ge//; exists X => //; rewrite fsbig_finite// ?set_fsetK//=. +rewrite esum_ge//; exists [set` X] => //; rewrite fsbig_finite// ?set_fsetK//=. rewrite lee_sum // => i _; rewrite ?AE// !hlength_itv/= ?lte_fin -?EFinD/=. do !case: ifPn => //= ?; do ?by rewrite ?adde_ge0 ?lee_fin// ?subr_ge0// ?ltW. by rewrite addrAC. diff --git a/theories/measure.v b/theories/measure.v index 5803e37a6e..a6aac32dbc 100644 --- a/theories/measure.v +++ b/theories/measure.v @@ -1536,7 +1536,7 @@ Lemma finite_card_dirac (A : set T) : finite_set A -> \esum_(i in A) \d_ i A = (#|` fset_set A|%:R)%:E :> \bar R. Proof. move=> finA. -rewrite -sum_fset_set// big_seq_cond (eq_bigr (fun=> 1)) -?big_seq_cond. +rewrite esum_fset// big_seq_cond (eq_bigr (fun=> 1)) -?big_seq_cond. by rewrite card_fset_sum1// natr_sum -sumEFin. by move=> i; rewrite andbT in_fset_set//= /dirac indicE => ->. Qed. @@ -1546,12 +1546,13 @@ Lemma infinite_card_dirac (A : set T) : infinite_set A -> Proof. move=> infA; apply/eq_pinftyP => r r0. have [B BA Br] := infinite_set_fset `|ceil r| infA. -apply: esum_ge; exists B => //; apply: (@le_trans _ _ `|ceil r|%:R%:E). +apply: esum_ge; exists [set` B] => //; apply: (@le_trans _ _ `|ceil r|%:R%:E). by rewrite lee_fin natr_absz gtr0_norm ?ceil_gt0// ceil_ge. move: Br; rewrite -(@ler_nat R) -lee_fin => /le_trans; apply. rewrite big_seq (eq_bigr (cst 1))/=; last first. - by move=> i Bi; rewrite /dirac indicE mem_set//; exact: BA. -by rewrite -big_seq card_fset_sum1 sumEFin natr_sum. + move=> i; rewrite in_fset_set// inE/= => Bi; rewrite /dirac indicE mem_set//. + exact: BA. +by rewrite -big_seq card_fset_sum1 sumEFin natr_sum// set_fsetK. Qed. End dirac_lemmas. From f50cfec041dc93a4390494d23bc6a8a4751627fa Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 27 Jul 2022 00:08:02 +0900 Subject: [PATCH 06/42] favor fsbig --- CHANGELOG_UNRELEASED.md | 16 ++++++++ theories/esum.v | 77 +++++++++++++++++++------------------ theories/fsbigop.v | 10 +++++ theories/lebesgue_measure.v | 1 + theories/measure.v | 4 +- 5 files changed, 68 insertions(+), 40 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 5c8a7e1ecd..884ed4e430 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -33,6 +33,22 @@ + lemma `measurable_funTS` - in `lebesgue_measure.v`: + lemma `measurable_fun_indic` + + lemma `big_const_idem` + + lemma `big_id_idem` + + lemma `big_rem_AC` + + lemma `bigD1_AC` + + lemma `big_mkcond_idem` + + lemma `big_split_idem` + + lemma `big_id_idem_AC` + + lemma `bigID_idem` +- in `mathcomp_extra.v`: + + lemmas `bigmax_le` and `bigmax_lt` + + lemma `bigmin_idr` + + lemma `bigmax_idr` +- in `classical_sets.v`: + + lemma `subset_refl` +- in `fsbigop.v`: + + lemma `lee_fsum_nneg_subset` ### Changed diff --git a/theories/esum.v b/theories/esum.v index b137796212..f90a049d79 100644 --- a/theories/esum.v +++ b/theories/esum.v @@ -54,15 +54,15 @@ Section esum. Variables (R : realFieldType) (T : choiceType). Implicit Types (S : set T) (a : T -> \bar R). -Definition esum S a := ereal_sup [set \sum_(x <- fset_set A) a x | A in fsets S]. +Definition esum S a := ereal_sup [set \sum_(x \in A) a x | A in fsets S]. Local Notation "\esum_ ( i 'in' P ) A" := (esum P (fun i => A)). Lemma esum_set0 a : \esum_(i in set0) a i = 0. Proof. rewrite /esum fsets0 [X in ereal_sup X](_ : _ = [set 0%E]) ?ereal_sup1//. -apply/seteqP; split=> [x [_ /= ->]|x]; first by rewrite fset_set0 big_seq_fset0. -by move=> -> /=; exists set0 => //; rewrite fset_set0 big_seq_fset0. +apply/seteqP; split=> [x [_ /= ->]|x]; first by rewrite fsbig_set0. +by move=> -> /=; exists set0 => //; rewrite fsbig_set0. Qed. End esum. @@ -77,33 +77,33 @@ Lemma esum_ge0 (S : set T) a : (forall x, S x -> 0 <= a x) -> 0 <= \esum_(i in S) a i. Proof. move=> a0; apply: ereal_sup_ub. -by exists set0; [exact: fsets_set0|rewrite fset_set0 big_nil]. +by exists set0; [exact: fsets_set0|rewrite fsbig_set0]. Qed. Lemma esum_fset (F : set T) a : finite_set F -> (forall i, i \in F -> 0 <= a i) -> - \esum_(i in F) a i = \sum_(i <- fset_set F) a i. + \esum_(i in F) a i = \sum_(i \in F) a i. Proof. move=> finF f0; apply/eqP; rewrite eq_le; apply/andP; split; last first. by apply ereal_sup_ub; exists F => //; exact: fsets_self. -apply ub_ereal_sup => /= ? -[F' [finF' F'F] <-]; apply/lee_sum_nneg_subfset. - by apply/fsubsetP; rewrite -fset_set_sub. -by move=> t; rewrite inE/= !in_fset_set// => /andP[_] /f0. +apply ub_ereal_sup => /= ? -[F' [finF' F'F] <-]. +apply/lee_fsum_nneg_subset => //; first exact/subsetP. +by move=> t; rewrite inE/= => /andP[_] /f0. Qed. Lemma esum_set1 t a : 0 <= a t -> \esum_(i in [set t]) a i = a t. Proof. -by move=> ?; rewrite esum_fset// ?fset_set1// ?big_seq_fset1// => t' /[!inE] ->. +by move=> ?; rewrite esum_fset// ?fset_set1// ?fsbig_set1// => t' /[!inE] ->. Qed. Lemma fsbig_esum (A : set T) a : finite_set A -> (forall x, 0 <= a x) -> \sum_(x \in A) (a x) = \esum_(x in A) a x. -Proof. by move=> *; rewrite fsbig_finite//= -esum_fset. Qed. +Proof. by move=> *; rewrite esum_fset. Qed. End esum_realType. Lemma esum_ge [R : realType] [T : choiceType] (I : set T) (a : T -> \bar R) x : - (exists2 X : set T, fsets I X & x <= \sum_(i <- fset_set X) a i) -> + (exists2 X : set T, fsets I X & x <= \sum_(i \in X) a i) -> x <= \esum_(i in I) a i. Proof. by move=> [X IX /le_trans->//]; apply: ereal_sup_ub => /=; exists X. Qed. @@ -112,8 +112,8 @@ Lemma esum0 [R : realFieldType] [I : choiceType] (D : set I) (a : I -> \bar R) : Proof. move=> a0; rewrite /esum (_ : [set _ | _ in _] = [set 0]) ?ereal_sup1//. apply/seteqP; split=> x //= => [[X [finX XI]] <-|->]. - by rewrite big_seq big1// => i; rewrite in_fset_set// inE=> /XI/a0. -by exists set0; rewrite ?fset_set0 ?big_seq_fset0//; exact: fsets_set0. + by rewrite fsbig1// => i /XI/a0. +by exists set0; rewrite ?fsbig_set0//; exact: fsets_set0. Qed. Lemma le_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : @@ -121,8 +121,8 @@ Lemma le_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : \esum_(i in I) a i <= \esum_(i in I) b i. Proof. move=> le_ab; rewrite ub_ereal_sup => //= _ [X [finX XI]] <-; rewrite esum_ge//. -exists X => //; rewrite big_seq [x in _ <= x]big_seq lee_sum => // i. -by rewrite in_fset_set// inE => /XI /le_ab. +exists X => //; rewrite !fsbig_finite// big_seq [x in _ <= x]big_seq lee_sum //. +by move=> i; rewrite in_fset_set// inE => /XI/le_ab. Qed. Lemma eq_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : @@ -135,7 +135,7 @@ Lemma esumD [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : \esum_(i in I) (a i + b i) = \esum_(i in I) a i + \esum_(i in I) b i. Proof. move=> ag0 bg0; apply/eqP; rewrite eq_le; apply/andP; split. - rewrite ub_ereal_sup//= => x [X XI] <-; rewrite big_split/=. + rewrite ub_ereal_sup//= => x [X [finX XI]] <-; rewrite fsbig_split//=. by rewrite lee_add// ereal_sup_ub//=; exists X. wlog : a b ag0 bg0 / \esum_(i in I) a i \isn't a fin_num => [saoo|]; last first. move=> /fin_numPn[->|/[dup] aoo ->]; first by rewrite leNye. @@ -143,28 +143,26 @@ wlog : a b ag0 bg0 / \esum_(i in I) a i \isn't a fin_num => [saoo|]; last first. rewrite leye_eq; apply/eqP/eq_infty => y; rewrite esum_ge//. have : y%:E < \esum_(i in I) a i by rewrite aoo// ltey. move=> /ereal_sup_gt[_ [X [finX XI]] <-] /ltW yle; exists X => //=. - rewrite (le_trans yle)// big_split lee_addl// big_seq sume_ge0 => // i. - by rewrite in_fset_set// inE => /XI; exact: bg0. + rewrite (le_trans yle)// fsbig_split// lee_addl// fsume_ge0// => // i. + by move=> /XI; exact: bg0. case: (boolP (\esum_(i in I) a i \is a fin_num)) => sa; last exact: saoo. case: (boolP (\esum_(i in I) b i \is a fin_num)) => sb; last first. by rewrite addeC (eq_esum (fun _ _ => addeC _ _)) saoo. rewrite -lee_subr_addr// ub_ereal_sup//= => _ [X [finX XI]] <-. -have saX : \sum_(i <- fset_set X) a i \is a fin_num. +have saX : \sum_(i \in X) a i \is a fin_num. apply: contraTT sa => /fin_numPn[] sa. - suff : \sum_(i <- fset_set X) a i >= 0 by rewrite sa. - by rewrite big_seq sume_ge0// => t; rewrite in_fset_set// inE => /XI/ag0. + suff : \sum_(i \in X) a i >= 0 by rewrite sa. + by rewrite fsume_ge0// => i /XI/ag0. apply/fin_numPn; right; apply/eqP; rewrite -leye_eq esum_ge//. by exists X; rewrite // sa. rewrite lee_subr_addr// addeC -lee_subr_addr// ub_ereal_sup//= => _ [Y [finY YI]] <-. rewrite lee_subr_addr// addeC esum_ge//; exists (X `|` Y). by split; [rewrite finite_setU|rewrite subUset]. -rewrite big_split/= lee_add//= lee_sum_nneg_subfset//=. -- by apply/fsubsetP; rewrite -fset_set_sub// finite_setU. -- move=> x; rewrite !inE fset_setU// in_fsetU !in_fset_set// andb_orr andNb/=. - by move=> /andP[_] /[!inE] /YI/ag0. -- by apply/fsubsetP; rewrite -fset_set_sub// finite_setU. -- move=> x; rewrite !inE fset_setU// in_fsetU !in_fset_set// andb_orr andNb/=. - by rewrite orbF => /andP[_] /[!inE] /XI/bg0. +rewrite fsbig_split ?finite_setU//= lee_add// lee_fsum_nneg_subset//= ?finite_setU//. +- exact/subsetP/subsetUl. +- by move=> x; rewrite !inE in_setU andb_orr andNb/= => /andP[_] /[!inE] /YI/ag0. +- exact/subsetP/subsetUr. +- by move=> x; rewrite !inE in_setU andb_orr andNb/= orbF => /andP[_] /[!inE] /XI/bg0. Qed. Lemma esum_mkcond [R : realType] [T : choiceType] (I : set T) @@ -173,11 +171,10 @@ Lemma esum_mkcond [R : realType] [T : choiceType] (I : set T) Proof. apply/eqP; rewrite eq_le !ub_ereal_sup//= => _ [X [finX XI]] <-. rewrite -big_mkcond/= big_fset_condE/=; set Y := [fset _ | _ in _ & _]%fset. - rewrite ereal_sup_ub//=; exists [set` Y]; last by rewrite set_fsetK. + rewrite ereal_sup_ub//=; exists [set` Y]; last by rewrite fsbig_finite// set_fsetK. by split => // i/=; rewrite !inE/= => /andP[_]; rewrite inE. -rewrite ereal_sup_ub//; exists X => //; rewrite -big_mkcond/=. -rewrite big_seq_cond [RHS]big_seq; apply: eq_bigl => i. -by rewrite in_fset_set// andb_idr// 2!inE => /XI. +rewrite ereal_sup_ub//; exists X => //; apply: eq_fsbigr => x; rewrite inE => Xx. +by rewrite ifT// inE; exact: XI. Qed. Lemma esum_mkcondr [R : realType] [T : choiceType] (I J : set T) (a : T -> \bar R) : @@ -227,6 +224,7 @@ Lemma esum_esum [R : realType] [T1 T2 : choiceType] Proof. move=> a_ge0; apply/eqP; rewrite eq_le; apply/andP; split. apply: ub_ereal_sup => /= _ [X [finX IX]] <-. + rewrite fsbig_finite//=. under eq_bigr do rewrite esum_mkcond. rewrite -esum_sum; last by move=> i j _ _; case: ifP. under eq_esum do rewrite -big_mkcond/=. @@ -236,7 +234,8 @@ move=> a_ge0; apply/eqP; rewrite eq_le; apply/andP; split. by rewrite in_setM => -[/andP[] /[!inE]]. exists [set z | z \in X `*` Y /\ z.2 \in J z.1] => /=. by split => //= z/=; rewrite !inE/= => -[[/IX]]. - rewrite (exchange_big_dep xpredT)//= pair_big_dep_cond/=. + rewrite fsbig_finite//=. + rewrite [in RHS]fsbig_finite// (exchange_big_dep xpredT)//= pair_big_dep_cond/=. apply: eq_fbigl => -[/= k1 k2]; rewrite in_fset_set//; apply/idP/imfset2P. rewrite !inE/= !inE/= -andA => -[kX [kY kJ]]. exists k1; first by rewrite !inE/= andbT/= in_fset_set// inE. @@ -251,7 +250,7 @@ apply: (@le_trans _ _ rewrite [leRHS](big_fsetID _ (mem X))/=. rewrite (_ : [fset x | x in Y & x \in X] = Y `&` fset_set X)%fset; last first. by apply/fsetP => x; rewrite 2!inE/= in_fset_set. - rewrite (fsetIidPr _); first by rewrite lee_addl// sume_ge0. + rewrite (fsetIidPr _); first by rewrite fsbig_finite// lee_addl// sume_ge0. apply/fsubsetP => -[i j]; rewrite in_fset_set// inE => Xij; apply/imfset2P. exists i => /=. rewrite !inE/= in_fset_set//; last exact: finite_set_fst. @@ -259,12 +258,13 @@ apply: (@le_trans _ _ exists j => //; rewrite !inE/=; have /XIJ[/= _ Jij] := Xij. rewrite in_fset_set; last exact: finite_set_snd. by apply/andP; split; rewrite ?inE//; exists i. +rewrite fsbig_finite; last exact: finite_set_fst. rewrite big_mkcond [leRHS]big_mkcond. apply: lee_sum => i Xi; rewrite ereal_sup_ub => //=. have ? : finite_set (X.`2 `&` J i). by apply: finite_setI; left; apply: finite_set_snd. exists (X.`2 `&` J i) => //. -rewrite [in RHS]big_fset_condE/=; apply eq_fbigl => j. +rewrite [in RHS]big_fset_condE/= fsbig_finite//; apply eq_fbigl => j. by rewrite in_fset_set// !inE/= in_setI in_fset_set//; exact: finite_set_snd. Qed. @@ -306,10 +306,10 @@ move=> a0; apply/eqP; rewrite eq_le; apply/andP; split. apply: ereal_sup_ub => /=; exists [set` [fset val i | i in 'I_n & P i]%fset]. split; first exact: finite_fset. by move=> /= k /imfsetP[/= i]; rewrite inE => + ->. - rewrite set_fsetK big_imfset/=; last by move=> ? ? ? ? /val_inj. + rewrite fsbig_finite//= set_fsetK big_imfset/=; last by move=> ? ? ? ? /val_inj. by rewrite big_filter big_enum_cond/= big_mkord. apply: ub_ereal_sup => _ [/= F [finF PF] <-]. -rewrite -(big_rmcond_in P)/=; last first. +rewrite fsbig_finite//= -(big_rmcond_in P)/=; last first. by move=> k; rewrite in_fset_set// inE => /PF ->. by apply: lee_sum_fset_lim. Qed. @@ -334,9 +334,10 @@ rewrite ub_ereal_sup => //= _ [X [finX XQ] <-]; rewrite ereal_sup_ub => //=. exists [set` (e^-1 @` (fset_set X))%fset]. split=> [|t /= /imfsetP[t'/=]]; first exact: finite_fset. by rewrite in_fset_set// inE => /XQ Qt' ->; exact: funS. -rewrite set_fsetK big_imfset => //=; last first. +rewrite fsbig_finite//= set_fsetK big_imfset => //=; last first. move=> x y/=; rewrite !in_fset_set// !inE => /XQ Qx /XQ Qy /(congr1 e). by rewrite !invK ?inE. +rewrite fsbig_finite//=. apply: eq_big_seq => i; rewrite in_fset_set// inE => /XQ Qi. by rewrite invK ?inE. Qed. diff --git a/theories/fsbigop.v b/theories/fsbigop.v index c1f0ec7281..e5c4c2a8b9 100644 --- a/theories/fsbigop.v +++ b/theories/fsbigop.v @@ -425,6 +425,16 @@ Arguments reindex_fsbigT {R idx op I J} _ _. #[deprecated(note="use reindex_fsbigT instead")] Notation reindex_inside_setT := reindex_fsbigT. +Lemma lee_fsum_nneg_subset {R : realDomainType} [T : choiceType] [A B : set T] + [f : T -> \bar R] : finite_set A -> finite_set B -> + {subset A <= B} -> {in [predD B & A], forall t : T, 0 <= f t}%E -> + (\sum_(t \in A) f t <= \sum_(t \in B) f t)%E. +Proof. +move=> finA finB AB f0; rewrite !fsbig_finite//=; apply: lee_sum_nneg_subfset. + by apply/fsubsetP; rewrite -fset_set_sub//; apply/subsetP. +by move=> t; rewrite !inE !in_fset_set// => /f0. +Qed. + Lemma ge0_mule_fsumr (T : choiceType) (R : realDomainType) (x : \bar R) (F : T -> \bar R) (P : set T) : (forall i : T, 0 <= F i)%E -> (x * (\sum_(i \in P) F i) = \sum_(i \in P) x * F i)%E. diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 5de4b68350..db42eff25b 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -371,6 +371,7 @@ move=> /(_ _ _ _)/Box[]//=; apply: le_le_trans. by case: ltP => //; rewrite lee_fin subr_le0. rewrite nneseries_esum//; last by move=> *; rewrite adde_ge0//= ?lee_fin. rewrite esum_ge//; exists [set` X] => //; rewrite fsbig_finite// ?set_fsetK//=. +rewrite fsbig_finite//= set_fsetK//. rewrite lee_sum // => i _; rewrite ?AE// !hlength_itv/= ?lte_fin -?EFinD/=. do !case: ifPn => //= ?; do ?by rewrite ?adde_ge0 ?lee_fin// ?subr_ge0// ?ltW. by rewrite addrAC. diff --git a/theories/measure.v b/theories/measure.v index a6aac32dbc..37aee93992 100644 --- a/theories/measure.v +++ b/theories/measure.v @@ -1536,7 +1536,7 @@ Lemma finite_card_dirac (A : set T) : finite_set A -> \esum_(i in A) \d_ i A = (#|` fset_set A|%:R)%:E :> \bar R. Proof. move=> finA. -rewrite esum_fset// big_seq_cond (eq_bigr (fun=> 1)) -?big_seq_cond. +rewrite esum_fset// fsbig_finite// big_seq_cond (eq_bigr (fun=> 1)) -?big_seq_cond. by rewrite card_fset_sum1// natr_sum -sumEFin. by move=> i; rewrite andbT in_fset_set//= /dirac indicE => ->. Qed. @@ -1549,7 +1549,7 @@ have [B BA Br] := infinite_set_fset `|ceil r| infA. apply: esum_ge; exists [set` B] => //; apply: (@le_trans _ _ `|ceil r|%:R%:E). by rewrite lee_fin natr_absz gtr0_norm ?ceil_gt0// ceil_ge. move: Br; rewrite -(@ler_nat R) -lee_fin => /le_trans; apply. -rewrite big_seq (eq_bigr (cst 1))/=; last first. +rewrite fsbig_finite// big_seq (eq_bigr (cst 1))/=; last first. move=> i; rewrite in_fset_set// inE/= => Bi; rewrite /dirac indicE mem_set//. exact: BA. by rewrite -big_seq card_fset_sum1 sumEFin natr_sum// set_fsetK. From 757fa18b45aa856ca9234edf98ad1f6e0a331d31 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Thu, 4 Aug 2022 14:20:02 +0900 Subject: [PATCH 07/42] minor improvements --- CHANGELOG_UNRELEASED.md | 65 ++++++---------------------------- theories/classical_sets.v | 47 +++++++++++++++++++++---- theories/esum.v | 68 ++++++++++++++++-------------------- theories/fsbigop.v | 9 +++++ theories/lebesgue_integral.v | 52 +++++++++------------------ theories/measure.v | 2 +- 6 files changed, 108 insertions(+), 135 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 884ed4e430..05db641bda 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -33,64 +33,22 @@ + lemma `measurable_funTS` - in `lebesgue_measure.v`: + lemma `measurable_fun_indic` - + lemma `big_const_idem` - + lemma `big_id_idem` - + lemma `big_rem_AC` - + lemma `bigD1_AC` - + lemma `big_mkcond_idem` - + lemma `big_split_idem` - + lemma `big_id_idem_AC` - + lemma `bigID_idem` -- in `mathcomp_extra.v`: - + lemmas `bigmax_le` and `bigmax_lt` - + lemma `bigmin_idr` - + lemma `bigmax_idr` -- in `classical_sets.v`: - + lemma `subset_refl` - in `fsbigop.v`: - + lemma `lee_fsum_nneg_subset` + + lemmas `lee_fsum_nneg_subset`, `lee_fsum` +- in `classical_sets.v`: + + lemmas `subset_fst_set`, `subset_snd_set`, `fst_set_fst`, `snd_set_snd`, + `fset_setM`, `snd_setM`, `fst_setMR` + + lemmas `xsection_snd_set`, `ysection_fst_set` ### Changed - in `measure.v`: + generalize `measurable_uncurry` - + generalize `pushforward` -- in `lebesgue_integral.v` - + change `Arguments` of `eq_integrable` -- in `lebesgue_integral.v`: - + fix pretty-printing of `{mfun _ >-> _}`, `{sfun _ >-> _}`, `{nnfun _ >-> _}` -- in `lebesgue_integral.v` - + minor generalization of `eq_measure_integral` -- from `topology.v` to `mathcomp_extra.v`: - + generalize `ltr_bigminr` to `porderType` and rename to `bigmin_lt` - + generalize `bigminr_ler` to `orderType` and rename to `bigmin_le` -- moved out of module `Bigminr` in `normedtype.v` to `mathcomp_extra.v` and generalized to `orderType`: - + lemma `bigminr_ler_cond`, renamed to `bigmin_le_cond` -- moved out of module `Bigminr` in `normedtype.v` to `mathcomp_extra.v`: - + lemma `bigminr_maxr` -- moved from from module `Bigminr` in `normedtype.v` - + to `mathcomp_extra.v` and generalized to `orderType` - * `bigminr_mkcond` -> `bigmin_mkcond` - * `bigminr_split` -> `bigmin_split` - * `bigminr_idl` -> `bigmin_idl` - * `bigminrID` -> `bigminID` - * `bigminrD1` -> `bigminD1` - * `bigminr_inf` -> `bigmin_inf` - * `bigminr_gerP` -> `bigmin_geP` - * `bigminr_gtrP` -> ``bigmin_gtP`` - * `bigminr_eq_arg` -> `bigmin_eq_arg` - * `eq_bigminr` -> `eq_bigmin` - + to `topology.v` and generalized to `orderType` - * `bigminr_lerP` -> `bigmin_leP` - * `bigminr_ltrP` -> `bigmin_ltP` -- moved from `topology.v` to `mathcomp_extra.v`: - + `bigmax_lerP` -> `bigmax_leP` - + `bigmax_ltrP` -> `bigmax_ltP` - + `ler_bigmax_cond` -> `le_bigmax_cond` - + `ler_bigmax` -> `le_bigmax` - + `le_bigmax` -> `homo_le_bigmax` - in `esum.v`: + definition `esum` +- moved from `lebesgue_integral.v` to `classical_sets.v`: + + `mem_set_pair1` -> `mem_xsection` + + `mem_set_pair2` -> `mem_ysection` ### Renamed @@ -113,14 +71,11 @@ + `muleindic_ge0` -> `nnfun_muleindic_ge0` + `mulem_ge0` -> `mulemu_ge0` + `nnfun_mulem_ge0` -> `nnsfun_mulemu_ge0` +- in `esum.v`: + + `esum0` -> `esum1` ### Removed -- in `normedtype.v` (module `Bigminr`) - + `bigminr_ler_cond`, `bigminr_ler`. - + `bigminr_seq1`, `bigminr_pred1_eq`, `bigminr_pred1` -- in `topology.v`: - + `bigmax_seq1`, `bigmax_pred1_eq`, `bigmax_pred1` - in `esum.v`: + lemma `fsetsP`, `sum_fset_set` diff --git a/theories/classical_sets.v b/theories/classical_sets.v index e14ea7e3e9..4bd42aff8a 100644 --- a/theories/classical_sets.v +++ b/theories/classical_sets.v @@ -2907,6 +2907,31 @@ End Exports. End SetOrder. Export SetOrder.Exports. +Section product. +Variables (T1 T2 : Type). +Implicit Type A B : set (T1 * T2). + +Lemma subset_fst_set : {homo @fst_set T1 T2 : A B / A `<=` B}. +Proof. by move=> A B AB x [y Axy]; exists y; exact/AB. Qed. + +Lemma subset_snd_set : {homo @snd_set T1 T2 : A B / A `<=` B}. +Proof. by move=> A B AB x [y Axy]; exists y; exact/AB. Qed. + +Lemma fst_set_fst A : A `<=` A.`1 \o fst. Proof. by move=> [x y]; exists y. Qed. + +Lemma snd_set_snd A: A `<=` A.`2 \o snd. Proof. by move=> [x y]; exists x. Qed. + +Lemma fst_setM (X : set T1) (Y : set T2) : (X `*` Y).`1 `<=` X. +Proof. by move=> x [y [//]]. Qed. + +Lemma snd_setM (X : set T1) (Y : set T2) : (X `*` Y).`2 `<=` Y. +Proof. by move=> x [y [//]]. Qed. + +Lemma fst_setMR (X : set T1) (Y : T1 -> set T2) : (X `*`` Y).`1 `<=` X. +Proof. by move=> x [y [//]]. Qed. + +End product. + Section section. Variables (T1 T2 : Type). Implicit Types (A : set (T1 * T2)) (x : T1) (y : T2). @@ -2915,6 +2940,18 @@ Definition xsection A x := [set y | (x, y) \in A]. Definition ysection A y := [set x | (x, y) \in A]. +Lemma xsection_snd_set A x : xsection A x `<=` A.`2. +Proof. by move=> y Axy; exists x; rewrite /xsection/= inE in Axy. Qed. + +Lemma ysection_fst_set A y : ysection A y `<=` A.`1. +Proof. by move=> x Axy; exists y; rewrite /ysection/= inE in Axy. Qed. + +Lemma mem_xsection x y A : (y \in xsection A x) = ((x, y) \in A). +Proof. by apply/idP/idP => [|]; [rewrite inE|rewrite /xsection !inE /= inE]. Qed. + +Lemma mem_ysection x y A : (x \in ysection A y) = ((x, y) \in A). +Proof. by apply/idP/idP => [|]; [rewrite inE|rewrite /ysection !inE /= inE]. Qed. + Lemma xsection0 x : xsection set0 x = set0. Proof. by rewrite predeqE /xsection => y; split => //=; rewrite inE. Qed. @@ -2923,16 +2960,14 @@ Proof. by rewrite predeqE /ysection => x; split => //=; rewrite inE. Qed. Lemma in_xsectionM X1 X2 x : x \in X1 -> xsection (X1 `*` X2) x = X2. Proof. -move=> xX1; rewrite /xsection predeqE => y /=; split; rewrite inE. - by move=> []. -by move=> X2y; split => //=; rewrite inE in xX1. +move=> xX1; apply/seteqP; split=> [y /xsection_snd_set|]; first exact: snd_setM. +by move=> y X2y; rewrite /xsection/= inE; split=> //=; rewrite inE in xX1. Qed. Lemma in_ysectionM X1 X2 y : y \in X2 -> ysection (X1 `*` X2) y = X1. Proof. -move=> yX2; rewrite /ysection predeqE => x /=; split; rewrite inE. - by move=> []. -by move=> X1x; split => //=; rewrite inE in yX2. +move=> yX2; apply/seteqP; split=> [x /ysection_fst_set|]; first exact: fst_setM. +by move=> x X1x; rewrite /ysection/= inE; split=> //=; rewrite inE in yX2. Qed. Lemma notin_xsectionM X1 X2 x : x \notin X1 -> xsection (X1 `*` X2) x = set0. diff --git a/theories/esum.v b/theories/esum.v index f90a049d79..88cebfb0b8 100644 --- a/theories/esum.v +++ b/theories/esum.v @@ -107,7 +107,7 @@ Lemma esum_ge [R : realType] [T : choiceType] (I : set T) (a : T -> \bar R) x : x <= \esum_(i in I) a i. Proof. by move=> [X IX /le_trans->//]; apply: ereal_sup_ub => /=; exists X. Qed. -Lemma esum0 [R : realFieldType] [I : choiceType] (D : set I) (a : I -> \bar R) : +Lemma esum1 [R : realFieldType] [I : choiceType] (D : set I) (a : I -> \bar R) : (forall i, D i -> a i = 0) -> \esum_(i in D) a i = 0. Proof. move=> a0; rewrite /esum (_ : [set _ | _ in _] = [set 0]) ?ereal_sup1//. @@ -121,8 +121,7 @@ Lemma le_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : \esum_(i in I) a i <= \esum_(i in I) b i. Proof. move=> le_ab; rewrite ub_ereal_sup => //= _ [X [finX XI]] <-; rewrite esum_ge//. -exists X => //; rewrite !fsbig_finite// big_seq [x in _ <= x]big_seq lee_sum //. -by move=> i; rewrite in_fset_set// inE => /XI/le_ab. +by exists X => //; apply: lee_fsum => // t /XI /le_ab. Qed. Lemma eq_esum [R : realType] [T : choiceType] (I : set T) (a b : T -> \bar R) : @@ -209,7 +208,7 @@ Lemma esum_sum [R : realType] [T1 T2 : choiceType] \sum_(j <- r | P j) \esum_(i in I) a i j. Proof. move=> a_ge0; elim: r => [|j r IHr]; rewrite ?(big_nil, big_cons)// -?IHr. - by rewrite esum0// => i; rewrite big_nil. + by rewrite esum1// => i; rewrite big_nil. case: (boolP (P j)) => Pj; last first. by apply: eq_esum => i Ii; rewrite big_cons (negPf Pj). have aj_ge0 i : I i -> a i j >= 0 by move=> ?; apply: a_ge0. @@ -223,25 +222,24 @@ Lemma esum_esum [R : realType] [T1 T2 : choiceType] \esum_(i in I) \esum_(j in J i) a i j = \esum_(k in I `*`` J) a k.1 k.2. Proof. move=> a_ge0; apply/eqP; rewrite eq_le; apply/andP; split. - apply: ub_ereal_sup => /= _ [X [finX IX]] <-. - rewrite fsbig_finite//=. - under eq_bigr do rewrite esum_mkcond. - rewrite -esum_sum; last by move=> i j _ _; case: ifP. + apply: ub_ereal_sup => /= _ [X [finX XI]] <-. + under eq_fsbigr do rewrite esum_mkcond. + rewrite fsbig_finite//= -esum_sum; last by move=> i j _ _; case: ifP. under eq_esum do rewrite -big_mkcond/=. apply: ub_ereal_sup => /= _ [Y [finY _] <-]; apply: ereal_sup_ub => /=. - have ? : finite_set [set z | z \in X `*` Y /\ z.2 \in J z.1]. + set XYJ := [set z | z \in X `*` Y /\ z.2 \in J z.1]. + have ? : finite_set XYJ. apply: sub_finite_set (finite_setM finX finY) => z/=. - by rewrite in_setM => -[/andP[] /[!inE]]. - exists [set z | z \in X `*` Y /\ z.2 \in J z.1] => /=. - by split => //= z/=; rewrite !inE/= => -[[/IX]]. - rewrite fsbig_finite//=. - rewrite [in RHS]fsbig_finite// (exchange_big_dep xpredT)//= pair_big_dep_cond/=. - apply: eq_fbigl => -[/= k1 k2]; rewrite in_fset_set//; apply/idP/imfset2P. - rewrite !inE/= !inE/= -andA => -[kX [kY kJ]]. - exists k1; first by rewrite !inE/= andbT/= in_fset_set// inE. - by exists k2 => //; rewrite !inE in_fset_set//; apply/andP; split=> /[!inE]. - move=> [t1]; rewrite !inE andbT/= !inE/= in_fset_set// inE => Xt1. - by move=> [t2]; rewrite !inE in_fset_set// =>/andP[/[!inE] Yt1 Jt1t2] [-> ->]. + by rewrite /XYJ/= in_setM => -[/andP[] /[!inE]]. + exists XYJ => /=; first by split => //= z; rewrite /XYJ/= 2!inE=> -[[/XI]]. + rewrite [in RHS]fsbig_finite//= (exchange_big_dep xpredT)// pair_big_dep_cond. + rewrite fsbig_finite//; apply: eq_fbigl => -[/= x y]; rewrite in_fset_set//. + apply/idP/imfset2P. + rewrite /XYJ !inE/= !inE/= -andA => -[Xx [Yy Jxy]]. + exists x; first by rewrite !inE in_fset_set// mem_set. + by exists y => //; rewrite !inE mem_set// in_fset_set// mem_set. + move=> [t1]; rewrite !inE andbT/= in_fset_set// inE => Xt1. + by move=> [t2]; rewrite !inE in_fset_set /XYJ//= =>/andP[/[!inE] ? ?] [-> ->]. apply: ub_ereal_sup => _ /= [X/= [finX XIJ]] <-; apply: esum_ge. exists X.`1; first by split=> [|x [y /XIJ[]//]]; exact: finite_set_fst. apply: (@le_trans _ _ @@ -254,15 +252,14 @@ apply: (@le_trans _ _ apply/fsubsetP => -[i j]; rewrite in_fset_set// inE => Xij; apply/imfset2P. exists i => /=. rewrite !inE/= in_fset_set//; last exact: finite_set_fst. - by rewrite andbT inE; exists j. - exists j => //; rewrite !inE/=; have /XIJ[/= _ Jij] := Xij. - rewrite in_fset_set; last exact: finite_set_snd. - by apply/andP; split; rewrite ?inE//; exists i. -rewrite fsbig_finite; last exact: finite_set_fst. -rewrite big_mkcond [leRHS]big_mkcond. -apply: lee_sum => i Xi; rewrite ereal_sup_ub => //=. -have ? : finite_set (X.`2 `&` J i). - by apply: finite_setI; left; apply: finite_set_snd. + by rewrite andbT mem_set//; move/fst_set_fst : Xij. + exists j => //; rewrite !inE/= in_fset_set; last exact: finite_set_snd. + rewrite mem_set/=; last by move/snd_set_snd : Xij. + by rewrite mem_set//; move/XIJ : Xij => []. +rewrite -fsbig_finite; last exact: finite_set_fst. +apply lee_fsum=> [|i Xi]; first exact: finite_set_fst. +rewrite ereal_sup_ub => //=; have ? : finite_set (X.`2 `&` J i). + by apply: finite_setI; left; exact: finite_set_snd. exists (X.`2 `&` J i) => //. rewrite [in RHS]big_fset_condE/= fsbig_finite//; apply eq_fbigl => j. by rewrite in_fset_set// !inE/= in_setI in_fset_set//; exact: finite_set_snd. @@ -309,9 +306,8 @@ move=> a0; apply/eqP; rewrite eq_le; apply/andP; split. rewrite fsbig_finite//= set_fsetK big_imfset/=; last by move=> ? ? ? ? /val_inj. by rewrite big_filter big_enum_cond/= big_mkord. apply: ub_ereal_sup => _ [/= F [finF PF] <-]. -rewrite fsbig_finite//= -(big_rmcond_in P)/=; last first. - by move=> k; rewrite in_fset_set// inE => /PF ->. -by apply: lee_sum_fset_lim. +rewrite fsbig_finite//= -(big_rmcond_in P)/=; first exact: lee_sum_fset_lim. +by move=> k; rewrite in_fset_set// inE => /PF ->. Qed. Lemma reindex_esum (R : realType) (T T' : choiceType) @@ -335,11 +331,9 @@ exists [set` (e^-1 @` (fset_set X))%fset]. split=> [|t /= /imfsetP[t'/=]]; first exact: finite_fset. by rewrite in_fset_set// inE => /XQ Qt' ->; exact: funS. rewrite fsbig_finite//= set_fsetK big_imfset => //=; last first. - move=> x y/=; rewrite !in_fset_set// !inE => /XQ Qx /XQ Qy /(congr1 e). + move=> x y; rewrite !in_fset_set// !inE => /XQ ? /XQ ? /(congr1 e). by rewrite !invK ?inE. -rewrite fsbig_finite//=. -apply: eq_big_seq => i; rewrite in_fset_set// inE => /XQ Qi. -by rewrite invK ?inE. +by rewrite -fsbig_finite//; apply eq_fsbigr=> x /[!inE]/XQ ?; rewrite invK ?inE. Qed. Arguments reindex_esum {R T T'} P Q e a. @@ -608,7 +602,7 @@ Let ge0_esum_posneg D f : (forall x, D x -> 0 <= f x) -> esum_posneg D f = \esum_(x in D) f x. Proof. move=> Sa; rewrite /esum_posneg [X in _ - X](_ : _ = 0) ?sube0; last first. - by rewrite esum0// => x Sx; rewrite -[LHS]/(f^\- x) (ge0_funenegE Sa)// inE. + by rewrite esum1// => x Sx; rewrite -[LHS]/(f^\- x) (ge0_funenegE Sa)// inE. by apply: eq_esum => t St; apply/max_idPl; exact: Sa. Qed. diff --git a/theories/fsbigop.v b/theories/fsbigop.v index e5c4c2a8b9..e812a13765 100644 --- a/theories/fsbigop.v +++ b/theories/fsbigop.v @@ -435,6 +435,15 @@ move=> finA finB AB f0; rewrite !fsbig_finite//=; apply: lee_sum_nneg_subfset. by move=> t; rewrite !inE !in_fset_set// => /f0. Qed. +Lemma lee_fsum [R : realDomainType] [T : choiceType] (I : set T) + (a b : T -> \bar R) : finite_set I -> + (forall i, I i -> a i <= b i)%E -> (\sum_(i \in I) a i <= \sum_(i \in I) b i)%E. +Proof. +move=> finI ab. +rewrite !fsbig_finite// big_seq [in leRHS]big_seq lee_sum //. +by move=> i; rewrite in_fset_set// inE; exact: ab. +Qed. + Lemma ge0_mule_fsumr (T : choiceType) (R : realDomainType) (x : \bar R) (F : T -> \bar R) (P : set T) : (forall i : T, 0 <= F i)%E -> (x * (\sum_(i \in P) F i) = \sum_(i \in P) x * F i)%E. diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index b27a2b902f..7145f93626 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -3738,7 +3738,7 @@ Proof. have -> : \sum_(n \bar R. rewrite nneseries_esum// (_ : [set _ | _] = setT); last exact/seteqP. rewrite [in LHS](esumID A)// !setTI [X in _ + X](_ : _ = 0) ?adde0//. - by apply esum0 => i Ai; rewrite /= /dirac indicE memNset. + by apply esum1 => i Ai; rewrite /= /dirac indicE memNset. rewrite /counting/=; case: ifPn => /asboolP finA. by rewrite -finite_card_dirac. by rewrite infinite_card_dirac. @@ -4082,29 +4082,19 @@ Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2). Implicit Types (A : set (T1 * T2)). -Lemma mem_set_pair1 x y A : - (y \in [set y' | (x, y') \in A]) = ((x, y) \in A). -Proof. by apply/idP/idP => [|]; [rewrite inE|rewrite !inE /= inE]. Qed. - -Lemma mem_set_pair2 x y A : - (x \in [set x' | (x', y) \in A]) = ((x, y) \in A). -Proof. by apply/idP/idP => [|]; [rewrite inE|rewrite 2!inE /= inE]. Qed. - Variable R : realType. Lemma measurable_xsection A x : measurable A -> measurable (xsection A x). Proof. move=> mA. pose f : T1 * T2 -> \bar R := EFin \o indic_nnsfun R mA. -have mf : measurable_fun setT f by apply/EFin_measurable_fun/measurable_funP. +have mf : measurable_fun setT f by exact/EFin_measurable_fun/measurable_funP. have _ : (fun y => (y \in xsection A x)%:R%:E) = f \o (fun y => (x, y)). - rewrite funeqE => y /=; rewrite /xsection /f. - by rewrite /= /mindic indicE/= mem_set_pair1. + by apply/funext => y/=; rewrite mem_xsection. have -> : xsection A x = (fun y => f (x, y)) @^-1` [set 1%E]. - rewrite predeqE => y; split; rewrite /xsection /preimage /= /f. - by rewrite /= /mindic indicE/= => ->. - rewrite /= /mindic indicE. - by case: (_ \in _) => //= -[] /eqP; rewrite eq_sym oner_eq0. + apply/funext => y/=; rewrite -(in_setE (xsection A x)) mem_xsection. + rewrite /f/= mindicE//; apply/propext; split=> [->//|[]]. + by case: (_ \in _)=> // /esym/eqP; rewrite oner_eq0. by rewrite -(setTI (_ @^-1` _)); exact: measurable_fun_prod1. Qed. @@ -4114,13 +4104,11 @@ move=> mA. pose f : T1 * T2 -> \bar R := EFin \o indic_nnsfun R mA. have mf : measurable_fun setT f by apply/EFin_measurable_fun/measurable_funP. have _ : (fun x => (x \in ysection A y)%:R%:E) = f \o (fun x => (x, y)). - rewrite funeqE => x /=; rewrite /ysection /f. - by rewrite /= /mindic indicE mem_set_pair2. + by apply/funext => x/=; rewrite mem_ysection. have -> : ysection A y = (fun x => f (x, y)) @^-1` [set 1%E]. - rewrite predeqE => x; split; rewrite /ysection /preimage /= /f. - by rewrite /= /mindic indicE => ->. - rewrite /= /mindic indicE. - by case: (_ \in _) => //= -[] /eqP; rewrite eq_sym oner_eq0. + apply/funext => x/=; rewrite -(in_setE (ysection A y)) mem_ysection. + rewrite /f/= mindicE//; apply/propext; split=> [->//|[]]. + by case: (_ \in _)=> // /esym/eqP; rewrite oner_eq0. by rewrite -(setTI (_ @^-1` _)); exact: measurable_fun_prod2. Qed. @@ -4144,9 +4132,8 @@ have phiF x : (fun i => phi (F i) x) --> phi (\bigcup_i F i) x. rewrite /phi /= xsection_bigcup; apply: nondecreasing_cvg_mu. - by move=> n; apply: measurable_xsection; case: (BF n). - by apply: bigcupT_measurable => i; apply: measurable_xsection; case: (BF i). - - move=> m n mn; apply/subsetPset => y; rewrite /xsection/= !inE. - by have /subsetPset FmFn := ndF _ _ mn; exact: FmFn. -apply: (emeasurable_fun_cvg (phi \o F)). + - by move=> m n mn; exact/subsetPset/le_xsection/subsetPset/ndF. +apply: (emeasurable_fun_cvg (phi \o F)) => //. - by move=> i; have [] := BF i. - by move=> x _; exact: phiF. Qed. @@ -4165,9 +4152,8 @@ have psiF x : (fun i => psi (F i) x) --> psi (\bigcup_i F i) x. rewrite /psi /= ysection_bigcup; apply: nondecreasing_cvg_mu. - by move=> n; apply: measurable_ysection; case: (BF n). - by apply: bigcupT_measurable => i; apply: measurable_ysection; case: (BF i). - - move=> m n mn; apply/subsetPset => y; rewrite /ysection/= !inE. - by have /subsetPset FmFn := ndF _ _ mn; exact: FmFn. -apply: (emeasurable_fun_cvg (psi \o F)). + - by move=> m n mn; exact/subsetPset/le_ysection/subsetPset/ndF. +apply: (emeasurable_fun_cvg (psi \o F)) => //. - by move=> i; have [] := BF i. - by move=> x _; exact: psiF. Qed. @@ -4580,10 +4566,7 @@ rewrite funeqE => x; rewrite /= -(setTI (xsection _ _)). rewrite -integral_indic//; last exact: measurable_xsection. rewrite /F /fubini_F -(setTI (xsection _ _)). rewrite integral_setI_indic; [|exact: measurable_xsection|by []]. -apply: eq_integral => y _ /=; rewrite indicT mul1e /f !indicE. -have [|] /= := boolP (y \in xsection _ _). - by rewrite inE /xsection /= => ->. -by rewrite /xsection /= notin_set /= => /negP/negbTE ->. +by apply: eq_integral => y _ /=; rewrite indicT mul1e /f !indicE mem_xsection. Qed. Lemma indic_fubini_tonelli_GE : G = m1 \o ysection A. @@ -4592,10 +4575,7 @@ rewrite funeqE => y; rewrite /= -(setTI (ysection _ _)). rewrite -integral_indic//; last exact: measurable_ysection. rewrite /F /fubini_F -(setTI (ysection _ _)). rewrite integral_setI_indic; [|exact: measurable_ysection|by []]. -apply: eq_integral => x _ /=; rewrite indicT mul1e /f 2!indicE. -have [|] /= := boolP (x \in ysection _ _). - by rewrite inE /xsection /= => ->. -by rewrite /xsection /= notin_set /= => /negP/negbTE ->. +by apply: eq_integral => x _ /=; rewrite indicT mul1e /f 2!indicE mem_ysection. Qed. Lemma indic_measurable_fun_fubini_tonelli_F : measurable_fun setT F. diff --git a/theories/measure.v b/theories/measure.v index 37aee93992..ac6555b1c5 100644 --- a/theories/measure.v +++ b/theories/measure.v @@ -1550,7 +1550,7 @@ apply: esum_ge; exists [set` B] => //; apply: (@le_trans _ _ `|ceil r|%:R%:E). by rewrite lee_fin natr_absz gtr0_norm ?ceil_gt0// ceil_ge. move: Br; rewrite -(@ler_nat R) -lee_fin => /le_trans; apply. rewrite fsbig_finite// big_seq (eq_bigr (cst 1))/=; last first. - move=> i; rewrite in_fset_set// inE/= => Bi; rewrite /dirac indicE mem_set//. + move=> i; rewrite in_fset_set// inE/= => Bi; rewrite diracE mem_set//. exact: BA. by rewrite -big_seq card_fset_sum1 sumEFin natr_sum// set_fsetK. Qed. From eb333477ed9396d3e65c2697f891631fc7adae13 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 26 Aug 2022 11:42:52 +0900 Subject: [PATCH 08/42] transfer lemma --- CHANGELOG_UNRELEASED.md | 8 ++++++ theories/lebesgue_integral.v | 52 ++++++++++++++++++++++++++++++++++++ theories/lebesgue_measure.v | 3 +++ theories/measure.v | 16 ++++++----- 4 files changed, 72 insertions(+), 7 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 05db641bda..2445dd9a97 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -39,6 +39,10 @@ + lemmas `subset_fst_set`, `subset_snd_set`, `fst_set_fst`, `snd_set_snd`, `fset_setM`, `snd_setM`, `fst_setMR` + lemmas `xsection_snd_set`, `ysection_fst_set` + + Hint about `measurable_fun_normr` +- in `lebesgue_integral.v`: + + lemma `integral_pushforward` + ### Changed @@ -49,6 +53,10 @@ - moved from `lebesgue_integral.v` to `classical_sets.v`: + `mem_set_pair1` -> `mem_xsection` + `mem_set_pair2` -> `mem_ysection` +- in `lebesgue_measure.v`: + + `pushforward` requires a proof that its argument is measurable +- in `lebesgue_integral.v`: + + change implicits of `integralM_indic` ### Renamed diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index 7145f93626..224f829307 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -2196,6 +2196,7 @@ by rewrite preimage_nnfun0. Qed. End integralM_indic. +Arguments integralM_indic {d T R m D} mD f. Section integral_mscale. Local Open Scope ereal_scope. @@ -2389,6 +2390,57 @@ Qed. End integral_cst. +Section transfer. +Local Open Scope ereal_scope. +Variables (d1 d2 : _) (X : measurableType d1) (Y : measurableType d2). +Variables (phi : X -> Y) (mphi : measurable_fun setT phi). +Variables (R : realType) (mu : {measure set X -> \bar R}). + +Lemma integral_pushforward (f : Y -> \bar R) : + measurable_fun setT f -> (forall y, 0 <= f y) -> + \int[pushforward mu mphi]_y f y = \int[mu]_x (f \o phi) x. +Proof. +move=> mf f0. +have [f_ [ndf_ f_f]] := approximation measurableT mf (fun t _ => f0 t). +transitivity (lim (fun n => \int[pushforward mu mphi]_x (f_ n x)%:E)). + rewrite -monotone_convergence//. + - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: f_f. + - by move=> n; exact/EFin_measurable_fun. + - by move=> n y _; rewrite lee_fin. + - by move=> y _ m n mn; rewrite lee_fin; apply/lefP/ndf_. +rewrite (_ : (fun _ => _) = (fun n => \int[mu]_x (EFin \o f_ n \o phi) x)). + rewrite -monotone_convergence//; last 3 first. + - move=> n /=; apply: measurable_fun_comp; first exact: measurable_fun_EFin. + by apply: measurable_fun_comp => //; exact: measurable_sfun. + - by move=> n x _ /=; rewrite lee_fin. + - by move=> x _ m n mn; rewrite lee_fin; exact/lefP/ndf_. + by apply: eq_integral => x _ /=; apply/cvg_lim => //; exact: f_f. +apply/funext => n. +have mfnphi r : measurable (f_ n @^-1` [set r] \o phi). + rewrite -[_ \o _]/(phi @^-1` (f_ n @^-1` [set r])) -(setTI (_ @^-1` _)). + exact/mphi. +transitivity (\sum_(k <- fset_set (range (f_ n))) + \int[mu]_x (k * \1_((f_ n @^-1` [set k]) \o phi) x)%:E). + under eq_integral do rewrite fimfunE -sumEFin. + rewrite ge0_integral_sum//; last 2 first. + - move=> y; apply/EFin_measurable_fun; apply: measurable_funM. + exact: measurable_fun_cst. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (f_ n) y)). + - by move=> y x _; rewrite nnfun_muleindic_ge0. + apply eq_bigr => r _; rewrite integralM_indic_nnsfun// integral_indic//=. + rewrite (integralM_indic _ (fun r => f_ n @^-1` [set r] \o phi))//. + by congr (_ * _); rewrite [RHS](@integral_indic). + by move=> r0; rewrite preimage_nnfun0. +rewrite -ge0_integral_sum//; last 2 first. + - move=> r; apply/EFin_measurable_fun; apply: measurable_funM. + exact: measurable_fun_cst. + by rewrite (_ : \1_ _ = mindic R (mfnphi r)). + - by move=> r x _; rewrite nnfun_muleindic_ge0. +by apply eq_integral => x _; rewrite sumEFin -fimfunE. +Qed. + +End transfer. + Section integral_dirac. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (a : T) (R : realType). diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index db42eff25b..409d8208bc 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -1521,6 +1521,9 @@ Qed. End standard_measurable_fun. +#[global] Hint Extern 0 (measurable_fun _ normr) => + solve [exact: measurable_fun_normr] : core. + Section measurable_fun_realType. Variables (d : measure_display) (T : measurableType d) (R : realType). Implicit Types (D : set T) (f g : T -> R). diff --git a/theories/measure.v b/theories/measure.v index ac6555b1c5..5b3f2d9dc6 100644 --- a/theories/measure.v +++ b/theories/measure.v @@ -84,7 +84,8 @@ From HB Require Import structures. (* isMeasure == factory corresponding to the type of measures *) (* Measure == structure corresponding to measures *) (* *) -(* pushforward f m == pushforward/image measure of m by f *) +(* pushforward mf m == pushforward/image measure of m by f, where mf is a *) +(* proof that f is measurable *) (* \d_a == Dirac measure *) (* msum mu n == the measure corresponding to the sum of the measures *) (* mu_0, ..., mu_{n-1} *) @@ -1461,18 +1462,19 @@ Section pushforward_measure. Local Open Scope ereal_scope. Variables (d d' : measure_display). Variables (T1 : measurableType d) (T2 : measurableType d') (f : T1 -> T2). -Hypothesis mf : measurable_fun setT f. Variables (R : realFieldType) (m : {measure set T1 -> \bar R}). -Definition pushforward A := m (f @^-1` A). +Definition pushforward (mf : measurable_fun setT f) A := m (f @^-1` A). + +Hypothesis mf : measurable_fun setT f. -Let pushforward0 : pushforward set0 = 0. +Let pushforward0 : pushforward mf set0 = 0. Proof. by rewrite /pushforward preimage_set0 measure0. Qed. -Let pushforward_ge0 A : 0 <= pushforward A. +Let pushforward_ge0 A : 0 <= pushforward mf A. Proof. by apply: measure_ge0; rewrite -[X in measurable X]setIT; apply: mf. Qed. -Let pushforward_sigma_additive : semi_sigma_additive pushforward. +Let pushforward_sigma_additive : semi_sigma_additive (pushforward mf). Proof. move=> F mF tF mUF; rewrite /pushforward preimage_bigcup. apply: measure_semi_sigma_additive. @@ -1483,7 +1485,7 @@ apply: measure_semi_sigma_additive. Qed. HB.instance Definition _ := isMeasure.Build _ _ _ - pushforward pushforward0 pushforward_ge0 pushforward_sigma_additive. + (pushforward mf) pushforward0 pushforward_ge0 pushforward_sigma_additive. End pushforward_measure. From feb7526a64b2f5eda7d9b82a45b9e9c676370614 Mon Sep 17 00:00:00 2001 From: Cyril Cohen Date: Tue, 30 Aug 2022 11:42:02 +0200 Subject: [PATCH 09/42] Missing results in derive Co-Authored-By: Abastro --- CHANGELOG_UNRELEASED.md | 4 ++++ theories/derive.v | 17 ++++++++++++----- 2 files changed, 16 insertions(+), 5 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 2445dd9a97..04ccebf953 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -43,6 +43,8 @@ - in `lebesgue_integral.v`: + lemma `integral_pushforward` +- in `derive.v`: + + lemma `diff_derivable` ### Changed @@ -57,6 +59,8 @@ + `pushforward` requires a proof that its argument is measurable - in `lebesgue_integral.v`: + change implicits of `integralM_indic` +- in `derive.v`: + + generalized `is_diff_scalel` ### Renamed diff --git a/theories/derive.v b/theories/derive.v index eeedb6157b..f9695ab301 100644 --- a/theories/derive.v +++ b/theories/derive.v @@ -655,13 +655,13 @@ apply: DiffDef; first exact/linear_differentiable/scaler_continuous. by rewrite diff_lin //; apply: scaler_continuous. Qed. -Global Instance is_diff_scalel (x k : R) : +Global Instance is_diff_scalel (k : R) (x : V) : is_diff k ( *:%R ^~ x) ( *:%R ^~ x). Proof. -have -> : *:%R ^~ x = GRing.scale_linear R x. - by rewrite funeqE => ? /=; rewrite [_ *: _]mulrC. -apply: DiffDef; first exact/linear_differentiable/scaler_continuous. -by rewrite diff_lin //; apply: scaler_continuous. +have sx_lin : linear ( *:%R ^~ x) by move=> u y z; rewrite scalerDl scalerA. +have -> : *:%R ^~ x = Linear sx_lin by rewrite funeqE. +apply: DiffDef; first exact/linear_differentiable/scalel_continuous. +by rewrite diff_lin//; apply: scalel_continuous. Qed. Lemma differentiable_coord m n (M : 'M[R]_(m.+1, n.+1)) i j : @@ -1064,6 +1064,13 @@ Lemma derivableP (U : normedModType R) (f : V -> U) x v : derivable f x v -> is_derive x v f ('D_v f x). Proof. by move=> df; apply: DeriveDef. Qed. +Lemma diff_derivable (f : V -> W) a v : + differentiable f a -> derivable f a v. +Proof. +move=> dfa; apply/derivable1P/derivable1_diffP. +by apply: differentiable_comp; rewrite ?scale0r ?add0r. +Qed. + Global Instance is_derive_cst (U : normedModType R) (a : U) (x v : V) : is_derive x v (cst a) 0. Proof. From abc88b5fe9606aa2c0514eb2f1126dde9efa71ee Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 31 Aug 2022 16:04:58 +0900 Subject: [PATCH 10/42] minor shortenings and sectioning --- theories/derive.v | 159 ++++++++++++++++++++++++---------------------- 1 file changed, 82 insertions(+), 77 deletions(-) diff --git a/theories/derive.v b/theories/derive.v index f9695ab301..30279a41ad 100644 --- a/theories/derive.v +++ b/theories/derive.v @@ -1007,10 +1007,11 @@ Proof. by move=> /differentiableP df; rewrite diff_val. Qed. End DifferentialR3_numFieldType. -Section Derive. -Variables (R : numFieldType) (V W : normedModType R). +Section DeriveRU. +Variables (R : numFieldType) (U : normedModType R). +Implicit Types f : R -> U. -Let der1 (U : normedModType R) (f : R -> U) x : derivable f x 1 -> +Let der1 f x : derivable f x 1 -> f \o shift x = cst (f x) + ( *:%R^~ (f^`() x)) +o_ (0 : R) id. Proof. move=> df; apply/eqaddoE; have /derivable_nbhsP := df. @@ -1019,20 +1020,19 @@ have -> : (fun h => (f \o shift x) h%:A) = f \o shift x. by rewrite derive1E =>->. Qed. -Lemma deriv1E (U : normedModType R) (f : R -> U) x : - derivable f x 1 -> 'd f x = ( *:%R^~ (f^`() x)) :> (R -> U). +Lemma deriv1E f x : derivable f x 1 -> 'd f x = ( *:%R^~ (f^`() x)) :> (R -> U). Proof. move=> df; have lin_scal : linear (fun h : R => h *: f^`() x). - by move=> ???; rewrite scalerDl scalerA. + by move=> ? ? ?; rewrite scalerDl scalerA. have -> : (fun h => h *: f^`() x) = Linear lin_scal by []. by apply: diff_unique; [apply: scalel_continuous|apply: der1]. Qed. -Lemma diff1E (U : normedModType R) (f : R -> U) x : +Lemma diff1E f x : differentiable f x -> 'd f x = (fun h => h *: f^`() x) :> (R -> U). Proof. move=> df; have lin_scal : linear (fun h : R => h *: 'd f x 1). - by move=> ???; rewrite scalerDl scalerA. + by move=> ? ? ?; rewrite scalerDl scalerA. have -> : (fun h => h *: f^`() x) = Linear lin_scal. by rewrite derive1E'. apply: diff_unique; first exact: scalel_continuous. @@ -1040,8 +1040,7 @@ apply/eqaddoE; have /diff_locally -> := df; congr (_ + _ + _). by rewrite funeqE => h /=; rewrite -{1}[h]mulr1 linearZ. Qed. -Lemma derivable1_diffP (U : normedModType R) (f : R -> U) x : - derivable f x 1 <-> differentiable f x. +Lemma derivable1_diffP f x : derivable f x 1 <-> differentiable f x. Proof. split=> dfx. by apply/diff_locallyP; rewrite deriv1E //; split; @@ -1052,7 +1051,13 @@ have -> : (fun h => (f \o shift x) h%:A) = f \o shift x. by have /diff_locally := dfx; rewrite diff1E // derive1E =>->. Qed. -Lemma derivable1P (U : normedModType R) (f : V -> U) x v : +End DeriveRU. + +Section DeriveVW. +Variables (R : numFieldType) (V W : normedModType R). +Implicit Types f : V -> W. + +Lemma derivable1P f x v : derivable f x v <-> derivable (fun h : R => f (h *: v + x)) 0 1. Proof. rewrite /derivable; set g1 := fun h => h^-1 *: _; set g2 := fun h => h^-1 *: _. @@ -1060,26 +1065,32 @@ suff -> : g1 = g2 by []. by rewrite funeqE /g1 /g2 => h /=; rewrite addr0 scale0r add0r [_%:A]mulr1. Qed. -Lemma derivableP (U : normedModType R) (f : V -> U) x v : - derivable f x v -> is_derive x v f ('D_v f x). +Lemma derivableP f x v : derivable f x v -> is_derive x v f ('D_v f x). Proof. by move=> df; apply: DeriveDef. Qed. -Lemma diff_derivable (f : V -> W) a v : - differentiable f a -> derivable f a v. +Lemma diff_derivable f a v : differentiable f a -> derivable f a v. Proof. move=> dfa; apply/derivable1P/derivable1_diffP. by apply: differentiable_comp; rewrite ?scale0r ?add0r. Qed. -Global Instance is_derive_cst (U : normedModType R) (a : U) (x v : V) : - is_derive x v (cst a) 0. +Global Instance is_derive_cst (a : W) (x v : V) : is_derive x v (cst a) 0. Proof. apply: DeriveDef; last by rewrite deriveE // diff_val. -apply/derivable1P/derivable1_diffP. -by have -> : (fun h => cst a (h *: v + x)) = cst a by rewrite funeqE. +exact/diff_derivable. Qed. -Fact der_add (f g : V -> W) (x v : V) : derivable f x v -> derivable g x v -> +Lemma is_derive_eq f (x v : V) (df f' : W) : + is_derive x v f f' -> f' = df -> is_derive x v f df. +Proof. by move=> ? <-. Qed. + +End DeriveVW. + +Section Derive_lemmasVW. +Variables (R : numFieldType) (V W : normedModType R). +Implicit Types f g : V -> W. + +Fact der_add f g (x v : V) : derivable f x v -> derivable g x v -> (fun h => h^-1 *: (((f + g) \o shift x) (h *: v) - (f + g) x)) @ 0^' --> 'D_v f x + 'D_v g x. Proof. @@ -1090,50 +1101,50 @@ evar (fg : R -> W); rewrite [X in X @ _](_ : _ = fg) /=; last first. exact: cvgD. Qed. -Lemma deriveD (f g : V -> W) (x v : V) : derivable f x v -> derivable g x v -> +Lemma deriveD f g (x v : V) : derivable f x v -> derivable g x v -> 'D_v (f + g) x = 'D_v f x + 'D_v g x. Proof. by move=> df dg; apply: cvg_map_lim (der_add df dg). Qed. -Lemma derivableD (f g : V -> W) (x v : V) : +Lemma derivableD f g (x v : V) : derivable f x v -> derivable g x v -> derivable (f + g) x v. Proof. move=> df dg; apply/cvg_ex; exists (derive f x v + derive g x v). exact: der_add. Qed. -Global Instance is_deriveD (f g : V -> W) (x v : V) (df dg : W) : +Global Instance is_deriveD f g (x v : V) (df dg : W) : is_derive x v f df -> is_derive x v g dg -> is_derive x v (f + g) (df + dg). Proof. move=> dfx dgx; apply: DeriveDef; first exact: derivableD. by rewrite deriveD // !derive_val. Qed. -Global Instance is_derive_sum n (f : 'I_n -> V -> W) (x v : V) - (df : 'I_n -> W) : (forall i, is_derive x v (f i) (df i)) -> - is_derive x v (\sum_(i < n) f i) (\sum_(i < n) df i). +Global Instance is_derive_sum n (h : 'I_n -> V -> W) (x v : V) + (dh : 'I_n -> W) : (forall i, is_derive x v (h i) (dh i)) -> + is_derive x v (\sum_(i < n) h i) (\sum_(i < n) dh i). Proof. -elim: n f df => [f df dfx|f df dfx n ihn]. +elim: n h dh => [h dh dhx|h dh dhx n ihn]. by rewrite !big_ord0 //; apply: is_derive_cst. by rewrite !big_ord_recr /=; apply: is_deriveD. Qed. -Lemma derivable_sum n (f : 'I_n -> V -> W) (x v : V) : - (forall i, derivable (f i) x v) -> derivable (\sum_(i < n) f i) x v. +Lemma derivable_sum n (h : 'I_n -> V -> W) (x v : V) : + (forall i, derivable (h i) x v) -> derivable (\sum_(i < n) h i) x v. Proof. -move=> df; suff : forall i, is_derive x v (f i) ('D_v (f i) x) by []. +move=> df; suff : forall i, is_derive x v (h i) ('D_v (h i) x) by []. by move=> ?; apply: derivableP. Qed. -Lemma derive_sum n (f : 'I_n -> V -> W) (x v : V) : - (forall i, derivable (f i) x v) -> - 'D_v (\sum_(i < n) f i) x = \sum_(i < n) 'D_v (f i) x. +Lemma derive_sum n (h : 'I_n -> V -> W) (x v : V) : + (forall i, derivable (h i) x v) -> + 'D_v (\sum_(i < n) h i) x = \sum_(i < n) 'D_v (h i) x. Proof. -move=> df; suff dfx : forall i, is_derive x v (f i) ('D_v (f i) x). +move=> df; suff dfx : forall i, is_derive x v (h i) ('D_v (h i) x). by rewrite derive_val. by move=> ?; apply: derivableP. Qed. -Fact der_opp (f : V -> W) (x v : V) : derivable f x v -> +Fact der_opp f (x v : V) : derivable f x v -> (fun h => h^-1 *: (((- f) \o shift x) (h *: v) - (- f) x)) @ 0^' --> - 'D_v f x. Proof. @@ -1142,65 +1153,67 @@ move=> df; evar (g : R -> W); rewrite [X in X @ _](_ : _ = g) /=; last first. exact: cvgN. Qed. -Lemma deriveN (f : V -> W) (x v : V) : derivable f x v -> +Lemma deriveN f (x v : V) : derivable f x v -> 'D_v (- f) x = - 'D_v f x. Proof. by move=> df; apply: cvg_map_lim (der_opp df). Qed. -Lemma derivableN (f : V -> W) (x v : V) : +Lemma derivableN f (x v : V) : derivable f x v -> derivable (- f) x v. Proof. by move=> df; apply/cvg_ex; exists (- 'D_v f x); apply: der_opp. Qed. -Global Instance is_deriveN (f : V -> W) (x v : V) (df : W) : +Global Instance is_deriveN f (x v : V) (df : W) : is_derive x v f df -> is_derive x v (- f) (- df). Proof. move=> dfx; apply: DeriveDef; first exact: derivableN. by rewrite deriveN // derive_val. Qed. -Lemma is_derive_eq (V' W' : normedModType R) (f : V' -> W') (x v : V') - (df f' : W') : is_derive x v f f' -> f' = df -> is_derive x v f df. -Proof. by move=> ? <-. Qed. - -Global Instance is_deriveB (f g : V -> W) (x v : V) (df dg : W) : +Global Instance is_deriveB f g (x v : V) (df dg : W) : is_derive x v f df -> is_derive x v g dg -> is_derive x v (f - g) (df - dg). Proof. by move=> ??; apply: is_derive_eq. Qed. -Lemma deriveB (f g : V -> W) (x v : V) : derivable f x v -> derivable g x v -> +Lemma deriveB f g (x v : V) : derivable f x v -> derivable g x v -> 'D_v (f - g) x = 'D_v f x - 'D_v g x. Proof. by move=> /derivableP df /derivableP dg; rewrite derive_val. Qed. -Lemma derivableB (f g : V -> W) (x v : V) : +Lemma derivableB f g (x v : V) : derivable f x v -> derivable g x v -> derivable (f - g) x v. Proof. by move=> /derivableP df /derivableP dg. Qed. -Fact der_scal (f : V -> W) (k : R) (x v : V) : derivable f x v -> +Fact der_scal f (k : R) (x v : V) : derivable f x v -> (fun h => h^-1 *: ((k \*: f \o shift x) (h *: v) - (k \*: f) x)) @ (0 : R)^' --> k *: 'D_v f x. Proof. -move=> df; evar (g : R -> W); rewrite [X in X @ _](_ : _ = g) /=; last first. - rewrite funeqE => h. - by rewrite scalerBr !scalerA mulrC -!scalerA -!scalerBr /g. +move=> df; evar (h : R -> W); rewrite [X in X @ _](_ : _ = h) /=; last first. + rewrite funeqE => r. + by rewrite scalerBr !scalerA mulrC -!scalerA -!scalerBr /h. exact: cvgZr. Qed. -Lemma deriveZ (f : V -> W) (k : R) (x v : V) : derivable f x v -> +Lemma deriveZ f (k : R) (x v : V) : derivable f x v -> 'D_v (k \*: f) x = k *: 'D_v f x. Proof. by move=> df; apply: cvg_map_lim (der_scal df). Qed. -Lemma derivableZ (f : V -> W) (k : R) (x v : V) : +Lemma derivableZ f (k : R) (x v : V) : derivable f x v -> derivable (k \*: f) x v. Proof. by move=> df; apply/cvg_ex; exists (k *: 'D_v f x); apply: der_scal. Qed. -Global Instance is_deriveZ (f : V -> W) (k : R) (x v : V) (df : W) : +Global Instance is_deriveZ f (k : R) (x v : V) (df : W) : is_derive x v f df -> is_derive x v (k \*: f) (k *: df). Proof. move=> dfx; apply: DeriveDef; first exact: derivableZ. by rewrite deriveZ // derive_val. Qed. -Fact der_mult (f g : V -> R) (x v : V) : +End Derive_lemmasVW. + +Section Derive_lemmasVR. +Variables (R : numFieldType) (V : normedModType R). +Implicit Types f g : V -> R. + +Fact der_mult f g (x v : V) : derivable f x v -> derivable g x v -> (fun h => h^-1 *: (((f * g) \o shift x) (h *: v) - (f * g) x)) @ (0 : R)^' --> f x *: 'D_v g x + g x *: 'D_v f x. @@ -1217,22 +1230,21 @@ apply: cvgD; last exact: cvgZr df. apply: cvg_comp2 (@mul_continuous _ (_, _)) => /=; last exact: dg. suff : {for 0, continuous (fun h : R => f(h *: v + x))}. by move=> /continuous_withinNx; rewrite scale0r add0r. -exact/differentiable_continuous/derivable1_diffP/derivable1P. +exact/differentiable_continuous/derivable1_diffP/(derivable1P _ _ _).1. Qed. -Lemma deriveM (f g : V -> R) (x v : V) : - derivable f x v -> derivable g x v -> +Lemma deriveM f g (x v : V) : derivable f x v -> derivable g x v -> 'D_v (f * g) x = f x *: 'D_v g x + g x *: 'D_v f x. Proof. by move=> df dg; apply: cvg_map_lim (der_mult df dg). Qed. -Lemma derivableM (f g : V -> R) (x v : V) : +Lemma derivableM f g (x v : V) : derivable f x v -> derivable g x v -> derivable (f * g) x v. Proof. move=> df dg; apply/cvg_ex; exists (f x *: 'D_v g x + g x *: 'D_v f x). exact: der_mult. Qed. -Global Instance is_deriveM (f g : V -> R) (x v : V) (df dg : R) : +Global Instance is_deriveM f g (x v : V) (df dg : R) : is_derive x v f df -> is_derive x v g dg -> is_derive x v (f * g) (f x *: dg + g x *: df). Proof. @@ -1240,7 +1252,7 @@ move=> dfx dgx; apply: DeriveDef; first exact: derivableM. by rewrite deriveM // !derive_val. Qed. -Global Instance is_deriveX (f : V -> R) n (x v : V) (df : R) : +Global Instance is_deriveX f n (x v : V) (df : R) : is_derive x v f df -> is_derive x v (f ^+ n.+1) ((n.+1%:R * f x ^+n) *: df). Proof. move=> dfx; elim: n => [|n ihn]; first by rewrite expr1 expr0 mulr1 scale1r. @@ -1249,17 +1261,14 @@ rewrite scalerA -scalerDl mulrCA -[f x * _]exprS. by rewrite [in LHS]mulr_natl exprfctE -mulrSr mulr_natl. Qed. -Lemma derivableX (f : V -> R) n (x v : V) : - derivable f x v -> derivable (f ^+ n.+1) x v. +Lemma derivableX f n (x v : V) : derivable f x v -> derivable (f ^+ n.+1) x v. Proof. by move/derivableP. Qed. -Lemma deriveX (f : V -> R) n (x v : V) : - derivable f x v -> +Lemma deriveX f n (x v : V) : derivable f x v -> 'D_v (f ^+ n.+1) x = (n.+1%:R * f x ^+ n) *: 'D_v f x. Proof. by move=> /derivableP df; rewrite derive_val. Qed. -Fact der_inv (f : V -> R) (x v : V) : - f x != 0 -> derivable f x v -> +Fact der_inv f (x v : V) : f x != 0 -> derivable f x v -> (fun h => h^-1 *: (((fun y => (f y)^-1) \o shift x) (h *: v) - (f x)^-1)) @ (0 : R)^' --> - (f x) ^-2 *: 'D_v f x. Proof. @@ -1288,23 +1297,22 @@ rewrite -scalerA [_ *: f _]mulVf // [_%:A]mulr1. by rewrite mulrC -scalerA [_ *: f _]mulVf // [_%:A]mulr1. Unshelve. all: by end_near. Qed. -Lemma deriveV (f : V -> R) x v : f x != 0 -> derivable f x v -> +Lemma deriveV f x v : f x != 0 -> derivable f x v -> 'D_v (fun y => (f y)^-1) x = - (f x) ^- 2 *: 'D_v f x. Proof. by move=> fxn0 df; apply: cvg_map_lim (der_inv fxn0 df). Qed. -Lemma derivableV (f : V -> R) (x v : V) : +Lemma derivableV f (x v : V) : f x != 0 -> derivable f x v -> derivable (fun y => (f y)^-1) x v. Proof. move=> df dg; apply/cvg_ex; exists (- (f x) ^- 2 *: 'D_v f x). exact: der_inv. Qed. -End Derive. - -Lemma derive1_cst (R : numFieldType) (V : normedModType R) (k : V) t : - (cst k)^`() t = 0. +Lemma derive1_cst (k : V) t : (cst k)^`() t = 0. Proof. by rewrite derive1E derive_val. Qed. +End Derive_lemmasVR. + Lemma EVT_max (R : realType) (f : R -> R) (a b : R) : (* TODO : Filter not infered *) a <= b -> {within `[a, b], continuous f} -> exists2 c, c \in `[a, b]%R & forall t, t \in `[a, b]%R -> f t <= f c. @@ -1576,13 +1584,10 @@ Section is_derive_instances. Variables (R : numFieldType) (V : normedModType R). Lemma derivable_cst (x : V) : derivable (fun=> x) 0 1. -Proof. exact/derivable1_diffP/differentiable_cst. Qed. +Proof. exact/diff_derivable. Qed. Lemma derivable_id (x v : V) : derivable id x v. -Proof. -apply/derivable1P/derivableD; last exact/derivable_cst. -exact/derivable1_diffP/differentiableZl. -Qed. +Proof. exact/diff_derivable. Qed. Global Instance is_derive_id (x v : V) : is_derive x v id v. Proof. @@ -1591,7 +1596,7 @@ by rewrite deriveE// (@diff_lin _ _ _ [linear of idfun]). Qed. Global Instance is_deriveNid (x v : V) : is_derive x v -%R (- v). -Proof. by apply: is_deriveN. Qed. +Proof. exact: is_deriveN. Qed. End is_derive_instances. From ab7110a3217ac98392ff1aede300a7e87b74ec73 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 22 Aug 2022 23:14:58 +0900 Subject: [PATCH 11/42] generalization of continuous_measurable_fun --- CHANGELOG_UNRELEASED.md | 9 +++++++++ theories/lebesgue_measure.v | 18 ++++++++++++------ theories/topology.v | 25 +++++++++++++++++++------ 3 files changed, 40 insertions(+), 12 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 04ccebf953..959afdda25 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -61,6 +61,10 @@ + change implicits of `integralM_indic` - in `derive.v`: + generalized `is_diff_scalel` +- in `topology.v`: + + generalize `continuousP` +- in `lebesgue_measure.v`: + + generalize `continuous_measurable_fun` ### Renamed @@ -86,6 +90,11 @@ - in `esum.v`: + `esum0` -> `esum1` +- in `topology.v`: + + `continuousP` -> `continuousTP` +- in `lebesgue_measure.v`: + + `continuous_measurable_fun` -> `continuousT_measurable_fun` + ### Removed - in `esum.v`: diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 409d8208bc..f2fbbfe196 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -1487,14 +1487,20 @@ move=> q; case: ifPn => // qfab; apply: is_interval_measurable => //. exact: is_interval_bigcup_ointsub. Qed. -Lemma continuous_measurable_fun (f : R -> R) : continuous f -> - measurable_fun setT f. +Lemma continuous_measurable_fun (f : R -> R) D : + open D -> {in D, continuous f} -> measurable_fun D f. Proof. -move=> /continuousP cf; apply: (measurability (RGenOpens.measurableE R)). -move=> _ [_ [a [b ->] <-]]; rewrite setTI. +move=> oD /(continuousP _ oD) cf. +apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. by apply: open_measurable; exact/cf/interval_open. Qed. +Lemma continuousT_measurable_fun (f : R -> R) : + continuous f -> measurable_fun setT f. +Proof. +by move=> cf; apply: continuous_measurable_fun => //; exact: openT. +Qed. + End coutinuous_measurable. Section standard_measurable_fun. @@ -1550,7 +1556,7 @@ Lemma measurable_funrM D f (k : R) : measurable_fun D f -> measurable_fun D (fun x => k * f x). Proof. apply: (@measurable_fun_comp _ _ _ _ _ _ ( *%R k)). -by apply: continuous_measurable_fun; apply: mulrl_continuous. +by apply: continuousT_measurable_fun; apply: mulrl_continuous. Qed. Lemma measurable_funN D f : measurable_fun D f -> measurable_fun D (-%R \o f). @@ -1570,7 +1576,7 @@ Lemma measurable_fun_exprn D n f : measurable_fun D f -> measurable_fun D (fun x => f x ^+ n). Proof. apply: measurable_fun_comp ((@GRing.exp R)^~ n) _ _ _. -by apply: continuous_measurable_fun; apply: exprn_continuous. +by apply: continuousT_measurable_fun; apply: exprn_continuous. Qed. Lemma measurable_fun_sqr D f : diff --git a/theories/topology.v b/theories/topology.v index eba7e2d981..ecce594c96 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -1731,12 +1731,25 @@ Notation "A ^°" := (interior A) : classical_set_scope. Notation continuous f := (forall x, f%function @ x --> f%function x). -Lemma continuousP (S T : topologicalType) (f : S -> T) : - continuous f <-> forall A, open A -> open (f @^-1` A). +Lemma continuousP (S T : topologicalType) (f : S -> T) (D : set S) : + open D -> + {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). Proof. -split=> fcont; first by rewrite !openE => A Aop ? /Aop /fcont. -move=> s A; rewrite nbhs_simpl /= !nbhsE => - [B [[Bop Bfs] sBA]]. -by exists (f @^-1` B); split; [split=> //; apply/fcont|move=> ? /sBA]. +move=> oD; split=> [fcont|fcont s /[!inE] sD A]. + rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. + by rewrite interiorI; split => //; move: oD; rewrite openE; exact. +rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. +by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. +Qed. + +Lemma continuousTP (S T : topologicalType) (f : S -> T) : + continuous f <-> (forall A, open A -> open (f @^-1` A)). +Proof. +split=> [cf A oA|cf]. + rewrite -(setTI (_ @^-1` _)). + by move: A oA; apply/continuousP => //; exact/openT. +apply: in1TT. +by apply/continuousP; [exact: openT|move=> ? ?; rewrite setTI; exact: cf]. Qed. Lemma continuous_comp (R S T : topologicalType) (f : R -> S) (g : S -> T) x : @@ -2236,7 +2249,7 @@ Definition weak_topologicalType := weak_topologicalTypeMixin). Lemma weak_continuous : continuous (f : weak_topologicalType -> T). -Proof. by apply/continuousP => A ?; exists A. Qed. +Proof. by apply/continuousTP => A ?; exists A. Qed. Lemma cvg_image (F : set (set S)) (s : S) : Filter F -> f @` setT = setT -> From 95d342050559c3dd01d7ef4e605f5f1292c4ea95 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Tue, 30 Aug 2022 16:55:39 +0900 Subject: [PATCH 12/42] change naming --- CHANGELOG_UNRELEASED.md | 4 +--- theories/lebesgue_measure.v | 2 +- theories/topology.v | 10 +++++----- 3 files changed, 7 insertions(+), 9 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 959afdda25..a82bd6bb63 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -62,7 +62,7 @@ - in `derive.v`: + generalized `is_diff_scalel` - in `topology.v`: - + generalize `continuousP` + + generalize `continuousP` and rename to `continuous_inP` - in `lebesgue_measure.v`: + generalize `continuous_measurable_fun` @@ -90,8 +90,6 @@ - in `esum.v`: + `esum0` -> `esum1` -- in `topology.v`: - + `continuousP` -> `continuousTP` - in `lebesgue_measure.v`: + `continuous_measurable_fun` -> `continuousT_measurable_fun` diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index f2fbbfe196..7ccd906b2e 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -1490,7 +1490,7 @@ Qed. Lemma continuous_measurable_fun (f : R -> R) D : open D -> {in D, continuous f} -> measurable_fun D f. Proof. -move=> oD /(continuousP _ oD) cf. +move=> oD /(continuous_inP _ oD) cf. apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. by apply: open_measurable; exact/cf/interval_open. Qed. diff --git a/theories/topology.v b/theories/topology.v index ecce594c96..5364109e44 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -1731,7 +1731,7 @@ Notation "A ^°" := (interior A) : classical_set_scope. Notation continuous f := (forall x, f%function @ x --> f%function x). -Lemma continuousP (S T : topologicalType) (f : S -> T) (D : set S) : +Lemma continuous_inP (S T : topologicalType) (f : S -> T) (D : set S) : open D -> {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). Proof. @@ -1742,14 +1742,14 @@ rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. Qed. -Lemma continuousTP (S T : topologicalType) (f : S -> T) : +Lemma continuousP (S T : topologicalType) (f : S -> T) : continuous f <-> (forall A, open A -> open (f @^-1` A)). Proof. split=> [cf A oA|cf]. rewrite -(setTI (_ @^-1` _)). - by move: A oA; apply/continuousP => //; exact/openT. + by move: A oA; apply/continuous_inP => //; exact/openT. apply: in1TT. -by apply/continuousP; [exact: openT|move=> ? ?; rewrite setTI; exact: cf]. +by apply/continuous_inP; [exact: openT|move=> ? ?; rewrite setTI; exact: cf]. Qed. Lemma continuous_comp (R S T : topologicalType) (f : R -> S) (g : S -> T) x : @@ -2249,7 +2249,7 @@ Definition weak_topologicalType := weak_topologicalTypeMixin). Lemma weak_continuous : continuous (f : weak_topologicalType -> T). -Proof. by apply/continuousTP => A ?; exists A. Qed. +Proof. by apply/continuousP => A ?; exists A. Qed. Lemma cvg_image (F : set (set S)) (s : S) : Filter F -> f @` setT = setT -> From 2e17ab0403bbb55e44a698d14a13860adff0fbca Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Tue, 30 Aug 2022 17:02:46 +0900 Subject: [PATCH 13/42] generalize continuous_measurable_fun further Co-authored-by: Zachary Stone --- CHANGELOG_UNRELEASED.md | 6 +++++- theories/lebesgue_measure.v | 24 +++++++++++++++++++----- theories/topology.v | 4 ++-- 3 files changed, 26 insertions(+), 8 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index a82bd6bb63..6c86864b1b 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -45,6 +45,11 @@ - in `derive.v`: + lemma `diff_derivable` +- in `topology.v`: + + lemma `continuous_inP` +- in `lebesgue_measure.v`: + + lemma `open_measurable_subspace` + + corollary `open_continuous_measurable_fun` ### Changed @@ -89,7 +94,6 @@ + `nnfun_mulem_ge0` -> `nnsfun_mulemu_ge0` - in `esum.v`: + `esum0` -> `esum1` - - in `lebesgue_measure.v`: + `continuous_measurable_fun` -> `continuousT_measurable_fun` diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 7ccd906b2e..e03162fa19 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -1487,18 +1487,32 @@ move=> q; case: ifPn => // qfab; apply: is_interval_measurable => //. exact: is_interval_bigcup_ointsub. Qed. -Lemma continuous_measurable_fun (f : R -> R) D : +Lemma open_measurable_subspace (D : set R) (U : set (subspace D)) : + measurable D -> open U -> measurable (D `&` U). +Proof. +move=> mD /open_subspaceP [V [oV] VD]; rewrite setIC -VD. +by apply: measurableI => //; exact: open_measurable. +Qed. + +Lemma continuous_measurable_fun (D : set R) (f : subspace D -> R) : + measurable D -> continuous f -> measurable_fun D f. +Proof. +move=> mD /continuousP cf; apply: (measurability (RGenOpens.measurableE R)). +move=> _ [_ [a [b ->] <-]]; apply: open_measurable_subspace => //. +by exact/cf/interval_open. +Qed. + +Corollary open_continuous_measurable_fun (D : set R) (f : R -> R) : open D -> {in D, continuous f} -> measurable_fun D f. Proof. -move=> oD /(continuous_inP _ oD) cf. -apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. -by apply: open_measurable; exact/cf/interval_open. +move=> oD; rewrite -(continuous_open_subspace f oD). +by apply :continuous_measurable_fun; exact: open_measurable. Qed. Lemma continuousT_measurable_fun (f : R -> R) : continuous f -> measurable_fun setT f. Proof. -by move=> cf; apply: continuous_measurable_fun => //; exact: openT. +by move=> cf; apply: open_continuous_measurable_fun => //; exact: openT. Qed. End coutinuous_measurable. diff --git a/theories/topology.v b/theories/topology.v index 5364109e44..685878325e 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -1733,7 +1733,7 @@ Notation continuous f := (forall x, f%function @ x --> f%function x). Lemma continuous_inP (S T : topologicalType) (f : S -> T) (D : set S) : open D -> - {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). + {in D, continuous f} <-> forall A, open A -> open (D `&` f @^-1` A). Proof. move=> oD; split=> [fcont|fcont s /[!inE] sD A]. rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. @@ -1743,7 +1743,7 @@ by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. Qed. Lemma continuousP (S T : topologicalType) (f : S -> T) : - continuous f <-> (forall A, open A -> open (f @^-1` A)). + continuous f <-> forall A, open A -> open (f @^-1` A). Proof. split=> [cf A oA|cf]. rewrite -(setTI (_ @^-1` _)). From d101109984eb8abfdd6c880a4f73b28fde52e3cf Mon Sep 17 00:00:00 2001 From: Cyril Cohen Date: Wed, 31 Aug 2022 18:39:00 +0200 Subject: [PATCH 14/42] continuous_inP as a consequence of continuousP --- theories/topology.v | 209 ++++++++++++++++++++++++-------------------- 1 file changed, 113 insertions(+), 96 deletions(-) diff --git a/theories/topology.v b/theories/topology.v index 685878325e..51f793cf9f 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -1731,25 +1731,12 @@ Notation "A ^°" := (interior A) : classical_set_scope. Notation continuous f := (forall x, f%function @ x --> f%function x). -Lemma continuous_inP (S T : topologicalType) (f : S -> T) (D : set S) : - open D -> - {in D, continuous f} <-> forall A, open A -> open (D `&` f @^-1` A). -Proof. -move=> oD; split=> [fcont|fcont s /[!inE] sD A]. - rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. - by rewrite interiorI; split => //; move: oD; rewrite openE; exact. -rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. -by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. -Qed. - Lemma continuousP (S T : topologicalType) (f : S -> T) : continuous f <-> forall A, open A -> open (f @^-1` A). Proof. -split=> [cf A oA|cf]. - rewrite -(setTI (_ @^-1` _)). - by move: A oA; apply/continuous_inP => //; exact/openT. -apply: in1TT. -by apply/continuous_inP; [exact: openT|move=> ? ?; rewrite setTI; exact: cf]. +split=> fcont; first by rewrite !openE => A Aop ? /Aop /fcont. +move=> s A; rewrite nbhs_simpl /= !nbhsE => - [B [[Bop Bfs] sBA]]. +by exists (f @^-1` B); split; [split=> //; apply/fcont|move=> ? /sBA]. Qed. Lemma continuous_comp (R S T : topologicalType) (f : R -> S) (g : S -> T) x : @@ -2140,7 +2127,7 @@ move=> N_gt0 P [n _ Pn]; exists (n * N)%N => //= m. by rewrite /= -leq_divRL//; apply: Pn. Qed. -Lemma near_inftyS (P : set nat) : +Lemma near_inftyS (P : set nat) : (\forall x \near \oo, P (S x)) -> (\forall x \near \oo, P x). Proof. case=> N _ NPS; exists (S N) => // [[]]; rewrite /= ?ltn0 //. Qed. @@ -5716,6 +5703,29 @@ Lemma closed_subspaceW (U : set T) : closed (U : set T) -> closed (U : set (subspace A)). Proof. by move=> /closed_openC/open_subspaceW/open_closedC; rewrite setCK. Qed. +Lemma open_setIS (U : set (subspace A)) : open A -> + open (U `&` A : set T) = open U. +Proof. +move=> oA; apply/propext; rewrite open_subspaceP. +split=> [|[V [oV <-]]]; last exact: openI. +by move=> oUA; exists (U `&` A); rewrite -setIA setIid. +Qed. + +Lemma open_setSI (U : set (subspace A)) : open A -> open (A `&` U) = open U. +Proof. by move=> oA; rewrite -setIC open_setIS. Qed. + +Lemma closed_setIS (U : set (subspace A)) : closed A -> + closed (U `&` A : set T) = closed U. +Proof. +move=> oA; apply/propext; rewrite closed_subspaceP. +split=> [|[V [oV <-]]]; last exact: closedI. +by move=> oUA; exists (U `&` A); rewrite -setIA setIid. +Qed. + +Lemma closed_setSI (U : set (subspace A)) : + closed A -> closed (A `&` U) = closed U. +Proof. by move=> oA; rewrite -setIC closed_setIS. Qed. + Lemma closure_subspaceW (U : set T) : U `<=` A -> closure (U : set (subspace A)) = closure (U : set T) `&` A. Proof. @@ -5754,6 +5764,92 @@ Qed. End Subspace. +Global Instance subspace_filter {T : topologicalType} + (A : set T) (x : subspace A) : + Filter (nbhs_subspace x) := nbhs_subspace_filter x. + +Global Instance subspace_proper_filter {T : topologicalType} + (A : set T) (x : subspace A) : + ProperFilter (nbhs_subspace x) := nbhs_subspace_filter x. + +Notation "{ 'within' A , 'continuous' f }" := + (continuous (f : (subspace A) -> _)). + +Section SubspaceRelative. +Context {T : topologicalType}. +Implicit Types (U : topologicalType) (A B : set T). + +Lemma nbhs_subspace_subset A B (x : T) : + A `<=` B -> nbhs (x : subspace B) `<=` nbhs (x : subspace A). +Proof. +rewrite /nbhs //= => AB; case: (nbhs_subspaceP A); case: (nbhs_subspaceP B). +- by move=> ? ?; apply: within_subset => //=; exact: (nbhs_filter x). +- by move=> ? /AB. +- by move=> Bx ? W /nbhs_singleton /(_ Bx) ? ? ->. +- by move=> ? ?. +Qed. + +Lemma continuous_subspaceW {U} A B (f : T -> U) : + A `<=` B -> + {within B, continuous f} -> {within A, continuous f}. +Proof. +by move=> ? ctsF ? ? ?; apply: (@nbhs_subspace_subset A B) => //; exact: ctsF. +Qed. + +Lemma nbhs_subspaceT (x : T) : nbhs (x : subspace setT) = nbhs (x) . +Proof. +rewrite {1}/nbhs //=; have [_|] := nbhs_subspaceP (@setT T); last by cbn. +rewrite eqEsubset withinE; split => [W [V nbhsV]|W ?]; last by exists W. +by rewrite 2!setIT => ->. +Qed. + +Lemma continuous_subspaceT_for {U} A (f : T -> U) (x : T) : + A x -> {for x, continuous f} -> {for x, continuous (f : subspace A -> U)}. +Proof. +rewrite /filter_of/nbhs/=/prop_for => inA ctsf. +have [_|//] := nbhs_subspaceP A x. +apply: (cvg_trans _ ctsf); apply: cvg_fmap2; apply: cvg_within. +by rewrite /subspace; exact: nbhs_filter. +Qed. + +Lemma continuous_subspaceT {U} A (f : T -> U) : + {in A, continuous f} -> {within A, continuous f}. +Proof. +rewrite continuous_subspace_in ?in_setP => ctsf t At. +by apply continuous_subspaceT_for => //=; apply: ctsf. +Qed. + +Lemma continuous_open_subspace {U} A (f : T -> U) : + open A -> {within A, continuous f} = {in A, continuous f}. +Proof. +rewrite openE continuous_subspace_in /= => oA; rewrite propeqE ?in_setP. +by split => + x /[dup] Ax /oA Aox; rewrite /filter_of /= => /(_ _ Ax); + rewrite -(nbhs_subspace_interior Aox). +Qed. + +Lemma continuous_inP {U} A (f : T -> U) : open A -> + {in A, continuous f} <-> forall X, open X -> open (A `&` f @^-1` X). +Proof. +move=> oA; rewrite -continuous_open_subspace// continuousP. +by under eq_forall do rewrite -open_setSI//. +Qed. + +Lemma pasting {U} A B (f : T -> U) : closed A -> closed B -> + {within A, continuous f} -> {within B, continuous f} -> + {within A `|` B, continuous f}. +Proof. +move=> ? ? ctsA ctsB; apply/continuous_closedP => W oW. +case/continuous_closedP/(_ _ oW)/closed_subspaceP: ctsA => V1 [? V1W]. +case/continuous_closedP/(_ _ oW)/closed_subspaceP: ctsB => V2 [? V2W]. +apply/closed_subspaceP; exists ((V1 `&` A) `|` (V2 `&` B)); split. + by apply: closedU; exact: closedI. +rewrite [RHS]setIUr -V2W -V1W eqEsubset; split=> ?. + by case=> [[][]] ? ? [] ?; [left | left | right | right]; split. +by case=> [][] ? ?; split=> []; [left; split | left | right; split | right]. +Qed. + +End SubspaceRelative. + Section SubspaceUniform. Local Notation "A ^-1" := ([set xy | A (xy.2, xy.1)]) : classical_set_scope. Context {X : uniformType} (A : set X). @@ -5862,85 +5958,6 @@ Canonical subspace_pseudoMetricType := End SubspacePseudoMetric. -Global Instance subspace_filter {T : topologicalType} - (A : set T) (x : subspace A) : - Filter (nbhs_subspace x) := nbhs_subspace_filter x. - -Global Instance subspace_proper_filter {T : topologicalType} - (A : set T) (x : subspace A) : - ProperFilter (nbhs_subspace x) := nbhs_subspace_filter x. - -Notation "{ 'within' A , 'continuous' f }" := - (continuous (f : (subspace A) -> _)). - -Section SubspaceRelative. -Context {T : topologicalType}. -Implicit Types (U : topologicalType) (A B : set T). - -Lemma nbhs_subspace_subset A B (x : T) : - A `<=` B -> nbhs (x : subspace B) `<=` nbhs (x : subspace A). -Proof. -rewrite /nbhs //= => AB; case: (nbhs_subspaceP A); case: (nbhs_subspaceP B). -- by move=> ? ?; apply: within_subset => //=; exact: (nbhs_filter x). -- by move=> ? /AB. -- by move=> Bx ? W /nbhs_singleton /(_ Bx) ? ? ->. -- by move=> ? ?. -Qed. - -Lemma continuous_subspaceW {U} A B (f : T -> U) : - A `<=` B -> - {within B, continuous f} -> {within A, continuous f}. -Proof. -by move=> ? ctsF ? ? ?; apply: (@nbhs_subspace_subset A B) => //; exact: ctsF. -Qed. - -Lemma nbhs_subspaceT (x : T) : nbhs (x : subspace setT) = nbhs (x) . -Proof. -rewrite {1}/nbhs //=; have [_|] := nbhs_subspaceP (@setT T); last by cbn. -rewrite eqEsubset withinE; split => [W [V nbhsV]|W ?]; last by exists W. -by rewrite 2!setIT => ->. -Qed. - -Lemma continuous_subspaceT_for {U} A (f : T -> U) (x : T) : - A x -> {for x, continuous f} -> {for x, continuous (f : subspace A -> U)}. -Proof. -rewrite /filter_of/nbhs/=/prop_for => inA ctsf. -have [_|//] := nbhs_subspaceP A x. -apply: (cvg_trans _ ctsf); apply: cvg_fmap2; apply: cvg_within. -by rewrite /subspace; exact: nbhs_filter. -Qed. - -Lemma continuous_subspaceT {U} A (f : T -> U) : - {in A, continuous f} -> {within A, continuous f}. -Proof. -rewrite continuous_subspace_in ?in_setP => ctsf t At. -by apply continuous_subspaceT_for => //=; apply: ctsf. -Qed. - -Lemma continuous_open_subspace {U} A (f : T -> U) : - @open T A -> {within A, continuous f} = {in A, continuous f}. -Proof. -rewrite openE continuous_subspace_in /= => oA; rewrite propeqE ?in_setP. -by split => + x /[dup] Ax /oA Aox; rewrite /filter_of /= => /(_ _ Ax); - rewrite -(nbhs_subspace_interior Aox). -Qed. - -Lemma pasting {U} A B (f : T -> U) : closed A -> closed B -> - {within A, continuous f} -> {within B, continuous f} -> - {within A `|` B, continuous f}. -Proof. -move=> ? ? ctsA ctsB; apply/continuous_closedP => W oW. -case/continuous_closedP/(_ _ oW)/closed_subspaceP: ctsA => V1 [? V1W]. -case/continuous_closedP/(_ _ oW)/closed_subspaceP: ctsB => V2 [? V2W]. -apply/closed_subspaceP; exists ((V1 `&` A) `|` (V2 `&` B)); split. - by apply: closedU; exact: closedI. -rewrite [RHS]setIUr -V2W -V1W eqEsubset; split=> ?. - by case=> [[][]] ? ? [] ?; [left | left | right | right]; split. -by case=> [][] ? ?; split=> []; [left; split | left | right; split | right]. -Qed. - -End SubspaceRelative. - Lemma continuous_compact {T U : topologicalType} (f : T -> U) A : {within A, continuous f} -> compact A -> compact (f @` A). Proof. From df62972a2814bed19664ed8e44247f09972e6ef9 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Thu, 1 Sep 2022 15:44:52 +0900 Subject: [PATCH 15/42] fix naming, upd changelog --- CHANGELOG_UNRELEASED.md | 10 ++++------ theories/lebesgue_measure.v | 10 +++++----- theories/topology.v | 2 +- 3 files changed, 10 insertions(+), 12 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 6c86864b1b..d9c218315f 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -49,7 +49,11 @@ + lemma `continuous_inP` - in `lebesgue_measure.v`: + lemma `open_measurable_subspace` + + lemma ``subspace_continuous_measurable_fun`` + corollary `open_continuous_measurable_fun` + + lemma `continuous_inP` +- in `topology.v`: + + lemmas `open_setIS`, `open_setSI`, `closed_setIS`, `closed_setSI` ### Changed @@ -66,10 +70,6 @@ + change implicits of `integralM_indic` - in `derive.v`: + generalized `is_diff_scalel` -- in `topology.v`: - + generalize `continuousP` and rename to `continuous_inP` -- in `lebesgue_measure.v`: - + generalize `continuous_measurable_fun` ### Renamed @@ -94,8 +94,6 @@ + `nnfun_mulem_ge0` -> `nnsfun_mulemu_ge0` - in `esum.v`: + `esum0` -> `esum1` -- in `lebesgue_measure.v`: - + `continuous_measurable_fun` -> `continuousT_measurable_fun` ### Removed diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index e03162fa19..666c91fd5e 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -1494,7 +1494,7 @@ move=> mD /open_subspaceP [V [oV] VD]; rewrite setIC -VD. by apply: measurableI => //; exact: open_measurable. Qed. -Lemma continuous_measurable_fun (D : set R) (f : subspace D -> R) : +Lemma subspace_continuous_measurable_fun (D : set R) (f : subspace D -> R) : measurable D -> continuous f -> measurable_fun D f. Proof. move=> mD /continuousP cf; apply: (measurability (RGenOpens.measurableE R)). @@ -1506,10 +1506,10 @@ Corollary open_continuous_measurable_fun (D : set R) (f : R -> R) : open D -> {in D, continuous f} -> measurable_fun D f. Proof. move=> oD; rewrite -(continuous_open_subspace f oD). -by apply :continuous_measurable_fun; exact: open_measurable. +by apply: subspace_continuous_measurable_fun; exact: open_measurable. Qed. -Lemma continuousT_measurable_fun (f : R -> R) : +Lemma continuous_measurable_fun (f : R -> R) : continuous f -> measurable_fun setT f. Proof. by move=> cf; apply: open_continuous_measurable_fun => //; exact: openT. @@ -1570,7 +1570,7 @@ Lemma measurable_funrM D f (k : R) : measurable_fun D f -> measurable_fun D (fun x => k * f x). Proof. apply: (@measurable_fun_comp _ _ _ _ _ _ ( *%R k)). -by apply: continuousT_measurable_fun; apply: mulrl_continuous. +by apply: continuous_measurable_fun; apply: mulrl_continuous. Qed. Lemma measurable_funN D f : measurable_fun D f -> measurable_fun D (-%R \o f). @@ -1590,7 +1590,7 @@ Lemma measurable_fun_exprn D n f : measurable_fun D f -> measurable_fun D (fun x => f x ^+ n). Proof. apply: measurable_fun_comp ((@GRing.exp R)^~ n) _ _ _. -by apply: continuousT_measurable_fun; apply: exprn_continuous. +by apply: continuous_measurable_fun; apply: exprn_continuous. Qed. Lemma measurable_fun_sqr D f : diff --git a/theories/topology.v b/theories/topology.v index 51f793cf9f..033d201996 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -5796,7 +5796,7 @@ Proof. by move=> ? ctsF ? ? ?; apply: (@nbhs_subspace_subset A B) => //; exact: ctsF. Qed. -Lemma nbhs_subspaceT (x : T) : nbhs (x : subspace setT) = nbhs (x) . +Lemma nbhs_subspaceT (x : T) : nbhs (x : subspace setT) = nbhs x. Proof. rewrite {1}/nbhs //=; have [_|] := nbhs_subspaceP (@setT T); last by cbn. rewrite eqEsubset withinE; split => [W [V nbhsV]|W ?]; last by exists W. From 9acbfd0924721e9d70bb87aeeed33ee086faba8f Mon Sep 17 00:00:00 2001 From: Cyril Cohen Date: Thu, 1 Sep 2022 11:57:05 +0200 Subject: [PATCH 16/42] Update CHANGELOG_UNRELEASED.md --- CHANGELOG_UNRELEASED.md | 1 - 1 file changed, 1 deletion(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index d9c218315f..85cb758c45 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -51,7 +51,6 @@ + lemma `open_measurable_subspace` + lemma ``subspace_continuous_measurable_fun`` + corollary `open_continuous_measurable_fun` - + lemma `continuous_inP` - in `topology.v`: + lemmas `open_setIS`, `open_setSI`, `closed_setIS`, `closed_setSI` From fd48780144cefab7bb206986aedf9fad4488adc8 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 2 Sep 2022 11:55:57 +0900 Subject: [PATCH 17/42] natrS --- CHANGELOG_UNRELEASED.md | 2 ++ theories/ereal.v | 4 ++-- theories/exp.v | 8 ++++---- theories/lebesgue_integral.v | 17 ++++++++--------- theories/lebesgue_measure.v | 5 ++--- theories/mathcomp_extra.v | 6 ++++++ theories/reals.v | 2 +- theories/sequences.v | 2 +- theories/set_interval.v | 2 +- 9 files changed, 27 insertions(+), 21 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 85cb758c45..e82105c4f6 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -53,6 +53,8 @@ + corollary `open_continuous_measurable_fun` - in `topology.v`: + lemmas `open_setIS`, `open_setSI`, `closed_setIS`, `closed_setSI` +- in `mathcomp_extra.v`: + + lemmas `natrS`, `natSr` ### Changed diff --git a/theories/ereal.v b/theories/ereal.v index 9bcf26b7ca..72617f6785 100644 --- a/theories/ereal.v +++ b/theories/ereal.v @@ -1127,13 +1127,13 @@ case: x => /= [x [_/posnumP[d] dP] |[d [dreal dP]] |[d [dreal dP]]]; last 2 firs exists N.+1 => // n ltNn; apply: dP. have /le_lt_trans : (d <= Num.max d 0)%R by rewrite le_maxr lexx. apply; apply: lt_le_trans (lt_succ_floor _) _; rewrite Nfloor. - by rewrite -(@natrD R N 1) ler_nat addn1. + by rewrite -natrS ler_nat. have /ZnatP [N Nfloor] : floor (Num.max (- d)%R 0%R) \is a Znat. by rewrite Znat_def floor_ge0 le_maxr lexx orbC. exists N.+1 => // n ltNn; apply: dP; rewrite lte_fin ltr_oppl. have /le_lt_trans : (- d <= Num.max (- d) 0)%R by rewrite le_maxr lexx. apply; apply: lt_le_trans (lt_succ_floor _) _; rewrite Nfloor. - by rewrite -(@natrD R N 1) ler_nat addn1. + by rewrite -natrS ler_nat. have /ZnatP [N Nfloor] : floor (d%:num^-1) \is a Znat. by rewrite Znat_def floor_ge0. exists N => // n leNn; have gt0Sn : (0 < n%:R + 1 :> R)%R. diff --git a/theories/exp.v b/theories/exp.v index 0731429b43..01e0268a89 100644 --- a/theories/exp.v +++ b/theories/exp.v @@ -132,9 +132,9 @@ rewrite mulrBr mulrC divfK //. case: n => [|n]. by rewrite !expr0 !(mul0r, mulr0, subr0, subrr, big_geq). rewrite subrXX addrK -mulrBr; congr (_ * _). -rewrite -(big_mkord xpredT (fun i : nat => (h + z) ^+ (n - i) * z ^+ i)). +rewrite -(big_mkord xpredT (fun i => (h + z) ^+ (n - i) * z ^+ i)). rewrite big_nat_recr //= subnn expr0 -addrA -mulrBl. -rewrite -add1n natrD opprD addrA subrr sub0r mulNr. +rewrite natSr opprD addrA subrr sub0r mulNr. rewrite mulr_natl -[in X in _ *+ X](subn0 n) -sumr_const_nat -sumrB. rewrite pseries_diffs_P1 mulr_sumr !big_mkord; apply: eq_bigr => i _. rewrite mulrCA; congr (_ * _). @@ -394,7 +394,7 @@ Qed. Lemma expRMm n x : expR (n%:R * x) = expR x ^+ n. Proof. elim: n x => [x|n IH x] /=; first by rewrite mul0r expr0 expR0. -by rewrite exprS -add1n natrD mulrDl mul1r expRD IH. +by rewrite exprS natSr mulrDl mul1r expRD IH. Qed. Lemma expR_gt1 x: (1 < expR x) = (0 < x). @@ -606,7 +606,7 @@ Qed. Lemma exp_fun_mulrn a n : 0 < a -> exp_fun a n%:R = a ^+ n. Proof. move=> a0; elim: n => [|n ih]; first by rewrite mulr0n expr0 exp_funr0. -by rewrite -addn1 natrD exp_funD // exprD ih exp_funr1. +by rewrite natrS exprSr exp_funD// ih exp_funr1. Qed. End ExpFun. diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index 224f829307..0d2bb99082 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -1228,7 +1228,7 @@ rewrite predeqE => r; split => [/= /[!in_itv]/= /andP[nr rn1]|]. rewrite ler_pdivr_mulr// (le_trans _ (floor_le _))//. by rewrite -(@gez0_abs (floor _))// floor_ge0 mulr_ge0// (le_trans _ nr). rewrite ltr_pdivl_mulr// (lt_le_trans (lt_succ_floor _))//. - rewrite -[in leRHS]addn1 natrD ler_add2r// -(@gez0_abs (floor _))// floor_ge0. + rewrite [in leRHS]natrS ler_add2r// -(@gez0_abs (floor _))// floor_ge0. by rewrite mulr_ge0// (le_trans _ nr). - rewrite -bigcup_set => -[/= k] /[!mem_index_iota] /andP[nk kn]. rewrite in_itv /= => /andP[knr rkn]; rewrite in_itv /=; apply/andP; split. @@ -1381,7 +1381,7 @@ have xAnK : x \in A n (Ordinal K). rewrite ler_pdivr_mulr// (le_trans _ (floor_le _))//. by rewrite -(@gez0_abs (floor _))// floor_ge0 mulr_ge0// ltW. rewrite ltr_pdivl_mulr// (lt_le_trans (lt_succ_floor _))//. - rewrite -[in leRHS]addn1 natrD ler_add2r// -{1}(@gez0_abs (floor _))//. + rewrite [in leRHS]natrS ler_add2r// -{1}(@gez0_abs (floor _))//. by rewrite floor_ge0// mulr_ge0// ltW. have /[!mem_index_enum]/(_ isT) := An0 (Ordinal K). apply/negP. @@ -1497,7 +1497,7 @@ have : fine (f x) < n%:R. near: n. exists `|floor (fine (f x))|.+1%N => //= p /=. rewrite -(@ler_nat R); apply: lt_le_trans. - rewrite -addn1 natrD (_ : `| _ |%:R = (floor (fine (f x)))%:~R); last first. + rewrite natrS (_ : `| _ |%:R = (floor (fine (f x)))%:~R); last first. by rewrite -[in RHS](@gez0_abs (floor _))// floor_ge0//; exact/fine_ge0/f0. by rewrite lt_succ_floor. rewrite -lte_fin (fineK fxfin) => fxn. @@ -1509,8 +1509,7 @@ rewrite (@le_lt_trans _ _ (1 / 2 ^+ n)) //. rewrite ler_subr_addl -mulrBl -lee_fin (fineK fxfin) -rfx lee_fin. rewrite (le_trans _ k1)// ler_pmul2r// -(@natrB _ _ 1) // ler_nat subn1. by rewrite leq_pred. - rewrite ler_subl_addr -mulrDl -lee_fin -(natrD _ 1) add1n. - by rewrite fineK// ltW// -rfx lte_fin. + by rewrite ler_subl_addr -mulrDl -lee_fin -natSr fineK// ltW// -rfx lte_fin. by near: n; exact: near_infty_natSinv_expn_lt. Unshelve. all: by end_near. Qed. @@ -3114,7 +3113,7 @@ apply/negP; rewrite -ltNge. rewrite -[X in _ * X](@fineK _ (mu (E `&` D))); last first. by rewrite fin_numElt muEDoo andbT (lt_le_trans _ (measure_ge0 _ _)). rewrite lte_fin -ltr_pdivr_mulr. - rewrite -addn1 natrD natr_absz ger0_norm. + rewrite natrS natr_absz ger0_norm. by rewrite (le_lt_trans (ceil_ge _))// ltr_addl. by rewrite ceil_ge0// divr_ge0//; apply/le0R/measure_ge0; exact: measurableI. rewrite -lte_fin fineK. @@ -3442,7 +3441,7 @@ move=> mf; split=> [iDf0|Df0]. - rewrite inE unitfE fine_eq0 // abse_eq0 ft0/=; apply/fine_gt0. by rewrite lt_neqAle abse_ge0 -ge0_fin_numE// eq_sym abse_eq0 ft0. - by rewrite inE unitfE invr_eq0 pnatr_eq0 /= invr_gt0. - rewrite invrK /m -addn1 natrD natr_absz ger0_norm ?ceil_ge0//. + rewrite invrK /m natrS natr_absz ger0_norm ?ceil_ge0//. apply: (@le_trans _ _ ((fine `|f t|)^-1 + 1)%R); first by rewrite ler_addl. by rewrite ler_add2r// ceil_ge. by split => //; apply: contraTN nft => /eqP ->; rewrite abse0 -ltNge. @@ -3463,9 +3462,9 @@ have -> : (fun x => `|f x|) = (fun x => lim (f_^~ x)). rewrite /= (@ger0_norm _ n%:R)// ger0_norm; last first. by rewrite subr_ge0 ler_pdivr_mulr ?mul1r ?ler_addr. rewrite -{1}(@divrr _ (1 + n%:R)%R) ?unitfE; last first. - by rewrite gt_eqF// {1}(_ : 1 = 1%:R)%R // -natrD add1n. + by rewrite gt_eqF// {1}(_ : 1 = 1%:R)%R // natrS. rewrite -mulrBl addrK ltr_pdivr_mulr; last first. - by rewrite {1}(_ : 1 = 1%:R)%R // -natrD add1n. + by rewrite {1}(_ : 1 = 1%:R)%R // natrS. rewrite mulrDr mulr1 ltr_spsaddl// -ltr_pdivr_mull// mulr1. near: n. exists `|ceil (1 + e%:num^-1)|%N => // n /=. diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 666c91fd5e..597d194512 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -562,8 +562,7 @@ move=> x rx; apply/esym/eqP; rewrite eq_le (itvP (rx 0%N _))// andbT. apply/ler_addgt0Pl => e e_gt0; rewrite -ler_subl_addl ltW//. have := rx `|floor e^-1%R|%N I; rewrite /= in_itv => /andP[/le_lt_trans->]//. rewrite ler_add2l ler_opp2 -lef_pinv ?invrK//; last by rewrite qualifE. -rewrite -addn1 natrD natr_absz ger0_norm ?floor_ge0 ?invr_ge0 1?ltW//. -by rewrite lt_succ_floor. +by rewrite natrS natr_absz ger0_norm ?floor_ge0 ?invr_ge0 1?ltW// lt_succ_floor. Qed. Lemma itv_bnd_open_bigcup (R : realType) b (r s : R) : @@ -576,7 +575,7 @@ apply/seteqP; split => [x/=|]; last first. rewrite in_itv/= => /andP[sx xs]; exists `|ceil ((s - x)^-1)|%N => //=. rewrite in_itv/= sx/= ler_subr_addl addrC -ler_subr_addl. rewrite -[in X in _ <= X](invrK (s - x)) ler_pinv. -- rewrite -addn1 natrD natr_absz ger0_norm; last first. +- rewrite natrS natr_absz ger0_norm; last first. by rewrite ceil_ge0// invr_ge0 subr_ge0 ltW. by rewrite (@le_trans _ _ (ceil (s - x)^-1)%:~R)// ?ler_addl// ceil_ge. - by rewrite inE unitfE ltr0n andbT pnatr_eq0. diff --git a/theories/mathcomp_extra.v b/theories/mathcomp_extra.v index 9a16c6f757..0aeb46336b 100644 --- a/theories/mathcomp_extra.v +++ b/theories/mathcomp_extra.v @@ -148,6 +148,12 @@ Proof. by move->. Qed. Lemma eqbRL (b1 b2 : bool) : b1 = b2 -> b2 -> b1. Proof. by move->. Qed. +Lemma natrS (R : ringType) (n : nat) : (n.+1%:R = n%:R + 1 :> R)%R. +Proof. by rewrite -addn1 GRing.natrD. Qed. + +Lemma natSr (R : ringType) (n : nat) : (n.+1%:R = 1 + n%:R :> R)%R. +Proof. by rewrite -add1n GRing.natrD. Qed. + (***************************) (* MathComp 1.15 additions *) (***************************) diff --git a/theories/reals.v b/theories/reals.v index c301b5a46a..ca0b9c57bf 100644 --- a/theories/reals.v +++ b/theories/reals.v @@ -726,7 +726,7 @@ Proof. move=> yx; exists `|floor (x - y)^-1|%N. rewrite -ltr_subr_addl -{2}(invrK (x - y)%R) ltf_pinv ?qualifE ?ltr0n//. by rewrite invr_gt0 subr_gt0. -rewrite -addn1 natrD natr_absz ger0_norm. +rewrite natrS natr_absz ger0_norm. by rewrite floor_ge0 invr_ge0 subr_ge0 ltW. by rewrite -RfloorE lt_succ_Rfloor. Qed. diff --git a/theories/sequences.v b/theories/sequences.v index 61275f94f5..00e2197b78 100644 --- a/theories/sequences.v +++ b/theories/sequences.v @@ -1204,7 +1204,7 @@ have /uoo[N _ NuA] : \oo [set m | `|ceil A|.+1 <= m]%N by exists `|ceil A|.+1. near=> n; have /NuA : (N <= n)%N by near: n; exact: nbhs_infty_ge. rewrite /= -(ler_nat R); apply: le_trans. have [A0|A0] := leP 0%R A; last by rewrite (le_trans (ltW A0)). -by rewrite -addn1 natrD natr_absz ger0_norm// ?ceil_ge0// ler_paddr// ceil_ge. +by rewrite natrS natr_absz ger0_norm// ?ceil_ge0// ler_paddr// ceil_ge. Unshelve. all: by end_near. Qed. Lemma nat_cvgPpinfty (u : nat^nat) : diff --git a/theories/set_interval.v b/theories/set_interval.v index dcd8df8d9a..16d5d7a164 100644 --- a/theories/set_interval.v +++ b/theories/set_interval.v @@ -413,7 +413,7 @@ Proof. rewrite predeqE => y; split=> /=; last first. by move=> [n _]/=; rewrite in_itv => /andP[xy yn]; rewrite in_itv /= xy. rewrite in_itv /= andbT => xy; exists (`|floor y|%N.+1) => //=. -rewrite in_itv /= xy /= -addn1 natrD. +rewrite in_itv /= xy /= natrS. have [y0|y0] := ltP 0 y; last by rewrite (le_lt_trans y0)// ltr_spaddr. by rewrite natr_absz ger0_norm ?lt_succ_floor// floor_ge0 ltW. Qed. From ce7f6b2e6916579f9986e679c97cb0a6c0434f19 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 5 Sep 2022 09:01:19 +0900 Subject: [PATCH 18/42] fix --- theories/mathcomp_extra.v | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/theories/mathcomp_extra.v b/theories/mathcomp_extra.v index 0aeb46336b..567d181c14 100644 --- a/theories/mathcomp_extra.v +++ b/theories/mathcomp_extra.v @@ -149,10 +149,10 @@ Lemma eqbRL (b1 b2 : bool) : b1 = b2 -> b2 -> b1. Proof. by move->. Qed. Lemma natrS (R : ringType) (n : nat) : (n.+1%:R = n%:R + 1 :> R)%R. -Proof. by rewrite -addn1 GRing.natrD. Qed. +Proof. by rewrite GRing.mulrSr. Qed. Lemma natSr (R : ringType) (n : nat) : (n.+1%:R = 1 + n%:R :> R)%R. -Proof. by rewrite -add1n GRing.natrD. Qed. +Proof. by rewrite GRing.mulrS. Qed. (***************************) (* MathComp 1.15 additions *) From 7127fc77a0bdebf86119b7c34bf5d0f6e4445182 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Tue, 6 Sep 2022 00:36:47 +0900 Subject: [PATCH 19/42] natrS -> natr1 --- CHANGELOG_UNRELEASED.md | 2 +- theories/ereal.v | 12 ++++-------- theories/exp.v | 6 +++--- theories/lebesgue_integral.v | 12 ++++++------ theories/lebesgue_measure.v | 4 ++-- theories/mathcomp_extra.v | 4 ++-- theories/reals.v | 2 +- theories/sequences.v | 2 +- theories/set_interval.v | 2 +- 9 files changed, 21 insertions(+), 25 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index e82105c4f6..1e13dea62b 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -54,7 +54,7 @@ - in `topology.v`: + lemmas `open_setIS`, `open_setSI`, `closed_setIS`, `closed_setSI` - in `mathcomp_extra.v`: - + lemmas `natrS`, `natSr` + + lemmas `natr1`, `nat1r` ### Changed diff --git a/theories/ereal.v b/theories/ereal.v index 72617f6785..8aec800004 100644 --- a/theories/ereal.v +++ b/theories/ereal.v @@ -1126,20 +1126,16 @@ case: x => /= [x [_/posnumP[d] dP] |[d [dreal dP]] |[d [dreal dP]]]; last 2 firs by rewrite Znat_def floor_ge0 le_maxr lexx orbC. exists N.+1 => // n ltNn; apply: dP. have /le_lt_trans : (d <= Num.max d 0)%R by rewrite le_maxr lexx. - apply; apply: lt_le_trans (lt_succ_floor _) _; rewrite Nfloor. - by rewrite -natrS ler_nat. + by apply; rewrite (lt_le_trans (lt_succ_floor _))// Nfloor natr1 ler_nat. have /ZnatP [N Nfloor] : floor (Num.max (- d)%R 0%R) \is a Znat. by rewrite Znat_def floor_ge0 le_maxr lexx orbC. exists N.+1 => // n ltNn; apply: dP; rewrite lte_fin ltr_oppl. have /le_lt_trans : (- d <= Num.max (- d) 0)%R by rewrite le_maxr lexx. - apply; apply: lt_le_trans (lt_succ_floor _) _; rewrite Nfloor. - by rewrite -natrS ler_nat. + by apply; rewrite (lt_le_trans (lt_succ_floor _))// Nfloor natr1 ler_nat. have /ZnatP [N Nfloor] : floor (d%:num^-1) \is a Znat. by rewrite Znat_def floor_ge0. -exists N => // n leNn; have gt0Sn : (0 < n%:R + 1 :> R)%R. - by apply: ltr_spaddr => //; exact/ler0n. -apply: dP; last first. - by rewrite eq_sym addrC -subr_eq subrr eq_sym; apply/invr_neq0/lt0r_neq0. +exists N => // n leNn; apply: dP; last first. + by rewrite eq_sym addrC -subr_eq subrr eq_sym; exact/invr_neq0/lt0r_neq0. rewrite /= opprD addrA subrr distrC subr0 gtr0_norm; last by rewrite invr_gt0. rewrite -[ltLHS]mulr1 ltr_pdivr_mull // -ltr_pdivr_mulr // div1r. by rewrite (lt_le_trans (lt_succ_floor _))// Nfloor ler_add// ler_nat. diff --git a/theories/exp.v b/theories/exp.v index 01e0268a89..e99f0a64e8 100644 --- a/theories/exp.v +++ b/theories/exp.v @@ -134,7 +134,7 @@ case: n => [|n]. rewrite subrXX addrK -mulrBr; congr (_ * _). rewrite -(big_mkord xpredT (fun i => (h + z) ^+ (n - i) * z ^+ i)). rewrite big_nat_recr //= subnn expr0 -addrA -mulrBl. -rewrite natSr opprD addrA subrr sub0r mulNr. +rewrite -nat1r opprD addrA subrr sub0r mulNr. rewrite mulr_natl -[in X in _ *+ X](subn0 n) -sumr_const_nat -sumrB. rewrite pseries_diffs_P1 mulr_sumr !big_mkord; apply: eq_bigr => i _. rewrite mulrCA; congr (_ * _). @@ -394,7 +394,7 @@ Qed. Lemma expRMm n x : expR (n%:R * x) = expR x ^+ n. Proof. elim: n x => [x|n IH x] /=; first by rewrite mul0r expr0 expR0. -by rewrite exprS natSr mulrDl mul1r expRD IH. +by rewrite exprS -nat1r mulrDl mul1r expRD IH. Qed. Lemma expR_gt1 x: (1 < expR x) = (0 < x). @@ -606,7 +606,7 @@ Qed. Lemma exp_fun_mulrn a n : 0 < a -> exp_fun a n%:R = a ^+ n. Proof. move=> a0; elim: n => [|n ih]; first by rewrite mulr0n expr0 exp_funr0. -by rewrite natrS exprSr exp_funD// ih exp_funr1. +by rewrite -natr1 exprSr exp_funD// ih exp_funr1. Qed. End ExpFun. diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index 0d2bb99082..b40449688d 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -1228,7 +1228,7 @@ rewrite predeqE => r; split => [/= /[!in_itv]/= /andP[nr rn1]|]. rewrite ler_pdivr_mulr// (le_trans _ (floor_le _))//. by rewrite -(@gez0_abs (floor _))// floor_ge0 mulr_ge0// (le_trans _ nr). rewrite ltr_pdivl_mulr// (lt_le_trans (lt_succ_floor _))//. - rewrite [in leRHS]natrS ler_add2r// -(@gez0_abs (floor _))// floor_ge0. + rewrite -[in leRHS]natr1 ler_add2r// -(@gez0_abs (floor _))// floor_ge0. by rewrite mulr_ge0// (le_trans _ nr). - rewrite -bigcup_set => -[/= k] /[!mem_index_iota] /andP[nk kn]. rewrite in_itv /= => /andP[knr rkn]; rewrite in_itv /=; apply/andP; split. @@ -1381,7 +1381,7 @@ have xAnK : x \in A n (Ordinal K). rewrite ler_pdivr_mulr// (le_trans _ (floor_le _))//. by rewrite -(@gez0_abs (floor _))// floor_ge0 mulr_ge0// ltW. rewrite ltr_pdivl_mulr// (lt_le_trans (lt_succ_floor _))//. - rewrite [in leRHS]natrS ler_add2r// -{1}(@gez0_abs (floor _))//. + rewrite -[in leRHS]natr1 ler_add2r// -{1}(@gez0_abs (floor _))//. by rewrite floor_ge0// mulr_ge0// ltW. have /[!mem_index_enum]/(_ isT) := An0 (Ordinal K). apply/negP. @@ -1497,7 +1497,7 @@ have : fine (f x) < n%:R. near: n. exists `|floor (fine (f x))|.+1%N => //= p /=. rewrite -(@ler_nat R); apply: lt_le_trans. - rewrite natrS (_ : `| _ |%:R = (floor (fine (f x)))%:~R); last first. + rewrite -natr1 (_ : `| _ |%:R = (floor (fine (f x)))%:~R); last first. by rewrite -[in RHS](@gez0_abs (floor _))// floor_ge0//; exact/fine_ge0/f0. by rewrite lt_succ_floor. rewrite -lte_fin (fineK fxfin) => fxn. @@ -1509,7 +1509,7 @@ rewrite (@le_lt_trans _ _ (1 / 2 ^+ n)) //. rewrite ler_subr_addl -mulrBl -lee_fin (fineK fxfin) -rfx lee_fin. rewrite (le_trans _ k1)// ler_pmul2r// -(@natrB _ _ 1) // ler_nat subn1. by rewrite leq_pred. - by rewrite ler_subl_addr -mulrDl -lee_fin -natSr fineK// ltW// -rfx lte_fin. + by rewrite ler_subl_addr -mulrDl -lee_fin nat1r fineK// ltW// -rfx lte_fin. by near: n; exact: near_infty_natSinv_expn_lt. Unshelve. all: by end_near. Qed. @@ -3113,7 +3113,7 @@ apply/negP; rewrite -ltNge. rewrite -[X in _ * X](@fineK _ (mu (E `&` D))); last first. by rewrite fin_numElt muEDoo andbT (lt_le_trans _ (measure_ge0 _ _)). rewrite lte_fin -ltr_pdivr_mulr. - rewrite natrS natr_absz ger0_norm. + rewrite -natr1 natr_absz ger0_norm. by rewrite (le_lt_trans (ceil_ge _))// ltr_addl. by rewrite ceil_ge0// divr_ge0//; apply/le0R/measure_ge0; exact: measurableI. rewrite -lte_fin fineK. @@ -3441,7 +3441,7 @@ move=> mf; split=> [iDf0|Df0]. - rewrite inE unitfE fine_eq0 // abse_eq0 ft0/=; apply/fine_gt0. by rewrite lt_neqAle abse_ge0 -ge0_fin_numE// eq_sym abse_eq0 ft0. - by rewrite inE unitfE invr_eq0 pnatr_eq0 /= invr_gt0. - rewrite invrK /m natrS natr_absz ger0_norm ?ceil_ge0//. + rewrite invrK /m -natr1 natr_absz ger0_norm ?ceil_ge0//. apply: (@le_trans _ _ ((fine `|f t|)^-1 + 1)%R); first by rewrite ler_addl. by rewrite ler_add2r// ceil_ge. by split => //; apply: contraTN nft => /eqP ->; rewrite abse0 -ltNge. diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 597d194512..6c5421d8a2 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -562,7 +562,7 @@ move=> x rx; apply/esym/eqP; rewrite eq_le (itvP (rx 0%N _))// andbT. apply/ler_addgt0Pl => e e_gt0; rewrite -ler_subl_addl ltW//. have := rx `|floor e^-1%R|%N I; rewrite /= in_itv => /andP[/le_lt_trans->]//. rewrite ler_add2l ler_opp2 -lef_pinv ?invrK//; last by rewrite qualifE. -by rewrite natrS natr_absz ger0_norm ?floor_ge0 ?invr_ge0 1?ltW// lt_succ_floor. +by rewrite -natr1 natr_absz ger0_norm ?floor_ge0 ?invr_ge0 1?ltW// lt_succ_floor. Qed. Lemma itv_bnd_open_bigcup (R : realType) b (r s : R) : @@ -575,7 +575,7 @@ apply/seteqP; split => [x/=|]; last first. rewrite in_itv/= => /andP[sx xs]; exists `|ceil ((s - x)^-1)|%N => //=. rewrite in_itv/= sx/= ler_subr_addl addrC -ler_subr_addl. rewrite -[in X in _ <= X](invrK (s - x)) ler_pinv. -- rewrite natrS natr_absz ger0_norm; last first. +- rewrite -natr1 natr_absz ger0_norm; last first. by rewrite ceil_ge0// invr_ge0 subr_ge0 ltW. by rewrite (@le_trans _ _ (ceil (s - x)^-1)%:~R)// ?ler_addl// ceil_ge. - by rewrite inE unitfE ltr0n andbT pnatr_eq0. diff --git a/theories/mathcomp_extra.v b/theories/mathcomp_extra.v index 567d181c14..47bf46425f 100644 --- a/theories/mathcomp_extra.v +++ b/theories/mathcomp_extra.v @@ -148,10 +148,10 @@ Proof. by move->. Qed. Lemma eqbRL (b1 b2 : bool) : b1 = b2 -> b2 -> b1. Proof. by move->. Qed. -Lemma natrS (R : ringType) (n : nat) : (n.+1%:R = n%:R + 1 :> R)%R. +Lemma natr1 (R : ringType) (n : nat) : (n%:R + 1 = n.+1%:R :> R)%R. Proof. by rewrite GRing.mulrSr. Qed. -Lemma natSr (R : ringType) (n : nat) : (n.+1%:R = 1 + n%:R :> R)%R. +Lemma nat1r (R : ringType) (n : nat) : (1 + n%:R = n.+1%:R :> R)%R. Proof. by rewrite GRing.mulrS. Qed. (***************************) diff --git a/theories/reals.v b/theories/reals.v index ca0b9c57bf..0ff1b78eaa 100644 --- a/theories/reals.v +++ b/theories/reals.v @@ -726,7 +726,7 @@ Proof. move=> yx; exists `|floor (x - y)^-1|%N. rewrite -ltr_subr_addl -{2}(invrK (x - y)%R) ltf_pinv ?qualifE ?ltr0n//. by rewrite invr_gt0 subr_gt0. -rewrite natrS natr_absz ger0_norm. +rewrite -natr1 natr_absz ger0_norm. by rewrite floor_ge0 invr_ge0 subr_ge0 ltW. by rewrite -RfloorE lt_succ_Rfloor. Qed. diff --git a/theories/sequences.v b/theories/sequences.v index 00e2197b78..5f76422e15 100644 --- a/theories/sequences.v +++ b/theories/sequences.v @@ -1204,7 +1204,7 @@ have /uoo[N _ NuA] : \oo [set m | `|ceil A|.+1 <= m]%N by exists `|ceil A|.+1. near=> n; have /NuA : (N <= n)%N by near: n; exact: nbhs_infty_ge. rewrite /= -(ler_nat R); apply: le_trans. have [A0|A0] := leP 0%R A; last by rewrite (le_trans (ltW A0)). -by rewrite natrS natr_absz ger0_norm// ?ceil_ge0// ler_paddr// ceil_ge. +by rewrite -natr1 natr_absz ger0_norm// ?ceil_ge0// ler_paddr// ceil_ge. Unshelve. all: by end_near. Qed. Lemma nat_cvgPpinfty (u : nat^nat) : diff --git a/theories/set_interval.v b/theories/set_interval.v index 16d5d7a164..558a3b143e 100644 --- a/theories/set_interval.v +++ b/theories/set_interval.v @@ -413,7 +413,7 @@ Proof. rewrite predeqE => y; split=> /=; last first. by move=> [n _]/=; rewrite in_itv => /andP[xy yn]; rewrite in_itv /= xy. rewrite in_itv /= andbT => xy; exists (`|floor y|%N.+1) => //=. -rewrite in_itv /= xy /= natrS. +rewrite in_itv /= xy /= -natr1. have [y0|y0] := ltP 0 y; last by rewrite (le_lt_trans y0)// ltr_spaddr. by rewrite natr_absz ger0_norm ?lt_succ_floor// floor_ge0 ltW. Qed. From 5e8b1a896435d8f7a4942edb392e9cfe61bceafa Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 5 Sep 2022 09:10:00 +0900 Subject: [PATCH 20/42] deprecated warnings for Rstruct's bigmax --- theories/Rstruct.v | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) diff --git a/theories/Rstruct.v b/theories/Rstruct.v index 0a7d9c6a01..18941ddfe8 100644 --- a/theories/Rstruct.v +++ b/theories/Rstruct.v @@ -503,15 +503,19 @@ Section bigmaxr. Context {R : realDomainType}. (* bigop pour le max pour des listes non vides ? *) +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Definition bigmaxr (r : R) s := \big[Num.max/head r s]_(i <- s) i. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_nil (x0 : R) : bigmaxr x0 [::] = x0. Proof. by rewrite /bigmaxr /= big_nil. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_un (x0 x : R) : bigmaxr x0 [:: x] = x. Proof. by rewrite /bigmaxr /= big_cons big_nil maxxx. Qed. (* previous definition *) +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxrE (r : R) s : bigmaxr r s = foldr Num.max (head r s) (behead s). Proof. rewrite (_ : bigmaxr _ _ = if s isn't h :: t then r else \big[Num.max/h]_(i <- s) i). @@ -520,6 +524,7 @@ rewrite (_ : bigmaxr _ _ = if s isn't h :: t then r else \big[Num.max/h]_(i <- s by case: s => //=; rewrite /bigmaxr big_nil. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigrmax_dflt (x y : R) s : Num.max x (\big[Num.max/x]_(j <- y :: s) j) = Num.max x (\big[Num.max/y]_(i <- y :: s) i). Proof. @@ -528,10 +533,12 @@ by rewrite !big_cons !big_nil maxxx maxCA maxxx maxC. by rewrite big_cons maxCA IH maxCA [in RHS]big_cons IH. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_cons (x0 x y : R) lr : bigmaxr x0 (x :: y :: lr) = Num.max x (bigmaxr x0 (y :: lr)). Proof. by rewrite [y :: lr]lock /bigmaxr /= -lock big_cons bigrmax_dflt. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_ler (x0 : R) s i : (i < size s)%N -> (nth x0 s i) <= (bigmaxr x0 s). Proof. @@ -542,6 +549,7 @@ by rewrite big_cons bigrmax_dflt le_maxr orbC IH. Qed. (* Compatibilité avec l'addition *) +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_addr (x0 : R) lr (x : R) : bigmaxr (x0 + x) (map (fun y : R => y + x) lr) = (bigmaxr x0 lr) + x. Proof. @@ -550,6 +558,7 @@ elim: t h => /= [|h' t IH] h; first by rewrite ?(big_cons,big_nil) -addr_maxl. by rewrite [in RHS]big_cons bigrmax_dflt addr_maxl -IH big_cons bigrmax_dflt. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_mem (x0 : R) lr : (0 < size lr)%N -> bigmaxr x0 lr \in lr. Proof. rewrite /bigmaxr; case: lr => // h t _. @@ -560,6 +569,7 @@ rewrite big_cons bigrmax_dflt inE eq_le; case: lerP => /=. Qed. (* TODO: bigmaxr_morph? *) +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_mulr (A : finType) (s : seq A) (k : R) (x : A -> R) : 0 <= k -> bigmaxr 0 (map (fun i => k * x i) s) = k * bigmaxr 0 (map x s). Proof. @@ -569,6 +579,7 @@ by rewrite !bigmaxr_un. by rewrite bigmaxr_cons {}ih bigmaxr_cons maxr_pmulr. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_index (x0 : R) lr : (0 < size lr)%N -> (index (bigmaxr x0 lr) lr < size lr)%N. Proof. @@ -577,6 +588,7 @@ move: (@bigmaxr_mem x0 (h :: t) isT). by rewrite ltnS index_mem inE /= eq_sym H. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_lerP (x0 : R) lr (x : R) : (0 < size lr)%N -> reflect (forall i, (i < size lr)%N -> (nth x0 lr i) <= x) ((bigmaxr x0 lr) <= x). @@ -586,6 +598,7 @@ move=> lr_size; apply: (iffP idP) => [le_x i i_size | H]. by move/(nthP x0): (bigmaxr_mem x0 lr_size) => [i i_size <-]; apply: H. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_ltrP (x0 : R) lr (x : R) : (0 < size lr)%N -> reflect (forall i, (i < size lr)%N -> (nth x0 lr i) < x) ((bigmaxr x0 lr) < x). @@ -595,6 +608,7 @@ move=> lr_size; apply: (iffP idP) => [lt_x i i_size | H]. by move/(nthP x0): (bigmaxr_mem x0 lr_size) => [i i_size <-]; apply: H. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxrP (x0 : R) lr (x : R) : (x \in lr /\ forall i, (i < size lr) %N -> (nth x0 lr i) <= x) -> (bigmaxr x0 lr = x). Proof. @@ -614,6 +628,7 @@ apply/negP => /eqP H; apply: neq_i; rewrite -H eq_sym; apply/eqP. by apply: index_uniq. Qed. *) +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bigmaxr_lerif (x0 : R) lr : uniq lr -> forall i, (i < size lr)%N -> (nth x0 lr i) <= (bigmaxr x0 lr) ?= iff (i == index (bigmaxr x0 lr) lr). @@ -625,9 +640,11 @@ by apply: bigmaxr_mem; apply: (leq_trans _ i_size). Qed. (* bigop pour le max pour des listes non vides ? *) +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Definition bmaxrf n (f : {ffun 'I_n.+1 -> R}) := bigmaxr (f ord0) (codom f). +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bmaxrf_ler n (f : {ffun 'I_n.+1 -> R}) i : (f i) <= (bmaxrf f). Proof. @@ -637,6 +654,7 @@ suff -> : nth (f ord0) (codom f) i = f i; first by []. by rewrite /codom (nth_map ord0) ?size_enum_ord // nth_ord_enum. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bmaxrf_index n (f : {ffun 'I_n.+1 -> R}) : (index (bmaxrf f) (codom f) < n.+1)%N. Proof. @@ -646,11 +664,14 @@ rewrite [in X in (_ < X)%N](_ : n.+1 = size (codom f)); last first. by apply: bigmaxr_index; rewrite size_codom card_ord. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Definition index_bmaxrf n f := Ordinal (@bmaxrf_index n f). +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma ordnat i n (ord_i : (i < n)%N) : i = Ordinal ord_i :> nat. Proof. by []. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma eq_index_bmaxrf n (f : {ffun 'I_n.+1 -> R}) : f (index_bmaxrf f) = bmaxrf f. Proof. @@ -661,6 +682,7 @@ move/(nth_index (f ord0)) => <-; rewrite (nth_map ord0). by rewrite size_enum_ord; apply: bmaxrf_index. Qed. +#[deprecated(note="To be removed. Use topology.v's bigmax/min lemmas instead.")] Lemma bmaxrf_lerif n (f : {ffun 'I_n.+1 -> R}) : injective f -> forall i, (f i) <= (bmaxrf f) ?= iff (i == index_bmaxrf f). From d2ef896b1120fc2ea8e0f1c7cfaa7a8df34ce8d8 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 5 Sep 2022 22:46:42 +0900 Subject: [PATCH 21/42] changelog for version 0.5.4 --- CHANGELOG.md | 92 +++++++++++++++++++++++++++++++++++++- CHANGELOG_UNRELEASED.md | 91 ------------------------------------- INSTALL.md | 4 +- README.md | 2 +- coq-mathcomp-analysis.opam | 2 +- meta.yml | 4 +- 6 files changed, 97 insertions(+), 98 deletions(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 4a5b57ca10..62d5ebc18f 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -1,6 +1,96 @@ # Changelog -Lastest releases: [[0.5.3] - 2022-08-10](#053---2022-08-10) and [[0.5.2] - 2022-06-08](#052---2022-07-08) +Lastest releases: [[0.5.4] - 2022-09-07](#055---2022-09-07) and [[0.5.3] - 2022-08-10](#053---2022-08-10) + +## [0.5.4] - 2022-09-07 + +### Added + +- in `mathcomp_extra.v`: + + defintion `onem` and notation ``` `1- ``` + + lemmas `onem0`, `onem1`, `onemK`, `onem_gt0`, `onem_ge0`, `onem_le1`, `onem_lt1`, + `onemX_ge0`, `onemX_lt1`, `onemD`, `onemMr`, `onemM` + + lemmas `natr1`, `nat1r` +- in `classical_sets.v`: + + lemmas `subset_fst_set`, `subset_snd_set`, `fst_set_fst`, `snd_set_snd`, + `fset_setM`, `snd_setM`, `fst_setMR` + + lemmas `xsection_snd_set`, `ysection_fst_set` +- in `constructive_ereal.v`: + + lemmas `ltNye_eq`, `sube_lt0`, `subre_lt0`, `suber_lt0`, `sube_ge0` + + lemmas `dsubre_gt0`, `dsuber_gt0`, `dsube_gt0`, `dsube_le0` +- in `signed.v`: + + lemmas `onem_PosNum`, `onemX_NngNum` +- in `fsbigop.v`: + + lemmas `lee_fsum_nneg_subset`, `lee_fsum` +- in `topology.v`: + + lemma `near_inftyS` + + lemma `continuous_closedP`, `closedU`, `pasting` + + lemma `continuous_inP` + + lemmas `open_setIS`, `open_setSI`, `closed_setIS`, `closed_setSI` +- in `normedtype.v`: + + definitions `contraction` and `is_contraction` + + lemma `contraction_fixpoint_unique` +- in `sequences.v`: + + lemmas `contraction_dist`, `contraction_cvg`, + `contraction_cvg_fixed`, `banach_fixed_point`, + `contraction_unique` +- in `derive.v`: + + lemma `diff_derivable` +- in `measure.v`: + + lemma `measurable_funTS` +- in `lebesgue_measure.v`: + + lemma `measurable_fun_fine` + + lemma `open_measurable_subspace` + + lemma ``subspace_continuous_measurable_fun`` + + corollary `open_continuous_measurable_fun` + + Hint about `measurable_fun_normr` +- in `lebesgue_integral.v`: + + lemma `measurable_fun_indic` + + lemma `ge0_integral_mscale` + + lemma `integral_pushforward` + +### Changed + +- in `esum.v`: + + definition `esum` +- moved from `lebesgue_integral.v` to `classical_sets.v`: + + `mem_set_pair1` -> `mem_xsection` + + `mem_set_pair2` -> `mem_ysection` +- in `derive.v`: + + generalized `is_diff_scalel` +- in `measure.v`: + + generalize `measurable_uncurry` +- in `lebesgue_measure.v`: + + `pushforward` requires a proof that its argument is measurable +- in `lebesgue_integral.v`: + + change implicits of `integralM_indic` + +### Renamed + +- in `constructive_ereal.v`: + + `lte_pinfty_eq` -> `ltey_eq` + + `le0R` -> `fine_ge0` + + `lt0R` -> `fine_gt0` +- in `ereal.v`: + + `lee_pinfty_eq` -> `leye_eq` + + `lee_ninfty_eq` -> `leeNy_eq` +- in `esum.v`: + + `esum0` -> `esum1` +- in `sequences.v`: + + `nneseries_lim_ge0` -> `nneseries_ge0` +- in `measure.v`: + + `ring_fsets` -> `ring_finite_set` + + `discrete_measurable` -> `discrete_measurable_nat` + + `cvg_mu_inc` -> `nondecreasing_cvg_mu` +- in `lebesgue_integral.v`: + + `muleindic_ge0` -> `nnfun_muleindic_ge0` + + `mulem_ge0` -> `mulemu_ge0` + + `nnfun_mulem_ge0` -> `nnsfun_mulemu_ge0` + +### Removed + +- in `esum.v`: + + lemma `fsetsP`, `sum_fset_set` ## [0.5.3] - 2022-08-10 diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index 1e13dea62b..c1ee5bdc82 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -3,104 +3,13 @@ ## [Unreleased] ### Added -- in `normedtype.v`: - + definitions `contraction` and `is_contraction` - + lemma `contraction_fixpoint_unique` - -- in `constructive_ereal.v`: - + lemmas `ltNye_eq`, `sube_lt0`, `subre_lt0`, `suber_lt0`, `sube_ge0` - + lemmas `dsubre_gt0`, `dsuber_gt0`, `dsube_gt0`, `dsube_le0` - -- in `topology.v`: - + lemma `near_inftyS` - + lemma `continuous_closedP`, `closedU`, `pasting` - -- in `sequences.v`: - + lemmas `contraction_dist`, `contraction_cvg`, - `contraction_cvg_fixed`, `banach_fixed_point`, - `contraction_unique` -- in `mathcomp_extra.v`: - + defintion `onem` and notation ``` `1- ``` - + lemmas `onem0`, `onem1`, `onemK`, `onem_gt0`, `onem_ge0`, `onem_le1`, `onem_lt1`, - `onemX_ge0`, `onemX_lt1`, `onemD`, `onemMr`, `onemM` -- in `signed.v`: - + lemmas `onem_PosNum`, `onemX_NngNum` -- in `lebesgue_measure.v`: - + lemma `measurable_fun_fine` -- in `lebesgue_integral.v`: - + lemma `ge0_integral_mscale` -- in `measure.v`: - + lemma `measurable_funTS` -- in `lebesgue_measure.v`: - + lemma `measurable_fun_indic` -- in `fsbigop.v`: - + lemmas `lee_fsum_nneg_subset`, `lee_fsum` -- in `classical_sets.v`: - + lemmas `subset_fst_set`, `subset_snd_set`, `fst_set_fst`, `snd_set_snd`, - `fset_setM`, `snd_setM`, `fst_setMR` - + lemmas `xsection_snd_set`, `ysection_fst_set` - + Hint about `measurable_fun_normr` -- in `lebesgue_integral.v`: - + lemma `integral_pushforward` - -- in `derive.v`: - + lemma `diff_derivable` -- in `topology.v`: - + lemma `continuous_inP` -- in `lebesgue_measure.v`: - + lemma `open_measurable_subspace` - + lemma ``subspace_continuous_measurable_fun`` - + corollary `open_continuous_measurable_fun` -- in `topology.v`: - + lemmas `open_setIS`, `open_setSI`, `closed_setIS`, `closed_setSI` -- in `mathcomp_extra.v`: - + lemmas `natr1`, `nat1r` ### Changed -- in `measure.v`: - + generalize `measurable_uncurry` -- in `esum.v`: - + definition `esum` -- moved from `lebesgue_integral.v` to `classical_sets.v`: - + `mem_set_pair1` -> `mem_xsection` - + `mem_set_pair2` -> `mem_ysection` -- in `lebesgue_measure.v`: - + `pushforward` requires a proof that its argument is measurable -- in `lebesgue_integral.v`: - + change implicits of `integralM_indic` -- in `derive.v`: - + generalized `is_diff_scalel` - ### Renamed -- in `constructive_ereal.v`: - + `lte_pinfty_eq` -> `ltey_eq` -- in `sequences.v`: - + `nneseries_lim_ge0` -> `nneseries_ge0` -- in `constructive_ereal.v`: - + `le0R` -> `fine_ge0` - + `lt0R` -> `fine_gt0` -- in `measure.v`: - + `ring_fsets` -> `ring_finite_set` - + `discrete_measurable` -> `discrete_measurable_nat` -- in `ereal.v`: - + `lee_pinfty_eq` -> `leye_eq` - + `lee_ninfty_eq` -> `leeNy_eq` -- in `measure.v`: - + `cvg_mu_inc` -> `nondecreasing_cvg_mu` -- in `lebesgue_integral.v`: - + `muleindic_ge0` -> `nnfun_muleindic_ge0` - + `mulem_ge0` -> `mulemu_ge0` - + `nnfun_mulem_ge0` -> `nnsfun_mulemu_ge0` -- in `esum.v`: - + `esum0` -> `esum1` - ### Removed -- in `esum.v`: - + lemma `fsetsP`, `sum_fset_set` - ### Infrastructure ### Misc diff --git a/INSTALL.md b/INSTALL.md index 5b0c85aac5..19435cf871 100644 --- a/INSTALL.md +++ b/INSTALL.md @@ -5,7 +5,7 @@ - [The Coq Proof Assistant version ≥ 8.13](https://coq.inria.fr) - [Mathematical Components version ≥ 1.13.0](https://github.com/math-comp/math-comp) - [Finmap library version ≥ 1.5.1](https://github.com/math-comp/finmap) -- [Hierarchy builder version >= 1.2.0](https://github.com/math-comp/hierarchy-builder) +- [Hierarchy builder version >= 1.3.0](https://github.com/math-comp/hierarchy-builder) These requirements can be installed in a custom way, or through [opam](https://opam.ocaml.org/) (the recommended way) using @@ -47,7 +47,7 @@ $ opam install coq-mathcomp-analysis ``` To install a precise version, type, say ``` -$ opam install coq-mathcomp-analysis.0.5.3 +$ opam install coq-mathcomp-analysis.0.5.4 ``` 4. Everytime you want to work in this same context, you need to type ``` diff --git a/README.md b/README.md index 723f921bcd..c7748564f9 100644 --- a/README.md +++ b/README.md @@ -33,7 +33,7 @@ the Coq proof-assistant and using the Mathematical Components library. - Pierre-Yves Strub (initial) - Laurent Théry - License: [CeCILL-C](LICENSE) -- Compatible Coq versions: Coq 8.14 to 8.15 (or dev) +- Compatible Coq versions: Coq 8.14 to 8.16 (or dev) - Additional dependencies: - [MathComp ssreflect 1.13 or later](https://math-comp.github.io) - [MathComp fingroup 1.13 or later](https://math-comp.github.io) diff --git a/coq-mathcomp-analysis.opam b/coq-mathcomp-analysis.opam index dacb81fa12..3e8ca66e90 100644 --- a/coq-mathcomp-analysis.opam +++ b/coq-mathcomp-analysis.opam @@ -18,7 +18,7 @@ the Coq proof-assistant and using the Mathematical Components library.""" build: [make "-j%{jobs}%"] install: [make "install"] depends: [ - "coq" { (>= "8.14" & < "8.16~") | (= "dev") } + "coq" { (>= "8.14" & < "8.17~") | (= "dev") } "coq-mathcomp-ssreflect" { (>= "1.13.0" & < "1.16~") | (= "dev") } "coq-mathcomp-fingroup" { (>= "1.13.0" & < "1.16~") | (= "dev") } "coq-mathcomp-algebra" { (>= "1.13.0" & < "1.16~") | (= "dev") } diff --git a/meta.yml b/meta.yml index 9690e63d3d..9223c20791 100644 --- a/meta.yml +++ b/meta.yml @@ -49,8 +49,8 @@ license: file: LICENSE supported_coq_versions: - text: Coq 8.14 to 8.15 (or dev) - opam: '{ (>= "8.14" & < "8.16~") | (= "dev") }' + text: Coq 8.14 to 8.16 (or dev) + opam: '{ (>= "8.14" & < "8.17~") | (= "dev") }' tested_coq_opam_versions: - version: '1.13.0-coq-8.14' From f9a37af9405ebd6bd610582ed3e2f067059a419d Mon Sep 17 00:00:00 2001 From: zstone1 Date: Wed, 14 Sep 2022 22:57:57 -0400 Subject: [PATCH 22/42] Subspace topology restrictions (#739) * changing susbapce names * updating changelog * linting * changing names * fixing names * update changelog * changelog fix * fix within-continuous notation - various minor shortenings * name improvements Co-authored-by: Reynald Affeldt --- CHANGELOG_UNRELEASED.md | 5 +++++ theories/derive.v | 2 +- theories/exp.v | 2 +- theories/realfun.v | 8 ++++---- theories/topology.v | 34 +++++++++++++++++++++++-------- theories/trigo.v | 44 +++++++++++++++++++---------------------- 6 files changed, 57 insertions(+), 38 deletions(-) diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md index c1ee5bdc82..a3ee55697f 100644 --- a/CHANGELOG_UNRELEASED.md +++ b/CHANGELOG_UNRELEASED.md @@ -3,11 +3,16 @@ ## [Unreleased] ### Added +- in `topology.v`: + + lemmas `continuous_subspaceT`, `subspaceT_continuous` ### Changed ### Renamed +- in `topology.v`: + + renamed `continuous_subspaceT` to `continuous_in_subspaceT` + ### Removed ### Infrastructure diff --git a/theories/derive.v b/theories/derive.v index 30279a41ad..f7e6aa6321 100644 --- a/theories/derive.v +++ b/theories/derive.v @@ -1507,7 +1507,7 @@ have gdrvbl x : x \in `]a, b[%R -> derivable g x 1. by move=> /fdrvbl dfx; apply: derivableB => //; exact/derivable1_diffP. have gcont : {within `[a, b], continuous g}. move=> x; apply: continuousD _ ; first by move=>?; exact: fcont. - by apply/continuousN/continuous_subspaceT => ? ?; exact: scalel_continuous. + by apply/continuousN/continuous_subspaceT=> ?; exact: scalel_continuous. have gaegb : g a = g b. rewrite /g -![(_ - _ : _ -> _) _]/(_ - _). apply/eqP; rewrite -subr_eq /= opprK addrAC -addrA -scalerBl. diff --git a/theories/exp.v b/theories/exp.v index e99f0a64e8..3bfbcde94c 100644 --- a/theories/exp.v +++ b/theories/exp.v @@ -442,7 +442,7 @@ Proof. move=> x_ge1; have x_ge0 : 0 <= x by apply: le_trans x_ge1. have [x1 x1Ix| |x1 _ /eqP] := @IVT _ (fun y => expR y - x) _ _ 0 x_ge0. - apply: continuousB => // y1; last exact: cst_continuous. - by apply/continuous_subspaceT=> ? _; exact: continuous_expR. + by apply/continuous_subspaceT=> ?; exact: continuous_expR. - rewrite expR0; have [_| |] := ltrgtP (1- x) (expR x - x). + by rewrite subr_le0 x_ge1 subr_ge0 (le_trans _ (expR_ge1Dx _)) ?ler_addr. + by rewrite ltr_add2r expR_lt1 ltNge x_ge0. diff --git a/theories/realfun.v b/theories/realfun.v index 91af1dc519..f32caccf13 100644 --- a/theories/realfun.v +++ b/theories/realfun.v @@ -32,7 +32,7 @@ Implicit Types (a b : R) (f g : R -> R). Lemma continuous_subspace_itv (I : interval R) (f : R -> R) : {in I, continuous f} -> {within [set` I], continuous f}. Proof. -move=> ctsf; apply: continuous_subspaceT => x Ix; apply: ctsf. +move=> ctsf; apply: continuous_in_subspaceT => x Ix; apply: ctsf. by move: Ix; rewrite inE. Qed. @@ -51,10 +51,10 @@ gen have main : f / forall c, {in I, continuous f} -> {in I &, injective f} -> have intP := interval_is_interval aI bI. have cI : c \in I by rewrite intP// (ltW aLc) ltW. have ctsACf : {within `[a, c], continuous f}. - apply: continuous_subspaceT => x; rewrite inE => /itvP axc; apply: fC. + apply: continuous_in_subspaceT => x; rewrite inE => /itvP axc; apply: fC. by rewrite intP// axc/= (le_trans _ (ltW cLb))// axc. have ctsCBf : {within `[c,b], continuous f}. - apply: continuous_subspaceT => x; rewrite inE => /itvP axc; apply: fC. + apply: continuous_in_subspaceT => x; rewrite inE => /itvP axc; apply: fC. by rewrite intP// axc andbT (le_trans (ltW aLc)) ?axc. have [aLb alb'] : a < b /\ a <= b by rewrite ltW (lt_trans aLc). have [faLfc|fcLfa|/eqP faEfc] /= := ltrgtP (f a) (f c). @@ -487,7 +487,7 @@ move=> Hd. wlog xLy : x y / x <= y by move=> H; case: (leP x y) => [/H |/ltW /H]. rewrite -(subKr (f y) (f x)). have [| _ _] := MVT_segment xLy; last by rewrite mul0r => ->; rewrite subr0. -apply/continuous_subspaceT => r _. +apply/continuous_subspaceT=> r. exact/differentiable_continuous/derivable1_diffP. Qed. diff --git a/theories/topology.v b/theories/topology.v index 033d201996..8a3f69ccbf 100644 --- a/theories/topology.v +++ b/theories/topology.v @@ -5772,8 +5772,8 @@ Global Instance subspace_proper_filter {T : topologicalType} (A : set T) (x : subspace A) : ProperFilter (nbhs_subspace x) := nbhs_subspace_filter x. -Notation "{ 'within' A , 'continuous' f }" := - (continuous (f : (subspace A) -> _)). +Notation "{ 'within' A , 'continuous' f }" := + (continuous (f : subspace A -> _)). Section SubspaceRelative. Context {T : topologicalType}. @@ -5812,19 +5812,26 @@ apply: (cvg_trans _ ctsf); apply: cvg_fmap2; apply: cvg_within. by rewrite /subspace; exact: nbhs_filter. Qed. -Lemma continuous_subspaceT {U} A (f : T -> U) : +Lemma continuous_in_subspaceT {U} A (f : T -> U) : {in A, continuous f} -> {within A, continuous f}. Proof. rewrite continuous_subspace_in ?in_setP => ctsf t At. by apply continuous_subspaceT_for => //=; apply: ctsf. Qed. +Lemma continuous_subspaceT{U} A (f : T -> U) : + continuous f -> {within A, continuous f}. +Proof. +move=> ctsf; rewrite continuous_subspace_in => ? ?. +exact: continuous_in_subspaceT => ? ?. +Qed. + Lemma continuous_open_subspace {U} A (f : T -> U) : open A -> {within A, continuous f} = {in A, continuous f}. Proof. rewrite openE continuous_subspace_in /= => oA; rewrite propeqE ?in_setP. -by split => + x /[dup] Ax /oA Aox; rewrite /filter_of /= => /(_ _ Ax); - rewrite -(nbhs_subspace_interior Aox). +by split => + x /[dup] Ax /oA Aox => /(_ _ Ax); + rewrite /filter_of -(nbhs_subspace_interior Aox). Qed. Lemma continuous_inP {U} A (f : T -> U) : open A -> @@ -5848,6 +5855,17 @@ rewrite [RHS]setIUr -V2W -V1W eqEsubset; split=> ?. by case=> [][] ? ?; split=> []; [left; split | left | right; split | right]. Qed. +Lemma subspaceT_continuous {U} A (B : set U) (f : {fun A >-> B}) : + {within A, continuous f} -> continuous (f : subspace A -> subspace B). +Proof. +move=> /continuousP ctsf; apply/continuousP => O /open_subspaceP [V [oV VBOB]]. +rewrite -open_subspaceIT; apply/open_subspaceP. +case/open_subspaceP: (ctsf _ oV) => W [oW fVA]; exists W; split => //. +rewrite fVA -setIA setIid eqEsubset; split => x [fVx Ax]; split => //. +- by have /[!VBOB]-[] : (V `&` B) (f x) by split => //; exact: funS. +- by have /[!esym VBOB]-[] : (O `&` B) (f x) by split => //; exact: funS. +Qed. + End SubspaceRelative. Section SubspaceUniform. @@ -6051,7 +6069,7 @@ Lemma continuous_localP {X Y : topologicalType} (f : X -> Y) : forall (x : X), \forall U \near powerset_filter_from (nbhs x), {within U, continuous f}. Proof. -split; first by move=> ? ?; near=> U; apply: continuous_subspaceT => ? ?; exact. +split; first by move=> ? ?; near=> U; apply: continuous_subspaceT=> ?; exact. move=> + x => /(_ x)/near_powerset_filter_fromP. case; first by move=> ? ?; exact: continuous_subspaceW. move=> U nbhsU wctsf; wlog oU : U wctsf nbhsU / open U. @@ -6177,7 +6195,7 @@ Lemma precompact_pointwise_precompact (W : set {family compact, X -> Y}) : Proof. move=> + x; rewrite ?precompactE => pcptW. have : compact (prod_topo_apply x @` (closure W)). - apply: continuous_compact => //; apply: continuous_subspaceT => g _. + apply: continuous_compact => //; apply: continuous_subspaceT=> g. move=> E nbhsE; have := (@prod_topo_apply_continuous _ _ x g E nbhsE). exact: (@pointwise_cvg_compact_family _ _ (nbhs g)). move=> /[dup]/(compact_closed hsdf)/closure_id -> /subclosed_compact. @@ -6308,7 +6326,7 @@ apply/continuous_localP => x'; apply/near_powerset_filter_fromP. by move=> ? ?; exact: continuous_subspaceW. case: (@lcptX x') => // U; rewrite withinET => nbhsU [cptU _]. exists U => //; apply: (uniform_limit_continuous_subspace PG _ _). - by near=> g; apply: continuous_subspaceT => + _; near: g; exact: GW. + by near=> g; apply: continuous_subspaceT; near: g; exact: GW. by move/fam_cvgP/(_ _ cptU) : Gf. Unshelve. end_near. Qed. diff --git a/theories/trigo.v b/theories/trigo.v index 3a50493a07..d2bd4069bd 100644 --- a/theories/trigo.v +++ b/theories/trigo.v @@ -560,10 +560,9 @@ Qed. Lemma cos_exists : exists2 pih : R, 1 <= pih <= 2 & cos pih = 0. Proof. have /IVT[] : minr (cos 1) (cos 2) <= (0 : R) <= maxr (cos 1) (cos 2). - - rewrite /minr /maxr ltNge (ltW (lt_trans cos2_lt0 cos1_gt0))/=. - by rewrite (ltW cos2_lt0)/= (ltW cos1_gt0). + - by rewrite le_maxr (ltW cos1_gt0) le_minl (ltW cos2_lt0) orbC. - by rewrite ler1n. - - by move=> *; apply/continuous_subspaceT=> ? _; exact: continuous_cos. + - by apply/continuous_subspaceT => ?; exact: continuous_cos. by move=> pih /itvP pihI chpi_eq0; exists pih; rewrite ?pihI. Qed. @@ -577,7 +576,7 @@ case: (x =P y) => // /eqP xDy. have xLLs : x < y by rewrite le_eqVlt (negPf xDy) in xLy. have /(Rolle xLLs)[x1 _|x1|x1 x1I [_ x1D]] : cos x = cos y by rewrite cy0. - exact: derivable_cos. -- by apply/continuous_subspaceT=> ? _; exact: continuous_cos. +- by apply/continuous_subspaceT => ?; exact: continuous_cos. - have [_ /esym/eqP] := is_derive_cos x1; rewrite x1D oppr_eq0 => /eqP Hs. suff : 0 < sin x1 by rewrite Hs ltxx. apply/sin2_gt0/andP; split. @@ -641,13 +640,13 @@ wlog : x / 0 <= x => [Hw|x_ge0]. move=> /andP[x_gt0 xLpi2]; case: (ler0P (cos x)) => // cx_le0. have /IVT[]// : minr (cos 0) (cos x) <= 0 <= maxr (cos 0) (cos x). by rewrite cos0 /minr /maxr !ifN ?cx_le0 //= -leNgt (le_trans cx_le0). -- by move=> *; apply/continuous_subspaceT=> ? _; apply: continuous_cos. -move=> x1 /itvP Hx1 cx1_eq0. +- by apply/continuous_subspaceT => ?; exact: continuous_cos. +move=> x1 /itvP xx1 cx1_eq0. suff x1E : x1 = pi/2. - have : x1 < pi / 2 by apply: le_lt_trans xLpi2; rewrite Hx1. + have : x1 < pi / 2 by apply: le_lt_trans xLpi2; rewrite xx1. by rewrite x1E ltxx. apply: cos_02_uniq=> //; last by case pihalf_02_cos_pihalf => _ ->. - by rewrite Hx1 ltW // (lt_trans _ pihalf_lt2) // (le_lt_trans _ xLpi2) // Hx1. + by rewrite xx1 ltW // (lt_trans _ pihalf_lt2) // (le_lt_trans _ xLpi2) // xx1. by rewrite divr_ge0 ?(ltW pihalf_lt2)// pi_ge0. Qed. @@ -732,7 +731,7 @@ move=> x y; rewrite !in_itv/= le_eqVlt; case: eqP => [<- _|_] /=. rewrite y_gt0; apply/idP. suff : cos y != 1 by case: ltrgtP (cos_le1 y). rewrite -cos0 eq_sym; apply/eqP => /Rolle [||x1|x1 /itvP x1I [_ x1D]] //. - by apply/continuous_subspaceT=> ? _; exact: continuous_cos. + by apply/continuous_subspaceT => ?; exact: continuous_cos. case: (is_derive_cos x1) => _ /eqP; rewrite x1D eq_sym oppr_eq0 => /eqP s_eq0. suff : 0 < sin x1 by rewrite s_eq0 ltxx. by apply: sin_gt0_pi; rewrite x1I /= (lt_le_trans (_ : _ < y)) ?x1I // yI. @@ -747,7 +746,7 @@ rewrite le_eqVlt; case: eqP => /= [-> _ | _ /andP[y_gt0 y_ltpi]]. rewrite cospi x_ltpi; apply/idP. suff : cos x != -1 by case: ltrgtP (cos_geN1 x). rewrite -cospi; apply/eqP => /Rolle [||x1|x1 /itvP x1I [_ x1D]] //. - by apply/continuous_subspaceT=> ? _; exact: continuous_cos. + by apply/continuous_subspaceT => ?; exact: continuous_cos. case: (is_derive_cos x1) => _ /eqP; rewrite x1D eq_sym oppr_eq0 => /eqP s_eq0. suff : 0 < sin x1 by rewrite s_eq0 ltxx. by apply: sin_gt0_pi; rewrite x1I /= (lt_le_trans (_ : _ < x)) ?x1I. @@ -757,8 +756,8 @@ wlog xLy : x y x_gt0 x_ltpi y_gt0 y_ltpi / x <= y => [H | ]. case: (x =P y) => [->| /eqP xDy]; first by rewrite ltxx. have xLLs : x < y by rewrite le_eqVlt (negPf xDy) in xLy. rewrite xLLs -subr_gt0 -opprB; rewrite -subr_gt0 in xLLs; apply/idP. -have [x1|z /itvP zI ->] := @MVT_segment _ cos (-sin) _ _ xLy. - by apply/continuous_subspaceT=> ? _; exact: continuous_cos. +have [x1|z /itvP zI ->] := @MVT_segment _ cos (- sin) _ _ xLy. + by apply/continuous_subspaceT => ?; exact: continuous_cos. rewrite -mulNr opprK mulr_gt0 //; apply: sin_gt0_pi. by rewrite (lt_le_trans x_gt0) ?zI //= (le_lt_trans _ y_ltpi) ?zI. Qed. @@ -870,8 +869,7 @@ Proof. by move=> /is_derive_tan[]. Qed. Lemma ltr_tan : {in `](- (pi/2)), (pi/2)[ &, {mono tan : x y / x < y}}. Proof. -move=> x y. -wlog xLy : x y / x <= y => [H | ] xB yB. +move=> x y; wlog xLy : x y / x <= y => [H xB yB|/itvP xB /itvP yB]. case: (lerP x y) => [/H //->//|yLx]. by rewrite !ltNge ltW ?(ltW yLx) // H // ltW. case: (x =P y) => [->| /eqP xDy]; first by rewrite ltxx. @@ -879,16 +877,14 @@ have xLLs : x < y by rewrite le_eqVlt (negPf xDy) in xLy. rewrite -subr_gt0 xLLs; rewrite -subr_gt0 in xLLs; apply/idP. have [x1 /itvP x1I|z |] := @MVT_segment _ tan (fun x => (cos x) ^-2) _ _ xLy. - apply: is_derive_tan. - rewrite gt_eqF // cos_gt0_pihalf // (@lt_le_trans _ _ x) ?x1I ?(itvP xB)//=. - by rewrite (@le_lt_trans _ _ y) ?x1I ?(itvP yB). -- apply/continuous_subspaceT=> ? inI; apply: continuous_tan. - rewrite /= inE /<=%O/= in inI; move/andP: inI => /= [? ?]. - rewrite gt_eqF // cos_gt0_pihalf // (@lt_le_trans _ _ x) ?zI ?(itvP xB)//=. - rewrite (@le_lt_trans _ _ y) ?zI ?(itvP yB) //. + rewrite gt_eqF // cos_gt0_pihalf // (@lt_le_trans _ _ x) ?x1I ?xB//=. + by rewrite (@le_lt_trans _ _ y) ?x1I ?yB. +- apply/continuous_in_subspaceT => ? -/[!(@mem_setE R)] /itvP inI. + apply: continuous_tan; rewrite gt_eqF// cos_gt0_pihalf//. + by rewrite (@lt_le_trans _ _ x) ?xB ?inI// (@le_lt_trans _ _ y) ?yB ?inI. - move=> x1 /itvP x1I ->. rewrite mulr_gt0 // invr_gt0 // exprn_gte0 // cos_gt0_pihalf //. - rewrite (@lt_le_trans _ _ x) ?x1I ?(itvP xB)//=. - by rewrite (@le_lt_trans _ _ y) ?x1I ?(itvP yB). + by rewrite (@lt_le_trans _ _ x) ?x1I ?xB//= (@le_lt_trans _ _ y) ?x1I ?yB. Qed. Lemma tan_inj : {in `](- (pi/2)), (pi/2)[ &, injective tan}. @@ -917,7 +913,7 @@ have /(IVT (@pi_ge0 _))[] // : minr (f 0) (f pi) <= 0 <= maxr (f 0) (f pi). rewrite /f cos0 cospi /minr /maxr ltr_add2r -subr_lt0 opprK (_ : 1 + 1 = 2)//. by rewrite ltrn0 subr_le0 subr_ge0. - move=> y y0pi. - by apply: continuousB; apply/continuous_subspaceT=> ? ?; + by apply: continuousB; apply/continuous_in_subspaceT => ? ?; [exact: continuous_cos|exact: cst_continuous]. - rewrite /f => x1 /itvP x1I /eqP; rewrite subr_eq0 => /eqP cosx1E. by case: (He x1); rewrite !x1I. @@ -1043,7 +1039,7 @@ have /IVT[] // : rewrite /f sinN sin_pihalf /minr /maxr ltr_add2r -subr_gt0 opprK. by rewrite (_ : 1 + 1 = 2)// ltr0n/= subr_le0 subr_ge0. - by rewrite -subr_ge0 opprK -splitr pi_ge0. -- by move=> *; apply: continuousB; apply/continuous_subspaceT=> ? ?; +- by move=> *; apply: continuousB; apply/continuous_in_subspaceT => ? ?; [exact: continuous_sin| exact: cst_continuous]. - rewrite /f => x1 /itvP x1I /eqP; rewrite subr_eq0 => /eqP sinx1E. by case: (He x1); rewrite !x1I. From 42e7d3def432ee4c81ab440ab54cf7ed6d27e830 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 8 Jun 2022 13:07:28 +0900 Subject: [PATCH 23/42] tentative definition of kernel --- theories/kernel.v | 67 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 67 insertions(+) create mode 100644 theories/kernel.v diff --git a/theories/kernel.v b/theories/kernel.v new file mode 100644 index 0000000000..ff9806d79c --- /dev/null +++ b/theories/kernel.v @@ -0,0 +1,67 @@ +(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. +Require Import mathcomp_extra boolp classical_sets signed functions cardinality. +Require Import reals ereal topology normedtype sequences esum measure. +Require Import lebesgue_measure fsbigop numfun lebesgue_integral. + +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. +Local Open Scope ereal_scope. + +HB.mixin Record isKernel + (R : realType) (X Y : measurableType) + (k : X -> {measure set Y -> \bar R}) := { + kernel_measurable_fun : + forall U, measurable_fun setT (k ^~ U) +}. + +#[short(type=kernel)] +HB.structure Definition Kernel + (R : realType) (X Y : measurableType) := + {k & isKernel R X Y k}. +Notation "X ^^> Y" := (kernel _ X Y) (at level 42). + +HB.mixin Record isProbabilityKernel + (R : realType) (X Y : measurableType) + (k : X -> {measure set Y -> \bar R}) + of isKernel R X Y k := { + prob_kernel : forall x : X, k x setT = 1 +}. + +HB.structure Definition ProbKernel + (R : realType) (X Y : measurableType) := + {k & isProbabilityKernel R X Y k }. +(* TODO: warning *) + +Definition sum_of_kernels + (R : realType) (X Y : measurableType) + (k : (X ^^> Y)^nat) : X -> {measure set Y -> \bar R} := + fun x => [the {measure _ -> _} of mseries (k ^~ x) 0]. + +Lemma kernel_measurable_fun_sum_of_kernels + (R : realType) (X Y : measurableType) + (k : (kernel R X Y)^nat) : + forall U, measurable_fun setT ((sum_of_kernels k) ^~ U). +Proof. +Admitted. + +HB.instance Definition _ + (R : realType) (X Y : measurableType) + (k : (kernel R X Y)^nat) := + isKernel.Build R X Y (sum_of_kernels k) + (kernel_measurable_fun_sum_of_kernels k). + +Lemma proposition1 + (R : realType) (X Y : measurableType) + (k : (kernel R X Y)^nat) (f : Y -> \bar R) x : + \int[sum_of_kernels k x]_y (f y) = \sum_(i Date: Wed, 15 Jun 2022 12:53:38 +0900 Subject: [PATCH 24/42] tentative statement of lemma 3 --- theories/kernel.v | 54 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 54 insertions(+) diff --git a/theories/kernel.v b/theories/kernel.v index ff9806d79c..8ecefaa538 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -50,6 +50,15 @@ Lemma kernel_measurable_fun_sum_of_kernels (k : (kernel R X Y)^nat) : forall U, measurable_fun setT ((sum_of_kernels k) ^~ U). Proof. +move=> U; rewrite /sum_of_kernels /= /mseries. +rewrite [X in measurable_fun _ X](_ : _ = + (fun x => elim_sup (fun n => \sum_(i < n) k i x U))); last first. + apply/funext => x. + rewrite -lim_mkord. + (* TODO: see cvg_lim_supE *) + admit. +apply: measurable_fun_elim_sup => n. +(*TODO: use measurable_funD *) Admitted. HB.instance Definition _ @@ -63,5 +72,50 @@ Lemma proposition1 (k : (kernel R X Y)^nat) (f : Y -> \bar R) x : \int[sum_of_kernels k x]_y (f y) = \sum_(i {measure set Y -> \bar R}) + of isKernel R X Y k := { + finite_kernel : exists r : R, forall x : X, k x setT < r%:E +}. + +HB.structure Definition FiniteKernel + (R : realType) (X Y : measurableType) := + {k & isFiniteKernel R X Y k }. + +HB.mixin Record isSFiniteKernel + (R : realType) (X Y : measurableType) + (k : X -> {measure set Y -> \bar R}) + of isKernel R X Y k := { + finite_kernel : exists k_ : (X ^^> Y)^nat, forall x U, + k x U = \sum_(i set Z -> \bar R := + fun x => fun U => \int[l x]_y k (x, y) U. + +Definition star_kernel (R : realType) (X Y Z : measurableType) + (k : sfinitekernel R [the measurableType of (X * Y)%type] Z) + (l : sfinitekernel R X Y) : X -> {measure set Z -> \bar R}. +(* TODO *) +Admitted. + +Lemma lemma3 (R : realType) (X Y Z : measurableType) + (k : sfinitekernel R [the measurableType of (X * Y)%type] Z) + (l : sfinitekernel R X Y) : forall x f, + \int[star_kernel k l x]_z f z = + \int[l x]_y (\int[k (x, y)]_z f z). +Proof. +(* TODO *) +Admitted. From 16a856283ad1ec1fdf429e0e787dd9c9552dd5b3 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 22 Jun 2022 11:27:55 +0900 Subject: [PATCH 25/42] complete infinite sum of kernels is a kernel --- theories/kernel.v | 21 ++++++++++++++------- 1 file changed, 14 insertions(+), 7 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 8ecefaa538..5bb8488bf5 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -45,6 +45,14 @@ Definition sum_of_kernels (k : (X ^^> Y)^nat) : X -> {measure set Y -> \bar R} := fun x => [the {measure _ -> _} of mseries (k ^~ x) 0]. +(* PR in progress *) +Lemma preimage_cst (aT rT : Type) (x : aT) (A : set aT) : + @cst rT _ x @^-1` A = if x \in A then setT else set0. +Proof. +apply/seteqP; rewrite /preimage; split; first by move=> *; rewrite mem_set. +by case: ifPn => [/[!inE] ?//|_]; exact: sub0set. +Qed. + Lemma kernel_measurable_fun_sum_of_kernels (R : realType) (X Y : measurableType) (k : (kernel R X Y)^nat) : @@ -52,14 +60,13 @@ Lemma kernel_measurable_fun_sum_of_kernels Proof. move=> U; rewrite /sum_of_kernels /= /mseries. rewrite [X in measurable_fun _ X](_ : _ = - (fun x => elim_sup (fun n => \sum_(i < n) k i x U))); last first. - apply/funext => x. - rewrite -lim_mkord. - (* TODO: see cvg_lim_supE *) - admit. + (fun x => elim_sup (fun n => \sum_(0 <= i < n) k i x U))); last first. + apply/funext => x; rewrite -lim_mkord is_cvg_elim_supE. + by rewrite -lim_mkord. + exact: is_cvg_nneseries. apply: measurable_fun_elim_sup => n. -(*TODO: use measurable_funD *) -Admitted. +by apply: measurable_fun_sum => *; exact/kernel_measurable_fun. +Qed. HB.instance Definition _ (R : realType) (X Y : measurableType) From f0670da890d41f6bada62aa2782f4d47dcf54d0f Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 22 Jun 2022 17:40:06 +0900 Subject: [PATCH 26/42] prove that star_kernel is a measure --- theories/kernel.v | 49 +++++++++++++++++++++++++++++++---------------- 1 file changed, 33 insertions(+), 16 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 5bb8488bf5..ecd265a5cb 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -45,14 +45,6 @@ Definition sum_of_kernels (k : (X ^^> Y)^nat) : X -> {measure set Y -> \bar R} := fun x => [the {measure _ -> _} of mseries (k ^~ x) 0]. -(* PR in progress *) -Lemma preimage_cst (aT rT : Type) (x : aT) (A : set aT) : - @cst rT _ x @^-1` A = if x \in A then setT else set0. -Proof. -apply/seteqP; rewrite /preimage; split; first by move=> *; rewrite mem_set. -by case: ifPn => [/[!inE] ?//|_]; exact: sub0set. -Qed. - Lemma kernel_measurable_fun_sum_of_kernels (R : realType) (X Y : measurableType) (k : (kernel R X Y)^nat) : @@ -107,16 +99,41 @@ HB.structure Definition SFiniteKernel (R : realType) (X Y : measurableType) := {k & isSFiniteKernel R X Y k}. -Definition star_kernel' (R : realType) (X Y Z : measurableType) - (k : sfinitekernel R [the measurableType of (X * Y)%type] Z) - (l : sfinitekernel R X Y) : X -> set Z -> \bar R := +Section starkernel. +Variables (R : realType) (X Y Z : measurableType). +Variable k : sfinitekernel R [the measurableType of (X * Y)%type] Z. +Variable l : sfinitekernel R X Y. + +Definition star_kernel' : X -> set Z -> \bar R := fun x => fun U => \int[l x]_y k (x, y) U. -Definition star_kernel (R : realType) (X Y Z : measurableType) - (k : sfinitekernel R [the measurableType of (X * Y)%type] Z) - (l : sfinitekernel R X Y) : X -> {measure set Z -> \bar R}. -(* TODO *) -Admitted. +Let star_kernel'0 (x : X) : star_kernel' x set0 = 0. +Proof. +rewrite /star_kernel' (eq_integral (cst 0)) ?integral0// => y _. +by rewrite measure0. +Qed. + +Let star_kernel'_ge0 (x : X) (U : set Z) : 0 <= star_kernel' x U. +Proof. by apply: integral_ge0 => y _; exact: measure_ge0. Qed. + +Let star_kernel'_sigma_additive (x : X) : semi_sigma_additive (star_kernel' x). +Proof. +move=> F mF tF mUF; rewrite [X in _ --> X](_ : _ = + \int[l x]_y (\sum_(n U _. + by apply/esym/cvg_lim => //; apply/measure_semi_sigma_additive. +apply/cvg_closeP; split. + by apply: is_cvg_nneseries => n _; exact: integral_ge0. +rewrite closeE// integral_sum// => n. +move: (@kernel_measurable_fun R _ _ k (F n)) => /measurable_fun_prod1. +exact. +Qed. + +Canonical star_kernel : X -> {measure set Z -> \bar R} := + fun x => Measure.Pack _ (Measure.Axioms (star_kernel'0 x) (star_kernel'_ge0 x) + (@star_kernel'_sigma_additive x)). + +End starkernel. Lemma lemma3 (R : realType) (X Y Z : measurableType) (k : sfinitekernel R [the measurableType of (X * Y)%type] Z) From ce366a819ce4df5eee5cc8790cb0de6b5d2e8d82 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Thu, 23 Jun 2022 00:37:03 +0900 Subject: [PATCH 27/42] proposition 1 --- theories/kernel.v | 22 ++++++++++++++++++---- 1 file changed, 18 insertions(+), 4 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index ecd265a5cb..61958577b0 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -66,14 +66,28 @@ HB.instance Definition _ isKernel.Build R X Y (sum_of_kernels k) (kernel_measurable_fun_sum_of_kernels k). +(* PR in progress *) +Section ge0_integral_measure_series. +Local Open Scope ereal_scope. +Variables (T : measurableType) (R : realType) (m_ : {measure set T -> \bar R}^nat). +Let m := measure_series m_ O. + +Lemma ge0_integral_measure_series (D : set T) (mD : measurable D) (f : T -> \bar R) : + (forall t, D t -> 0 <= f t) -> + measurable_fun D f -> + \int[m]_(x in D) f x = \sum_(n \bar R) x : - \int[sum_of_kernels k x]_y (f y) = \sum_(i + measurable_fun setT f -> + \int[sum_of_kernels k x]_y (f y) = \sum_(i f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. +Qed. HB.mixin Record isFiniteKernel (R : realType) (X Y : measurableType) From ff441916eda2f764264e9de1e2e07a2f446bdd3e Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 11 Jul 2022 17:46:33 +0900 Subject: [PATCH 28/42] tentative first part of lemma 3 (admit pending) - tentative example of semantics --- theories/kernel.v | 902 +++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 816 insertions(+), 86 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 61958577b0..6568e49d5b 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -15,73 +15,62 @@ Local Open Scope classical_set_scope. Local Open Scope ring_scope. Local Open Scope ereal_scope. -HB.mixin Record isKernel - (R : realType) (X Y : measurableType) +HB.mixin Record isKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) := { - kernel_measurable_fun : - forall U, measurable_fun setT (k ^~ U) + kernelP : forall U, measurable U -> measurable_fun setT (k ^~ U) }. #[short(type=kernel)] -HB.structure Definition Kernel - (R : realType) (X Y : measurableType) := - {k & isKernel R X Y k}. +HB.structure Definition Kernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') := + {k & isKernel d d' R X Y k}. Notation "X ^^> Y" := (kernel _ X Y) (at level 42). -HB.mixin Record isProbabilityKernel - (R : realType) (X Y : measurableType) +HB.mixin Record isProbabilityKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) - of isKernel R X Y k := { - prob_kernel : forall x : X, k x setT = 1 + of isKernel d d' R X Y k := { + prob_kernelP : forall x : X, k x [set: Y] = 1 }. -HB.structure Definition ProbKernel - (R : realType) (X Y : measurableType) := - {k & isProbabilityKernel R X Y k }. -(* TODO: warning *) +#[short(type=probability_kernel)] +HB.structure Definition ProbabilityKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') := + {k of isProbabilityKernel d d' R X Y k & isKernel d d' R X Y k}. -Definition sum_of_kernels - (R : realType) (X Y : measurableType) - (k : (X ^^> Y)^nat) : X -> {measure set Y -> \bar R} := - fun x => [the {measure _ -> _} of mseries (k ^~ x) 0]. +Section sum_of_kernels. +Variables (d d' : measure_display) (R : realType). +Variables (X : measurableType d) (Y : measurableType d'). +Variable k : (kernel R X Y)^nat. -Lemma kernel_measurable_fun_sum_of_kernels - (R : realType) (X Y : measurableType) - (k : (kernel R X Y)^nat) : - forall U, measurable_fun setT ((sum_of_kernels k) ^~ U). +Definition sum_of_kernels : X -> {measure set Y -> \bar R} := + fun x => [the measure _ _ of mseries (k ^~ x) 0]. + +Lemma kernel_measurable_fun_sum_of_kernels (U : set Y) : + measurable U -> + measurable_fun setT (sum_of_kernels ^~ U). Proof. -move=> U; rewrite /sum_of_kernels /= /mseries. +move=> mU; rewrite /sum_of_kernels /= /mseries. rewrite [X in measurable_fun _ X](_ : _ = (fun x => elim_sup (fun n => \sum_(0 <= i < n) k i x U))); last first. apply/funext => x; rewrite -lim_mkord is_cvg_elim_supE. by rewrite -lim_mkord. exact: is_cvg_nneseries. apply: measurable_fun_elim_sup => n. -by apply: measurable_fun_sum => *; exact/kernel_measurable_fun. +apply: emeasurable_fun_sum => *. +by apply/kernelP. Qed. -HB.instance Definition _ - (R : realType) (X Y : measurableType) - (k : (kernel R X Y)^nat) := - isKernel.Build R X Y (sum_of_kernels k) - (kernel_measurable_fun_sum_of_kernels k). +HB.instance Definition _ := + isKernel.Build d d' R X Y sum_of_kernels + kernel_measurable_fun_sum_of_kernels. -(* PR in progress *) -Section ge0_integral_measure_series. -Local Open Scope ereal_scope. -Variables (T : measurableType) (R : realType) (m_ : {measure set T -> \bar R}^nat). -Let m := measure_series m_ O. +End sum_of_kernels. -Lemma ge0_integral_measure_series (D : set T) (mD : measurable D) (f : T -> \bar R) : - (forall t, D t -> 0 <= f t) -> - measurable_fun D f -> - \int[m]_(x in D) f x = \sum_(n \bar R) x : +Lemma integral_sum_of_kernels (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') + (k : (X ^^> Y)^nat) (f : Y -> \bar R) x : (forall y, 0 <= f y) -> measurable_fun setT f -> \int[sum_of_kernels k x]_y (f y) = \sum_(i f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. Qed. -HB.mixin Record isFiniteKernel - (R : realType) (X Y : measurableType) +HB.mixin Record isFiniteKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) - of isKernel R X Y k := { - finite_kernel : exists r : R, forall x : X, k x setT < r%:E + of isKernel d d' R X Y k := { + finite_kernelP : exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E }. -HB.structure Definition FiniteKernel - (R : realType) (X Y : measurableType) := - {k & isFiniteKernel R X Y k }. +#[short(type=finite_kernel)] +HB.structure Definition FiniteKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') := + {k of isFiniteKernel d d' R X Y k & isKernel d d' R X Y k}. -HB.mixin Record isSFiniteKernel - (R : realType) (X Y : measurableType) +HB.mixin Record isSFiniteKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) - of isKernel R X Y k := { - finite_kernel : exists k_ : (X ^^> Y)^nat, forall x U, - k x U = \sum_(i + k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U }. -#[short(type=sfinitekernel)] -HB.structure Definition SFiniteKernel - (R : realType) (X Y : measurableType) := - {k & isSFiniteKernel R X Y k}. +#[short(type=sfinite_kernel)] +HB.structure Definition SFiniteKernel (d d' : measure_display) + (R : realType) (X : measurableType d) (Y : measurableType d') := + {k of isSFiniteKernel d d' R X Y k & + isFiniteKernel d d' R X Y k & + isKernel d d' R X Y k}. -Section starkernel. -Variables (R : realType) (X Y Z : measurableType). -Variable k : sfinitekernel R [the measurableType of (X * Y)%type] Z. -Variable l : sfinitekernel R X Y. +Section star_is_kernel. +Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). +Variable k : kernel R [the measurableType _ of (X * Y)%type] Z. +Variable l : kernel R X Y. -Definition star_kernel' : X -> set Z -> \bar R := - fun x => fun U => \int[l x]_y k (x, y) U. +Definition star : X -> set Z -> \bar R := fun x U => \int[l x]_y k (x, y) U. -Let star_kernel'0 (x : X) : star_kernel' x set0 = 0. +Let star0 (x : X) : star x set0 = 0. Proof. -rewrite /star_kernel' (eq_integral (cst 0)) ?integral0// => y _. -by rewrite measure0. +by rewrite /star (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. Qed. -Let star_kernel'_ge0 (x : X) (U : set Z) : 0 <= star_kernel' x U. +Let star_ge0 (x : X) (U : set Z) : 0 <= star x U. Proof. by apply: integral_ge0 => y _; exact: measure_ge0. Qed. -Let star_kernel'_sigma_additive (x : X) : semi_sigma_additive (star_kernel' x). +Let star_sigma_additive (x : X) : semi_sigma_additive (star x). Proof. -move=> F mF tF mUF; rewrite [X in _ --> X](_ : _ = - \int[l x]_y (\sum_(n U _. - by apply/esym/cvg_lim => //; apply/measure_semi_sigma_additive. +move=> U mU tU mUU. +rewrite [X in _ --> X](_ : _ = + \int[l x]_y (\sum_(n V _. + by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. apply/cvg_closeP; split. by apply: is_cvg_nneseries => n _; exact: integral_ge0. rewrite closeE// integral_sum// => n. -move: (@kernel_measurable_fun R _ _ k (F n)) => /measurable_fun_prod1. +move: (@kernelP _ _ R _ _ k (U n) (mU n)) => /measurable_fun_prod1. exact. Qed. -Canonical star_kernel : X -> {measure set Z -> \bar R} := - fun x => Measure.Pack _ (Measure.Axioms (star_kernel'0 x) (star_kernel'_ge0 x) - (@star_kernel'_sigma_additive x)). +HB.instance Definition _ (x : X) := + isMeasure.Build _ R _ (star x) (star0 x) (star_ge0 x) (@star_sigma_additive x). + +Definition mstar : X -> {measure set Z -> \bar R} := fun x => [the measure _ _ of star x]. + +End star_is_kernel. + +(* TODO: PR *) +Section integralM_indic. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). + +Lemma integralM_indic_new (f : R -> T -> R) (k : R) + (f0 : forall r t, D t -> (0 <= f r t)%R) : + ((k < 0)%R -> f k = cst 0%R) -> measurable_fun setT (f k) -> + \int[m]_(x in D) (k * (f k) x)%:E = k%:E * \int[m]_(x in D) ((f k) x)%:E. +Proof. +move=> fk0 mfk; have [k0|k0] := ltP k 0%R. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. + by move=> x _; rewrite fk0// mulr0. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. + by rewrite fk0// indic0. +under eq_integral do rewrite EFinM. +rewrite ge0_integralM//. +- apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. +- by move=> y Dy; rewrite lee_fin f0. +Qed. + +End integralM_indic. + +Section test. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). + +Lemma integralM_indic_test (f : R -> set T) (k : R) : + ((k < 0)%R -> f k = set0) -> measurable (f k) -> + \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. +Proof. +move=> fk0 mfk. +apply: (@integralM_indic_new _ _ _ _ _ _ (fun k x => \1_(f k) x)) => //=. + move/fk0 => -> /=. + apply/funext => x. + by rewrite indicE in_set0. +by rewrite (_ : \1_(f k) = mindic R mfk). +Qed. + +End test. + + +Lemma muleCA (R : realType) : left_commutative ( *%E : _ -> _ -> \bar R). +Proof. by move=> x y z; rewrite muleC (muleC x) muleA. Qed. + +Section integral_mscale. +Variables (R : realType) (k : {nonneg R}). +Variables (d : measure_display) (T : measurableType d). +Variable (m : {measure set T -> \bar R}) (D : set T) (f : T -> \bar R). +Hypotheses (mD : measurable D). + +Let integral_mscale_indic (E : set T) (mE : measurable E) : + \int[mscale k m]_(x in D) (\1_E x)%:E = + k%:num%:E * \int[m]_(x in D) (\1_E x)%:E. +Proof. by rewrite !integral_indic. Qed. + +(*NB: notation { mfun aT >-> rT} broken? *) +Let integral_mscale_nnsfun (h : {nnsfun T >-> R}) : + \int[mscale k m]_(x in D) (h x)%:E = k%:num%:E * \int[m]_(x in D) (h x)%:E. +Proof. +rewrite -ge0_integralM//; last 2 first. +apply/EFin_measurable_fun. + exact: measurable_funS (@measurable_funP _ _ _ h). + by move=> x _; rewrite lee_fin. +under eq_integral do rewrite fimfunE -sumEFin. +under [LHS]eq_integral do rewrite fimfunE -sumEFin. +rewrite /=. +rewrite ge0_integral_sum//; last 2 first. + move=> r. + apply/EFin_measurable_fun/measurable_funrM. + apply: (@measurable_funS _ _ _ _ setT) => //. + have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + by move=> n x Dx; rewrite EFinM muleindic_ge0. +under eq_integral. + move=> x xD. + rewrite ge0_sume_distrr//; last first. + by move=> r _; rewrite EFinM muleindic_ge0. + over. +rewrite /=. +rewrite ge0_integral_sum//; last 2 first. + move=> r. + apply/EFin_measurable_fun/measurable_funrM/measurable_funrM. + apply: (@measurable_funS _ _ _ _ setT) => //. + have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + move=> n x Dx. + by rewrite EFinM mule_ge0// muleindic_ge0. +apply eq_bigr => r _. +rewrite ge0_integralM//; last 2 first. + apply/EFin_measurable_fun/measurable_funrM. + apply: (@measurable_funS _ _ _ _ setT) => //. + have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + by move=> t Dt; rewrite muleindic_ge0. +rewrite (@integralM_indic_new _ _ _ _ _ _ (fun r x => \1_(h @^-1` [set r]) x))//; last 2 first. + move=> r0. + by rewrite preimage_nnfun0// indic0. + have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). +rewrite /=. +rewrite (@integralM_indic_new _ _ _ _ _ _ (fun r x => \1_(h @^-1` [set r]) x))//; last 2 first. + move=> r0. + by rewrite preimage_nnfun0// indic0. + have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). +rewrite integral_mscale_indic//. +by rewrite muleCA. +Qed. + +Lemma ge0_integral_mscale (mf : measurable_fun D f) : + (forall x, D x -> 0 <= f x) -> + \int[mscale k m]_(x in D) f x = k%:num%:E * \int[m]_(x in D) f x. +Proof. +move=> f0; have [f_ [ndf_ f_f]] := approximation mD mf f0. +transitivity (lim (fun n => \int[mscale k m]_(x in D) (f_ n x)%:E)). + rewrite -monotone_convergence//=; last 3 first. + move=> n; apply/EFin_measurable_fun. + by apply: (@measurable_funS _ _ _ _ setT). + by move=> n x Dx; rewrite lee_fin. + by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. + apply eq_integral => x /[!inE] xD; apply/esym/cvg_lim => //=. + exact: f_f. +rewrite (_ : \int[m]_(x in D) _ = lim (fun n => \int[m]_(x in D) (f_ n x)%:E)); last first. + rewrite -monotone_convergence//. + apply: eq_integral => x /[!inE] xD. + apply/esym/cvg_lim => //. + exact: f_f. + move=> n. + apply/EFin_measurable_fun. + by apply: (@measurable_funS _ _ _ _ setT). + by move=> n x Dx; rewrite lee_fin. + by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. +rewrite -ereal_limrM//; last first. + apply/ereal_nondecreasing_is_cvg => a b ab. + apply ge0_le_integral => //. + by move=> x Dx; rewrite lee_fin. + apply/EFin_measurable_fun. + by apply: (@measurable_funS _ _ _ _ setT). + by move=> x Dx; rewrite lee_fin. + apply/EFin_measurable_fun. + by apply: (@measurable_funS _ _ _ _ setT). + move=> x Dx. + rewrite lee_fin. + by move/ndf_ : ab => /lefP. +congr (lim _). +apply/funext => n /=. +by rewrite integral_mscale_nnsfun//. +Qed. + +End integral_mscale. + +(* TODO: rename emeasurable_funeM? *) + +Section ndseq_closed_B. +Variables (d1 d2 : measure_display). +Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Implicit Types A : set (T1 * T2). + +Section xsection. +Variables (pt2 : T2) (m2 : T1 -> {measure set T2 -> \bar R}). +Let phi A x := m2 x (xsection A x). +Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. + +Lemma xsection_ndseq_closed : ndseq_closed B. +Proof. +move=> F ndF; rewrite /B /= => BF; split. + by apply: bigcupT_measurable => n; have [] := BF n. +have phiF x : (fun i => phi (F i) x) --> phi (\bigcup_i F i) x. + rewrite /phi /= xsection_bigcup; apply: cvg_mu_inc => //. + - by move=> n; apply: measurable_xsection; case: (BF n). + - by apply: bigcupT_measurable => i; apply: measurable_xsection; case: (BF i). + - move=> m n mn; apply/subsetPset => y; rewrite /xsection/= !inE. + by have /subsetPset FmFn := ndF _ _ mn; exact: FmFn. +apply: (emeasurable_fun_cvg (phi \o F)) => //. +- by move=> i; have [] := BF i. +- by move=> x _; exact: phiF. +Qed. +End xsection. + +End ndseq_closed_B. + +Section measurable_prod_subset. +Variables (d1 d2 : measure_display). +Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Implicit Types A : set (T1 * T2). + +Section xsection. +Variable (m2 : T1 -> {measure set T2 -> \bar R}) (D : set T2) (mD : measurable D). +Let m2D x := mrestr (m2 x) mD. +HB.instance Definition _ x := Measure.on (m2D x). +Let phi A := fun x => m2D x (xsection A x). +Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. + +Hypothesis H1 : forall X2, measurable X2 -> measurable_fun [set: T1] (m2D^~ X2). + +Lemma measurable_prod_subset_xsection + (m2D_bounded : forall x, exists M, forall X, measurable X -> (m2D x X < M%:E)%E) : + measurable `<=` B. +Proof. +rewrite measurable_prod_measurableType. +set C := [set A1 `*` A2 | A1 in measurable & A2 in measurable]. +have CI : setI_closed C. + move=> X Y [X1 mX1 [X2 mX2 <-{X}]] [Y1 mY1 [Y2 mY2 <-{Y}]]. + exists (X1 `&` Y1); first exact: measurableI. + by exists (X2 `&` Y2); [exact: measurableI|rewrite setMI]. +have CT : C setT by exists setT => //; exists setT => //; rewrite setMTT. +have CB : C `<=` B. + move=> X [X1 mX1 [X2 mX2 <-{X}]]; split; first exact: measurableM. + have -> : phi (X1 `*` X2) = (fun x => m2D x X2 * (\1_X1 x)%:E)%E. + rewrite funeqE => x; rewrite indicE /phi /m2/= /mrestr. + have [xX1|xX1] := boolP (x \in X1); first by rewrite mule1 in_xsectionM. + by rewrite mule0 notin_xsectionM// set0I measure0. + apply: emeasurable_funM => //. + by apply: H1. + apply/EFin_measurable_fun. + by rewrite (_ : \1_ _ = mindic R mX1). +suff monoB : monotone_class setT B by exact: monotone_class_subset. +split => //; [exact: CB| |exact: xsection_ndseq_closed]. +move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. +have -> : phi (X `\` Y) = (fun x => phi X x - phi Y x)%E. + rewrite funeqE => x; rewrite /phi/= xsectionD// /m2D measureD. + - by rewrite setIidr//; exact: le_xsection. + - exact: measurable_xsection. + - exact: measurable_xsection. + - move: (m2D_bounded x) => [M m2M]. + rewrite (lt_le_trans (m2M (xsection X x) _))// ?leey//. + exact: measurable_xsection. +exact: emeasurable_funB. +Qed. + +End xsection. + +End measurable_prod_subset. + +(*NB: measurable_xsection as a superfluous parameter*) + +Section measurable_fun_xsection. +Variables (d1 d2 : measure_display). +Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Variables (m2 : T1 -> {measure set T2 -> \bar R}). +Implicit Types A : set (T1 * T2). +Hypotheses (sm2 : exists r : {posnum R}, forall x, m2 x [set: T2] < r%:num%:E). + +Hypothesis H1 : forall X2, measurable X2 -> measurable_fun [set: T1] ((fun x => mrestr (m2 x) measurableT)^~ X2). + +Let phi A := (fun x => m2 x (xsection A x)). +Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. + +Lemma measurable_fun_xsection A : + A \in measurable -> measurable_fun setT (phi A). +Proof. +move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. +move=> X mX. +(*move/sigma_finiteP : sf_m2 => [F F_T [F_nd F_oo]] X mX.*) +(*have -> : X = \bigcup_n (X `&` (setT `*` F n)). + by rewrite -setI_bigcupr -setM_bigcupr -F_T setMTT setIT. +apply: xsection_ndseq_closed. + move=> m n mn; apply/subsetPset; apply: setIS; apply: setSM => //. + exact/subsetPset/F_nd. +move=> n; rewrite -/B; have [? ?] := F_oo n.*) +(*pose m2Fn := [the measure _ _ of mrestr m2 (F_oo n).1].*) +rewrite /B/=; split => //. +rewrite /phi. +rewrite -(_ : (fun x : T1 => mrestr (m2 x) measurableT (xsection X x)) = (fun x => (m2 x) (xsection X x)))//; last first. + apply/funext => x//=. + by rewrite /mrestr setIT. +apply measurable_prod_subset_xsection => //; last first. + move=> x. + case: sm2 => r hr. + exists r%:num => Y mY. + apply: (le_lt_trans _ (hr x)) => //. + rewrite /mrestr. + apply le_measure => //. + rewrite inE. + apply: measurableI => //. + by rewrite inE. + +Qed. + +End measurable_fun_xsection. + +Section fubini_F_dep. +Local Open Scope ereal_scope. +Variables (d1 d2 : measure_display). +Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Variables (m2 : T1 -> {measure set T2 -> \bar R}). +Variable f : T1 * T2 -> \bar R. + +Definition fubini_F_dep x := \int[m2 x]_y f (x, y). + +End fubini_F_dep. + +Section fubini_tonelli. +Local Open Scope ereal_scope. +Variables (d1 d2 : measure_display). +Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Variables (m1 : {measure set T1 -> \bar R}) (m2 : T1 -> {measure set T2 -> \bar R}). +Hypotheses (sm2 : exists r : {posnum R}, forall x, m2 x [set: T2] < r%:num%:E). + +Section indic_fubini_tonelli. +Variables (A : set (T1 * T2)) (mA : measurable A). +Implicit Types A : set (T1 * T2). +Let f : (T1 * T2) -> R := \1_A. + +Let F := fubini_F_dep m2 (EFin \o f). + +Lemma indic_fubini_tonelli_FE : F = (fun x => m2 x (xsection A x)). +Proof. +rewrite funeqE => x; rewrite /= -(setTI (xsection _ _)). +rewrite -integral_indic//; last exact: measurable_xsection. +rewrite /F /fubini_F -(setTI (xsection _ _)). +rewrite integral_setI_indic; [|exact: measurable_xsection|by []]. +apply: eq_integral => y _ /=; rewrite indicT mul1e /f !indicE. +have [|] /= := boolP (y \in xsection _ _). + by rewrite inE /xsection /= => ->. +by rewrite /xsection /= notin_set /= => /negP/negbTE ->. +Qed. + +Hypothesis H1 : forall X2, measurable X2 -> + measurable_fun [set: T1] ((fun x => mrestr (m2 x) measurableT)^~ X2). + +Lemma indic_measurable_fun_fubini_tonelli_F_dep : measurable_fun setT F. +Proof. +rewrite indic_fubini_tonelli_FE//. +apply: measurable_fun_xsection => //. +by rewrite inE. +Qed. + +End indic_fubini_tonelli. + +End fubini_tonelli. + +Lemma pollard (d d' : measure_display) + (R : realType) + (X : measurableType d) + (Y : measurableType d') + (k : (X * Y)%type -> \bar R) + (k0 : (forall t : X * Y, True -> 0 <= k t)) + (mk : measurable_fun setT k) + (l : finite_kernel R X Y) : +measurable_fun [set: X] (fun x : X => \int[l x]_y k (x, y)). +Proof. +have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. +simpl in *. +rewrite (_ : (fun x => \int[l x]_y k (x, y)) = + (fun x => elim_sup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. + apply/funeqP => x. + transitivity (lim (fun n => \int[l x]_y (k_ n (x, y))%:E)); last first. + rewrite is_cvg_elim_supE//. + apply: ereal_nondecreasing_is_cvg => m n mn. + apply: ge0_le_integral => //. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun/measurable_fun_prod1. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun/measurable_fun_prod1. + - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. + rewrite -monotone_convergence//. + - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. + - by move=> n; exact/EFin_measurable_fun/measurable_fun_prod1. + - by move=> n y' _; rewrite lee_fin. + - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. +apply: measurable_fun_elim_sup => n. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l x0]_y + ((\sum_(r <- fset_set (range (k_ n))) + r * \1_(k_ n @^-1` [set r]) (x0, y)))%:E)); last first. + by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) + (\int[l x0]_y + (r * \1_(k_ n @^-1` [set r]) (x0, y))%:E))); last first. + apply/funext => x; rewrite -ge0_integral_sum//. + - by apply: eq_integral => y _; rewrite sumEFin. + - move=> r. + apply/EFin_measurable_fun/measurable_funrM/measurable_fun_prod1 => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). + - by move=> m y _; rewrite muleindic_ge0. +apply emeasurable_fun_sum => r. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * + \int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. + apply/funext => x. + rewrite (@integralM_indic_new _ _ _ _ _ _ (fun k y => \1_(k_ n @^-1` [set r]) (x, y)))//. + - move=> r_lt0; apply/funext => y. + by rewrite preimage_nnfun0// ?indicE ?in_set0. + - apply/measurable_fun_prod1 => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). +apply: emeasurable_funeM. +apply: indic_measurable_fun_fubini_tonelli_F_dep. +- by apply/finite_kernelP. +- by apply/measurable_sfunP. +- move=> X2. + rewrite (_ : (fun x : X => mrestr (l x) measurableT X2) = (fun x : X => (l x) X2))//. + by apply/kernelP. + apply/funeqP => x. + by rewrite /mrestr setIT. +Qed. + +Section star_is_kernel2. +Variables (d d' : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType (d, d').-prod). +Variable k : finite_kernel R [the measurableType _ of (X * Y)%type] Z. +Variable l : finite_kernel R X Y. -End starkernel. +Lemma star_measurable U : measurable U -> measurable_fun setT (mstar k l ^~ U). +Proof. +(* k is a bounded measurable function *) +(* l is a finite kernel *) +move=> mU. +rewrite /star. +apply: (@pollard _ _ R X Y (fun xy => k xy U)) => //. +by apply: (@kernelP _ _ R [the measurableType (d, d').-prod of (X * Y)%type] Z k U) => //. +Qed. + +HB.instance Definition _ := + isKernel.Build _ _ R X Z (mstar k l) star_measurable. + +End star_is_kernel2. + +Section star_is_finite_kernel. +Variables (d d' : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType (d, d').-prod). +Variable k : finite_kernel R [the measurableType _ of (X * Y)%type] Z. +Variable l : finite_kernel R X Y. + +Lemma star_finite : exists r : {posnum R}, forall x, star k l x [set: Z] < r%:num%:E. +Proof. +have [r hr] := @finite_kernelP _ _ _ _ _ k. +have [s hs] := @finite_kernelP _ _ _ _ _ l. +exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x. +rewrite /star. +apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). + apply ge0_le_integral => //. + - have := @kernelP _ _ _ _ _ k setT measurableT. + exact/measurable_fun_prod1. + - exact/measurable_fun_cst. + - by move=> y _; apply/ltW/hr. +by rewrite integral_cst//= EFinM lte_pmul2l. +Qed. + +HB.instance Definition _ := + isFiniteKernel.Build _ _ R X Z (mstar k l) star_finite. + +End star_is_finite_kernel. + +Lemma eq_measure (d : measure_display) (T : measurableType d) (R : realType) + (m1 m2 : {measure set T -> \bar R}) : + (forall U, measurable U -> m1 U = m2 U) -> m1 = m2. +Proof. +Abort. + +Section eq_measure_integral_new. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType) + (D : set T). +Implicit Types m : {measure set T -> \bar R}. + +Let eq_measure_integral0 m2 m1 (f : T -> \bar R) : + (forall A, measurable A -> A `<=` D -> m1 A = m2 A) -> + [set sintegral m1 h | h in + [set h : {nnsfun T >-> R} | (forall x, (h x)%:E <= (f \_ D) x)]] `<=` + [set sintegral m2 h | h in + [set h : {nnsfun T >-> R} | (forall x, (h x)%:E <= (f \_ D) x)]]. +Proof. +move=> m12 _ [h hfD <-] /=; exists h => //; apply: eq_fsbigr => r _. +have [hrD|hrD] := pselect (h @^-1` [set r] `<=` D); first by rewrite m12. +suff : r = 0%R by move=> ->; rewrite !mul0e. +apply: contra_notP hrD => /eqP r0 t/= htr. +have := hfD t. +rewrite /patch/=; case: ifPn; first by rewrite inE. +move=> tD. +move: r0; rewrite -htr => ht0. +by rewrite le_eqVlt eqe (negbTE ht0)/= lte_fin// ltNge// fun_ge0. +Qed. + +Lemma eq_measure_integral_new m2 m1 (f : T -> \bar R) : + (forall A, measurable A -> A `<=` D -> m1 A = m2 A) -> + \int[m1]_(x in D) f x = \int[m2]_(x in D) f x. +Proof. +move=> m12; rewrite /integral funepos_restrict funeneg_restrict. +congr (ereal_sup _ - ereal_sup _)%E; rewrite eqEsubset; split; + apply: eq_measure_integral0 => A /m12 //. +by move=> /[apply]. +by move=> /[apply]. +Qed. + +End eq_measure_integral_new. +Arguments eq_measure_integral_new {d T R D} m2 {m1 f}. + +Section star_is_sfinite_kernel. +Variables (d d' : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType (d, d').-prod). +Variable k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z. +Variable l : sfinite_kernel R X Y. + +Lemma star_sfinite : exists k_ : (finite_kernel R X Z)^nat, forall x U, measurable U -> + mstar k l x U = [the measure _ _ of mseries (k_ ^~ x) O] U. +Proof. +have [k_ hk_] := @sfinite_kernelP _ _ _ _ _ k. +have [l_ hl_] := @sfinite_kernelP _ _ _ _ _ l. +pose K := [the kernel _ _ _ of sum_of_kernels k_]. +pose L := [the kernel _ _ _ of sum_of_kernels l_]. +have H1 x U : measurable U -> star k l x U = star K L x U. + move=> mU. + rewrite /star /L /K /=. + transitivity (\int[ + [the measure _ _ of mseries (fun x0 : nat => l_ x0 x) 0] +]_y k (x, y) U). + apply eq_measure_integral_new => A mA _ . + by rewrite hl_. + apply eq_integral => y _. + by rewrite hk_//. +have H2 x U : star K L x U = + \int[mseries (l_ ^~ x) 0]_y (\sum_(i y _. +have H3 x U : measurable U -> + \int[mseries (l_ ^~ x) 0]_y (\sum_(i mU. + rewrite integral_sum//= => n. + have := @kernelP _ _ _ _ _ (k_ n) _ mU. + by move/measurable_fun_prod1; exact. +have H4 x U : measurable U -> + \sum_(i mU. + apply: eq_nneseries => i _. + rewrite integral_sum_of_kernels//. + have := @kernelP _ _ _ _ _ (k_ i) _ mU. + by move/measurable_fun_prod1; exact. +have H5 x U : \sum_(i i _; exact: eq_nneseries. +suff: exists k_0 : (finite_kernel R X Z) ^nat, forall x U, + \esum_(i in setT) star (k_ i.1) (l_ i.2) x U = \sum_(i [kl_ hkl_]. + exists kl_ => x U mU. + rewrite /=. + rewrite /mstar/= /mseries H1// H2 H3//. + rewrite H4//. + rewrite H5// -hkl_ /=. + rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split. + rewrite -(@esum_esum _ _ _ _ _ (fun i j => star (k_ i) (l_ j) x U))//. + rewrite nneseries_esum; last by move=> n _; exact: nneseries_lim_ge0(* TODO: rename this lemma *). + by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true. +rewrite /=. +have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card. + by rewrite card_eq_sym; exact: card_nat2. +exists (fun i => [the finite_kernel _ _ _ of mstar (k_ (f i).1) (l_ (f i).2)]) => x U. +rewrite (reindex_esum [set: nat] [set: nat * nat] f)//. +by rewrite nneseries_esum// fun_true. +Qed. + +HB.instance Definition _ := + isSFiniteKernel.Build d ((d, d').-prod)%mdisp R X Z (mstar k l) star_sfinite. + +End star_is_sfinite_kernel. -Lemma lemma3 (R : realType) (X Y Z : measurableType) - (k : sfinitekernel R [the measurableType of (X * Y)%type] Z) - (l : sfinitekernel R X Y) : forall x f, - \int[star_kernel k l x]_z f z = - \int[l x]_y (\int[k (x, y)]_z f z). +Lemma lemma3_indic d d' (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType (d, d').-prod) + (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) + (l : sfinite_kernel R X Y) x (E : set _) (mE : measurable E) : + \int[mstar k l x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). Proof. +rewrite integral_indic// /mstar/= /star/=. +by apply eq_integral => y _; rewrite integral_indic. +Qed. + +Lemma lemma3_nnsfun d d' (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType (d, d').-prod) + (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) + (l : sfinite_kernel R X Y) x (f : {nnsfun Z >-> R}) : + \int[mstar k l x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). +Proof. +under eq_integral do rewrite fimfunE -sumEFin. +rewrite ge0_integral_sum//; last 2 first. + move=> r. + apply/EFin_measurable_fun/measurable_funrM. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + by move=> r z _; rewrite EFinM muleindic_ge0. +under eq_bigr. + move=> r _. + rewrite /=. + rewrite (@integralM_indic_new _ _ _ _ _ _ (fun r x0 => \1_(f @^-1` [set r]) x0))//; last 2 first. + move=> r0. + apply/funext => z/=. + by rewrite indicE memNset// preimage_nnfun0. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + rewrite /=. + rewrite lemma3_indic//. + over. +rewrite /=. +apply/esym. +Admitted. + +Lemma lemma3 d d' (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType (d, d').-prod) + (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) + (l : sfinite_kernel R X Y) x f : (forall z, 0 <= f z) -> measurable_fun setT f -> + \int[mstar k l x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). +Proof. +move=> f0 mf. +have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). +simpl in *. (* TODO *) Admitted. + +HB.mixin Record isProbability (d : measure_display) (T : measurableType d) + (R : realType) (P : set T -> \bar R) of isMeasure d R T P := + { probability_setT : P setT = 1%E }. + +#[short(type=probability)] +HB.structure Definition Probability (d : measure_display) (T : measurableType d) + (R : realType) := + {P of isProbability d T R P & isMeasure d R T P }. + +Section discrete_measurable2. + +Definition discrete_measurable_bool : set (set bool) := [set: set bool]. + +Let discrete_measurable0 : discrete_measurable_bool set0. Proof. by []. Qed. + +Let discrete_measurableC X : + discrete_measurable_bool X -> discrete_measurable_bool (~` X). +Proof. by []. Qed. + +Let discrete_measurableU (F : (set bool)^nat) : + (forall i, discrete_measurable_bool (F i)) -> + discrete_measurable_bool (\bigcup_i F i). +Proof. by []. Qed. + +HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Pointed.class _) + discrete_measurable_bool discrete_measurable0 discrete_measurableC + discrete_measurableU. + +End discrete_measurable2. + +Definition twoseven (R : realType) : {nonneg R}. +Admitted. + +Definition fiveseven (R : realType) : {nonneg R}. +Admitted. + +Definition bernoulli (R : realType) : {measure set _ -> \bar R} := + [the measure _ _ of measure_add + [the measure _ _ of mscale (twoseven R) [the measure _ _ of dirac true]] + [the measure _ _ of mscale (fiveseven R) [the measure _ _ of dirac false]]]. + +Canonical unit_pointedType := PointedType unit tt. + +Section unit_measurable. + +Definition unit_measurable : set (set unit) := [set: set unit]. + +Let unit_measurable0 : unit_measurable set0. Proof. by []. Qed. + +Let unit_measurableC X : unit_measurable X -> unit_measurable (~` X). +Proof. by []. Qed. + +Let unit_measurableU (F : (set unit)^nat) : + (forall i, unit_measurable (F i)) -> unit_measurable (\bigcup_i F i). +Proof. by []. Qed. + +HB.instance Definition _ := @isMeasurable.Build default_measure_display unit (Pointed.class _) + unit_measurable unit_measurable0 unit_measurableC + unit_measurableU. + +End unit_measurable. + +(* semantics for a sample operation? *) +Section kernel_from_measure. +Variables (d : measure_display) (R : realType) (X : measurableType d). +Variable m : {measure set X -> \bar R}. (* measure, probability measure *) + +Definition kernel_from_measure : unit -> {measure set X -> \bar R} := + fun _ : unit => m. + +Lemma kernel_from_measureP : forall U, measurable U -> measurable_fun setT (kernel_from_measure ^~ U). +Proof. by []. Qed. + +HB.instance Definition _ := + @isKernel.Build default_measure_display d R _ X kernel_from_measure + kernel_from_measureP. +End kernel_from_measure. + +(* semantics for return? *) +Section kernel_from_dirac. +Variables (R : realType) (d : _) (T : measurableType d). + +Definition kernel_from_dirac : T -> {measure set T -> \bar R} := + fun x => [the measure _ _ of dirac x]. + +Lemma kernel_from_diracP : forall U, measurable U -> measurable_fun setT (kernel_from_dirac ^~ U). +Proof. +move=> U mU. +rewrite /kernel_from_dirac. +rewrite /=. +rewrite /dirac/=. +apply/EFin_measurable_fun. +rewrite [X in measurable_fun _ X](_ : _ = mindic R mU)//. +Qed. + +HB.instance Definition _ := + isKernel.Build _ _ R _ _ kernel_from_dirac kernel_from_diracP. +End kernel_from_dirac. + +(* let x = sample (bernoulli 2/7) in + return x *) + +Definition letin (d d' d3 : measure_display) (R : realType) + (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) + (l : X ^^> Y) (k : _ ^^> Z) : X -> {measure set Z -> \bar R}:= + @mstar _ _ _ R _ _ _ k l. + +Section sample_program. +Variables (R : realType). + +Definition sample_bernoulli27 (*NB: 1 ^^> bool *) := + [the kernel _ _ _ of kernel_from_measure (bernoulli R)] . + +Definition Return : kernel R _ [the measurableType (default_measure_display,default_measure_display).-prod of (Datatypes_unit__canonical__measure_SemiRingOfSets * Datatypes_bool__canonical__measure_SemiRingOfSets)%type] (* NB: 1 * bool ^^> 1 * bool *) := + [the kernel _ _ _ of @kernel_from_dirac R _ _]. + +Definition program : unit -> set (unit * bool) -> \bar R (* NB: 1 ^^> 1 * bool *) := + letin + sample_bernoulli27 + Return. + +Lemma programE : forall U, program tt U = + ((twoseven R)%:num)%:E * \d_(tt, true) U + + ((fiveseven R)%:num)%:E * \d_(tt, false) U. +Proof. +move=> U. +rewrite /program/= /star/=. +rewrite ge0_integral_measure_sum// 2!big_ord_recl/= big_ord0 adde0/=. +rewrite !ge0_integral_mscale//=. +rewrite !integral_dirac//=. +by rewrite indicE in_setT mul1e indicE in_setT mul1e. +Qed. + +End sample_program. From 49c3156b9a81c6ccd79d8d69ac1f8e663ee028be Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 22 Jul 2022 19:20:17 +0900 Subject: [PATCH 29/42] complete lemma 3 and s-finite proofs - s-finite proofs for bernoulli, return, score - various fixes --- theories/kernel.v | 1168 +++++++++++++++++++++++++++++++++++---------- 1 file changed, 910 insertions(+), 258 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 6568e49d5b..1e1abeb33c 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -15,6 +15,26 @@ Local Open Scope classical_set_scope. Local Open Scope ring_scope. Local Open Scope ereal_scope. +(* PR 516 in progress *) +HB.mixin Record isProbability (d : measure_display) (T : measurableType d) + (R : realType) (P : set T -> \bar R) of isMeasure d R T P := + { probability_setT : P setT = 1%E }. + +#[short(type=probability)] +HB.structure Definition Probability (d : measure_display) (T : measurableType d) + (R : realType) := + {P of isProbability d T R P & isMeasure d R T P }. + +Section probability_lemmas. +Variables (d : _) (T : measurableType d) (R : realType) (P : probability T R). + +Lemma probability_le1 (A : set T) : measurable A -> (P A <= 1)%E. +Proof. +Admitted. + +End probability_lemmas. +(* /PR 516 in progress *) + HB.mixin Record isKernel (d d' : measure_display) (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) := { @@ -27,6 +47,7 @@ HB.structure Definition Kernel (d d' : measure_display) {k & isKernel d d' R X Y k}. Notation "X ^^> Y" := (kernel _ X Y) (at level 42). +(* TODO: define using the probability type *) HB.mixin Record isProbabilityKernel (d d' : measure_display) (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) @@ -58,8 +79,7 @@ rewrite [X in measurable_fun _ X](_ : _ = by rewrite -lim_mkord. exact: is_cvg_nneseries. apply: measurable_fun_elim_sup => n. -apply: emeasurable_fun_sum => *. -by apply/kernelP. +by apply: emeasurable_fun_sum => *; exact/kernelP. Qed. HB.instance Definition _ := @@ -78,12 +98,18 @@ Proof. by move=> f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. Qed. +Section kernel_uub. +Variables (d d' : measure_display) (R : numFieldType) (X : measurableType d) + (Y : measurableType d') (k : X -> set Y -> \bar R). + +Definition kernel_uub := exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E. + +End kernel_uub. + HB.mixin Record isFiniteKernel (d d' : measure_display) (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) - of isKernel d d' R X Y k := { - finite_kernelP : exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E -}. + := { finite_kernelP : kernel_uub k }. #[short(type=finite_kernel)] HB.structure Definition FiniteKernel (d d' : measure_display) @@ -93,63 +119,85 @@ HB.structure Definition FiniteKernel (d d' : measure_display) HB.mixin Record isSFiniteKernel (d d' : measure_display) (R : realType) (X : measurableType d) (Y : measurableType d') (k : X -> {measure set Y -> \bar R}) - of isKernel d d' R X Y k := { - sfinite_kernelP : exists k_ : (finite_kernel R X Y)^nat, forall x U, - measurable U -> - k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U + := { + sfinite_kernelP : exists k_ : (finite_kernel R X Y)^nat, + forall x U, measurable U -> + k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U }. #[short(type=sfinite_kernel)] HB.structure Definition SFiniteKernel (d d' : measure_display) (R : realType) (X : measurableType d) (Y : measurableType d') := {k of isSFiniteKernel d d' R X Y k & - isFiniteKernel d d' R X Y k & isKernel d d' R X Y k}. -Section star_is_kernel. -Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3). +Section star_is_measure. +Variables (d1 d2 d3 : _) (R : realType) (X : measurableType d1) + (Y : measurableType d2) (Z : measurableType d3). Variable k : kernel R [the measurableType _ of (X * Y)%type] Z. Variable l : kernel R X Y. Definition star : X -> set Z -> \bar R := fun x U => \int[l x]_y k (x, y) U. -Let star0 (x : X) : star x set0 = 0. +Let star0 x : star x set0 = 0. Proof. by rewrite /star (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. Qed. -Let star_ge0 (x : X) (U : set Z) : 0 <= star x U. +Let star_ge0 x U : 0 <= star x U. Proof. by apply: integral_ge0 => y _; exact: measure_ge0. Qed. -Let star_sigma_additive (x : X) : semi_sigma_additive (star x). +Let star_sigma_additive x : semi_sigma_additive (star x). Proof. -move=> U mU tU mUU. -rewrite [X in _ --> X](_ : _ = +move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = \int[l x]_y (\sum_(n V _. by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. apply/cvg_closeP; split. by apply: is_cvg_nneseries => n _; exact: integral_ge0. rewrite closeE// integral_sum// => n. -move: (@kernelP _ _ R _ _ k (U n) (mU n)) => /measurable_fun_prod1. -exact. +have := @kernelP _ _ R _ _ k (U n) (mU n). +exact/measurable_fun_prod1. Qed. -HB.instance Definition _ (x : X) := - isMeasure.Build _ R _ (star x) (star0 x) (star_ge0 x) (@star_sigma_additive x). +HB.instance Definition _ x := isMeasure.Build _ R _ + (star x) (star0 x) (star_ge0 x) (@star_sigma_additive x). -Definition mstar : X -> {measure set Z -> \bar R} := fun x => [the measure _ _ of star x]. +Definition mstar : X -> {measure set Z -> \bar R} := + fun x => [the measure _ _ of star x]. -End star_is_kernel. +End star_is_measure. (* TODO: PR *) -Section integralM_indic. +Section integralM_0ifneg. +Local Open Scope ereal_scope. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). + +Lemma integralM_0ifneg (f : R -> T -> \bar R) (k : R) + (f0 : forall r t, D t -> (0 <= f r t)) : + ((k < 0)%R -> f k = cst 0%E) -> measurable_fun setT (f k) -> + \int[m]_(x in D) (k%:E * (f k) x) = k%:E * \int[m]_(x in D) ((f k) x). +Proof. +move=> fk0 mfk; have [k0|k0] := ltP k 0%R. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. + by move=> x _; rewrite fk0// mule0. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. + by rewrite fk0// indic0. +rewrite ge0_integralM//. +- by apply/(@measurable_funS _ _ _ _ setT) => //. +- by move=> y Dy; rewrite f0. +Qed. + +End integralM_0ifneg. +Arguments integralM_0ifneg {d T R} m {D} mD f. + +(*Section integralM_0ifneg. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). -Lemma integralM_indic_new (f : R -> T -> R) (k : R) +Lemma integralM_0ifneg (f : R -> T -> R) (k : R) (f0 : forall r t, D t -> (0 <= f r t)%R) : ((k < 0)%R -> f k = cst 0%R) -> measurable_fun setT (f k) -> \int[m]_(x in D) (k * (f k) x)%:E = k%:E * \int[m]_(x in D) ((f k) x)%:E. @@ -165,31 +213,31 @@ rewrite ge0_integralM//. - by move=> y Dy; rewrite lee_fin f0. Qed. -End integralM_indic. +End integralM_0ifneg. +Arguments integralM_0ifneg {d T R} m {D} mD f.*) -Section test. +Section integralM_indic. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). -Lemma integralM_indic_test (f : R -> set T) (k : R) : +Let integralM_indic (f : R -> set T) (k : R) : ((k < 0)%R -> f k = set0) -> measurable (f k) -> \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. Proof. move=> fk0 mfk. -apply: (@integralM_indic_new _ _ _ _ _ _ (fun k x => \1_(f k) x)) => //=. - move/fk0 => -> /=. - apply/funext => x. - by rewrite indicE in_set0. -by rewrite (_ : \1_(f k) = mindic R mfk). +under eq_integral do rewrite EFinM. +apply: (integralM_0ifneg _ _ (fun k x => (\1_(f k) x)%:E)) => //=. +- by move=> r t Dt; rewrite lee_fin. +- by move/fk0 => -> /=; apply/funext => x; rewrite indicE in_set0. +- apply/EFin_measurable_fun. + by rewrite (_ : \1_(f k) = mindic R mfk). Qed. -End test. - - -Lemma muleCA (R : realType) : left_commutative ( *%E : _ -> _ -> \bar R). -Proof. by move=> x y z; rewrite muleC (muleC x) muleA. Qed. +End integralM_indic. +Arguments integralM_indic {d T R} m {D} mD f. +(* NB: PR in progress *) Section integral_mscale. Variables (R : realType) (k : {nonneg R}). Variables (d : measure_display) (T : measurableType d). @@ -201,7 +249,6 @@ Let integral_mscale_indic (E : set T) (mE : measurable E) : k%:num%:E * \int[m]_(x in D) (\1_E x)%:E. Proof. by rewrite !integral_indic. Qed. -(*NB: notation { mfun aT >-> rT} broken? *) Let integral_mscale_nnsfun (h : {nnsfun T >-> R}) : \int[mscale k m]_(x in D) (h x)%:E = k%:num%:E * \int[m]_(x in D) (h x)%:E. Proof. @@ -240,19 +287,7 @@ rewrite ge0_integralM//; last 2 first. have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. by rewrite (_ : \1__ = mindic R fr). by move=> t Dt; rewrite muleindic_ge0. -rewrite (@integralM_indic_new _ _ _ _ _ _ (fun r x => \1_(h @^-1` [set r]) x))//; last 2 first. - move=> r0. - by rewrite preimage_nnfun0// indic0. - have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). -rewrite /=. -rewrite (@integralM_indic_new _ _ _ _ _ _ (fun r x => \1_(h @^-1` [set r]) x))//; last 2 first. - move=> r0. - by rewrite preimage_nnfun0// indic0. - have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). -rewrite integral_mscale_indic//. -by rewrite muleCA. +by rewrite !integralM_indic_nnsfun//= integral_mscale_indic// muleCA. Qed. Lemma ge0_integral_mscale (mf : measurable_fun D f) : @@ -290,15 +325,12 @@ rewrite -ereal_limrM//; last first. move=> x Dx. rewrite lee_fin. by move/ndf_ : ab => /lefP. -congr (lim _). -apply/funext => n /=. -by rewrite integral_mscale_nnsfun//. +congr (lim _); apply/funext => n /=. +by rewrite integral_mscale_nnsfun. Qed. End integral_mscale. -(* TODO: rename emeasurable_funeM? *) - Section ndseq_closed_B. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). @@ -309,7 +341,7 @@ Variables (pt2 : T2) (m2 : T1 -> {measure set T2 -> \bar R}). Let phi A x := m2 x (xsection A x). Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. -Lemma xsection_ndseq_closed : ndseq_closed B. +Lemma xsection_ndseq_closed_dep : ndseq_closed B. Proof. move=> F ndF; rewrite /B /= => BF; split. by apply: bigcupT_measurable => n; have [] := BF n. @@ -341,7 +373,7 @@ Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. Hypothesis H1 : forall X2, measurable X2 -> measurable_fun [set: T1] (m2D^~ X2). -Lemma measurable_prod_subset_xsection +Lemma measurable_prod_subset_xsection_dep (m2D_bounded : forall x, exists M, forall X, measurable X -> (m2D x X < M%:E)%E) : measurable `<=` B. Proof. @@ -363,7 +395,7 @@ have CB : C `<=` B. apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R mX1). suff monoB : monotone_class setT B by exact: monotone_class_subset. -split => //; [exact: CB| |exact: xsection_ndseq_closed]. +split => //; [exact: CB| |exact: xsection_ndseq_closed_dep]. move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. have -> : phi (X `\` Y) = (fun x => phi X x - phi Y x)%E. rewrite funeqE => x; rewrite /phi/= xsectionD// /m2D measureD. @@ -380,49 +412,39 @@ End xsection. End measurable_prod_subset. -(*NB: measurable_xsection as a superfluous parameter*) - Section measurable_fun_xsection. -Variables (d1 d2 : measure_display). -Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Variables (d1 d2 : measure_display) (T1 : measurableType d1) + (T2 : measurableType d2) (R : realType). Variables (m2 : T1 -> {measure set T2 -> \bar R}). Implicit Types A : set (T1 * T2). -Hypotheses (sm2 : exists r : {posnum R}, forall x, m2 x [set: T2] < r%:num%:E). +Hypotheses m2_ub : kernel_uub m2. -Hypothesis H1 : forall X2, measurable X2 -> measurable_fun [set: T1] ((fun x => mrestr (m2 x) measurableT)^~ X2). +Hypothesis H1 : forall X2, measurable X2 -> + measurable_fun [set: T1] ((fun x => mrestr (m2 x) measurableT)^~ X2). Let phi A := (fun x => m2 x (xsection A x)). Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. -Lemma measurable_fun_xsection A : +Lemma measurable_fun_xsection_dep A : A \in measurable -> measurable_fun setT (phi A). Proof. move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. move=> X mX. -(*move/sigma_finiteP : sf_m2 => [F F_T [F_nd F_oo]] X mX.*) -(*have -> : X = \bigcup_n (X `&` (setT `*` F n)). - by rewrite -setI_bigcupr -setM_bigcupr -F_T setMTT setIT. -apply: xsection_ndseq_closed. - move=> m n mn; apply/subsetPset; apply: setIS; apply: setSM => //. - exact/subsetPset/F_nd. -move=> n; rewrite -/B; have [? ?] := F_oo n.*) -(*pose m2Fn := [the measure _ _ of mrestr m2 (F_oo n).1].*) rewrite /B/=; split => //. rewrite /phi. rewrite -(_ : (fun x : T1 => mrestr (m2 x) measurableT (xsection X x)) = (fun x => (m2 x) (xsection X x)))//; last first. apply/funext => x//=. by rewrite /mrestr setIT. -apply measurable_prod_subset_xsection => //; last first. - move=> x. - case: sm2 => r hr. - exists r%:num => Y mY. - apply: (le_lt_trans _ (hr x)) => //. - rewrite /mrestr. - apply le_measure => //. - rewrite inE. - apply: measurableI => //. - by rewrite inE. - +apply measurable_prod_subset_xsection_dep => //. +move=> x. +case: m2_ub => r hr. +exists r%:num => Y mY. +apply: (le_lt_trans _ (hr x)) => //. +rewrite /mrestr. +apply le_measure => //. +rewrite inE. +apply: measurableI => //. +by rewrite inE. Qed. End measurable_fun_xsection. @@ -443,7 +465,7 @@ Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : T1 -> {measure set T2 -> \bar R}). -Hypotheses (sm2 : exists r : {posnum R}, forall x, m2 x [set: T2] < r%:num%:E). +Hypotheses m2_ub : kernel_uub m2. Section indic_fubini_tonelli. Variables (A : set (T1 * T2)) (mA : measurable A). @@ -452,7 +474,7 @@ Let f : (T1 * T2) -> R := \1_A. Let F := fubini_F_dep m2 (EFin \o f). -Lemma indic_fubini_tonelli_FE : F = (fun x => m2 x (xsection A x)). +Lemma indic_fubini_tonelli_FE_dep : F = (fun x => m2 x (xsection A x)). Proof. rewrite funeqE => x; rewrite /= -(setTI (xsection _ _)). rewrite -integral_indic//; last exact: measurable_xsection. @@ -469,8 +491,7 @@ Hypothesis H1 : forall X2, measurable X2 -> Lemma indic_measurable_fun_fubini_tonelli_F_dep : measurable_fun setT F. Proof. -rewrite indic_fubini_tonelli_FE//. -apply: measurable_fun_xsection => //. +rewrite indic_fubini_tonelli_FE_dep//; apply: measurable_fun_xsection_dep => //. by rewrite inE. Qed. @@ -478,15 +499,11 @@ End indic_fubini_tonelli. End fubini_tonelli. -Lemma pollard (d d' : measure_display) - (R : realType) - (X : measurableType d) - (Y : measurableType d') - (k : (X * Y)%type -> \bar R) - (k0 : (forall t : X * Y, True -> 0 <= k t)) - (mk : measurable_fun setT k) - (l : finite_kernel R X Y) : -measurable_fun [set: X] (fun x : X => \int[l x]_y k (x, y)). +Lemma pollard_finite (d d' : measure_display) (R : realType) + (X : measurableType d) (Y : measurableType d') + (k : (X * Y)%type -> \bar R) (k0 : (forall t : X * Y, True -> 0 <= k t)) + (mk : measurable_fun setT k) (l : finite_kernel R X Y) : + measurable_fun [set: X] (fun x : X => \int[l x]_y k (x, y)). Proof. have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. simpl in *. @@ -525,12 +542,13 @@ apply emeasurable_fun_sum => r. rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * \int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. apply/funext => x. - rewrite (@integralM_indic_new _ _ _ _ _ _ (fun k y => \1_(k_ n @^-1` [set r]) (x, y)))//. - - move=> r_lt0; apply/funext => y. - by rewrite preimage_nnfun0// ?indicE ?in_set0. - - apply/measurable_fun_prod1 => /=. + under eq_integral do rewrite EFinM. + rewrite (integralM_0ifneg _ _ (fun k y => (\1_(k_ n @^-1` [set r]) (x, y))%:E))//. + - by move=> _ t _; rewrite lee_fin. + - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. + - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). -apply: emeasurable_funeM. +apply: measurable_funeM. apply: indic_measurable_fun_fubini_tonelli_F_dep. - by apply/finite_kernelP. - by apply/measurable_sfunP. @@ -541,41 +559,43 @@ apply: indic_measurable_fun_fubini_tonelli_F_dep. by rewrite /mrestr setIT. Qed. -Section star_is_kernel2. -Variables (d d' : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType (d, d').-prod). -Variable k : finite_kernel R [the measurableType _ of (X * Y)%type] Z. +Module STAR_IS_FINITE_KERNEL. + +Section star_is_kernel_finite. +Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). +Variable k : kernel R [the measurableType _ of (X * Y)%type] Z. Variable l : finite_kernel R X Y. -Lemma star_measurable U : measurable U -> measurable_fun setT (mstar k l ^~ U). +Lemma star_measurable_finite U : measurable U -> measurable_fun setT (star k l ^~ U). Proof. (* k is a bounded measurable function *) (* l is a finite kernel *) move=> mU. rewrite /star. -apply: (@pollard _ _ R X Y (fun xy => k xy U)) => //. +apply: (@pollard_finite _ _ R X Y (fun xy => k xy U)) => //. by apply: (@kernelP _ _ R [the measurableType (d, d').-prod of (X * Y)%type] Z k U) => //. Qed. HB.instance Definition _ := - isKernel.Build _ _ R X Z (mstar k l) star_measurable. + isKernel.Build _ _ R X Z (mstar k l) star_measurable_finite. -End star_is_kernel2. +End star_is_kernel_finite. Section star_is_finite_kernel. -Variables (d d' : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType (d, d').-prod). +Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). Variable k : finite_kernel R [the measurableType _ of (X * Y)%type] Z. Variable l : finite_kernel R X Y. -Lemma star_finite : exists r : {posnum R}, forall x, star k l x [set: Z] < r%:num%:E. +Lemma star_finite : kernel_uub (mstar k l). Proof. have [r hr] := @finite_kernelP _ _ _ _ _ k. have [s hs] := @finite_kernelP _ _ _ _ _ l. exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x. rewrite /star. apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). - apply ge0_le_integral => //. + apply: ge0_le_integral => //. - have := @kernelP _ _ _ _ _ k setT measurableT. exact/measurable_fun_prod1. - exact/measurable_fun_cst. @@ -587,57 +607,159 @@ HB.instance Definition _ := isFiniteKernel.Build _ _ R X Z (mstar k l) star_finite. End star_is_finite_kernel. - -Lemma eq_measure (d : measure_display) (T : measurableType d) (R : realType) - (m1 m2 : {measure set T -> \bar R}) : - (forall U, measurable U -> m1 U = m2 U) -> m1 = m2. +End STAR_IS_FINITE_KERNEL. + +Lemma pollard_sfinite (d d' d3 : measure_display) (R : realType) + (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) + (k : Z -> \bar R) (k0 : (forall z, True -> 0 <= k z)) + (mk : measurable_fun setT k) + (l : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) c : + measurable_fun [set: Y] (fun x0 : Y => \int[l (c, x0)]_z k z). Proof. -Abort. +have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. +simpl in *. +rewrite (_ : (fun x0 => \int[l (c, x0)]_z k z) = + (fun x0 => elim_sup (fun n => \int[l (c, x0)]_z (k_ n z)%:E))); last first. + apply/funeqP => x. + transitivity (lim (fun n => \int[l (c, x)]_z (k_ n z)%:E)); last first. + rewrite is_cvg_elim_supE//. + apply: ereal_nondecreasing_is_cvg => m n mn. + apply: ge0_le_integral => //. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun. + - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. + rewrite -monotone_convergence//. + - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. + - by move=> n; exact/EFin_measurable_fun. + - by move=> n y' _; rewrite lee_fin. + - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. +apply: measurable_fun_elim_sup => n. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l (c, x0)]_z + ((\sum_(r <- fset_set (range (k_ n))) + r * \1_(k_ n @^-1` [set r]) z))%:E)); last first. + by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) + (\int[l (c, x0)]_z + (r * \1_(k_ n @^-1` [set r]) z)%:E))); last first. + apply/funext => x; rewrite -ge0_integral_sum//. + - by apply: eq_integral => y _; rewrite sumEFin. + - move=> r. + apply/EFin_measurable_fun/measurable_funrM => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). + - by move=> m y _; rewrite muleindic_ge0. +apply emeasurable_fun_sum => r. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * + \int[l (c ,x)]_z (\1_(k_ n @^-1` [set r]) z)%:E)); last first. + apply/funext => x. + under eq_integral do rewrite EFinM. + rewrite (integralM_0ifneg _ _ (fun k z => (\1_(k_ n @^-1` [set r]) z)%:E))//. + - by move=> _ t _; rewrite lee_fin. + - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. + - apply/EFin_measurable_fun => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). +apply: measurable_funeM. +rewrite (_ : (fun x : Y => \int[l (c, x)]_z (\1_(k_ n @^-1` [set r]) z)%:E) = + (fun x : Y => l (c, x) (k_ n @^-1` [set r]))); last first. + apply/funext => y. + by rewrite integral_indic// setIT. +have := @kernelP _ _ R _ _ l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). +rewrite /=. +move/measurable_fun_prod1. +exact. +Qed. -Section eq_measure_integral_new. -Local Open Scope ereal_scope. -Variables (d : measure_display) (T : measurableType d) (R : realType) - (D : set T). -Implicit Types m : {measure set T -> \bar R}. - -Let eq_measure_integral0 m2 m1 (f : T -> \bar R) : - (forall A, measurable A -> A `<=` D -> m1 A = m2 A) -> - [set sintegral m1 h | h in - [set h : {nnsfun T >-> R} | (forall x, (h x)%:E <= (f \_ D) x)]] `<=` - [set sintegral m2 h | h in - [set h : {nnsfun T >-> R} | (forall x, (h x)%:E <= (f \_ D) x)]]. -Proof. -move=> m12 _ [h hfD <-] /=; exists h => //; apply: eq_fsbigr => r _. -have [hrD|hrD] := pselect (h @^-1` [set r] `<=` D); first by rewrite m12. -suff : r = 0%R by move=> ->; rewrite !mul0e. -apply: contra_notP hrD => /eqP r0 t/= htr. -have := hfD t. -rewrite /patch/=; case: ifPn; first by rewrite inE. -move=> tD. -move: r0; rewrite -htr => ht0. -by rewrite le_eqVlt eqe (negbTE ht0)/= lte_fin// ltNge// fun_ge0. -Qed. - -Lemma eq_measure_integral_new m2 m1 (f : T -> \bar R) : - (forall A, measurable A -> A `<=` D -> m1 A = m2 A) -> - \int[m1]_(x in D) f x = \int[m2]_(x in D) f x. -Proof. -move=> m12; rewrite /integral funepos_restrict funeneg_restrict. -congr (ereal_sup _ - ereal_sup _)%E; rewrite eqEsubset; split; - apply: eq_measure_integral0 => A /m12 //. -by move=> /[apply]. -by move=> /[apply]. -Qed. - -End eq_measure_integral_new. -Arguments eq_measure_integral_new {d T R D} m2 {m1 f}. +Lemma pollard_sfinite2 (d d' : measure_display) (R : realType) + (X : measurableType d) (Y : measurableType d') + (k : (X * Y)%type -> \bar R) (k0 : (forall (t : X * Y), True -> 0 <= k t)) + (l : sfinite_kernel R X Y) + (mk : measurable_fun setT k) : + measurable_fun [set: X] (fun x : X => \int[l x]_y k (x, y)). +Proof. +have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. +simpl in *. +rewrite (_ : (fun x => \int[l x]_y k (x, y)) = + (fun x => elim_sup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. + apply/funeqP => x. + transitivity (lim (fun n => \int[l x]_y (k_ n (x, y))%:E)); last first. + rewrite is_cvg_elim_supE//. + apply: ereal_nondecreasing_is_cvg => m n mn. + apply: ge0_le_integral => //. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun/measurable_fun_prod1. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun/measurable_fun_prod1. + - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. + rewrite -monotone_convergence//. + - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. + - by move=> n; exact/EFin_measurable_fun/measurable_fun_prod1. + - by move=> n y' _; rewrite lee_fin. + - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. +apply: measurable_fun_elim_sup => n. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l x0]_y + ((\sum_(r <- fset_set (range (k_ n))) + r * \1_(k_ n @^-1` [set r]) (x0, y)))%:E)); last first. + by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) + (\int[l x0]_y + (r * \1_(k_ n @^-1` [set r]) (x0, y))%:E))); last first. + apply/funext => x; rewrite -ge0_integral_sum//. + - by apply: eq_integral => y _; rewrite sumEFin. + - move=> r. + apply/EFin_measurable_fun/measurable_funrM/measurable_fun_prod1 => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). + - by move=> m y _; rewrite muleindic_ge0. +apply emeasurable_fun_sum => r. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * + \int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. + apply/funext => x. + under eq_integral do rewrite EFinM. + rewrite (integralM_0ifneg _ _ (fun k y => (\1_(k_ n @^-1` [set r]) (x, y))%:E))//. + - by move=> _ t _; rewrite lee_fin. + - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. + - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). +apply: measurable_funeM. +rewrite (_ : (fun x : X => \int[l x]_z (\1_(k_ n @^-1` [set r]) (x, z))%:E) = + (fun x : X => l x (xsection (k_ n @^-1` [set r]) x))); last first. + apply/funext => y. + rewrite integral_indic//; last first. + rewrite (_ : (fun x : Y => (k_ n @^-1` [set r]) (y, x)) = xsection (k_ n @^-1` [set r]) y); last first. + apply/seteqP; split. + by move=> y2/=; rewrite /xsection/= inE//. + by rewrite /xsection/= => y2/=; rewrite inE /preimage/=. + by apply: measurable_xsection. + congr (l y _). + apply/funext => y1/=. + rewrite /xsection/= inE. + by apply/propext; tauto. +have [l_ hl_] := @sfinite_kernelP _ _ _ _ _ l. +rewrite (_ : (fun x : X => _) = + (fun x : X => mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x)) +); last first. + apply/funext => x. + rewrite hl_//. + by apply/measurable_xsection. +rewrite /mseries/=. +apply: ge0_emeasurable_fun_sum => // k1. +apply: measurable_fun_xsection_dep => //. +by have := @finite_kernelP _ _ _ _ _ (l_ k1). +move=> X2 mX2. +rewrite /mrestr. +apply/kernelP. +by rewrite setIT. +by rewrite inE. +Qed. Section star_is_sfinite_kernel. -Variables (d d' : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType (d, d').-prod). +Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). Variable k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z. Variable l : sfinite_kernel R X Y. +Import STAR_IS_FINITE_KERNEL. + Lemma star_sfinite : exists k_ : (finite_kernel R X Z)^nat, forall x U, measurable U -> mstar k l x U = [the measure _ _ of mseries (k_ ^~ x) O] U. Proof. @@ -649,9 +771,8 @@ have H1 x U : measurable U -> star k l x U = star K L x U. move=> mU. rewrite /star /L /K /=. transitivity (\int[ - [the measure _ _ of mseries (fun x0 : nat => l_ x0 x) 0] -]_y k (x, y) U). - apply eq_measure_integral_new => A mA _ . + [the measure _ _ of mseries (fun x0 : nat => l_ x0 x) 0] ]_y k (x, y) U). + apply eq_measure_integral => A mA _ . by rewrite hl_. apply eq_integral => y _. by rewrite hk_//. @@ -697,13 +818,36 @@ rewrite (reindex_esum [set: nat] [set: nat * nat] f)//. by rewrite nneseries_esum// fun_true. Qed. +Lemma star_measurable_sfinite U : measurable U -> measurable_fun setT (star k l ^~ U). +Proof. +move=> mU. +rewrite /star. +apply: (@pollard_sfinite2 _ _ _ _ _ (k ^~ U)) => //. +by apply/kernelP. +Qed. + +End star_is_sfinite_kernel. + +Module STAR_IS_SFINITE_KERNEL. +Section star_is_sfinite_kernel. +Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). +Variable k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z. +Variable l : sfinite_kernel R X Y. + HB.instance Definition _ := - isSFiniteKernel.Build d ((d, d').-prod)%mdisp R X Z (mstar k l) star_sfinite. + isKernel.Build _ _ R X Z (mstar k l) (star_measurable_sfinite k l). + +#[export] +HB.instance Definition _ := + isSFiniteKernel.Build d d3 R X Z (mstar k l) (star_sfinite k l). End star_is_sfinite_kernel. +End STAR_IS_SFINITE_KERNEL. +HB.export STAR_IS_SFINITE_KERNEL. -Lemma lemma3_indic d d' (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType (d, d').-prod) +Lemma lemma3_indic d d' d3 (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType d3) (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) (l : sfinite_kernel R X Y) x (E : set _) (mE : measurable E) : \int[mstar k l x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). @@ -712,57 +856,125 @@ rewrite integral_indic// /mstar/= /star/=. by apply eq_integral => y _; rewrite integral_indic. Qed. -Lemma lemma3_nnsfun d d' (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType (d, d').-prod) +Lemma lemma3_nnsfun d d' d3 (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType d3) (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) (l : sfinite_kernel R X Y) x (f : {nnsfun Z >-> R}) : \int[mstar k l x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). Proof. -under eq_integral do rewrite fimfunE -sumEFin. +under [in LHS]eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//; last 2 first. - move=> r. - apply/EFin_measurable_fun/measurable_funrM. + move=> r; apply/EFin_measurable_fun/measurable_funrM. have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. by rewrite (_ : \1__ = mindic R fr). by move=> r z _; rewrite EFinM muleindic_ge0. -under eq_bigr. - move=> r _. - rewrite /=. - rewrite (@integralM_indic_new _ _ _ _ _ _ (fun r x0 => \1_(f @^-1` [set r]) x0))//; last 2 first. - move=> r0. - apply/funext => z/=. - by rewrite indicE memNset// preimage_nnfun0. - have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - rewrite /=. - rewrite lemma3_indic//. +under [in RHS]eq_integral. + move=> y _. + under eq_integral. + move=> z _. + rewrite fimfunE -sumEFin. + over. + rewrite /= ge0_integral_sum//; last 2 first. + move=> r; apply/EFin_measurable_fun/measurable_funrM. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + by move=> r z _; rewrite EFinM muleindic_ge0. + under eq_bigr. + move=> r _. + rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. + by move=> r0; rewrite preimage_nnfun0. + rewrite integral_indic// setIT. + over. over. -rewrite /=. -apply/esym. -Admitted. +rewrite /= ge0_integral_sum//; last 2 first. + move=> r; apply: measurable_funeM. + have := @kernelP _ _ _ _ _ k (f @^-1` [set r]) (measurable_sfunP f r). + by move/measurable_fun_prod1; exact. + move=> n y _. + have := @mulem_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). + apply. + exact: preimage_nnfun0. +apply eq_bigr => r _. +rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. + exact: preimage_nnfun0. +rewrite /= lemma3_indic; last exact/measurable_sfunP. +rewrite (@integralM_0ifneg _ _ _ _ _ _ (fun r t => k (x, t) (f @^-1` [set r])))//; last 2 first. + move=> r0. + apply/funext => y. + by rewrite preimage_nnfun0// measure0. + have := @kernelP _ _ _ _ _ k (f @^-1` [set r]) (measurable_sfunP f r). + by move/measurable_fun_prod1; exact. +congr (_ * _). +apply eq_integral => y _. +by rewrite integral_indic// setIT. +Qed. -Lemma lemma3 d d' (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType (d, d').-prod) +Lemma lemma3 d d' d3 (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType d3) (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) (l : sfinite_kernel R X Y) x f : (forall z, 0 <= f z) -> measurable_fun setT f -> \int[mstar k l x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). Proof. move=> f0 mf. have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). -simpl in *. -(* TODO *) -Admitted. +transitivity (\int[mstar k l x]_z (lim (EFin \o (f_^~ z)))). + apply/eq_integral => z _. + apply/esym/cvg_lim => //=. + exact: f_f. +rewrite monotone_convergence//; last 3 first. + by move=> n; apply/EFin_measurable_fun. + by move=> n z _; rewrite lee_fin. + by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin. +rewrite (_ : (fun _ => _) = (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first. + by apply/funext => n; rewrite lemma3_nnsfun. +transitivity (\int[l x]_y lim (fun n => \int[k (x, y)]_z (f_ n z)%:E)). + rewrite -monotone_convergence//; last 3 first. + move=> n. + apply: pollard_sfinite => //. + - by move=> z; rewrite lee_fin. + - by apply/EFin_measurable_fun. + - move=> n y _. + by apply integral_ge0 => // z _; rewrite lee_fin. + - move=> y _ a b ab. + apply: ge0_le_integral => //. + + by move=> z _; rewrite lee_fin. + + exact/EFin_measurable_fun. + + by move=> z _; rewrite lee_fin. + + exact/EFin_measurable_fun. + + move: ab => /ndf_ /lefP ab z _. + by rewrite lee_fin. +apply eq_integral => y _. +rewrite -monotone_convergence//; last 3 first. + move=> n; exact/EFin_measurable_fun. + by move=> n z _; rewrite lee_fin. + by move=> z _ a b /ndf_ /lefP; rewrite lee_fin. +apply eq_integral => z _. +apply/cvg_lim => //. +exact: f_f. +Qed. -HB.mixin Record isProbability (d : measure_display) (T : measurableType d) - (R : realType) (P : set T -> \bar R) of isMeasure d R T P := - { probability_setT : P setT = 1%E }. +Canonical unit_pointedType := PointedType unit tt. -#[short(type=probability)] -HB.structure Definition Probability (d : measure_display) (T : measurableType d) - (R : realType) := - {P of isProbability d T R P & isMeasure d R T P }. +Section discrete_measurable_unit. + +Definition discrete_measurable_unit : set (set unit) := [set: set unit]. -Section discrete_measurable2. +Let discrete_measurable0 : discrete_measurable_unit set0. Proof. by []. Qed. + +Let discrete_measurableC X : discrete_measurable_unit X -> discrete_measurable_unit (~` X). +Proof. by []. Qed. + +Let discrete_measurableU (F : (set unit)^nat) : + (forall i, discrete_measurable_unit (F i)) -> discrete_measurable_unit (\bigcup_i F i). +Proof. by []. Qed. + +HB.instance Definition _ := @isMeasurable.Build default_measure_display unit (Pointed.class _) + discrete_measurable_unit discrete_measurable0 discrete_measurableC + discrete_measurableU. + +End discrete_measurable_unit. + +Section discrete_measurable_bool. Definition discrete_measurable_bool : set (set bool) := [set: set bool]. @@ -781,109 +993,549 @@ HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Po discrete_measurable_bool discrete_measurable0 discrete_measurableC discrete_measurableU. -End discrete_measurable2. +End discrete_measurable_bool. -Definition twoseven (R : realType) : {nonneg R}. -Admitted. +Section nonneg_constants. +Variable R : realType. +Let twoseven_proof : (0 <= 2 / 7 :> R)%R. +Proof. by rewrite divr_ge0// ler0n. Qed. -Definition fiveseven (R : realType) : {nonneg R}. -Admitted. +Definition twoseven : {nonneg R} := NngNum twoseven_proof. -Definition bernoulli (R : realType) : {measure set _ -> \bar R} := - [the measure _ _ of measure_add +Let fiveseven_proof : (0 <= 5 / 7 :> R)%R. +Proof. by rewrite divr_ge0// ler0n. Qed. + +Definition fiveseven : {nonneg R} := NngNum fiveseven_proof. + +End nonneg_constants. + +Lemma measure_diract_setT_true (R : realType) : + [the measure _ _ of dirac true] [set: bool] = 1 :> \bar R. +Proof. by rewrite /= diracE in_setT. Qed. + +Lemma measure_diract_setT_false (R : realType) : + [the measure _ _ of dirac false] [set: bool] = 1 :> \bar R. +Proof. by rewrite /= diracE in_setT. Qed. + +Section bernoulli27. +Variable R : realType. + +Definition bernoulli27 : set _ -> \bar R := + measure_add [the measure _ _ of mscale (twoseven R) [the measure _ _ of dirac true]] - [the measure _ _ of mscale (fiveseven R) [the measure _ _ of dirac false]]]. + [the measure _ _ of mscale (fiveseven R) [the measure _ _ of dirac false]]. -Canonical unit_pointedType := PointedType unit tt. +HB.instance Definition _ := Measure.on bernoulli27. -Section unit_measurable. +Lemma bernoulli27_setT : bernoulli27 [set: _] = 1. +Proof. +rewrite /bernoulli27/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. +rewrite /mscale/= !diracE !in_setT !mule1 -EFinD. +by rewrite -mulrDl -natrD divrr// unitfE pnatr_eq0. +Qed. -Definition unit_measurable : set (set unit) := [set: set unit]. +HB.instance Definition _ := @isProbability.Build _ _ R bernoulli27 bernoulli27_setT. -Let unit_measurable0 : unit_measurable set0. Proof. by []. Qed. +End bernoulli27. -Let unit_measurableC X : unit_measurable X -> unit_measurable (~` X). -Proof. by []. Qed. +Section kernel_from_mzero. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (d' : measure_display) (T' : measurableType d'). -Let unit_measurableU (F : (set unit)^nat) : - (forall i, unit_measurable (F i)) -> unit_measurable (\bigcup_i F i). -Proof. by []. Qed. +Definition kernel_from_mzero : T' -> {measure set T -> \bar R} := + fun _ : T' => [the measure _ _ of mzero]. -HB.instance Definition _ := @isMeasurable.Build default_measure_display unit (Pointed.class _) - unit_measurable unit_measurable0 unit_measurableC - unit_measurableU. +Lemma kernel_from_mzeroP : forall U, measurable U -> + measurable_fun setT (kernel_from_mzero ^~ U). +Proof. by move=> U mU/=; exact: measurable_fun_cst. Qed. + +HB.instance Definition _ := + @isKernel.Build d' d R T' T kernel_from_mzero + kernel_from_mzeroP. -End unit_measurable. +Lemma kernel_from_mzero_uub : kernel_uub kernel_from_mzero. +Proof. +exists (PosNum ltr01) => /= t. +by rewrite /mzero/=. +Qed. + +HB.instance Definition _ := + @isFiniteKernel.Build d' d R _ T kernel_from_mzero + kernel_from_mzero_uub. + +End kernel_from_mzero. + +(* a finite kernel is always an s-finite kernel *) +Lemma finite_kernel_sfinite_kernelP (d : measure_display) + (R : realType) (X : measurableType d) (d' : measure_display) (T : measurableType d') + (k : finite_kernel R T X) : + exists k_ : (finite_kernel R _ _)^nat, forall x U, measurable U -> + k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +exists (fun n => if n is O then + k + else + [the finite_kernel _ _ _ of @kernel_from_mzero _ X R _ T] + ). +move=> t U mU/=. +rewrite /mseries. +rewrite (nneseries_split 1%N)// big_ord_recl/= big_ord0 adde0. +rewrite ereal_series (@eq_nneseries _ _ (fun=> 0%E)); last first. + by case. +by rewrite nneseries0// adde0. +Qed. (* semantics for a sample operation? *) -Section kernel_from_measure. +Section kernel_probability. Variables (d : measure_display) (R : realType) (X : measurableType d). -Variable m : {measure set X -> \bar R}. (* measure, probability measure *) +Variables (d' : _) (T' : measurableType d'). +Variable m : probability X R. -Definition kernel_from_measure : unit -> {measure set X -> \bar R} := - fun _ : unit => m. +Definition kernel_probability : T' -> {measure set X -> \bar R} := + fun _ : T' => m. -Lemma kernel_from_measureP : forall U, measurable U -> measurable_fun setT (kernel_from_measure ^~ U). -Proof. by []. Qed. +Lemma kernel_probabilityP : forall U, measurable U -> + measurable_fun setT (kernel_probability ^~ U). +Proof. +move=> U mU. +rewrite /kernel_probability. +exact: measurable_fun_cst. +Qed. + +HB.instance Definition _ := + @isKernel.Build _ d R _ X kernel_probability + kernel_probabilityP. + +Lemma kernel_probability_uub : kernel_uub kernel_probability. +Proof. +(*NB: shouldn't this work? exists 2%:pos. *) +exists (PosNum (addr_gt0 ltr01 ltr01)) => /= ?. +rewrite (le_lt_trans (probability_le1 m measurableT))//. +by rewrite lte_fin ltr_addr. +Qed. + +HB.instance Definition _ := + @isFiniteKernel.Build _ d R _ X kernel_probability + kernel_probability_uub. + +Lemma kernel_probability_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, + forall x U, measurable U -> + kernel_probability x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. exact: finite_kernel_sfinite_kernelP. Qed. HB.instance Definition _ := - @isKernel.Build default_measure_display d R _ X kernel_from_measure - kernel_from_measureP. -End kernel_from_measure. + @isSFiniteKernel.Build _ d R _ X kernel_probability + kernel_probability_sfinite_kernelP. + +End kernel_probability. (* semantics for return? *) -Section kernel_from_dirac. +Section kernel_dirac. Variables (R : realType) (d : _) (T : measurableType d). -Definition kernel_from_dirac : T -> {measure set T -> \bar R} := +Definition kernel_dirac : T -> {measure set T -> \bar R} := fun x => [the measure _ _ of dirac x]. -Lemma kernel_from_diracP : forall U, measurable U -> measurable_fun setT (kernel_from_dirac ^~ U). +Lemma kernel_diracP U : measurable U -> measurable_fun setT (kernel_dirac ^~ U). Proof. -move=> U mU. -rewrite /kernel_from_dirac. -rewrite /=. -rewrite /dirac/=. -apply/EFin_measurable_fun. -rewrite [X in measurable_fun _ X](_ : _ = mindic R mU)//. +move=> mU; apply/EFin_measurable_fun. +by rewrite [X in measurable_fun _ X](_ : _ = mindic R mU). +Qed. + +HB.instance Definition _ := isKernel.Build _ _ R _ _ kernel_dirac kernel_diracP. + +Lemma kernel_dirac_uub : kernel_uub kernel_dirac. +Proof. +exists (PosNum (addr_gt0 ltr01 ltr01)) => t/=. +by rewrite diracE in_setT lte_fin ltr_addr. Qed. HB.instance Definition _ := - isKernel.Build _ _ R _ _ kernel_from_dirac kernel_from_diracP. -End kernel_from_dirac. + @isFiniteKernel.Build d d R _ T kernel_dirac kernel_dirac_uub. -(* let x = sample (bernoulli 2/7) in - return x *) +Lemma kernel_dirac_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, + forall x U, measurable U -> + kernel_dirac x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. exact: finite_kernel_sfinite_kernelP. Qed. + +HB.instance Definition _ := + @isSFiniteKernel.Build d d R T T kernel_dirac kernel_dirac_sfinite_kernelP. + +End kernel_dirac. + +Section kernel_dirac2. +Variables (R : realType) (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variable (f : T -> T'). + +Definition kernel_dirac2 (mf : measurable_fun setT f) : T -> {measure set T' -> \bar R} := + fun x => [the measure _ _ of dirac (f x)]. + +Variable (mf : measurable_fun setT f). + +Lemma kernel_dirac2P U : measurable U -> measurable_fun setT (kernel_dirac2 mf ^~ U). +Proof. +move=> mU; apply/EFin_measurable_fun. +have mTU : measurable (f @^-1` U). + have := mf measurableT mU. + by rewrite setTI. +by rewrite [X in measurable_fun _ X](_ : _ = mindic R mTU). +Qed. + +HB.instance Definition _ := + isKernel.Build _ _ R _ _ (kernel_dirac2 mf) kernel_dirac2P. + +Lemma kernel_dirac2_uub : kernel_uub (kernel_dirac2 mf). +Proof. +exists (PosNum (addr_gt0 ltr01 ltr01)) => t/=. +by rewrite diracE in_setT lte_fin ltr_addr. +Qed. + +HB.instance Definition _ := + @isFiniteKernel.Build _ _ R _ _ (kernel_dirac2 mf) kernel_dirac2_uub. + +Lemma kernel_dirac2_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, + forall x U, measurable U -> + kernel_dirac2 mf x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. exact: finite_kernel_sfinite_kernelP. Qed. + +HB.instance Definition _ := + @isSFiniteKernel.Build _ _ R _ _ (kernel_dirac2 mf) kernel_dirac2_sfinite_kernelP. + +End kernel_dirac2. Definition letin (d d' d3 : measure_display) (R : realType) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) - (l : X ^^> Y) (k : _ ^^> Z) : X -> {measure set Z -> \bar R}:= - @mstar _ _ _ R _ _ _ k l. + (l : sfinite_kernel R X Y) + (k : sfinite_kernel R [the measurableType (d, d').-prod of (X * Y)%type] Z) + : sfinite_kernel R X Z := + [the sfinite_kernel _ _ _ of @mstar d d' d3 R X Y Z k l]. -Section sample_program. +(* semantics for score? *) + +Lemma set_unit (A : set unit) : A = set0 \/ A = setT. +Proof. +have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. +by apply/seteqP; split => [|] []. +Qed. + +Section score_measure. +Variables (R : realType). + +Definition mscore (r : R) (U : set unit) : \bar R := if U == set0 then 0 else `| r%:E |. + +Lemma mscore0 r : mscore r (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /mscore eqxx. Qed. + +Lemma mscore_ge0 r U : 0 <= mscore r U. +Proof. by rewrite /mscore; case: ifP. Qed. + +Lemma mscore_sigma_additive r : semi_sigma_additive (mscore r). +Proof. +move=> /= F mF tF mUF; rewrite /mscore; case: ifPn => [/eqP/bigcup0P F0|]. + rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. + apply/funext => k. + under eq_bigr do rewrite F0// eqxx. + by rewrite big1. +move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. +rewrite -(cvg_shiftn k.+1)/=. +rewrite (_ : (fun _ => _) = cst `|r%:E|); first exact: cvg_cst. +apply/funext => n. +rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. +rewrite (negbTE Fk0) big1 ?adde0// => i/= ik; rewrite ifT//. +have [/eqP//|Fitt] := set_unit (F i). +move/trivIsetP : tF => /(_ i k Logic.I Logic.I ik). +by rewrite Fitt setTI => /eqP; rewrite (negbTE Fk0). +Qed. + +HB.instance Definition _ (r : R) := isMeasure.Build _ _ _ + (mscore r) (mscore0 r) (mscore_ge0 r) (@mscore_sigma_additive r). + +End score_measure. + +(* NB: score r = observe 0 from exp r, + the density of the exponential distribution exp(r) at 0 is r = r e^(-r * 0) + more generally, score (r e^(-r * t)) = observe t from exp(r), + score (f(r)) = observe r from p where f is the density of p + +*) + +Module KERNEL_SCORE. +Section kernel_score. +Variable (R : realType) (d : _) (T : measurableType d). + +Definition k_' (r : R) (i : nat) : T -> set unit -> \bar R := + fun _ U => + if i%:R%:E <= mscore r U < i.+1%:R%:E then + mscore r U + else + 0. + +Lemma k_'0 (r : R) i (t : T) : k_' r i t (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /k_' measure0; case: ifP. Qed. + +Lemma k_'ge0 (r : R) i (t : T) B : 0 <= k_' r i t B. +Proof. by rewrite /k_'; case: ifP. Qed. + +Lemma k_'sigma_additive (r : R) i (t : T) : semi_sigma_additive (k_' r i t). +Proof. +move=> /= F mF tF mUF. +rewrite /k_' /=. +have [F0|] := eqVneq (\bigcup_n F n) set0. + rewrite [in X in _ --> X]/mscore F0 eqxx. + rewrite (_ : (fun _ => _) = cst 0). + by case: ifPn => _; exact: cvg_cst. + apply/funext => k; rewrite big1// => n _. + move : F0 => /bigcup0P F0. + by rewrite /mscore F0// eqxx; case: ifP. +move=> UF0; move: (UF0). +move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. +rewrite [in X in _ --> X]/mscore (negbTE UF0). +rewrite -(cvg_shiftn k.+1)/=. +case: ifPn => ir. + rewrite (_ : (fun _ => _) = cst `|r%:E|); first exact: cvg_cst. + apply/funext => n. + rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. + rewrite [in X in X + _]/mscore (negbTE Fk0) ir big1 ?adde0// => /= j jk. + rewrite /mscore. + have /eqP Fj0 : F j == set0. + have [/eqP//|Fjtt] := set_unit (F j). + move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). + by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). + rewrite Fj0 eqxx. + by case: ifP. +rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. +apply/funext => n. +rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. +rewrite [in X in if X then _ else _]/mscore (negbTE Fk0) (negbTE ir) add0e. +rewrite big1//= => j jk. +rewrite /mscore. +have /eqP Fj0 : F j == set0. + have [/eqP//|Fjtt] := set_unit (F j). + move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). + by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). +rewrite Fj0 eqxx. +by case: ifP. +Qed. + +HB.instance Definition _ (r : R) (i : nat) (t : T) := isMeasure.Build _ _ _ + (k_' r i t) (k_'0 r i t) (k_'ge0 r i t) (@k_'sigma_additive r i t). + +Lemma k_kernelP (r : R) (i : nat) : forall U, measurable U -> measurable_fun setT (k_' r i ^~ U). +Proof. +move=> /= U mU. +rewrite /k_'. +by case: ifPn => _; exact: measurable_fun_cst. +Qed. + +Definition mk_' (r : R) i (t : T) := [the measure _ _ of k_' r i t]. + +HB.instance Definition _ (r : R) (i : nat) := + isKernel.Build _ _ R _ _ (mk_' r i) (k_kernelP r i). + +Lemma k_uub (r : R) (i : nat) : kernel_uub (mk_' r i). +Proof. +exists (PosNum (ltr0Sn _ i)) => /= t. +rewrite /k_' /mscore setT_unit. +rewrite (_ : [set tt] == set0 = false); last first. + by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). +by case: ifPn => // /andP[]. +Qed. + +HB.instance Definition _ (r : R) (i : nat) := + @isFiniteKernel.Build _ _ R _ _ (mk_' r i) (k_uub r i). + +End kernel_score. +End KERNEL_SCORE. + +Section kernel_score_kernel. +Variables (R : realType) (d : _) (T : measurableType d). + +Definition kernel_score (r : R) : T -> {measure set _ -> \bar R} := + fun _ : T => [the measure _ _ of mscore r]. + +Lemma kernel_scoreP (r : R) : forall U, measurable U -> + measurable_fun setT (kernel_score r ^~ U). +Proof. +move=> /= U mU; rewrite /mscore; case: ifP => U0. + exact: measurable_fun_cst. +apply: measurable_fun_comp => //. +apply/EFin_measurable_fun. +exact: measurable_fun_cst. +Qed. + +HB.instance Definition _ (r : R) := + isKernel.Build _ _ R T + _ (*Datatypes_unit__canonical__measure_Measurable*) + (kernel_score r) (kernel_scoreP r). +End kernel_score_kernel. + +Section kernel_score_sfinite_kernel. +Variables (R : realType) (d : _) (T : measurableType d). + +Import KERNEL_SCORE. + +Lemma kernel_score_sfinite_kernelP (r : R) : exists k_ : (finite_kernel R T _)^nat, + forall x U, measurable U -> + kernel_score r x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +exists (fun i => [the finite_kernel _ _ _ of mk_' r i]) => /= r' U mU. +rewrite /mseries /mscore; case: ifPn => [/eqP U0|U0]. + by apply/esym/nneseries0 => i _; rewrite U0 measure0. +rewrite /mk_' /= /k_' /= /mscore (negbTE U0). +apply/esym/cvg_lim => //. +rewrite -(cvg_shiftn `|floor (fine `|r%:E|)|%N.+1)/=. +rewrite (_ : (fun _ => _) = cst `|r%:E|); first exact: cvg_cst. +apply/funext => n. +pose floor_r := widen_ord (leq_addl n `|floor `|r| |.+1) (Ordinal (ltnSn `|floor `|r| |)). +rewrite big_mkord (bigD1 floor_r)//= ifT; last first. + rewrite lee_fin lte_fin; apply/andP; split. + by rewrite natr_absz (@ger0_norm _ (floor `|r|)) ?floor_ge0 ?floor_le. + by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|r|)) ?floor_ge0 ?lt_succ_floor. +rewrite big1 ?adde0//= => j jk. +rewrite ifF// lte_fin lee_fin. +move: jk; rewrite neq_ltn/= => /orP[|] jr. +- suff : (j.+1%:R <= `|r|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. + rewrite (_ : j.+1%:R = j.+1%:~R)// floor_ge_int. + move: jr; rewrite -lez_nat => /le_trans; apply. + by rewrite -[leRHS](@ger0_norm _ (floor `|r|)) ?floor_ge0. +- suff : (`|r| < j%:R)%R by rewrite ltNge => /negbTE ->. + move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|r|)) ?floor_ge0// ltr_int. + by rewrite -floor_lt_int. +Qed. + +HB.instance Definition _ (r : R) := @isSFiniteKernel.Build _ _ _ _ _ + (kernel_score r) (kernel_score_sfinite_kernelP r). + +End kernel_score_sfinite_kernel. + +Section ite. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables + (u1 : sfinite_kernel R + [the measurableType _ of (T * bool)%type] + [the measurableType _ of T']) + (u2 : sfinite_kernel R + [the measurableType _ of (T * bool)%type] + [the measurableType _ of T']). + +Definition ite : T * bool -> set _ -> \bar R := + fun t => if t.2 then u1 t else u2 t. + +Lemma ite0 tb : ite tb set0 = 0. +Proof. by rewrite /ite; case: ifPn => //. Qed. + +Lemma ite_ge0 tb (U : set _) : 0 <= ite tb U. +Proof. by rewrite /ite; case: ifPn => //. Qed. + +Lemma ite_sigma_additive tb : semi_sigma_additive (ite tb). +Proof. +Admitted. + +HB.instance Definition _ tb := isMeasure.Build _ _ _ + (ite tb) + (ite0 tb) (ite_ge0 tb) (@ite_sigma_additive tb). + +Lemma ite_kernelP : forall U, measurable U -> measurable_fun setT (ite ^~ U). +Admitted. + +Definition mite tb := [the measure _ _ of ite tb]. + +HB.instance Definition _ := isKernel.Build _ _ R _ _ mite ite_kernelP. + +Lemma ite_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, + forall x U, measurable U -> + ite x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Admitted. + +HB.instance Definition _ := + @isSFiniteKernel.Build _ _ _ _ _ mite ite_sfinite_kernelP. + +End ite. + +Section insn. Variables (R : realType). -Definition sample_bernoulli27 (*NB: 1 ^^> bool *) := - [the kernel _ _ _ of kernel_from_measure (bernoulli R)] . +Definition sample_bernoulli27 (d : _) (T : measurableType d) := + [the sfinite_kernel _ T _ of + kernel_probability [the probability _ _ of bernoulli27 R]] . + +Definition Ite (d d' : _) (T : measurableType d) (T' : measurableType d') + (u1 : sfinite_kernel R [the measurableType _ of (T * bool)%type] + [the measurableType _ of T']) + (u2 : sfinite_kernel R [the measurableType _ of (T * bool)%type] + [the measurableType _ of T']) + : sfinite_kernel R [the measurableType _ of (T * bool)%type] _ := + [the sfinite_kernel R _ _ of mite u1 u2]. -Definition Return : kernel R _ [the measurableType (default_measure_display,default_measure_display).-prod of (Datatypes_unit__canonical__measure_SemiRingOfSets * Datatypes_bool__canonical__measure_SemiRingOfSets)%type] (* NB: 1 * bool ^^> 1 * bool *) := - [the kernel _ _ _ of @kernel_from_dirac R _ _]. +Definition Return (d : _) (T : measurableType d) : sfinite_kernel R T T := + [the sfinite_kernel _ _ _ of @kernel_dirac R _ _]. -Definition program : unit -> set (unit * bool) -> \bar R (* NB: 1 ^^> 1 * bool *) := +Definition Return2 (d d' : _) (T : measurableType d) (T' : measurableType d') + (f : T -> T') (mf : measurable_fun setT f) : sfinite_kernel R T T' := + [the sfinite_kernel _ _ _ of @kernel_dirac2 R _ _ T T' f mf]. + +Definition Score (d : _) (T : measurableType d) (r : R) : + sfinite_kernel R T Datatypes_unit__canonical__measure_Measurable := + [the sfinite_kernel R _ _ of @kernel_score R _ _ r]. + +End insn. + +Section program1. +Variables (R : realType) (d : _) (T : measurableType d). + +Lemma measurable_fun_snd : measurable_fun setT (snd : T * bool -> bool). Admitted. + +Definition program1 : sfinite_kernel R T + _ := letin - sample_bernoulli27 - Return. + (sample_bernoulli27 R T) (* T -> B *) + (Return2 R measurable_fun_snd) (* T * B -> B *). -Lemma programE : forall U, program tt U = - ((twoseven R)%:num)%:E * \d_(tt, true) U + - ((fiveseven R)%:num)%:E * \d_(tt, false) U. +Lemma program1E (t : T) (U : _) : program1 t U = + ((twoseven R)%:num)%:E * \d_true U + + ((fiveseven R)%:num)%:E * \d_false U. Proof. -move=> U. -rewrite /program/= /star/=. +rewrite /program1/= /star/=. rewrite ge0_integral_measure_sum// 2!big_ord_recl/= big_ord0 adde0/=. rewrite !ge0_integral_mscale//=. rewrite !integral_dirac//=. by rewrite indicE in_setT mul1e indicE in_setT mul1e. Qed. -End sample_program. +End program1. + +Section program2. +Variables (R : realType) (d : _) (T : measurableType d). + +Definition program2 : sfinite_kernel R T Datatypes_unit__canonical__measure_Measurable := + letin + (sample_bernoulli27 R T) (* T -> B *) + (Score _ (1%:R : R)). + +End program2. + +Section program3. +Variables (R : realType) (d : _) (T : measurableType d). + +(* let x = sample (bernoulli (2/7)) in + let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in + let _ = score (1/4! r^4 e^-r) in + return x *) + +Definition k3' : T * bool -> R := cst 3. +Definition k10' : T * bool -> R := cst 10. + +Lemma mk3 : measurable_fun setT k3'. +exact: measurable_fun_cst. +Qed. + +Lemma mk10 : measurable_fun setT k10'. +exact: measurable_fun_cst. +Qed. + +Definition program10 : sfinite_kernel R T _ := + letin + (sample_bernoulli27 R T) (* T -> B *) + (Return2 R mk3). + +End program3. From a407f090a3ef33ad109a4102fd8026a3528fad85 Mon Sep 17 00:00:00 2001 From: saito ayumu Date: Mon, 8 Aug 2022 12:05:23 +0900 Subject: [PATCH 30/42] nonneg 2/7 --- theories/kernel.v | 22 +++++++++++++++++----- 1 file changed, 17 insertions(+), 5 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 1e1abeb33c..34f48e4c1d 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -997,15 +997,20 @@ End discrete_measurable_bool. Section nonneg_constants. Variable R : realType. +(* Let twoseven_proof : (0 <= 2 / 7 :> R)%R. Proof. by rewrite divr_ge0// ler0n. Qed. +*) -Definition twoseven : {nonneg R} := NngNum twoseven_proof. +(* Check (2%:R / 7%:R)%:nng. *) +(* Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. *) +(* Let fiveseven_proof : (0 <= 5 / 7 :> R)%R. Proof. by rewrite divr_ge0// ler0n. Qed. Definition fiveseven : {nonneg R} := NngNum fiveseven_proof. + *) End nonneg_constants. @@ -1020,13 +1025,20 @@ Proof. by rewrite /= diracE in_setT. Qed. Section bernoulli27. Variable R : realType. +Local Open Scope ring_scope. +Notation "'2/7'" := (2%:R / 7%:R)%:nng. +Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. +Definition fiveseven : {nonneg R} := (5%:R / 7%:R)%:nng. + Definition bernoulli27 : set _ -> \bar R := measure_add - [the measure _ _ of mscale (twoseven R) [the measure _ _ of dirac true]] - [the measure _ _ of mscale (fiveseven R) [the measure _ _ of dirac false]]. + [the measure _ _ of mscale twoseven [the measure _ _ of dirac true]] + [the measure _ _ of mscale fiveseven [the measure _ _ of dirac false]]. HB.instance Definition _ := Measure.on bernoulli27. +Local Close Scope ring_scope. + Lemma bernoulli27_setT : bernoulli27 [set: _] = 1. Proof. rewrite /bernoulli27/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. @@ -1522,8 +1534,8 @@ Variables (R : realType) (d : _) (T : measurableType d). let _ = score (1/4! r^4 e^-r) in return x *) -Definition k3' : T * bool -> R := cst 3. -Definition k10' : T * bool -> R := cst 10. +Definition k3' : T * bool -> R := cst 3%:R. +Definition k10' : T * bool -> R := cst 10%:R. Lemma mk3 : measurable_fun setT k3'. exact: measurable_fun_cst. From b33db3e958d3ebf79ab9f6344032d3f6545f6f06 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Mon, 8 Aug 2022 17:31:23 +0900 Subject: [PATCH 31/42] s-finite kernels for ite and examples - some rewriting laws for programs --- _CoqProject | 2 + theories/kernel.v | 1730 +++++++++++++++++----------------- theories/lebesgue_integral.v | 12 +- theories/prob_lang.v | 357 +++++++ 4 files changed, 1221 insertions(+), 880 deletions(-) create mode 100644 theories/prob_lang.v diff --git a/_CoqProject b/_CoqProject index 9adc71f2ff..b5a4ba0fef 100644 --- a/_CoqProject +++ b/_CoqProject @@ -32,6 +32,8 @@ theories/derive.v theories/measure.v theories/numfun.v theories/lebesgue_integral.v +theories/kernel.v +theories/prob_lang.v theories/summability.v theories/functions.v theories/signed.v diff --git a/theories/kernel.v b/theories/kernel.v index 34f48e4c1d..1ea424ee32 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -5,6 +5,20 @@ Require Import mathcomp_extra boolp classical_sets signed functions cardinality. Require Import reals ereal topology normedtype sequences esum measure. Require Import lebesgue_measure fsbigop numfun lebesgue_integral. +(******************************************************************************) +(* Kernels *) +(* *) +(* R.-ker X ~> Y == kernel *) +(* R.-fker X ~> Y == finite kernel *) +(* R.-sfker X ~> Y == s-finite kernel *) +(* sum_of_kernels == *) +(* l \; k == composition of kernels *) +(* kernel_mfun == kernel defined by a measurable function *) +(* mscore == *) +(* ite_true/ite_false == *) +(* add_of_kernels == *) +(******************************************************************************) + Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. @@ -35,147 +49,14 @@ Admitted. End probability_lemmas. (* /PR 516 in progress *) -HB.mixin Record isKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') - (k : X -> {measure set Y -> \bar R}) := { - kernelP : forall U, measurable U -> measurable_fun setT (k ^~ U) -}. - -#[short(type=kernel)] -HB.structure Definition Kernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') := - {k & isKernel d d' R X Y k}. -Notation "X ^^> Y" := (kernel _ X Y) (at level 42). - -(* TODO: define using the probability type *) -HB.mixin Record isProbabilityKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') - (k : X -> {measure set Y -> \bar R}) - of isKernel d d' R X Y k := { - prob_kernelP : forall x : X, k x [set: Y] = 1 -}. - -#[short(type=probability_kernel)] -HB.structure Definition ProbabilityKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') := - {k of isProbabilityKernel d d' R X Y k & isKernel d d' R X Y k}. - -Section sum_of_kernels. -Variables (d d' : measure_display) (R : realType). -Variables (X : measurableType d) (Y : measurableType d'). -Variable k : (kernel R X Y)^nat. - -Definition sum_of_kernels : X -> {measure set Y -> \bar R} := - fun x => [the measure _ _ of mseries (k ^~ x) 0]. - -Lemma kernel_measurable_fun_sum_of_kernels (U : set Y) : - measurable U -> - measurable_fun setT (sum_of_kernels ^~ U). -Proof. -move=> mU; rewrite /sum_of_kernels /= /mseries. -rewrite [X in measurable_fun _ X](_ : _ = - (fun x => elim_sup (fun n => \sum_(0 <= i < n) k i x U))); last first. - apply/funext => x; rewrite -lim_mkord is_cvg_elim_supE. - by rewrite -lim_mkord. - exact: is_cvg_nneseries. -apply: measurable_fun_elim_sup => n. -by apply: emeasurable_fun_sum => *; exact/kernelP. -Qed. - -HB.instance Definition _ := - isKernel.Build d d' R X Y sum_of_kernels - kernel_measurable_fun_sum_of_kernels. - -End sum_of_kernels. - -Lemma integral_sum_of_kernels (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') - (k : (X ^^> Y)^nat) (f : Y -> \bar R) x : - (forall y, 0 <= f y) -> - measurable_fun setT f -> - \int[sum_of_kernels k x]_y (f y) = \sum_(i f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. -Qed. - -Section kernel_uub. -Variables (d d' : measure_display) (R : numFieldType) (X : measurableType d) - (Y : measurableType d') (k : X -> set Y -> \bar R). - -Definition kernel_uub := exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E. - -End kernel_uub. - -HB.mixin Record isFiniteKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') - (k : X -> {measure set Y -> \bar R}) - := { finite_kernelP : kernel_uub k }. - -#[short(type=finite_kernel)] -HB.structure Definition FiniteKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') := - {k of isFiniteKernel d d' R X Y k & isKernel d d' R X Y k}. - -HB.mixin Record isSFiniteKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') - (k : X -> {measure set Y -> \bar R}) - := { - sfinite_kernelP : exists k_ : (finite_kernel R X Y)^nat, - forall x U, measurable U -> - k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U -}. - -#[short(type=sfinite_kernel)] -HB.structure Definition SFiniteKernel (d d' : measure_display) - (R : realType) (X : measurableType d) (Y : measurableType d') := - {k of isSFiniteKernel d d' R X Y k & - isKernel d d' R X Y k}. - -Section star_is_measure. -Variables (d1 d2 d3 : _) (R : realType) (X : measurableType d1) - (Y : measurableType d2) (Z : measurableType d3). -Variable k : kernel R [the measurableType _ of (X * Y)%type] Z. -Variable l : kernel R X Y. - -Definition star : X -> set Z -> \bar R := fun x U => \int[l x]_y k (x, y) U. - -Let star0 x : star x set0 = 0. -Proof. -by rewrite /star (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. -Qed. - -Let star_ge0 x U : 0 <= star x U. -Proof. by apply: integral_ge0 => y _; exact: measure_ge0. Qed. - -Let star_sigma_additive x : semi_sigma_additive (star x). -Proof. -move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = - \int[l x]_y (\sum_(n V _. - by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. -apply/cvg_closeP; split. - by apply: is_cvg_nneseries => n _; exact: integral_ge0. -rewrite closeE// integral_sum// => n. -have := @kernelP _ _ R _ _ k (U n) (mU n). -exact/measurable_fun_prod1. -Qed. - -HB.instance Definition _ x := isMeasure.Build _ R _ - (star x) (star0 x) (star_ge0 x) (@star_sigma_additive x). - -Definition mstar : X -> {measure set Z -> \bar R} := - fun x => [the measure _ _ of star x]. - -End star_is_measure. - -(* TODO: PR *) +(* TODO: PR? *) Section integralM_0ifneg. Local Open Scope ereal_scope. -Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables (d : _) (T : measurableType d) (R : realType). Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). Lemma integralM_0ifneg (f : R -> T -> \bar R) (k : R) - (f0 : forall r t, D t -> (0 <= f r t)) : + (f0 : forall r t, D t -> 0 <= f r t) : ((k < 0)%R -> f k = cst 0%E) -> measurable_fun setT (f k) -> \int[m]_(x in D) (k%:E * (f k) x) = k%:E * \int[m]_(x in D) ((f k) x). Proof. @@ -192,30 +73,6 @@ Qed. End integralM_0ifneg. Arguments integralM_0ifneg {d T R} m {D} mD f. -(*Section integralM_0ifneg. -Local Open Scope ereal_scope. -Variables (d : measure_display) (T : measurableType d) (R : realType). -Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). - -Lemma integralM_0ifneg (f : R -> T -> R) (k : R) - (f0 : forall r t, D t -> (0 <= f r t)%R) : - ((k < 0)%R -> f k = cst 0%R) -> measurable_fun setT (f k) -> - \int[m]_(x in D) (k * (f k) x)%:E = k%:E * \int[m]_(x in D) ((f k) x)%:E. -Proof. -move=> fk0 mfk; have [k0|k0] := ltP k 0%R. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. - by move=> x _; rewrite fk0// mulr0. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. - by rewrite fk0// indic0. -under eq_integral do rewrite EFinM. -rewrite ge0_integralM//. -- apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. -- by move=> y Dy; rewrite lee_fin f0. -Qed. - -End integralM_0ifneg. -Arguments integralM_0ifneg {d T R} m {D} mD f.*) - Section integralM_indic. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). @@ -331,49 +188,276 @@ Qed. End integral_mscale. -Section ndseq_closed_B. -Variables (d1 d2 : measure_display). -Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). -Implicit Types A : set (T1 * T2). +(* TODO: PR *) +Canonical unit_pointedType := PointedType unit tt. -Section xsection. -Variables (pt2 : T2) (m2 : T1 -> {measure set T2 -> \bar R}). -Let phi A x := m2 x (xsection A x). -Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. +Section discrete_measurable_unit. + +Definition discrete_measurable_unit : set (set unit) := [set: set unit]. + +Let discrete_measurable0 : discrete_measurable_unit set0. Proof. by []. Qed. + +Let discrete_measurableC X : discrete_measurable_unit X -> discrete_measurable_unit (~` X). +Proof. by []. Qed. + +Let discrete_measurableU (F : (set unit)^nat) : + (forall i, discrete_measurable_unit (F i)) -> discrete_measurable_unit (\bigcup_i F i). +Proof. by []. Qed. + +HB.instance Definition _ := @isMeasurable.Build default_measure_display unit (Pointed.class _) + discrete_measurable_unit discrete_measurable0 discrete_measurableC + discrete_measurableU. + +End discrete_measurable_unit. + +Section discrete_measurable_bool. + +Definition discrete_measurable_bool : set (set bool) := [set: set bool]. + +Let discrete_measurable0 : discrete_measurable_bool set0. Proof. by []. Qed. + +Let discrete_measurableC X : + discrete_measurable_bool X -> discrete_measurable_bool (~` X). +Proof. by []. Qed. + +Let discrete_measurableU (F : (set bool)^nat) : + (forall i, discrete_measurable_bool (F i)) -> + discrete_measurable_bool (\bigcup_i F i). +Proof. by []. Qed. + +HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Pointed.class _) + discrete_measurable_bool discrete_measurable0 discrete_measurableC + discrete_measurableU. + +End discrete_measurable_bool. + +Lemma measurable_fun_fst (d1 d2 : _) (T1 : measurableType d1) + (T2 : measurableType d2) : measurable_fun setT (@fst T1 T2). +Proof. +have := @measurable_fun_id _ [the measurableType _ of (T1 * T2)%type] setT. +by move=> /prod_measurable_funP[]. +Qed. + +Lemma measurable_fun_snd (d1 d2 : _) (T1 : measurableType d1) + (T2 : measurableType d2) : measurable_fun setT (@snd T1 T2). +Proof. +have := @measurable_fun_id _ [the measurableType _ of (T1 * T2)%type] setT. +by move=> /prod_measurable_funP[]. +Qed. + +Lemma measurable_uncurry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) + (G : T1 -> T2 -> set T) (x : T1 * T2) : + measurable (G x.1 x.2) <-> measurable (uncurry G x). +Proof. by case: x. Qed. + +Lemma measurable_curry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) + (G : T1 * T2 -> set T) (x : T1 * T2) : + measurable (G x) <-> measurable (curry G x.1 x.2). +Proof. by case: x. Qed. + +Lemma measurable_fun_if (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') : + measurable_fun setT x -> + measurable_fun setT y -> + measurable_fun setT (fun b : T * bool => if b.2 then x b.1 else y b.1). +Proof. +move=> mx my /= _ Y mY. +rewrite setTI. +have := mx measurableT Y mY. +rewrite setTI => xY. +have := my measurableT Y mY. +rewrite setTI => yY. +rewrite (_ : _ @^-1` Y = (x @^-1` Y) `*` [set true] `|` (y @^-1` Y) `*` [set false]); last first. + apply/seteqP; split. + move=> [t [|]]/=. + by left. + by right. + move=> [t [|]]/=. + by case=> [[]//|[]]. + by case=> [[]//|[]]. +by apply: measurableU; apply: measurableM => //. +Qed. + +(*/ PR*) + +Reserved Notation "R .-ker X ~> Y" (at level 42). +Reserved Notation "R .-fker X ~> Y" (at level 42). +Reserved Notation "R .-sfker X ~> Y" (at level 42). + +HB.mixin Record isKernel d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := + { measurable_kernel : forall U, measurable U -> measurable_fun setT (k ^~ U) }. + +#[short(type=kernel)] +HB.structure Definition Kernel + d d' (X : measurableType d) (Y : measurableType d') (R : realType) := + { k & isKernel _ _ X Y R k }. +Notation "R .-ker X ~> Y" := (kernel X Y R). + +Arguments measurable_kernel {_ _ _ _ _} _. + +Section sum_of_kernels. +Variables (d d' : measure_display) (R : realType). +Variables (X : measurableType d) (Y : measurableType d'). +Variable k : (R.-ker X ~> Y)^nat. + +Definition sum_of_kernels : X -> {measure set Y -> \bar R} := + fun x => [the measure _ _ of mseries (k ^~ x) 0]. + +Lemma kernel_measurable_fun_sum_of_kernels (U : set Y) : + measurable U -> + measurable_fun setT (sum_of_kernels ^~ U). +Proof. +move=> mU; rewrite /sum_of_kernels /= /mseries. +rewrite [X in measurable_fun _ X](_ : _ = + (fun x => elim_sup (fun n => \sum_(0 <= i < n) k i x U))); last first. + apply/funext => x; rewrite -lim_mkord is_cvg_elim_supE. + by rewrite -lim_mkord. + exact: is_cvg_nneseries. +apply: measurable_fun_elim_sup => n. +by apply: emeasurable_fun_sum => *; exact/measurable_kernel. +Qed. + +HB.instance Definition _ := + isKernel.Build _ _ _ _ _ sum_of_kernels + kernel_measurable_fun_sum_of_kernels. -Lemma xsection_ndseq_closed_dep : ndseq_closed B. +End sum_of_kernels. + +Lemma integral_sum_of_kernels + (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType) (k : (R.-ker X ~> Y)^nat) (f : Y -> \bar R) x : + (forall y, 0 <= f y) -> + measurable_fun setT f -> + \int[sum_of_kernels k x]_y (f y) = \sum_(i f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. +Qed. + +(* TODO: define using the probability type *) +HB.mixin Record isProbabilityKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) + of isKernel _ _ X Y R k := { + prob_kernelP : forall x, k x [set: Y] = 1 +}. + +#[short(type=probability_kernel)] +HB.structure Definition ProbabilityKernel + (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of isProbabilityKernel _ _ X Y R k & isKernel _ _ X Y R k}. + +Section measure_uub. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : numFieldType) (k : X -> {measure set Y -> \bar R}). + +Definition measure_uub := exists r, forall x, k x [set: Y] < r%:E. + +Lemma measure_uubP : measure_uub <-> + exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E. +Proof. +split => [|] [r kr]; last by exists r%:num. +suff r_gt0 : (0 < r)%R by exists (PosNum r_gt0). +by rewrite -lte_fin; apply: (le_lt_trans _ (kr point)). +Qed. + +End measure_uub. + +HB.mixin Record isFiniteKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := + { kernel_uub : measure_uub k }. + +#[short(type=finite_kernel)] +HB.structure Definition FiniteKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of isFiniteKernel _ _ X Y R k & isKernel _ _ X Y R k}. +Notation "R .-fker X ~> Y" := (finite_kernel X Y R). + +Arguments kernel_uub {_ _ _ _ _} _. + +Section kernel_from_mzero. +Variables (d : _) (T : measurableType d) (R : realType). +Variables (d' : _) (T' : measurableType d'). + +Definition kernel_from_mzero : T' -> {measure set T -> \bar R} := + fun _ : T' => [the measure _ _ of mzero]. + +Lemma kernel_from_mzeroP : forall U, measurable U -> + measurable_fun setT (kernel_from_mzero ^~ U). +Proof. by move=> U mU/=; exact: measurable_fun_cst. Qed. + +HB.instance Definition _ := + @isKernel.Build _ _ T' T R kernel_from_mzero + kernel_from_mzeroP. + +Lemma kernel_from_mzero_uub : measure_uub kernel_from_mzero. +Proof. +exists 1%R => /= t. +by rewrite /mzero/=. +Qed. + +HB.instance Definition _ := + @isFiniteKernel.Build _ _ _ T R kernel_from_mzero + kernel_from_mzero_uub. + +End kernel_from_mzero. + +HB.mixin Record isSFiniteKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := { + sfinite : exists s : (R.-fker X ~> Y)^nat, + forall x U, measurable U -> + k x U = [the measure _ _ of mseries (s ^~ x) 0] U }. + +#[short(type=sfinite_kernel)] +HB.structure Definition SFiniteKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of isSFiniteKernel _ _ X Y R k & + isKernel _ _ X Y _ k}. +Notation "R .-sfker X ~> Y" := (sfinite_kernel X Y R). + +Arguments sfinite {_ _ _ _ _} _. + +(* a finite kernel is always an s-finite kernel *) +Section finite_is_sfinite. +Variables (d d' : _) (X : measurableType d) (T : measurableType d'). +Variables (R : realType) (k : R.-fker T ~> X). + +Lemma sfinite_finite : + exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> + k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -move=> F ndF; rewrite /B /= => BF; split. - by apply: bigcupT_measurable => n; have [] := BF n. -have phiF x : (fun i => phi (F i) x) --> phi (\bigcup_i F i) x. - rewrite /phi /= xsection_bigcup; apply: cvg_mu_inc => //. - - by move=> n; apply: measurable_xsection; case: (BF n). - - by apply: bigcupT_measurable => i; apply: measurable_xsection; case: (BF i). - - move=> m n mn; apply/subsetPset => y; rewrite /xsection/= !inE. - by have /subsetPset FmFn := ndF _ _ mn; exact: FmFn. -apply: (emeasurable_fun_cvg (phi \o F)) => //. -- by move=> i; have [] := BF i. -- by move=> x _; exact: phiF. +exists (fun n => if n is O then k else + [the finite_kernel _ _ _ of @kernel_from_mzero _ X R _ T]). +move=> t U mU/=. +rewrite /mseries. +rewrite (nneseries_split 1%N)// big_ord_recl/= big_ord0 adde0. +rewrite ereal_series (@eq_nneseries _ _ (fun=> 0%E)); last by case. +by rewrite nneseries0// adde0. Qed. -End xsection. -End ndseq_closed_B. +End finite_is_sfinite. -Section measurable_prod_subset. -Variables (d1 d2 : measure_display). -Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +(* see measurable_prod_subset in lebesgue_integral.v; + the differences between the two are: + - m2 is a kernel instead of a measure + - m2D_bounded holds for all x *) +Section measurable_prod_subset_kernel. +Variables (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) + (R : realType). Implicit Types A : set (T1 * T2). -Section xsection. -Variable (m2 : T1 -> {measure set T2 -> \bar R}) (D : set T2) (mD : measurable D). +Section xsection_kernel. +Variable (m2 : R.-ker T1 ~> T2) (D : set T2) (mD : measurable D). Let m2D x := mrestr (m2 x) mD. HB.instance Definition _ x := Measure.on (m2D x). Let phi A := fun x => m2D x (xsection A x). Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. -Hypothesis H1 : forall X2, measurable X2 -> measurable_fun [set: T1] (m2D^~ X2). - -Lemma measurable_prod_subset_xsection_dep +Lemma measurable_prod_subset_xsection_kernel (m2D_bounded : forall x, exists M, forall X, measurable X -> (m2D x X < M%:E)%E) : measurable `<=` B. Proof. @@ -390,123 +474,68 @@ have CB : C `<=` B. rewrite funeqE => x; rewrite indicE /phi /m2/= /mrestr. have [xX1|xX1] := boolP (x \in X1); first by rewrite mule1 in_xsectionM. by rewrite mule0 notin_xsectionM// set0I measure0. - apply: emeasurable_funM => //. - by apply: H1. + apply: emeasurable_funM => //; first exact/measurable_kernel/measurableI. apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R mX1). suff monoB : monotone_class setT B by exact: monotone_class_subset. -split => //; [exact: CB| |exact: xsection_ndseq_closed_dep]. +split => //; [exact: CB| |exact: xsection_ndseq_closed]. move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. -have -> : phi (X `\` Y) = (fun x => phi X x - phi Y x)%E. - rewrite funeqE => x; rewrite /phi/= xsectionD// /m2D measureD. - - by rewrite setIidr//; exact: le_xsection. - - exact: measurable_xsection. - - exact: measurable_xsection. - - move: (m2D_bounded x) => [M m2M]. - rewrite (lt_le_trans (m2M (xsection X x) _))// ?leey//. - exact: measurable_xsection. -exact: emeasurable_funB. +suff : phi (X `\` Y) = (fun x => phi X x - phi Y x)%E. + by move=> ->; exact: emeasurable_funB. +rewrite funeqE => x; rewrite /phi/= xsectionD// /m2D measureD. +- by rewrite setIidr//; exact: le_xsection. +- exact: measurable_xsection. +- exact: measurable_xsection. +- move: (m2D_bounded x) => [M m2M]. + rewrite (lt_le_trans (m2M (xsection X x) _))// ?leey//. + exact: measurable_xsection. Qed. -End xsection. +End xsection_kernel. -End measurable_prod_subset. +End measurable_prod_subset_kernel. -Section measurable_fun_xsection. -Variables (d1 d2 : measure_display) (T1 : measurableType d1) - (T2 : measurableType d2) (R : realType). -Variables (m2 : T1 -> {measure set T2 -> \bar R}). +(* see measurable_fun_xsection in lebesgue_integral.v + the difference is that this section uses a finite kernel m2 + instead of a sigma-finite measure m2 *) +Section measurable_fun_xsection_finite_kernel. +Variables (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) + (R : realType). +Variable m2 : R.-fker T1 ~> T2. Implicit Types A : set (T1 * T2). -Hypotheses m2_ub : kernel_uub m2. -Hypothesis H1 : forall X2, measurable X2 -> - measurable_fun [set: T1] ((fun x => mrestr (m2 x) measurableT)^~ X2). - -Let phi A := (fun x => m2 x (xsection A x)). +Let phi A := fun x => m2 x (xsection A x). Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. -Lemma measurable_fun_xsection_dep A : +Lemma measurable_fun_xsection_finite_kernel A : A \in measurable -> measurable_fun setT (phi A). Proof. move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. move=> X mX. rewrite /B/=; split => //. rewrite /phi. -rewrite -(_ : (fun x : T1 => mrestr (m2 x) measurableT (xsection X x)) = (fun x => (m2 x) (xsection X x)))//; last first. - apply/funext => x//=. - by rewrite /mrestr setIT. -apply measurable_prod_subset_xsection_dep => //. +rewrite -(_ : (fun x => mrestr (m2 x) measurableT (xsection X x)) = + (fun x => (m2 x) (xsection X x)))//; last first. + by apply/funext => x//=; rewrite /mrestr setIT. +apply measurable_prod_subset_xsection_kernel => //. move=> x. -case: m2_ub => r hr. -exists r%:num => Y mY. +have [r hr] := kernel_uub m2. +exists r => Y mY. apply: (le_lt_trans _ (hr x)) => //. rewrite /mrestr. -apply le_measure => //. -rewrite inE. -apply: measurableI => //. -by rewrite inE. +by apply le_measure => //; rewrite inE//; exact: measurableI. Qed. -End measurable_fun_xsection. +End measurable_fun_xsection_finite_kernel. -Section fubini_F_dep. -Local Open Scope ereal_scope. -Variables (d1 d2 : measure_display). -Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). -Variables (m2 : T1 -> {measure set T2 -> \bar R}). -Variable f : T1 * T2 -> \bar R. - -Definition fubini_F_dep x := \int[m2 x]_y f (x, y). - -End fubini_F_dep. - -Section fubini_tonelli. -Local Open Scope ereal_scope. -Variables (d1 d2 : measure_display). -Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). -Variables (m1 : {measure set T1 -> \bar R}) (m2 : T1 -> {measure set T2 -> \bar R}). -Hypotheses m2_ub : kernel_uub m2. - -Section indic_fubini_tonelli. -Variables (A : set (T1 * T2)) (mA : measurable A). -Implicit Types A : set (T1 * T2). -Let f : (T1 * T2) -> R := \1_A. - -Let F := fubini_F_dep m2 (EFin \o f). - -Lemma indic_fubini_tonelli_FE_dep : F = (fun x => m2 x (xsection A x)). +(* pollard *) +Lemma measurable_fun_integral_finite_kernel + (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType) (l : R.-fker X ~> Y) (k : (X * Y)%type -> \bar R) + (k0 : (forall z, True -> 0 <= k z)) (mk : measurable_fun setT k) : + measurable_fun setT (fun x => \int[l x]_y k (x, y)). Proof. -rewrite funeqE => x; rewrite /= -(setTI (xsection _ _)). -rewrite -integral_indic//; last exact: measurable_xsection. -rewrite /F /fubini_F -(setTI (xsection _ _)). -rewrite integral_setI_indic; [|exact: measurable_xsection|by []]. -apply: eq_integral => y _ /=; rewrite indicT mul1e /f !indicE. -have [|] /= := boolP (y \in xsection _ _). - by rewrite inE /xsection /= => ->. -by rewrite /xsection /= notin_set /= => /negP/negbTE ->. -Qed. - -Hypothesis H1 : forall X2, measurable X2 -> - measurable_fun [set: T1] ((fun x => mrestr (m2 x) measurableT)^~ X2). - -Lemma indic_measurable_fun_fubini_tonelli_F_dep : measurable_fun setT F. -Proof. -rewrite indic_fubini_tonelli_FE_dep//; apply: measurable_fun_xsection_dep => //. -by rewrite inE. -Qed. - -End indic_fubini_tonelli. - -End fubini_tonelli. - -Lemma pollard_finite (d d' : measure_display) (R : realType) - (X : measurableType d) (Y : measurableType d') - (k : (X * Y)%type -> \bar R) (k0 : (forall t : X * Y, True -> 0 <= k t)) - (mk : measurable_fun setT k) (l : finite_kernel R X Y) : - measurable_fun [set: X] (fun x : X => \int[l x]_y k (x, y)). -Proof. -have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. -simpl in *. +have [k_ [ndk_ k_k]] := approximation measurableT mk k0. rewrite (_ : (fun x => \int[l x]_y k (x, y)) = (fun x => elim_sup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. apply/funeqP => x. @@ -525,13 +554,12 @@ rewrite (_ : (fun x => \int[l x]_y k (x, y)) = - by move=> n y' _; rewrite lee_fin. - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. apply: measurable_fun_elim_sup => n. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l x0]_y - ((\sum_(r <- fset_set (range (k_ n))) - r * \1_(k_ n @^-1` [set r]) (x0, y)))%:E)); last first. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y + (\sum_(r <- fset_set (range (k_ n))) + r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) - (\int[l x0]_y - (r * \1_(k_ n @^-1` [set r]) (x0, y))%:E))); last first. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => \sum_(r <- fset_set (range (k_ n))) + (\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first. apply/funext => x; rewrite -ge0_integral_sum//. - by apply: eq_integral => y _; rewrite sumEFin. - move=> r. @@ -549,135 +577,126 @@ rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). apply: measurable_funeM. -apply: indic_measurable_fun_fubini_tonelli_F_dep. -- by apply/finite_kernelP. -- by apply/measurable_sfunP. -- move=> X2. - rewrite (_ : (fun x : X => mrestr (l x) measurableT X2) = (fun x : X => (l x) X2))//. - by apply/kernelP. - apply/funeqP => x. - by rewrite /mrestr setIT. +rewrite (_ : (fun x => _) = (fun x => l x (xsection (k_ n @^-1` [set r]) x))); last first. + apply/funext => y. + rewrite integral_indic//; last first. + rewrite (_ : (fun x => _) = xsection (k_ n @^-1` [set r]) y); last first. + apply/seteqP; split. + by move=> y2/=; rewrite /xsection/= inE//. + by rewrite /xsection/= => y2/=; rewrite inE. + exact: measurable_xsection. + congr (l y _). + apply/funext => y1/=. + rewrite /xsection/= inE. + by apply/propext; tauto. +have [l_ hl_] := kernel_uub l. +by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. Qed. -Module STAR_IS_FINITE_KERNEL. +Section kcomp_def. +Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) + (Z : measurableType d3) (R : realType). +Variable l : X -> {measure set Y -> \bar R}. +Variable k : (X * Y)%type -> {measure set Z -> \bar R}. -Section star_is_kernel_finite. -Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3). -Variable k : kernel R [the measurableType _ of (X * Y)%type] Z. -Variable l : finite_kernel R X Y. +Definition kcomp x U := \int[l x]_y k (x, y) U. -Lemma star_measurable_finite U : measurable U -> measurable_fun setT (star k l ^~ U). -Proof. -(* k is a bounded measurable function *) -(* l is a finite kernel *) -move=> mU. -rewrite /star. -apply: (@pollard_finite _ _ R X Y (fun xy => k xy U)) => //. -by apply: (@kernelP _ _ R [the measurableType (d, d').-prod of (X * Y)%type] Z k U) => //. -Qed. +End kcomp_def. -HB.instance Definition _ := - isKernel.Build _ _ R X Z (mstar k l) star_measurable_finite. +Section kcomp_is_measure. +Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) + (Z : measurableType d3) (R : realType). +Variable l : R.-ker X ~> Y. +Variable k : R.-ker [the measurableType _ of (X * Y)%type] ~> Z. -End star_is_kernel_finite. +Local Notation "l \; k" := (kcomp l k). -Section star_is_finite_kernel. -Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3). -Variable k : finite_kernel R [the measurableType _ of (X * Y)%type] Z. -Variable l : finite_kernel R X Y. +Let kcomp0 x : (l \; k) x set0 = 0. +Proof. +by rewrite /kcomp (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. +Qed. -Lemma star_finite : kernel_uub (mstar k l). +Let kcomp_ge0 x U : 0 <= (l \; k) x U. Proof. exact: integral_ge0. Qed. + +Let kcomp_sigma_additive x : semi_sigma_additive ((l \; k) x). Proof. -have [r hr] := @finite_kernelP _ _ _ _ _ k. -have [s hs] := @finite_kernelP _ _ _ _ _ l. -exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x. -rewrite /star. -apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). - apply: ge0_le_integral => //. - - have := @kernelP _ _ _ _ _ k setT measurableT. - exact/measurable_fun_prod1. - - exact/measurable_fun_cst. - - by move=> y _; apply/ltW/hr. -by rewrite integral_cst//= EFinM lte_pmul2l. +move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = + \int[l x]_y (\sum_(n V _. + by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. +apply/cvg_closeP; split. + by apply: is_cvg_nneseries => n _; exact: integral_ge0. +rewrite closeE// integral_sum// => n. +by have /measurable_fun_prod1 := measurable_kernel k (U n) (mU n). Qed. -HB.instance Definition _ := - isFiniteKernel.Build _ _ R X Z (mstar k l) star_finite. +HB.instance Definition _ x := isMeasure.Build _ R _ + ((l \; k) x) (kcomp0 x) (kcomp_ge0 x) (@kcomp_sigma_additive x). + +Definition mkcomp : X -> {measure set Z -> \bar R} := + fun x => [the measure _ _ of (l \; k) x]. + +End kcomp_is_measure. + +Notation "l \; k" := (mkcomp l k). + +Module KCOMP_FINITE_KERNEL. -End star_is_finite_kernel. -End STAR_IS_FINITE_KERNEL. +Section kcomp_finite_kernel_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-fker X ~> Y) + (k : R.-ker [the measurableType _ of (X * Y)%type] ~> Z). -Lemma pollard_sfinite (d d' d3 : measure_display) (R : realType) - (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) - (k : Z -> \bar R) (k0 : (forall z, True -> 0 <= k z)) - (mk : measurable_fun setT k) - (l : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) c : - measurable_fun [set: Y] (fun x0 : Y => \int[l (c, x0)]_z k z). +Lemma measurable_fun_kcomp_finite U : + measurable U -> measurable_fun setT ((l \; k) ^~ U). Proof. -have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. -simpl in *. -rewrite (_ : (fun x0 => \int[l (c, x0)]_z k z) = - (fun x0 => elim_sup (fun n => \int[l (c, x0)]_z (k_ n z)%:E))); last first. - apply/funeqP => x. - transitivity (lim (fun n => \int[l (c, x)]_z (k_ n z)%:E)); last first. - rewrite is_cvg_elim_supE//. - apply: ereal_nondecreasing_is_cvg => m n mn. - apply: ge0_le_integral => //. - - by move=> y' _; rewrite lee_fin. - - exact/EFin_measurable_fun. - - by move=> y' _; rewrite lee_fin. - - exact/EFin_measurable_fun. - - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. - rewrite -monotone_convergence//. - - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. - - by move=> n; exact/EFin_measurable_fun. - - by move=> n y' _; rewrite lee_fin. - - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. -apply: measurable_fun_elim_sup => n. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l (c, x0)]_z - ((\sum_(r <- fset_set (range (k_ n))) - r * \1_(k_ n @^-1` [set r]) z))%:E)); last first. - by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) - (\int[l (c, x0)]_z - (r * \1_(k_ n @^-1` [set r]) z)%:E))); last first. - apply/funext => x; rewrite -ge0_integral_sum//. - - by apply: eq_integral => y _; rewrite sumEFin. - - move=> r. - apply/EFin_measurable_fun/measurable_funrM => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). - - by move=> m y _; rewrite muleindic_ge0. -apply emeasurable_fun_sum => r. -rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * - \int[l (c ,x)]_z (\1_(k_ n @^-1` [set r]) z)%:E)); last first. - apply/funext => x. - under eq_integral do rewrite EFinM. - rewrite (integralM_0ifneg _ _ (fun k z => (\1_(k_ n @^-1` [set r]) z)%:E))//. - - by move=> _ t _; rewrite lee_fin. - - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. - - apply/EFin_measurable_fun => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). -apply: measurable_funeM. -rewrite (_ : (fun x : Y => \int[l (c, x)]_z (\1_(k_ n @^-1` [set r]) z)%:E) = - (fun x : Y => l (c, x) (k_ n @^-1` [set r]))); last first. - apply/funext => y. - by rewrite integral_indic// setIT. -have := @kernelP _ _ R _ _ l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). +move=> mU. +rewrite /kcomp. +apply: (@measurable_fun_integral_finite_kernel _ _ _ _ _ _ (k ^~ U)) => //=. +exact/measurable_kernel. +Qed. + +HB.instance Definition _ := + isKernel.Build _ _ X Z R (l \; k) measurable_fun_kcomp_finite. + +End kcomp_finite_kernel_kernel. + +Section kcomp_finite_kernel_finite. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable l : R.-fker X ~> Y. +Variable k : R.-fker [the measurableType _ of (X * Y)%type] ~> Z. + +Lemma mkcomp_finite : measure_uub (l \; k). +Proof. +have /measure_uubP[r hr] := kernel_uub k. +have /measure_uubP[s hs] := kernel_uub l. +apply/measure_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x. rewrite /=. -move/measurable_fun_prod1. -exact. +apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). + apply: ge0_le_integral => //. + - have /measurable_fun_prod1 := measurable_kernel k setT measurableT. + exact. + - exact/measurable_fun_cst. + - by move=> y _; exact/ltW/hr. +by rewrite integral_cst//= EFinM lte_pmul2l. Qed. -Lemma pollard_sfinite2 (d d' : measure_display) (R : realType) - (X : measurableType d) (Y : measurableType d') - (k : (X * Y)%type -> \bar R) (k0 : (forall (t : X * Y), True -> 0 <= k t)) - (l : sfinite_kernel R X Y) - (mk : measurable_fun setT k) : - measurable_fun [set: X] (fun x : X => \int[l x]_y k (x, y)). +HB.instance Definition _ := + isFiniteKernel.Build _ _ X Z R (l \; k) mkcomp_finite. + +End kcomp_finite_kernel_finite. +End KCOMP_FINITE_KERNEL. + +(* pollard *) +Lemma measurable_fun_integral_sfinite_kernel + (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType) + (l : R.-sfker X ~> Y) + (k : (X * Y)%type -> \bar R) (k0 : (forall t, True -> 0 <= k t)) + (mk : measurable_fun setT k) : + measurable_fun [set: X] (fun x => \int[l x]_y k (x, y)). Proof. -have [k_ [ndk_ k_k]] := @approximation _ _ _ _ measurableT k mk k0. +have [k_ [ndk_ k_k]] := approximation measurableT mk k0. simpl in *. rewrite (_ : (fun x => \int[l x]_y k (x, y)) = (fun x => elim_sup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. @@ -698,8 +717,8 @@ rewrite (_ : (fun x => \int[l x]_y k (x, y)) = - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. apply: measurable_fun_elim_sup => n. rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l x0]_y - ((\sum_(r <- fset_set (range (k_ n))) - r * \1_(k_ n @^-1` [set r]) (x0, y)))%:E)); last first. + (\sum_(r <- fset_set (range (k_ n))) + r * \1_(k_ n @^-1` [set r]) (x0, y))%:E)); last first. by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) (\int[l x0]_y @@ -721,146 +740,194 @@ rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). apply: measurable_funeM. -rewrite (_ : (fun x : X => \int[l x]_z (\1_(k_ n @^-1` [set r]) (x, z))%:E) = - (fun x : X => l x (xsection (k_ n @^-1` [set r]) x))); last first. +rewrite (_ : (fun x => \int[l x]_z (\1_(k_ n @^-1` [set r]) (x, z))%:E) = + (fun x => l x (xsection (k_ n @^-1` [set r]) x))); last first. apply/funext => y. rewrite integral_indic//; last first. - rewrite (_ : (fun x : Y => (k_ n @^-1` [set r]) (y, x)) = xsection (k_ n @^-1` [set r]) y); last first. + rewrite (_ : (fun x => (k_ n @^-1` [set r]) (y, x)) = xsection (k_ n @^-1` [set r]) y); last first. apply/seteqP; split. by move=> y2/=; rewrite /xsection/= inE//. by rewrite /xsection/= => y2/=; rewrite inE /preimage/=. - by apply: measurable_xsection. + exact: measurable_xsection. congr (l y _). apply/funext => y1/=. rewrite /xsection/= inE. by apply/propext; tauto. -have [l_ hl_] := @sfinite_kernelP _ _ _ _ _ l. -rewrite (_ : (fun x : X => _) = - (fun x : X => mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x)) -); last first. +have [l_ hl_] := sfinite l. +rewrite (_ : (fun x => _) = (fun x => mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x))); last first. apply/funext => x. rewrite hl_//. - by apply/measurable_xsection. + exact/measurable_xsection. rewrite /mseries/=. apply: ge0_emeasurable_fun_sum => // k1. -apply: measurable_fun_xsection_dep => //. -by have := @finite_kernelP _ _ _ _ _ (l_ k1). -move=> X2 mX2. -rewrite /mrestr. -apply/kernelP. -by rewrite setIT. +apply: measurable_fun_xsection_finite_kernel => //. by rewrite inE. Qed. -Section star_is_sfinite_kernel. -Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3). -Variable k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z. -Variable l : sfinite_kernel R X Y. +Section kcomp_sfinite_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable l : R.-sfker X ~> Y. +Variable k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. -Import STAR_IS_FINITE_KERNEL. +Import KCOMP_FINITE_KERNEL. -Lemma star_sfinite : exists k_ : (finite_kernel R X Z)^nat, forall x U, measurable U -> - mstar k l x U = [the measure _ _ of mseries (k_ ^~ x) O] U. +Lemma mkcomp_sfinite : exists k_ : (R.-fker X ~> Z)^nat, forall x U, measurable U -> + (l \; k) x U = [the measure _ _ of mseries (k_ ^~ x) O] U. Proof. -have [k_ hk_] := @sfinite_kernelP _ _ _ _ _ k. -have [l_ hl_] := @sfinite_kernelP _ _ _ _ _ l. +have [k_ hk_] := sfinite k. +have [l_ hl_] := sfinite l. pose K := [the kernel _ _ _ of sum_of_kernels k_]. pose L := [the kernel _ _ _ of sum_of_kernels l_]. -have H1 x U : measurable U -> star k l x U = star K L x U. - move=> mU. - rewrite /star /L /K /=. +have H1 x U : measurable U -> (l \; k) x U = (L \; K) x U. + move=> mU /=. + rewrite /kcomp /L /K /=. transitivity (\int[ - [the measure _ _ of mseries (fun x0 : nat => l_ x0 x) 0] ]_y k (x, y) U). - apply eq_measure_integral => A mA _ . - by rewrite hl_. - apply eq_integral => y _. - by rewrite hk_//. -have H2 x U : star K L x U = + [the measure _ _ of mseries (l_ ^~ x) 0] ]_y k (x, y) U). + by apply eq_measure_integral => A mA _; rewrite hl_. + by apply eq_integral => y _; rewrite hk_. +have H2 x U : (L \; K) x U = \int[mseries (l_ ^~ x) 0]_y (\sum_(i y _. + exact: eq_integral. have H3 x U : measurable U -> \int[mseries (l_ ^~ x) 0]_y (\sum_(i mU. rewrite integral_sum//= => n. - have := @kernelP _ _ _ _ _ (k_ n) _ mU. - by move/measurable_fun_prod1; exact. + have := measurable_kernel (k_ n) _ mU. + by move=> /measurable_fun_prod1; exact. have H4 x U : measurable U -> \sum_(i mU. apply: eq_nneseries => i _. rewrite integral_sum_of_kernels//. - have := @kernelP _ _ _ _ _ (k_ i) _ mU. - by move/measurable_fun_prod1; exact. + have := measurable_kernel (k_ i) _ mU. + by move=> /measurable_fun_prod1; exact. have H5 x U : \sum_(i i _; exact: eq_nneseries. -suff: exists k_0 : (finite_kernel R X Z) ^nat, forall x U, - \esum_(i in setT) star (k_ i.1) (l_ i.2) x U = \sum_(i Z) ^nat, forall x U, + \esum_(i in setT) ((l_ i.2) \; (k_ i.1)) x U = \sum_(i [kl_ hkl_]. exists kl_ => x U mU. - rewrite /=. - rewrite /mstar/= /mseries H1// H2 H3//. - rewrite H4//. - rewrite H5// -hkl_ /=. + rewrite /= H1// H2 H3// H4// H5// /mseries -hkl_/=. rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split. - rewrite -(@esum_esum _ _ _ _ _ (fun i j => star (k_ i) (l_ j) x U))//. - rewrite nneseries_esum; last by move=> n _; exact: nneseries_lim_ge0(* TODO: rename this lemma *). + rewrite -(@esum_esum _ _ _ _ _ (fun i j => (l_ j \; k_ i) x U))//. + rewrite nneseries_esum; last by move=> n _; exact: nneseries_lim_ge0. by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true. rewrite /=. have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card. by rewrite card_eq_sym; exact: card_nat2. -exists (fun i => [the finite_kernel _ _ _ of mstar (k_ (f i).1) (l_ (f i).2)]) => x U. +exists (fun i => [the _.-fker _ ~> _ of (l_ (f i).2) \; (k_ (f i).1)]) => x U. rewrite (reindex_esum [set: nat] [set: nat * nat] f)//. by rewrite nneseries_esum// fun_true. Qed. -Lemma star_measurable_sfinite U : measurable U -> measurable_fun setT (star k l ^~ U). +Lemma measurable_fun_mkcomp_sfinite U : measurable U -> measurable_fun setT ((l \; k) ^~ U). Proof. move=> mU. -rewrite /star. -apply: (@pollard_sfinite2 _ _ _ _ _ (k ^~ U)) => //. -by apply/kernelP. +apply: (@measurable_fun_integral_sfinite_kernel _ _ _ _ _ _ (k ^~ U)) => //. +exact/measurable_kernel. Qed. -End star_is_sfinite_kernel. +End kcomp_sfinite_kernel. -Module STAR_IS_SFINITE_KERNEL. -Section star_is_sfinite_kernel. -Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3). -Variable k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z. -Variable l : sfinite_kernel R X Y. +Module KCOMP_SFINITE_KERNEL. +Section kcomp_sfinite_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable l : R.-sfker X ~> Y. +Variable k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. HB.instance Definition _ := - isKernel.Build _ _ R X Z (mstar k l) (star_measurable_sfinite k l). + isKernel.Build _ _ X Z R (l \; k) (measurable_fun_mkcomp_sfinite l k). #[export] HB.instance Definition _ := - isSFiniteKernel.Build d d3 R X Z (mstar k l) (star_sfinite k l). + isSFiniteKernel.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). + +End kcomp_sfinite_kernel. +End KCOMP_SFINITE_KERNEL. +HB.export KCOMP_SFINITE_KERNEL. + +(* pollard *) +Lemma measurable_fun_integral_sfinite_kernel_prod + (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) + (l : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) c + (k : Z -> \bar R) (k0 : (forall z, True -> 0 <= k z)) (mk : measurable_fun setT k) : + measurable_fun [set: Y] (fun y => \int[l (c, y)]_z k z). +Proof. +have [k_ [ndk_ k_k]] := approximation measurableT mk k0. +simpl in *. +rewrite (_ : (fun x0 => \int[l (c, x0)]_z k z) = + (fun x0 => elim_sup (fun n => \int[l (c, x0)]_z (k_ n z)%:E))); last first. + apply/funeqP => x. + transitivity (lim (fun n => \int[l (c, x)]_z (k_ n z)%:E)); last first. + rewrite is_cvg_elim_supE//. + apply: ereal_nondecreasing_is_cvg => m n mn. + apply: ge0_le_integral => //. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun. + - by move=> y' _; rewrite lee_fin. + - exact/EFin_measurable_fun. + - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. + rewrite -monotone_convergence//. + - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. + - by move=> n; exact/EFin_measurable_fun. + - by move=> n y' _; rewrite lee_fin. + - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. +apply: measurable_fun_elim_sup => n. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l (c, x0)]_z + ((\sum_(r <- fset_set (range (k_ n))) + r * \1_(k_ n @^-1` [set r]) z))%:E)); last first. + by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. +rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) + (\int[l (c, x0)]_z + (r * \1_(k_ n @^-1` [set r]) z)%:E))); last first. + apply/funext => x; rewrite -ge0_integral_sum//. + - by apply: eq_integral => y _; rewrite sumEFin. + - move=> r. + apply/EFin_measurable_fun/measurable_funrM => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). + - by move=> m y _; rewrite muleindic_ge0. +apply emeasurable_fun_sum => r. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * + \int[l (c ,x)]_z (\1_(k_ n @^-1` [set r]) z)%:E)); last first. + apply/funext => x. + under eq_integral do rewrite EFinM. + rewrite (integralM_0ifneg _ _ (fun k z => (\1_(k_ n @^-1` [set r]) z)%:E))//. + - by move=> _ t _; rewrite lee_fin. + - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. + - apply/EFin_measurable_fun => /=. + by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). +apply: measurable_funeM. +rewrite (_ : (fun x : Y => \int[l (c, x)]_z (\1_(k_ n @^-1` [set r]) z)%:E) = + (fun x : Y => l (c, x) (k_ n @^-1` [set r]))); last first. + apply/funext => y. + by rewrite integral_indic// setIT. +have := measurable_kernel l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). +by move=> /measurable_fun_prod1; exact. +Qed. -End star_is_sfinite_kernel. -End STAR_IS_SFINITE_KERNEL. -HB.export STAR_IS_SFINITE_KERNEL. +Section integral_kcomp. -Lemma lemma3_indic d d' d3 (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType d3) - (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) - (l : sfinite_kernel R X Y) x (E : set _) (mE : measurable E) : - \int[mstar k l x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). +Let integral_kcomp_indic d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x (E : set _) (mE : measurable E) : + \int[(l \; k) x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). Proof. -rewrite integral_indic// /mstar/= /star/=. +rewrite integral_indic//= /kcomp. by apply eq_integral => y _; rewrite integral_indic. Qed. -Lemma lemma3_nnsfun d d' d3 (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType d3) - (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) - (l : sfinite_kernel R X Y) x (f : {nnsfun Z >-> R}) : - \int[mstar k l x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). +Let integral_kcomp_nnsfun d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x (f : {nnsfun Z >-> R}) : + \int[(l \; k) x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). Proof. under [in LHS]eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//; last 2 first. @@ -887,37 +954,35 @@ under [in RHS]eq_integral. over. over. rewrite /= ge0_integral_sum//; last 2 first. - move=> r; apply: measurable_funeM. - have := @kernelP _ _ _ _ _ k (f @^-1` [set r]) (measurable_sfunP f r). - by move/measurable_fun_prod1; exact. - move=> n y _. - have := @mulem_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). - apply. - exact: preimage_nnfun0. + - move=> r; apply: measurable_funeM. + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). + by move=> /measurable_fun_prod1; exact. + - move=> n y _. + have := @mulem_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). + by apply; exact: preimage_nnfun0. apply eq_bigr => r _. rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. exact: preimage_nnfun0. -rewrite /= lemma3_indic; last exact/measurable_sfunP. +rewrite /= integral_kcomp_indic; last exact/measurable_sfunP. rewrite (@integralM_0ifneg _ _ _ _ _ _ (fun r t => k (x, t) (f @^-1` [set r])))//; last 2 first. move=> r0. apply/funext => y. by rewrite preimage_nnfun0// measure0. - have := @kernelP _ _ _ _ _ k (f @^-1` [set r]) (measurable_sfunP f r). + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). by move/measurable_fun_prod1; exact. -congr (_ * _). -apply eq_integral => y _. +congr (_ * _); apply eq_integral => y _. by rewrite integral_indic// setIT. Qed. -Lemma lemma3 d d' d3 (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType d3) - (k : sfinite_kernel R [the measurableType _ of (X * Y)%type] Z) - (l : sfinite_kernel R X Y) x f : (forall z, 0 <= f z) -> measurable_fun setT f -> - \int[mstar k l x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). +Lemma integral_kcomp d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x f : (forall z, 0 <= f z) -> measurable_fun setT f -> + \int[(l \; k) x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). Proof. move=> f0 mf. have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). -transitivity (\int[mstar k l x]_z (lim (EFin \o (f_^~ z)))). +transitivity (\int[(l \; k) x]_z (lim (EFin \o (f_^~ z)))). apply/eq_integral => z _. apply/esym/cvg_lim => //=. exact: f_f. @@ -926,11 +991,11 @@ rewrite monotone_convergence//; last 3 first. by move=> n z _; rewrite lee_fin. by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin. rewrite (_ : (fun _ => _) = (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first. - by apply/funext => n; rewrite lemma3_nnsfun. + by apply/funext => n; rewrite integral_kcomp_nnsfun. transitivity (\int[l x]_y lim (fun n => \int[k (x, y)]_z (f_ n z)%:E)). rewrite -monotone_convergence//; last 3 first. move=> n. - apply: pollard_sfinite => //. + apply: measurable_fun_integral_sfinite_kernel_prod => //. - by move=> z; rewrite lee_fin. - by apply/EFin_measurable_fun. - move=> n y _. @@ -953,153 +1018,11 @@ apply/cvg_lim => //. exact: f_f. Qed. -Canonical unit_pointedType := PointedType unit tt. - -Section discrete_measurable_unit. - -Definition discrete_measurable_unit : set (set unit) := [set: set unit]. - -Let discrete_measurable0 : discrete_measurable_unit set0. Proof. by []. Qed. - -Let discrete_measurableC X : discrete_measurable_unit X -> discrete_measurable_unit (~` X). -Proof. by []. Qed. - -Let discrete_measurableU (F : (set unit)^nat) : - (forall i, discrete_measurable_unit (F i)) -> discrete_measurable_unit (\bigcup_i F i). -Proof. by []. Qed. - -HB.instance Definition _ := @isMeasurable.Build default_measure_display unit (Pointed.class _) - discrete_measurable_unit discrete_measurable0 discrete_measurableC - discrete_measurableU. - -End discrete_measurable_unit. - -Section discrete_measurable_bool. - -Definition discrete_measurable_bool : set (set bool) := [set: set bool]. - -Let discrete_measurable0 : discrete_measurable_bool set0. Proof. by []. Qed. - -Let discrete_measurableC X : - discrete_measurable_bool X -> discrete_measurable_bool (~` X). -Proof. by []. Qed. - -Let discrete_measurableU (F : (set bool)^nat) : - (forall i, discrete_measurable_bool (F i)) -> - discrete_measurable_bool (\bigcup_i F i). -Proof. by []. Qed. - -HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Pointed.class _) - discrete_measurable_bool discrete_measurable0 discrete_measurableC - discrete_measurableU. - -End discrete_measurable_bool. - -Section nonneg_constants. -Variable R : realType. -(* -Let twoseven_proof : (0 <= 2 / 7 :> R)%R. -Proof. by rewrite divr_ge0// ler0n. Qed. -*) - -(* Check (2%:R / 7%:R)%:nng. *) - -(* Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. *) -(* -Let fiveseven_proof : (0 <= 5 / 7 :> R)%R. -Proof. by rewrite divr_ge0// ler0n. Qed. - -Definition fiveseven : {nonneg R} := NngNum fiveseven_proof. - *) - -End nonneg_constants. - -Lemma measure_diract_setT_true (R : realType) : - [the measure _ _ of dirac true] [set: bool] = 1 :> \bar R. -Proof. by rewrite /= diracE in_setT. Qed. - -Lemma measure_diract_setT_false (R : realType) : - [the measure _ _ of dirac false] [set: bool] = 1 :> \bar R. -Proof. by rewrite /= diracE in_setT. Qed. - -Section bernoulli27. -Variable R : realType. - -Local Open Scope ring_scope. -Notation "'2/7'" := (2%:R / 7%:R)%:nng. -Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. -Definition fiveseven : {nonneg R} := (5%:R / 7%:R)%:nng. - -Definition bernoulli27 : set _ -> \bar R := - measure_add - [the measure _ _ of mscale twoseven [the measure _ _ of dirac true]] - [the measure _ _ of mscale fiveseven [the measure _ _ of dirac false]]. - -HB.instance Definition _ := Measure.on bernoulli27. - -Local Close Scope ring_scope. - -Lemma bernoulli27_setT : bernoulli27 [set: _] = 1. -Proof. -rewrite /bernoulli27/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. -rewrite /mscale/= !diracE !in_setT !mule1 -EFinD. -by rewrite -mulrDl -natrD divrr// unitfE pnatr_eq0. -Qed. - -HB.instance Definition _ := @isProbability.Build _ _ R bernoulli27 bernoulli27_setT. +End integral_kcomp. -End bernoulli27. - -Section kernel_from_mzero. -Variables (d : measure_display) (T : measurableType d) (R : realType). -Variables (d' : measure_display) (T' : measurableType d'). - -Definition kernel_from_mzero : T' -> {measure set T -> \bar R} := - fun _ : T' => [the measure _ _ of mzero]. - -Lemma kernel_from_mzeroP : forall U, measurable U -> - measurable_fun setT (kernel_from_mzero ^~ U). -Proof. by move=> U mU/=; exact: measurable_fun_cst. Qed. - -HB.instance Definition _ := - @isKernel.Build d' d R T' T kernel_from_mzero - kernel_from_mzeroP. - -Lemma kernel_from_mzero_uub : kernel_uub kernel_from_mzero. -Proof. -exists (PosNum ltr01) => /= t. -by rewrite /mzero/=. -Qed. - -HB.instance Definition _ := - @isFiniteKernel.Build d' d R _ T kernel_from_mzero - kernel_from_mzero_uub. - -End kernel_from_mzero. - -(* a finite kernel is always an s-finite kernel *) -Lemma finite_kernel_sfinite_kernelP (d : measure_display) - (R : realType) (X : measurableType d) (d' : measure_display) (T : measurableType d') - (k : finite_kernel R T X) : - exists k_ : (finite_kernel R _ _)^nat, forall x U, measurable U -> - k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -exists (fun n => if n is O then - k - else - [the finite_kernel _ _ _ of @kernel_from_mzero _ X R _ T] - ). -move=> t U mU/=. -rewrite /mseries. -rewrite (nneseries_split 1%N)// big_ord_recl/= big_ord0 adde0. -rewrite ereal_series (@eq_nneseries _ _ (fun=> 0%E)); last first. - by case. -by rewrite nneseries0// adde0. -Qed. - -(* semantics for a sample operation? *) +(* semantics for a sample operation *) Section kernel_probability. -Variables (d : measure_display) (R : realType) (X : measurableType d). +Variables (d : _) (R : realType) (X : measurableType d). Variables (d' : _) (T' : measurableType d'). Variable m : probability X R. @@ -1115,115 +1038,69 @@ exact: measurable_fun_cst. Qed. HB.instance Definition _ := - @isKernel.Build _ d R _ X kernel_probability + @isKernel.Build _ _ _ X R kernel_probability kernel_probabilityP. -Lemma kernel_probability_uub : kernel_uub kernel_probability. +Lemma kernel_probability_uub : measure_uub kernel_probability. Proof. (*NB: shouldn't this work? exists 2%:pos. *) -exists (PosNum (addr_gt0 ltr01 ltr01)) => /= ?. +exists 2%R => /= ?. rewrite (le_lt_trans (probability_le1 m measurableT))//. by rewrite lte_fin ltr_addr. Qed. HB.instance Definition _ := - @isFiniteKernel.Build _ d R _ X kernel_probability + @isFiniteKernel.Build _ _ _ X R kernel_probability kernel_probability_uub. -Lemma kernel_probability_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, +Lemma sfinite_kernel_probability : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> kernel_probability x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: finite_kernel_sfinite_kernelP. Qed. +Proof. exact: sfinite_finite. Qed. HB.instance Definition _ := - @isSFiniteKernel.Build _ d R _ X kernel_probability - kernel_probability_sfinite_kernelP. + @isSFiniteKernel.Build _ _ _ X R kernel_probability + sfinite_kernel_probability. End kernel_probability. -(* semantics for return? *) -Section kernel_dirac. -Variables (R : realType) (d : _) (T : measurableType d). - -Definition kernel_dirac : T -> {measure set T -> \bar R} := - fun x => [the measure _ _ of dirac x]. - -Lemma kernel_diracP U : measurable U -> measurable_fun setT (kernel_dirac ^~ U). -Proof. -move=> mU; apply/EFin_measurable_fun. -by rewrite [X in measurable_fun _ X](_ : _ = mindic R mU). -Qed. - -HB.instance Definition _ := isKernel.Build _ _ R _ _ kernel_dirac kernel_diracP. - -Lemma kernel_dirac_uub : kernel_uub kernel_dirac. -Proof. -exists (PosNum (addr_gt0 ltr01 ltr01)) => t/=. -by rewrite diracE in_setT lte_fin ltr_addr. -Qed. - -HB.instance Definition _ := - @isFiniteKernel.Build d d R _ T kernel_dirac kernel_dirac_uub. - -Lemma kernel_dirac_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, - forall x U, measurable U -> - kernel_dirac x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: finite_kernel_sfinite_kernelP. Qed. - -HB.instance Definition _ := - @isSFiniteKernel.Build d d R T T kernel_dirac kernel_dirac_sfinite_kernelP. - -End kernel_dirac. - -Section kernel_dirac2. -Variables (R : realType) (d d' : _) (T : measurableType d) (T' : measurableType d'). -Variable (f : T -> T'). +Section kernel_of_mfun. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (f : T -> T'). -Definition kernel_dirac2 (mf : measurable_fun setT f) : T -> {measure set T' -> \bar R} := - fun x => [the measure _ _ of dirac (f x)]. +Definition kernel_mfun (mf : measurable_fun setT f) : T -> {measure set T' -> \bar R} := + fun t => [the measure _ _ of dirac (f t)]. -Variable (mf : measurable_fun setT f). +Hypothesis mf : measurable_fun setT f. -Lemma kernel_dirac2P U : measurable U -> measurable_fun setT (kernel_dirac2 mf ^~ U). +Lemma measurable_kernel_mfun U : measurable U -> measurable_fun setT (kernel_mfun mf ^~ U). Proof. -move=> mU; apply/EFin_measurable_fun. -have mTU : measurable (f @^-1` U). - have := mf measurableT mU. - by rewrite setTI. -by rewrite [X in measurable_fun _ X](_ : _ = mindic R mTU). +move=> mU. +apply/EFin_measurable_fun. +rewrite (_ : (fun x => _) = mindic R mU \o f)//. +exact/measurable_fun_comp. Qed. -HB.instance Definition _ := - isKernel.Build _ _ R _ _ (kernel_dirac2 mf) kernel_dirac2P. +HB.instance Definition _ := isKernel.Build _ _ _ _ R (kernel_mfun mf) + measurable_kernel_mfun. -Lemma kernel_dirac2_uub : kernel_uub (kernel_dirac2 mf). -Proof. -exists (PosNum (addr_gt0 ltr01 ltr01)) => t/=. -by rewrite diracE in_setT lte_fin ltr_addr. -Qed. +Lemma kernel_mfun_uub : measure_uub (kernel_mfun mf). +Proof. by exists 2%R => t/=; rewrite diracE in_setT lte_fin ltr_addr. Qed. -HB.instance Definition _ := - @isFiniteKernel.Build _ _ R _ _ (kernel_dirac2 mf) kernel_dirac2_uub. +HB.instance Definition _ := isFiniteKernel.Build _ _ _ _ R (kernel_mfun mf) + kernel_mfun_uub. -Lemma kernel_dirac2_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, - forall x U, measurable U -> - kernel_dirac2 mf x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: finite_kernel_sfinite_kernelP. Qed. +Lemma sfinite_kernel_mfun : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + kernel_mfun mf x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. exact: sfinite_finite. Qed. HB.instance Definition _ := - @isSFiniteKernel.Build _ _ R _ _ (kernel_dirac2 mf) kernel_dirac2_sfinite_kernelP. - -End kernel_dirac2. - -Definition letin (d d' d3 : measure_display) (R : realType) - (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) - (l : sfinite_kernel R X Y) - (k : sfinite_kernel R [the measurableType (d, d').-prod of (X * Y)%type] Z) - : sfinite_kernel R X Z := - [the sfinite_kernel _ _ _ of @mstar d d' d3 R X Y Z k l]. + @isSFiniteKernel.Build _ _ _ _ _ (kernel_mfun mf) sfinite_kernel_mfun. -(* semantics for score? *) +End kernel_of_mfun. +(* semantics for score *) Lemma set_unit (A : set unit) : A = set0 \/ A = setT. Proof. have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. @@ -1231,17 +1108,19 @@ by apply/seteqP; split => [|] []. Qed. Section score_measure. -Variables (R : realType). +Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R) (mr : measurable_fun setT r). -Definition mscore (r : R) (U : set unit) : \bar R := if U == set0 then 0 else `| r%:E |. +Definition mscore (t : T) (U : set unit) : \bar R := + if U == set0 then 0 else `| (r t)%:E |. -Lemma mscore0 r : mscore r (set0 : set unit) = 0 :> \bar R. +Lemma mscore0 t : mscore t (set0 : set unit) = 0 :> \bar R. Proof. by rewrite /mscore eqxx. Qed. -Lemma mscore_ge0 r U : 0 <= mscore r U. +Lemma mscore_ge0 t U : 0 <= mscore t U. Proof. by rewrite /mscore; case: ifP. Qed. -Lemma mscore_sigma_additive r : semi_sigma_additive (mscore r). +Lemma mscore_sigma_additive t : semi_sigma_additive (mscore t). Proof. move=> /= F mF tF mUF; rewrite /mscore; case: ifPn => [/eqP/bigcup0P F0|]. rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. @@ -1250,7 +1129,7 @@ move=> /= F mF tF mUF; rewrite /mscore; case: ifPn => [/eqP/bigcup0P F0|]. by rewrite big1. move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. rewrite -(cvg_shiftn k.+1)/=. -rewrite (_ : (fun _ => _) = cst `|r%:E|); first exact: cvg_cst. +rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. apply/funext => n. rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. rewrite (negbTE Fk0) big1 ?adde0// => i/= ik; rewrite ifT//. @@ -1259,36 +1138,32 @@ move/trivIsetP : tF => /(_ i k Logic.I Logic.I ik). by rewrite Fitt setTI => /eqP; rewrite (negbTE Fk0). Qed. -HB.instance Definition _ (r : R) := isMeasure.Build _ _ _ - (mscore r) (mscore0 r) (mscore_ge0 r) (@mscore_sigma_additive r). +HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ + (mscore t) (mscore0 t) (mscore_ge0 t) (@mscore_sigma_additive t). End score_measure. -(* NB: score r = observe 0 from exp r, - the density of the exponential distribution exp(r) at 0 is r = r e^(-r * 0) - more generally, score (r e^(-r * t)) = observe t from exp(r), - score (f(r)) = observe r from p where f is the density of p - -*) - Module KERNEL_SCORE. Section kernel_score. -Variable (R : realType) (d : _) (T : measurableType d). +Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R). -Definition k_' (r : R) (i : nat) : T -> set unit -> \bar R := - fun _ U => - if i%:R%:E <= mscore r U < i.+1%:R%:E then - mscore r U +Definition k_' (mr : measurable_fun setT r) (i : nat) : T -> set unit -> \bar R := + fun t U => + if i%:R%:E <= mscore r t U < i.+1%:R%:E then + mscore r t U else 0. -Lemma k_'0 (r : R) i (t : T) : k_' r i t (set0 : set unit) = 0 :> \bar R. +Variable (mr : measurable_fun setT r). + +Lemma k_'0 i (t : T) : k_' mr i t (set0 : set unit) = 0 :> \bar R. Proof. by rewrite /k_' measure0; case: ifP. Qed. -Lemma k_'ge0 (r : R) i (t : T) B : 0 <= k_' r i t B. +Lemma k_'ge0 i (t : T) B : 0 <= k_' mr i t B. Proof. by rewrite /k_'; case: ifP. Qed. -Lemma k_'sigma_additive (r : R) i (t : T) : semi_sigma_additive (k_' r i t). +Lemma k_'sigma_additive i (t : T) : semi_sigma_additive (k_' mr i t). Proof. move=> /= F mF tF mUF. rewrite /k_' /=. @@ -1304,7 +1179,7 @@ move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. rewrite [in X in _ --> X]/mscore (negbTE UF0). rewrite -(cvg_shiftn k.+1)/=. case: ifPn => ir. - rewrite (_ : (fun _ => _) = cst `|r%:E|); first exact: cvg_cst. + rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. apply/funext => n. rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. rewrite [in X in X + _]/mscore (negbTE Fk0) ir big1 ?adde0// => /= j jk. @@ -1329,225 +1204,328 @@ rewrite Fj0 eqxx. by case: ifP. Qed. -HB.instance Definition _ (r : R) (i : nat) (t : T) := isMeasure.Build _ _ _ - (k_' r i t) (k_'0 r i t) (k_'ge0 r i t) (@k_'sigma_additive r i t). +HB.instance Definition _ (i : nat) (t : T) := isMeasure.Build _ _ _ + (k_' mr i t) (k_'0 i t) (k_'ge0 i t) (@k_'sigma_additive i t). + +Lemma emeasurable_itv (i : nat) : + measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set \bar R). +Proof. +rewrite -[X in measurable X]setCK. +apply: measurableC. +rewrite set_interval.setCitv /=. +apply: measurableU. +exact: emeasurable_itv_ninfty_bnd. +exact: emeasurable_itv_bnd_pinfty. +Qed. -Lemma k_kernelP (r : R) (i : nat) : forall U, measurable U -> measurable_fun setT (k_' r i ^~ U). +Lemma k_kernelP (i : nat) : forall U, measurable U -> measurable_fun setT (k_' mr i ^~ U). Proof. move=> /= U mU. -rewrite /k_'. -by case: ifPn => _; exact: measurable_fun_cst. +rewrite /k_' /=. +rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o (fun x => mscore r x U)) //. +apply: measurable_fun_comp; last first. + rewrite /mscore. + have [U0|U0] := eqVneq U set0. + exact: measurable_fun_cst. + apply: measurable_fun_comp => //. + by apply/EFin_measurable_fun. +rewrite /=. +pose A : _ -> \bar R := (fun x : \bar R => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). +rewrite (_ : (fun x => _) = A); last first. + apply/funext => x; rewrite /A; case: ifPn => ix. + by rewrite indicE/= mem_set ?mule1//. + rewrite indicE/= memNset ?mule0//. + rewrite /= in_itv/=. + exact/negP. +rewrite /A. +apply emeasurable_funM => /=. + exact: measurable_fun_id. +apply/EFin_measurable_fun. +have mi : measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set (\bar R)). + exact: emeasurable_itv. +by rewrite (_ : \1__ = mindic R mi)//. Qed. -Definition mk_' (r : R) i (t : T) := [the measure _ _ of k_' r i t]. +Definition mk_' i (t : T) := [the measure _ _ of k_' mr i t]. -HB.instance Definition _ (r : R) (i : nat) := - isKernel.Build _ _ R _ _ (mk_' r i) (k_kernelP r i). +HB.instance Definition _ (i : nat) := + isKernel.Build _ _ _ _ R (mk_' i) (k_kernelP i). -Lemma k_uub (r : R) (i : nat) : kernel_uub (mk_' r i). +Lemma k_uub (i : nat) : measure_uub (mk_' i). Proof. -exists (PosNum (ltr0Sn _ i)) => /= t. +exists i.+1%:R => /= t. rewrite /k_' /mscore setT_unit. rewrite (_ : [set tt] == set0 = false); last first. by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). by case: ifPn => // /andP[]. Qed. -HB.instance Definition _ (r : R) (i : nat) := - @isFiniteKernel.Build _ _ R _ _ (mk_' r i) (k_uub r i). +HB.instance Definition _ (i : nat) := + @isFiniteKernel.Build _ _ _ _ R (mk_' i) (k_uub i). End kernel_score. End KERNEL_SCORE. Section kernel_score_kernel. Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R). -Definition kernel_score (r : R) : T -> {measure set _ -> \bar R} := - fun _ : T => [the measure _ _ of mscore r]. +Definition kernel_score (mr : measurable_fun setT r) : T -> {measure set Datatypes_unit__canonical__measure_Measurable -> \bar R} := + fun t : T => [the measure _ _ of mscore r t]. -Lemma kernel_scoreP (r : R) : forall U, measurable U -> - measurable_fun setT (kernel_score r ^~ U). +Variable (mr : measurable_fun setT r). + +Lemma kernel_scoreP : forall U, measurable U -> + measurable_fun setT (kernel_score mr ^~ U). Proof. -move=> /= U mU; rewrite /mscore; case: ifP => U0. +move=> /= U mU. +rewrite /mscore. +have [U0|U0] := eqVneq U set0. exact: measurable_fun_cst. apply: measurable_fun_comp => //. -apply/EFin_measurable_fun. -exact: measurable_fun_cst. +by apply/EFin_measurable_fun. Qed. -HB.instance Definition _ (r : R) := - isKernel.Build _ _ R T - _ (*Datatypes_unit__canonical__measure_Measurable*) - (kernel_score r) (kernel_scoreP r). +HB.instance Definition _ := + isKernel.Build _ _ T + _ (*Datatypes_unit__canonical__measure_Measurable*) R + (kernel_score mr) (kernel_scoreP). End kernel_score_kernel. Section kernel_score_sfinite_kernel. Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R) (mr : measurable_fun setT r). Import KERNEL_SCORE. -Lemma kernel_score_sfinite_kernelP (r : R) : exists k_ : (finite_kernel R T _)^nat, +Lemma kernel_score_sfinite_kernelP : exists k_ : (R.-fker T ~> _)^nat, forall x U, measurable U -> - kernel_score r x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. + kernel_score mr x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -exists (fun i => [the finite_kernel _ _ _ of mk_' r i]) => /= r' U mU. +rewrite /=. +exists (fun i => [the finite_kernel _ _ _ of mk_' mr i]) => /= r' U mU. rewrite /mseries /mscore; case: ifPn => [/eqP U0|U0]. by apply/esym/nneseries0 => i _; rewrite U0 measure0. rewrite /mk_' /= /k_' /= /mscore (negbTE U0). apply/esym/cvg_lim => //. -rewrite -(cvg_shiftn `|floor (fine `|r%:E|)|%N.+1)/=. -rewrite (_ : (fun _ => _) = cst `|r%:E|); first exact: cvg_cst. +rewrite -(cvg_shiftn `|floor (fine `|(r r')%:E|)|%N.+1)/=. +rewrite (_ : (fun _ => _) = cst `|(r r')%:E|); first exact: cvg_cst. apply/funext => n. -pose floor_r := widen_ord (leq_addl n `|floor `|r| |.+1) (Ordinal (ltnSn `|floor `|r| |)). +pose floor_r := widen_ord (leq_addl n `|floor `|(r r')| |.+1) (Ordinal (ltnSn `|floor `|(r r')| |)). rewrite big_mkord (bigD1 floor_r)//= ifT; last first. rewrite lee_fin lte_fin; apply/andP; split. - by rewrite natr_absz (@ger0_norm _ (floor `|r|)) ?floor_ge0 ?floor_le. - by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|r|)) ?floor_ge0 ?lt_succ_floor. + by rewrite natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?floor_le. + by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?lt_succ_floor. rewrite big1 ?adde0//= => j jk. rewrite ifF// lte_fin lee_fin. move: jk; rewrite neq_ltn/= => /orP[|] jr. -- suff : (j.+1%:R <= `|r|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. +- suff : (j.+1%:R <= `|(r r')|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. rewrite (_ : j.+1%:R = j.+1%:~R)// floor_ge_int. move: jr; rewrite -lez_nat => /le_trans; apply. - by rewrite -[leRHS](@ger0_norm _ (floor `|r|)) ?floor_ge0. -- suff : (`|r| < j%:R)%R by rewrite ltNge => /negbTE ->. - move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|r|)) ?floor_ge0// ltr_int. + by rewrite -[leRHS](@ger0_norm _ (floor `|(r r')|)) ?floor_ge0. +- suff : (`|(r r')| < j%:R)%R by rewrite ltNge => /negbTE ->. + move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|(r r')|)) ?floor_ge0// ltr_int. by rewrite -floor_lt_int. Qed. -HB.instance Definition _ (r : R) := @isSFiniteKernel.Build _ _ _ _ _ - (kernel_score r) (kernel_score_sfinite_kernelP r). +HB.instance Definition _ := @isSFiniteKernel.Build _ _ _ _ _ + (kernel_score mr) (kernel_score_sfinite_kernelP). End kernel_score_sfinite_kernel. -Section ite. +Section ite_true_kernel. Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables - (u1 : sfinite_kernel R - [the measurableType _ of (T * bool)%type] - [the measurableType _ of T']) - (u2 : sfinite_kernel R - [the measurableType _ of (T * bool)%type] - [the measurableType _ of T']). - -Definition ite : T * bool -> set _ -> \bar R := - fun t => if t.2 then u1 t else u2 t. +Variables (u1 : R.-ker T ~> T'). -Lemma ite0 tb : ite tb set0 = 0. -Proof. by rewrite /ite; case: ifPn => //. Qed. +Definition ite_true : T * bool -> {measure set T' -> \bar R} := + fun b => if b.2 then u1 b.1 else [the measure _ _ of mzero]. -Lemma ite_ge0 tb (U : set _) : 0 <= ite tb U. -Proof. by rewrite /ite; case: ifPn => //. Qed. - -Lemma ite_sigma_additive tb : semi_sigma_additive (ite tb). +Lemma measurable_ite_true U : measurable U -> measurable_fun setT (ite_true ^~ U). Proof. -Admitted. +move=> /= mcU. +rewrite /ite_true. +rewrite (_ : (fun x : T * bool => _) = (fun x => if x.2 then u1 x.1 U else [the {measure set T' -> \bar R} of mzero] U)); last first. + apply/funext => -[t b]/=. + by case: ifPn. +apply: (@measurable_fun_if _ _ _ _ (u1 ^~ U) (fun=> mzero U)). + exact/measurable_kernel. +exact: measurable_fun_cst. +Qed. -HB.instance Definition _ tb := isMeasure.Build _ _ _ - (ite tb) - (ite0 tb) (ite_ge0 tb) (@ite_sigma_additive tb). +HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_true measurable_ite_true. +End ite_true_kernel. -Lemma ite_kernelP : forall U, measurable U -> measurable_fun setT (ite ^~ U). -Admitted. +Section ite_true_finite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u1 : R.-fker T ~> T'). + +Lemma ite_true_uub : measure_uub (ite_true u1). +Proof. +have /measure_uubP[M hM] := kernel_uub u1. +exists M%:num => /= -[]; rewrite /ite_true => t [|]/=. + exact: hM. +by rewrite /= /mzero. +Qed. -Definition mite tb := [the measure _ _ of ite tb]. +HB.instance Definition _ t := + isFiniteKernel.Build _ _ _ _ R (ite_true u1) ite_true_uub. +End ite_true_finite_kernel. -HB.instance Definition _ := isKernel.Build _ _ R _ _ mite ite_kernelP. +Section ite_true_sfinite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u1 : R.-sfker T ~> T'). -Lemma ite_sfinite_kernelP : exists k_ : (finite_kernel R _ _)^nat, - forall x U, measurable U -> - ite x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Admitted. +Lemma sfinite_ite_true : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + ite_true u1 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +have [k hk /=] := sfinite u1. +rewrite /ite_true. +exists (fun n => [the finite_kernel _ _ _ of ite_true (k n)]) => b U mU. +case: ifPn => hb. + rewrite /mseries hk//= /mseries. + apply: eq_nneseries => n _. + by rewrite /ite_true hb. +rewrite /= /mseries nneseries0// => n _. +by rewrite /ite_true (negbTE hb). +Qed. -HB.instance Definition _ := - @isSFiniteKernel.Build _ _ _ _ _ mite ite_sfinite_kernelP. +HB.instance Definition _ t := + @isSFiniteKernel.Build _ _ _ _ _ (ite_true u1) sfinite_ite_true. -End ite. +End ite_true_sfinite_kernel. -Section insn. -Variables (R : realType). +Section ite_false_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u2 : R.-ker T ~> T'). -Definition sample_bernoulli27 (d : _) (T : measurableType d) := - [the sfinite_kernel _ T _ of - kernel_probability [the probability _ _ of bernoulli27 R]] . +Definition ite_false : T * bool -> {measure set T' -> \bar R} := + fun b => if ~~ b.2 then u2 b.1 else [the measure _ _ of mzero]. -Definition Ite (d d' : _) (T : measurableType d) (T' : measurableType d') - (u1 : sfinite_kernel R [the measurableType _ of (T * bool)%type] - [the measurableType _ of T']) - (u2 : sfinite_kernel R [the measurableType _ of (T * bool)%type] - [the measurableType _ of T']) - : sfinite_kernel R [the measurableType _ of (T * bool)%type] _ := - [the sfinite_kernel R _ _ of mite u1 u2]. +Lemma measurable_ite_false U : measurable U -> measurable_fun setT (ite_false ^~ U). +Proof. +move=> /= mcU. +rewrite /ite_false. +rewrite (_ : (fun x => _) = (fun x => if x.2 then [the {measure set T' -> \bar R} of mzero] U else u2 x.1 U)); last first. + apply/funext => -[t b]/=. + rewrite if_neg/=. + by case: b. +apply: (@measurable_fun_if _ _ _ _ (fun=> mzero U) (u2 ^~ U)). + exact: measurable_fun_cst. +exact/measurable_kernel. +Qed. -Definition Return (d : _) (T : measurableType d) : sfinite_kernel R T T := - [the sfinite_kernel _ _ _ of @kernel_dirac R _ _]. +HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_false measurable_ite_false. -Definition Return2 (d d' : _) (T : measurableType d) (T' : measurableType d') - (f : T -> T') (mf : measurable_fun setT f) : sfinite_kernel R T T' := - [the sfinite_kernel _ _ _ of @kernel_dirac2 R _ _ T T' f mf]. +End ite_false_kernel. -Definition Score (d : _) (T : measurableType d) (r : R) : - sfinite_kernel R T Datatypes_unit__canonical__measure_Measurable := - [the sfinite_kernel R _ _ of @kernel_score R _ _ r]. +Section ite_false_finite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u2 : R.-fker T ~> T'). -End insn. +Lemma ite_false_uub : measure_uub (ite_false u2). +Proof. +have /measure_uubP[M hM] := kernel_uub u2. +exists M%:num => /= -[]; rewrite /ite_false/= => t b. +case: b => //=. +by rewrite /mzero. +Qed. -Section program1. -Variables (R : realType) (d : _) (T : measurableType d). +HB.instance Definition _ := + isFiniteKernel.Build _ _ _ _ R (ite_false u2) ite_false_uub. -Lemma measurable_fun_snd : measurable_fun setT (snd : T * bool -> bool). Admitted. +End ite_false_finite_kernel. -Definition program1 : sfinite_kernel R T - _ := - letin - (sample_bernoulli27 R T) (* T -> B *) - (Return2 R measurable_fun_snd) (* T * B -> B *). +Section ite_false_sfinite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u2 : R.-sfker T ~> T'). -Lemma program1E (t : T) (U : _) : program1 t U = - ((twoseven R)%:num)%:E * \d_true U + - ((fiveseven R)%:num)%:E * \d_false U. +Lemma sfinite_ite_false : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + ite_false u2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -rewrite /program1/= /star/=. -rewrite ge0_integral_measure_sum// 2!big_ord_recl/= big_ord0 adde0/=. -rewrite !ge0_integral_mscale//=. -rewrite !integral_dirac//=. -by rewrite indicE in_setT mul1e indicE in_setT mul1e. +have [k hk] := sfinite u2. +rewrite /= /ite_false. +exists (fun n => [the finite_kernel _ _ _ of ite_false (k n)]) => b U mU. +case: ifPn => hb. + rewrite /mseries hk//= /mseries/=. + apply: eq_nneseries => // n _. + by rewrite /ite_false hb. +rewrite /= /mseries nneseries0// => n _. +rewrite negbK in hb. +by rewrite /ite_false hb/=. Qed. -End program1. +HB.instance Definition _ := + @isSFiniteKernel.Build _ _ _ _ _ (ite_false u2) sfinite_ite_false. -Section program2. -Variables (R : realType) (d : _) (T : measurableType d). +End ite_false_sfinite_kernel. -Definition program2 : sfinite_kernel R T Datatypes_unit__canonical__measure_Measurable := - letin - (sample_bernoulli27 R T) (* T -> B *) - (Score _ (1%:R : R)). +Section add_of_kernels. +Variables (d d' : measure_display) (R : realType). +Variables (X : measurableType d) (Y : measurableType d'). +Variables (u1 u2 : R.-ker X ~> Y). -End program2. +Definition add_of_kernels : X -> {measure set Y -> \bar R} := + fun t => [the measure _ _ of measure_add (u1 t) (u2 t)]. -Section program3. -Variables (R : realType) (d : _) (T : measurableType d). +Lemma measurable_add_of_kernels U : measurable U -> measurable_fun setT (add_of_kernels ^~ U). +Proof. +move=> mU. +rewrite /add_of_kernels. +rewrite (_ : (fun x : X => _) = (fun x => (u1 x U) + (u2 x U))); last first. + apply/funext => x. + by rewrite -measure_addE. +by apply: emeasurable_funD; exact/measurable_kernel. +Qed. -(* let x = sample (bernoulli (2/7)) in - let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in - let _ = score (1/4! r^4 e^-r) in - return x *) +HB.instance Definition _ := + @isKernel.Build _ _ _ _ _ add_of_kernels measurable_add_of_kernels. +End add_of_kernels. -Definition k3' : T * bool -> R := cst 3%:R. -Definition k10' : T * bool -> R := cst 10%:R. +Section add_of_finite_kernels. +Variables (d d' : measure_display) (R : realType). +Variables (X : measurableType d) (Y : measurableType d'). +Variables (u1 u2 : R.-fker X ~> Y). -Lemma mk3 : measurable_fun setT k3'. -exact: measurable_fun_cst. +Lemma add_of_finite_kernels_uub : measure_uub (add_of_kernels u1 u2). +Proof. +have [k1 hk1] := kernel_uub u1. +have [k2 hk2] := kernel_uub u2. +exists (k1 + k2)%R => x. +rewrite /add_of_kernels/=. +rewrite -/(measure_add (u1 x) (u2 x)). +rewrite measure_addE. +rewrite EFinD. +exact: lte_add. Qed. -Lemma mk10 : measurable_fun setT k10'. -exact: measurable_fun_cst. -Qed. +HB.instance Definition _ t := + isFiniteKernel.Build _ _ _ _ R (add_of_kernels u1 u2) add_of_finite_kernels_uub. +End add_of_finite_kernels. + +Section add_of_sfinite_kernels. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (u1 u2 : R.-sfker X ~> Y). -Definition program10 : sfinite_kernel R T _ := - letin - (sample_bernoulli27 R T) (* T -> B *) - (Return2 R mk3). +Lemma sfinite_add_of_kernels : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + add_of_kernels u1 u2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +have [k1 hk1] := sfinite u1. +have [k2 hk2] := sfinite u2. +exists (fun n => [the finite_kernel _ _ _ of add_of_kernels (k1 n) (k2 n)]) => x U mU. +rewrite /add_of_kernels/=. +rewrite -/(measure_add (u1 x) (u2 x)). +rewrite measure_addE. +rewrite /mseries. +rewrite hk1//= hk2//= /mseries. +rewrite -nneseriesD//. +apply: eq_nneseries => n _. +rewrite -/(measure_add (k1 n x) (k2 n x)). +by rewrite measure_addE. +Qed. -End program3. +HB.instance Definition _ t := + isSFiniteKernel.Build _ _ _ _ R (add_of_kernels u1 u2) sfinite_add_of_kernels. +End add_of_sfinite_kernels. diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index b40449688d..8cf3524e02 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -4166,13 +4166,17 @@ Qed. End measurable_section. Section ndseq_closed_B. -Variables (d1 d2 : measure_display). -Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Variables (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) + (R : realType). Implicit Types A : set (T1 * T2). Section xsection. -Variables (pt2 : T2) (m2 : {measure set T2 -> \bar R}). -Let phi A := m2 \o xsection A. +Variables (pt2 : T2) (m2 : T1 -> {measure set T2 -> \bar R}). +(* the generalization from m2 : {measure set T2 -> \bar R}t to + T1 -> {measure set T2 -> \bar R} is needed to develop the theory + of kernels; the original type was sufficient for the the development + of the theory of integration *) +Let phi A x := m2 x (xsection A x). Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. Lemma xsection_ndseq_closed : ndseq_closed B. diff --git a/theories/prob_lang.v b/theories/prob_lang.v new file mode 100644 index 0000000000..375ca61078 --- /dev/null +++ b/theories/prob_lang.v @@ -0,0 +1,357 @@ +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. +Require Import mathcomp_extra boolp classical_sets signed functions cardinality. +Require Import reals ereal topology normedtype sequences esum measure. +Require Import lebesgue_measure fsbigop numfun lebesgue_integral kernel. + +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. +Local Open Scope ereal_scope. + +Section ite. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variables (R : realType) (f : T -> bool) (u1 u2 : R.-sfker T ~> T'). + +Definition ite (mf : measurable_fun setT f) : T -> set T' -> \bar R := + fun t => if f t then u1 t else u2 t. + +Variables mf : measurable_fun setT f. + +Lemma ite0 tb : ite mf tb set0 = 0. +Proof. by rewrite /ite; case: ifPn => //. Qed. + +Lemma ite_ge0 tb (U : set _) : 0 <= ite mf tb U. +Proof. by rewrite /ite; case: ifPn => //. Qed. + +Lemma ite_sigma_additive tb : semi_sigma_additive (ite mf tb). +Proof. +rewrite /ite. +case: ifPn => ftb. + exact: measure_semi_sigma_additive. +exact: measure_semi_sigma_additive. +Qed. + +HB.instance Definition _ tb := isMeasure.Build _ _ _ (ite mf tb) + (ite0 tb) (ite_ge0 tb) (@ite_sigma_additive tb). + +Definition ite' : R.-sfker + [the measurableType _ of (T * bool)%type] ~> T' := + [the R.-sfker _ ~> _ of add_of_kernels + [the R.-sfker _ ~> T' of ite_true u1] + [the R.-sfker _ ~> T' of ite_false u2] ]. + +Definition mite := [the sfinite_kernel _ _ _ of kernel_mfun R mf] \; ite'. + +End ite. + +Section bernoulli27. +Variable R : realType. + +Local Open Scope ring_scope. +Notation "'2/7'" := (2%:R / 7%:R)%:nng. +Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. +Definition fiveseven : {nonneg R} := (5%:R / 7%:R)%:nng. + +Definition bernoulli27 : set _ -> \bar R := + measure_add + [the measure _ _ of mscale twoseven [the measure _ _ of dirac true]] + [the measure _ _ of mscale fiveseven [the measure _ _ of dirac false]]. + +HB.instance Definition _ := Measure.on bernoulli27. + +Local Close Scope ring_scope. + +Lemma bernoulli27_setT : bernoulli27 [set: _] = 1. +Proof. +rewrite /bernoulli27/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. +rewrite /mscale/= !diracE !in_setT !mule1 -EFinD. +by rewrite -mulrDl -natrD divrr// unitfE pnatr_eq0. +Qed. + +HB.instance Definition _ := @isProbability.Build _ _ R bernoulli27 bernoulli27_setT. + +End bernoulli27. + +Section insn. +Variables (R : realType). + +Definition letin (d d' d3 : _) + (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) + (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) + : R.-sfker X ~> Z := + [the sfinite_kernel _ _ _ of (l \; k)]. + +Definition Return (d d' : _) (T : measurableType d) (T' : measurableType d') + (f : T -> T') (mf : measurable_fun setT f) : R.-sfker T ~> T' := + [the sfinite_kernel _ _ _ of @kernel_mfun _ _ T T' R f mf]. + +Definition sample_bernoulli27 (d : _) (T : measurableType d) := + [the sfinite_kernel T _ _ of + kernel_probability [the probability _ _ of bernoulli27 R]] . + +(* NB: score r = observe 0 from exp r, + the density of the exponential distribution exp(r) at 0 is r = r e^(-r * 0) + more generally, score (r e^(-r * t)) = observe t from exp(r), + score (f(r)) = observe r from p where f is the density of p *) +Definition Score (d : _) (T : measurableType d) (r : T -> R) (mr : measurable_fun setT r) : + R.-sfker T ~> Datatypes_unit__canonical__measure_Measurable := + [the sfinite_kernel _ _ R of @kernel_score R _ _ r mr]. + +Definition Ite (d d' : _) (T : measurableType d) (T' : measurableType d') + (f : T -> bool) (mf : measurable_fun setT f) + (u1 u2 : R.-sfker T ~> T') + : R.-sfker T ~> T' := + [the R.-sfker _ ~> _ of mite u1 u2 mf]. + +Lemma IteE (d d' : _) (T : measurableType d) (T' : measurableType d') + (f : T -> bool) (mf : measurable_fun setT f) + (u1 u2 : R.-sfker T ~> T') tb U : + Ite mf u1 u2 tb U = ite u1 u2 mf tb U. +Proof. +rewrite /= /kcomp /ite. +rewrite integral_dirac//=. +rewrite indicT /cst. +rewrite mul1e. +rewrite -/(measure_add (ite_true u1 (tb, f tb)) + (ite_false u2 (tb, f tb))). +rewrite measure_addE. +rewrite /ite_true /ite_false/=. +case: (ifPn (f tb)) => /=. + by rewrite /mzero adde0. +by rewrite /mzero add0e. +Qed. + +End insn. + +(* a few laws *) + +Section letin_return. +Variables (d d' d3 : _) (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType d3). + +Lemma letin_ureturn (u : R.-sfker X ~> Y) + (f : _ -> Z) (mf : measurable_fun setT f) : + forall x U, measurable U -> letin u (Return R mf) x U = u x ((fun y => f (x, y)) @^-1` U). +Proof. +move=> x U mU. +rewrite /letin/= /kcomp/= integral_indic// ?setIT//. +move/measurable_fun_prod1 : mf => /(_ x)/(_ measurableT U mU). +by rewrite setTI. +Qed. + +Lemma letin_returnu + (u : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) + (f : _ -> Y) (mf : measurable_fun setT f) : + forall x U, measurable U -> letin (Return R mf) u x U = u (x, f x) U. +Proof. +move=> x U mU. +rewrite /letin/= /kcomp/= integral_dirac//. + by rewrite indicE mem_set// mul1e. +have /measurable_fun_prod1 := measurable_kernel u _ mU. +exact. +Qed. + +End letin_return. + +Section letin_ite. +Variables (R : realType) (d d2 d3 : _) (T : measurableType d) + (T2 : measurableType d2) (T3 : measurableType d3) + (u1 u2 : R.-sfker T ~> T3) (u : R.-sfker [the measurableType _ of (T * T3)%type] ~> T2) + (f : T -> bool) (mf : measurable_fun setT f) + (t : T) (U : set T2). + +Lemma letin_ite_true : f t -> letin (Ite mf u1 u2) u t U = letin u1 u t U. +Proof. +move=> ftT. +rewrite /letin/= /kcomp. +apply eq_measure_integral => V mV _. +by rewrite IteE /ite ftT. +Qed. + +Lemma letin_ite_false : ~~ f t -> letin (Ite mf u1 u2) u t U = letin u2 u t U. +Proof. +move=> ftF. +rewrite /letin/= /kcomp. +apply eq_measure_integral => V mV _. +by rewrite IteE/= /ite (negbTE ftF). +Qed. + +End letin_ite. + +(* sample programs *) + +Require Import exp. + +Definition poisson (R : realType) (r : R) (k : nat) := (r ^+ k / k%:R^-1 * expR (- r))%R. + +Definition poisson3 (R : realType) := poisson (3%:R : R) 4. (* 0.168 *) +Definition poisson10 (R : realType) := poisson (10%:R : R) 4. (* 0.019 *) + +Lemma poisson_ge0 (R : realType) (r : R) k : (0 <= r)%R -> (0 <= poisson r k)%R. +Proof. +move=> r0; rewrite /poisson mulr_ge0//. + by rewrite mulr_ge0// exprn_ge0//. +by rewrite ltW// expR_gt0. +Qed. + +Lemma mpoisson (R : realType) k : measurable_fun setT (@poisson R ^~ k). +Proof. +apply: measurable_funM => /=. + apply: measurable_funM => //=; last exact: measurable_fun_cst. + exact/measurable_fun_exprn/measurable_fun_id. +apply: measurable_fun_comp. + apply: continuous_measurable_fun. + exact: continuous_expR. +apply: continuous_measurable_fun. +by have := (@opp_continuous R [the normedModType R of R^o]). +Qed. + +Section cst_fun. +Variables (R : realType) (d : _) (T : measurableType d). + +Definition kn (n : nat) := @measurable_fun_cst _ _ T _ setT (n%:R : R). +Definition k3 : measurable_fun _ _ := kn 3. +Definition k10 : measurable_fun _ _ := kn 10. + +End cst_fun. + +Lemma ScoreE (R : realType) (d : _) (T : measurableType d) (t : T) (U : set bool) (n : nat) (b : bool) + (f : R -> R) (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) (mf : measurable_fun setT f) : + Score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) + (t, b, cst n%:R (t, b)) + ((fun y : unit => (snd \o fst) (t, b, y)) @^-1` U) = + (f n%:R)%:E * \d_b U. +Proof. +rewrite /Score/= /mscore/= diracE. +have [U0|U0] := set_unit ((fun=> b) @^-1` U). +- rewrite U0 eqxx memNset ?mule0//. + move=> Ub. + move: U0. + move/seteqP => [/(_ tt)] /=. + by move/(_ Ub). +- rewrite U0 setT_unit ifF//; last first. + by apply/negbTE/negP => /eqP/seteqP[/(_ tt erefl)]. + rewrite /= mem_set//; last first. + by move: U0 => /seteqP[_]/(_ tt)/=; exact. + by rewrite mule1 ger0_norm// f0. +Qed. + +Lemma letin_sample_bernoulli27 (R : realType) (d d' : _) (T : measurableType d) + (T' : measurableType d') + (u : R.-sfker [the measurableType _ of (T * bool)%type] ~> T') x y : + letin (sample_bernoulli27 R T) u x y = + (2 / 7)%:E * u (x, true) y + (5 / 7)%:E * u (x, false) y. +Proof. +rewrite {1}/letin/= {1}/kcomp/=. +rewrite ge0_integral_measure_sum//. +rewrite 2!big_ord_recl/= big_ord0 adde0/=. +rewrite !ge0_integral_mscale//=. +rewrite !integral_dirac//=. +by rewrite indicE in_setT mul1e indicE in_setT mul1e. +Qed. + +(* *) + +Section program1. +Variables (R : realType) (d : _) (T : measurableType d). + +Definition program1 : R.-sfker T ~> _ := + letin + (sample_bernoulli27 R T) (* T -> B *) + (Return R (@measurable_fun_snd _ _ _ _)) (* T * B -> B *). + +Lemma program1E (t : T) (U : _) : program1 t U = + ((twoseven R)%:num)%:E * \d_true U + + ((fiveseven R)%:num)%:E * \d_false U. +Proof. +rewrite /program1. +by rewrite letin_sample_bernoulli27. +Qed. + +End program1. + +Section program2. +Variables (R : realType) (d : _) (T : measurableType d). + +Definition program2 : R.-sfker T ~> _ := + letin + (sample_bernoulli27 R T) (* T -> B *) + (Score (measurable_fun_cst (1%:R : R))). + +End program2. + +Section program3. +Variables (R : realType) (d : _) (T : measurableType d). + +(* let x = sample (bernoulli (2/7)) in + let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in + return r *) + +Definition program3 : + R.-sfker T ~> [the measurableType default_measure_display of Real_sort__canonical__measure_Measurable R] := + letin + (sample_bernoulli27 R T) (* T -> B *) + (Ite (@measurable_fun_snd _ _ _ _) + (Return R (@k3 _ _ [the measurableType _ of (T * bool)%type])) + (Return R (@k10 _ _ [the measurableType _ of (T * bool)%type]))). + +Lemma program3E (t : T) (U : _) : program3 t U = + ((twoseven R)%:num)%:E * \d_(3%:R : R) U + + ((fiveseven R)%:num)%:E * \d_(10%:R : R) U. +Proof. +rewrite /program3 letin_sample_bernoulli27. +congr (_ * _ + _ * _). +by rewrite IteE. +by rewrite IteE. +Qed. + +End program3. + +Section program4. +Variables (R : realType) (d : _) (T : measurableType d). + +(* let x = sample (bernoulli (2/7)) in + let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in + let _ = score (1/4! r^4 e^-r) in + return x *) + +Definition program4 : R.-sfker T ~> Datatypes_bool__canonical__measure_Measurable := + letin + (sample_bernoulli27 R T) (* T -> B *) + (letin + (letin (* T * B -> unit *) + (Ite (@measurable_fun_snd _ _ _ _) + (Return R (@k3 _ _ [the measurableType _ of (T * bool)%type])) + (Return R (@k10 _ _ [the measurableType _ of (T * bool)%type]))) (* T * B -> R *) + (Score (measurable_fun_comp (@mpoisson R 4) (@measurable_fun_snd _ _ _ _))) (* B * R -> unit *)) + (Return R (measurable_fun_comp (@measurable_fun_snd _ _ _ _) (@measurable_fun_fst _ _ _ _)))). + +(* true -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) +(* false -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) + +Lemma program4E (t : T) (U : _) : program4 t U = + ((twoseven R)%:num)%:E * (poisson 3%:R 4)%:E * \d_(true) U + + ((fiveseven R)%:num)%:E * (poisson 10%:R 4)%:E * \d_(false) U. +Proof. +rewrite /program4. +rewrite letin_sample_bernoulli27. +rewrite -!muleA. +congr (_ * _ + _ * _). + rewrite letin_ureturn //. + rewrite letin_ite_true//. + rewrite letin_returnu//. + by rewrite ScoreE// => r r0; exact: poisson_ge0. +rewrite letin_ureturn //. +rewrite letin_ite_false//. +rewrite letin_returnu//. +by rewrite ScoreE// => r r0; exact: poisson_ge0. +Qed. + +End program4. From 067b47e4ce89f8de87df9bb13b2a07fdd222ff1a Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Thu, 18 Aug 2022 20:57:33 +0900 Subject: [PATCH 32/42] factorization of code, normalize, cleaning --- theories/kernel.v | 800 +++++++++++++------------------------------ theories/prob_lang.v | 560 +++++++++++++++++++++++++----- 2 files changed, 720 insertions(+), 640 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 1ea424ee32..266960f527 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -14,8 +14,6 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (* sum_of_kernels == *) (* l \; k == composition of kernels *) (* kernel_mfun == kernel defined by a measurable function *) -(* mscore == *) -(* ite_true/ite_false == *) (* add_of_kernels == *) (******************************************************************************) @@ -255,10 +253,11 @@ Lemma measurable_curry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) measurable (G x) <-> measurable (curry G x.1 x.2). Proof. by case: x. Qed. -Lemma measurable_fun_if (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') : +Lemma measurable_fun_if0 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') + (f : T -> bool) (mf : measurable_fun setT f) : measurable_fun setT x -> measurable_fun setT y -> - measurable_fun setT (fun b : T * bool => if b.2 then x b.1 else y b.1). + measurable_fun setT (fun b : T => if f b then x b else y b). Proof. move=> mx my /= _ Y mY. rewrite setTI. @@ -266,17 +265,53 @@ have := mx measurableT Y mY. rewrite setTI => xY. have := my measurableT Y mY. rewrite setTI => yY. -rewrite (_ : _ @^-1` Y = (x @^-1` Y) `*` [set true] `|` (y @^-1` Y) `*` [set false]); last first. +rewrite (_ : _ @^-1` Y = ((x @^-1` Y) `&` (f @^-1` [set true])) `|` + ((y @^-1` Y) `&` (f @^-1` [set false]))); last first. apply/seteqP; split. - move=> [t [|]]/=. + move=> t/=; case: ifPn => ft. by left. by right. - move=> [t [|]]/=. - by case=> [[]//|[]]. - by case=> [[]//|[]]. -by apply: measurableU; apply: measurableM => //. + by move=> t/=; case: ifPn => ft; case=> -[]. +apply: measurableU; apply: measurableI => //. + have := mf measurableT [set true]. + by rewrite setTI; exact. +have := mf measurableT [set false]. +by rewrite setTI; exact. +Qed. + +Lemma measurable_fun_if (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') : + measurable_fun setT x -> + measurable_fun setT y -> + measurable_fun setT (fun b : T * bool => if b.2 then x b.1 else y b.1). +Proof. +move=> mx my. +have {}mx : measurable_fun [set: T * bool] (x \o fst). + apply: measurable_fun_comp => //. + exact: measurable_fun_fst. +have {}my : measurable_fun [set: T * bool] (y \o fst). + apply: measurable_fun_comp => //. + exact: measurable_fun_fst. +rewrite /=. +apply: measurable_fun_if0 => //=. +exact: measurable_fun_snd. +Qed. + +Lemma emeasurable_itv (R : realType) (i : nat) : + measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set \bar R). +Proof. +rewrite -[X in measurable X]setCK. +apply: measurableC. +rewrite set_interval.setCitv /=. +apply: measurableU. + exact: emeasurable_itv_ninfty_bnd. +exact: emeasurable_itv_bnd_pinfty. Qed. +Lemma set_unit (A : set unit) : A = set0 \/ A = setT. +Proof. +have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. +by apply/seteqP; split => [|] []. +Qed. (*/ PR*) Reserved Notation "R .-ker X ~> Y" (at level 42). @@ -347,13 +382,13 @@ HB.structure Definition ProbabilityKernel (R : realType) := {k of isProbabilityKernel _ _ X Y R k & isKernel _ _ X Y R k}. -Section measure_uub. +Section measure_fam_uub. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variables (R : numFieldType) (k : X -> {measure set Y -> \bar R}). -Definition measure_uub := exists r, forall x, k x [set: Y] < r%:E. +Definition measure_fam_uub := exists r, forall x, k x [set: Y] < r%:E. -Lemma measure_uubP : measure_uub <-> +Lemma measure_fam_uubP : measure_fam_uub <-> exists r : {posnum R}, forall x, k x [set: Y] < r%:num%:E. Proof. split => [|] [r kr]; last by exists r%:num. @@ -361,12 +396,12 @@ suff r_gt0 : (0 < r)%R by exists (PosNum r_gt0). by rewrite -lte_fin; apply: (le_lt_trans _ (kr point)). Qed. -End measure_uub. +End measure_fam_uub. HB.mixin Record isFiniteKernel d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := - { kernel_uub : measure_uub k }. + { kernel_uub : measure_fam_uub k }. #[short(type=finite_kernel)] HB.structure Definition FiniteKernel @@ -392,11 +427,8 @@ HB.instance Definition _ := @isKernel.Build _ _ T' T R kernel_from_mzero kernel_from_mzeroP. -Lemma kernel_from_mzero_uub : measure_uub kernel_from_mzero. -Proof. -exists 1%R => /= t. -by rewrite /mzero/=. -Qed. +Lemma kernel_from_mzero_uub : measure_fam_uub kernel_from_mzero. +Proof. by exists 1%R => /= t; rewrite /mzero/=. Qed. HB.instance Definition _ := @isFiniteKernel.Build _ _ _ T R kernel_from_mzero @@ -529,42 +561,51 @@ Qed. End measurable_fun_xsection_finite_kernel. (* pollard *) -Lemma measurable_fun_integral_finite_kernel - (d d' : _) (X : measurableType d) (Y : measurableType d') - (R : realType) (l : R.-fker X ~> Y) (k : (X * Y)%type -> \bar R) - (k0 : (forall z, True -> 0 <= k z)) (mk : measurable_fun setT k) : +Section measurable_fun_integral_finite_sfinite. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType). + +Lemma measurable_fun_xsection_integral + (l : X -> {measure set Y -> \bar R}) + (k : X * Y -> \bar R) + (k_ : ({nnsfun [the measurableType _ of (X * Y)%type] >-> R})^nat) + (ndk_ : nondecreasing_seq (k_ : (X * Y -> R)^nat)) + (k_k : forall z, EFin \o (k_ ^~ z) --> k z) : + (forall n r, measurable_fun setT (fun x => l x (xsection (k_ n @^-1` [set r]) x))) -> measurable_fun setT (fun x => \int[l x]_y k (x, y)). Proof. -have [k_ [ndk_ k_k]] := approximation measurableT mk k0. -rewrite (_ : (fun x => \int[l x]_y k (x, y)) = +move=> h. +rewrite (_ : (fun x => _) = (fun x => elim_sup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. - apply/funeqP => x. + apply/funext => x. transitivity (lim (fun n => \int[l x]_y (k_ n (x, y))%:E)); last first. rewrite is_cvg_elim_supE//. apply: ereal_nondecreasing_is_cvg => m n mn. apply: ge0_le_integral => //. - - by move=> y' _; rewrite lee_fin. + - by move=> y _; rewrite lee_fin. - exact/EFin_measurable_fun/measurable_fun_prod1. - - by move=> y' _; rewrite lee_fin. + - by move=> y _; rewrite lee_fin. - exact/EFin_measurable_fun/measurable_fun_prod1. - - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. + - by move=> y _; rewrite lee_fin; exact/lefP/ndk_. rewrite -monotone_convergence//. - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. - by move=> n; exact/EFin_measurable_fun/measurable_fun_prod1. - - by move=> n y' _; rewrite lee_fin. - - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. + - by move=> n y _; rewrite lee_fin. + - by move=> y _ m n mn; rewrite lee_fin; exact/lefP/ndk_. apply: measurable_fun_elim_sup => n. rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y (\sum_(r <- fset_set (range (k_ n))) r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. -rewrite [X in measurable_fun _ X](_ : _ = (fun x => \sum_(r <- fset_set (range (k_ n))) - (\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => + \sum_(r <- fset_set (range (k_ n))) + (\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first. apply/funext => x; rewrite -ge0_integral_sum//. - by apply: eq_integral => y _; rewrite sumEFin. - move=> r. apply/EFin_measurable_fun/measurable_funrM/measurable_fun_prod1 => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). + rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r))//. + exact/measurable_funP. - by move=> m y _; rewrite muleindic_ge0. apply emeasurable_fun_sum => r. rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * @@ -572,27 +613,53 @@ rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * apply/funext => x. under eq_integral do rewrite EFinM. rewrite (integralM_0ifneg _ _ (fun k y => (\1_(k_ n @^-1` [set r]) (x, y))%:E))//. - - by move=> _ t _; rewrite lee_fin. - - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. + - by move=> _ y _; rewrite lee_fin. + - by move=> r0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). -apply: measurable_funeM. -rewrite (_ : (fun x => _) = (fun x => l x (xsection (k_ n @^-1` [set r]) x))); last first. - apply/funext => y. - rewrite integral_indic//; last first. - rewrite (_ : (fun x => _) = xsection (k_ n @^-1` [set r]) y); last first. - apply/seteqP; split. - by move=> y2/=; rewrite /xsection/= inE//. - by rewrite /xsection/= => y2/=; rewrite inE. + rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r))//. + exact/measurable_funP. +apply/measurable_funeM. +rewrite (_ : (fun x => _) = (fun x => l x (xsection (k_ n @^-1` [set r]) x))). + exact/h. +apply/funext => x; rewrite integral_indic//; last first. + rewrite (_ : (fun x => _) = xsection (k_ n @^-1` [set r]) x). exact: measurable_xsection. - congr (l y _). - apply/funext => y1/=. - rewrite /xsection/= inE. - by apply/propext; tauto. + by rewrite /xsection; apply/seteqP; split=> y/= /[!inE]. +congr (l x _); apply/funext => y1/=; rewrite /xsection/= inE. +by apply/propext; tauto. +Qed. + +Lemma measurable_fun_integral_finite_kernel + (l : R.-fker X ~> Y) + (k : X * Y -> \bar R) (k0 : forall z, 0 <= k z) (mk : measurable_fun setT k) : + measurable_fun setT (fun x => \int[l x]_y k (x, y)). +Proof. +have [k_ [ndk_ k_k]] := approximation measurableT mk (fun x _ => k0 x). +apply: (measurable_fun_xsection_integral ndk_ (k_k ^~ Logic.I)) => n r. have [l_ hl_] := kernel_uub l. by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. Qed. +Lemma measurable_fun_integral_sfinite_kernel + (l : R.-sfker X ~> Y) + (k : X * Y -> \bar R) (k0 : forall t, 0 <= k t) (mk : measurable_fun setT k) : + measurable_fun setT (fun x => \int[l x]_y k (x, y)). +Proof. +have [k_ [ndk_ k_k]] := approximation measurableT mk (fun xy _ => k0 xy). +apply: (measurable_fun_xsection_integral ndk_ (k_k ^~ Logic.I)) => n r. +have [l_ hl_] := sfinite l. +rewrite (_ : (fun x => _) = + (fun x => mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x))); last first. + by apply/funext => x; rewrite hl_//; exact/measurable_xsection. +apply: ge0_emeasurable_fun_sum => // m. +by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. +Qed. + +End measurable_fun_integral_finite_sfinite. +Arguments measurable_fun_xsection_integral {_ _ _ _ _} l k. +Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} l k. +Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} l k. + Section kcomp_def. Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) (Z : measurableType d3) (R : realType). @@ -650,9 +717,7 @@ Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') Lemma measurable_fun_kcomp_finite U : measurable U -> measurable_fun setT ((l \; k) ^~ U). Proof. -move=> mU. -rewrite /kcomp. -apply: (@measurable_fun_integral_finite_kernel _ _ _ _ _ _ (k ^~ U)) => //=. +move=> mU; apply: (measurable_fun_integral_finite_kernel _ (k ^~ U)) => //=. exact/measurable_kernel. Qed. @@ -667,12 +732,11 @@ Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') Variable l : R.-fker X ~> Y. Variable k : R.-fker [the measurableType _ of (X * Y)%type] ~> Z. -Lemma mkcomp_finite : measure_uub (l \; k). +Lemma mkcomp_finite : measure_fam_uub (l \; k). Proof. -have /measure_uubP[r hr] := kernel_uub k. -have /measure_uubP[s hs] := kernel_uub l. -apply/measure_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x. -rewrite /=. +have /measure_fam_uubP[r hr] := kernel_uub k. +have /measure_fam_uubP[s hs] := kernel_uub l. +apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=. apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). apply: ge0_le_integral => //. - have /measurable_fun_prod1 := measurable_kernel k setT measurableT. @@ -688,82 +752,6 @@ HB.instance Definition _ := End kcomp_finite_kernel_finite. End KCOMP_FINITE_KERNEL. -(* pollard *) -Lemma measurable_fun_integral_sfinite_kernel - (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType) - (l : R.-sfker X ~> Y) - (k : (X * Y)%type -> \bar R) (k0 : (forall t, True -> 0 <= k t)) - (mk : measurable_fun setT k) : - measurable_fun [set: X] (fun x => \int[l x]_y k (x, y)). -Proof. -have [k_ [ndk_ k_k]] := approximation measurableT mk k0. -simpl in *. -rewrite (_ : (fun x => \int[l x]_y k (x, y)) = - (fun x => elim_sup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. - apply/funeqP => x. - transitivity (lim (fun n => \int[l x]_y (k_ n (x, y))%:E)); last first. - rewrite is_cvg_elim_supE//. - apply: ereal_nondecreasing_is_cvg => m n mn. - apply: ge0_le_integral => //. - - by move=> y' _; rewrite lee_fin. - - exact/EFin_measurable_fun/measurable_fun_prod1. - - by move=> y' _; rewrite lee_fin. - - exact/EFin_measurable_fun/measurable_fun_prod1. - - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. - rewrite -monotone_convergence//. - - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. - - by move=> n; exact/EFin_measurable_fun/measurable_fun_prod1. - - by move=> n y' _; rewrite lee_fin. - - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. -apply: measurable_fun_elim_sup => n. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l x0]_y - (\sum_(r <- fset_set (range (k_ n))) - r * \1_(k_ n @^-1` [set r]) (x0, y))%:E)); last first. - by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) - (\int[l x0]_y - (r * \1_(k_ n @^-1` [set r]) (x0, y))%:E))); last first. - apply/funext => x; rewrite -ge0_integral_sum//. - - by apply: eq_integral => y _; rewrite sumEFin. - - move=> r. - apply/EFin_measurable_fun/measurable_funrM/measurable_fun_prod1 => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). - - by move=> m y _; rewrite muleindic_ge0. -apply emeasurable_fun_sum => r. -rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * - \int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. - apply/funext => x. - under eq_integral do rewrite EFinM. - rewrite (integralM_0ifneg _ _ (fun k y => (\1_(k_ n @^-1` [set r]) (x, y))%:E))//. - - by move=> _ t _; rewrite lee_fin. - - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. - - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). -apply: measurable_funeM. -rewrite (_ : (fun x => \int[l x]_z (\1_(k_ n @^-1` [set r]) (x, z))%:E) = - (fun x => l x (xsection (k_ n @^-1` [set r]) x))); last first. - apply/funext => y. - rewrite integral_indic//; last first. - rewrite (_ : (fun x => (k_ n @^-1` [set r]) (y, x)) = xsection (k_ n @^-1` [set r]) y); last first. - apply/seteqP; split. - by move=> y2/=; rewrite /xsection/= inE//. - by rewrite /xsection/= => y2/=; rewrite inE /preimage/=. - exact: measurable_xsection. - congr (l y _). - apply/funext => y1/=. - rewrite /xsection/= inE. - by apply/propext; tauto. -have [l_ hl_] := sfinite l. -rewrite (_ : (fun x => _) = (fun x => mseries (l_ ^~ x) 0 (xsection (k_ n @^-1` [set r]) x))); last first. - apply/funext => x. - rewrite hl_//. - exact/measurable_xsection. -rewrite /mseries/=. -apply: ge0_emeasurable_fun_sum => // k1. -apply: measurable_fun_xsection_finite_kernel => //. -by rewrite inE. -Qed. - Section kcomp_sfinite_kernel. Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) (R : realType). @@ -824,10 +812,10 @@ rewrite (reindex_esum [set: nat] [set: nat * nat] f)//. by rewrite nneseries_esum// fun_true. Qed. -Lemma measurable_fun_mkcomp_sfinite U : measurable U -> measurable_fun setT ((l \; k) ^~ U). +Lemma measurable_fun_mkcomp_sfinite U : measurable U -> + measurable_fun setT ((l \; k) ^~ U). Proof. -move=> mU. -apply: (@measurable_fun_integral_sfinite_kernel _ _ _ _ _ _ (k ^~ U)) => //. +move=> mU; apply: (measurable_fun_integral_sfinite_kernel _ (k ^~ U)) => //. exact/measurable_kernel. Qed. @@ -851,62 +839,77 @@ End kcomp_sfinite_kernel. End KCOMP_SFINITE_KERNEL. HB.export KCOMP_SFINITE_KERNEL. -(* pollard *) -Lemma measurable_fun_integral_sfinite_kernel_prod +(* pollard? *) +Section measurable_fun_integral_kernel'. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType). +Variables (l : X -> {measure set Y -> \bar R}) + (k : Y -> \bar R) + (k_ : ({nnsfun Y >-> R}) ^nat) + (ndk_ : nondecreasing_seq (k_ : (Y -> R)^nat)) + (k_k : forall z, setT z -> EFin \o (k_ ^~ z) --> k z). + +Let p : (X * Y -> R)^nat := fun n xy => k_ n xy.2. + +Let p_ge0 n x : (0 <= p n x)%R. Proof. by []. Qed. + +HB.instance Definition _ n := @IsNonNegFun.Build _ R (p n) (p_ge0 n). + +Let mp n : measurable_fun setT (p n). +Proof. +rewrite /p => _ /= B mB; rewrite setTI. +have mk_n : measurable_fun setT (k_ n) by []. +rewrite (_ : _ @^-1` _ = setT `*` (k_ n @^-1` B)); last first. + by apply/seteqP; split => xy /=; tauto. +apply: measurableM => //. +have := mk_n measurableT _ mB. +by rewrite setTI. +Qed. + +HB.instance Definition _ n := @IsMeasurableFun.Build _ _ R (p n) (mp n). + +Let fp n : finite_set (range (p n)). +Proof. +have := @fimfunP _ _ (k_ n). +suff : range (k_ n) = range (p n) by move=> <-. +by apply/seteqP; split => r [y ?] <-; [exists (point, y)|exists y.2]. +Qed. + +HB.instance Definition _ n := @FiniteImage.Build _ _ (p n) (fp n). + +Lemma measurable_fun_preimage_integral : + (forall n r, measurable_fun setT (fun x => l x (k_ n @^-1` [set r]))) -> + measurable_fun setT (fun x => \int[l x]_z k z). +Proof. +move=> h. +apply: (measurable_fun_xsection_integral l (fun xy => k xy.2) + (fun n => [the {nnsfun _ >-> _} of p n])) => /=. +- by rewrite /p => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_. +- by move=> [x y]; exact: k_k. +- move=> n r _ /= B mB. + have := h n r measurableT B mB. + rewrite !setTI. + suff : ((fun x => l x (k_ n @^-1` [set r])) @^-1` B) = + ((fun x => l x (xsection (p n @^-1` [set r]) x)) @^-1` B) by move=> ->. + apply/seteqP; split => x/=. + suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. + by apply/seteqP; split; move=> y/=; + rewrite /xsection/= /p /preimage/= inE/=. + suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. + by apply/seteqP; split; move=> y/=; rewrite /xsection/= /p /preimage/= inE/=. +Qed. + +End measurable_fun_integral_kernel'. + +Lemma measurable_fun_integral_kernel (d d' d3 : _) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) (R : realType) - (l : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) c - (k : Z -> \bar R) (k0 : (forall z, True -> 0 <= k z)) (mk : measurable_fun setT k) : - measurable_fun [set: Y] (fun y => \int[l (c, y)]_z k z). + (l : R.-ker [the measurableType _ of (X * Y)%type] ~> Z) c + (k : Z -> \bar R) (k0 : forall z, True -> 0 <= k z) (mk : measurable_fun setT k) : + measurable_fun setT (fun y => \int[l (c, y)]_z k z). Proof. have [k_ [ndk_ k_k]] := approximation measurableT mk k0. -simpl in *. -rewrite (_ : (fun x0 => \int[l (c, x0)]_z k z) = - (fun x0 => elim_sup (fun n => \int[l (c, x0)]_z (k_ n z)%:E))); last first. - apply/funeqP => x. - transitivity (lim (fun n => \int[l (c, x)]_z (k_ n z)%:E)); last first. - rewrite is_cvg_elim_supE//. - apply: ereal_nondecreasing_is_cvg => m n mn. - apply: ge0_le_integral => //. - - by move=> y' _; rewrite lee_fin. - - exact/EFin_measurable_fun. - - by move=> y' _; rewrite lee_fin. - - exact/EFin_measurable_fun. - - by move=> y' _; rewrite lee_fin; apply/lefP/ndk_. - rewrite -monotone_convergence//. - - by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: k_k. - - by move=> n; exact/EFin_measurable_fun. - - by move=> n y' _; rewrite lee_fin. - - by move=> y' _ m n mn; rewrite lee_fin; apply/lefP/ndk_. -apply: measurable_fun_elim_sup => n. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \int[l (c, x0)]_z - ((\sum_(r <- fset_set (range (k_ n))) - r * \1_(k_ n @^-1` [set r]) z))%:E)); last first. - by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. -rewrite [X in measurable_fun _ X](_ : _ = (fun x0 => \sum_(r <- fset_set (range (k_ n))) - (\int[l (c, x0)]_z - (r * \1_(k_ n @^-1` [set r]) z)%:E))); last first. - apply/funext => x; rewrite -ge0_integral_sum//. - - by apply: eq_integral => y _; rewrite sumEFin. - - move=> r. - apply/EFin_measurable_fun/measurable_funrM => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). - - by move=> m y _; rewrite muleindic_ge0. -apply emeasurable_fun_sum => r. -rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * - \int[l (c ,x)]_z (\1_(k_ n @^-1` [set r]) z)%:E)); last first. - apply/funext => x. - under eq_integral do rewrite EFinM. - rewrite (integralM_0ifneg _ _ (fun k z => (\1_(k_ n @^-1` [set r]) z)%:E))//. - - by move=> _ t _; rewrite lee_fin. - - by move=> r_lt0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. - - apply/EFin_measurable_fun => /=. - by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r)). -apply: measurable_funeM. -rewrite (_ : (fun x : Y => \int[l (c, x)]_z (\1_(k_ n @^-1` [set r]) z)%:E) = - (fun x : Y => l (c, x) (k_ n @^-1` [set r]))); last first. - apply/funext => y. - by rewrite integral_indic// setIT. +apply: (measurable_fun_preimage_integral ndk_ k_k) => n r. have := measurable_kernel l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). by move=> /measurable_fun_prod1; exact. Qed. @@ -995,7 +998,7 @@ rewrite (_ : (fun _ => _) = (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E) transitivity (\int[l x]_y lim (fun n => \int[k (x, y)]_z (f_ n z)%:E)). rewrite -monotone_convergence//; last 3 first. move=> n. - apply: measurable_fun_integral_sfinite_kernel_prod => //. + apply: measurable_fun_integral_kernel => //. - by move=> z; rewrite lee_fin. - by apply/EFin_measurable_fun. - move=> n y _. @@ -1041,7 +1044,7 @@ HB.instance Definition _ := @isKernel.Build _ _ _ X R kernel_probability kernel_probabilityP. -Lemma kernel_probability_uub : measure_uub kernel_probability. +Lemma kernel_probability_uub : measure_fam_uub kernel_probability. Proof. (*NB: shouldn't this work? exists 2%:pos. *) exists 2%R => /= ?. @@ -1084,7 +1087,7 @@ Qed. HB.instance Definition _ := isKernel.Build _ _ _ _ R (kernel_mfun mf) measurable_kernel_mfun. -Lemma kernel_mfun_uub : measure_uub (kernel_mfun mf). +Lemma kernel_mfun_uub : measure_fam_uub (kernel_mfun mf). Proof. by exists 2%R => t/=; rewrite diracE in_setT lte_fin ltr_addr. Qed. HB.instance Definition _ := isFiniteKernel.Build _ _ _ _ R (kernel_mfun mf) @@ -1100,367 +1103,6 @@ HB.instance Definition _ := End kernel_of_mfun. -(* semantics for score *) -Lemma set_unit (A : set unit) : A = set0 \/ A = setT. -Proof. -have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. -by apply/seteqP; split => [|] []. -Qed. - -Section score_measure. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R) (mr : measurable_fun setT r). - -Definition mscore (t : T) (U : set unit) : \bar R := - if U == set0 then 0 else `| (r t)%:E |. - -Lemma mscore0 t : mscore t (set0 : set unit) = 0 :> \bar R. -Proof. by rewrite /mscore eqxx. Qed. - -Lemma mscore_ge0 t U : 0 <= mscore t U. -Proof. by rewrite /mscore; case: ifP. Qed. - -Lemma mscore_sigma_additive t : semi_sigma_additive (mscore t). -Proof. -move=> /= F mF tF mUF; rewrite /mscore; case: ifPn => [/eqP/bigcup0P F0|]. - rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. - apply/funext => k. - under eq_bigr do rewrite F0// eqxx. - by rewrite big1. -move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. -rewrite -(cvg_shiftn k.+1)/=. -rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. -apply/funext => n. -rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. -rewrite (negbTE Fk0) big1 ?adde0// => i/= ik; rewrite ifT//. -have [/eqP//|Fitt] := set_unit (F i). -move/trivIsetP : tF => /(_ i k Logic.I Logic.I ik). -by rewrite Fitt setTI => /eqP; rewrite (negbTE Fk0). -Qed. - -HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ - (mscore t) (mscore0 t) (mscore_ge0 t) (@mscore_sigma_additive t). - -End score_measure. - -Module KERNEL_SCORE. -Section kernel_score. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R). - -Definition k_' (mr : measurable_fun setT r) (i : nat) : T -> set unit -> \bar R := - fun t U => - if i%:R%:E <= mscore r t U < i.+1%:R%:E then - mscore r t U - else - 0. - -Variable (mr : measurable_fun setT r). - -Lemma k_'0 i (t : T) : k_' mr i t (set0 : set unit) = 0 :> \bar R. -Proof. by rewrite /k_' measure0; case: ifP. Qed. - -Lemma k_'ge0 i (t : T) B : 0 <= k_' mr i t B. -Proof. by rewrite /k_'; case: ifP. Qed. - -Lemma k_'sigma_additive i (t : T) : semi_sigma_additive (k_' mr i t). -Proof. -move=> /= F mF tF mUF. -rewrite /k_' /=. -have [F0|] := eqVneq (\bigcup_n F n) set0. - rewrite [in X in _ --> X]/mscore F0 eqxx. - rewrite (_ : (fun _ => _) = cst 0). - by case: ifPn => _; exact: cvg_cst. - apply/funext => k; rewrite big1// => n _. - move : F0 => /bigcup0P F0. - by rewrite /mscore F0// eqxx; case: ifP. -move=> UF0; move: (UF0). -move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. -rewrite [in X in _ --> X]/mscore (negbTE UF0). -rewrite -(cvg_shiftn k.+1)/=. -case: ifPn => ir. - rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. - apply/funext => n. - rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. - rewrite [in X in X + _]/mscore (negbTE Fk0) ir big1 ?adde0// => /= j jk. - rewrite /mscore. - have /eqP Fj0 : F j == set0. - have [/eqP//|Fjtt] := set_unit (F j). - move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). - by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). - rewrite Fj0 eqxx. - by case: ifP. -rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. -apply/funext => n. -rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. -rewrite [in X in if X then _ else _]/mscore (negbTE Fk0) (negbTE ir) add0e. -rewrite big1//= => j jk. -rewrite /mscore. -have /eqP Fj0 : F j == set0. - have [/eqP//|Fjtt] := set_unit (F j). - move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). - by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). -rewrite Fj0 eqxx. -by case: ifP. -Qed. - -HB.instance Definition _ (i : nat) (t : T) := isMeasure.Build _ _ _ - (k_' mr i t) (k_'0 i t) (k_'ge0 i t) (@k_'sigma_additive i t). - -Lemma emeasurable_itv (i : nat) : - measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set \bar R). -Proof. -rewrite -[X in measurable X]setCK. -apply: measurableC. -rewrite set_interval.setCitv /=. -apply: measurableU. -exact: emeasurable_itv_ninfty_bnd. -exact: emeasurable_itv_bnd_pinfty. -Qed. - -Lemma k_kernelP (i : nat) : forall U, measurable U -> measurable_fun setT (k_' mr i ^~ U). -Proof. -move=> /= U mU. -rewrite /k_' /=. -rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o (fun x => mscore r x U)) //. -apply: measurable_fun_comp; last first. - rewrite /mscore. - have [U0|U0] := eqVneq U set0. - exact: measurable_fun_cst. - apply: measurable_fun_comp => //. - by apply/EFin_measurable_fun. -rewrite /=. -pose A : _ -> \bar R := (fun x : \bar R => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). -rewrite (_ : (fun x => _) = A); last first. - apply/funext => x; rewrite /A; case: ifPn => ix. - by rewrite indicE/= mem_set ?mule1//. - rewrite indicE/= memNset ?mule0//. - rewrite /= in_itv/=. - exact/negP. -rewrite /A. -apply emeasurable_funM => /=. - exact: measurable_fun_id. -apply/EFin_measurable_fun. -have mi : measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set (\bar R)). - exact: emeasurable_itv. -by rewrite (_ : \1__ = mindic R mi)//. -Qed. - -Definition mk_' i (t : T) := [the measure _ _ of k_' mr i t]. - -HB.instance Definition _ (i : nat) := - isKernel.Build _ _ _ _ R (mk_' i) (k_kernelP i). - -Lemma k_uub (i : nat) : measure_uub (mk_' i). -Proof. -exists i.+1%:R => /= t. -rewrite /k_' /mscore setT_unit. -rewrite (_ : [set tt] == set0 = false); last first. - by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). -by case: ifPn => // /andP[]. -Qed. - -HB.instance Definition _ (i : nat) := - @isFiniteKernel.Build _ _ _ _ R (mk_' i) (k_uub i). - -End kernel_score. -End KERNEL_SCORE. - -Section kernel_score_kernel. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R). - -Definition kernel_score (mr : measurable_fun setT r) : T -> {measure set Datatypes_unit__canonical__measure_Measurable -> \bar R} := - fun t : T => [the measure _ _ of mscore r t]. - -Variable (mr : measurable_fun setT r). - -Lemma kernel_scoreP : forall U, measurable U -> - measurable_fun setT (kernel_score mr ^~ U). -Proof. -move=> /= U mU. -rewrite /mscore. -have [U0|U0] := eqVneq U set0. - exact: measurable_fun_cst. -apply: measurable_fun_comp => //. -by apply/EFin_measurable_fun. -Qed. - -HB.instance Definition _ := - isKernel.Build _ _ T - _ (*Datatypes_unit__canonical__measure_Measurable*) R - (kernel_score mr) (kernel_scoreP). -End kernel_score_kernel. - -Section kernel_score_sfinite_kernel. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R) (mr : measurable_fun setT r). - -Import KERNEL_SCORE. - -Lemma kernel_score_sfinite_kernelP : exists k_ : (R.-fker T ~> _)^nat, - forall x U, measurable U -> - kernel_score mr x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -rewrite /=. -exists (fun i => [the finite_kernel _ _ _ of mk_' mr i]) => /= r' U mU. -rewrite /mseries /mscore; case: ifPn => [/eqP U0|U0]. - by apply/esym/nneseries0 => i _; rewrite U0 measure0. -rewrite /mk_' /= /k_' /= /mscore (negbTE U0). -apply/esym/cvg_lim => //. -rewrite -(cvg_shiftn `|floor (fine `|(r r')%:E|)|%N.+1)/=. -rewrite (_ : (fun _ => _) = cst `|(r r')%:E|); first exact: cvg_cst. -apply/funext => n. -pose floor_r := widen_ord (leq_addl n `|floor `|(r r')| |.+1) (Ordinal (ltnSn `|floor `|(r r')| |)). -rewrite big_mkord (bigD1 floor_r)//= ifT; last first. - rewrite lee_fin lte_fin; apply/andP; split. - by rewrite natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?floor_le. - by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?lt_succ_floor. -rewrite big1 ?adde0//= => j jk. -rewrite ifF// lte_fin lee_fin. -move: jk; rewrite neq_ltn/= => /orP[|] jr. -- suff : (j.+1%:R <= `|(r r')|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. - rewrite (_ : j.+1%:R = j.+1%:~R)// floor_ge_int. - move: jr; rewrite -lez_nat => /le_trans; apply. - by rewrite -[leRHS](@ger0_norm _ (floor `|(r r')|)) ?floor_ge0. -- suff : (`|(r r')| < j%:R)%R by rewrite ltNge => /negbTE ->. - move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|(r r')|)) ?floor_ge0// ltr_int. - by rewrite -floor_lt_int. -Qed. - -HB.instance Definition _ := @isSFiniteKernel.Build _ _ _ _ _ - (kernel_score mr) (kernel_score_sfinite_kernelP). - -End kernel_score_sfinite_kernel. - -Section ite_true_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u1 : R.-ker T ~> T'). - -Definition ite_true : T * bool -> {measure set T' -> \bar R} := - fun b => if b.2 then u1 b.1 else [the measure _ _ of mzero]. - -Lemma measurable_ite_true U : measurable U -> measurable_fun setT (ite_true ^~ U). -Proof. -move=> /= mcU. -rewrite /ite_true. -rewrite (_ : (fun x : T * bool => _) = (fun x => if x.2 then u1 x.1 U else [the {measure set T' -> \bar R} of mzero] U)); last first. - apply/funext => -[t b]/=. - by case: ifPn. -apply: (@measurable_fun_if _ _ _ _ (u1 ^~ U) (fun=> mzero U)). - exact/measurable_kernel. -exact: measurable_fun_cst. -Qed. - -HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_true measurable_ite_true. -End ite_true_kernel. - -Section ite_true_finite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u1 : R.-fker T ~> T'). - -Lemma ite_true_uub : measure_uub (ite_true u1). -Proof. -have /measure_uubP[M hM] := kernel_uub u1. -exists M%:num => /= -[]; rewrite /ite_true => t [|]/=. - exact: hM. -by rewrite /= /mzero. -Qed. - -HB.instance Definition _ t := - isFiniteKernel.Build _ _ _ _ R (ite_true u1) ite_true_uub. -End ite_true_finite_kernel. - -Section ite_true_sfinite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u1 : R.-sfker T ~> T'). - -Lemma sfinite_ite_true : exists k_ : (R.-fker _ ~> _)^nat, - forall x U, measurable U -> - ite_true u1 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -have [k hk /=] := sfinite u1. -rewrite /ite_true. -exists (fun n => [the finite_kernel _ _ _ of ite_true (k n)]) => b U mU. -case: ifPn => hb. - rewrite /mseries hk//= /mseries. - apply: eq_nneseries => n _. - by rewrite /ite_true hb. -rewrite /= /mseries nneseries0// => n _. -by rewrite /ite_true (negbTE hb). -Qed. - -HB.instance Definition _ t := - @isSFiniteKernel.Build _ _ _ _ _ (ite_true u1) sfinite_ite_true. - -End ite_true_sfinite_kernel. - -Section ite_false_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u2 : R.-ker T ~> T'). - -Definition ite_false : T * bool -> {measure set T' -> \bar R} := - fun b => if ~~ b.2 then u2 b.1 else [the measure _ _ of mzero]. - -Lemma measurable_ite_false U : measurable U -> measurable_fun setT (ite_false ^~ U). -Proof. -move=> /= mcU. -rewrite /ite_false. -rewrite (_ : (fun x => _) = (fun x => if x.2 then [the {measure set T' -> \bar R} of mzero] U else u2 x.1 U)); last first. - apply/funext => -[t b]/=. - rewrite if_neg/=. - by case: b. -apply: (@measurable_fun_if _ _ _ _ (fun=> mzero U) (u2 ^~ U)). - exact: measurable_fun_cst. -exact/measurable_kernel. -Qed. - -HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_false measurable_ite_false. - -End ite_false_kernel. - -Section ite_false_finite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u2 : R.-fker T ~> T'). - -Lemma ite_false_uub : measure_uub (ite_false u2). -Proof. -have /measure_uubP[M hM] := kernel_uub u2. -exists M%:num => /= -[]; rewrite /ite_false/= => t b. -case: b => //=. -by rewrite /mzero. -Qed. - -HB.instance Definition _ := - isFiniteKernel.Build _ _ _ _ R (ite_false u2) ite_false_uub. - -End ite_false_finite_kernel. - -Section ite_false_sfinite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u2 : R.-sfker T ~> T'). - -Lemma sfinite_ite_false : exists k_ : (R.-fker _ ~> _)^nat, - forall x U, measurable U -> - ite_false u2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -have [k hk] := sfinite u2. -rewrite /= /ite_false. -exists (fun n => [the finite_kernel _ _ _ of ite_false (k n)]) => b U mU. -case: ifPn => hb. - rewrite /mseries hk//= /mseries/=. - apply: eq_nneseries => // n _. - by rewrite /ite_false hb. -rewrite /= /mseries nneseries0// => n _. -rewrite negbK in hb. -by rewrite /ite_false hb/=. -Qed. - -HB.instance Definition _ := - @isSFiniteKernel.Build _ _ _ _ _ (ite_false u2) sfinite_ite_false. - -End ite_false_sfinite_kernel. - Section add_of_kernels. Variables (d d' : measure_display) (R : realType). Variables (X : measurableType d) (Y : measurableType d'). @@ -1488,7 +1130,7 @@ Variables (d d' : measure_display) (R : realType). Variables (X : measurableType d) (Y : measurableType d'). Variables (u1 u2 : R.-fker X ~> Y). -Lemma add_of_finite_kernels_uub : measure_uub (add_of_kernels u1 u2). +Lemma add_of_finite_kernels_uub : measure_fam_uub (add_of_kernels u1 u2). Proof. have [k1 hk1] := kernel_uub u1. have [k2 hk2] := kernel_uub u2. @@ -1529,3 +1171,53 @@ Qed. HB.instance Definition _ t := isSFiniteKernel.Build _ _ _ _ R (add_of_kernels u1 u2) sfinite_add_of_kernels. End add_of_sfinite_kernels. + +Section normalize_measure. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variables (R : realType) (f : T -> {measure set T' -> \bar R}) (P : probability T' R). + +Definition normalize (t : T) (U : set T') := + let evidence := f t setT in + if (evidence == 0%E) || (evidence == +oo) then P U + else f t U * (fine evidence)^-1%:E. + +Lemma normalize0 t : normalize t set0 = 0. +Proof. +rewrite /normalize. +case: ifPn => // _. +by rewrite measure0 mul0e. +Qed. + +Lemma normalize_ge0 t U : 0 <= normalize t U. +Proof. +by rewrite /normalize; case: ifPn. +Qed. + +Lemma normalize_sigma_additive t : semi_sigma_additive (normalize t). +Proof. +move=> F mF tF mUF. +rewrite /normalize/=. +case: ifPn => [_|_]. + exact: measure_semi_sigma_additive. +rewrite (_ : (fun n => _) = ((fun n=> \sum_(0 <= i < n) f t (F i)) \* cst ((fine (f t [set: T']))^-1)%:E)); last first. + by apply/funext => n; rewrite -ge0_sume_distrl. +by apply: ereal_cvgMr => //; exact: measure_semi_sigma_additive. +Qed. + +HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ + (normalize t) (normalize0 t) (normalize_ge0 t) (@normalize_sigma_additive t). + +Lemma normalize1 t : normalize t setT = 1. +Proof. +rewrite /normalize; case: ifPn. + by rewrite probability_setT. +rewrite negb_or => /andP[ft0 ftoo]. +have ? : f t [set: T'] \is a fin_num. + by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. +rewrite -{1}(@fineK _ (f t setT))//. +rewrite -EFinM divrr// ?unitfE fine_eq0//. +Qed. + +HB.instance Definition _ t := isProbability.Build _ _ _ (normalize t) (normalize1 t). + +End normalize_measure. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index 375ca61078..bf33aa724e 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -4,6 +4,14 @@ Require Import mathcomp_extra boolp classical_sets signed functions cardinality. Require Import reals ereal topology normedtype sequences esum measure. Require Import lebesgue_measure fsbigop numfun lebesgue_integral kernel. +(******************************************************************************) +(* Semantics of a PPL using s-finite kernels *) +(* *) +(* bernoulli == *) +(* score == *) +(* ite_true/ite_false == *) +(******************************************************************************) + Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. @@ -14,6 +22,383 @@ Local Open Scope classical_set_scope. Local Open Scope ring_scope. Local Open Scope ereal_scope. +Definition onem (R : numDomainType) (p : R) := (1 - p)%R. + +Lemma onem1 (R : numDomainType) (p : R) : (p + onem p = 1)%R. +Proof. by rewrite /onem addrCA subrr addr0. Qed. + +Lemma onem_nonneg_proof (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) : + (0 <= onem p%:num)%R. +Proof. by rewrite /onem/= subr_ge0. Qed. + +Definition onem_nonneg (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) := + NngNum (onem_nonneg_proof p1). + +Section bernoulli. +Variables (R : realType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R). +Local Open Scope ring_scope. + +Definition bernoulli : set _ -> \bar R := + measure_add + [the measure _ _ of mscale p [the measure _ _ of dirac true]] + [the measure _ _ of mscale (onem_nonneg p1) [the measure _ _ of dirac false]]. + +HB.instance Definition _ := Measure.on bernoulli. + +Local Close Scope ring_scope. + +Lemma bernoulli_setT : bernoulli [set: _] = 1. +Proof. +rewrite /bernoulli/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. +by rewrite /mscale/= !diracE !in_setT !mule1 -EFinD onem1. +Qed. + +HB.instance Definition _ := @isProbability.Build _ _ R bernoulli bernoulli_setT. + +End bernoulli. + +Section score_measure. +Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R). + +Definition score (t : T) (U : set unit) : \bar R := + if U == set0 then 0 else `| (r t)%:E |. + +Let score0 t : score t (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /score eqxx. Qed. + +Let score_ge0 t U : 0 <= score t U. +Proof. by rewrite /score; case: ifP. Qed. + +Let score_sigma_additive t : semi_sigma_additive (score t). +Proof. +move=> /= F mF tF mUF; rewrite /score; case: ifPn => [/eqP/bigcup0P F0|]. + rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. + apply/funext => k. + under eq_bigr do rewrite F0// eqxx. + by rewrite big1. +move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. +rewrite -(cvg_shiftn k.+1)/=. +rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. +apply/funext => n. +rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. +rewrite (negbTE Fk0) big1 ?adde0// => i/= ik; rewrite ifT//. +have [/eqP//|Fitt] := set_unit (F i). +move/trivIsetP : tF => /(_ i k Logic.I Logic.I ik). +by rewrite Fitt setTI => /eqP; rewrite (negbTE Fk0). +Qed. + +HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ + (score t) (score0 t) (score_ge0 t) (@score_sigma_additive t). + +End score_measure. + +(* decomposition of score into finite kernels *) +Module SCORE. +Section score. +Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R). + +Definition k_ (mr : measurable_fun setT r) (i : nat) : T -> set unit -> \bar R := + fun t U => + if i%:R%:E <= score r t U < i.+1%:R%:E then + score r t U + else + 0. + +Hypothesis mr : measurable_fun setT r. + +Lemma k_0 i (t : T) : k_ mr i t (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /k_ measure0; case: ifP. Qed. + +Lemma k_ge0 i (t : T) B : 0 <= k_ mr i t B. +Proof. by rewrite /k_; case: ifP. Qed. + +Lemma k_sigma_additive i (t : T) : semi_sigma_additive (k_ mr i t). +Proof. +move=> /= F mF tF mUF. +rewrite /k_ /=. +have [F0|] := eqVneq (\bigcup_n F n) set0. + rewrite [in X in _ --> X]/score F0 eqxx. + rewrite (_ : (fun _ => _) = cst 0). + by case: ifPn => _; exact: cvg_cst. + apply/funext => k; rewrite big1// => n _. + move : F0 => /bigcup0P F0. + by rewrite /score F0// eqxx; case: ifP. +move=> UF0; move: (UF0). +move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. +rewrite [in X in _ --> X]/score (negbTE UF0). +rewrite -(cvg_shiftn k.+1)/=. +case: ifPn => ir. + rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. + apply/funext => n. + rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. + rewrite [in X in X + _]/score (negbTE Fk0) ir big1 ?adde0// => /= j jk. + rewrite /score. + have /eqP Fj0 : F j == set0. + have [/eqP//|Fjtt] := set_unit (F j). + move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). + by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). + rewrite Fj0 eqxx. + by case: ifP. +rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. +apply/funext => n. +rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. +rewrite [in X in if X then _ else _]/score (negbTE Fk0) (negbTE ir) add0e. +rewrite big1//= => j jk. +rewrite /score. +have /eqP Fj0 : F j == set0. + have [/eqP//|Fjtt] := set_unit (F j). + move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). + by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). +rewrite Fj0 eqxx. +by case: ifP. +Qed. + +HB.instance Definition _ (i : nat) (t : T) := isMeasure.Build _ _ _ + (k_ mr i t) (k_0 i t) (k_ge0 i t) (@k_sigma_additive i t). + +Lemma measurable_fun_k_ (i : nat) U : measurable U -> measurable_fun setT (k_ mr i ^~ U). +Proof. +move=> /= mU. +rewrite /k_ /=. +rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o (fun x => score r x U)) //. +apply: measurable_fun_comp; last first. + rewrite /score. + have [U0|U0] := eqVneq U set0. + exact: measurable_fun_cst. + apply: measurable_fun_comp => //. + by apply/EFin_measurable_fun. +rewrite /=. +pose A : _ -> \bar R := (fun x : \bar R => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). +rewrite (_ : (fun x => _) = A); last first. + apply/funext => x; rewrite /A; case: ifPn => ix. + by rewrite indicE/= mem_set ?mule1//. + rewrite indicE/= memNset ?mule0//. + rewrite /= in_itv/=. + exact/negP. +rewrite /A. +apply emeasurable_funM => /=. + exact: measurable_fun_id. +apply/EFin_measurable_fun. +have mi : measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set (\bar R)). + exact: emeasurable_itv. +by rewrite (_ : \1__ = mindic R mi)//. +Qed. + +Definition mk_ i (t : T) := [the measure _ _ of k_ mr i t]. + +HB.instance Definition _ (i : nat) := + isKernel.Build _ _ _ _ R (mk_ i) (measurable_fun_k_ i). + +Lemma mk_uub (i : nat) : measure_fam_uub (mk_ i). +Proof. +exists i.+1%:R => /= t. +rewrite /k_ /score setT_unit. +rewrite (_ : [set tt] == set0 = false); last first. + by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). +by case: ifPn => // /andP[]. +Qed. + +HB.instance Definition _ (i : nat) := + @isFiniteKernel.Build _ _ _ _ R (mk_ i) (mk_uub i). + +End score. +End SCORE. + +Section score_kernel. +Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R). + +Definition kernel_score (mr : measurable_fun setT r) + : T -> {measure set Datatypes_unit__canonical__measure_Measurable -> \bar R} := + fun t => [the measure _ _ of score r t]. + +Variable (mr : measurable_fun setT r). + +Let measurable_fun_score U : measurable U -> measurable_fun setT (kernel_score mr ^~ U). +Proof. +move=> /= mU; rewrite /score. +have [U0|U0] := eqVneq U set0; first exact: measurable_fun_cst. +by apply: measurable_fun_comp => //; exact/EFin_measurable_fun. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ T _ + (*Datatypes_unit__canonical__measure_Measurable*) R (kernel_score mr) measurable_fun_score. +End score_kernel. + +Section score_sfinite_kernel. +Variables (R : realType) (d : _) (T : measurableType d). +Variables (r : T -> R) (mr : measurable_fun setT r). + +Import SCORE. + +Let sfinite_score : exists k_ : (R.-fker T ~> _)^nat, + forall x U, measurable U -> + kernel_score mr x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +rewrite /=. +exists (fun i => [the finite_kernel _ _ _ of mk_ mr i]) => /= r' U mU. +rewrite /mseries /score; case: ifPn => [/eqP U0|U0]. + by apply/esym/nneseries0 => i _; rewrite U0 measure0. +rewrite /mk_ /= /k_ /= /score (negbTE U0). +apply/esym/cvg_lim => //. +rewrite -(cvg_shiftn `|floor (fine `|(r r')%:E|)|%N.+1)/=. +rewrite (_ : (fun _ => _) = cst `|(r r')%:E|); first exact: cvg_cst. +apply/funext => n. +pose floor_r := widen_ord (leq_addl n `|floor `|(r r')| |.+1) (Ordinal (ltnSn `|floor `|(r r')| |)). +rewrite big_mkord (bigD1 floor_r)//= ifT; last first. + rewrite lee_fin lte_fin; apply/andP; split. + by rewrite natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?floor_le. + by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?lt_succ_floor. +rewrite big1 ?adde0//= => j jk. +rewrite ifF// lte_fin lee_fin. +move: jk; rewrite neq_ltn/= => /orP[|] jr. +- suff : (j.+1%:R <= `|(r r')|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. + rewrite (_ : j.+1%:R = j.+1%:~R)// floor_ge_int. + move: jr; rewrite -lez_nat => /le_trans; apply. + by rewrite -[leRHS](@ger0_norm _ (floor `|(r r')|)) ?floor_ge0. +- suff : (`|(r r')| < j%:R)%R by rewrite ltNge => /negbTE ->. + move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|(r r')|)) ?floor_ge0// ltr_int. + by rewrite -floor_lt_int. +Qed. + +HB.instance Definition _ := @isSFiniteKernel.Build _ _ _ _ _ + (kernel_score mr) sfinite_score. + +End score_sfinite_kernel. + +(* decomposition of if-then-else *) +Module ITE. +Section ite_true_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u1 : R.-ker T ~> T'). + +Definition ite_true : T * bool -> {measure set T' -> \bar R} := + fun b => if b.2 then u1 b.1 else [the measure _ _ of mzero]. + +Lemma measurable_ite_true U : measurable U -> measurable_fun setT (ite_true ^~ U). +Proof. +move=> /= mcU. +rewrite /ite_true. +rewrite (_ : (fun x : T * bool => _) = (fun x => if x.2 then u1 x.1 U else [the {measure set T' -> \bar R} of mzero] U)); last first. + apply/funext => -[t b]/=. + by case: ifPn. +apply: (@measurable_fun_if _ _ _ _ (u1 ^~ U) (fun=> mzero U)). + exact/measurable_kernel. +exact: measurable_fun_cst. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_true measurable_ite_true. +End ite_true_kernel. + +Section ite_true_finite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u1 : R.-fker T ~> T'). + +Lemma ite_true_uub : measure_fam_uub (ite_true u1). +Proof. +have /measure_fam_uubP[M hM] := kernel_uub u1. +exists M%:num => /= -[]; rewrite /ite_true => t [|]/=. + exact: hM. +by rewrite /= /mzero. +Qed. + +HB.instance Definition _ t := + isFiniteKernel.Build _ _ _ _ R (ite_true u1) ite_true_uub. +End ite_true_finite_kernel. + +Section ite_true_sfinite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u1 : R.-sfker T ~> T'). + +Let sfinite_ite_true : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + ite_true u1 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +have [k hk /=] := sfinite u1. +rewrite /ite_true. +exists (fun n => [the _.-fker _ ~> _ of ite_true (k n)]) => b U mU. +case: ifPn => hb. + rewrite /mseries hk//= /mseries. + apply: eq_nneseries => n _. + by rewrite /ite_true hb. +rewrite /= /mseries nneseries0// => n _. +by rewrite /ite_true (negbTE hb). +Qed. + +HB.instance Definition _ t := + @isSFiniteKernel.Build _ _ _ _ _ (ite_true u1) sfinite_ite_true. + +End ite_true_sfinite_kernel. + +Section ite_false_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u2 : R.-ker T ~> T'). + +Definition ite_false : T * bool -> {measure set T' -> \bar R} := + fun b => if ~~ b.2 then u2 b.1 else [the measure _ _ of mzero]. + +Let measurable_ite_false U : measurable U -> measurable_fun setT (ite_false ^~ U). +Proof. +move=> /= mcU. +rewrite /ite_false. +rewrite (_ : (fun x => _) = (fun x => if x.2 then [the {measure set T' -> \bar R} of mzero] U else u2 x.1 U)); last first. + apply/funext => -[t b]/=. + rewrite if_neg/=. + by case: b. +apply: (@measurable_fun_if _ _ _ _ (fun=> mzero U) (u2 ^~ U)). + exact: measurable_fun_cst. +exact/measurable_kernel. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_false measurable_ite_false. + +End ite_false_kernel. + +Section ite_false_finite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u2 : R.-fker T ~> T'). + +Let ite_false_uub : measure_fam_uub (ite_false u2). +Proof. +have /measure_fam_uubP[M hM] := kernel_uub u2. +exists M%:num => /= -[]; rewrite /ite_false/= => t b. +case: b => //=. +by rewrite /mzero. +Qed. + +HB.instance Definition _ := + isFiniteKernel.Build _ _ _ _ R (ite_false u2) ite_false_uub. + +End ite_false_finite_kernel. + +Section ite_false_sfinite_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (u2 : R.-sfker T ~> T'). + +Let sfinite_ite_false : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + ite_false u2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +have [k hk] := sfinite u2. +rewrite /= /ite_false. +exists (fun n => [the finite_kernel _ _ _ of ite_false (k n)]) => b U mU. +case: ifPn => hb. + rewrite /mseries hk//= /mseries/=. + apply: eq_nneseries => // n _. + by rewrite /ite_false hb. +rewrite /= /mseries nneseries0// => n _. +rewrite negbK in hb. +by rewrite /ite_false hb/=. +Qed. + +HB.instance Definition _ := + @isSFiniteKernel.Build _ _ _ _ _ (ite_false u2) sfinite_ite_false. + +End ite_false_sfinite_kernel. +End ITE. + Section ite. Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). Variables (R : realType) (f : T -> bool) (u1 u2 : R.-sfker T ~> T'). @@ -40,7 +425,9 @@ Qed. HB.instance Definition _ tb := isMeasure.Build _ _ _ (ite mf tb) (ite0 tb) (ite_ge0 tb) (@ite_sigma_additive tb). -Definition ite' : R.-sfker +Import ITE. + +Let ite' : R.-sfker [the measurableType _ of (T * bool)%type] ~> T' := [the R.-sfker _ ~> _ of add_of_kernels [the R.-sfker _ ~> T' of ite_true u1] @@ -52,29 +439,17 @@ End ite. Section bernoulli27. Variable R : realType. - Local Open Scope ring_scope. -Notation "'2/7'" := (2%:R / 7%:R)%:nng. Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. Definition fiveseven : {nonneg R} := (5%:R / 7%:R)%:nng. -Definition bernoulli27 : set _ -> \bar R := - measure_add - [the measure _ _ of mscale twoseven [the measure _ _ of dirac true]] - [the measure _ _ of mscale fiveseven [the measure _ _ of dirac false]]. - -HB.instance Definition _ := Measure.on bernoulli27. - -Local Close Scope ring_scope. +Lemma onem_twoseven : onem (2 / 7) = fiveseven%:num. +Proof. by apply/eqP; rewrite subr_eq/= -mulrDl -natrD divrr// unitfE. Qed. -Lemma bernoulli27_setT : bernoulli27 [set: _] = 1. -Proof. -rewrite /bernoulli27/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. -rewrite /mscale/= !diracE !in_setT !mule1 -EFinD. -by rewrite -mulrDl -natrD divrr// unitfE pnatr_eq0. -Qed. +Lemma twoseven_proof : (twoseven%:num <= 1 :> R)%R. +Proof. by rewrite /= lter_pdivr_mulr// mul1r ler_nat. Qed. -HB.instance Definition _ := @isProbability.Build _ _ R bernoulli27 bernoulli27_setT. +Definition bernoulli27 : set _ -> \bar R := bernoulli twoseven_proof. End bernoulli27. @@ -86,7 +461,7 @@ Definition letin (d d' d3 : _) (l : R.-sfker X ~> Y) (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) : R.-sfker X ~> Z := - [the sfinite_kernel _ _ _ of (l \; k)]. + [the sfinite_kernel _ _ _ of l \; k]. Definition Return (d d' : _) (T : measurableType d) (T' : measurableType d') (f : T -> T') (mf : measurable_fun setT f) : R.-sfker T ~> T' := @@ -104,6 +479,27 @@ Definition Score (d : _) (T : measurableType d) (r : T -> R) (mr : measurable_fu R.-sfker T ~> Datatypes_unit__canonical__measure_Measurable := [the sfinite_kernel _ _ R of @kernel_score R _ _ r mr]. +Lemma ScoreE (d : _) (T : measurableType d) (t : T) (U : set bool) (n : nat) (b : bool) + (f : R -> R) (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) (mf : measurable_fun setT f) : + Score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) + (t, b, cst n%:R (t, b)) + ((fun y : unit => (snd \o fst) (t, b, y)) @^-1` U) = + (f n%:R)%:E * \d_b U. +Proof. +rewrite /Score/= /score/= diracE. +have [U0|U0] := set_unit ((fun=> b) @^-1` U). +- rewrite U0 eqxx memNset ?mule0//. + move=> Ub. + move: U0. + move/seteqP => [/(_ tt)] /=. + by move/(_ Ub). +- rewrite U0 setT_unit ifF//; last first. + by apply/negbTE/negP => /eqP/seteqP[/(_ tt erefl)]. + rewrite /= mem_set//; last first. + by move: U0 => /seteqP[_]/(_ tt)/=; exact. + by rewrite mule1 ger0_norm// f0. +Qed. + Definition Ite (d d' : _) (T : measurableType d) (T' : measurableType d') (f : T -> bool) (mf : measurable_fun setT f) (u1 u2 : R.-sfker T ~> T') @@ -119,10 +515,10 @@ rewrite /= /kcomp /ite. rewrite integral_dirac//=. rewrite indicT /cst. rewrite mul1e. -rewrite -/(measure_add (ite_true u1 (tb, f tb)) - (ite_false u2 (tb, f tb))). +rewrite -/(measure_add (ITE.ite_true u1 (tb, f tb)) + (ITE.ite_false u2 (tb, f tb))). rewrite measure_addE. -rewrite /ite_true /ite_false/=. +rewrite /ITE.ite_true /ITE.ite_false/=. case: (ifPn (f tb)) => /=. by rewrite /mzero adde0. by rewrite /mzero add0e. @@ -222,27 +618,6 @@ Definition k10 : measurable_fun _ _ := kn 10. End cst_fun. -Lemma ScoreE (R : realType) (d : _) (T : measurableType d) (t : T) (U : set bool) (n : nat) (b : bool) - (f : R -> R) (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) (mf : measurable_fun setT f) : - Score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) - (t, b, cst n%:R (t, b)) - ((fun y : unit => (snd \o fst) (t, b, y)) @^-1` U) = - (f n%:R)%:E * \d_b U. -Proof. -rewrite /Score/= /mscore/= diracE. -have [U0|U0] := set_unit ((fun=> b) @^-1` U). -- rewrite U0 eqxx memNset ?mule0//. - move=> Ub. - move: U0. - move/seteqP => [/(_ tt)] /=. - by move/(_ Ub). -- rewrite U0 setT_unit ifF//; last first. - by apply/negbTE/negP => /eqP/seteqP[/(_ tt erefl)]. - rewrite /= mem_set//; last first. - by move: U0 => /seteqP[_]/(_ tt)/=; exact. - by rewrite mule1 ger0_norm// f0. -Qed. - Lemma letin_sample_bernoulli27 (R : realType) (d d' : _) (T : measurableType d) (T' : measurableType d') (u : R.-sfker [the measurableType _ of (T * bool)%type] ~> T') x y : @@ -254,47 +629,43 @@ rewrite ge0_integral_measure_sum//. rewrite 2!big_ord_recl/= big_ord0 adde0/=. rewrite !ge0_integral_mscale//=. rewrite !integral_dirac//=. -by rewrite indicE in_setT mul1e indicE in_setT mul1e. +rewrite indicE in_setT mul1e indicE in_setT mul1e. +by rewrite onem_twoseven. Qed. -(* *) - -Section program1. +Section sample_and_return. Variables (R : realType) (d : _) (T : measurableType d). -Definition program1 : R.-sfker T ~> _ := +Definition sample_and_return : R.-sfker T ~> _ := letin (sample_bernoulli27 R T) (* T -> B *) (Return R (@measurable_fun_snd _ _ _ _)) (* T * B -> B *). -Lemma program1E (t : T) (U : _) : program1 t U = - ((twoseven R)%:num)%:E * \d_true U + - ((fiveseven R)%:num)%:E * \d_false U. -Proof. -rewrite /program1. -by rewrite letin_sample_bernoulli27. -Qed. +Lemma sample_and_returnE t U : sample_and_return t U = + (twoseven R)%:num%:E * \d_true U + + (fiveseven R)%:num%:E * \d_false U. +Proof. by rewrite letin_sample_bernoulli27. Qed. -End program1. +End sample_and_return. -Section program2. +Section sample_and_score. Variables (R : realType) (d : _) (T : measurableType d). -Definition program2 : R.-sfker T ~> _ := +Definition sample_and_score : R.-sfker T ~> _ := letin (sample_bernoulli27 R T) (* T -> B *) - (Score (measurable_fun_cst (1%:R : R))). + (Score (measurable_fun_cst (1%R : R))). -End program2. +End sample_and_score. -Section program3. +Section sample_and_branch. Variables (R : realType) (d : _) (T : measurableType d). (* let x = sample (bernoulli (2/7)) in let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in return r *) -Definition program3 : +Definition sample_and_branch : R.-sfker T ~> [the measurableType default_measure_display of Real_sort__canonical__measure_Measurable R] := letin (sample_bernoulli27 R T) (* T -> B *) @@ -302,19 +673,14 @@ Definition program3 : (Return R (@k3 _ _ [the measurableType _ of (T * bool)%type])) (Return R (@k10 _ _ [the measurableType _ of (T * bool)%type]))). -Lemma program3E (t : T) (U : _) : program3 t U = - ((twoseven R)%:num)%:E * \d_(3%:R : R) U + - ((fiveseven R)%:num)%:E * \d_(10%:R : R) U. -Proof. -rewrite /program3 letin_sample_bernoulli27. -congr (_ * _ + _ * _). -by rewrite IteE. -by rewrite IteE. -Qed. +Lemma sample_and_branchE t U : sample_and_branch t U = + (twoseven R)%:num%:E * \d_(3%R : R) U + + (fiveseven R)%:num%:E * \d_(10%R : R) U. +Proof. by rewrite /sample_and_branch letin_sample_bernoulli27 !IteE. Qed. -End program3. +End sample_and_branch. -Section program4. +Section staton_bus. Variables (R : realType) (d : _) (T : measurableType d). (* let x = sample (bernoulli (2/7)) in @@ -322,25 +688,47 @@ Variables (R : realType) (d : _) (T : measurableType d). let _ = score (1/4! r^4 e^-r) in return x *) -Definition program4 : R.-sfker T ~> Datatypes_bool__canonical__measure_Measurable := +Let mR := Real_sort__canonical__measure_Measurable R. +Let munit := Datatypes_unit__canonical__measure_Measurable. +Let mbool := Datatypes_bool__canonical__measure_Measurable. + +Notation var2_of2 := (@measurable_fun_snd _ _ _ _). +Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) (@measurable_fun_fst _ _ _ _)). +Notation var3_of3 := (@measurable_fun_snd _ _ _ _). + +Definition staton_bus' : R.-sfker T ~> mbool := letin - (sample_bernoulli27 R T) (* T -> B *) + (sample_bernoulli27 R T : _.-sfker T ~> mbool) (letin - (letin (* T * B -> unit *) - (Ite (@measurable_fun_snd _ _ _ _) - (Return R (@k3 _ _ [the measurableType _ of (T * bool)%type])) - (Return R (@k10 _ _ [the measurableType _ of (T * bool)%type]))) (* T * B -> R *) - (Score (measurable_fun_comp (@mpoisson R 4) (@measurable_fun_snd _ _ _ _))) (* B * R -> unit *)) - (Return R (measurable_fun_comp (@measurable_fun_snd _ _ _ _) (@measurable_fun_fst _ _ _ _)))). + (letin + (Ite var2_of2 + (Return R (@k3 _ _ _)) + (Return R (@k10 _ _ _)) + : _.-sfker [the measurableType _ of (T * bool)%type] ~> mR) + (Score (measurable_fun_comp (@mpoisson R 4) var3_of3) + : _.-sfker [the measurableType _ of (T * bool* mR)%type] ~> munit) + : _.-sfker [the measurableType _ of (T * bool)%type] ~> munit) + (Return R var2_of3 + : _.-sfker [the measurableType _ of (T * bool * munit)%type] ~> mbool) + : _.-sfker [the measurableType _ of (T * bool)%type] ~> mbool). + +Definition staton_bus : R.-sfker T ~> mbool := + letin (sample_bernoulli27 R T) + (letin + (letin (Ite var2_of2 + (Return R (@k3 _ _ _)) + (Return R (@k10 _ _ _))) + (Score (measurable_fun_comp (@mpoisson R 4) var3_of3))) + (Return R var2_of3)). (* true -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) (* false -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) -Lemma program4E (t : T) (U : _) : program4 t U = - ((twoseven R)%:num)%:E * (poisson 3%:R 4)%:E * \d_(true) U + - ((fiveseven R)%:num)%:E * (poisson 10%:R 4)%:E * \d_(false) U. +Lemma staton_busE t U : staton_bus t U = + (twoseven R)%:num%:E * (poisson 3%:R 4)%:E * \d_true U + + (fiveseven R)%:num%:E * (poisson 10%:R 4)%:E * \d_false U. Proof. -rewrite /program4. +rewrite /staton_bus. rewrite letin_sample_bernoulli27. rewrite -!muleA. congr (_ * _ + _ * _). @@ -354,4 +742,4 @@ rewrite letin_returnu//. by rewrite ScoreE// => r r0; exact: poisson_ge0. Qed. -End program4. +End staton_bus. From 984f9865f3ea7dcf9c37cd4d12f1e3387a3dc40a Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 19 Aug 2022 15:13:01 +0900 Subject: [PATCH 33/42] complete normalize, finite fubini, improve hier with pker --- theories/kernel.v | 583 ++++++++++++++++++++++++++++++++++++++----- theories/prob_lang.v | 134 +++++++++- 2 files changed, 645 insertions(+), 72 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 266960f527..3bb04fa77a 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -9,8 +9,9 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (* Kernels *) (* *) (* R.-ker X ~> Y == kernel *) -(* R.-fker X ~> Y == finite kernel *) (* R.-sfker X ~> Y == s-finite kernel *) +(* R.-fker X ~> Y == finite kernel *) +(* R.-pker X ~> Y == probability kernel *) (* sum_of_kernels == *) (* l \; k == composition of kernels *) (* kernel_mfun == kernel defined by a measurable function *) @@ -42,7 +43,9 @@ Variables (d : _) (T : measurableType d) (R : realType) (P : probability T R). Lemma probability_le1 (A : set T) : measurable A -> (P A <= 1)%E. Proof. -Admitted. +move=> mA; rewrite -(@probability_setT _ _ _ P). +by apply: le_measure => //; rewrite ?in_setE. +Qed. End probability_lemmas. (* /PR 516 in progress *) @@ -253,30 +256,96 @@ Lemma measurable_curry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) measurable (G x) <-> measurable (curry G x.1 x.2). Proof. by case: x. Qed. -Lemma measurable_fun_if0 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') +Lemma measurable_fun_if000 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') + D (md : measurable D) (f : T -> bool) (mf : measurable_fun setT f) : + measurable_fun (D `&` [set b | f b ]) x -> + measurable_fun (D `&` [set b | ~~ f b]) y -> + measurable_fun D (fun b : T => if f b then x b else y b). +Proof. +move=> mx my /= _ Y mY. +have H1 : measurable (D `&` [set b | f b]). + apply: measurableI => //. + rewrite [X in measurable X](_ : _ = f @^-1` [set true])//. + have := mf measurableT [set true]. + rewrite setTI. + exact. +have := mx H1 Y mY. +have H0 : [set t | ~~ f t] = [set t | f t = false]. + by apply/seteqP; split => [t/= /negbTE//|t/= ->]. +have H2 : measurable (D `&` [set b | ~~ f b]). + apply: measurableI => //. + have := mf measurableT [set false]. + rewrite setTI. + rewrite /preimage/=. + by rewrite H0; exact. +have := my H2 Y mY. +move=> yY xY. +rewrite (_ : _ @^-1` Y = ([set b | f b = true] `&` (x @^-1` Y) `&` (f @^-1` [set true])) `|` + ([set b | f b = false] `&` (y @^-1` Y) `&` (f @^-1` [set false]))); last first. + apply/seteqP; split. + move=> t/=; case: ifPn => ft. + by left. + by right. + by move=> t/= [|]; case: ifPn => ft; case=> -[]. +rewrite setIUr. +apply: measurableU. + rewrite -(setIid D). + rewrite -(setIA D). + rewrite setICA. + rewrite setIA. + apply: measurableI => //. + by rewrite setIA. + + rewrite -(setIid D). + rewrite -(setIA D). + rewrite setICA. + rewrite setIA. + rewrite /preimage/=. + rewrite -H0. + apply: measurableI => //. + by rewrite setIA. +Qed. + +Lemma measurable_fun_if00 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') (f : T -> bool) (mf : measurable_fun setT f) : - measurable_fun setT x -> - measurable_fun setT y -> + measurable_fun [set b | f b = true] x -> + measurable_fun [set b | f b = false] y -> measurable_fun setT (fun b : T => if f b then x b else y b). Proof. move=> mx my /= _ Y mY. rewrite setTI. -have := mx measurableT Y mY. -rewrite setTI => xY. -have := my measurableT Y mY. -rewrite setTI => yY. -rewrite (_ : _ @^-1` Y = ((x @^-1` Y) `&` (f @^-1` [set true])) `|` - ((y @^-1` Y) `&` (f @^-1` [set false]))); last first. +have H1 : measurable [set b | f b = true]. +rewrite [X in measurable X](_ : _ = f @^-1` [set true])//. + have := mf measurableT [set true]. + rewrite setTI. + exact. +have := mx H1 Y mY. +have H2 : measurable [set b | f b = false]. + have := mf measurableT [set false]. + rewrite setTI. + exact. +have := my H2 Y mY. +move=> yY xY. +rewrite (_ : _ @^-1` Y = ([set b | f b = true] `&` (x @^-1` Y) `&` (f @^-1` [set true])) `|` + ([set b | f b = false] `&` (y @^-1` Y) `&` (f @^-1` [set false]))); last first. apply/seteqP; split. move=> t/=; case: ifPn => ft. by left. by right. - by move=> t/=; case: ifPn => ft; case=> -[]. -apply: measurableU; apply: measurableI => //. - have := mf measurableT [set true]. - by rewrite setTI; exact. -have := mf measurableT [set false]. -by rewrite setTI; exact. + by move=> t/= [|]; case: ifPn => ft; case=> -[]. +by apply: measurableU; apply: measurableI => //. +Qed. + +Lemma measurable_fun_if0 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') + (f : T -> bool) (mf : measurable_fun setT f) : + measurable_fun setT x -> + measurable_fun setT y -> + measurable_fun setT (fun b : T => if f b then x b else y b). +Proof. +move=> mx my. +apply: measurable_fun_if000 => //. +by apply: measurable_funS mx. +by apply: measurable_funS my. Qed. Lemma measurable_fun_if (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') : @@ -317,6 +386,7 @@ Qed. Reserved Notation "R .-ker X ~> Y" (at level 42). Reserved Notation "R .-fker X ~> Y" (at level 42). Reserved Notation "R .-sfker X ~> Y" (at level 42). +Reserved Notation "R .-pker X ~> Y" (at level 42). HB.mixin Record isKernel d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := @@ -368,20 +438,6 @@ Proof. by move=> f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. Qed. -(* TODO: define using the probability type *) -HB.mixin Record isProbabilityKernel - d d' (X : measurableType d) (Y : measurableType d') - (R : realType) (k : X -> {measure set Y -> \bar R}) - of isKernel _ _ X Y R k := { - prob_kernelP : forall x, k x [set: Y] = 1 -}. - -#[short(type=probability_kernel)] -HB.structure Definition ProbabilityKernel - (d d' : _) (X : measurableType d) (Y : measurableType d') - (R : realType) := - {k of isProbabilityKernel _ _ X Y R k & isKernel _ _ X Y R k}. - Section measure_fam_uub. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variables (R : numFieldType) (k : X -> {measure set Y -> \bar R}). @@ -473,6 +529,70 @@ Qed. End finite_is_sfinite. +(* TODO: define using the probability type *) +HB.mixin Record isProbabilityFam + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) + := { + prob_kernelP : forall x, k x [set: Y] = 1 +}. + +#[short(type=probability_kernel)] +HB.structure Definition ProbabilityKernel + (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of isProbabilityFam _ _ X Y R k & isKernel _ _ X Y R k & isFiniteKernel _ _ X Y R k & isSFiniteKernel _ _ X Y R k}. +Notation "R .-pker X ~> Y" := (probability_kernel X Y R). + +HB.factory Record isProbabilityKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := { + prob_kernelP2 : forall x, k x [set: Y] = 1 +}. + +HB.builders Context d d' (X : measurableType d) (Y : measurableType d') + (R : realType) k of isProbabilityKernel d d' X Y R k. + +Lemma is_finite_kernel : measure_fam_uub k. +Proof. +exists 2%R => /= ?. +rewrite (@le_lt_trans _ _ 1%:E)//. +rewrite prob_kernelP2//. +by rewrite lte_fin ltr1n. +Qed. + +HB.instance Definition _ := @isFiniteKernel.Build _ _ _ _ _ _ is_finite_kernel. + +Lemma is_sfinite_kernel : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> + k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +exact: sfinite_finite. +Qed. + +HB.instance Definition _ := @isSFiniteKernel.Build _ _ _ _ _ _ is_sfinite_kernel. + +Lemma is_probability_kernel : forall x, k x setT = 1. +Proof. +exact/prob_kernelP2. +Qed. + +HB.instance Definition _ := @isProbabilityFam.Build _ _ _ _ _ _ is_probability_kernel. + +HB.end. + +(*Section tmp. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType) + (f : R.-fker T ~> T'). + +Let tmp : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + f x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. exact: sfinite_finite. Qed. + +HB.instance Definition _ := + @isSFiniteKernel.Build d d' T T' R f tmp. +End tmp.*) + (* see measurable_prod_subset in lebesgue_integral.v; the differences between the two are: - m2 is a kernel instead of a measure @@ -770,6 +890,7 @@ pose L := [the kernel _ _ _ of sum_of_kernels l_]. have H1 x U : measurable U -> (l \; k) x U = (L \; K) x U. move=> mU /=. rewrite /kcomp /L /K /=. + (* TODO: lemma so that we can get away with a rewrite *) transitivity (\int[ [the measure _ _ of mseries (l_ ^~ x) 0] ]_y k (x, y) U). by apply eq_measure_integral => A mA _; rewrite hl_. @@ -1023,47 +1144,158 @@ Qed. End integral_kcomp. -(* semantics for a sample operation *) -Section kernel_probability. -Variables (d : _) (R : realType) (X : measurableType d). -Variables (d' : _) (T' : measurableType d'). -Variable m : probability X R. +Definition finite_measure d (T : measurableType d) (R : realType) (mu : set T -> \bar R) := + mu setT < +oo. -Definition kernel_probability : T' -> {measure set X -> \bar R} := - fun _ : T' => m. +Lemma finite_kernel_finite_measure d (T : measurableType d) (R : realType) + (mu : R.-fker Datatypes_unit__canonical__measure_Measurable ~> T) : + finite_measure (mu tt). +Proof. +have [M muM] := kernel_uub mu. +by rewrite /finite_measure (lt_le_trans (muM tt))// leey. +Qed. -Lemma kernel_probabilityP : forall U, measurable U -> - measurable_fun setT (kernel_probability ^~ U). +Lemma finite_measure_sigma_finite d (T : measurableType d) (R : realType) + (mu : {measure set T -> \bar R}) : + finite_measure mu -> sigma_finite setT mu. Proof. -move=> U mU. -rewrite /kernel_probability. -exact: measurable_fun_cst. +rewrite /finite_measure => muoo. +exists (fun i => if i \in [set 0%N] then setT else set0). + by rewrite -bigcup_mkcondr setTI bigcup_const//; exists 0%N. +move=> n; split; first by case: ifPn. +by case: ifPn => // _; rewrite measure0. Qed. -HB.instance Definition _ := - @isKernel.Build _ _ _ X R kernel_probability - kernel_probabilityP. +Section finite_fubini. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). +Variables (mu : {measure set X -> \bar R}) (fmu : finite_measure mu). +Variables (la : {measure set Y -> \bar R}) (fla : finite_measure la). +Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). +Variables (mf : measurable_fun setT f). -Lemma kernel_probability_uub : measure_fam_uub kernel_probability. +Lemma finite_fubini : + \int[mu]_x \int[la]_y f (x, y) = \int[la]_y \int[mu]_x f (x, y). Proof. -(*NB: shouldn't this work? exists 2%:pos. *) -exists 2%R => /= ?. -rewrite (le_lt_trans (probability_le1 m measurableT))//. -by rewrite lte_fin ltr_addr. +rewrite -fubini_tonelli1//. + exact: finite_measure_sigma_finite. +move=> H. +rewrite fubini_tonelli2//. +exact: finite_measure_sigma_finite. +Qed. + +End finite_fubini. + +Section sfinite_fubini. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). +Variables (mu : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X). +Variables (la : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> Y). +Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). +Variable (mf : measurable_fun setT f). + +Lemma sfinite_fubini : + \int[mu tt]_x \int[la tt]_y f (x, y) = \int[la tt]_y \int[mu tt]_x f (x, y). +Proof. +have [mu_ mu_E] := sfinite mu. +have [la_ la_E] := sfinite la. +transitivity ( + \int[[the measure _ _ of mseries (fun i => mu_ i tt) 0]]_x + \int[la tt]_y f (x, y)). + apply: eq_measure_integral => U mU _. (* TODO: awkward *) + by rewrite mu_E. +transitivity ( + \int[[the measure _ _ of mseries (fun i => mu_ i tt) 0]]_x + \int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y f (x, y)). + apply eq_integral => x _. + apply: eq_measure_integral => U mU _. (* TODO: awkward *) + by rewrite la_E. +transitivity (\sum_(n t _; exact: integral_ge0 => x _. +(* have := @measurable_fun_integral_sfinite_kernel _ _ _ Y R la. + rewrite /=.*) + rewrite /=. + rewrite [X in measurable_fun _ X](_ : _ = + fun x => \sum_(n x. + rewrite ge0_integral_measure_series//. + exact/measurable_fun_prod1. + apply: ge0_emeasurable_fun_sum => //. + move=> k x. + by apply: integral_ge0. + move=> k. + apply: measurable_fun_fubini_tonelli_F => //=. + apply: finite_measure_sigma_finite. + exact: finite_kernel_finite_measure. + apply: eq_nneseries => n _; apply eq_integral => x _. + rewrite ge0_integral_measure_series//. + exact/measurable_fun_prod1. +transitivity (\sum_(n n _. + rewrite integral_sum(*TODO: ge0_integral_sum*)//. + move=> m. + apply: measurable_fun_fubini_tonelli_F => //=. + apply: finite_measure_sigma_finite. + exact: finite_kernel_finite_measure. + by move=> m x _; exact: integral_ge0. +transitivity (\sum_(n n _; apply eq_nneseries => m _. + rewrite finite_fubini//. + exact: finite_kernel_finite_measure. + exact: finite_kernel_finite_measure. +transitivity (\sum_(n la_ i tt) 0]]_y \int[mu_ n tt]_x f (x, y)). + apply eq_nneseries => n _. + rewrite /= ge0_integral_measure_series//. + by move=> y _; exact: integral_ge0. + apply: measurable_fun_fubini_tonelli_G => //=. + apply: finite_measure_sigma_finite. + exact: finite_kernel_finite_measure. +rewrite /=. +transitivity (\int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y \sum_(n n. + apply: measurable_fun_fubini_tonelli_G => //=. + apply: finite_measure_sigma_finite. + exact: finite_kernel_finite_measure. + by move=> n y _; exact: integral_ge0. +rewrite /=. +transitivity (\int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y \int[[the measure _ _ of mseries (fun i => mu_ i tt) 0]]_x f (x, y)). + apply eq_integral => y _. + rewrite ge0_integral_measure_series//. + exact/measurable_fun_prod2. +rewrite /=. +transitivity ( + \int[la tt]_y \int[mseries (fun i : nat => mu_ i tt) 0]_x f (x, y) +). + apply eq_measure_integral => A mA _ /=. + by rewrite la_E. +apply eq_integral => y _. +apply eq_measure_integral => A mA _ /=. +by rewrite mu_E. Qed. +End sfinite_fubini. + +(* semantics for a sample operation *) +Section kernel_probability. +Variables (d d' : _) (R : realType) (X : measurableType d) (T' : measurableType d'). +Variable m : probability X R. + +Definition kernel_probability : T' -> {measure set X -> \bar R} := + fun _ : T' => m. + +Lemma kernel_probabilityP U : measurable U -> + measurable_fun setT (kernel_probability ^~ U). +Proof. by move=> mU; exact: measurable_fun_cst. Qed. + HB.instance Definition _ := - @isFiniteKernel.Build _ _ _ X R kernel_probability - kernel_probability_uub. + @isKernel.Build _ _ _ X R kernel_probability kernel_probabilityP. -Lemma sfinite_kernel_probability : exists k_ : (R.-fker _ ~> _)^nat, - forall x U, measurable U -> - kernel_probability x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: sfinite_finite. Qed. +Lemma kernel_probability' x : kernel_probability x [set: X] = 1. +Proof. by rewrite /kernel_probability/= probability_setT. Qed. HB.instance Definition _ := - @isSFiniteKernel.Build _ _ _ X R kernel_probability - sfinite_kernel_probability. + @isProbabilityKernel.Build _ _ _ X R kernel_probability kernel_probability'. End kernel_probability. @@ -1178,25 +1410,29 @@ Variables (R : realType) (f : T -> {measure set T' -> \bar R}) (P : probability Definition normalize (t : T) (U : set T') := let evidence := f t setT in - if (evidence == 0%E) || (evidence == +oo) then P U + if evidence == 0%E then P U + else if evidence == +oo then P U else f t U * (fine evidence)^-1%:E. Lemma normalize0 t : normalize t set0 = 0. Proof. rewrite /normalize. case: ifPn => // _. +case: ifPn => // _. by rewrite measure0 mul0e. Qed. Lemma normalize_ge0 t U : 0 <= normalize t U. Proof. -by rewrite /normalize; case: ifPn. +by rewrite /normalize; case: ifPn => //; case: ifPn. Qed. Lemma normalize_sigma_additive t : semi_sigma_additive (normalize t). Proof. move=> F mF tF mUF. rewrite /normalize/=. +case: ifPn => [_|_]. + exact: measure_semi_sigma_additive. case: ifPn => [_|_]. exact: measure_semi_sigma_additive. rewrite (_ : (fun n => _) = ((fun n=> \sum_(0 <= i < n) f t (F i)) \* cst ((fine (f t [set: T']))^-1)%:E)); last first. @@ -1211,13 +1447,234 @@ Lemma normalize1 t : normalize t setT = 1. Proof. rewrite /normalize; case: ifPn. by rewrite probability_setT. -rewrite negb_or => /andP[ft0 ftoo]. +case: ifPn. + by rewrite probability_setT. +move=> ftoo ft0. have ? : f t [set: T'] \is a fin_num. by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. rewrite -{1}(@fineK _ (f t setT))//. -rewrite -EFinM divrr// ?unitfE fine_eq0//. +by rewrite -EFinM divrr// ?unitfE fine_eq0. Qed. HB.instance Definition _ t := isProbability.Build _ _ _ (normalize t) (normalize1 t). End normalize_measure. + +Section measurable_fun_comp. +Variables (d1 d2 d3 : measure_display). +Variables (T1 : measurableType d1). +Variables (T2 : measurableType d2). +Variables (T3 : measurableType d3). + +Lemma measurable_fun_comp_new F (f : T2 -> T3) E (g : T1 -> T2) : + measurable F -> + g @` E `<=` F -> + measurable_fun F f -> measurable_fun E g -> measurable_fun E (f \o g). +Proof. +move=> mF FgE mf mg /= mE A mA. +rewrite comp_preimage. +rewrite [X in measurable X](_ : _ = (E `&` g @^-1` (F `&` f @^-1` A))); last first. + apply/seteqP; split. + move=> x/= [Ex Afgx]; split => //; split => //. + by apply: FgE => //. + by move=> x/= [Ex] [Fgx Afgx]. +apply/mg => //. +by apply: mf => //. +Qed. + +End measurable_fun_comp. + +Lemma open_continuousP (S T : topologicalType) (f : S -> T) (D : set S) : + open D -> + {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). +Proof. +move=> oD; split=> [fcont|fcont s /[!inE] sD A]. + rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. + by rewrite interiorI; split => //; move: oD; rewrite openE; exact. +rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. +by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. +Qed. + +Lemma open_continuous_measurable_fun (R : realType) (f : R -> R) D : + open D -> {in D, continuous f} -> measurable_fun D f. +Proof. +move=> oD /(open_continuousP _ oD) cf. +apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. +by apply: open_measurable; exact/cf/interval_open. +Qed. + +Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. +Proof. +have [Bt|Bt] := boolP (true \in B). + have [Bf|Bf] := boolP (false \in B). + have -> : B = setT. + by apply/seteqP; split => // -[] _; [rewrite inE in Bt| rewrite inE in Bf]. + apply/or4P. + by rewrite eqxx/= !orbT. + have -> : B = [set true]. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bf. + by rewrite inE in Bt. + apply/or4P. + by rewrite eqxx/=. +have [Bf|Bf] := boolP (false \in B). + have -> : B = [set false]. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bt. + by rewrite inE in Bf. + apply/or4P. + by rewrite eqxx/= orbT. +have -> : B = set0. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bt. + by rewrite notin_set in Bf. +apply/or4P. +by rewrite eqxx/= !orbT. +Qed. + +Lemma measurable_eq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') + (R : realType) (f : R.-ker T ~> T') k : + measurable [set t | f t setT == k]. +Proof. +rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set k]); last first. + by apply/seteqP; split => t/= /eqP. +rewrite /=. +have := measurable_kernel f setT measurableT. +rewrite /=. +move/(_ measurableT [set k]). +rewrite setTI. +exact. +Qed. + +Lemma measurable_neq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') + (R : realType) (f : R.-ker T ~> T') k : measurable [set t | f t setT != k]. +Proof. +rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` (setT `\` [set k])); last first. + apply/seteqP; split => t/=. + by move/eqP; tauto. + by move=> []? /eqP; tauto. +rewrite /=. +have := measurable_kernel f setT measurableT. +rewrite /=. +move/(_ measurableT (setT `\` [set k])). +rewrite setTI. +apply => //. +exact: measurableD. +Qed. + +Lemma measurable_fun_eq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') + (R : realType) (f : R.-ker T ~> T') k : measurable_fun [set: T] (fun b : T => f b setT == k). +Proof. +move=> _ /= B mB. +rewrite setTI. +have [/eqP->|/eqP->|/eqP->|/eqP->] := set_boolE B. +- exact: measurable_eq_cst. +- rewrite (_ : _ @^-1` _ = [set b | f b setT != k]); last first. + apply/seteqP; split => t/=. + by move/negbT. + by move/negbTE. + exact: measurable_neq_cst. +- by rewrite preimage_set0. +- by rewrite preimage_setT. +Qed. + +(* TODO: PR *) +Lemma measurable_fun_fine (R : realType) : measurable_fun [set: \bar R] fine. +Proof. +move=> _ /= B mB. +rewrite setTI [X in measurable X](_ : _ @^-1` _ = + if 0%R \in B then (EFin @` B) `|` [set -oo; +oo] else EFin @` B); last first. + apply/seteqP; split=> [[r Br|B0|B0]|]. + case: ifPn => //= B0. + by left; exists r. + by exists r. + by rewrite mem_set//=; tauto. + by rewrite mem_set//=; tauto. + move=> [r| |]//=; case: ifPn => B0 /=. + case; last first. + by case. + by move=> [r' Br' [<-]]. + by move=> [r' Br' [<-]]. + by rewrite inE in B0. + by case => //. + case=> //. + by case=> //. + by rewrite inE in B0. + by case=> //. +case: ifPn => B0. + apply: measurableU. + by apply: measurable_EFin. + by apply: measurableU. +by apply: measurable_EFin. +Qed. + +Section normalize_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variables (R : realType) (f : R.-ker T ~> T'). + +Definition normalize_kernel (P : probability T' R) := + fun t => [the measure _ _ of normalize f P t]. + +Variable P : probability T' R. + +Lemma measurable_fun_normalize U : measurable U -> measurable_fun setT (normalize_kernel P ^~ U). +Proof. +move=> mU. +rewrite /normalize_kernel/= /normalize /=. +apply: measurable_fun_if000 => //. +- exact: measurable_fun_eq_cst. +- exact: measurable_fun_cst. +- apply: measurable_fun_if000 => //. + + rewrite setTI. + exact: measurable_neq_cst. + + exact: measurable_fun_eq_cst. + + exact: measurable_fun_cst. + + apply: emeasurable_funM. + have := (measurable_kernel f U mU). + by apply: measurable_funS => //. + apply/EFin_measurable_fun. + rewrite /=. + apply: (measurable_fun_comp_new (F := [set r : R | r != 0%R])) => //. + exact: open_measurable. + move=> /= r [t] [] [_ H1] H2 H3. + apply/eqP => H4; subst r. + move/eqP : H4. + rewrite fine_eq0 ?(negbTE H1)//. + rewrite ge0_fin_numE//. + by rewrite lt_neqAle leey H2. + apply: open_continuous_measurable_fun => //. + apply/in_setP => x /= x0. + by apply: inv_continuous. + apply: measurable_fun_comp => /=. + exact: measurable_fun_fine. + have := (measurable_kernel f setT measurableT). + by apply: measurable_funS => //. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ R (normalize_kernel P) + measurable_fun_normalize. + +End normalize_kernel. + +Section normalize_prob_kernel. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variables (R : realType) (f : R.-ker T ~> T') (P : probability T' R). + +Lemma normalize_prob_kernelP x : normalize_kernel f P x [set: T'] = 1. +Proof. +rewrite /normalize_kernel/= /normalize. +case: ifPn => [_|fx0]. + by rewrite probability_setT. +case: ifPn => [_|fxoo]. + by rewrite probability_setT. +have ? : f x [set: _] \is a fin_num. + by rewrite ge0_fin_numE// lt_neqAle fxoo/= leey. +rewrite -{1}(@fineK _ (f x setT))//=. +by rewrite -EFinM divrr// ?lte_fin ?ltr1n// ?unitfE fine_eq0. +Qed. + +HB.instance Definition _ := + @isProbabilityKernel.Build _ _ _ _ _ (normalize_kernel f P) + normalize_prob_kernelP. + +End normalize_prob_kernel. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index bf33aa724e..b23d89b09a 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -437,6 +437,19 @@ Definition mite := [the sfinite_kernel _ _ _ of kernel_mfun R mf] \; ite'. End ite. +Section normalize. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d') + (R : realType) (f : R.-sfker T ~> T') (Pdef : probability T' R). + +Definition Normalize := [the R.-pker T ~> T' of normalize_kernel f Pdef]. + +Lemma NormalizeE x U : Normalize x U = normalize_kernel f Pdef x U. +Proof. +by []. +Qed. + +End normalize. + Section bernoulli27. Variable R : realType. Local Open Scope ring_scope. @@ -463,6 +476,15 @@ Definition letin (d d' d3 : _) : R.-sfker X ~> Z := [the sfinite_kernel _ _ _ of l \; k]. +Lemma letinE (d d' d3 : _) + (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) + (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) + : forall x U, letin l k x U = \int[l x]_y k (x, y) U. +Proof. +by []. +Qed. + Definition Return (d d' : _) (T : measurableType d) (T' : measurableType d') (f : T -> T') (mf : measurable_fun setT f) : R.-sfker T ~> T' := [the sfinite_kernel _ _ _ of @kernel_mfun _ _ T T' R f mf]. @@ -597,6 +619,13 @@ move=> r0; rewrite /poisson mulr_ge0//. by rewrite ltW// expR_gt0. Qed. +Lemma poisson_gt0 (R : realType) (r : R) k : (0 < r)%R -> (0 < poisson r k.+1)%R. +Proof. +move=> r0; rewrite /poisson mulr_gt0//. + by rewrite mulr_gt0// exprn_gt0. +by rewrite expR_gt0. +Qed. + Lemma mpoisson (R : realType) k : measurable_fun setT (@poisson R ^~ k). Proof. apply: measurable_funM => /=. @@ -693,11 +722,14 @@ Let munit := Datatypes_unit__canonical__measure_Measurable. Let mbool := Datatypes_bool__canonical__measure_Measurable. Notation var2_of2 := (@measurable_fun_snd _ _ _ _). -Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) (@measurable_fun_fst _ _ _ _)). +Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) + (@measurable_fun_fst _ _ _ _)). Notation var3_of3 := (@measurable_fun_snd _ _ _ _). -Definition staton_bus' : R.-sfker T ~> mbool := - letin +Variable Pdef : probability mbool R. + +Definition staton_bus_measure' : R.-sfker T ~> mbool := + (letin (sample_bernoulli27 R T : _.-sfker T ~> mbool) (letin (letin @@ -710,25 +742,25 @@ Definition staton_bus' : R.-sfker T ~> mbool := : _.-sfker [the measurableType _ of (T * bool)%type] ~> munit) (Return R var2_of3 : _.-sfker [the measurableType _ of (T * bool * munit)%type] ~> mbool) - : _.-sfker [the measurableType _ of (T * bool)%type] ~> mbool). + : _.-sfker [the measurableType _ of (T * bool)%type] ~> mbool)). -Definition staton_bus : R.-sfker T ~> mbool := - letin (sample_bernoulli27 R T) +Definition staton_bus_measure : R.-sfker T ~> mbool := + (letin (sample_bernoulli27 R T) (letin (letin (Ite var2_of2 (Return R (@k3 _ _ _)) (Return R (@k10 _ _ _))) (Score (measurable_fun_comp (@mpoisson R 4) var3_of3))) - (Return R var2_of3)). + (Return R var2_of3))). (* true -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) (* false -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) -Lemma staton_busE t U : staton_bus t U = +Lemma staton_bus_measureE t U : staton_bus_measure t U = (twoseven R)%:num%:E * (poisson 3%:R 4)%:E * \d_true U + (fiveseven R)%:num%:E * (poisson 10%:R 4)%:E * \d_false U. Proof. -rewrite /staton_bus. +rewrite /staton_bus_measure. rewrite letin_sample_bernoulli27. rewrite -!muleA. congr (_ * _ + _ * _). @@ -742,4 +774,88 @@ rewrite letin_returnu//. by rewrite ScoreE// => r r0; exact: poisson_ge0. Qed. +Definition staton_bus : R.-pker T ~> mbool := + Normalize staton_bus_measure Pdef. + +Lemma staton_busE t U : + let N := (fine (((twoseven R)%:num)%:E * (poisson 3 4)%:E + ((fiveseven R)%:num)%:E * (poisson 10 4)%:E)) in + staton_bus t U = + ((twoseven R)%:num%:E * (poisson 3%:R 4)%:E * \d_true U + + (fiveseven R)%:num%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. +Proof. +rewrite /staton_bus. +rewrite NormalizeE /=. +rewrite /normalize. +rewrite !staton_bus_measureE. +rewrite diracE mem_set// mule1. +rewrite diracE mem_set// mule1. +rewrite ifF //. +apply/negbTE. +by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= poisson_gt0. +Qed. + End staton_bus. + +(* wip *) + +Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1). + +Section letinC_example. + +Variables (d d' d3 d4 : _) (R : realType) (X : measurableType d) + (Y : measurableType d') (Z : measurableType d3) (U : measurableType d4). +Let f (xyz : unit * X * X) := (xyz.1.2, xyz.2). +Lemma mf : measurable_fun setT f. +Proof. +rewrite /=. +apply/prod_measurable_funP => /=; split. + rewrite /f. + rewrite (_ : _ \o _ = (fun xyz : unit * X * X => xyz.1.2))//. + apply: measurable_fun_comp => /=. + exact: measurable_fun_snd. + exact: measurable_fun_fst. +rewrite (_ : _ \o _ = (fun xyz : unit * X * X => xyz.2))//. +apply: measurable_fun_comp => /=. + exact: measurable_fun_snd. +exact: measurable_fun_id. +Qed. + +Let measurable_fun_swap : measurable_fun [set: X * X] (swap (T2:=X)). +Proof. +apply/prod_measurable_funP => /=; split. + exact: measurable_fun_snd. +exact: measurable_fun_fst. +Qed. + +Let f' := @swap _ _ \o f. +Lemma mf' : measurable_fun setT f'. +Proof. +rewrite /=. +apply: measurable_fun_comp => /=. + exact: measurable_fun_swap. +exact: mf. +Qed. + +Variables (t : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X) + (t' : R.-sfker [the measurableType _ of (unit * X)%type] ~> X) + (u : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X) + (u' : R.-sfker [the measurableType _ of (unit * X)%type] ~> X) + (H1 : forall y, u tt = u' (tt, y)) + (H2 : forall y, t tt = t' (tt, y)). +Lemma letinC x A : measurable A -> + letin t (letin u' (Return R mf)) x A = letin u (letin t' (Return R mf')) x A. +Proof. +move=> mA. +rewrite /letin /= /kcomp /= /kcomp /=. +destruct x. +rewrite /f/=. +under eq_integral do rewrite -H1. +rewrite (@sfinite_fubini _ _ X X R t u (fun x => (\d_(x.1, x.2) A)))//=. +apply eq_integral => x _. + by rewrite -H2. +apply/EFin_measurable_fun => /=. +rewrite (_ : (fun x => _) = mindic R mA)//. +by apply/funext => -[a b] /=. +Qed. + +End letinC_example. From e98fd85a82f0cc3f01f157188d6ecdc135ea7b91 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 24 Aug 2022 22:09:28 +0900 Subject: [PATCH 34/42] more uniform naming, kdirac is pker, etc. --- theories/kernel.v | 582 +++++++++++++----------------- theories/prob_lang.v | 838 ++++++++++++++++++++++--------------------- 2 files changed, 679 insertions(+), 741 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 3bb04fa77a..6decb7a906 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -14,8 +14,8 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (* R.-pker X ~> Y == probability kernel *) (* sum_of_kernels == *) (* l \; k == composition of kernels *) -(* kernel_mfun == kernel defined by a measurable function *) -(* add_of_kernels == *) +(* kdirac mf == kernel defined by a measurable function *) +(* kadd k1 k2 == *) (******************************************************************************) Set Implicit Arguments. @@ -232,6 +232,24 @@ HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Po End discrete_measurable_bool. +(* NB: PR in progress *) +Lemma measurable_fun_fine (R : realType) (D : set (\bar R)) : measurable D -> + measurable_fun D fine. +Proof. +move=> mD _ /= B mB; rewrite [X in measurable X](_ : _ `&` _ = if 0%R \in B then + D `&` ((EFin @` B) `|` [set -oo; +oo]) else D `&` EFin @` B); last first. + apply/seteqP; split=> [[r [Dr Br]|[Doo B0]|[Doo B0]]|[r| |]]. + - by case: ifPn => _; split => //; left; exists r. + - by rewrite mem_set//; split => //; right; right. + - by rewrite mem_set//; split => //; right; left. + - by case: ifPn => [_ [Dr [[s + [sr]]|[]//]]|_ [Dr [s + [sr]]]]; rewrite sr. + - by case: ifPn => [/[!inE] B0 [Doo [[]//|]] [//|_]|B0 [Doo//] []]. + - by case: ifPn => [/[!inE] B0 [Doo [[]//|]] [//|_]|B0 [Doo//] []]. +case: ifPn => B0; apply/measurableI => //; last exact: measurable_EFin. +by apply: measurableU; [exact: measurable_EFin|exact: measurableU]. +Qed. + +(* TODO: PR *) Lemma measurable_fun_fst (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) : measurable_fun setT (@fst T1 T2). Proof. @@ -246,16 +264,83 @@ have := @measurable_fun_id _ [the measurableType _ of (T1 * T2)%type] setT. by move=> /prod_measurable_funP[]. Qed. -Lemma measurable_uncurry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) - (G : T1 -> T2 -> set T) (x : T1 * T2) : - measurable (G x.1 x.2) <-> measurable (uncurry G x). -Proof. by case: x. Qed. - Lemma measurable_curry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) (G : T1 * T2 -> set T) (x : T1 * T2) : measurable (G x) <-> measurable (curry G x.1 x.2). Proof. by case: x. Qed. +Section measurable_fun_comp. +Variables (d1 d2 d3 : measure_display). +Variables (T1 : measurableType d1). +Variables (T2 : measurableType d2). +Variables (T3 : measurableType d3). + +Lemma measurable_fun_comp_new F (f : T2 -> T3) E (g : T1 -> T2) : + measurable F -> + g @` E `<=` F -> + measurable_fun F f -> measurable_fun E g -> measurable_fun E (f \o g). +Proof. +move=> mF FgE mf mg /= mE A mA. +rewrite comp_preimage. +rewrite [X in measurable X](_ : _ = (E `&` g @^-1` (F `&` f @^-1` A))); last first. + apply/seteqP; split. + move=> x/= [Ex Afgx]; split => //; split => //. + by apply: FgE => //. + by move=> x/= [Ex] [Fgx Afgx]. +apply/mg => //. +by apply: mf => //. +Qed. + +End measurable_fun_comp. + +Lemma open_continuousP (S T : topologicalType) (f : S -> T) (D : set S) : + open D -> + {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). +Proof. +move=> oD; split=> [fcont|fcont s /[!inE] sD A]. + rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. + by rewrite interiorI; split => //; move: oD; rewrite openE; exact. +rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. +by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. +Qed. + +Lemma open_continuous_measurable_fun (R : realType) (f : R -> R) D : + open D -> {in D, continuous f} -> measurable_fun D f. +Proof. +move=> oD /(open_continuousP _ oD) cf. +apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. +by apply: open_measurable; exact/cf/interval_open. +Qed. + +Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. +Proof. +have [Bt|Bt] := boolP (true \in B). + have [Bf|Bf] := boolP (false \in B). + have -> : B = setT. + by apply/seteqP; split => // -[] _; [rewrite inE in Bt| rewrite inE in Bf]. + apply/or4P. + by rewrite eqxx/= !orbT. + have -> : B = [set true]. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bf. + by rewrite inE in Bt. + apply/or4P. + by rewrite eqxx/=. +have [Bf|Bf] := boolP (false \in B). + have -> : B = [set false]. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bt. + by rewrite inE in Bf. + apply/or4P. + by rewrite eqxx/= orbT. +have -> : B = set0. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bt. + by rewrite notin_set in Bf. +apply/or4P. +by rewrite eqxx/= !orbT. +Qed. + Lemma measurable_fun_if000 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') D (md : measurable D) (f : T -> bool) (mf : measurable_fun setT f) : measurable_fun (D `&` [set b | f b ]) x -> @@ -306,36 +391,6 @@ apply: measurableU. by rewrite setIA. Qed. -Lemma measurable_fun_if00 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') - (f : T -> bool) (mf : measurable_fun setT f) : - measurable_fun [set b | f b = true] x -> - measurable_fun [set b | f b = false] y -> - measurable_fun setT (fun b : T => if f b then x b else y b). -Proof. -move=> mx my /= _ Y mY. -rewrite setTI. -have H1 : measurable [set b | f b = true]. -rewrite [X in measurable X](_ : _ = f @^-1` [set true])//. - have := mf measurableT [set true]. - rewrite setTI. - exact. -have := mx H1 Y mY. -have H2 : measurable [set b | f b = false]. - have := mf measurableT [set false]. - rewrite setTI. - exact. -have := my H2 Y mY. -move=> yY xY. -rewrite (_ : _ @^-1` Y = ([set b | f b = true] `&` (x @^-1` Y) `&` (f @^-1` [set true])) `|` - ([set b | f b = false] `&` (y @^-1` Y) `&` (f @^-1` [set false]))); last first. - apply/seteqP; split. - move=> t/=; case: ifPn => ft. - by left. - by right. - by move=> t/= [|]; case: ifPn => ft; case=> -[]. -by apply: measurableU; apply: measurableI => //. -Qed. - Lemma measurable_fun_if0 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') (f : T -> bool) (mf : measurable_fun setT f) : measurable_fun setT x -> @@ -454,19 +509,19 @@ Qed. End measure_fam_uub. -HB.mixin Record isFiniteKernel +HB.mixin Record isFiniteFam d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := - { kernel_uub : measure_fam_uub k }. + { measure_uub : measure_fam_uub k }. #[short(type=finite_kernel)] HB.structure Definition FiniteKernel d d' (X : measurableType d) (Y : measurableType d') (R : realType) := - {k of isFiniteKernel _ _ X Y R k & isKernel _ _ X Y R k}. + {k of isFiniteFam _ _ X Y R k & isKernel _ _ X Y R k}. Notation "R .-fker X ~> Y" := (finite_kernel X Y R). -Arguments kernel_uub {_ _ _ _ _} _. +Arguments measure_uub {_ _ _ _ _} _. Section kernel_from_mzero. Variables (d : _) (T : measurableType d) (R : realType). @@ -483,16 +538,16 @@ HB.instance Definition _ := @isKernel.Build _ _ T' T R kernel_from_mzero kernel_from_mzeroP. -Lemma kernel_from_mzero_uub : measure_fam_uub kernel_from_mzero. +Let kernel_from_mzero_uub : measure_fam_uub kernel_from_mzero. Proof. by exists 1%R => /= t; rewrite /mzero/=. Qed. HB.instance Definition _ := - @isFiniteKernel.Build _ _ _ T R kernel_from_mzero + @isFiniteFam.Build _ _ _ T R kernel_from_mzero kernel_from_mzero_uub. End kernel_from_mzero. -HB.mixin Record isSFiniteKernel +HB.mixin Record isSFinite d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { sfinite : exists s : (R.-fker X ~> Y)^nat, @@ -503,8 +558,7 @@ HB.mixin Record isSFiniteKernel HB.structure Definition SFiniteKernel d d' (X : measurableType d) (Y : measurableType d') (R : realType) := - {k of isSFiniteKernel _ _ X Y R k & - isKernel _ _ X Y _ k}. + {k of isSFinite _ _ X Y R k & isKernel _ _ X Y _ k}. Notation "R .-sfker X ~> Y" := (sfinite_kernel X Y R). Arguments sfinite {_ _ _ _ _} _. @@ -529,52 +583,43 @@ Qed. End finite_is_sfinite. -(* TODO: define using the probability type *) HB.mixin Record isProbabilityFam d d' (X : measurableType d) (Y : measurableType d') - (R : realType) (k : X -> {measure set Y -> \bar R}) - := { - prob_kernelP : forall x, k x [set: Y] = 1 -}. + (R : realType) (k : X -> {measure set Y -> \bar R}) := + { prob_kernel : forall x, k x [set: Y] = 1}. #[short(type=probability_kernel)] HB.structure Definition ProbabilityKernel (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType) := - {k of isProbabilityFam _ _ X Y R k & isKernel _ _ X Y R k & isFiniteKernel _ _ X Y R k & isSFiniteKernel _ _ X Y R k}. + {k of isProbabilityFam _ _ X Y R k & isKernel _ _ X Y R k & + isFiniteFam _ _ X Y R k & isSFinite _ _ X Y R k}. Notation "R .-pker X ~> Y" := (probability_kernel X Y R). HB.factory Record isProbabilityKernel d d' (X : measurableType d) (Y : measurableType d') - (R : realType) (k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := { - prob_kernelP2 : forall x, k x [set: Y] = 1 -}. + (R : realType) (k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := + { prob_kernel' : forall x, k x setT = 1 }. HB.builders Context d d' (X : measurableType d) (Y : measurableType d') (R : realType) k of isProbabilityKernel d d' X Y R k. -Lemma is_finite_kernel : measure_fam_uub k. +Let is_finite_kernel : measure_fam_uub k. Proof. exists 2%R => /= ?. -rewrite (@le_lt_trans _ _ 1%:E)//. -rewrite prob_kernelP2//. -by rewrite lte_fin ltr1n. +by rewrite (@le_lt_trans _ _ 1%:E) ?lte_fin ?ltr1n// prob_kernel'. Qed. -HB.instance Definition _ := @isFiniteKernel.Build _ _ _ _ _ _ is_finite_kernel. +HB.instance Definition _ := @isFiniteFam.Build _ _ _ _ _ _ is_finite_kernel. -Lemma is_sfinite_kernel : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> - k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -exact: sfinite_finite. -Qed. +Lemma is_sfinite_kernel : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> + k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. exact: sfinite_finite. Qed. -HB.instance Definition _ := @isSFiniteKernel.Build _ _ _ _ _ _ is_sfinite_kernel. +HB.instance Definition _ := @isSFinite.Build _ _ _ _ _ _ is_sfinite_kernel. Lemma is_probability_kernel : forall x, k x setT = 1. -Proof. -exact/prob_kernelP2. -Qed. + exact/prob_kernel'. Qed. HB.instance Definition _ := @isProbabilityFam.Build _ _ _ _ _ _ is_probability_kernel. @@ -671,7 +716,7 @@ rewrite -(_ : (fun x => mrestr (m2 x) measurableT (xsection X x)) = by apply/funext => x//=; rewrite /mrestr setIT. apply measurable_prod_subset_xsection_kernel => //. move=> x. -have [r hr] := kernel_uub m2. +have [r hr] := measure_uub m2. exists r => Y mY. apply: (le_lt_trans _ (hr x)) => //. rewrite /mrestr. @@ -756,7 +801,7 @@ Lemma measurable_fun_integral_finite_kernel Proof. have [k_ [ndk_ k_k]] := approximation measurableT mk (fun x _ => k0 x). apply: (measurable_fun_xsection_integral ndk_ (k_k ^~ Logic.I)) => n r. -have [l_ hl_] := kernel_uub l. +have [l_ hl_] := measure_uub l. by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. Qed. @@ -852,10 +897,10 @@ Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') Variable l : R.-fker X ~> Y. Variable k : R.-fker [the measurableType _ of (X * Y)%type] ~> Z. -Lemma mkcomp_finite : measure_fam_uub (l \; k). +Let mkcomp_finite : measure_fam_uub (l \; k). Proof. -have /measure_fam_uubP[r hr] := kernel_uub k. -have /measure_fam_uubP[s hs] := kernel_uub l. +have /measure_fam_uubP[r hr] := measure_uub k. +have /measure_fam_uubP[s hs] := measure_uub l. apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=. apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). apply: ge0_le_integral => //. @@ -867,7 +912,7 @@ by rewrite integral_cst//= EFinM lte_pmul2l. Qed. HB.instance Definition _ := - isFiniteKernel.Build _ _ X Z R (l \; k) mkcomp_finite. + isFiniteFam.Build _ _ X Z R (l \; k) mkcomp_finite. End kcomp_finite_kernel_finite. End KCOMP_FINITE_KERNEL. @@ -923,7 +968,7 @@ suff: exists k_0 : (R.-fker X ~> Z) ^nat, forall x U, rewrite /= H1// H2 H3// H4// H5// /mseries -hkl_/=. rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split. rewrite -(@esum_esum _ _ _ _ _ (fun i j => (l_ j \; k_ i) x U))//. - rewrite nneseries_esum; last by move=> n _; exact: nneseries_lim_ge0. + rewrite nneseries_esum; last by move=> n _; exact: nneseries_ge0. by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true. rewrite /=. have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card. @@ -954,7 +999,7 @@ HB.instance Definition _ := #[export] HB.instance Definition _ := - isSFiniteKernel.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). + isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). End kcomp_sfinite_kernel. End KCOMP_SFINITE_KERNEL. @@ -1151,7 +1196,7 @@ Lemma finite_kernel_finite_measure d (T : measurableType d) (R : realType) (mu : R.-fker Datatypes_unit__canonical__measure_Measurable ~> T) : finite_measure (mu tt). Proof. -have [M muM] := kernel_uub mu. +have [M muM] := measure_uub mu. by rewrite /finite_measure (lt_le_trans (muM tt))// leey. Qed. @@ -1276,261 +1321,113 @@ Qed. End sfinite_fubini. -(* semantics for a sample operation *) -Section kernel_probability. -Variables (d d' : _) (R : realType) (X : measurableType d) (T' : measurableType d'). -Variable m : probability X R. +Section kprobability. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (m : probability Y R). -Definition kernel_probability : T' -> {measure set X -> \bar R} := - fun _ : T' => m. +Definition kprobability : X -> {measure set Y -> \bar R} := fun _ : X => m. -Lemma kernel_probabilityP U : measurable U -> - measurable_fun setT (kernel_probability ^~ U). +Let measurable_fun_kprobability U : measurable U -> + measurable_fun setT (kprobability ^~ U). Proof. by move=> mU; exact: measurable_fun_cst. Qed. HB.instance Definition _ := - @isKernel.Build _ _ _ X R kernel_probability kernel_probabilityP. + @isKernel.Build _ _ X Y R kprobability measurable_fun_kprobability. -Lemma kernel_probability' x : kernel_probability x [set: X] = 1. -Proof. by rewrite /kernel_probability/= probability_setT. Qed. +Let kprobability_prob x : kprobability x setT = 1. +Proof. by rewrite /kprobability/= probability_setT. Qed. HB.instance Definition _ := - @isProbabilityKernel.Build _ _ _ X R kernel_probability kernel_probability'. + @isProbabilityKernel.Build _ _ X Y R kprobability kprobability_prob. -End kernel_probability. +End kprobability. -Section kernel_of_mfun. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (f : T -> T'). +Section kdirac. +Variables (d d' : _) (T : measurableType d) (Y : measurableType d'). +Variables (R : realType) (f : T -> Y). -Definition kernel_mfun (mf : measurable_fun setT f) : T -> {measure set T' -> \bar R} := +Definition kdirac (mf : measurable_fun setT f) : T -> {measure set Y -> \bar R} := fun t => [the measure _ _ of dirac (f t)]. Hypothesis mf : measurable_fun setT f. -Lemma measurable_kernel_mfun U : measurable U -> measurable_fun setT (kernel_mfun mf ^~ U). +Let measurable_fun_kdirac U : measurable U -> measurable_fun setT (kdirac mf ^~ U). Proof. -move=> mU. -apply/EFin_measurable_fun. +move=> mU; apply/EFin_measurable_fun. rewrite (_ : (fun x => _) = mindic R mU \o f)//. exact/measurable_fun_comp. Qed. -HB.instance Definition _ := isKernel.Build _ _ _ _ R (kernel_mfun mf) - measurable_kernel_mfun. +HB.instance Definition _ := isKernel.Build _ _ _ _ R (kdirac mf) + measurable_fun_kdirac. -Lemma kernel_mfun_uub : measure_fam_uub (kernel_mfun mf). -Proof. by exists 2%R => t/=; rewrite diracE in_setT lte_fin ltr_addr. Qed. - -HB.instance Definition _ := isFiniteKernel.Build _ _ _ _ R (kernel_mfun mf) - kernel_mfun_uub. - -Lemma sfinite_kernel_mfun : exists k_ : (R.-fker _ ~> _)^nat, - forall x U, measurable U -> - kernel_mfun mf x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: sfinite_finite. Qed. +Let kdirac_prob x : kdirac mf x setT = 1. +Proof. by rewrite /kdirac/= diracE in_setT. Qed. HB.instance Definition _ := - @isSFiniteKernel.Build _ _ _ _ _ (kernel_mfun mf) sfinite_kernel_mfun. + @isProbabilityKernel.Build _ _ _ _ _ (kdirac mf) kdirac_prob. -End kernel_of_mfun. +End kdirac. +Arguments kdirac {d d' T Y R f}. -Section add_of_kernels. -Variables (d d' : measure_display) (R : realType). -Variables (X : measurableType d) (Y : measurableType d'). -Variables (u1 u2 : R.-ker X ~> Y). +Section kadd. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k1 k2 : R.-ker X ~> Y). -Definition add_of_kernels : X -> {measure set Y -> \bar R} := - fun t => [the measure _ _ of measure_add (u1 t) (u2 t)]. +Definition kadd : X -> {measure set Y -> \bar R} := + fun t => [the measure _ _ of measure_add (k1 t) (k2 t)]. -Lemma measurable_add_of_kernels U : measurable U -> measurable_fun setT (add_of_kernels ^~ U). +Let measurable_fun_kadd U : measurable U -> measurable_fun setT (kadd ^~ U). Proof. -move=> mU. -rewrite /add_of_kernels. -rewrite (_ : (fun x : X => _) = (fun x => (u1 x U) + (u2 x U))); last first. - apply/funext => x. - by rewrite -measure_addE. +move=> mU; rewrite /kadd. +rewrite (_ : (fun _ => _) = (fun x => k1 x U + k2 x U)); last first. + by apply/funext => x; rewrite -measure_addE. by apply: emeasurable_funD; exact/measurable_kernel. Qed. HB.instance Definition _ := - @isKernel.Build _ _ _ _ _ add_of_kernels measurable_add_of_kernels. -End add_of_kernels. + @isKernel.Build _ _ _ _ _ kadd measurable_fun_kadd. +End kadd. -Section add_of_finite_kernels. -Variables (d d' : measure_display) (R : realType). -Variables (X : measurableType d) (Y : measurableType d'). -Variables (u1 u2 : R.-fker X ~> Y). +Section fkadd. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k1 k2 : R.-fker X ~> Y). -Lemma add_of_finite_kernels_uub : measure_fam_uub (add_of_kernels u1 u2). +Let kadd_finite_uub : measure_fam_uub (kadd k1 k2). Proof. -have [k1 hk1] := kernel_uub u1. -have [k2 hk2] := kernel_uub u2. -exists (k1 + k2)%R => x. -rewrite /add_of_kernels/=. -rewrite -/(measure_add (u1 x) (u2 x)). -rewrite measure_addE. -rewrite EFinD. -exact: lte_add. +have [f1 hk1] := measure_uub k1; have [f2 hk2] := measure_uub k2. +exists (f1 + f2)%R => x; rewrite /kadd /=. +rewrite -/(measure_add (k1 x) (k2 x)). +by rewrite measure_addE EFinD; exact: lte_add. Qed. HB.instance Definition _ t := - isFiniteKernel.Build _ _ _ _ R (add_of_kernels u1 u2) add_of_finite_kernels_uub. -End add_of_finite_kernels. + isFiniteFam.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. +End fkadd. -Section add_of_sfinite_kernels. +Section sfkadd. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (u1 u2 : R.-sfker X ~> Y). +Variables (R : realType) (k1 k2 : R.-sfker X ~> Y). -Lemma sfinite_add_of_kernels : exists k_ : (R.-fker _ ~> _)^nat, +Let sfinite_kadd : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> - add_of_kernels u1 u2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. + kadd k1 k2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -have [k1 hk1] := sfinite u1. -have [k2 hk2] := sfinite u2. -exists (fun n => [the finite_kernel _ _ _ of add_of_kernels (k1 n) (k2 n)]) => x U mU. -rewrite /add_of_kernels/=. -rewrite -/(measure_add (u1 x) (u2 x)). -rewrite measure_addE. +have [f1 hk1] := sfinite k1. +have [f2 hk2] := sfinite k2. +exists (fun n => [the finite_kernel _ _ _ of kadd (f1 n) (f2 n)]) => x U mU. +rewrite /kadd/=. +rewrite -/(measure_add (k1 x) (k2 x)) measure_addE. rewrite /mseries. rewrite hk1//= hk2//= /mseries. rewrite -nneseriesD//. apply: eq_nneseries => n _. -rewrite -/(measure_add (k1 n x) (k2 n x)). -by rewrite measure_addE. +by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE. Qed. HB.instance Definition _ t := - isSFiniteKernel.Build _ _ _ _ R (add_of_kernels u1 u2) sfinite_add_of_kernels. -End add_of_sfinite_kernels. - -Section normalize_measure. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). -Variables (R : realType) (f : T -> {measure set T' -> \bar R}) (P : probability T' R). - -Definition normalize (t : T) (U : set T') := - let evidence := f t setT in - if evidence == 0%E then P U - else if evidence == +oo then P U - else f t U * (fine evidence)^-1%:E. - -Lemma normalize0 t : normalize t set0 = 0. -Proof. -rewrite /normalize. -case: ifPn => // _. -case: ifPn => // _. -by rewrite measure0 mul0e. -Qed. - -Lemma normalize_ge0 t U : 0 <= normalize t U. -Proof. -by rewrite /normalize; case: ifPn => //; case: ifPn. -Qed. - -Lemma normalize_sigma_additive t : semi_sigma_additive (normalize t). -Proof. -move=> F mF tF mUF. -rewrite /normalize/=. -case: ifPn => [_|_]. - exact: measure_semi_sigma_additive. -case: ifPn => [_|_]. - exact: measure_semi_sigma_additive. -rewrite (_ : (fun n => _) = ((fun n=> \sum_(0 <= i < n) f t (F i)) \* cst ((fine (f t [set: T']))^-1)%:E)); last first. - by apply/funext => n; rewrite -ge0_sume_distrl. -by apply: ereal_cvgMr => //; exact: measure_semi_sigma_additive. -Qed. - -HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ - (normalize t) (normalize0 t) (normalize_ge0 t) (@normalize_sigma_additive t). - -Lemma normalize1 t : normalize t setT = 1. -Proof. -rewrite /normalize; case: ifPn. - by rewrite probability_setT. -case: ifPn. - by rewrite probability_setT. -move=> ftoo ft0. -have ? : f t [set: T'] \is a fin_num. - by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. -rewrite -{1}(@fineK _ (f t setT))//. -by rewrite -EFinM divrr// ?unitfE fine_eq0. -Qed. - -HB.instance Definition _ t := isProbability.Build _ _ _ (normalize t) (normalize1 t). - -End normalize_measure. - -Section measurable_fun_comp. -Variables (d1 d2 d3 : measure_display). -Variables (T1 : measurableType d1). -Variables (T2 : measurableType d2). -Variables (T3 : measurableType d3). - -Lemma measurable_fun_comp_new F (f : T2 -> T3) E (g : T1 -> T2) : - measurable F -> - g @` E `<=` F -> - measurable_fun F f -> measurable_fun E g -> measurable_fun E (f \o g). -Proof. -move=> mF FgE mf mg /= mE A mA. -rewrite comp_preimage. -rewrite [X in measurable X](_ : _ = (E `&` g @^-1` (F `&` f @^-1` A))); last first. - apply/seteqP; split. - move=> x/= [Ex Afgx]; split => //; split => //. - by apply: FgE => //. - by move=> x/= [Ex] [Fgx Afgx]. -apply/mg => //. -by apply: mf => //. -Qed. - -End measurable_fun_comp. - -Lemma open_continuousP (S T : topologicalType) (f : S -> T) (D : set S) : - open D -> - {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). -Proof. -move=> oD; split=> [fcont|fcont s /[!inE] sD A]. - rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. - by rewrite interiorI; split => //; move: oD; rewrite openE; exact. -rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. -by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. -Qed. - -Lemma open_continuous_measurable_fun (R : realType) (f : R -> R) D : - open D -> {in D, continuous f} -> measurable_fun D f. -Proof. -move=> oD /(open_continuousP _ oD) cf. -apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. -by apply: open_measurable; exact/cf/interval_open. -Qed. - -Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. -Proof. -have [Bt|Bt] := boolP (true \in B). - have [Bf|Bf] := boolP (false \in B). - have -> : B = setT. - by apply/seteqP; split => // -[] _; [rewrite inE in Bt| rewrite inE in Bf]. - apply/or4P. - by rewrite eqxx/= !orbT. - have -> : B = [set true]. - apply/seteqP; split => -[]//=. - by rewrite notin_set in Bf. - by rewrite inE in Bt. - apply/or4P. - by rewrite eqxx/=. -have [Bf|Bf] := boolP (false \in B). - have -> : B = [set false]. - apply/seteqP; split => -[]//=. - by rewrite notin_set in Bt. - by rewrite inE in Bf. - apply/or4P. - by rewrite eqxx/= orbT. -have -> : B = set0. - apply/seteqP; split => -[]//=. - by rewrite notin_set in Bt. - by rewrite notin_set in Bf. -apply/or4P. -by rewrite eqxx/= !orbT. -Qed. + isSFinite.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. +End sfkadd. Lemma measurable_eq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k : @@ -1578,49 +1475,73 @@ have [/eqP->|/eqP->|/eqP->|/eqP->] := set_boolE B. - by rewrite preimage_setT. Qed. -(* TODO: PR *) -Lemma measurable_fun_fine (R : realType) : measurable_fun [set: \bar R] fine. +Section mnormalize. +Variables (d d' : _) (T : measurableType d) (Y : measurableType d'). +Variables (R : realType) (f : T -> {measure set Y -> \bar R}) (P : probability Y R). + +Definition mnormalize t U := + let evidence := f t setT in + if (evidence == 0) || (evidence == +oo) then P U + else f t U * (fine evidence)^-1%:E. + +Let mnormalize0 t : mnormalize t set0 = 0. Proof. -move=> _ /= B mB. -rewrite setTI [X in measurable X](_ : _ @^-1` _ = - if 0%R \in B then (EFin @` B) `|` [set -oo; +oo] else EFin @` B); last first. - apply/seteqP; split=> [[r Br|B0|B0]|]. - case: ifPn => //= B0. - by left; exists r. - by exists r. - by rewrite mem_set//=; tauto. - by rewrite mem_set//=; tauto. - move=> [r| |]//=; case: ifPn => B0 /=. - case; last first. - by case. - by move=> [r' Br' [<-]]. - by move=> [r' Br' [<-]]. - by rewrite inE in B0. - by case => //. - case=> //. - by case=> //. - by rewrite inE in B0. - by case=> //. -case: ifPn => B0. - apply: measurableU. - by apply: measurable_EFin. - by apply: measurableU. -by apply: measurable_EFin. +rewrite /mnormalize; case: ifPn => // _. +by rewrite measure0 mul0e. Qed. -Section normalize_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). -Variables (R : realType) (f : R.-ker T ~> T'). +Let mnormalize_ge0 t U : 0 <= mnormalize t U. +Proof. by rewrite /mnormalize; case: ifPn => //; case: ifPn. Qed. -Definition normalize_kernel (P : probability T' R) := - fun t => [the measure _ _ of normalize f P t]. +Lemma mnormalize_sigma_additive t : semi_sigma_additive (mnormalize t). +Proof. +move=> F mF tF mUF; rewrite /mnormalize/=. +case: ifPn => [_|_]. + exact: measure_semi_sigma_additive. +rewrite (_ : (fun n => _) = ((fun n=> \sum_(0 <= i < n) f t (F i)) \* + cst ((fine (f t setT))^-1)%:E)); last first. + by apply/funext => n; rewrite -ge0_sume_distrl. +by apply: ereal_cvgMr => //; exact: measure_semi_sigma_additive. +Qed. -Variable P : probability T' R. +HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ + (mnormalize t) (mnormalize0 t) (mnormalize_ge0 t) (@mnormalize_sigma_additive t). -Lemma measurable_fun_normalize U : measurable U -> measurable_fun setT (normalize_kernel P ^~ U). +Lemma mnormalize1 t : mnormalize t setT = 1. +Proof. +rewrite /mnormalize; case: ifPn; first by rewrite probability_setT. +rewrite negb_or => /andP[ft0 ftoo]. +have ? : f t setT \is a fin_num. + by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. +rewrite -{1}(@fineK _ (f t setT))//. +by rewrite -EFinM divrr// ?unitfE fine_eq0. +Qed. + +HB.instance Definition _ t := + isProbability.Build _ _ _ (mnormalize t) (mnormalize1 t). + +End mnormalize. + +Section knormalize. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (f : R.-ker X ~> Y). + +Definition knormalize (P : probability Y R) := + fun t => [the measure _ _ of mnormalize f P t]. + +Variable P : probability Y R. + +Let measurable_fun_knormalize U : + measurable U -> measurable_fun setT (knormalize P ^~ U). Proof. move=> mU. -rewrite /normalize_kernel/= /normalize /=. +rewrite /knormalize/= /mnormalize /=. +rewrite (_ : (fun _ => _) = (fun x => + if f x [set: Y] == 0 then P U else if f x [set: Y] == +oo then P U + else f x U * ((fine (f x [set: Y]))^-1)%:E)); last first. + apply/funext => x; case: ifPn => [/orP[->//|->]|]. + by case: ifPn. + by rewrite negb_or=> /andP[/negbTE -> /negbTE ->]. apply: measurable_fun_if000 => //. - exact: measurable_fun_eq_cst. - exact: measurable_fun_cst. @@ -1651,22 +1572,14 @@ apply: measurable_fun_if000 => //. by apply: measurable_funS => //. Qed. -HB.instance Definition _ := isKernel.Build _ _ _ _ R (normalize_kernel P) - measurable_fun_normalize. - -End normalize_kernel. - -Section normalize_prob_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). -Variables (R : realType) (f : R.-ker T ~> T') (P : probability T' R). +HB.instance Definition _ := isKernel.Build _ _ _ _ R (knormalize P) + measurable_fun_knormalize. -Lemma normalize_prob_kernelP x : normalize_kernel f P x [set: T'] = 1. +Let knormalize1 x : knormalize P x setT = 1. Proof. -rewrite /normalize_kernel/= /normalize. -case: ifPn => [_|fx0]. - by rewrite probability_setT. -case: ifPn => [_|fxoo]. - by rewrite probability_setT. +rewrite /knormalize/= /mnormalize. +case: ifPn => [_|]; first by rewrite probability_setT. +rewrite negb_or => /andP[fx0 fxoo]. have ? : f x [set: _] \is a fin_num. by rewrite ge0_fin_numE// lt_neqAle fxoo/= leey. rewrite -{1}(@fineK _ (f x setT))//=. @@ -1674,7 +1587,6 @@ by rewrite -EFinM divrr// ?lte_fin ?ltr1n// ?unitfE fine_eq0. Qed. HB.instance Definition _ := - @isProbabilityKernel.Build _ _ _ _ _ (normalize_kernel f P) - normalize_prob_kernelP. + @isProbabilityKernel.Build _ _ _ _ _ (knormalize P) knormalize1. -End normalize_prob_kernel. +End knormalize. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index b23d89b09a..e644a49fd2 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -5,11 +5,21 @@ Require Import reals ereal topology normedtype sequences esum measure. Require Import lebesgue_measure fsbigop numfun lebesgue_integral kernel. (******************************************************************************) -(* Semantics of a PPL using s-finite kernels *) +(* Semantics of a programming language PPL using s-finite kernels *) (* *) -(* bernoulli == *) -(* score == *) -(* ite_true/ite_false == *) +(* bernoulli r1 == Bernoulli probability *) +(* *) +(* sample P == sample according to the probability P *) +(* letin l k == execute l, augment the context, and execute k *) +(* ret mf == access the context with f and return the result *) +(* score mf == observe t from d, where f is the density of d and *) +(* t occurs in f *) +(* e.g., score (r e^(-r * t)) = observe t from exp(r) *) +(* normalize k P == normalize the kernel k into a probability kernel, *) +(* P is a default probability in case normalization is *) +(* not possible *) +(* ite mf k1 k2 == access the context with the boolean function f and *) +(* behaves as k1 or k2 according to the result *) (******************************************************************************) Set Implicit Arguments. @@ -22,13 +32,21 @@ Local Open Scope classical_set_scope. Local Open Scope ring_scope. Local Open Scope ereal_scope. -Definition onem (R : numDomainType) (p : R) := (1 - p)%R. +(* TODO: PR *) +Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1). -Lemma onem1 (R : numDomainType) (p : R) : (p + onem p = 1)%R. +Lemma measurable_fun_swap d (X : measurableType d) : measurable_fun [set: X * X] (swap (T2:=X)). +Proof. +apply/prod_measurable_funP => /=; split. + exact: measurable_fun_snd. +exact: measurable_fun_fst. +Qed. + +Lemma onem1 (R : numDomainType) (p : R) : (p + `1- p = 1)%R. Proof. by rewrite /onem addrCA subrr addr0. Qed. Lemma onem_nonneg_proof (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) : - (0 <= onem p%:num)%R. + (0 <= `1-(p%:num))%R. Proof. by rewrite /onem/= subr_ge0. Qed. Definition onem_nonneg (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) := @@ -38,48 +56,50 @@ Section bernoulli. Variables (R : realType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R). Local Open Scope ring_scope. -Definition bernoulli : set _ -> \bar R := +Definition mbernoulli : set _ -> \bar R := measure_add [the measure _ _ of mscale p [the measure _ _ of dirac true]] [the measure _ _ of mscale (onem_nonneg p1) [the measure _ _ of dirac false]]. -HB.instance Definition _ := Measure.on bernoulli. +HB.instance Definition _ := Measure.on mbernoulli. Local Close Scope ring_scope. -Lemma bernoulli_setT : bernoulli [set: _] = 1. +Let mbernoulli_setT : mbernoulli [set: _] = 1. Proof. -rewrite /bernoulli/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. +rewrite /mbernoulli/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. by rewrite /mscale/= !diracE !in_setT !mule1 -EFinD onem1. Qed. -HB.instance Definition _ := @isProbability.Build _ _ R bernoulli bernoulli_setT. +HB.instance Definition _ := @isProbability.Build _ _ R mbernoulli mbernoulli_setT. + +Definition bernoulli := [the probability _ _ of mbernoulli]. End bernoulli. -Section score_measure. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R). +Section mscore. +Variables (d : _) (T : measurableType d). +Variables (R : realType) (f : T -> R). -Definition score (t : T) (U : set unit) : \bar R := - if U == set0 then 0 else `| (r t)%:E |. +Definition mscore t (U : set unit) : \bar R := + if U == set0 then 0 else `| (f t)%:E |. -Let score0 t : score t (set0 : set unit) = 0 :> \bar R. -Proof. by rewrite /score eqxx. Qed. +Let mscore0 t : mscore t (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /mscore eqxx. Qed. -Let score_ge0 t U : 0 <= score t U. -Proof. by rewrite /score; case: ifP. Qed. +Let mscore_ge0 t U : 0 <= mscore t U. +Proof. by rewrite /mscore; case: ifP. Qed. -Let score_sigma_additive t : semi_sigma_additive (score t). +Let mscore_sigma_additive t : semi_sigma_additive (mscore t). Proof. -move=> /= F mF tF mUF; rewrite /score; case: ifPn => [/eqP/bigcup0P F0|]. +move=> /= F mF tF mUF; rewrite /mscore; case: ifPn => [/eqP/bigcup0P F0|]. rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. apply/funext => k. under eq_bigr do rewrite F0// eqxx. by rewrite big1. move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. rewrite -(cvg_shiftn k.+1)/=. -rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. +rewrite (_ : (fun _ => _) = cst `|(f t)%:E|); first exact: cvg_cst. apply/funext => n. rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. rewrite (negbTE Fk0) big1 ?adde0// => i/= ik; rewrite ifT//. @@ -89,9 +109,9 @@ by rewrite Fitt setTI => /eqP; rewrite (negbTE Fk0). Qed. HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ - (score t) (score0 t) (score_ge0 t) (@score_sigma_additive t). + (mscore t) (mscore0 t) (mscore_ge0 t) (@mscore_sigma_additive t). -End score_measure. +End mscore. (* decomposition of score into finite kernels *) Module SCORE. @@ -99,454 +119,421 @@ Section score. Variables (R : realType) (d : _) (T : measurableType d). Variables (r : T -> R). -Definition k_ (mr : measurable_fun setT r) (i : nat) : T -> set unit -> \bar R := +Definition k (mr : measurable_fun setT r) (i : nat) : T -> set unit -> \bar R := fun t U => - if i%:R%:E <= score r t U < i.+1%:R%:E then - score r t U + if i%:R%:E <= mscore r t U < i.+1%:R%:E then + mscore r t U else 0. Hypothesis mr : measurable_fun setT r. -Lemma k_0 i (t : T) : k_ mr i t (set0 : set unit) = 0 :> \bar R. -Proof. by rewrite /k_ measure0; case: ifP. Qed. +Lemma k0 i t : k mr i t (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /k measure0; case: ifP. Qed. -Lemma k_ge0 i (t : T) B : 0 <= k_ mr i t B. -Proof. by rewrite /k_; case: ifP. Qed. +Lemma k_ge0 i t B : 0 <= k mr i t B. +Proof. by rewrite /k; case: ifP. Qed. -Lemma k_sigma_additive i (t : T) : semi_sigma_additive (k_ mr i t). +Lemma k_sigma_additive i t : semi_sigma_additive (k mr i t). Proof. -move=> /= F mF tF mUF. -rewrite /k_ /=. +move=> /= F mF tF mUF; rewrite /k /=. have [F0|] := eqVneq (\bigcup_n F n) set0. - rewrite [in X in _ --> X]/score F0 eqxx. + rewrite [in X in _ --> X]/mscore F0 eqxx. rewrite (_ : (fun _ => _) = cst 0). by case: ifPn => _; exact: cvg_cst. apply/funext => k; rewrite big1// => n _. move : F0 => /bigcup0P F0. - by rewrite /score F0// eqxx; case: ifP. + by rewrite /mscore F0// eqxx; case: ifP. move=> UF0; move: (UF0). -move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. -rewrite [in X in _ --> X]/score (negbTE UF0). -rewrite -(cvg_shiftn k.+1)/=. +move=> /eqP/bigcup0P/existsNP[m /not_implyP[_ /eqP Fm0]]. +rewrite [in X in _ --> X]/mscore (negbTE UF0). +rewrite -(cvg_shiftn m.+1)/=. case: ifPn => ir. rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. apply/funext => n. - rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. - rewrite [in X in X + _]/score (negbTE Fk0) ir big1 ?adde0// => /= j jk. - rewrite /score. + rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn m))))//=. + rewrite [in X in X + _]/mscore (negbTE Fm0) ir big1 ?adde0// => /= j jk. + rewrite /mscore. have /eqP Fj0 : F j == set0. have [/eqP//|Fjtt] := set_unit (F j). - move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). - by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). + move/trivIsetP : tF => /(_ j m Logic.I Logic.I jk). + by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fm0). rewrite Fj0 eqxx. by case: ifP. rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. apply/funext => n. -rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. -rewrite [in X in if X then _ else _]/score (negbTE Fk0) (negbTE ir) add0e. -rewrite big1//= => j jk. -rewrite /score. +rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn m))))//=. +rewrite [in X in if X then _ else _]/mscore (negbTE Fm0) (negbTE ir) add0e. +rewrite big1//= => j jm. +rewrite /mscore. have /eqP Fj0 : F j == set0. have [/eqP//|Fjtt] := set_unit (F j). - move/trivIsetP : tF => /(_ j k Logic.I Logic.I jk). - by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fk0). -rewrite Fj0 eqxx. -by case: ifP. -Qed. - -HB.instance Definition _ (i : nat) (t : T) := isMeasure.Build _ _ _ - (k_ mr i t) (k_0 i t) (k_ge0 i t) (@k_sigma_additive i t). - -Lemma measurable_fun_k_ (i : nat) U : measurable U -> measurable_fun setT (k_ mr i ^~ U). -Proof. -move=> /= mU. -rewrite /k_ /=. -rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o (fun x => score r x U)) //. -apply: measurable_fun_comp; last first. - rewrite /score. - have [U0|U0] := eqVneq U set0. - exact: measurable_fun_cst. - apply: measurable_fun_comp => //. - by apply/EFin_measurable_fun. -rewrite /=. -pose A : _ -> \bar R := (fun x : \bar R => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). + move/trivIsetP : tF => /(_ j m Logic.I Logic.I jm). + by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fm0). +by rewrite Fj0 eqxx; case: ifP. +Qed. + +HB.instance Definition _ i t := isMeasure.Build _ _ _ + (k mr i t) (k0 i t) (k_ge0 i t) (@k_sigma_additive i t). + +Lemma measurable_fun_k i U : measurable U -> measurable_fun setT (k mr i ^~ U). +Proof. +move=> /= mU; rewrite /k /=. +rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o + (mscore r ^~ U)) //. +apply: measurable_fun_comp => /=; last first. + rewrite /mscore. + have [U0|U0] := eqVneq U set0; first exact: measurable_fun_cst. + by apply: measurable_fun_comp => //; exact/EFin_measurable_fun. +pose A : _ -> \bar R := (fun x => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). rewrite (_ : (fun x => _) = A); last first. apply/funext => x; rewrite /A; case: ifPn => ix. by rewrite indicE/= mem_set ?mule1//. rewrite indicE/= memNset ?mule0//. - rewrite /= in_itv/=. - exact/negP. -rewrite /A. -apply emeasurable_funM => /=. - exact: measurable_fun_id. + by rewrite /= in_itv/=; exact/negP. +rewrite {}/A. +apply emeasurable_funM => /=; first exact: measurable_fun_id. apply/EFin_measurable_fun. have mi : measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set (\bar R)). exact: emeasurable_itv. -by rewrite (_ : \1__ = mindic R mi)//. +by rewrite (_ : \1__ = mindic R mi). Qed. -Definition mk_ i (t : T) := [the measure _ _ of k_ mr i t]. +Definition mk i t := [the measure _ _ of k mr i t]. -HB.instance Definition _ (i : nat) := - isKernel.Build _ _ _ _ R (mk_ i) (measurable_fun_k_ i). +HB.instance Definition _ i := + isKernel.Build _ _ _ _ R (mk i) (measurable_fun_k i). -Lemma mk_uub (i : nat) : measure_fam_uub (mk_ i). +Lemma mk_uub (i : nat) : measure_fam_uub (mk i). Proof. exists i.+1%:R => /= t. -rewrite /k_ /score setT_unit. +rewrite /k /mscore setT_unit. rewrite (_ : [set tt] == set0 = false); last first. by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). by case: ifPn => // /andP[]. Qed. -HB.instance Definition _ (i : nat) := - @isFiniteKernel.Build _ _ _ _ R (mk_ i) (mk_uub i). +HB.instance Definition _ i := @isFiniteFam.Build _ _ _ _ R (mk i) (mk_uub i). End score. End SCORE. -Section score_kernel. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R). +Section kscore. +Variables (R : realType) (d : _) (T : measurableType d) (r : T -> R). -Definition kernel_score (mr : measurable_fun setT r) - : T -> {measure set Datatypes_unit__canonical__measure_Measurable -> \bar R} := - fun t => [the measure _ _ of score r t]. +Definition kscore (mr : measurable_fun setT r) + : T -> {measure set _ -> \bar R} := + fun t => [the measure _ _ of mscore r t]. Variable (mr : measurable_fun setT r). -Let measurable_fun_score U : measurable U -> measurable_fun setT (kernel_score mr ^~ U). +Let measurable_fun_kscore U : measurable U -> measurable_fun setT (kscore mr ^~ U). Proof. -move=> /= mU; rewrite /score. +move=> /= mU; rewrite /mscore. have [U0|U0] := eqVneq U set0; first exact: measurable_fun_cst. by apply: measurable_fun_comp => //; exact/EFin_measurable_fun. Qed. HB.instance Definition _ := isKernel.Build _ _ T _ - (*Datatypes_unit__canonical__measure_Measurable*) R (kernel_score mr) measurable_fun_score. -End score_kernel. - -Section score_sfinite_kernel. -Variables (R : realType) (d : _) (T : measurableType d). -Variables (r : T -> R) (mr : measurable_fun setT r). + (*Datatypes_unit__canonical__measure_Measurable*) R (kscore mr) measurable_fun_kscore. Import SCORE. -Let sfinite_score : exists k_ : (R.-fker T ~> _)^nat, +Let sfinite_kscore : exists k : (R.-fker T ~> _)^nat, forall x U, measurable U -> - kernel_score mr x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. + kscore mr x U = [the measure _ _ of mseries (k ^~ x) 0] U. Proof. rewrite /=. -exists (fun i => [the finite_kernel _ _ _ of mk_ mr i]) => /= r' U mU. -rewrite /mseries /score; case: ifPn => [/eqP U0|U0]. +exists (fun i => [the R.-fker _ ~> _ of mk mr i]) => /= t U mU. +rewrite /mseries /mscore; case: ifPn => [/eqP U0|U0]. by apply/esym/nneseries0 => i _; rewrite U0 measure0. -rewrite /mk_ /= /k_ /= /score (negbTE U0). +rewrite /mk /= /k /= /mscore (negbTE U0). apply/esym/cvg_lim => //. -rewrite -(cvg_shiftn `|floor (fine `|(r r')%:E|)|%N.+1)/=. -rewrite (_ : (fun _ => _) = cst `|(r r')%:E|); first exact: cvg_cst. +rewrite -(cvg_shiftn `|floor (fine `|(r t)%:E|)|%N.+1)/=. +rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. apply/funext => n. -pose floor_r := widen_ord (leq_addl n `|floor `|(r r')| |.+1) (Ordinal (ltnSn `|floor `|(r r')| |)). +pose floor_r := widen_ord (leq_addl n `|floor `|r t| |.+1) (Ordinal (ltnSn `|floor `|r t| |)). rewrite big_mkord (bigD1 floor_r)//= ifT; last first. rewrite lee_fin lte_fin; apply/andP; split. - by rewrite natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?floor_le. - by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|(r r')|)) ?floor_ge0 ?lt_succ_floor. + by rewrite natr_absz (@ger0_norm _ (floor `|r t|)) ?floor_ge0 ?floor_le. + by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|r t|)) ?floor_ge0 ?lt_succ_floor. rewrite big1 ?adde0//= => j jk. rewrite ifF// lte_fin lee_fin. move: jk; rewrite neq_ltn/= => /orP[|] jr. -- suff : (j.+1%:R <= `|(r r')|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. +- suff : (j.+1%:R <= `|r t|)%R by rewrite leNgt => /negbTE ->; rewrite andbF. rewrite (_ : j.+1%:R = j.+1%:~R)// floor_ge_int. move: jr; rewrite -lez_nat => /le_trans; apply. - by rewrite -[leRHS](@ger0_norm _ (floor `|(r r')|)) ?floor_ge0. -- suff : (`|(r r')| < j%:R)%R by rewrite ltNge => /negbTE ->. - move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|(r r')|)) ?floor_ge0// ltr_int. + by rewrite -[leRHS](@ger0_norm _ (floor `|r t|)) ?floor_ge0. +- suff : (`|r t| < j%:R)%R by rewrite ltNge => /negbTE ->. + move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|r t|)) ?floor_ge0// ltr_int. by rewrite -floor_lt_int. Qed. -HB.instance Definition _ := @isSFiniteKernel.Build _ _ _ _ _ - (kernel_score mr) sfinite_score. +HB.instance Definition _ := @isSFinite.Build _ _ _ _ _ (kscore mr) sfinite_kscore. -End score_sfinite_kernel. +End kscore. -(* decomposition of if-then-else *) +(* decomposition of ite into s-finite kernels *) Module ITE. -Section ite_true_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u1 : R.-ker T ~> T'). +Section kiteT. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-ker X ~> Y). -Definition ite_true : T * bool -> {measure set T' -> \bar R} := - fun b => if b.2 then u1 b.1 else [the measure _ _ of mzero]. +Definition kiteT : X * bool -> {measure set Y -> \bar R} := + fun xb => if xb.2 then k xb.1 else [the measure _ _ of mzero]. -Lemma measurable_ite_true U : measurable U -> measurable_fun setT (ite_true ^~ U). +Let measurable_fun_kiteT U : measurable U -> measurable_fun setT (kiteT ^~ U). Proof. -move=> /= mcU. -rewrite /ite_true. -rewrite (_ : (fun x : T * bool => _) = (fun x => if x.2 then u1 x.1 U else [the {measure set T' -> \bar R} of mzero] U)); last first. - apply/funext => -[t b]/=. - by case: ifPn. -apply: (@measurable_fun_if _ _ _ _ (u1 ^~ U) (fun=> mzero U)). +move=> /= mcU; rewrite /kiteT. +rewrite (_ : (fun _ => _) = (fun x => if x.2 then k x.1 U + else [the {measure set Y -> \bar R} of mzero] U)); last first. + by apply/funext => -[t b]/=; case: ifPn. +apply: (@measurable_fun_if _ _ _ _ (k ^~ U) (fun=> mzero U)). exact/measurable_kernel. exact: measurable_fun_cst. Qed. -HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_true measurable_ite_true. -End ite_true_kernel. +HB.instance Definition _ := isKernel.Build _ _ _ _ R kiteT measurable_fun_kiteT. +End kiteT. -Section ite_true_finite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u1 : R.-fker T ~> T'). +Section fkiteT. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-fker X ~> Y). -Lemma ite_true_uub : measure_fam_uub (ite_true u1). +Let kiteT_uub : measure_fam_uub (kiteT k). Proof. -have /measure_fam_uubP[M hM] := kernel_uub u1. -exists M%:num => /= -[]; rewrite /ite_true => t [|]/=. - exact: hM. +have /measure_fam_uubP[M hM] := measure_uub k. +exists M%:num => /= -[]; rewrite /kiteT => t [|]/=; first exact: hM. by rewrite /= /mzero. Qed. -HB.instance Definition _ t := - isFiniteKernel.Build _ _ _ _ R (ite_true u1) ite_true_uub. -End ite_true_finite_kernel. +HB.instance Definition _ t := isFiniteFam.Build _ _ _ _ R (kiteT k) kiteT_uub. +End fkiteT. -Section ite_true_sfinite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u1 : R.-sfker T ~> T'). +Section sfkiteT. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-sfker X ~> Y). -Let sfinite_ite_true : exists k_ : (R.-fker _ ~> _)^nat, +Let sfinite_kiteT : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> - ite_true u1 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. + kiteT k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -have [k hk /=] := sfinite u1. -rewrite /ite_true. -exists (fun n => [the _.-fker _ ~> _ of ite_true (k n)]) => b U mU. -case: ifPn => hb. +have [k_ hk /=] := sfinite k. +exists (fun n => [the _.-fker _ ~> _ of kiteT (k_ n)]) => b U mU. +rewrite /kiteT; case: ifPn => hb. rewrite /mseries hk//= /mseries. apply: eq_nneseries => n _. - by rewrite /ite_true hb. + by rewrite /kiteT hb. rewrite /= /mseries nneseries0// => n _. -by rewrite /ite_true (negbTE hb). +by rewrite /kiteT (negbTE hb). Qed. -HB.instance Definition _ t := - @isSFiniteKernel.Build _ _ _ _ _ (ite_true u1) sfinite_ite_true. +HB.instance Definition _ t := @isSFinite.Build _ _ _ _ _ (kiteT k) sfinite_kiteT. -End ite_true_sfinite_kernel. +End sfkiteT. -Section ite_false_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u2 : R.-ker T ~> T'). +Section kiteF. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-ker X ~> Y). -Definition ite_false : T * bool -> {measure set T' -> \bar R} := - fun b => if ~~ b.2 then u2 b.1 else [the measure _ _ of mzero]. +Definition kiteF : X * bool -> {measure set Y -> \bar R} := + fun xb => if ~~ xb.2 then k xb.1 else [the measure _ _ of mzero]. -Let measurable_ite_false U : measurable U -> measurable_fun setT (ite_false ^~ U). +Let measurable_fun_kiteF U : measurable U -> measurable_fun setT (kiteF ^~ U). Proof. -move=> /= mcU. -rewrite /ite_false. -rewrite (_ : (fun x => _) = (fun x => if x.2 then [the {measure set T' -> \bar R} of mzero] U else u2 x.1 U)); last first. +move=> /= mcU; rewrite /kiteF. +rewrite (_ : (fun x => _) = (fun x => if x.2 then [the measure _ _ of mzero] U else k x.1 U)); last first. apply/funext => -[t b]/=. - rewrite if_neg/=. - by case: b. -apply: (@measurable_fun_if _ _ _ _ (fun=> mzero U) (u2 ^~ U)). + by rewrite if_neg//; case: ifPn. +apply: (@measurable_fun_if _ _ _ _ (fun=> mzero U) (k ^~ U)). exact: measurable_fun_cst. exact/measurable_kernel. Qed. -HB.instance Definition _ := isKernel.Build _ _ _ _ R ite_false measurable_ite_false. +HB.instance Definition _ := isKernel.Build _ _ _ _ R kiteF measurable_fun_kiteF. -End ite_false_kernel. +End kiteF. -Section ite_false_finite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u2 : R.-fker T ~> T'). +Section fkiteF. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-fker X ~> Y). -Let ite_false_uub : measure_fam_uub (ite_false u2). +Let kiteF_uub : measure_fam_uub (kiteF k). Proof. -have /measure_fam_uubP[M hM] := kernel_uub u2. -exists M%:num => /= -[]; rewrite /ite_false/= => t b. -case: b => //=. -by rewrite /mzero. +have /measure_fam_uubP[M hM] := measure_uub k. +exists M%:num => /= -[]; rewrite /kiteF/= => t. +by case => //=; rewrite /mzero. Qed. -HB.instance Definition _ := - isFiniteKernel.Build _ _ _ _ R (ite_false u2) ite_false_uub. +HB.instance Definition _ := isFiniteFam.Build _ _ _ _ R (kiteF k) kiteF_uub. -End ite_false_finite_kernel. +End fkiteF. -Section ite_false_sfinite_kernel. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType). -Variables (u2 : R.-sfker T ~> T'). +Section sfkiteF. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-sfker X ~> Y). -Let sfinite_ite_false : exists k_ : (R.-fker _ ~> _)^nat, +Let sfinite_kiteF : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> - ite_false u2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -have [k hk] := sfinite u2. -rewrite /= /ite_false. -exists (fun n => [the finite_kernel _ _ _ of ite_false (k n)]) => b U mU. -case: ifPn => hb. - rewrite /mseries hk//= /mseries/=. - apply: eq_nneseries => // n _. - by rewrite /ite_false hb. -rewrite /= /mseries nneseries0// => n _. -rewrite negbK in hb. -by rewrite /ite_false hb/=. + kiteF k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +have [k_ hk] := sfinite k. +exists (fun n => [the finite_kernel _ _ _ of kiteF (k_ n)]) => b U mU. +rewrite /= /kiteF /=; case: ifPn => hb. + by rewrite /mseries hk//= /mseries/=. +by rewrite /= /mseries nneseries0. Qed. -HB.instance Definition _ := - @isSFiniteKernel.Build _ _ _ _ _ (ite_false u2) sfinite_ite_false. +HB.instance Definition _ := @isSFinite.Build _ _ _ _ _ (kiteF k) sfinite_kiteF. -End ite_false_sfinite_kernel. +End sfkiteF. End ITE. Section ite. Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). Variables (R : realType) (f : T -> bool) (u1 u2 : R.-sfker T ~> T'). -Definition ite (mf : measurable_fun setT f) : T -> set T' -> \bar R := +Definition mite (mf : measurable_fun setT f) : T -> set T' -> \bar R := fun t => if f t then u1 t else u2 t. Variables mf : measurable_fun setT f. -Lemma ite0 tb : ite mf tb set0 = 0. -Proof. by rewrite /ite; case: ifPn => //. Qed. +Let mite0 tb : mite mf tb set0 = 0. +Proof. by rewrite /mite; case: ifPn => //. Qed. -Lemma ite_ge0 tb (U : set _) : 0 <= ite mf tb U. -Proof. by rewrite /ite; case: ifPn => //. Qed. +Let mite_ge0 tb (U : set _) : 0 <= mite mf tb U. +Proof. by rewrite /mite; case: ifPn => //. Qed. -Lemma ite_sigma_additive tb : semi_sigma_additive (ite mf tb). +Let mite_sigma_additive tb : semi_sigma_additive (mite mf tb). Proof. -rewrite /ite. -case: ifPn => ftb. - exact: measure_semi_sigma_additive. -exact: measure_semi_sigma_additive. +by rewrite /mite; case: ifPn => ftb; exact: measure_semi_sigma_additive. Qed. -HB.instance Definition _ tb := isMeasure.Build _ _ _ (ite mf tb) - (ite0 tb) (ite_ge0 tb) (@ite_sigma_additive tb). +HB.instance Definition _ tb := isMeasure.Build _ _ _ (mite mf tb) + (mite0 tb) (mite_ge0 tb) (@mite_sigma_additive tb). Import ITE. -Let ite' : R.-sfker - [the measurableType _ of (T * bool)%type] ~> T' := - [the R.-sfker _ ~> _ of add_of_kernels - [the R.-sfker _ ~> T' of ite_true u1] - [the R.-sfker _ ~> T' of ite_false u2] ]. - -Definition mite := [the sfinite_kernel _ _ _ of kernel_mfun R mf] \; ite'. +Definition kite := + [the R.-sfker _ ~> _ of kdirac mf] \; + [the R.-sfker _ ~> _ of kadd + [the R.-sfker _ ~> T' of kiteT u1] + [the R.-sfker _ ~> T' of kiteF u2] ]. End ite. -Section normalize. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') - (R : realType) (f : R.-sfker T ~> T') (Pdef : probability T' R). +Section insn1. +Variables (R : realType) (d : _) (X : measurableType d). + +Definition score (f : X -> R) (mf : measurable_fun setT f) := + [the R.-sfker X ~> _ of kscore mf]. -Definition Normalize := [the R.-pker T ~> T' of normalize_kernel f Pdef]. +End insn1. -Lemma NormalizeE x U : Normalize x U = normalize_kernel f Pdef x U. +Section insn1_lemmas. +Variables (R : realType) (d : _) (T : measurableType d). + +Lemma scoreE (t : T) (U : set bool) (n : nat) (b : bool) + (f : R -> R) + (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) + (mf : measurable_fun setT f) : + score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) + (t, b, n%:R) ((fun _ => (snd \o fst) (t, b, tt)) @^-1` U) = + (f n%:R)%:E * \d_b U. Proof. -by []. +rewrite /score/= /mscore/= diracE. +have [U0|U0] := set_unit ((fun=> b) @^-1` U). +- rewrite U0 eqxx memNset ?mule0// => Ub. + by move: U0 => /seteqP[/(_ tt)] /(_ Ub). +- rewrite U0 setT_unit ifF//; last first. + by apply/negbTE/negP => /eqP/seteqP[/(_ tt erefl)]. + rewrite /= mem_set//; last first. + by move: U0 => /seteqP[_]/(_ tt)/=; exact. + by rewrite mule1 ger0_norm// f0. Qed. -End normalize. +End insn1_lemmas. -Section bernoulli27. +Section insn2. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variable R : realType. -Local Open Scope ring_scope. -Definition twoseven : {nonneg R} := (2%:R / 7%:R)%:nng. -Definition fiveseven : {nonneg R} := (5%:R / 7%:R)%:nng. -Lemma onem_twoseven : onem (2 / 7) = fiveseven%:num. -Proof. by apply/eqP; rewrite subr_eq/= -mulrDl -natrD divrr// unitfE. Qed. +Definition ret (f : X -> Y) (mf : measurable_fun setT f) := + locked [the R.-sfker X ~> Y of kdirac mf]. -Lemma twoseven_proof : (twoseven%:num <= 1 :> R)%R. -Proof. by rewrite /= lter_pdivr_mulr// mul1r ler_nat. Qed. +Definition sample (P : probability Y R) := + locked [the R.-sfker X ~> Y of kprobability P] . -Definition bernoulli27 : set _ -> \bar R := bernoulli twoseven_proof. +Definition normalize (k : R.-sfker X ~> Y) P := + locked [the R.-pker X ~> Y of knormalize k P]. -End bernoulli27. +Definition ite (f : X -> bool) (mf : measurable_fun setT f) + (k1 k2 : R.-sfker X ~> Y):= + locked [the R.-sfker X ~> Y of kite k1 k2 mf]. -Section insn. -Variables (R : realType). +End insn2. +Arguments sample {d d' X Y R}. -Definition letin (d d' d3 : _) - (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) - (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) - : R.-sfker X ~> Z := - [the sfinite_kernel _ _ _ of l \; k]. - -Lemma letinE (d d' d3 : _) - (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) - (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) - : forall x U, letin l k x U = \int[l x]_y k (x, y) U. -Proof. -by []. -Qed. - -Definition Return (d d' : _) (T : measurableType d) (T' : measurableType d') - (f : T -> T') (mf : measurable_fun setT f) : R.-sfker T ~> T' := - [the sfinite_kernel _ _ _ of @kernel_mfun _ _ T T' R f mf]. - -Definition sample_bernoulli27 (d : _) (T : measurableType d) := - [the sfinite_kernel T _ _ of - kernel_probability [the probability _ _ of bernoulli27 R]] . - -(* NB: score r = observe 0 from exp r, - the density of the exponential distribution exp(r) at 0 is r = r e^(-r * 0) - more generally, score (r e^(-r * t)) = observe t from exp(r), - score (f(r)) = observe r from p where f is the density of p *) -Definition Score (d : _) (T : measurableType d) (r : T -> R) (mr : measurable_fun setT r) : - R.-sfker T ~> Datatypes_unit__canonical__measure_Measurable := - [the sfinite_kernel _ _ R of @kernel_score R _ _ r mr]. - -Lemma ScoreE (d : _) (T : measurableType d) (t : T) (U : set bool) (n : nat) (b : bool) - (f : R -> R) (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) (mf : measurable_fun setT f) : - Score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) - (t, b, cst n%:R (t, b)) - ((fun y : unit => (snd \o fst) (t, b, y)) @^-1` U) = - (f n%:R)%:E * \d_b U. +Section insn2_lemmas. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variable R : realType. + +Lemma retE (f : X -> Y) (mf : measurable_fun setT f) x : + ret R mf x = \d_(f x) :> (_ -> _). +Proof. by rewrite [in LHS]/ret; unlock. Qed. + +Lemma sampleE (P : probability Y R) (x : X) : sample P x = P. +Proof. by rewrite [in LHS]/sample; unlock. Qed. + +Lemma normalizeE (f : R.-sfker X ~> Y) P x U : + normalize f P x U = + if (f x [set: Y] == 0) || (f x [set: Y] == +oo) then P U + else f x U * ((fine (f x [set: Y]))^-1)%:E. Proof. -rewrite /Score/= /score/= diracE. -have [U0|U0] := set_unit ((fun=> b) @^-1` U). -- rewrite U0 eqxx memNset ?mule0//. - move=> Ub. - move: U0. - move/seteqP => [/(_ tt)] /=. - by move/(_ Ub). -- rewrite U0 setT_unit ifF//; last first. - by apply/negbTE/negP => /eqP/seteqP[/(_ tt erefl)]. - rewrite /= mem_set//; last first. - by move: U0 => /seteqP[_]/(_ tt)/=; exact. - by rewrite mule1 ger0_norm// f0. +by rewrite /normalize; unlock => /=; rewrite /mnormalize; case: ifPn. Qed. -Definition Ite (d d' : _) (T : measurableType d) (T' : measurableType d') - (f : T -> bool) (mf : measurable_fun setT f) - (u1 u2 : R.-sfker T ~> T') - : R.-sfker T ~> T' := - [the R.-sfker _ ~> _ of mite u1 u2 mf]. - -Lemma IteE (d d' : _) (T : measurableType d) (T' : measurableType d') - (f : T -> bool) (mf : measurable_fun setT f) - (u1 u2 : R.-sfker T ~> T') tb U : - Ite mf u1 u2 tb U = ite u1 u2 mf tb U. +Lemma iteE (f : X -> bool) (mf : measurable_fun setT f) + (k1 k2 : R.-sfker X ~> Y) x : + ite mf k1 k2 x = if f x then k1 x else k2 x. Proof. -rewrite /= /kcomp /ite. +apply/eq_measure/funext => U. +rewrite /ite; unlock => /=. +rewrite /kcomp/=. rewrite integral_dirac//=. -rewrite indicT /cst. +rewrite indicT. rewrite mul1e. -rewrite -/(measure_add (ITE.ite_true u1 (tb, f tb)) - (ITE.ite_false u2 (tb, f tb))). +rewrite -/(measure_add (ITE.kiteT k1 (x, f x)) + (ITE.kiteF k2 (x, f x))). rewrite measure_addE. -rewrite /ITE.ite_true /ITE.ite_false/=. -case: (ifPn (f tb)) => /=. - by rewrite /mzero adde0. -by rewrite /mzero add0e. +rewrite /ITE.kiteT /ITE.kiteF/=. +by case: ifPn => fx /=; rewrite /mzero ?(adde0,add0e). Qed. -End insn. +End insn2_lemmas. + +Section insn3. +Variables (R : realType). +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). + +Definition letin (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) := + locked [the R.-sfker X ~> Z of l \; k]. + +End insn3. + +Section insn3_lemmas. +Variables (R : realType). +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3). + +Lemma letinE (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) x U : + letin l k x U = \int[l x]_y k (x, y) U. +Proof. by rewrite /letin; unlock. Qed. + +End insn3_lemmas. (* a few laws *) @@ -554,25 +541,27 @@ Section letin_return. Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3). -Lemma letin_ureturn (u : R.-sfker X ~> Y) - (f : _ -> Z) (mf : measurable_fun setT f) : - forall x U, measurable U -> letin u (Return R mf) x U = u x ((fun y => f (x, y)) @^-1` U). +Lemma letin_kret (k : R.-sfker X ~> Y) + (f : _ -> Z) (mf : measurable_fun setT f) x U : + measurable U -> + letin k (ret R mf) x U = k x (curry f x @^-1` U). Proof. -move=> x U mU. -rewrite /letin/= /kcomp/= integral_indic// ?setIT//. +move=> mU. +rewrite letinE. +under eq_integral do rewrite retE. +rewrite integral_indic ?setIT//. move/measurable_fun_prod1 : mf => /(_ x)/(_ measurableT U mU). by rewrite setTI. Qed. -Lemma letin_returnu - (u : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) +Lemma letin_retk (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) (f : _ -> Y) (mf : measurable_fun setT f) : - forall x U, measurable U -> letin (Return R mf) u x U = u (x, f x) U. + forall x U, measurable U -> letin (ret R mf) k x U = k (x, f x) U. Proof. move=> x U mU. -rewrite /letin/= /kcomp/= integral_dirac//. +rewrite letinE retE integral_dirac//. by rewrite indicE mem_set// mul1e. -have /measurable_fun_prod1 := measurable_kernel u _ mU. +have /measurable_fun_prod1 := measurable_kernel k _ mU. exact. Qed. @@ -580,31 +569,44 @@ End letin_return. Section letin_ite. Variables (R : realType) (d d2 d3 : _) (T : measurableType d) - (T2 : measurableType d2) (T3 : measurableType d3) - (u1 u2 : R.-sfker T ~> T3) (u : R.-sfker [the measurableType _ of (T * T3)%type] ~> T2) + (T2 : measurableType d2) (Z : measurableType d3) + (k1 k2 : R.-sfker T ~> Z) (u : R.-sfker [the measurableType _ of (T * Z)%type] ~> T2) (f : T -> bool) (mf : measurable_fun setT f) (t : T) (U : set T2). -Lemma letin_ite_true : f t -> letin (Ite mf u1 u2) u t U = letin u1 u t U. +Lemma letin_iteT : f t -> letin (ite mf k1 k2) u t U = letin k1 u t U. Proof. move=> ftT. -rewrite /letin/= /kcomp. +rewrite !letinE/=. apply eq_measure_integral => V mV _. -by rewrite IteE /ite ftT. +by rewrite iteE ftT. Qed. -Lemma letin_ite_false : ~~ f t -> letin (Ite mf u1 u2) u t U = letin u2 u t U. +Lemma letin_iteF : ~~ f t -> letin (ite mf k1 k2) u t U = letin k2 u t U. Proof. move=> ftF. -rewrite /letin/= /kcomp. +rewrite !letinE/=. apply eq_measure_integral => V mV _. -by rewrite IteE/= /ite (negbTE ftF). +by rewrite iteE (negbTE ftF). Qed. End letin_ite. (* sample programs *) +Section constants. +Variable R : realType. +Local Open Scope ring_scope. + +Lemma onem27 : `1- (2 / 7%:R) = (5%:R / 7%:R)%:nng%:num :> R. +Proof. by apply/eqP; rewrite subr_eq/= -mulrDl -natrD divrr// unitfE. Qed. + +Lemma p27 : (2 / 7%:R)%:nng%:num <= 1 :> R. +Proof. by rewrite /= lter_pdivr_mulr// mul1r ler_nat. Qed. + +End constants. +Arguments p27 {R}. + Require Import exp. Definition poisson (R : realType) (r : R) (k : nat) := (r ^+ k / k%:R^-1 * expR (- r))%R. @@ -646,34 +648,53 @@ Definition k3 : measurable_fun _ _ := kn 3. Definition k10 : measurable_fun _ _ := kn 10. End cst_fun. +Arguments k3 {R d T}. +Arguments k10 {R d T}. -Lemma letin_sample_bernoulli27 (R : realType) (d d' : _) (T : measurableType d) - (T' : measurableType d') +Module Notations. + +Notation var1_of2 := (@measurable_fun_fst _ _ _ _). +Notation var2_of2 := (@measurable_fun_snd _ _ _ _). +Notation var1_of3 := (measurable_fun_comp (@measurable_fun_fst _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var3_of3 := (@measurable_fun_snd _ _ _ _). + +End Notations. + +Lemma letin_sample_bernoulli (R : realType) (d d' : _) (T : measurableType d) + (T' : measurableType d') (r : {nonneg R}) (r1 : (r%:num <= 1)%R) (u : R.-sfker [the measurableType _ of (T * bool)%type] ~> T') x y : - letin (sample_bernoulli27 R T) u x y = - (2 / 7)%:E * u (x, true) y + (5 / 7)%:E * u (x, false) y. + letin (sample (bernoulli r1)) u x y = + r%:num%:E * u (x, true) y + (`1- (r%:num : R))%:E * u (x, false) y. Proof. -rewrite {1}/letin/= {1}/kcomp/=. +rewrite letinE/= sampleE. rewrite ge0_integral_measure_sum//. rewrite 2!big_ord_recl/= big_ord0 adde0/=. rewrite !ge0_integral_mscale//=. rewrite !integral_dirac//=. -rewrite indicE in_setT mul1e indicE in_setT mul1e. -by rewrite onem_twoseven. +by rewrite indicE in_setT mul1e indicE in_setT mul1e. Qed. Section sample_and_return. Variables (R : realType) (d : _) (T : measurableType d). +Import Notations. + Definition sample_and_return : R.-sfker T ~> _ := letin - (sample_bernoulli27 R T) (* T -> B *) - (Return R (@measurable_fun_snd _ _ _ _)) (* T * B -> B *). + (sample (bernoulli p27)) (* T -> B *) + (ret R var2_of2) (* T * B -> B *). Lemma sample_and_returnE t U : sample_and_return t U = - (twoseven R)%:num%:E * \d_true U + - (fiveseven R)%:num%:E * \d_false U. -Proof. by rewrite letin_sample_bernoulli27. Qed. + (2 / 7%:R)%:E * \d_true U + (5%:R / 7%:R)%:E * \d_false U. +Proof. +rewrite /sample_and_return. +rewrite letin_sample_bernoulli/=. +rewrite !retE. +by rewrite onem27. +Qed. End sample_and_return. @@ -682,8 +703,8 @@ Variables (R : realType) (d : _) (T : measurableType d). Definition sample_and_score : R.-sfker T ~> _ := letin - (sample_bernoulli27 R T) (* T -> B *) - (Score (measurable_fun_cst (1%R : R))). + (sample (bernoulli p27)) (* T -> B *) + (score (measurable_fun_cst (1%R : R))). End sample_and_score. @@ -694,18 +715,26 @@ Variables (R : realType) (d : _) (T : measurableType d). let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in return r *) +Let mR := Real_sort__canonical__measure_Measurable R. + +Import Notations. + Definition sample_and_branch : - R.-sfker T ~> [the measurableType default_measure_display of Real_sort__canonical__measure_Measurable R] := + R.-sfker T ~> [the measurableType default_measure_display of mR] := letin - (sample_bernoulli27 R T) (* T -> B *) - (Ite (@measurable_fun_snd _ _ _ _) - (Return R (@k3 _ _ [the measurableType _ of (T * bool)%type])) - (Return R (@k10 _ _ [the measurableType _ of (T * bool)%type]))). + (sample (bernoulli p27)) (* T -> B *) + (ite var2_of2 + (ret R k3) + (ret R k10)). Lemma sample_and_branchE t U : sample_and_branch t U = - (twoseven R)%:num%:E * \d_(3%R : R) U + - (fiveseven R)%:num%:E * \d_(10%R : R) U. -Proof. by rewrite /sample_and_branch letin_sample_bernoulli27 !IteE. Qed. + (2 / 7%:R)%:E * \d_(3%:R : R) U + + (5%:R / 7%:R)%:E * \d_(10%:R : R) U. +Proof. +rewrite /sample_and_branch letin_sample_bernoulli/=. +rewrite !iteE/= !retE. +by rewrite onem27. +Qed. End sample_and_branch. @@ -721,72 +750,68 @@ Let mR := Real_sort__canonical__measure_Measurable R. Let munit := Datatypes_unit__canonical__measure_Measurable. Let mbool := Datatypes_bool__canonical__measure_Measurable. -Notation var2_of2 := (@measurable_fun_snd _ _ _ _). -Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var3_of3 := (@measurable_fun_snd _ _ _ _). +Variable P : probability mbool R. -Variable Pdef : probability mbool R. +Import Notations. -Definition staton_bus_measure' : R.-sfker T ~> mbool := - (letin - (sample_bernoulli27 R T : _.-sfker T ~> mbool) +Definition staton_bus_annotated : R.-sfker T ~> mbool := + normalize (letin + (sample (bernoulli p27) : _.-sfker T ~> mbool) (letin (letin - (Ite var2_of2 - (Return R (@k3 _ _ _)) - (Return R (@k10 _ _ _)) + (ite var2_of2 + (ret R k3) + (ret R k10) : _.-sfker [the measurableType _ of (T * bool)%type] ~> mR) - (Score (measurable_fun_comp (@mpoisson R 4) var3_of3) + (score (measurable_fun_comp (@mpoisson R 4) var3_of3) : _.-sfker [the measurableType _ of (T * bool* mR)%type] ~> munit) : _.-sfker [the measurableType _ of (T * bool)%type] ~> munit) - (Return R var2_of3 + (ret R var2_of3 : _.-sfker [the measurableType _ of (T * bool * munit)%type] ~> mbool) - : _.-sfker [the measurableType _ of (T * bool)%type] ~> mbool)). + : _.-sfker [the measurableType _ of (T * bool)%type] ~> mbool)) P. -Definition staton_bus_measure : R.-sfker T ~> mbool := - (letin (sample_bernoulli27 R T) +Let staton_bus' : R.-sfker T ~> _ := + (letin (sample (bernoulli p27)) (letin - (letin (Ite var2_of2 - (Return R (@k3 _ _ _)) - (Return R (@k10 _ _ _))) - (Score (measurable_fun_comp (@mpoisson R 4) var3_of3))) - (Return R var2_of3))). + (letin (ite var2_of2 + (ret R k3) + (ret R k10)) + (score (measurable_fun_comp (@mpoisson R 4) var3_of3))) + (ret R var2_of3))). (* true -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) (* false -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) -Lemma staton_bus_measureE t U : staton_bus_measure t U = - (twoseven R)%:num%:E * (poisson 3%:R 4)%:E * \d_true U + - (fiveseven R)%:num%:E * (poisson 10%:R 4)%:E * \d_false U. +Let staton_bus'E t U : staton_bus' t U = + (2 / 7%:R)%:E * (poisson 3%:R 4)%:E * \d_true U + + (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U. Proof. -rewrite /staton_bus_measure. -rewrite letin_sample_bernoulli27. -rewrite -!muleA. -congr (_ * _ + _ * _). - rewrite letin_ureturn //. - rewrite letin_ite_true//. - rewrite letin_returnu//. - by rewrite ScoreE// => r r0; exact: poisson_ge0. -rewrite letin_ureturn //. -rewrite letin_ite_false//. -rewrite letin_returnu//. -by rewrite ScoreE// => r r0; exact: poisson_ge0. +rewrite /staton_bus'. +rewrite letin_sample_bernoulli. +rewrite -!muleA; congr (_ * _ + _ * _). +- rewrite letin_kret//. + rewrite letin_iteT//. + rewrite letin_retk//. + by rewrite scoreE// => r r0; exact: poisson_ge0. +- by rewrite onem27. + rewrite letin_kret//. + rewrite letin_iteF//. + rewrite letin_retk//. + by rewrite scoreE// => r r0; exact: poisson_ge0. Qed. -Definition staton_bus : R.-pker T ~> mbool := - Normalize staton_bus_measure Pdef. +Definition staton_bus : R.-pker T ~> mbool := normalize staton_bus' P. Lemma staton_busE t U : - let N := (fine (((twoseven R)%:num)%:E * (poisson 3 4)%:E + ((fiveseven R)%:num)%:E * (poisson 10 4)%:E)) in + let N := ((2 / 7%:R) * poisson 3%:R 4 + + (5%:R / 7%:R) * poisson 10%:R 4)%R in staton_bus t U = - ((twoseven R)%:num%:E * (poisson 3%:R 4)%:E * \d_true U + - (fiveseven R)%:num%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. + ((2 / 7%:R)%:E * (poisson 3%:R 4)%:E * \d_true U + + (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. Proof. rewrite /staton_bus. -rewrite NormalizeE /=. -rewrite /normalize. -rewrite !staton_bus_measureE. +rewrite normalizeE /=. +rewrite !staton_bus'E. rewrite diracE mem_set// mule1. rewrite diracE mem_set// mule1. rewrite ifF //. @@ -798,9 +823,7 @@ End staton_bus. (* wip *) -Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1). - -Section letinC_example. +Section letinC. Variables (d d' d3 d4 : _) (R : realType) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) (U : measurableType d4). @@ -820,13 +843,6 @@ apply: measurable_fun_comp => /=. exact: measurable_fun_id. Qed. -Let measurable_fun_swap : measurable_fun [set: X * X] (swap (T2:=X)). -Proof. -apply/prod_measurable_funP => /=; split. - exact: measurable_fun_snd. -exact: measurable_fun_fst. -Qed. - Let f' := @swap _ _ \o f. Lemma mf' : measurable_fun setT f'. Proof. @@ -842,20 +858,30 @@ Variables (t : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X) (u' : R.-sfker [the measurableType _ of (unit * X)%type] ~> X) (H1 : forall y, u tt = u' (tt, y)) (H2 : forall y, t tt = t' (tt, y)). + Lemma letinC x A : measurable A -> - letin t (letin u' (Return R mf)) x A = letin u (letin t' (Return R mf')) x A. + letin t (letin u' (ret R mf)) x A = letin u (letin t' (ret R mf')) x A. Proof. move=> mA. -rewrite /letin /= /kcomp /= /kcomp /=. +rewrite !letinE. destruct x. rewrite /f/=. -under eq_integral do rewrite -H1. -rewrite (@sfinite_fubini _ _ X X R t u (fun x => (\d_(x.1, x.2) A)))//=. +under eq_integral. + move=> x _. + rewrite letinE/=. + rewrite -H1. + under eq_integral do rewrite retE /=. + over. +rewrite /=. +rewrite (@sfinite_fubini _ _ X X R t u (fun x => \d_(x.1, x.2) A ))//=. apply eq_integral => x _. - by rewrite -H2. + rewrite letinE/=. + rewrite -H2. + apply eq_integral => // x' _. + by rewrite retE. apply/EFin_measurable_fun => /=. rewrite (_ : (fun x => _) = mindic R mA)//. by apply/funext => -[a b] /=. Qed. -End letinC_example. +End letinC. From 85bcada36de45183586c15137dd57df424307dd4 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Thu, 1 Sep 2022 20:31:03 +0900 Subject: [PATCH 35/42] mscore using mscale and dirac --- theories/kernel.v | 159 ++----------------------------------------- theories/prob_lang.v | 95 ++++++++++---------------- 2 files changed, 41 insertions(+), 213 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 6decb7a906..5f0d7c2e9f 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -74,121 +74,6 @@ Qed. End integralM_0ifneg. Arguments integralM_0ifneg {d T R} m {D} mD f. -Section integralM_indic. -Local Open Scope ereal_scope. -Variables (d : measure_display) (T : measurableType d) (R : realType). -Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). - -Let integralM_indic (f : R -> set T) (k : R) : - ((k < 0)%R -> f k = set0) -> measurable (f k) -> - \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. -Proof. -move=> fk0 mfk. -under eq_integral do rewrite EFinM. -apply: (integralM_0ifneg _ _ (fun k x => (\1_(f k) x)%:E)) => //=. -- by move=> r t Dt; rewrite lee_fin. -- by move/fk0 => -> /=; apply/funext => x; rewrite indicE in_set0. -- apply/EFin_measurable_fun. - by rewrite (_ : \1_(f k) = mindic R mfk). -Qed. - -End integralM_indic. -Arguments integralM_indic {d T R} m {D} mD f. - -(* NB: PR in progress *) -Section integral_mscale. -Variables (R : realType) (k : {nonneg R}). -Variables (d : measure_display) (T : measurableType d). -Variable (m : {measure set T -> \bar R}) (D : set T) (f : T -> \bar R). -Hypotheses (mD : measurable D). - -Let integral_mscale_indic (E : set T) (mE : measurable E) : - \int[mscale k m]_(x in D) (\1_E x)%:E = - k%:num%:E * \int[m]_(x in D) (\1_E x)%:E. -Proof. by rewrite !integral_indic. Qed. - -Let integral_mscale_nnsfun (h : {nnsfun T >-> R}) : - \int[mscale k m]_(x in D) (h x)%:E = k%:num%:E * \int[m]_(x in D) (h x)%:E. -Proof. -rewrite -ge0_integralM//; last 2 first. -apply/EFin_measurable_fun. - exact: measurable_funS (@measurable_funP _ _ _ h). - by move=> x _; rewrite lee_fin. -under eq_integral do rewrite fimfunE -sumEFin. -under [LHS]eq_integral do rewrite fimfunE -sumEFin. -rewrite /=. -rewrite ge0_integral_sum//; last 2 first. - move=> r. - apply/EFin_measurable_fun/measurable_funrM. - apply: (@measurable_funS _ _ _ _ setT) => //. - have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - by move=> n x Dx; rewrite EFinM muleindic_ge0. -under eq_integral. - move=> x xD. - rewrite ge0_sume_distrr//; last first. - by move=> r _; rewrite EFinM muleindic_ge0. - over. -rewrite /=. -rewrite ge0_integral_sum//; last 2 first. - move=> r. - apply/EFin_measurable_fun/measurable_funrM/measurable_funrM. - apply: (@measurable_funS _ _ _ _ setT) => //. - have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - move=> n x Dx. - by rewrite EFinM mule_ge0// muleindic_ge0. -apply eq_bigr => r _. -rewrite ge0_integralM//; last 2 first. - apply/EFin_measurable_fun/measurable_funrM. - apply: (@measurable_funS _ _ _ _ setT) => //. - have fr : measurable (h @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - by move=> t Dt; rewrite muleindic_ge0. -by rewrite !integralM_indic_nnsfun//= integral_mscale_indic// muleCA. -Qed. - -Lemma ge0_integral_mscale (mf : measurable_fun D f) : - (forall x, D x -> 0 <= f x) -> - \int[mscale k m]_(x in D) f x = k%:num%:E * \int[m]_(x in D) f x. -Proof. -move=> f0; have [f_ [ndf_ f_f]] := approximation mD mf f0. -transitivity (lim (fun n => \int[mscale k m]_(x in D) (f_ n x)%:E)). - rewrite -monotone_convergence//=; last 3 first. - move=> n; apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT). - by move=> n x Dx; rewrite lee_fin. - by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. - apply eq_integral => x /[!inE] xD; apply/esym/cvg_lim => //=. - exact: f_f. -rewrite (_ : \int[m]_(x in D) _ = lim (fun n => \int[m]_(x in D) (f_ n x)%:E)); last first. - rewrite -monotone_convergence//. - apply: eq_integral => x /[!inE] xD. - apply/esym/cvg_lim => //. - exact: f_f. - move=> n. - apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT). - by move=> n x Dx; rewrite lee_fin. - by move=> x Dx a b /ndf_ /lefP; rewrite lee_fin. -rewrite -ereal_limrM//; last first. - apply/ereal_nondecreasing_is_cvg => a b ab. - apply ge0_le_integral => //. - by move=> x Dx; rewrite lee_fin. - apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT). - by move=> x Dx; rewrite lee_fin. - apply/EFin_measurable_fun. - by apply: (@measurable_funS _ _ _ _ setT). - move=> x Dx. - rewrite lee_fin. - by move/ndf_ : ab => /lefP. -congr (lim _); apply/funext => n /=. -by rewrite integral_mscale_nnsfun. -Qed. - -End integral_mscale. - (* TODO: PR *) Canonical unit_pointedType := PointedType unit tt. @@ -232,23 +117,6 @@ HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Po End discrete_measurable_bool. -(* NB: PR in progress *) -Lemma measurable_fun_fine (R : realType) (D : set (\bar R)) : measurable D -> - measurable_fun D fine. -Proof. -move=> mD _ /= B mB; rewrite [X in measurable X](_ : _ `&` _ = if 0%R \in B then - D `&` ((EFin @` B) `|` [set -oo; +oo]) else D `&` EFin @` B); last first. - apply/seteqP; split=> [[r [Dr Br]|[Doo B0]|[Doo B0]]|[r| |]]. - - by case: ifPn => _; split => //; left; exists r. - - by rewrite mem_set//; split => //; right; right. - - by rewrite mem_set//; split => //; right; left. - - by case: ifPn => [_ [Dr [[s + [sr]]|[]//]]|_ [Dr [s + [sr]]]]; rewrite sr. - - by case: ifPn => [/[!inE] B0 [Doo [[]//|]] [//|_]|B0 [Doo//] []]. - - by case: ifPn => [/[!inE] B0 [Doo [[]//|]] [//|_]|B0 [Doo//] []]. -case: ifPn => B0; apply/measurableI => //; last exact: measurable_EFin. -by apply: measurableU; [exact: measurable_EFin|exact: measurableU]. -Qed. - (* TODO: PR *) Lemma measurable_fun_fst (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) : measurable_fun setT (@fst T1 T2). @@ -293,25 +161,6 @@ Qed. End measurable_fun_comp. -Lemma open_continuousP (S T : topologicalType) (f : S -> T) (D : set S) : - open D -> - {in D, continuous f} <-> (forall A, open A -> open (D `&` f @^-1` A)). -Proof. -move=> oD; split=> [fcont|fcont s /[!inE] sD A]. - rewrite !openE => A Aop s [Ds] /Aop /fcont; rewrite inE => /(_ Ds) fsA. - by rewrite interiorI; split => //; move: oD; rewrite openE; exact. -rewrite nbhs_simpl /= !nbhsE => - [B [[oB Bfs] BA]]. -by exists (D `&` f @^-1` B); split=> [|t [Dt] /BA//]; split => //; exact/fcont. -Qed. - -Lemma open_continuous_measurable_fun (R : realType) (f : R -> R) D : - open D -> {in D, continuous f} -> measurable_fun D f. -Proof. -move=> oD /(open_continuousP _ oD) cf. -apply: (measurability (RGenOpens.measurableE R)) => _ [_ [a [b ->] <-]]. -by apply: open_measurable; exact/cf/interval_open. -Qed. - Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. Proof. have [Bt|Bt] := boolP (true \in B). @@ -771,7 +620,7 @@ rewrite [X in measurable_fun _ X](_ : _ = (fun x => apply/EFin_measurable_fun/measurable_funrM/measurable_fun_prod1 => /=. rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r))//. exact/measurable_funP. - - by move=> m y _; rewrite muleindic_ge0. + - by move=> m y _; rewrite nnfun_muleindic_ge0. apply emeasurable_fun_sum => r. rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * \int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. @@ -1103,7 +952,7 @@ rewrite ge0_integral_sum//; last 2 first. move=> r; apply/EFin_measurable_fun/measurable_funrM. have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. by rewrite (_ : \1__ = mindic R fr). - by move=> r z _; rewrite EFinM muleindic_ge0. + by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. under [in RHS]eq_integral. move=> y _. under eq_integral. @@ -1114,7 +963,7 @@ under [in RHS]eq_integral. move=> r; apply/EFin_measurable_fun/measurable_funrM. have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. by rewrite (_ : \1__ = mindic R fr). - by move=> r z _; rewrite EFinM muleindic_ge0. + by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. under eq_bigr. move=> r _. rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. @@ -1127,7 +976,7 @@ rewrite /= ge0_integral_sum//; last 2 first. have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). by move=> /measurable_fun_prod1; exact. - move=> n y _. - have := @mulem_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). + have := @mulemu_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). by apply; exact: preimage_nnfun0. apply eq_bigr => r _. rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index e644a49fd2..12a615e58a 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -42,7 +42,7 @@ apply/prod_measurable_funP => /=; split. exact: measurable_fun_fst. Qed. -Lemma onem1 (R : numDomainType) (p : R) : (p + `1- p = 1)%R. +Lemma onem1' (R : numDomainType) (p : R) : (p + `1- p = 1)%R. Proof. by rewrite /onem addrCA subrr addr0. Qed. Lemma onem_nonneg_proof (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) : @@ -68,7 +68,7 @@ Local Close Scope ring_scope. Let mbernoulli_setT : mbernoulli [set: _] = 1. Proof. rewrite /mbernoulli/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. -by rewrite /mscale/= !diracE !in_setT !mule1 -EFinD onem1. +by rewrite /mscale/= !diracE !in_setT !mule1 -EFinD onem1'. Qed. HB.instance Definition _ := @isProbability.Build _ _ R mbernoulli mbernoulli_setT. @@ -81,35 +81,25 @@ Section mscore. Variables (d : _) (T : measurableType d). Variables (R : realType) (f : T -> R). -Definition mscore t (U : set unit) : \bar R := - if U == set0 then 0 else `| (f t)%:E |. +Definition mscore t : {measure set _ -> \bar R} := + let p := NngNum (@normr_ge0 _ _ (`| f t |)%R) in + [the measure _ _ of mscale p [the measure _ _ of dirac tt]]. -Let mscore0 t : mscore t (set0 : set unit) = 0 :> \bar R. -Proof. by rewrite /mscore eqxx. Qed. - -Let mscore_ge0 t U : 0 <= mscore t U. -Proof. by rewrite /mscore; case: ifP. Qed. - -Let mscore_sigma_additive t : semi_sigma_additive (mscore t). +Lemma mscoreE t U : mscore t U = if U == set0 then 0 else `| (f t)%:E |. Proof. -move=> /= F mF tF mUF; rewrite /mscore; case: ifPn => [/eqP/bigcup0P F0|]. - rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. - apply/funext => k. - under eq_bigr do rewrite F0// eqxx. - by rewrite big1. -move=> /eqP/bigcup0P/existsNP[k /not_implyP[_ /eqP Fk0]]. -rewrite -(cvg_shiftn k.+1)/=. -rewrite (_ : (fun _ => _) = cst `|(f t)%:E|); first exact: cvg_cst. -apply/funext => n. -rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn k))))//=. -rewrite (negbTE Fk0) big1 ?adde0// => i/= ik; rewrite ifT//. -have [/eqP//|Fitt] := set_unit (F i). -move/trivIsetP : tF => /(_ i k Logic.I Logic.I ik). -by rewrite Fitt setTI => /eqP; rewrite (negbTE Fk0). +rewrite /mscore/= /mscale/=; have [->|->] := set_unit U. + by rewrite eqxx diracE in_set0 mule0. +rewrite diracE in_setT mule1 ifF// ?normr_id//. +by apply/negbTE/set0P; exists tt. Qed. -HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ - (mscore t) (mscore0 t) (mscore_ge0 t) (@mscore_sigma_additive t). +Lemma measurable_fun_mscore U : measurable_fun setT f -> + measurable_fun setT (mscore ^~ U). +Proof. +move=> mr; under eq_fun do rewrite mscoreE/=. +have [U0|U0] := eqVneq U set0; first exact: measurable_fun_cst. +by apply: measurable_fun_comp => //; exact: measurable_fun_comp. +Qed. End mscore. @@ -138,39 +128,34 @@ Lemma k_sigma_additive i t : semi_sigma_additive (k mr i t). Proof. move=> /= F mF tF mUF; rewrite /k /=. have [F0|] := eqVneq (\bigcup_n F n) set0. - rewrite [in X in _ --> X]/mscore F0 eqxx. - rewrite (_ : (fun _ => _) = cst 0). + rewrite F0 measure0 (_ : (fun _ => _) = cst 0). by case: ifPn => _; exact: cvg_cst. apply/funext => k; rewrite big1// => n _. - move : F0 => /bigcup0P F0. - by rewrite /mscore F0// eqxx; case: ifP. + by move: F0 => /bigcup0P -> //; rewrite measure0; case: ifPn. move=> UF0; move: (UF0). move=> /eqP/bigcup0P/existsNP[m /not_implyP[_ /eqP Fm0]]. -rewrite [in X in _ --> X]/mscore (negbTE UF0). +rewrite [in X in _ --> X]mscoreE (negbTE UF0). rewrite -(cvg_shiftn m.+1)/=. case: ifPn => ir. rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. apply/funext => n. rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn m))))//=. - rewrite [in X in X + _]/mscore (negbTE Fm0) ir big1 ?adde0// => /= j jk. - rewrite /mscore. - have /eqP Fj0 : F j == set0. + rewrite [in X in X + _]mscoreE (negbTE Fm0) ir big1 ?adde0// => /= j jk. + rewrite mscoreE; have /eqP -> : F j == set0. have [/eqP//|Fjtt] := set_unit (F j). move/trivIsetP : tF => /(_ j m Logic.I Logic.I jk). by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fm0). - rewrite Fj0 eqxx. - by case: ifP. + by rewrite eqxx; case: ifP. rewrite (_ : (fun _ => _) = cst 0); first exact: cvg_cst. apply/funext => n. rewrite big_mkord (bigD1 (widen_ord (leq_addl n _) (Ordinal (ltnSn m))))//=. -rewrite [in X in if X then _ else _]/mscore (negbTE Fm0) (negbTE ir) add0e. +rewrite [in X in if X then _ else _]mscoreE (negbTE Fm0) (negbTE ir) add0e. rewrite big1//= => j jm. -rewrite /mscore. -have /eqP Fj0 : F j == set0. +rewrite mscoreE; have /eqP -> : F j == set0. have [/eqP//|Fjtt] := set_unit (F j). move/trivIsetP : tF => /(_ j m Logic.I Logic.I jm). by rewrite Fjtt setTI => /eqP; rewrite (negbTE Fm0). -by rewrite Fj0 eqxx; case: ifP. +by rewrite eqxx; case: ifP. Qed. HB.instance Definition _ i t := isMeasure.Build _ _ _ @@ -181,16 +166,12 @@ Proof. move=> /= mU; rewrite /k /=. rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o (mscore r ^~ U)) //. -apply: measurable_fun_comp => /=; last first. - rewrite /mscore. - have [U0|U0] := eqVneq U set0; first exact: measurable_fun_cst. - by apply: measurable_fun_comp => //; exact/EFin_measurable_fun. +apply: measurable_fun_comp => /=; last exact/measurable_fun_mscore. pose A : _ -> \bar R := (fun x => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). rewrite (_ : (fun x => _) = A); last first. apply/funext => x; rewrite /A; case: ifPn => ix. by rewrite indicE/= mem_set ?mule1//. - rewrite indicE/= memNset ?mule0//. - by rewrite /= in_itv/=; exact/negP. + by rewrite indicE/= memNset ?mule0// /= in_itv/=; exact/negP. rewrite {}/A. apply emeasurable_funM => /=; first exact: measurable_fun_id. apply/EFin_measurable_fun. @@ -206,8 +187,7 @@ HB.instance Definition _ i := Lemma mk_uub (i : nat) : measure_fam_uub (mk i). Proof. -exists i.+1%:R => /= t. -rewrite /k /mscore setT_unit. +exists i.+1%:R => /= t; rewrite /k mscoreE setT_unit. rewrite (_ : [set tt] == set0 = false); last first. by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). by case: ifPn => // /andP[]. @@ -228,11 +208,7 @@ Definition kscore (mr : measurable_fun setT r) Variable (mr : measurable_fun setT r). Let measurable_fun_kscore U : measurable U -> measurable_fun setT (kscore mr ^~ U). -Proof. -move=> /= mU; rewrite /mscore. -have [U0|U0] := eqVneq U set0; first exact: measurable_fun_cst. -by apply: measurable_fun_comp => //; exact/EFin_measurable_fun. -Qed. +Proof. by move=> /= _; exact: measurable_fun_mscore. Qed. HB.instance Definition _ := isKernel.Build _ _ T _ (*Datatypes_unit__canonical__measure_Measurable*) R (kscore mr) measurable_fun_kscore. @@ -245,9 +221,9 @@ Let sfinite_kscore : exists k : (R.-fker T ~> _)^nat, Proof. rewrite /=. exists (fun i => [the R.-fker _ ~> _ of mk mr i]) => /= t U mU. -rewrite /mseries /mscore; case: ifPn => [/eqP U0|U0]. +rewrite /mseries /kscore/= mscoreE; case: ifPn => [/eqP U0|U0]. by apply/esym/nneseries0 => i _; rewrite U0 measure0. -rewrite /mk /= /k /= /mscore (negbTE U0). +rewrite /mk /= /k /= mscoreE (negbTE U0). apply/esym/cvg_lim => //. rewrite -(cvg_shiftn `|floor (fine `|(r t)%:E|)|%N.+1)/=. rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. @@ -438,10 +414,11 @@ Lemma scoreE (t : T) (U : set bool) (n : nat) (b : bool) (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) (mf : measurable_fun setT f) : score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) - (t, b, n%:R) ((fun _ => (snd \o fst) (t, b, tt)) @^-1` U) = + (t, b, n%:R) (curry (snd \o fst) (t, b) @^-1` U) = (f n%:R)%:E * \d_b U. Proof. -rewrite /score/= /mscore/= diracE. +set x := score _. +rewrite /score/= /kscore/= mscoreE diracE. have [U0|U0] := set_unit ((fun=> b) @^-1` U). - rewrite U0 eqxx memNset ?mule0// => Ub. by move: U0 => /seteqP[/(_ tt)] /(_ Ub). @@ -827,7 +804,9 @@ Section letinC. Variables (d d' d3 d4 : _) (R : realType) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) (U : measurableType d4). + Let f (xyz : unit * X * X) := (xyz.1.2, xyz.2). + Lemma mf : measurable_fun setT f. Proof. rewrite /=. From 4a278c15a23d73037a053ee5e101706eefa85ad8 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 2 Sep 2022 10:16:10 +0900 Subject: [PATCH 36/42] generalize mscoreE --- theories/kernel.v | 283 +++++++++++++++++-------------------------- theories/prob_lang.v | 117 +++++++++++------- 2 files changed, 186 insertions(+), 214 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 5f0d7c2e9f..ec15c77e3d 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -83,16 +83,18 @@ Definition discrete_measurable_unit : set (set unit) := [set: set unit]. Let discrete_measurable0 : discrete_measurable_unit set0. Proof. by []. Qed. -Let discrete_measurableC X : discrete_measurable_unit X -> discrete_measurable_unit (~` X). +Let discrete_measurableC X : + discrete_measurable_unit X -> discrete_measurable_unit (~` X). Proof. by []. Qed. Let discrete_measurableU (F : (set unit)^nat) : - (forall i, discrete_measurable_unit (F i)) -> discrete_measurable_unit (\bigcup_i F i). + (forall i, discrete_measurable_unit (F i)) -> + discrete_measurable_unit (\bigcup_i F i). Proof. by []. Qed. -HB.instance Definition _ := @isMeasurable.Build default_measure_display unit (Pointed.class _) - discrete_measurable_unit discrete_measurable0 discrete_measurableC - discrete_measurableU. +HB.instance Definition _ := @isMeasurable.Build default_measure_display unit + (Pointed.class _) discrete_measurable_unit discrete_measurable0 + discrete_measurableC discrete_measurableU. End discrete_measurable_unit. @@ -111,9 +113,9 @@ Let discrete_measurableU (F : (set bool)^nat) : discrete_measurable_bool (\bigcup_i F i). Proof. by []. Qed. -HB.instance Definition _ := @isMeasurable.Build default_measure_display bool (Pointed.class _) - discrete_measurable_bool discrete_measurable0 discrete_measurableC - discrete_measurableU. +HB.instance Definition _ := @isMeasurable.Build default_measure_display bool + (Pointed.class _) discrete_measurable_bool discrete_measurable0 + discrete_measurableC discrete_measurableU. End discrete_measurable_bool. @@ -143,130 +145,105 @@ Variables (T1 : measurableType d1). Variables (T2 : measurableType d2). Variables (T3 : measurableType d3). -Lemma measurable_fun_comp_new F (f : T2 -> T3) E (g : T1 -> T2) : +(* NB: this generalizes MathComp's measurable_fun_comp' *) +Lemma measurable_fun_comp' F (f : T2 -> T3) E (g : T1 -> T2) : measurable F -> g @` E `<=` F -> measurable_fun F f -> measurable_fun E g -> measurable_fun E (f \o g). Proof. move=> mF FgE mf mg /= mE A mA. rewrite comp_preimage. -rewrite [X in measurable X](_ : _ = (E `&` g @^-1` (F `&` f @^-1` A))); last first. - apply/seteqP; split. - move=> x/= [Ex Afgx]; split => //; split => //. - by apply: FgE => //. - by move=> x/= [Ex] [Fgx Afgx]. -apply/mg => //. -by apply: mf => //. +rewrite [X in measurable X](_ : _ = E `&` g @^-1` (F `&` f @^-1` A)); last first. + apply/seteqP; split=> [|? [?] []//]. + by move=> x/= [Ex Afgx]; split => //; split => //; exact: FgE. +by apply/mg => //; exact: mf. Qed. End measurable_fun_comp. +Lemma set_unit (A : set unit) : A = set0 \/ A = setT. +Proof. +have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. +by apply/seteqP; split => [|] []. +Qed. + Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. Proof. have [Bt|Bt] := boolP (true \in B). have [Bf|Bf] := boolP (false \in B). have -> : B = setT. by apply/seteqP; split => // -[] _; [rewrite inE in Bt| rewrite inE in Bf]. - apply/or4P. - by rewrite eqxx/= !orbT. + by apply/or4P; rewrite eqxx/= !orbT. have -> : B = [set true]. apply/seteqP; split => -[]//=. by rewrite notin_set in Bf. by rewrite inE in Bt. - apply/or4P. - by rewrite eqxx/=. + by apply/or4P; rewrite eqxx. have [Bf|Bf] := boolP (false \in B). have -> : B = [set false]. apply/seteqP; split => -[]//=. by rewrite notin_set in Bt. by rewrite inE in Bf. - apply/or4P. - by rewrite eqxx/= orbT. + by apply/or4P; rewrite eqxx/= orbT. have -> : B = set0. apply/seteqP; split => -[]//=. by rewrite notin_set in Bt. by rewrite notin_set in Bf. -apply/or4P. -by rewrite eqxx/= !orbT. +by apply/or4P; rewrite eqxx/= !orbT. Qed. -Lemma measurable_fun_if000 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') - D (md : measurable D) (f : T -> bool) (mf : measurable_fun setT f) : - measurable_fun (D `&` [set b | f b ]) x -> - measurable_fun (D `&` [set b | ~~ f b]) y -> - measurable_fun D (fun b : T => if f b then x b else y b). +Lemma measurable_fun_if (d d' : _) (T : measurableType d) + (T' : measurableType d') (x y : T -> T') D (md : measurable D) + (f : T -> bool) (mf : measurable_fun setT f) : + measurable_fun (D `&` (f @^-1` [set true])) x -> + measurable_fun (D `&` (f @^-1` [set false])) y -> + measurable_fun D (fun t => if f t then x t else y t). Proof. move=> mx my /= _ Y mY. -have H1 : measurable (D `&` [set b | f b]). +have mDf : measurable (D `&` [set b | f b]). apply: measurableI => //. rewrite [X in measurable X](_ : _ = f @^-1` [set true])//. - have := mf measurableT [set true]. - rewrite setTI. - exact. -have := mx H1 Y mY. -have H0 : [set t | ~~ f t] = [set t | f t = false]. - by apply/seteqP; split => [t/= /negbTE//|t/= ->]. -have H2 : measurable (D `&` [set b | ~~ f b]). + by have := mf measurableT [set true]; rewrite setTI; exact. +have := mx mDf Y mY. +have mDNf : measurable (D `&` f @^-1` [set false]). apply: measurableI => //. - have := mf measurableT [set false]. - rewrite setTI. - rewrite /preimage/=. - by rewrite H0; exact. -have := my H2 Y mY. + by have := mf measurableT [set false]; rewrite setTI; exact. +have := my mDNf Y mY. move=> yY xY. -rewrite (_ : _ @^-1` Y = ([set b | f b = true] `&` (x @^-1` Y) `&` (f @^-1` [set true])) `|` - ([set b | f b = false] `&` (y @^-1` Y) `&` (f @^-1` [set false]))); last first. - apply/seteqP; split. - move=> t/=; case: ifPn => ft. - by left. - by right. - by move=> t/= [|]; case: ifPn => ft; case=> -[]. -rewrite setIUr. -apply: measurableU. - rewrite -(setIid D). - rewrite -(setIA D). - rewrite setICA. - rewrite setIA. - apply: measurableI => //. - by rewrite setIA. - - rewrite -(setIid D). - rewrite -(setIA D). - rewrite setICA. - rewrite setIA. - rewrite /preimage/=. - rewrite -H0. - apply: measurableI => //. - by rewrite setIA. +rewrite (_ : _ @^-1` Y = + ((f @^-1` [set true]) `&` (x @^-1` Y) `&` (f @^-1` [set true])) `|` + ((f @^-1` [set false]) `&` (y @^-1` Y) `&` (f @^-1` [set false]))); last first. + apply/seteqP; split=> [t /=| t]. + by case: ifPn => ft; [left|right]. + by move=> /= [|]; case: ifPn => ft; case=> -[]. +rewrite setIUr; apply: measurableU. +- rewrite -(setIid D) -(setIA D) setICA setIA. + by apply: measurableI => //; rewrite setIA. +- rewrite -(setIid D) -(setIA D) setICA setIA. + by apply: measurableI => //; rewrite setIA. Qed. -Lemma measurable_fun_if0 (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') - (f : T -> bool) (mf : measurable_fun setT f) : - measurable_fun setT x -> - measurable_fun setT y -> - measurable_fun setT (fun b : T => if f b then x b else y b). +Lemma measurable_fun_ifT (d d' : _) (T : measurableType d) + (T' : measurableType d') (x y : T -> T') (f : T -> bool) + (mf : measurable_fun setT f) : + measurable_fun setT x -> measurable_fun setT y -> + measurable_fun setT (fun t => if f t then x t else y t). Proof. -move=> mx my. -apply: measurable_fun_if000 => //. -by apply: measurable_funS mx. -by apply: measurable_funS my. +by move=> mx my; apply: measurable_fun_if => //; + [exact: measurable_funS mx|exact: measurable_funS my]. Qed. -Lemma measurable_fun_if (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') : - measurable_fun setT x -> - measurable_fun setT y -> - measurable_fun setT (fun b : T * bool => if b.2 then x b.1 else y b.1). +Lemma measurable_fun_if_pair (d d' : _) (T : measurableType d) + (T' : measurableType d') (x y : T -> T') : + measurable_fun setT x -> measurable_fun setT y -> + measurable_fun setT (fun tb => if tb.2 then x tb.1 else y tb.1). Proof. move=> mx my. have {}mx : measurable_fun [set: T * bool] (x \o fst). - apply: measurable_fun_comp => //. - exact: measurable_fun_fst. + by apply: measurable_fun_comp => //; exact: measurable_fun_fst. have {}my : measurable_fun [set: T * bool] (y \o fst). - apply: measurable_fun_comp => //. - exact: measurable_fun_fst. -rewrite /=. -apply: measurable_fun_if0 => //=. -exact: measurable_fun_snd. + by apply: measurable_fun_comp => //; exact: measurable_fun_fst. +by apply: measurable_fun_ifT => //=; exact: measurable_fun_snd. Qed. Lemma emeasurable_itv (R : realType) (i : nat) : @@ -279,12 +256,6 @@ apply: measurableU. exact: emeasurable_itv_ninfty_bnd. exact: emeasurable_itv_bnd_pinfty. Qed. - -Lemma set_unit (A : set unit) : A = set0 \/ A = setT. -Proof. -have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. -by apply/seteqP; split => [|] []. -Qed. (*/ PR*) Reserved Notation "R .-ker X ~> Y" (at level 42). @@ -474,19 +445,6 @@ HB.instance Definition _ := @isProbabilityFam.Build _ _ _ _ _ _ is_probability_k HB.end. -(*Section tmp. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType) - (f : R.-fker T ~> T'). - -Let tmp : exists k_ : (R.-fker _ ~> _)^nat, - forall x U, measurable U -> - f x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: sfinite_finite. Qed. - -HB.instance Definition _ := - @isSFiniteKernel.Build d d' T T' R f tmp. -End tmp.*) - (* see measurable_prod_subset in lebesgue_integral.v; the differences between the two are: - m2 is a kernel instead of a measure @@ -1106,17 +1064,13 @@ transitivity (\sum_(n t _; exact: integral_ge0 => x _. -(* have := @measurable_fun_integral_sfinite_kernel _ _ _ Y R la. - rewrite /=.*) - rewrite /=. rewrite [X in measurable_fun _ X](_ : _ = fun x => \sum_(n x. rewrite ge0_integral_measure_series//. exact/measurable_fun_prod1. apply: ge0_emeasurable_fun_sum => //. - move=> k x. - by apply: integral_ge0. + by move=> k x; exact: integral_ge0. move=> k. apply: measurable_fun_fubini_tonelli_F => //=. apply: finite_measure_sigma_finite. @@ -1158,9 +1112,7 @@ transitivity (\int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y \int[[t rewrite ge0_integral_measure_series//. exact/measurable_fun_prod2. rewrite /=. -transitivity ( - \int[la tt]_y \int[mseries (fun i : nat => mu_ i tt) 0]_x f (x, y) -). +transitivity (\int[la tt]_y \int[mseries (fun i : nat => mu_ i tt) 0]_x f (x, y)). apply eq_measure_integral => A mA _ /=. by rewrite la_E. apply eq_integral => y _. @@ -1278,47 +1230,39 @@ HB.instance Definition _ t := isSFinite.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. End sfkadd. -Lemma measurable_eq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') - (R : realType) (f : R.-ker T ~> T') k : +Section kernel_measurable_preimage. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variable R : realType. + +Lemma measurable_eq_cst (f : R.-ker T ~> T') k : measurable [set t | f t setT == k]. Proof. rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set k]); last first. by apply/seteqP; split => t/= /eqP. -rewrite /=. -have := measurable_kernel f setT measurableT. -rewrite /=. -move/(_ measurableT [set k]). -rewrite setTI. -exact. +have /(_ measurableT [set k]) := measurable_kernel f setT measurableT. +by rewrite setTI; exact. Qed. -Lemma measurable_neq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') - (R : realType) (f : R.-ker T ~> T') k : measurable [set t | f t setT != k]. +Lemma measurable_neq_cst (f : R.-ker T ~> T') k : + measurable [set t | f t setT != k]. Proof. -rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` (setT `\` [set k])); last first. - apply/seteqP; split => t/=. - by move/eqP; tauto. - by move=> []? /eqP; tauto. -rewrite /=. -have := measurable_kernel f setT measurableT. -rewrite /=. -move/(_ measurableT (setT `\` [set k])). -rewrite setTI. -apply => //. -exact: measurableD. +rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set~ k]); last first. + by apply/seteqP; split => t /eqP. +have /(_ measurableT [set~ k]) := measurable_kernel f setT measurableT. +by rewrite setTI; apply => //; exact: measurableC. Qed. -Lemma measurable_fun_eq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') - (R : realType) (f : R.-ker T ~> T') k : measurable_fun [set: T] (fun b : T => f b setT == k). +End kernel_measurable_preimage. + +Lemma measurable_fun_eq_cst (d d' : _) (T : measurableType d) + (T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k : + measurable_fun setT (fun t => f t setT == k). Proof. -move=> _ /= B mB. -rewrite setTI. +move=> _ /= B mB; rewrite setTI. have [/eqP->|/eqP->|/eqP->|/eqP->] := set_boolE B. - exact: measurable_eq_cst. - rewrite (_ : _ @^-1` _ = [set b | f b setT != k]); last first. - apply/seteqP; split => t/=. - by move/negbT. - by move/negbTE. + by apply/seteqP; split => [t /negbT//|t /negbTE]. exact: measurable_neq_cst. - by rewrite preimage_set0. - by rewrite preimage_setT. @@ -1326,7 +1270,8 @@ Qed. Section mnormalize. Variables (d d' : _) (T : measurableType d) (Y : measurableType d'). -Variables (R : realType) (f : T -> {measure set Y -> \bar R}) (P : probability Y R). +Variables (R : realType) (f : T -> {measure set Y -> \bar R}). +Variable P : probability Y R. Definition mnormalize t U := let evidence := f t setT in @@ -1335,8 +1280,7 @@ Definition mnormalize t U := Let mnormalize0 t : mnormalize t set0 = 0. Proof. -rewrite /mnormalize; case: ifPn => // _. -by rewrite measure0 mul0e. +by rewrite /mnormalize; case: ifPn => // _; rewrite measure0 mul0e. Qed. Let mnormalize_ge0 t U : 0 <= mnormalize t U. @@ -1347,14 +1291,14 @@ Proof. move=> F mF tF mUF; rewrite /mnormalize/=. case: ifPn => [_|_]. exact: measure_semi_sigma_additive. -rewrite (_ : (fun n => _) = ((fun n=> \sum_(0 <= i < n) f t (F i)) \* +rewrite (_ : (fun n => _) = ((fun n => \sum_(0 <= i < n) f t (F i)) \* cst ((fine (f t setT))^-1)%:E)); last first. by apply/funext => n; rewrite -ge0_sume_distrl. by apply: ereal_cvgMr => //; exact: measure_semi_sigma_additive. Qed. -HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ - (mnormalize t) (mnormalize0 t) (mnormalize_ge0 t) (@mnormalize_sigma_additive t). +HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ (mnormalize t) + (mnormalize0 t) (mnormalize_ge0 t) (@mnormalize_sigma_additive t). Lemma mnormalize1 t : mnormalize t setT = 1. Proof. @@ -1362,8 +1306,7 @@ rewrite /mnormalize; case: ifPn; first by rewrite probability_setT. rewrite negb_or => /andP[ft0 ftoo]. have ? : f t setT \is a fin_num. by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. -rewrite -{1}(@fineK _ (f t setT))//. -by rewrite -EFinM divrr// ?unitfE fine_eq0. +by rewrite -{1}(@fineK _ (f t setT))// -EFinM divrr// ?unitfE fine_eq0. Qed. HB.instance Definition _ t := @@ -1383,42 +1326,39 @@ Variable P : probability Y R. Let measurable_fun_knormalize U : measurable U -> measurable_fun setT (knormalize P ^~ U). Proof. -move=> mU. -rewrite /knormalize/= /mnormalize /=. +move=> mU; rewrite /knormalize/= /mnormalize /=. rewrite (_ : (fun _ => _) = (fun x => - if f x [set: Y] == 0 then P U else if f x [set: Y] == +oo then P U - else f x U * ((fine (f x [set: Y]))^-1)%:E)); last first. + if f x setT == 0 then P U else if f x setT == +oo then P U + else f x U * ((fine (f x setT))^-1)%:E)); last first. apply/funext => x; case: ifPn => [/orP[->//|->]|]. by case: ifPn. by rewrite negb_or=> /andP[/negbTE -> /negbTE ->]. -apply: measurable_fun_if000 => //. +apply: measurable_fun_if => //. - exact: measurable_fun_eq_cst. - exact: measurable_fun_cst. -- apply: measurable_fun_if000 => //. - + rewrite setTI. +- apply: measurable_fun_if => //. + + rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]); last first. + by apply/seteqP; split => [x /negbT//|x /negbTE]. exact: measurable_neq_cst. + exact: measurable_fun_eq_cst. + exact: measurable_fun_cst. + apply: emeasurable_funM. - have := (measurable_kernel f U mU). - by apply: measurable_funS => //. + by have := measurable_kernel f U mU; exact: measurable_funS. apply/EFin_measurable_fun. - rewrite /=. - apply: (measurable_fun_comp_new (F := [set r : R | r != 0%R])) => //. - exact: open_measurable. - move=> /= r [t] [] [_ H1] H2 H3. + apply: (measurable_fun_comp' (F := [set r : R | r != 0%R])) => //. + * exact: open_measurable. + * move=> /= r [t] [] [_ H1] H2 H3. apply/eqP => H4; subst r. move/eqP : H4. - rewrite fine_eq0 ?(negbTE H1)//. + rewrite fine_eq0 ?H1//. rewrite ge0_fin_numE//. by rewrite lt_neqAle leey H2. - apply: open_continuous_measurable_fun => //. - apply/in_setP => x /= x0. - by apply: inv_continuous. - apply: measurable_fun_comp => /=. - exact: measurable_fun_fine. - have := (measurable_kernel f setT measurableT). - by apply: measurable_funS => //. + * apply: open_continuous_measurable_fun => //. + apply/in_setP => x /= x0. + by apply: inv_continuous. + * apply: measurable_fun_comp => /=. + exact: measurable_fun_fine. + by have := measurable_kernel f _ measurableT; exact: measurable_funS. Qed. HB.instance Definition _ := isKernel.Build _ _ _ _ R (knormalize P) @@ -1429,8 +1369,7 @@ Proof. rewrite /knormalize/= /mnormalize. case: ifPn => [_|]; first by rewrite probability_setT. rewrite negb_or => /andP[fx0 fxoo]. -have ? : f x [set: _] \is a fin_num. - by rewrite ge0_fin_numE// lt_neqAle fxoo/= leey. +have ? : f x setT \is a fin_num by rewrite ge0_fin_numE// lt_neqAle fxoo/= leey. rewrite -{1}(@fineK _ (f x setT))//=. by rewrite -EFinM divrr// ?lte_fin ?ltr1n// ?unitfE fine_eq0. Qed. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index 12a615e58a..0de50bd9db 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -33,13 +33,16 @@ Local Open Scope ring_scope. Local Open Scope ereal_scope. (* TODO: PR *) +Lemma setT0 (T1 : pointedType) : setT != set0 :> set T1. +Proof. by apply/eqP => /seteqP[] /(_ point) /(_ Logic.I). Qed. + Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1). -Lemma measurable_fun_swap d (X : measurableType d) : measurable_fun [set: X * X] (swap (T2:=X)). +Lemma measurable_fun_swap d d' (X : measurableType d) (Y : measurableType d') : + measurable_fun [set: X * Y] (@swap X Y). Proof. -apply/prod_measurable_funP => /=; split. - exact: measurable_fun_snd. -exact: measurable_fun_fst. +by apply/prod_measurable_funP => /=; split; + [exact: measurable_fun_snd|exact: measurable_fun_fst]. Qed. Lemma onem1' (R : numDomainType) (p : R) : (p + `1- p = 1)%R. @@ -89,8 +92,7 @@ Lemma mscoreE t U : mscore t U = if U == set0 then 0 else `| (f t)%:E |. Proof. rewrite /mscore/= /mscale/=; have [->|->] := set_unit U. by rewrite eqxx diracE in_set0 mule0. -rewrite diracE in_setT mule1 ifF// ?normr_id//. -by apply/negbTE/set0P; exists tt. +by rewrite diracE in_setT mule1 (negbTE (setT0 _)) normr_id. Qed. Lemma measurable_fun_mscore U : measurable_fun setT f -> @@ -264,7 +266,7 @@ move=> /= mcU; rewrite /kiteT. rewrite (_ : (fun _ => _) = (fun x => if x.2 then k x.1 U else [the {measure set Y -> \bar R} of mzero] U)); last first. by apply/funext => -[t b]/=; case: ifPn. -apply: (@measurable_fun_if _ _ _ _ (k ^~ U) (fun=> mzero U)). +apply: (@measurable_fun_if_pair _ _ _ _ (k ^~ U) (fun=> mzero U)). exact/measurable_kernel. exact: measurable_fun_cst. Qed. @@ -321,7 +323,7 @@ move=> /= mcU; rewrite /kiteF. rewrite (_ : (fun x => _) = (fun x => if x.2 then [the measure _ _ of mzero] U else k x.1 U)); last first. apply/funext => -[t b]/=. by rewrite if_neg//; case: ifPn. -apply: (@measurable_fun_if _ _ _ _ (fun=> mzero U) (k ^~ U)). +apply: (@measurable_fun_if_pair _ _ _ _ (fun=> mzero U) (k ^~ U)). exact: measurable_fun_cst. exact/measurable_kernel. Qed. @@ -398,39 +400,6 @@ Definition kite := End ite. -Section insn1. -Variables (R : realType) (d : _) (X : measurableType d). - -Definition score (f : X -> R) (mf : measurable_fun setT f) := - [the R.-sfker X ~> _ of kscore mf]. - -End insn1. - -Section insn1_lemmas. -Variables (R : realType) (d : _) (T : measurableType d). - -Lemma scoreE (t : T) (U : set bool) (n : nat) (b : bool) - (f : R -> R) - (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) - (mf : measurable_fun setT f) : - score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) - (t, b, n%:R) (curry (snd \o fst) (t, b) @^-1` U) = - (f n%:R)%:E * \d_b U. -Proof. -set x := score _. -rewrite /score/= /kscore/= mscoreE diracE. -have [U0|U0] := set_unit ((fun=> b) @^-1` U). -- rewrite U0 eqxx memNset ?mule0// => Ub. - by move: U0 => /seteqP[/(_ tt)] /(_ Ub). -- rewrite U0 setT_unit ifF//; last first. - by apply/negbTE/negP => /eqP/seteqP[/(_ tt erefl)]. - rewrite /= mem_set//; last first. - by move: U0 => /seteqP[_]/(_ tt)/=; exact. - by rewrite mule1 ger0_norm// f0. -Qed. - -End insn1_lemmas. - Section insn2. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variable R : realType. @@ -544,6 +513,70 @@ Qed. End letin_return. +Section insn1. +Variables (R : realType) (d : _) (X : measurableType d). + +Definition score (f : X -> R) (mf : measurable_fun setT f) := + [the R.-sfker X ~> _ of kscore mf]. + +End insn1. + +Section insn1_lemmas. +Variables (R : realType) (d : _) (T : measurableType d). + +Lemma scoreE' d' (T' : measurableType d') d2 (T2 : measurableType d2) (U : set T') + (g : R.-sfker [the measurableType _ of (T2 * unit)%type] ~> T') r fh (mh : measurable_fun setT fh) : + (score mh \; g) r U = + g (r, tt) U * `|fh r|%:E. +Proof. +rewrite [in LHS]/score [in LHS]/=. +rewrite /kcomp. +rewrite /kscore. +rewrite [in LHS]/=. +rewrite ge0_integral_mscale//=. +rewrite integral_dirac// normr_id muleC. +by rewrite indicE in_setT mul1e. +Qed. + +Lemma scoreE (t : T) (U : set bool) (n : nat) (b : bool) + (f : R -> R) + (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) + (mf : measurable_fun setT f) : + score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) + (t, b, n%:R) (curry (snd \o fst) (t, b) @^-1` U) = + (f n%:R)%:E * \d_b U. +Proof. +transitivity (letin ( + score (measurable_fun_comp mf (measurable_fun_snd (T2:=Real_sort__canonical__measure_Measurable R))) + ) ( + ret R (@measurable_fun_id _ _ _) +) (t, b, n%:R) (curry (snd \o fst) (t, b) @^-1` U)). + rewrite letin_kret//. + rewrite /curry/=. + rewrite preimage_cst. + by case: ifPn => //. +rewrite /letin. +unlock. +rewrite scoreE'//. +rewrite retE. +by rewrite ger0_norm// ?f0//= muleC. +Qed. + +(* example of property *) +Lemma score_score (f : R -> R) (g : R * unit -> R) (mf : measurable_fun setT f) (mg : measurable_fun setT g) x U : + letin (score mf) (score mg) x U = if U == set0 then 0 else `|g (x, tt)|%:E * `|f x|%:E. +Proof. +rewrite {1}/letin. +unlock. +rewrite scoreE'//=. +rewrite /mscale/= diracE !normr_id. +have [->|->]:= set_unit U. + by rewrite eqxx in_set0 mule0 mul0e. +by rewrite in_setT mule1 (negbTE (setT0 _)). +Qed. + +End insn1_lemmas. + Section letin_ite. Variables (R : realType) (d d2 d3 : _) (T : measurableType d) (T2 : measurableType d2) (Z : measurableType d3) @@ -731,7 +764,7 @@ Variable P : probability mbool R. Import Notations. -Definition staton_bus_annotated : R.-sfker T ~> mbool := +Definition staton_bus_annotated : R.-pker T ~> mbool := normalize (letin (sample (bernoulli p27) : _.-sfker T ~> mbool) (letin From b469dbb34cf2c2b065440775e3a12a53be3b7805 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Thu, 8 Sep 2022 10:13:33 +0900 Subject: [PATCH 37/42] various minor simplifications and generalizations --- theories/kernel.v | 1273 ++++++++++++++++++++---------------------- theories/prob_lang.v | 175 ++++-- 2 files changed, 735 insertions(+), 713 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index ec15c77e3d..646cc75c79 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -8,14 +8,21 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (******************************************************************************) (* Kernels *) (* *) -(* R.-ker X ~> Y == kernel *) -(* R.-sfker X ~> Y == s-finite kernel *) -(* R.-fker X ~> Y == finite kernel *) -(* R.-pker X ~> Y == probability kernel *) -(* sum_of_kernels == *) -(* l \; k == composition of kernels *) -(* kdirac mf == kernel defined by a measurable function *) -(* kadd k1 k2 == *) +(* This file provides a formation of kernels and extends the theory of *) +(* measure with, e.g., Fubini's theorem for s-finite measures. *) +(* *) +(* R.-ker X ~> Y == kernel *) +(* kseries == countable sum of kernels *) +(* R.-sfker X ~> Y == s-finite kernel *) +(* R.-fker X ~> Y == finite kernel *) +(* R.-pker X ~> Y == probability kernel *) +(* finite_measure mu == the measure mu is finite *) +(* sfinite_measure mu == the measure my is s-finite *) +(* kprobability m == kernel defined by a probability measure *) +(* kdirac mf == kernel defined by a measurable function *) +(* kadd k1 k2 == lifting of the addition of measures to kernels *) +(* mnormalize f == normalization of a kernel to a probability *) +(* l \; k == composition of kernels *) (******************************************************************************) Set Implicit Arguments. @@ -256,12 +263,27 @@ apply: measurableU. exact: emeasurable_itv_ninfty_bnd. exact: emeasurable_itv_bnd_pinfty. Qed. + +Section fubini_tonelli. (* TODO: move to lebesgue_integral.v *) +Local Open Scope ereal_scope. +Variables (d1 d2 : measure_display). +Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). +Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). +Hypotheses (sm1 : sigma_finite setT m1) (sm2 : sigma_finite setT m2). +Variables (f : T1 * T2 -> \bar R) (f0 : forall xy, 0 <= f xy). +Variables (mf : measurable_fun setT f). + +Lemma fubini_tonelli : + \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). +Proof. by rewrite -fubini_tonelli1// fubini_tonelli2. Qed. + +End fubini_tonelli. (*/ PR*) -Reserved Notation "R .-ker X ~> Y" (at level 42). -Reserved Notation "R .-fker X ~> Y" (at level 42). -Reserved Notation "R .-sfker X ~> Y" (at level 42). -Reserved Notation "R .-pker X ~> Y" (at level 42). +Reserved Notation "R .-ker X ~> Y" (at level 42, format "R .-ker X ~> Y"). +Reserved Notation "R .-fker X ~> Y" (at level 42, format "R .-fker X ~> Y"). +Reserved Notation "R .-sfker X ~> Y" (at level 42, format "R .-sfker X ~> Y"). +Reserved Notation "R .-pker X ~> Y" (at level 42, format "R .-pker X ~> Y"). HB.mixin Record isKernel d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := @@ -275,42 +297,35 @@ Notation "R .-ker X ~> Y" := (kernel X Y R). Arguments measurable_kernel {_ _ _ _ _} _. -Section sum_of_kernels. +Section kseries. Variables (d d' : measure_display) (R : realType). Variables (X : measurableType d) (Y : measurableType d'). Variable k : (R.-ker X ~> Y)^nat. -Definition sum_of_kernels : X -> {measure set Y -> \bar R} := +Definition kseries : X -> {measure set Y -> \bar R} := fun x => [the measure _ _ of mseries (k ^~ x) 0]. -Lemma kernel_measurable_fun_sum_of_kernels (U : set Y) : +Lemma measurable_fun_kseries (U : set Y) : measurable U -> - measurable_fun setT (sum_of_kernels ^~ U). + measurable_fun setT (kseries ^~ U). Proof. -move=> mU; rewrite /sum_of_kernels /= /mseries. -rewrite [X in measurable_fun _ X](_ : _ = - (fun x => elim_sup (fun n => \sum_(0 <= i < n) k i x U))); last first. - apply/funext => x; rewrite -lim_mkord is_cvg_elim_supE. - by rewrite -lim_mkord. - exact: is_cvg_nneseries. -apply: measurable_fun_elim_sup => n. -by apply: emeasurable_fun_sum => *; exact/measurable_kernel. +move=> mU; rewrite /kseries /= /mseries. +by apply: ge0_emeasurable_fun_sum => // n; apply/measurable_kernel. Qed. HB.instance Definition _ := - isKernel.Build _ _ _ _ _ sum_of_kernels - kernel_measurable_fun_sum_of_kernels. + isKernel.Build _ _ _ _ _ kseries measurable_fun_kseries. -End sum_of_kernels. +End kseries. -Lemma integral_sum_of_kernels +Lemma integral_kseries (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType) (k : (R.-ker X ~> Y)^nat) (f : Y -> \bar R) x : (forall y, 0 <= f y) -> measurable_fun setT f -> - \int[sum_of_kernels k x]_y (f y) = \sum_(i f0 mf; rewrite /sum_of_kernels/= ge0_integral_measure_series. +by move=> f0 mf; rewrite /kseries/= ge0_integral_measure_series. Qed. Section measure_fam_uub. @@ -371,8 +386,7 @@ HB.mixin Record isSFinite d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { sfinite : exists s : (R.-fker X ~> Y)^nat, - forall x U, measurable U -> - k x U = [the measure _ _ of mseries (s ^~ x) 0] U }. + forall x U, measurable U -> k x U = kseries s x U }. #[short(type=sfinite_kernel)] HB.structure Definition SFiniteKernel @@ -447,8 +461,9 @@ HB.end. (* see measurable_prod_subset in lebesgue_integral.v; the differences between the two are: - - m2 is a kernel instead of a measure - - m2D_bounded holds for all x *) + - m2 is a kernel instead of a measure (the proof uses the + measurability of each measure of the family) + - as a consequence, m2D_bounded holds for all x *) Section measurable_prod_subset_kernel. Variables (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). @@ -515,19 +530,14 @@ Lemma measurable_fun_xsection_finite_kernel A : A \in measurable -> measurable_fun setT (phi A). Proof. move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. -move=> X mX. -rewrite /B/=; split => //. -rewrite /phi. +move=> /= X mX; rewrite /B/=; split => //; rewrite /phi. rewrite -(_ : (fun x => mrestr (m2 x) measurableT (xsection X x)) = - (fun x => (m2 x) (xsection X x)))//; last first. + (fun x => m2 x (xsection X x)))//; last first. by apply/funext => x//=; rewrite /mrestr setIT. -apply measurable_prod_subset_xsection_kernel => //. -move=> x. -have [r hr] := measure_uub m2. -exists r => Y mY. -apply: (le_lt_trans _ (hr x)) => //. -rewrite /mrestr. -by apply le_measure => //; rewrite inE//; exact: measurableI. +apply measurable_prod_subset_xsection_kernel => // x. +have [r hr] := measure_uub m2; exists r => Y mY. +rewrite (le_lt_trans _ (hr x)) // /mrestr /= setIT. +by apply: le_measure => //; rewrite inE. Qed. End measurable_fun_xsection_finite_kernel. @@ -566,11 +576,11 @@ rewrite (_ : (fun x => _) = - by move=> y _ m n mn; rewrite lee_fin; exact/lefP/ndk_. apply: measurable_fun_elim_sup => n. rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y - (\sum_(r <- fset_set (range (k_ n))) + (\sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR is merged*) r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. rewrite [X in measurable_fun _ X](_ : _ = (fun x => - \sum_(r <- fset_set (range (k_ n))) + \sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR is merged*) (\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first. apply/funext => x; rewrite -ge0_integral_sum//. - by apply: eq_integral => y _; rewrite sumEFin. @@ -632,749 +642,702 @@ Arguments measurable_fun_xsection_integral {_ _ _ _ _} l k. Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} l k. Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} l k. -Section kcomp_def. -Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) - (Z : measurableType d3) (R : realType). -Variable l : X -> {measure set Y -> \bar R}. -Variable k : (X * Y)%type -> {measure set Z -> \bar R}. +(*HB.mixin Record isFiniteMeasure d (R : numFieldType) (T : semiRingOfSetsType d) + (mu : set T -> \bar R) := { + finite_measure : mu setT < +oo +}. -Definition kcomp x U := \int[l x]_y k (x, y) U. +#[short(type=fmeasure)] +HB.structure Definition FiniteMeasure d (R : realFieldType) + (T : semiRingOfSetsType d) := + {mu of isMeasure d R T mu & isFiniteMeasure d R T mu}. -End kcomp_def. +Notation "{ 'fmeasure' 'set' T '->' '\bar' R }" := (@fmeasure _ R T) + (at level 36, T, R at next level, + format "{ 'fmeasure' 'set' T '->' '\bar' R }") : ring_scope.*) -Section kcomp_is_measure. -Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) - (Z : measurableType d3) (R : realType). -Variable l : R.-ker X ~> Y. -Variable k : R.-ker [the measurableType _ of (X * Y)%type] ~> Z. +Definition finite_measure d (T : measurableType d) (R : realType) + (mu : set T -> \bar R) := + mu setT < +oo. -Local Notation "l \; k" := (kcomp l k). +Definition sfinite_measure d (T : measurableType d) (R : realType) + (mu : set T -> \bar R) := + exists mu_ : {measure set T -> \bar R}^nat, + (forall n, finite_measure (mu_ n)) /\ + (forall U, measurable U -> mu U = mseries mu_ 0 U). -Let kcomp0 x : (l \; k) x set0 = 0. +Lemma finite_measure_sigma_finite d (T : measurableType d) (R : realType) + (mu : {measure set T -> \bar R}) : + finite_measure mu -> sigma_finite setT mu. Proof. -by rewrite /kcomp (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. +exists (fun i => if i \in [set 0%N] then setT else set0). + by rewrite -bigcup_mkcondr setTI bigcup_const//; exists 0%N. +move=> n; split; first by case: ifPn. +by case: ifPn => // _; rewrite ?measure0//; exact: finite_measure. Qed. -Let kcomp_ge0 x U : 0 <= (l \; k) x U. Proof. exact: integral_ge0. Qed. +Section sfinite_fubini. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). +Variables (m1 : {measure set X -> \bar R}) (sfm1 : sfinite_measure m1). +Variables (m2 : {measure set Y -> \bar R}) (sfm2 : sfinite_measure m2). +Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). +Variable (mf : measurable_fun setT f). -Let kcomp_sigma_additive x : semi_sigma_additive ((l \; k) x). +Lemma sfinite_fubini : + \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). Proof. -move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = - \int[l x]_y (\sum_(n V _. - by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. -apply/cvg_closeP; split. - by apply: is_cvg_nneseries => n _; exact: integral_ge0. -rewrite closeE// integral_sum// => n. -by have /measurable_fun_prod1 := measurable_kernel k (U n) (mU n). +have [m1_ [fm1 m1E]] := sfm1. +have [m2_ [fm2 m2E]] := sfm2. +rewrite [LHS](eq_measure_integral [the measure _ _ of mseries m1_ 0]); last first. + by move=> A mA _; rewrite m1E. +transitivity (\int[[the measure _ _ of mseries m1_ 0]]_x + \int[[the measure _ _ of mseries m2_ 0]]_y f (x, y)). + by apply eq_integral => x _; apply: eq_measure_integral => U mA _; rewrite m2E. +transitivity (\sum_(n t _; exact: integral_ge0. + rewrite [X in measurable_fun _ X](_ : _ = + fun x => \sum_(n x. + by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod1. + apply: ge0_emeasurable_fun_sum; first by move=> k x; exact: integral_ge0. + move=> k; apply: measurable_fun_fubini_tonelli_F => //=. + exact: finite_measure_sigma_finite. + apply: eq_nneseries => n _; apply eq_integral => x _. + by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod1. +transitivity (\sum_(n n _. + rewrite integral_sum(*TODO: rename to ge0_integral_sum*)//. + move=> m; apply: measurable_fun_fubini_tonelli_F => //=. + exact: finite_measure_sigma_finite. + by move=> m x _; exact: integral_ge0. +transitivity (\sum_(n n _; apply eq_nneseries => m _. + by rewrite fubini_tonelli//; exact: finite_measure_sigma_finite. +transitivity (\sum_(n n _ /=. rewrite ge0_integral_measure_series//. + by move=> y _; exact: integral_ge0. + apply: measurable_fun_fubini_tonelli_G => //=. + by apply: finite_measure_sigma_finite; exact: fm1. +transitivity (\int[[the measure _ _ of mseries m2_ 0]]_y \sum_(n n; apply: measurable_fun_fubini_tonelli_G => //=. + by apply: finite_measure_sigma_finite; exact: fm1. + by move=> n y _; exact: integral_ge0. +transitivity (\int[[the measure _ _ of mseries m2_ 0]]_y + \int[[the measure _ _ of mseries m1_ 0]]_x f (x, y)). + apply eq_integral => y _. + by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod2. +transitivity (\int[m2]_y \int[mseries m1_ 0]_x f (x, y)). + by apply eq_measure_integral => A mA _ /=; rewrite m2E. +by apply eq_integral => y _; apply eq_measure_integral => A mA _ /=; rewrite m1E. Qed. -HB.instance Definition _ x := isMeasure.Build _ R _ - ((l \; k) x) (kcomp0 x) (kcomp_ge0 x) (@kcomp_sigma_additive x). +End sfinite_fubini. +Arguments sfinite_fubini {d d' X Y R m1} _ {m2} _ f. -Definition mkcomp : X -> {measure set Z -> \bar R} := - fun x => [the measure _ _ of (l \; k) x]. +Lemma finite_kernel_finite_measure d (T : measurableType d) (R : realType) + (mu : R.-fker Datatypes_unit__canonical__measure_Measurable ~> T) : + mu tt setT < +oo. +Proof. +have [M muM] := measure_uub mu. +by rewrite /finite_measure (lt_le_trans (muM tt))// leey. +Qed. -End kcomp_is_measure. +Section kprobability. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (m : probability Y R). -Notation "l \; k" := (mkcomp l k). +Definition kprobability : X -> {measure set Y -> \bar R} := fun _ : X => m. -Module KCOMP_FINITE_KERNEL. +Let measurable_fun_kprobability U : measurable U -> + measurable_fun setT (kprobability ^~ U). +Proof. by move=> mU; exact: measurable_fun_cst. Qed. -Section kcomp_finite_kernel_kernel. -Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-fker X ~> Y) - (k : R.-ker [the measurableType _ of (X * Y)%type] ~> Z). +HB.instance Definition _ := + @isKernel.Build _ _ X Y R kprobability measurable_fun_kprobability. -Lemma measurable_fun_kcomp_finite U : - measurable U -> measurable_fun setT ((l \; k) ^~ U). -Proof. -move=> mU; apply: (measurable_fun_integral_finite_kernel _ (k ^~ U)) => //=. -exact/measurable_kernel. -Qed. +Let kprobability_prob x : kprobability x setT = 1. +Proof. by rewrite /kprobability/= probability_setT. Qed. HB.instance Definition _ := - isKernel.Build _ _ X Z R (l \; k) measurable_fun_kcomp_finite. + @isProbabilityKernel.Build _ _ X Y R kprobability kprobability_prob. -End kcomp_finite_kernel_kernel. +End kprobability. -Section kcomp_finite_kernel_finite. -Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType). -Variable l : R.-fker X ~> Y. -Variable k : R.-fker [the measurableType _ of (X * Y)%type] ~> Z. +Section kdirac. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (f : X -> Y). -Let mkcomp_finite : measure_fam_uub (l \; k). +Definition kdirac (mf : measurable_fun setT f) + : X -> {measure set Y -> \bar R} := + fun x : X => [the measure _ _ of dirac (f x)]. + +Hypothesis mf : measurable_fun setT f. + +Let measurable_fun_kdirac U : measurable U -> + measurable_fun setT (kdirac mf ^~ U). Proof. -have /measure_fam_uubP[r hr] := measure_uub k. -have /measure_fam_uubP[s hs] := measure_uub l. -apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=. -apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). - apply: ge0_le_integral => //. - - have /measurable_fun_prod1 := measurable_kernel k setT measurableT. - exact. - - exact/measurable_fun_cst. - - by move=> y _; exact/ltW/hr. -by rewrite integral_cst//= EFinM lte_pmul2l. +move=> mU; apply/EFin_measurable_fun. +by rewrite (_ : (fun x => _) = mindic R mU \o f)//; exact/measurable_fun_comp. Qed. -HB.instance Definition _ := - isFiniteFam.Build _ _ X Z R (l \; k) mkcomp_finite. +HB.instance Definition _ := isKernel.Build _ _ _ _ _ (kdirac mf) + measurable_fun_kdirac. -End kcomp_finite_kernel_finite. -End KCOMP_FINITE_KERNEL. +Let kdirac_prob x : kdirac mf x setT = 1. +Proof. by rewrite /kdirac/= diracE in_setT. Qed. -Section kcomp_sfinite_kernel. -Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType). -Variable l : R.-sfker X ~> Y. -Variable k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. +HB.instance Definition _ := isProbabilityKernel.Build _ _ _ _ _ + (kdirac mf) kdirac_prob. -Import KCOMP_FINITE_KERNEL. +End kdirac. +Arguments kdirac {d d' X Y R f}. -Lemma mkcomp_sfinite : exists k_ : (R.-fker X ~> Z)^nat, forall x U, measurable U -> - (l \; k) x U = [the measure _ _ of mseries (k_ ^~ x) O] U. -Proof. -have [k_ hk_] := sfinite k. -have [l_ hl_] := sfinite l. -pose K := [the kernel _ _ _ of sum_of_kernels k_]. -pose L := [the kernel _ _ _ of sum_of_kernels l_]. -have H1 x U : measurable U -> (l \; k) x U = (L \; K) x U. - move=> mU /=. - rewrite /kcomp /L /K /=. - (* TODO: lemma so that we can get away with a rewrite *) - transitivity (\int[ - [the measure _ _ of mseries (l_ ^~ x) 0] ]_y k (x, y) U). - by apply eq_measure_integral => A mA _; rewrite hl_. - by apply eq_integral => y _; rewrite hk_. -have H2 x U : (L \; K) x U = - \int[mseries (l_ ^~ x) 0]_y (\sum_(i - \int[mseries (l_ ^~ x) 0]_y (\sum_(i mU. - rewrite integral_sum//= => n. - have := measurable_kernel (k_ n) _ mU. - by move=> /measurable_fun_prod1; exact. -have H4 x U : measurable U -> - \sum_(i mU. - apply: eq_nneseries => i _. - rewrite integral_sum_of_kernels//. - have := measurable_kernel (k_ i) _ mU. - by move=> /measurable_fun_prod1; exact. -have H5 x U : \sum_(i i _; exact: eq_nneseries. -suff: exists k_0 : (R.-fker X ~> Z) ^nat, forall x U, - \esum_(i in setT) ((l_ i.2) \; (k_ i.1)) x U = \sum_(i [kl_ hkl_]. - exists kl_ => x U mU. - rewrite /= H1// H2 H3// H4// H5// /mseries -hkl_/=. - rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split. - rewrite -(@esum_esum _ _ _ _ _ (fun i j => (l_ j \; k_ i) x U))//. - rewrite nneseries_esum; last by move=> n _; exact: nneseries_ge0. - by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true. -rewrite /=. -have /ppcard_eqP[f] : ([set: nat] #= [set: nat * nat])%card. - by rewrite card_eq_sym; exact: card_nat2. -exists (fun i => [the _.-fker _ ~> _ of (l_ (f i).2) \; (k_ (f i).1)]) => x U. -rewrite (reindex_esum [set: nat] [set: nat * nat] f)//. -by rewrite nneseries_esum// fun_true. -Qed. +Section kadd. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k1 k2 : R.-ker X ~> Y). -Lemma measurable_fun_mkcomp_sfinite U : measurable U -> - measurable_fun setT ((l \; k) ^~ U). +Definition kadd : X -> {measure set Y -> \bar R} := + fun x => [the measure _ _ of measure_add (k1 x) (k2 x)]. + +Let measurable_fun_kadd U : measurable U -> + measurable_fun setT (kadd ^~ U). Proof. -move=> mU; apply: (measurable_fun_integral_sfinite_kernel _ (k ^~ U)) => //. -exact/measurable_kernel. +move=> mU; rewrite /kadd. +rewrite (_ : (fun _ => _) = (fun x => k1 x U + k2 x U)); last first. + by apply/funext => x; rewrite -measure_addE. +by apply: emeasurable_funD; exact/measurable_kernel. Qed. -End kcomp_sfinite_kernel. - -Module KCOMP_SFINITE_KERNEL. -Section kcomp_sfinite_kernel. -Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType). -Variable l : R.-sfker X ~> Y. -Variable k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. - HB.instance Definition _ := - isKernel.Build _ _ X Z R (l \; k) (measurable_fun_mkcomp_sfinite l k). + @isKernel.Build _ _ _ _ _ kadd measurable_fun_kadd. +End kadd. -#[export] -HB.instance Definition _ := - isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). +Section fkadd. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k1 k2 : R.-fker X ~> Y). -End kcomp_sfinite_kernel. -End KCOMP_SFINITE_KERNEL. -HB.export KCOMP_SFINITE_KERNEL. +Let kadd_finite_uub : measure_fam_uub (kadd k1 k2). +Proof. +have [f1 hk1] := measure_uub k1; have [f2 hk2] := measure_uub k2. +exists (f1 + f2)%R => x; rewrite /kadd /=. +rewrite -/(measure_add (k1 x) (k2 x)). +by rewrite measure_addE EFinD; exact: lte_add. +Qed. -(* pollard? *) -Section measurable_fun_integral_kernel'. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d') - (R : realType). -Variables (l : X -> {measure set Y -> \bar R}) - (k : Y -> \bar R) - (k_ : ({nnsfun Y >-> R}) ^nat) - (ndk_ : nondecreasing_seq (k_ : (Y -> R)^nat)) - (k_k : forall z, setT z -> EFin \o (k_ ^~ z) --> k z). +HB.instance Definition _ t := + isFiniteFam.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. +End fkadd. -Let p : (X * Y -> R)^nat := fun n xy => k_ n xy.2. +Section sfkadd. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k1 k2 : R.-sfker X ~> Y). -Let p_ge0 n x : (0 <= p n x)%R. Proof. by []. Qed. +Let sfinite_kadd : exists k_ : (R.-fker _ ~> _)^nat, + forall x U, measurable U -> + kadd k1 k2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. +Proof. +have [f1 hk1] := sfinite k1. +have [f2 hk2] := sfinite k2. +exists (fun n => [the finite_kernel _ _ _ of kadd (f1 n) (f2 n)]) => x U mU. +rewrite /kadd/=. +rewrite -/(measure_add (k1 x) (k2 x)) measure_addE. +rewrite /mseries. +rewrite hk1//= hk2//= /mseries. +rewrite -nneseriesD//. +apply: eq_nneseries => n _. +by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE. +Qed. -HB.instance Definition _ n := @IsNonNegFun.Build _ R (p n) (p_ge0 n). +HB.instance Definition _ t := + isSFinite.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. +End sfkadd. -Let mp n : measurable_fun setT (p n). +Section kernel_measurable_preimage. +Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). +Variable R : realType. + +Lemma measurable_eq_cst (f : R.-ker T ~> T') k : + measurable [set t | f t setT == k]. Proof. -rewrite /p => _ /= B mB; rewrite setTI. -have mk_n : measurable_fun setT (k_ n) by []. -rewrite (_ : _ @^-1` _ = setT `*` (k_ n @^-1` B)); last first. - by apply/seteqP; split => xy /=; tauto. -apply: measurableM => //. -have := mk_n measurableT _ mB. -by rewrite setTI. +rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set k]); last first. + by apply/seteqP; split => t/= /eqP. +have /(_ measurableT [set k]) := measurable_kernel f setT measurableT. +by rewrite setTI; exact. Qed. -HB.instance Definition _ n := @IsMeasurableFun.Build _ _ R (p n) (mp n). - -Let fp n : finite_set (range (p n)). +Lemma measurable_neq_cst (f : R.-ker T ~> T') k : + measurable [set t | f t setT != k]. Proof. -have := @fimfunP _ _ (k_ n). -suff : range (k_ n) = range (p n) by move=> <-. -by apply/seteqP; split => r [y ?] <-; [exists (point, y)|exists y.2]. +rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set~ k]); last first. + by apply/seteqP; split => t /eqP. +have /(_ measurableT [set~ k]) := measurable_kernel f setT measurableT. +by rewrite setTI; apply => //; exact: measurableC. Qed. -HB.instance Definition _ n := @FiniteImage.Build _ _ (p n) (fp n). +End kernel_measurable_preimage. -Lemma measurable_fun_preimage_integral : - (forall n r, measurable_fun setT (fun x => l x (k_ n @^-1` [set r]))) -> - measurable_fun setT (fun x => \int[l x]_z k z). +Lemma measurable_fun_eq_cst (d d' : _) (T : measurableType d) + (T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k : + measurable_fun setT (fun t => f t setT == k). Proof. -move=> h. -apply: (measurable_fun_xsection_integral l (fun xy => k xy.2) - (fun n => [the {nnsfun _ >-> _} of p n])) => /=. -- by rewrite /p => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_. -- by move=> [x y]; exact: k_k. -- move=> n r _ /= B mB. - have := h n r measurableT B mB. - rewrite !setTI. - suff : ((fun x => l x (k_ n @^-1` [set r])) @^-1` B) = - ((fun x => l x (xsection (p n @^-1` [set r]) x)) @^-1` B) by move=> ->. - apply/seteqP; split => x/=. - suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. - by apply/seteqP; split; move=> y/=; - rewrite /xsection/= /p /preimage/= inE/=. - suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. - by apply/seteqP; split; move=> y/=; rewrite /xsection/= /p /preimage/= inE/=. +move=> _ /= B mB; rewrite setTI. +have [/eqP->|/eqP->|/eqP->|/eqP->] := set_boolE B. +- exact: measurable_eq_cst. +- rewrite (_ : _ @^-1` _ = [set b | f b setT != k]); last first. + by apply/seteqP; split => [t /negbT//|t /negbTE]. + exact: measurable_neq_cst. +- by rewrite preimage_set0. +- by rewrite preimage_setT. Qed. -End measurable_fun_integral_kernel'. +Section mnormalize. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (f : X -> {measure set Y -> \bar R}). +Variable P : probability Y R. -Lemma measurable_fun_integral_kernel - (d d' d3 : _) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) - (l : R.-ker [the measurableType _ of (X * Y)%type] ~> Z) c - (k : Z -> \bar R) (k0 : forall z, True -> 0 <= k z) (mk : measurable_fun setT k) : - measurable_fun setT (fun y => \int[l (c, y)]_z k z). +Definition mnormalize x U := + let evidence := f x [set: Y] in + if (evidence == 0) || (evidence == +oo) then P U + else f x U * (fine evidence)^-1%:E. + +Let mnormalize0 x : mnormalize x set0 = 0. Proof. -have [k_ [ndk_ k_k]] := approximation measurableT mk k0. -apply: (measurable_fun_preimage_integral ndk_ k_k) => n r. -have := measurable_kernel l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). -by move=> /measurable_fun_prod1; exact. +by rewrite /mnormalize; case: ifPn => // _; rewrite measure0 mul0e. Qed. -Section integral_kcomp. +Let mnormalize_ge0 x U : 0 <= mnormalize x U. +Proof. by rewrite /mnormalize; case: ifPn => //; case: ifPn. Qed. -Let integral_kcomp_indic d d' d3 (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) - x (E : set _) (mE : measurable E) : - \int[(l \; k) x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). +Lemma mnormalize_sigma_additive x : semi_sigma_additive (mnormalize x). Proof. -rewrite integral_indic//= /kcomp. -by apply eq_integral => y _; rewrite integral_indic. +move=> F mF tF mUF; rewrite /mnormalize/=. +case: ifPn => [_|_]. + exact: measure_semi_sigma_additive. +rewrite (_ : (fun n => _) = ((fun n => \sum_(0 <= i < n) f x (F i)) \* + cst ((fine (f x setT))^-1)%:E)); last first. + by apply/funext => n; rewrite -ge0_sume_distrl. +by apply: ereal_cvgMr => //; exact: measure_semi_sigma_additive. Qed. -Let integral_kcomp_nnsfun d d' d3 (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) - x (f : {nnsfun Z >-> R}) : - \int[(l \; k) x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). -Proof. -under [in LHS]eq_integral do rewrite fimfunE -sumEFin. -rewrite ge0_integral_sum//; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM. - have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. -under [in RHS]eq_integral. - move=> y _. - under eq_integral. - move=> z _. - rewrite fimfunE -sumEFin. - over. - rewrite /= ge0_integral_sum//; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM. - have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. - under eq_bigr. - move=> r _. - rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. - by move=> r0; rewrite preimage_nnfun0. - rewrite integral_indic// setIT. - over. - over. -rewrite /= ge0_integral_sum//; last 2 first. - - move=> r; apply: measurable_funeM. - have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). - by move=> /measurable_fun_prod1; exact. - - move=> n y _. - have := @mulemu_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). - by apply; exact: preimage_nnfun0. -apply eq_bigr => r _. -rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. - exact: preimage_nnfun0. -rewrite /= integral_kcomp_indic; last exact/measurable_sfunP. -rewrite (@integralM_0ifneg _ _ _ _ _ _ (fun r t => k (x, t) (f @^-1` [set r])))//; last 2 first. - move=> r0. - apply/funext => y. - by rewrite preimage_nnfun0// measure0. - have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). - by move/measurable_fun_prod1; exact. -congr (_ * _); apply eq_integral => y _. -by rewrite integral_indic// setIT. -Qed. +HB.instance Definition _ x := isMeasure.Build _ _ _ (mnormalize x) + (mnormalize0 x) (mnormalize_ge0 x) (@mnormalize_sigma_additive x). -Lemma integral_kcomp d d' d3 (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) - x f : (forall z, 0 <= f z) -> measurable_fun setT f -> - \int[(l \; k) x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). +Lemma mnormalize1 x : mnormalize x setT = 1. Proof. -move=> f0 mf. -have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). -transitivity (\int[(l \; k) x]_z (lim (EFin \o (f_^~ z)))). - apply/eq_integral => z _. - apply/esym/cvg_lim => //=. - exact: f_f. -rewrite monotone_convergence//; last 3 first. - by move=> n; apply/EFin_measurable_fun. - by move=> n z _; rewrite lee_fin. - by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin. -rewrite (_ : (fun _ => _) = (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first. - by apply/funext => n; rewrite integral_kcomp_nnsfun. -transitivity (\int[l x]_y lim (fun n => \int[k (x, y)]_z (f_ n z)%:E)). - rewrite -monotone_convergence//; last 3 first. - move=> n. - apply: measurable_fun_integral_kernel => //. - - by move=> z; rewrite lee_fin. - - by apply/EFin_measurable_fun. - - move=> n y _. - by apply integral_ge0 => // z _; rewrite lee_fin. - - move=> y _ a b ab. - apply: ge0_le_integral => //. - + by move=> z _; rewrite lee_fin. - + exact/EFin_measurable_fun. - + by move=> z _; rewrite lee_fin. - + exact/EFin_measurable_fun. - + move: ab => /ndf_ /lefP ab z _. - by rewrite lee_fin. -apply eq_integral => y _. -rewrite -monotone_convergence//; last 3 first. - move=> n; exact/EFin_measurable_fun. - by move=> n z _; rewrite lee_fin. - by move=> z _ a b /ndf_ /lefP; rewrite lee_fin. -apply eq_integral => z _. -apply/cvg_lim => //. -exact: f_f. +rewrite /mnormalize; case: ifPn; first by rewrite probability_setT. +rewrite negb_or => /andP[ft0 ftoo]. +have ? : f x setT \is a fin_num. + by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. +by rewrite -{1}(@fineK _ (f x setT))// -EFinM divrr// ?unitfE fine_eq0. Qed. -End integral_kcomp. +HB.instance Definition _ x := + isProbability.Build _ _ _ (mnormalize x) (mnormalize1 x). -Definition finite_measure d (T : measurableType d) (R : realType) (mu : set T -> \bar R) := - mu setT < +oo. +End mnormalize. -Lemma finite_kernel_finite_measure d (T : measurableType d) (R : realType) - (mu : R.-fker Datatypes_unit__canonical__measure_Measurable ~> T) : - finite_measure (mu tt). -Proof. -have [M muM] := measure_uub mu. -by rewrite /finite_measure (lt_le_trans (muM tt))// leey. -Qed. +Section knormalize. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (f : R.-ker X ~> Y). -Lemma finite_measure_sigma_finite d (T : measurableType d) (R : realType) - (mu : {measure set T -> \bar R}) : - finite_measure mu -> sigma_finite setT mu. -Proof. -rewrite /finite_measure => muoo. -exists (fun i => if i \in [set 0%N] then setT else set0). - by rewrite -bigcup_mkcondr setTI bigcup_const//; exists 0%N. -move=> n; split; first by case: ifPn. -by case: ifPn => // _; rewrite measure0. -Qed. +Definition knormalize (P : probability Y R) := + fun t => [the measure _ _ of mnormalize f P t]. -Section finite_fubini. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Variables (mu : {measure set X -> \bar R}) (fmu : finite_measure mu). -Variables (la : {measure set Y -> \bar R}) (fla : finite_measure la). -Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). -Variables (mf : measurable_fun setT f). +Variable P : probability Y R. -Lemma finite_fubini : - \int[mu]_x \int[la]_y f (x, y) = \int[la]_y \int[mu]_x f (x, y). +Let measurable_fun_knormalize U : + measurable U -> measurable_fun setT (knormalize P ^~ U). Proof. -rewrite -fubini_tonelli1//. - exact: finite_measure_sigma_finite. -move=> H. -rewrite fubini_tonelli2//. -exact: finite_measure_sigma_finite. +move=> mU; rewrite /knormalize/= /mnormalize /=. +rewrite (_ : (fun _ => _) = (fun x => + if f x setT == 0 then P U else if f x setT == +oo then P U + else f x U * ((fine (f x setT))^-1)%:E)); last first. + apply/funext => x; case: ifPn => [/orP[->//|->]|]. + by case: ifPn. + by rewrite negb_or=> /andP[/negbTE -> /negbTE ->]. +apply: measurable_fun_if => //. +- exact: measurable_fun_eq_cst. +- exact: measurable_fun_cst. +- apply: measurable_fun_if => //. + + rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]); last first. + by apply/seteqP; split => [x /negbT//|x /negbTE]. + exact: measurable_neq_cst. + + exact: measurable_fun_eq_cst. + + exact: measurable_fun_cst. + + apply: emeasurable_funM. + by have := measurable_kernel f U mU; exact: measurable_funS. + apply/EFin_measurable_fun. + apply: (measurable_fun_comp' (F := [set r : R | r != 0%R])) => //. + * exact: open_measurable. + * move=> /= r [t] [] [_ H1] H2 H3. + apply/eqP => H4; subst r. + move/eqP : H4. + rewrite fine_eq0 ?H1//. + rewrite ge0_fin_numE//. + by rewrite lt_neqAle leey H2. + * apply: open_continuous_measurable_fun => //. + apply/in_setP => x /= x0. + by apply: inv_continuous. + * apply: measurable_fun_comp => /=. + exact: measurable_fun_fine. + by have := measurable_kernel f _ measurableT; exact: measurable_funS. Qed. -End finite_fubini. - -Section sfinite_fubini. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Variables (mu : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X). -Variables (la : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> Y). -Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). -Variable (mf : measurable_fun setT f). +HB.instance Definition _ := isKernel.Build _ _ _ _ R (knormalize P) + measurable_fun_knormalize. -Lemma sfinite_fubini : - \int[mu tt]_x \int[la tt]_y f (x, y) = \int[la tt]_y \int[mu tt]_x f (x, y). +Let knormalize1 x : knormalize P x setT = 1. Proof. -have [mu_ mu_E] := sfinite mu. -have [la_ la_E] := sfinite la. -transitivity ( - \int[[the measure _ _ of mseries (fun i => mu_ i tt) 0]]_x - \int[la tt]_y f (x, y)). - apply: eq_measure_integral => U mU _. (* TODO: awkward *) - by rewrite mu_E. -transitivity ( - \int[[the measure _ _ of mseries (fun i => mu_ i tt) 0]]_x - \int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y f (x, y)). - apply eq_integral => x _. - apply: eq_measure_integral => U mU _. (* TODO: awkward *) - by rewrite la_E. -transitivity (\sum_(n t _; exact: integral_ge0 => x _. - rewrite [X in measurable_fun _ X](_ : _ = - fun x => \sum_(n x. - rewrite ge0_integral_measure_series//. - exact/measurable_fun_prod1. - apply: ge0_emeasurable_fun_sum => //. - by move=> k x; exact: integral_ge0. - move=> k. - apply: measurable_fun_fubini_tonelli_F => //=. - apply: finite_measure_sigma_finite. - exact: finite_kernel_finite_measure. - apply: eq_nneseries => n _; apply eq_integral => x _. - rewrite ge0_integral_measure_series//. - exact/measurable_fun_prod1. -transitivity (\sum_(n n _. - rewrite integral_sum(*TODO: ge0_integral_sum*)//. - move=> m. - apply: measurable_fun_fubini_tonelli_F => //=. - apply: finite_measure_sigma_finite. - exact: finite_kernel_finite_measure. - by move=> m x _; exact: integral_ge0. -transitivity (\sum_(n n _; apply eq_nneseries => m _. - rewrite finite_fubini//. - exact: finite_kernel_finite_measure. - exact: finite_kernel_finite_measure. -transitivity (\sum_(n la_ i tt) 0]]_y \int[mu_ n tt]_x f (x, y)). - apply eq_nneseries => n _. - rewrite /= ge0_integral_measure_series//. - by move=> y _; exact: integral_ge0. - apply: measurable_fun_fubini_tonelli_G => //=. - apply: finite_measure_sigma_finite. - exact: finite_kernel_finite_measure. -rewrite /=. -transitivity (\int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y \sum_(n n. - apply: measurable_fun_fubini_tonelli_G => //=. - apply: finite_measure_sigma_finite. - exact: finite_kernel_finite_measure. - by move=> n y _; exact: integral_ge0. -rewrite /=. -transitivity (\int[[the measure _ _ of mseries (fun i => la_ i tt) 0]]_y \int[[the measure _ _ of mseries (fun i => mu_ i tt) 0]]_x f (x, y)). - apply eq_integral => y _. - rewrite ge0_integral_measure_series//. - exact/measurable_fun_prod2. -rewrite /=. -transitivity (\int[la tt]_y \int[mseries (fun i : nat => mu_ i tt) 0]_x f (x, y)). - apply eq_measure_integral => A mA _ /=. - by rewrite la_E. -apply eq_integral => y _. -apply eq_measure_integral => A mA _ /=. -by rewrite mu_E. +rewrite /knormalize/= /mnormalize. +case: ifPn => [_|]; first by rewrite probability_setT. +rewrite negb_or => /andP[fx0 fxoo]. +have ? : f x setT \is a fin_num by rewrite ge0_fin_numE// lt_neqAle fxoo/= leey. +rewrite -{1}(@fineK _ (f x setT))//=. +by rewrite -EFinM divrr// ?lte_fin ?ltr1n// ?unitfE fine_eq0. Qed. -End sfinite_fubini. - -Section kprobability. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (m : probability Y R). +HB.instance Definition _ := + @isProbabilityKernel.Build _ _ _ _ _ (knormalize P) knormalize1. -Definition kprobability : X -> {measure set Y -> \bar R} := fun _ : X => m. +End knormalize. -Let measurable_fun_kprobability U : measurable U -> - measurable_fun setT (kprobability ^~ U). -Proof. by move=> mU; exact: measurable_fun_cst. Qed. +Section kcomp_def. +Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) + (Z : measurableType d3) (R : realType). +Variable l : X -> {measure set Y -> \bar R}. +Variable k : (X * Y)%type -> {measure set Z -> \bar R}. -HB.instance Definition _ := - @isKernel.Build _ _ X Y R kprobability measurable_fun_kprobability. +Definition kcomp x U := \int[l x]_y k (x, y) U. -Let kprobability_prob x : kprobability x setT = 1. -Proof. by rewrite /kprobability/= probability_setT. Qed. +End kcomp_def. -HB.instance Definition _ := - @isProbabilityKernel.Build _ _ X Y R kprobability kprobability_prob. - -End kprobability. +Section kcomp_is_measure. +Variables (d1 d2 d3 : _) (X : measurableType d1) (Y : measurableType d2) + (Z : measurableType d3) (R : realType). +Variable l : R.-ker X ~> Y. +Variable k : R.-ker [the measurableType _ of (X * Y)%type] ~> Z. -Section kdirac. -Variables (d d' : _) (T : measurableType d) (Y : measurableType d'). -Variables (R : realType) (f : T -> Y). +Local Notation "l \; k" := (kcomp l k). -Definition kdirac (mf : measurable_fun setT f) : T -> {measure set Y -> \bar R} := - fun t => [the measure _ _ of dirac (f t)]. +Let kcomp0 x : (l \; k) x set0 = 0. +Proof. +by rewrite /kcomp (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. +Qed. -Hypothesis mf : measurable_fun setT f. +Let kcomp_ge0 x U : 0 <= (l \; k) x U. Proof. exact: integral_ge0. Qed. -Let measurable_fun_kdirac U : measurable U -> measurable_fun setT (kdirac mf ^~ U). +Let kcomp_sigma_additive x : semi_sigma_additive ((l \; k) x). Proof. -move=> mU; apply/EFin_measurable_fun. -rewrite (_ : (fun x => _) = mindic R mU \o f)//. -exact/measurable_fun_comp. +move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = + \int[l x]_y (\sum_(n V _. + by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. +apply/cvg_closeP; split. + by apply: is_cvg_nneseries => n _; exact: integral_ge0. +rewrite closeE// integral_sum// => n. +by have /measurable_fun_prod1 := measurable_kernel k (U n) (mU n). Qed. -HB.instance Definition _ := isKernel.Build _ _ _ _ R (kdirac mf) - measurable_fun_kdirac. +HB.instance Definition _ x := isMeasure.Build _ R _ + ((l \; k) x) (kcomp0 x) (kcomp_ge0 x) (@kcomp_sigma_additive x). -Let kdirac_prob x : kdirac mf x setT = 1. -Proof. by rewrite /kdirac/= diracE in_setT. Qed. +Definition mkcomp : X -> {measure set Z -> \bar R} := + fun x => [the measure _ _ of (l \; k) x]. -HB.instance Definition _ := - @isProbabilityKernel.Build _ _ _ _ _ (kdirac mf) kdirac_prob. +End kcomp_is_measure. -End kdirac. -Arguments kdirac {d d' T Y R f}. +Notation "l \; k" := (mkcomp l k). -Section kadd. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k1 k2 : R.-ker X ~> Y). +Module KCOMP_FINITE_KERNEL. -Definition kadd : X -> {measure set Y -> \bar R} := - fun t => [the measure _ _ of measure_add (k1 t) (k2 t)]. +Section kcomp_finite_kernel_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-fker X ~> Y) + (k : R.-ker [the measurableType _ of (X * Y)%type] ~> Z). -Let measurable_fun_kadd U : measurable U -> measurable_fun setT (kadd ^~ U). +Lemma measurable_fun_kcomp_finite U : + measurable U -> measurable_fun setT ((l \; k) ^~ U). Proof. -move=> mU; rewrite /kadd. -rewrite (_ : (fun _ => _) = (fun x => k1 x U + k2 x U)); last first. - by apply/funext => x; rewrite -measure_addE. -by apply: emeasurable_funD; exact/measurable_kernel. +move=> mU; apply: (measurable_fun_integral_finite_kernel _ (k ^~ U)) => //=. +exact/measurable_kernel. Qed. HB.instance Definition _ := - @isKernel.Build _ _ _ _ _ kadd measurable_fun_kadd. -End kadd. + isKernel.Build _ _ X Z R (l \; k) measurable_fun_kcomp_finite. -Section fkadd. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k1 k2 : R.-fker X ~> Y). +End kcomp_finite_kernel_kernel. -Let kadd_finite_uub : measure_fam_uub (kadd k1 k2). +Section kcomp_finite_kernel_finite. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable l : R.-fker X ~> Y. +Variable k : R.-fker [the measurableType _ of (X * Y)%type] ~> Z. + +Let mkcomp_finite : measure_fam_uub (l \; k). Proof. -have [f1 hk1] := measure_uub k1; have [f2 hk2] := measure_uub k2. -exists (f1 + f2)%R => x; rewrite /kadd /=. -rewrite -/(measure_add (k1 x) (k2 x)). -by rewrite measure_addE EFinD; exact: lte_add. +have /measure_fam_uubP[r hr] := measure_uub k. +have /measure_fam_uubP[s hs] := measure_uub l. +apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=. +apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). + apply: ge0_le_integral => //. + - have /measurable_fun_prod1 := measurable_kernel k setT measurableT. + exact. + - exact/measurable_fun_cst. + - by move=> y _; exact/ltW/hr. +by rewrite integral_cst//= EFinM lte_pmul2l. Qed. -HB.instance Definition _ t := - isFiniteFam.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. -End fkadd. - -Section sfkadd. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k1 k2 : R.-sfker X ~> Y). +HB.instance Definition _ := + isFiniteFam.Build _ _ X Z R (l \; k) mkcomp_finite. -Let sfinite_kadd : exists k_ : (R.-fker _ ~> _)^nat, - forall x U, measurable U -> - kadd k1 k2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. -have [f1 hk1] := sfinite k1. -have [f2 hk2] := sfinite k2. -exists (fun n => [the finite_kernel _ _ _ of kadd (f1 n) (f2 n)]) => x U mU. -rewrite /kadd/=. -rewrite -/(measure_add (k1 x) (k2 x)) measure_addE. -rewrite /mseries. -rewrite hk1//= hk2//= /mseries. -rewrite -nneseriesD//. -apply: eq_nneseries => n _. -by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE. -Qed. +End kcomp_finite_kernel_finite. +End KCOMP_FINITE_KERNEL. -HB.instance Definition _ t := - isSFinite.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. -End sfkadd. +Section kcomp_sfinite_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable l : R.-sfker X ~> Y. +Variable k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. -Section kernel_measurable_preimage. -Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). -Variable R : realType. +Import KCOMP_FINITE_KERNEL. -Lemma measurable_eq_cst (f : R.-ker T ~> T') k : - measurable [set t | f t setT == k]. +Lemma mkcomp_sfinite : exists k_ : (R.-fker X ~> Z)^nat, forall x U, measurable U -> + (l \; k) x U = kseries k_ x U. Proof. -rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set k]); last first. - by apply/seteqP; split => t/= /eqP. -have /(_ measurableT [set k]) := measurable_kernel f setT measurableT. -by rewrite setTI; exact. +have [k_ hk_] := sfinite k; have [l_ hl_] := sfinite l. +have [kl hkl] : exists kl : (R.-fker X ~> Z) ^nat, forall x U, + \esum_(i in setT) (l_ i.2 \; k_ i.1) x U = \sum_(i [the _.-fker _ ~> _ of l_ (f i).2 \; k_ (f i).1]) => x U. + by rewrite (reindex_esum [set: nat] _ f)// nneseries_esum// fun_true. +exists kl => x U mU. +transitivity (([the _.-ker _ ~> _ of kseries l_] \; + [the _.-ker _ ~> _ of kseries k_]) x U). + rewrite /= /kcomp [in RHS](eq_measure_integral (l x)); last first. + by move=> *; rewrite hl_. + by apply: eq_integral => y _; rewrite hk_. +rewrite /= /kcomp/= integral_sum//=; last first. + by move=> n; have /measurable_fun_prod1 := measurable_kernel (k_ n) _ mU; exact. +transitivity (\sum_(i i _; rewrite integral_kseries//. + by have /measurable_fun_prod1 := measurable_kernel (k_ i) _ mU; exact. +rewrite /mseries -hkl/=. +rewrite (_ : setT = setT `*`` (fun=> setT)); last by apply/seteqP; split. +rewrite -(@esum_esum _ _ _ _ _ (fun i j => (l_ j \; k_ i) x U))//. +rewrite nneseries_esum; last by move=> n _; exact: nneseries_ge0. +by rewrite fun_true; apply: eq_esum => /= i _; rewrite nneseries_esum// fun_true. Qed. -Lemma measurable_neq_cst (f : R.-ker T ~> T') k : - measurable [set t | f t setT != k]. +Lemma measurable_fun_mkcomp_sfinite U : measurable U -> + measurable_fun setT ((l \; k) ^~ U). Proof. -rewrite [X in measurable X](_ : _ = (f ^~ setT) @^-1` [set~ k]); last first. - by apply/seteqP; split => t /eqP. -have /(_ measurableT [set~ k]) := measurable_kernel f setT measurableT. -by rewrite setTI; apply => //; exact: measurableC. +move=> mU; apply: (measurable_fun_integral_sfinite_kernel _ (k ^~ U)) => //. +exact/measurable_kernel. Qed. -End kernel_measurable_preimage. +End kcomp_sfinite_kernel. -Lemma measurable_fun_eq_cst (d d' : _) (T : measurableType d) - (T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k : - measurable_fun setT (fun t => f t setT == k). -Proof. -move=> _ /= B mB; rewrite setTI. -have [/eqP->|/eqP->|/eqP->|/eqP->] := set_boolE B. -- exact: measurable_eq_cst. -- rewrite (_ : _ @^-1` _ = [set b | f b setT != k]); last first. - by apply/seteqP; split => [t /negbT//|t /negbTE]. - exact: measurable_neq_cst. -- by rewrite preimage_set0. -- by rewrite preimage_setT. -Qed. +Module KCOMP_SFINITE_KERNEL. +Section kcomp_sfinite_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable l : R.-sfker X ~> Y. +Variable k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. -Section mnormalize. -Variables (d d' : _) (T : measurableType d) (Y : measurableType d'). -Variables (R : realType) (f : T -> {measure set Y -> \bar R}). -Variable P : probability Y R. +HB.instance Definition _ := + isKernel.Build _ _ X Z R (l \; k) (measurable_fun_mkcomp_sfinite l k). -Definition mnormalize t U := - let evidence := f t setT in - if (evidence == 0) || (evidence == +oo) then P U - else f t U * (fine evidence)^-1%:E. +#[export] +HB.instance Definition _ := + isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). -Let mnormalize0 t : mnormalize t set0 = 0. -Proof. -by rewrite /mnormalize; case: ifPn => // _; rewrite measure0 mul0e. -Qed. +End kcomp_sfinite_kernel. +End KCOMP_SFINITE_KERNEL. +HB.export KCOMP_SFINITE_KERNEL. -Let mnormalize_ge0 t U : 0 <= mnormalize t U. -Proof. by rewrite /mnormalize; case: ifPn => //; case: ifPn. Qed. +(* pollard? *) +Section measurable_fun_integral_kernel'. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType). +Variables (l : X -> {measure set Y -> \bar R}) + (k : Y -> \bar R) + (k_ : ({nnsfun Y >-> R}) ^nat) + (ndk_ : nondecreasing_seq (k_ : (Y -> R)^nat)) + (k_k : forall z, setT z -> EFin \o (k_ ^~ z) --> k z). + +Let p : (X * Y -> R)^nat := fun n xy => k_ n xy.2. + +Let p_ge0 n x : (0 <= p n x)%R. Proof. by []. Qed. + +HB.instance Definition _ n := @IsNonNegFun.Build _ R (p n) (p_ge0 n). -Lemma mnormalize_sigma_additive t : semi_sigma_additive (mnormalize t). +Let mp n : measurable_fun setT (p n). Proof. -move=> F mF tF mUF; rewrite /mnormalize/=. -case: ifPn => [_|_]. - exact: measure_semi_sigma_additive. -rewrite (_ : (fun n => _) = ((fun n => \sum_(0 <= i < n) f t (F i)) \* - cst ((fine (f t setT))^-1)%:E)); last first. - by apply/funext => n; rewrite -ge0_sume_distrl. -by apply: ereal_cvgMr => //; exact: measure_semi_sigma_additive. +rewrite /p => _ /= B mB; rewrite setTI. +have mk_n : measurable_fun setT (k_ n) by []. +rewrite (_ : _ @^-1` _ = setT `*` (k_ n @^-1` B)); last first. + by apply/seteqP; split => xy /=; tauto. +apply: measurableM => //. +have := mk_n measurableT _ mB. +by rewrite setTI. Qed. -HB.instance Definition _ (t : T) := isMeasure.Build _ _ _ (mnormalize t) - (mnormalize0 t) (mnormalize_ge0 t) (@mnormalize_sigma_additive t). +HB.instance Definition _ n := @IsMeasurableFun.Build _ _ R (p n) (mp n). -Lemma mnormalize1 t : mnormalize t setT = 1. +Let fp n : finite_set (range (p n)). Proof. -rewrite /mnormalize; case: ifPn; first by rewrite probability_setT. -rewrite negb_or => /andP[ft0 ftoo]. -have ? : f t setT \is a fin_num. - by rewrite ge0_fin_numE// lt_neqAle ftoo/= leey. -by rewrite -{1}(@fineK _ (f t setT))// -EFinM divrr// ?unitfE fine_eq0. +have := @fimfunP _ _ (k_ n). +suff : range (k_ n) = range (p n) by move=> <-. +by apply/seteqP; split => r [y ?] <-; [exists (point, y)|exists y.2]. Qed. -HB.instance Definition _ t := - isProbability.Build _ _ _ (mnormalize t) (mnormalize1 t). +HB.instance Definition _ n := @FiniteImage.Build _ _ (p n) (fp n). -End mnormalize. +Lemma measurable_fun_preimage_integral : + (forall n r, measurable_fun setT (fun x => l x (k_ n @^-1` [set r]))) -> + measurable_fun setT (fun x => \int[l x]_z k z). +Proof. +move=> h. +apply: (measurable_fun_xsection_integral l (fun xy => k xy.2) + (fun n => [the {nnsfun _ >-> _} of p n])) => /=. +- by rewrite /p => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_. +- by move=> [x y]; exact: k_k. +- move=> n r _ /= B mB. + have := h n r measurableT B mB. + rewrite !setTI. + suff : ((fun x => l x (k_ n @^-1` [set r])) @^-1` B) = + ((fun x => l x (xsection (p n @^-1` [set r]) x)) @^-1` B) by move=> ->. + apply/seteqP; split => x/=. + suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. + by apply/seteqP; split; move=> y/=; + rewrite /xsection/= /p /preimage/= inE/=. + suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. + by apply/seteqP; split; move=> y/=; rewrite /xsection/= /p /preimage/= inE/=. +Qed. -Section knormalize. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (f : R.-ker X ~> Y). +End measurable_fun_integral_kernel'. -Definition knormalize (P : probability Y R) := - fun t => [the measure _ _ of mnormalize f P t]. +Lemma measurable_fun_integral_kernel + (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) + (l : R.-ker [the measurableType _ of (X * Y)%type] ~> Z) c + (k : Z -> \bar R) (k0 : forall z, True -> 0 <= k z) (mk : measurable_fun setT k) : + measurable_fun setT (fun y => \int[l (c, y)]_z k z). +Proof. +have [k_ [ndk_ k_k]] := approximation measurableT mk k0. +apply: (measurable_fun_preimage_integral ndk_ k_k) => n r. +have := measurable_kernel l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). +by move=> /measurable_fun_prod1; exact. +Qed. -Variable P : probability Y R. +Section integral_kcomp. -Let measurable_fun_knormalize U : - measurable U -> measurable_fun setT (knormalize P ^~ U). +Let integral_kcomp_indic d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x (E : set _) (mE : measurable E) : + \int[(l \; k) x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). Proof. -move=> mU; rewrite /knormalize/= /mnormalize /=. -rewrite (_ : (fun _ => _) = (fun x => - if f x setT == 0 then P U else if f x setT == +oo then P U - else f x U * ((fine (f x setT))^-1)%:E)); last first. - apply/funext => x; case: ifPn => [/orP[->//|->]|]. - by case: ifPn. - by rewrite negb_or=> /andP[/negbTE -> /negbTE ->]. -apply: measurable_fun_if => //. -- exact: measurable_fun_eq_cst. -- exact: measurable_fun_cst. -- apply: measurable_fun_if => //. - + rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]); last first. - by apply/seteqP; split => [x /negbT//|x /negbTE]. - exact: measurable_neq_cst. - + exact: measurable_fun_eq_cst. - + exact: measurable_fun_cst. - + apply: emeasurable_funM. - by have := measurable_kernel f U mU; exact: measurable_funS. - apply/EFin_measurable_fun. - apply: (measurable_fun_comp' (F := [set r : R | r != 0%R])) => //. - * exact: open_measurable. - * move=> /= r [t] [] [_ H1] H2 H3. - apply/eqP => H4; subst r. - move/eqP : H4. - rewrite fine_eq0 ?H1//. - rewrite ge0_fin_numE//. - by rewrite lt_neqAle leey H2. - * apply: open_continuous_measurable_fun => //. - apply/in_setP => x /= x0. - by apply: inv_continuous. - * apply: measurable_fun_comp => /=. - exact: measurable_fun_fine. - by have := measurable_kernel f _ measurableT; exact: measurable_funS. +rewrite integral_indic//= /kcomp. +by apply eq_integral => y _; rewrite integral_indic. Qed. -HB.instance Definition _ := isKernel.Build _ _ _ _ R (knormalize P) - measurable_fun_knormalize. - -Let knormalize1 x : knormalize P x setT = 1. +Let integral_kcomp_nnsfun d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x (f : {nnsfun Z >-> R}) : + \int[(l \; k) x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). Proof. -rewrite /knormalize/= /mnormalize. -case: ifPn => [_|]; first by rewrite probability_setT. -rewrite negb_or => /andP[fx0 fxoo]. -have ? : f x setT \is a fin_num by rewrite ge0_fin_numE// lt_neqAle fxoo/= leey. -rewrite -{1}(@fineK _ (f x setT))//=. -by rewrite -EFinM divrr// ?lte_fin ?ltr1n// ?unitfE fine_eq0. +under [in LHS]eq_integral do rewrite fimfunE -sumEFin. +rewrite ge0_integral_sum//; last 2 first. + move=> r; apply/EFin_measurable_fun/measurable_funrM. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. +under [in RHS]eq_integral. + move=> y _. + under eq_integral. + move=> z _. + rewrite fimfunE -sumEFin. + over. + rewrite /= ge0_integral_sum//; last 2 first. + move=> r; apply/EFin_measurable_fun/measurable_funrM. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. + under eq_bigr. + move=> r _. + rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. + by move=> r0; rewrite preimage_nnfun0. + rewrite integral_indic// setIT. + over. + over. +rewrite /= ge0_integral_sum//; last 2 first. + - move=> r; apply: measurable_funeM. + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). + by move=> /measurable_fun_prod1; exact. + - move=> n y _. + have := @mulemu_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). + by apply; exact: preimage_nnfun0. +apply eq_bigr => r _. +rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. + exact: preimage_nnfun0. +rewrite /= integral_kcomp_indic; last exact/measurable_sfunP. +rewrite (@integralM_0ifneg _ _ _ _ _ _ (fun r t => k (x, t) (f @^-1` [set r])))//; last 2 first. + move=> r0. + apply/funext => y. + by rewrite preimage_nnfun0// measure0. + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). + by move/measurable_fun_prod1; exact. +congr (_ * _); apply eq_integral => y _. +by rewrite integral_indic// setIT. Qed. -HB.instance Definition _ := - @isProbabilityKernel.Build _ _ _ _ _ (knormalize P) knormalize1. +Lemma integral_kcomp d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x f : (forall z, 0 <= f z) -> measurable_fun setT f -> + \int[(l \; k) x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). +Proof. +move=> f0 mf. +have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). +transitivity (\int[(l \; k) x]_z (lim (EFin \o (f_^~ z)))). + apply/eq_integral => z _. + apply/esym/cvg_lim => //=. + exact: f_f. +rewrite monotone_convergence//; last 3 first. + by move=> n; apply/EFin_measurable_fun. + by move=> n z _; rewrite lee_fin. + by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin. +rewrite (_ : (fun _ => _) = (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first. + by apply/funext => n; rewrite integral_kcomp_nnsfun. +transitivity (\int[l x]_y lim (fun n => \int[k (x, y)]_z (f_ n z)%:E)). + rewrite -monotone_convergence//; last 3 first. + move=> n. + apply: measurable_fun_integral_kernel => //. + - by move=> z; rewrite lee_fin. + - by apply/EFin_measurable_fun. + - move=> n y _. + by apply integral_ge0 => // z _; rewrite lee_fin. + - move=> y _ a b ab. + apply: ge0_le_integral => //. + + by move=> z _; rewrite lee_fin. + + exact/EFin_measurable_fun. + + by move=> z _; rewrite lee_fin. + + exact/EFin_measurable_fun. + + move: ab => /ndf_ /lefP ab z _. + by rewrite lee_fin. +apply eq_integral => y _. +rewrite -monotone_convergence//; last 3 first. + move=> n; exact/EFin_measurable_fun. + by move=> n z _; rewrite lee_fin. + by move=> z _ a b /ndf_ /lefP; rewrite lee_fin. +apply eq_integral => z _. +apply/cvg_lim => //. +exact: f_f. +Qed. -End knormalize. +End integral_kcomp. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index 0de50bd9db..708c5780d9 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -15,7 +15,7 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral kernel. (* score mf == observe t from d, where f is the density of d and *) (* t occurs in f *) (* e.g., score (r e^(-r * t)) = observe t from exp(r) *) -(* normalize k P == normalize the kernel k into a probability kernel, *) +(* pnormalize k P == normalize the kernel k into a probability kernel, *) (* P is a default probability in case normalization is *) (* not possible *) (* ite mf k1 k2 == access the context with the boolean function f and *) @@ -408,11 +408,14 @@ Definition ret (f : X -> Y) (mf : measurable_fun setT f) := locked [the R.-sfker X ~> Y of kdirac mf]. Definition sample (P : probability Y R) := - locked [the R.-sfker X ~> Y of kprobability P] . + locked [the R.-pker X ~> Y of kprobability P] . -Definition normalize (k : R.-sfker X ~> Y) P := +Definition pnormalize (k : R.-sfker X ~> Y) P := locked [the R.-pker X ~> Y of knormalize k P]. +Definition dnormalize t (k : R.-sfker X ~> Y) P := + locked [the probability _ _ of mnormalize k P t]. + Definition ite (f : X -> bool) (mf : measurable_fun setT f) (k1 k2 : R.-sfker X ~> Y):= locked [the R.-sfker X ~> Y of kite k1 k2 mf]. @@ -431,12 +434,20 @@ Proof. by rewrite [in LHS]/ret; unlock. Qed. Lemma sampleE (P : probability Y R) (x : X) : sample P x = P. Proof. by rewrite [in LHS]/sample; unlock. Qed. -Lemma normalizeE (f : R.-sfker X ~> Y) P x U : - normalize f P x U = +Lemma pnormalizeE (f : R.-sfker X ~> Y) P x U : + pnormalize f P x U = + if (f x [set: Y] == 0) || (f x [set: Y] == +oo) then P U + else f x U * ((fine (f x [set: Y]))^-1)%:E. +Proof. +by rewrite /pnormalize; unlock => /=; rewrite /mnormalize; case: ifPn. +Qed. + +Lemma dnormalizeE (f : R.-sfker X ~> Y) P x U : + dnormalize x f P U = if (f x [set: Y] == 0) || (f x [set: Y] == +oo) then P U else f x U * ((fine (f x [set: Y]))^-1)%:E. Proof. -by rewrite /normalize; unlock => /=; rewrite /mnormalize; case: ifPn. +by rewrite /dnormalize; unlock => /=; rewrite /mnormalize; case: ifPn. Qed. Lemma iteE (f : X -> bool) (mf : measurable_fun setT f) @@ -563,8 +574,10 @@ by rewrite ger0_norm// ?f0//= muleC. Qed. (* example of property *) -Lemma score_score (f : R -> R) (g : R * unit -> R) (mf : measurable_fun setT f) (mg : measurable_fun setT g) x U : - letin (score mf) (score mg) x U = if U == set0 then 0 else `|g (x, tt)|%:E * `|f x|%:E. +Lemma score_score (f : R -> R) (g : R * unit -> R) (mf : measurable_fun setT f) + (mg : measurable_fun setT g) x U : + letin (score mf) (score mg) x U = + if U == set0 then 0 else `|f x|%:E * `|g (x, tt)|%:E. Proof. rewrite {1}/letin. unlock. @@ -572,7 +585,7 @@ rewrite scoreE'//=. rewrite /mscale/= diracE !normr_id. have [->|->]:= set_unit U. by rewrite eqxx in_set0 mule0 mul0e. -by rewrite in_setT mule1 (negbTE (setT0 _)). +by rewrite in_setT mule1 (negbTE (setT0 _)) muleC. Qed. End insn1_lemmas. @@ -765,7 +778,7 @@ Variable P : probability mbool R. Import Notations. Definition staton_bus_annotated : R.-pker T ~> mbool := - normalize (letin + pnormalize (letin (sample (bernoulli p27) : _.-sfker T ~> mbool) (letin (letin @@ -810,7 +823,7 @@ rewrite -!muleA; congr (_ * _ + _ * _). by rewrite scoreE// => r r0; exact: poisson_ge0. Qed. -Definition staton_bus : R.-pker T ~> mbool := normalize staton_bus' P. +Definition staton_bus : R.-pker T ~> mbool := pnormalize staton_bus' P. Lemma staton_busE t U : let N := ((2 / 7%:R) * poisson 3%:R 4 + @@ -820,7 +833,7 @@ Lemma staton_busE t U : (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. Proof. rewrite /staton_bus. -rewrite normalizeE /=. +rewrite pnormalizeE /=. rewrite !staton_bus'E. rewrite diracE mem_set// mule1. rewrite diracE mem_set// mule1. @@ -829,71 +842,117 @@ apply/negbTE. by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= poisson_gt0. Qed. -End staton_bus. +Definition dstaton_bus (t : T) : probability mbool R := dnormalize t staton_bus' P. -(* wip *) +Lemma dstaton_busE t U : + let N := ((2 / 7%:R) * poisson 3%:R 4 + + (5%:R / 7%:R) * poisson 10%:R 4)%R in + dstaton_bus t U = + ((2 / 7%:R)%:E * (poisson 3%:R 4)%:E * \d_true U + + (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. +Proof. +rewrite /staton_bus. +rewrite dnormalizeE /=. +rewrite !staton_bus'E. +rewrite diracE mem_set// mule1. +rewrite diracE mem_set// mule1. +rewrite ifF //. +apply/negbTE. +by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= poisson_gt0. +Qed. -Section letinC. +End staton_bus. + +(* TODO: move *) +Section measurable_fun_pair. +Variables (d d' d3 : _) (X : measurableType d) + (Y : measurableType d') (Z : measurableType d3). -Variables (d d' d3 d4 : _) (R : realType) (X : measurableType d) - (Y : measurableType d') (Z : measurableType d3) (U : measurableType d4). +Lemma measurable_fun_pair (f : X -> Y) (g : X -> Z) : + measurable_fun setT f -> + measurable_fun setT g -> + measurable_fun setT (fun x => (f x, g x)). +Proof. +by move=> mf mg; apply/prod_measurable_funP. +Qed. -Let f (xyz : unit * X * X) := (xyz.1.2, xyz.2). +End measurable_fun_pair. -Lemma mf : measurable_fun setT f. +(* TODO: move *) +Lemma finite_kernel_measure (d d' : _) (X : measurableType d) + (Y : measurableType d') (R : realType) (k : R.-fker X ~> Y) (x : X) : + finite_measure (k x). Proof. -rewrite /=. -apply/prod_measurable_funP => /=; split. - rewrite /f. - rewrite (_ : _ \o _ = (fun xyz : unit * X * X => xyz.1.2))//. - apply: measurable_fun_comp => /=. - exact: measurable_fun_snd. - exact: measurable_fun_fst. -rewrite (_ : _ \o _ = (fun xyz : unit * X * X => xyz.2))//. -apply: measurable_fun_comp => /=. - exact: measurable_fun_snd. -exact: measurable_fun_id. +have [r k_r] := measure_uub k. +by rewrite /finite_measure (@lt_trans _ _ r%:E) ?ltey. Qed. -Let f' := @swap _ _ \o f. -Lemma mf' : measurable_fun setT f'. +Lemma sfinite_kernel_measure (d d' : _) (X : measurableType d) + (Y : measurableType d') (R : realType) (k : R.-sfker X ~> Y) (x : X) : + sfinite_measure (k x). Proof. -rewrite /=. -apply: measurable_fun_comp => /=. - exact: measurable_fun_swap. -exact: mf. +have [k_ k_E] := sfinite k. +exists (fun n => k_ n x); split; last by move=> A mA; rewrite k_E. +by move=> n; exact: finite_kernel_measure. Qed. -Variables (t : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X) - (t' : R.-sfker [the measurableType _ of (unit * X)%type] ~> X) - (u : R.-sfker Datatypes_unit__canonical__measure_Measurable ~> X) - (u' : R.-sfker [the measurableType _ of (unit * X)%type] ~> X) - (H1 : forall y, u tt = u' (tt, y)) - (H2 : forall y, t tt = t' (tt, y)). +Section letinC. +Variables (d d1 : _) (X : measurableType d) (Y : measurableType d1). +Variables (R : realType) (d' : _) (Z : measurableType d'). + +Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var3_of3 := (@measurable_fun_snd _ _ _ _). -Lemma letinC x A : measurable A -> - letin t (letin u' (ret R mf)) x A = letin u (letin t' (ret R mf')) x A. +Variables (t : R.-sfker Z ~> X) + (t' : R.-sfker [the measurableType _ of (Z * Y)%type] ~> X) + (tt' : forall y, t =1 fun z => t' (z, y)) + (u : R.-sfker Z ~> Y) + (u' : R.-sfker [the measurableType _ of (Z * X)%type] ~> Y) + (uu' : forall x, u =1 fun z => u' (z, x)). + +Lemma letinC z A : measurable A -> + letin t + (letin u' + (ret R (measurable_fun_pair var2_of3 var3_of3))) z A = + letin u + (letin t' + (ret R (measurable_fun_pair var3_of3 var2_of3))) z A. Proof. move=> mA. rewrite !letinE. -destruct x. -rewrite /f/=. under eq_integral. move=> x _. - rewrite letinE/=. - rewrite -H1. + rewrite letinE/= -uu'. under eq_integral do rewrite retE /=. over. -rewrite /=. -rewrite (@sfinite_fubini _ _ X X R t u (fun x => \d_(x.1, x.2) A ))//=. -apply eq_integral => x _. - rewrite letinE/=. - rewrite -H2. - apply eq_integral => // x' _. - by rewrite retE. -apply/EFin_measurable_fun => /=. -rewrite (_ : (fun x => _) = mindic R mA)//. -by apply/funext => -[a b] /=. +rewrite (sfinite_fubini _ _ (fun x => \d_(x.1, x.2) A ))//; last 3 first. + exact: sfinite_kernel_measure. + exact: sfinite_kernel_measure. + apply/EFin_measurable_fun => /=; rewrite (_ : (fun x => _) = mindic R mA)//. + by apply/funext => -[]. +apply eq_integral => y _. +by rewrite letinE/= -tt'; apply eq_integral => // x _; rewrite retE. Qed. End letinC. + +Section dist_salgebra_instance. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables p0 : probability T R. + +Definition prob_pointed := Pointed.Class + (Choice.Class gen_eqMixin (Choice.Class gen_eqMixin gen_choiceMixin)) p0. + +Canonical probability_eqType := EqType (probability T R) prob_pointed. +Canonical probability_choiceType := ChoiceType (probability T R) prob_pointed. +Canonical probability_ptType := PointedType (probability T R) prob_pointed. + +Definition mset (U : set T) (r : R) := [set mu : probability T R | mu U < r%:E]. + +Definition pset : set (set (probability T R)) := + [set mset U r | r in `[0%R,1%R]%classic & U in @measurable d T]. + +Definition sset := [the measurableType pset.-sigma of salgebraType pset]. + +End dist_salgebra_instance. From 238bad43d673d0f2b6318fc88956faabd91647dc Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Tue, 13 Sep 2022 19:44:41 +0900 Subject: [PATCH 38/42] staton bus with exp --- theories/kernel.v | 378 ++++++++++++++-------------- theories/prob_lang.v | 574 ++++++++++++++++++++----------------------- 2 files changed, 471 insertions(+), 481 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 646cc75c79..08083a9df4 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -9,15 +9,15 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (* Kernels *) (* *) (* This file provides a formation of kernels and extends the theory of *) -(* measure with, e.g., Fubini's theorem for s-finite measures. *) +(* measures with, e.g., Tonelli-Fubini's theorem for s-finite measures. *) (* *) +(* finite_measure mu == the measure mu is finite *) +(* sfinite_measure mu == the measure mu is s-finite *) (* R.-ker X ~> Y == kernel *) (* kseries == countable sum of kernels *) -(* R.-sfker X ~> Y == s-finite kernel *) (* R.-fker X ~> Y == finite kernel *) +(* R.-sfker X ~> Y == s-finite kernel *) (* R.-pker X ~> Y == probability kernel *) -(* finite_measure mu == the measure mu is finite *) -(* sfinite_measure mu == the measure my is s-finite *) (* kprobability m == kernel defined by a probability measure *) (* kdirac mf == kernel defined by a measurable function *) (* kadd k1 k2 == lifting of the addition of measures to kernels *) @@ -57,31 +57,41 @@ Qed. End probability_lemmas. (* /PR 516 in progress *) -(* TODO: PR? *) -Section integralM_0ifneg. -Local Open Scope ereal_scope. -Variables (d : _) (T : measurableType d) (R : realType). -Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). +(* TODO: PR *) +Lemma setT0 (T : pointedType) : setT != set0 :> set T. +Proof. by apply/eqP => /seteqP[] /(_ point) /(_ Logic.I). Qed. -Lemma integralM_0ifneg (f : R -> T -> \bar R) (k : R) - (f0 : forall r t, D t -> 0 <= f r t) : - ((k < 0)%R -> f k = cst 0%E) -> measurable_fun setT (f k) -> - \int[m]_(x in D) (k%:E * (f k) x) = k%:E * \int[m]_(x in D) ((f k) x). +Lemma set_unit (A : set unit) : A = set0 \/ A = setT. Proof. -move=> fk0 mfk; have [k0|k0] := ltP k 0%R. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. - by move=> x _; rewrite fk0// mule0. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. - by rewrite fk0// indic0. -rewrite ge0_integralM//. -- by apply/(@measurable_funS _ _ _ _ setT) => //. -- by move=> y Dy; rewrite f0. +have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. +by apply/seteqP; split => [|] []. Qed. -End integralM_0ifneg. -Arguments integralM_0ifneg {d T R} m {D} mD f. +Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. +Proof. +have [Bt|Bt] := boolP (true \in B). + have [Bf|Bf] := boolP (false \in B). + have -> : B = setT. + by apply/seteqP; split => // -[] _; [rewrite inE in Bt| rewrite inE in Bf]. + by apply/or4P; rewrite eqxx/= !orbT. + have -> : B = [set true]. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bf. + by rewrite inE in Bt. + by apply/or4P; rewrite eqxx. +have [Bf|Bf] := boolP (false \in B). + have -> : B = [set false]. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bt. + by rewrite inE in Bf. + by apply/or4P; rewrite eqxx/= orbT. +have -> : B = set0. + apply/seteqP; split => -[]//=. + by rewrite notin_set in Bt. + by rewrite notin_set in Bf. +by apply/or4P; rewrite eqxx/= !orbT. +Qed. -(* TODO: PR *) Canonical unit_pointedType := PointedType unit tt. Section discrete_measurable_unit. @@ -126,7 +136,22 @@ HB.instance Definition _ := @isMeasurable.Build default_measure_display bool End discrete_measurable_bool. -(* TODO: PR *) +Lemma measurable_curry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) + (G : T1 * T2 -> set T) (x : T1 * T2) : + measurable (G x) <-> measurable (curry G x.1 x.2). +Proof. by case: x. Qed. + +Lemma emeasurable_itv (R : realType) (i : nat) : + measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set \bar R). +Proof. +rewrite -[X in measurable X]setCK. +apply: measurableC. +rewrite set_interval.setCitv /=. +apply: measurableU. + exact: emeasurable_itv_ninfty_bnd. +exact: emeasurable_itv_bnd_pinfty. +Qed. + Lemma measurable_fun_fst (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) : measurable_fun setT (@fst T1 T2). Proof. @@ -141,10 +166,25 @@ have := @measurable_fun_id _ [the measurableType _ of (T1 * T2)%type] setT. by move=> /prod_measurable_funP[]. Qed. -Lemma measurable_curry (T1 T2 : Type) (d : _) (T : semiRingOfSetsType d) - (G : T1 * T2 -> set T) (x : T1 * T2) : - measurable (G x) <-> measurable (curry G x.1 x.2). -Proof. by case: x. Qed. +Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1). + +Lemma measurable_fun_swap d d' (X : measurableType d) (Y : measurableType d') : + measurable_fun [set: X * Y] (@swap X Y). +Proof. +by apply/prod_measurable_funP => /=; split; + [exact: measurable_fun_snd|exact: measurable_fun_fst]. +Qed. + +Section measurable_fun_pair. +Variables (d d2 d3 : _) (X : measurableType d) (Y : measurableType d2) + (Z : measurableType d3). + +Lemma measurable_fun_pair (f : X -> Y) (g : X -> Z) : + measurable_fun setT f -> measurable_fun setT g -> + measurable_fun setT (fun x => (f x, g x)). +Proof. by move=> mf mg; apply/prod_measurable_funP. Qed. + +End measurable_fun_pair. Section measurable_fun_comp. Variables (d1 d2 d3 : measure_display). @@ -152,7 +192,7 @@ Variables (T1 : measurableType d1). Variables (T2 : measurableType d2). Variables (T3 : measurableType d3). -(* NB: this generalizes MathComp's measurable_fun_comp' *) +(* NB: this generalizes MathComp-Analysis' measurable_fun_comp *) Lemma measurable_fun_comp' F (f : T2 -> T3) E (g : T1 -> T2) : measurable F -> g @` E `<=` F -> @@ -168,37 +208,6 @@ Qed. End measurable_fun_comp. -Lemma set_unit (A : set unit) : A = set0 \/ A = setT. -Proof. -have [->|/set0P[[] Att]] := eqVneq A set0; [by left|right]. -by apply/seteqP; split => [|] []. -Qed. - -Lemma set_boolE (B : set bool) : [\/ B == [set true], B == [set false], B == set0 | B == setT]. -Proof. -have [Bt|Bt] := boolP (true \in B). - have [Bf|Bf] := boolP (false \in B). - have -> : B = setT. - by apply/seteqP; split => // -[] _; [rewrite inE in Bt| rewrite inE in Bf]. - by apply/or4P; rewrite eqxx/= !orbT. - have -> : B = [set true]. - apply/seteqP; split => -[]//=. - by rewrite notin_set in Bf. - by rewrite inE in Bt. - by apply/or4P; rewrite eqxx. -have [Bf|Bf] := boolP (false \in B). - have -> : B = [set false]. - apply/seteqP; split => -[]//=. - by rewrite notin_set in Bt. - by rewrite inE in Bf. - by apply/or4P; rewrite eqxx/= orbT. -have -> : B = set0. - apply/seteqP; split => -[]//=. - by rewrite notin_set in Bt. - by rewrite notin_set in Bf. -by apply/or4P; rewrite eqxx/= !orbT. -Qed. - Lemma measurable_fun_if (d d' : _) (T : measurableType d) (T' : measurableType d') (x y : T -> T') D (md : measurable D) (f : T -> bool) (mf : measurable_fun setT f) : @@ -253,18 +262,36 @@ have {}my : measurable_fun [set: T * bool] (y \o fst). by apply: measurable_fun_ifT => //=; exact: measurable_fun_snd. Qed. -Lemma emeasurable_itv (R : realType) (i : nat) : - measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set \bar R). +Lemma measurable_fun_opp (R : realType) : measurable_fun [set: R] -%R. Proof. -rewrite -[X in measurable X]setCK. -apply: measurableC. -rewrite set_interval.setCitv /=. -apply: measurableU. - exact: emeasurable_itv_ninfty_bnd. -exact: emeasurable_itv_bnd_pinfty. +apply: continuous_measurable_fun. +by have := (@opp_continuous R [the normedModType R of R^o]). Qed. -Section fubini_tonelli. (* TODO: move to lebesgue_integral.v *) +Section integralM_0ifneg. +Local Open Scope ereal_scope. +Variables (d : _) (T : measurableType d) (R : realType). +Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). + +Lemma integralM_0ifneg (f : R -> T -> \bar R) (k : R) + (f0 : forall r t, D t -> 0 <= f r t) : + ((k < 0)%R -> f k = cst 0%E) -> measurable_fun setT (f k) -> + \int[m]_(x in D) (k%:E * (f k) x) = k%:E * \int[m]_(x in D) ((f k) x). +Proof. +move=> fk0 mfk; have [k0|k0] := ltP k 0%R. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. + by move=> x _; rewrite fk0// mule0. + rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. + by rewrite fk0// indic0. +rewrite ge0_integralM//. +- by apply/(@measurable_funS _ _ _ _ setT) => //. +- by move=> y Dy; rewrite f0. +Qed. + +End integralM_0ifneg. +Arguments integralM_0ifneg {d T R} m {D} mD f. + +Section fubini_tonelli. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). @@ -278,7 +305,87 @@ Lemma fubini_tonelli : Proof. by rewrite -fubini_tonelli1// fubini_tonelli2. Qed. End fubini_tonelli. -(*/ PR*) +(* /TODO: PR *) + +Definition finite_measure d (T : measurableType d) (R : realType) + (mu : set T -> \bar R) := + mu setT < +oo. + +Definition sfinite_measure d (T : measurableType d) (R : realType) + (mu : set T -> \bar R) := + exists mu_ : {measure set T -> \bar R}^nat, + (forall n, finite_measure (mu_ n)) /\ + (forall U, measurable U -> mu U = mseries mu_ 0 U). + +Lemma finite_measure_sigma_finite d (T : measurableType d) (R : realType) + (mu : {measure set T -> \bar R}) : + finite_measure mu -> sigma_finite setT mu. +Proof. +exists (fun i => if i \in [set 0%N] then setT else set0). + by rewrite -bigcup_mkcondr setTI bigcup_const//; exists 0%N. +move=> n; split; first by case: ifPn. +by case: ifPn => // _; rewrite ?measure0//; exact: finite_measure. +Qed. + +Section sfinite_fubini. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). +Variables (m1 : {measure set X -> \bar R}) (sfm1 : sfinite_measure m1). +Variables (m2 : {measure set Y -> \bar R}) (sfm2 : sfinite_measure m2). +Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). +Variable (mf : measurable_fun setT f). + +Lemma sfinite_fubini : + \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). +Proof. +have [m1_ [fm1 m1E]] := sfm1. +have [m2_ [fm2 m2E]] := sfm2. +rewrite [LHS](eq_measure_integral [the measure _ _ of mseries m1_ 0]); last first. + by move=> A mA _; rewrite m1E. +transitivity (\int[[the measure _ _ of mseries m1_ 0]]_x + \int[[the measure _ _ of mseries m2_ 0]]_y f (x, y)). + by apply eq_integral => x _; apply: eq_measure_integral => U mA _; rewrite m2E. +transitivity (\sum_(n t _; exact: integral_ge0. + rewrite [X in measurable_fun _ X](_ : _ = + fun x => \sum_(n x. + by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod1. + apply: ge0_emeasurable_fun_sum; first by move=> k x; exact: integral_ge0. + move=> k; apply: measurable_fun_fubini_tonelli_F => //=. + exact: finite_measure_sigma_finite. + apply: eq_nneseries => n _; apply eq_integral => x _. + by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod1. +transitivity (\sum_(n n _. + rewrite integral_sum//. + move=> m; apply: measurable_fun_fubini_tonelli_F => //=. + exact: finite_measure_sigma_finite. + by move=> m x _; exact: integral_ge0. +transitivity (\sum_(n n _; apply eq_nneseries => m _. + by rewrite fubini_tonelli//; exact: finite_measure_sigma_finite. +transitivity (\sum_(n n _ /=. rewrite ge0_integral_measure_series//. + by move=> y _; exact: integral_ge0. + apply: measurable_fun_fubini_tonelli_G => //=. + by apply: finite_measure_sigma_finite; exact: fm1. +transitivity (\int[[the measure _ _ of mseries m2_ 0]]_y \sum_(n n; apply: measurable_fun_fubini_tonelli_G => //=. + by apply: finite_measure_sigma_finite; exact: fm1. + by move=> n y _; exact: integral_ge0. +transitivity (\int[[the measure _ _ of mseries m2_ 0]]_y + \int[[the measure _ _ of mseries m1_ 0]]_x f (x, y)). + apply eq_integral => y _. + by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod2. +transitivity (\int[m2]_y \int[mseries m1_ 0]_x f (x, y)). + by apply eq_measure_integral => A mA _ /=; rewrite m2E. +by apply eq_integral => y _; apply eq_measure_integral => A mA _ /=; rewrite m1E. +Qed. + +End sfinite_fubini. +Arguments sfinite_fubini {d d' X Y R m1} _ {m2} _ f. Reserved Notation "R .-ker X ~> Y" (at level 42, format "R .-ker X ~> Y"). Reserved Notation "R .-fker X ~> Y" (at level 42, format "R .-fker X ~> Y"). @@ -459,6 +566,23 @@ HB.instance Definition _ := @isProbabilityFam.Build _ _ _ _ _ _ is_probability_k HB.end. +Lemma finite_kernel_measure (d d' : _) (X : measurableType d) + (Y : measurableType d') (R : realType) (k : R.-fker X ~> Y) (x : X) : + finite_measure (k x). +Proof. +have [r k_r] := measure_uub k. +by rewrite /finite_measure (@lt_trans _ _ r%:E) ?ltey. +Qed. + +Lemma sfinite_kernel_measure (d d' : _) (X : measurableType d) + (Y : measurableType d') (R : realType) (k : R.-sfker X ~> Y) (x : X) : + sfinite_measure (k x). +Proof. +have [k_ k_E] := sfinite k. +exists (fun n => k_ n x); split; last by move=> A mA; rewrite k_E. +by move=> n; exact: finite_kernel_measure. +Qed. + (* see measurable_prod_subset in lebesgue_integral.v; the differences between the two are: - m2 is a kernel instead of a measure (the proof uses the @@ -542,7 +666,7 @@ Qed. End measurable_fun_xsection_finite_kernel. -(* pollard *) +(* pollard? *) Section measurable_fun_integral_finite_sfinite. Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). @@ -576,11 +700,11 @@ rewrite (_ : (fun x => _) = - by move=> y _ m n mn; rewrite lee_fin; exact/lefP/ndk_. apply: measurable_fun_elim_sup => n. rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y - (\sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR is merged*) + (\sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR 743 is merged*) r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. rewrite [X in measurable_fun _ X](_ : _ = (fun x => - \sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR is merged*) + \sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR 743 is merged*) (\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first. apply/funext => x; rewrite -ge0_integral_sum//. - by apply: eq_integral => y _; rewrite sumEFin. @@ -642,113 +766,11 @@ Arguments measurable_fun_xsection_integral {_ _ _ _ _} l k. Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} l k. Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} l k. -(*HB.mixin Record isFiniteMeasure d (R : numFieldType) (T : semiRingOfSetsType d) - (mu : set T -> \bar R) := { - finite_measure : mu setT < +oo -}. - -#[short(type=fmeasure)] -HB.structure Definition FiniteMeasure d (R : realFieldType) - (T : semiRingOfSetsType d) := - {mu of isMeasure d R T mu & isFiniteMeasure d R T mu}. - -Notation "{ 'fmeasure' 'set' T '->' '\bar' R }" := (@fmeasure _ R T) - (at level 36, T, R at next level, - format "{ 'fmeasure' 'set' T '->' '\bar' R }") : ring_scope.*) - -Definition finite_measure d (T : measurableType d) (R : realType) - (mu : set T -> \bar R) := - mu setT < +oo. - -Definition sfinite_measure d (T : measurableType d) (R : realType) - (mu : set T -> \bar R) := - exists mu_ : {measure set T -> \bar R}^nat, - (forall n, finite_measure (mu_ n)) /\ - (forall U, measurable U -> mu U = mseries mu_ 0 U). - -Lemma finite_measure_sigma_finite d (T : measurableType d) (R : realType) - (mu : {measure set T -> \bar R}) : - finite_measure mu -> sigma_finite setT mu. -Proof. -exists (fun i => if i \in [set 0%N] then setT else set0). - by rewrite -bigcup_mkcondr setTI bigcup_const//; exists 0%N. -move=> n; split; first by case: ifPn. -by case: ifPn => // _; rewrite ?measure0//; exact: finite_measure. -Qed. - -Section sfinite_fubini. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Variables (m1 : {measure set X -> \bar R}) (sfm1 : sfinite_measure m1). -Variables (m2 : {measure set Y -> \bar R}) (sfm2 : sfinite_measure m2). -Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). -Variable (mf : measurable_fun setT f). - -Lemma sfinite_fubini : - \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). -Proof. -have [m1_ [fm1 m1E]] := sfm1. -have [m2_ [fm2 m2E]] := sfm2. -rewrite [LHS](eq_measure_integral [the measure _ _ of mseries m1_ 0]); last first. - by move=> A mA _; rewrite m1E. -transitivity (\int[[the measure _ _ of mseries m1_ 0]]_x - \int[[the measure _ _ of mseries m2_ 0]]_y f (x, y)). - by apply eq_integral => x _; apply: eq_measure_integral => U mA _; rewrite m2E. -transitivity (\sum_(n t _; exact: integral_ge0. - rewrite [X in measurable_fun _ X](_ : _ = - fun x => \sum_(n x. - by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod1. - apply: ge0_emeasurable_fun_sum; first by move=> k x; exact: integral_ge0. - move=> k; apply: measurable_fun_fubini_tonelli_F => //=. - exact: finite_measure_sigma_finite. - apply: eq_nneseries => n _; apply eq_integral => x _. - by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod1. -transitivity (\sum_(n n _. - rewrite integral_sum(*TODO: rename to ge0_integral_sum*)//. - move=> m; apply: measurable_fun_fubini_tonelli_F => //=. - exact: finite_measure_sigma_finite. - by move=> m x _; exact: integral_ge0. -transitivity (\sum_(n n _; apply eq_nneseries => m _. - by rewrite fubini_tonelli//; exact: finite_measure_sigma_finite. -transitivity (\sum_(n n _ /=. rewrite ge0_integral_measure_series//. - by move=> y _; exact: integral_ge0. - apply: measurable_fun_fubini_tonelli_G => //=. - by apply: finite_measure_sigma_finite; exact: fm1. -transitivity (\int[[the measure _ _ of mseries m2_ 0]]_y \sum_(n n; apply: measurable_fun_fubini_tonelli_G => //=. - by apply: finite_measure_sigma_finite; exact: fm1. - by move=> n y _; exact: integral_ge0. -transitivity (\int[[the measure _ _ of mseries m2_ 0]]_y - \int[[the measure _ _ of mseries m1_ 0]]_x f (x, y)). - apply eq_integral => y _. - by rewrite ge0_integral_measure_series//; exact/measurable_fun_prod2. -transitivity (\int[m2]_y \int[mseries m1_ 0]_x f (x, y)). - by apply eq_measure_integral => A mA _ /=; rewrite m2E. -by apply eq_integral => y _; apply eq_measure_integral => A mA _ /=; rewrite m1E. -Qed. - -End sfinite_fubini. -Arguments sfinite_fubini {d d' X Y R m1} _ {m2} _ f. - -Lemma finite_kernel_finite_measure d (T : measurableType d) (R : realType) - (mu : R.-fker Datatypes_unit__canonical__measure_Measurable ~> T) : - mu tt setT < +oo. -Proof. -have [M muM] := measure_uub mu. -by rewrite /finite_measure (lt_le_trans (muM tt))// leey. -Qed. - Section kprobability. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (m : probability Y R). +Variables (R : realType) (P : probability Y R). -Definition kprobability : X -> {measure set Y -> \bar R} := fun _ : X => m. +Definition kprobability : X -> {measure set Y -> \bar R} := fun=> P. Let measurable_fun_kprobability U : measurable U -> measurable_fun setT (kprobability ^~ U). diff --git a/theories/prob_lang.v b/theories/prob_lang.v index 708c5780d9..3a6579a012 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -1,13 +1,16 @@ From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. +From mathcomp Require Import rat. Require Import mathcomp_extra boolp classical_sets signed functions cardinality. Require Import reals ereal topology normedtype sequences esum measure. -Require Import lebesgue_measure fsbigop numfun lebesgue_integral kernel. +Require Import lebesgue_measure fsbigop numfun lebesgue_integral exp kernel. +Require Import exp. (******************************************************************************) (* Semantics of a programming language PPL using s-finite kernels *) (* *) -(* bernoulli r1 == Bernoulli probability *) +(* bernoulli r1 == Bernoulli probability with r1 a proof that *) +(* r : {nonneg R} is smaller than 1 *) (* *) (* sample P == sample according to the probability P *) (* letin l k == execute l, augment the context, and execute k *) @@ -20,6 +23,10 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral kernel. (* not possible *) (* ite mf k1 k2 == access the context with the boolean function f and *) (* behaves as k1 or k2 according to the result *) +(* *) +(* poisson == Poisson distribution function *) +(* exp_density == density function for exponential distribution *) +(* *) (******************************************************************************) Set Implicit Arguments. @@ -33,18 +40,6 @@ Local Open Scope ring_scope. Local Open Scope ereal_scope. (* TODO: PR *) -Lemma setT0 (T1 : pointedType) : setT != set0 :> set T1. -Proof. by apply/eqP => /seteqP[] /(_ point) /(_ Logic.I). Qed. - -Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1). - -Lemma measurable_fun_swap d d' (X : measurableType d) (Y : measurableType d') : - measurable_fun [set: X * Y] (@swap X Y). -Proof. -by apply/prod_measurable_funP => /=; split; - [exact: measurable_fun_snd|exact: measurable_fun_fst]. -Qed. - Lemma onem1' (R : numDomainType) (p : R) : (p + `1- p = 1)%R. Proof. by rewrite /onem addrCA subrr addr0. Qed. @@ -55,6 +50,10 @@ Proof. by rewrite /onem/= subr_ge0. Qed. Definition onem_nonneg (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) := NngNum (onem_nonneg_proof p1). +Lemma expR_ge0 (R : realType) (x : R) : (0 <= expR x)%R. +Proof. by rewrite ltW// expR_gt0. Qed. +(* /TODO: PR *) + Section bernoulli. Variables (R : realType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R). Local Open Scope ring_scope. @@ -212,8 +211,8 @@ Variable (mr : measurable_fun setT r). Let measurable_fun_kscore U : measurable U -> measurable_fun setT (kscore mr ^~ U). Proof. by move=> /= _; exact: measurable_fun_mscore. Qed. -HB.instance Definition _ := isKernel.Build _ _ T _ - (*Datatypes_unit__canonical__measure_Measurable*) R (kscore mr) measurable_fun_kscore. +HB.instance Definition _ := isKernel.Build _ _ T _ R + (kscore mr) measurable_fun_kscore. Import SCORE. @@ -376,19 +375,19 @@ Definition mite (mf : measurable_fun setT f) : T -> set T' -> \bar R := Variables mf : measurable_fun setT f. -Let mite0 tb : mite mf tb set0 = 0. +Let mite0 t : mite mf t set0 = 0. Proof. by rewrite /mite; case: ifPn => //. Qed. -Let mite_ge0 tb (U : set _) : 0 <= mite mf tb U. +Let mite_ge0 t (U : set _) : 0 <= mite mf t U. Proof. by rewrite /mite; case: ifPn => //. Qed. -Let mite_sigma_additive tb : semi_sigma_additive (mite mf tb). +Let mite_sigma_additive t : semi_sigma_additive (mite mf t). Proof. -by rewrite /mite; case: ifPn => ftb; exact: measure_semi_sigma_additive. +by rewrite /mite; case: ifPn => ft; exact: measure_semi_sigma_additive. Qed. -HB.instance Definition _ tb := isMeasure.Build _ _ _ (mite mf tb) - (mite0 tb) (mite_ge0 tb) (@mite_sigma_additive tb). +HB.instance Definition _ t := isMeasure.Build _ _ _ (mite mf t) + (mite0 t) (mite_ge0 t) (@mite_sigma_additive t). Import ITE. @@ -400,6 +399,27 @@ Definition kite := End ite. +(* wip *) +Section dist_salgebra_instance. +Variables (d : measure_display) (T : measurableType d) (R : realType). +Variables p0 : probability T R. + +Definition prob_pointed := Pointed.Class + (Choice.Class gen_eqMixin (Choice.Class gen_eqMixin gen_choiceMixin)) p0. + +Canonical probability_eqType := EqType (probability T R) prob_pointed. +Canonical probability_choiceType := ChoiceType (probability T R) prob_pointed. +Canonical probability_ptType := PointedType (probability T R) prob_pointed. + +Definition mset (U : set T) (r : R) := [set mu : probability T R | mu U < r%:E]. + +Definition pset : set (set (probability T R)) := + [set mset U r | r in `[0%R,1%R]%classic & U in @measurable d T]. + +Definition sset := [the measurableType pset.-sigma of salgebraType pset]. + +End dist_salgebra_instance. + Section insn2. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variable R : realType. @@ -410,17 +430,15 @@ Definition ret (f : X -> Y) (mf : measurable_fun setT f) := Definition sample (P : probability Y R) := locked [the R.-pker X ~> Y of kprobability P] . -Definition pnormalize (k : R.-sfker X ~> Y) P := - locked [the R.-pker X ~> Y of knormalize k P]. - -Definition dnormalize t (k : R.-sfker X ~> Y) P := - locked [the probability _ _ of mnormalize k P t]. +Definition normalize (k : R.-sfker X ~> Y) P x := + locked [the probability _ _ of mnormalize k P x]. Definition ite (f : X -> bool) (mf : measurable_fun setT f) (k1 k2 : R.-sfker X ~> Y):= locked [the R.-sfker X ~> Y of kite k1 k2 mf]. End insn2. +Arguments ret {d d' X Y R f} mf. Arguments sample {d d' X Y R}. Section insn2_lemmas. @@ -428,26 +446,18 @@ Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variable R : realType. Lemma retE (f : X -> Y) (mf : measurable_fun setT f) x : - ret R mf x = \d_(f x) :> (_ -> _). + ret mf x = \d_(f x) :> (_ -> \bar R). Proof. by rewrite [in LHS]/ret; unlock. Qed. Lemma sampleE (P : probability Y R) (x : X) : sample P x = P. Proof. by rewrite [in LHS]/sample; unlock. Qed. -Lemma pnormalizeE (f : R.-sfker X ~> Y) P x U : - pnormalize f P x U = +Lemma normalizeE (f : R.-sfker X ~> Y) P x U : + normalize f P x U = if (f x [set: Y] == 0) || (f x [set: Y] == +oo) then P U else f x U * ((fine (f x [set: Y]))^-1)%:E. Proof. -by rewrite /pnormalize; unlock => /=; rewrite /mnormalize; case: ifPn. -Qed. - -Lemma dnormalizeE (f : R.-sfker X ~> Y) P x U : - dnormalize x f P U = - if (f x [set: Y] == 0) || (f x [set: Y] == +oo) then P U - else f x U * ((fine (f x [set: Y]))^-1)%:E. -Proof. -by rewrite /dnormalize; unlock => /=; rewrite /mnormalize; case: ifPn. +by rewrite /normalize; unlock => /=; rewrite /mnormalize; case: ifPn. Qed. Lemma iteE (f : X -> bool) (mf : measurable_fun setT f) @@ -492,31 +502,30 @@ Proof. by rewrite /letin; unlock. Qed. End insn3_lemmas. -(* a few laws *) - +(* rewriting laws *) Section letin_return. Variables (d d' d3 : _) (R : realType) (X : measurableType d) (Y : measurableType d') (Z : measurableType d3). Lemma letin_kret (k : R.-sfker X ~> Y) - (f : _ -> Z) (mf : measurable_fun setT f) x U : + (f : X * Y -> Z) (mf : measurable_fun setT f) x U : measurable U -> - letin k (ret R mf) x U = k x (curry f x @^-1` U). + letin k (ret mf) x U = k x (curry f x @^-1` U). Proof. -move=> mU. -rewrite letinE. +move=> mU; rewrite letinE. under eq_integral do rewrite retE. rewrite integral_indic ?setIT//. -move/measurable_fun_prod1 : mf => /(_ x)/(_ measurableT U mU). +move/measurable_fun_prod1 : mf => /(_ x measurableT U mU). by rewrite setTI. Qed. -Lemma letin_retk (k : R.-sfker [the measurableType (d, d').-prod of (X * Y)%type] ~> Z) - (f : _ -> Y) (mf : measurable_fun setT f) : - forall x U, measurable U -> letin (ret R mf) k x U = k (x, f x) U. +Lemma letin_retk + (f : X -> Y) (mf : measurable_fun setT f) + (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + x U : measurable U -> + letin (ret mf) k x U = k (x, f x) U. Proof. -move=> x U mU. -rewrite letinE retE integral_dirac//. +move=> mU; rewrite letinE retE integral_dirac//. by rewrite indicE mem_set// mul1e. have /measurable_fun_prod1 := measurable_kernel k _ mU. exact. @@ -532,60 +541,54 @@ Definition score (f : X -> R) (mf : measurable_fun setT f) := End insn1. +Module Notations. + +Notation var1_of2 := (@measurable_fun_fst _ _ _ _). +Notation var2_of2 := (@measurable_fun_snd _ _ _ _). +Notation var1_of3 := (measurable_fun_comp (@measurable_fun_fst _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var3_of3 := (@measurable_fun_snd _ _ _ _). + +Notation mR := Real_sort__canonical__measure_Measurable. +Notation munit := Datatypes_unit__canonical__measure_Measurable. +Notation mbool := Datatypes_bool__canonical__measure_Measurable. + +End Notations. + Section insn1_lemmas. +Import Notations. Variables (R : realType) (d : _) (T : measurableType d). -Lemma scoreE' d' (T' : measurableType d') d2 (T2 : measurableType d2) (U : set T') - (g : R.-sfker [the measurableType _ of (T2 * unit)%type] ~> T') r fh (mh : measurable_fun setT fh) : - (score mh \; g) r U = - g (r, tt) U * `|fh r|%:E. -Proof. -rewrite [in LHS]/score [in LHS]/=. -rewrite /kcomp. -rewrite /kscore. -rewrite [in LHS]/=. -rewrite ge0_integral_mscale//=. -rewrite integral_dirac// normr_id muleC. -by rewrite indicE in_setT mul1e. -Qed. - -Lemma scoreE (t : T) (U : set bool) (n : nat) (b : bool) - (f : R -> R) - (f0 : forall r, (0 <= r)%R -> (0 <= f r)%R) - (mf : measurable_fun setT f) : - score (measurable_fun_comp mf (@measurable_fun_snd _ _ _ _)) - (t, b, n%:R) (curry (snd \o fst) (t, b) @^-1` U) = - (f n%:R)%:E * \d_b U. -Proof. -transitivity (letin ( - score (measurable_fun_comp mf (measurable_fun_snd (T2:=Real_sort__canonical__measure_Measurable R))) - ) ( - ret R (@measurable_fun_id _ _ _) -) (t, b, n%:R) (curry (snd \o fst) (t, b) @^-1` U)). - rewrite letin_kret//. - rewrite /curry/=. - rewrite preimage_cst. - by case: ifPn => //. -rewrite /letin. -unlock. -rewrite scoreE'//. -rewrite retE. -by rewrite ger0_norm// ?f0//= muleC. +Let kcomp_scoreE d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) + (g : R.-sfker [the measurableType _ of (T1 * unit)%type] ~> T2) + f (mf : measurable_fun setT f) r U : + (score mf \; g) r U = `|f r|%:E * g (r, tt) U. +Proof. +rewrite /= /kcomp /kscore /= ge0_integral_mscale//= normr_id. +by rewrite integral_dirac// indicE in_setT mul1e. Qed. -(* example of property *) -Lemma score_score (f : R -> R) (g : R * unit -> R) (mf : measurable_fun setT f) +Lemma scoreE d' (T' : measurableType d') (x : T * T') (U : set T') (f : R -> R) + (r : R) (r0 : (0 <= r)%R) + (f0 : (forall r, 0 <= r -> 0 <= f r)%R) (mf : measurable_fun setT f) : + score (measurable_fun_comp mf var2_of2) + (x, r) (curry (snd \o fst) x @^-1` U) = + (f r)%:E * \d_x.2 U. +Proof. +by rewrite /score/= /mscale/= normr_id ger0_norm// f0. +Qed. + +Lemma score_score (f : R -> R) (g : R * unit -> R) + (mf : measurable_fun setT f) (mg : measurable_fun setT g) x U : letin (score mf) (score mg) x U = - if U == set0 then 0 else `|f x|%:E * `|g (x, tt)|%:E. + score (measurable_funM mf (measurable_fun_prod2 tt mg)) x U. Proof. rewrite {1}/letin. unlock. -rewrite scoreE'//=. -rewrite /mscale/= diracE !normr_id. -have [->|->]:= set_unit U. - by rewrite eqxx in_set0 mule0 mul0e. -by rewrite in_setT mule1 (negbTE (setT0 _)) muleC. +by rewrite kcomp_scoreE/= /mscale/= diracE !normr_id normrM muleA EFinM. Qed. End insn1_lemmas. @@ -615,6 +618,47 @@ Qed. End letin_ite. +Section letinC. +Variables (d d1 : _) (X : measurableType d) (Y : measurableType d1). +Variables (R : realType) (d' : _) (Z : measurableType d'). + +Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var3_of3 := (@measurable_fun_snd _ _ _ _). + +Variables (t : R.-sfker Z ~> X) + (t' : R.-sfker [the measurableType _ of (Z * Y)%type] ~> X) + (tt' : forall y, t =1 fun z => t' (z, y)) + (u : R.-sfker Z ~> Y) + (u' : R.-sfker [the measurableType _ of (Z * X)%type] ~> Y) + (uu' : forall x, u =1 fun z => u' (z, x)). + +Lemma letinC z A : measurable A -> + letin t + (letin u' + (ret (measurable_fun_pair var2_of3 var3_of3))) z A = + letin u + (letin t' + (ret (measurable_fun_pair var3_of3 var2_of3))) z A. +Proof. +move=> mA. +rewrite !letinE. +under eq_integral. + move=> x _. + rewrite letinE/= -uu'. + under eq_integral do rewrite retE /=. + over. +rewrite (sfinite_fubini _ _ (fun x => \d_(x.1, x.2) A ))//; last 3 first. + exact: sfinite_kernel_measure. + exact: sfinite_kernel_measure. + apply/EFin_measurable_fun => /=; rewrite (_ : (fun x => _) = mindic R mA)//. + by apply/funext => -[]. +apply eq_integral => y _. +by rewrite letinE/= -tt'; apply eq_integral => // x _; rewrite retE. +Qed. + +End letinC. + (* sample programs *) Section constants. @@ -630,39 +674,63 @@ Proof. by rewrite /= lter_pdivr_mulr// mul1r ler_nat. Qed. End constants. Arguments p27 {R}. -Require Import exp. - -Definition poisson (R : realType) (r : R) (k : nat) := (r ^+ k / k%:R^-1 * expR (- r))%R. +Section poisson. +Variable R : realType. +Local Open Scope ring_scope. -Definition poisson3 (R : realType) := poisson (3%:R : R) 4. (* 0.168 *) -Definition poisson10 (R : realType) := poisson (10%:R : R) 4. (* 0.019 *) +(* density function for Poisson *) +Definition poisson k r : R := r ^+ k / k`!%:R^-1 * expR (- r). -Lemma poisson_ge0 (R : realType) (r : R) k : (0 <= r)%R -> (0 <= poisson r k)%R. +Lemma poisson_ge0 k r : 0 <= r -> 0 <= poisson k r. Proof. -move=> r0; rewrite /poisson mulr_ge0//. - by rewrite mulr_ge0// exprn_ge0//. -by rewrite ltW// expR_gt0. +move=> r0; rewrite /poisson mulr_ge0 ?expR_ge0//. +by rewrite mulr_ge0// exprn_ge0. Qed. -Lemma poisson_gt0 (R : realType) (r : R) k : (0 < r)%R -> (0 < poisson r k.+1)%R. +Lemma poisson_gt0 k r : 0 < r -> 0 < poisson k.+1 r. Proof. -move=> r0; rewrite /poisson mulr_gt0//. - by rewrite mulr_gt0// exprn_gt0. -by rewrite expR_gt0. +move=> r0; rewrite /poisson mulr_gt0 ?expR_gt0//. +by rewrite divr_gt0// ?exprn_gt0// invr_gt0 ltr0n fact_gt0. Qed. -Lemma mpoisson (R : realType) k : measurable_fun setT (@poisson R ^~ k). +Lemma mpoisson k : measurable_fun setT (poisson k). Proof. apply: measurable_funM => /=. apply: measurable_funM => //=; last exact: measurable_fun_cst. exact/measurable_fun_exprn/measurable_fun_id. +apply: measurable_fun_comp; last exact: measurable_fun_opp. +by apply: continuous_measurable_fun; exact: continuous_expR. +Qed. + +Definition poisson3 := poisson 4 3. (* 0.168 *) +Definition poisson10 := poisson 4 10. (* 0.019 *) + +End poisson. + +Section exponential. +Variable R : realType. +Local Open Scope ring_scope. + +(* density function for exponential *) +Definition exp_density x r : R := r * expR (- r * x). + +Lemma exp_density_gt0 x r : 0 < r -> 0 < exp_density x r. +Proof. by move=> r0; rewrite /exp_density mulr_gt0// expR_gt0. Qed. + +Lemma exp_density_ge0 x r : 0 <= r -> 0 <= exp_density x r. +Proof. by move=> r0; rewrite /exp_density mulr_ge0// expR_ge0. Qed. + +Lemma mexp_density x : measurable_fun setT (exp_density x). +Proof. +apply: measurable_funM => /=; first exact: measurable_fun_id. apply: measurable_fun_comp. - apply: continuous_measurable_fun. - exact: continuous_expR. -apply: continuous_measurable_fun. -by have := (@opp_continuous R [the normedModType R of R^o]). + by apply: continuous_measurable_fun; exact: continuous_expR. +apply: measurable_funM => /=; first exact: measurable_fun_opp. +exact: measurable_fun_cst. Qed. +End exponential. + Section cst_fun. Variables (R : realType) (d : _) (T : measurableType d). @@ -674,18 +742,6 @@ End cst_fun. Arguments k3 {R d T}. Arguments k10 {R d T}. -Module Notations. - -Notation var1_of2 := (@measurable_fun_fst _ _ _ _). -Notation var2_of2 := (@measurable_fun_snd _ _ _ _). -Notation var1_of3 := (measurable_fun_comp (@measurable_fun_fst _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var3_of3 := (@measurable_fun_snd _ _ _ _). - -End Notations. - Lemma letin_sample_bernoulli (R : realType) (d d' : _) (T : measurableType d) (T' : measurableType d') (r : {nonneg R}) (r1 : (r%:num <= 1)%R) (u : R.-sfker [the measurableType _ of (T * bool)%type] ~> T') x y : @@ -701,17 +757,16 @@ by rewrite indicE in_setT mul1e indicE in_setT mul1e. Qed. Section sample_and_return. -Variables (R : realType) (d : _) (T : measurableType d). - Import Notations. +Variables (R : realType) (d : _) (T : measurableType d). Definition sample_and_return : R.-sfker T ~> _ := letin (sample (bernoulli p27)) (* T -> B *) - (ret R var2_of2) (* T * B -> B *). + (ret var2_of2) (* T * B -> B *). Lemma sample_and_returnE t U : sample_and_return t U = - (2 / 7%:R)%:E * \d_true U + (5%:R / 7%:R)%:E * \d_false U. + (2 / 7)%:E * \d_true U + (5 / 7)%:E * \d_false U. Proof. rewrite /sample_and_return. rewrite letin_sample_bernoulli/=. @@ -721,38 +776,24 @@ Qed. End sample_and_return. -Section sample_and_score. -Variables (R : realType) (d : _) (T : measurableType d). - -Definition sample_and_score : R.-sfker T ~> _ := - letin - (sample (bernoulli p27)) (* T -> B *) - (score (measurable_fun_cst (1%R : R))). - -End sample_and_score. - +(* trivial example *) Section sample_and_branch. +Import Notations. Variables (R : realType) (d : _) (T : measurableType d). (* let x = sample (bernoulli (2/7)) in let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in return r *) -Let mR := Real_sort__canonical__measure_Measurable R. - -Import Notations. - Definition sample_and_branch : - R.-sfker T ~> [the measurableType default_measure_display of mR] := + R.-sfker T ~> mR R := letin (sample (bernoulli p27)) (* T -> B *) - (ite var2_of2 - (ret R k3) - (ret R k10)). + (ite var2_of2 (ret k3) (ret k10)). Lemma sample_and_branchE t U : sample_and_branch t U = - (2 / 7%:R)%:E * \d_(3%:R : R) U + - (5%:R / 7%:R)%:E * \d_(10%:R : R) U. + (2 / 7)%:E * \d_(3 : R) U + + (5 / 7)%:E * \d_(10 : R) U. Proof. rewrite /sample_and_branch letin_sample_bernoulli/=. rewrite !iteE/= !retE. @@ -762,197 +803,124 @@ Qed. End sample_and_branch. Section staton_bus. -Variables (R : realType) (d : _) (T : measurableType d). +Import Notations. +Variables (R : realType) (d : _) (T : measurableType d) (density : R -> R). +Hypothesis mdensity : measurable_fun setT density. +Definition kstaton_bus : R.-sfker T ~> mbool := + letin (sample (bernoulli p27)) + (letin + (letin (ite var2_of2 (ret k3) (ret k10)) + (score (measurable_fun_comp mdensity var3_of3))) + (ret var2_of3)). + +Definition staton_bus := normalize kstaton_bus. + +End staton_bus. (* let x = sample (bernoulli (2/7)) in let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in let _ = score (1/4! r^4 e^-r) in return x *) - -Let mR := Real_sort__canonical__measure_Measurable R. -Let munit := Datatypes_unit__canonical__measure_Measurable. -Let mbool := Datatypes_bool__canonical__measure_Measurable. - -Variable P : probability mbool R. - +Section staton_bus_poisson. Import Notations. +Variables (R : realType) (d : _) (T : measurableType d). +Let poisson4 := @poisson R 4%N. +Let mpoisson4 := @mpoisson R 4%N. -Definition staton_bus_annotated : R.-pker T ~> mbool := - pnormalize (letin - (sample (bernoulli p27) : _.-sfker T ~> mbool) - (letin - (letin - (ite var2_of2 - (ret R k3) - (ret R k10) - : _.-sfker [the measurableType _ of (T * bool)%type] ~> mR) - (score (measurable_fun_comp (@mpoisson R 4) var3_of3) - : _.-sfker [the measurableType _ of (T * bool* mR)%type] ~> munit) - : _.-sfker [the measurableType _ of (T * bool)%type] ~> munit) - (ret R var2_of3 - : _.-sfker [the measurableType _ of (T * bool * munit)%type] ~> mbool) - : _.-sfker [the measurableType _ of (T * bool)%type] ~> mbool)) P. - -Let staton_bus' : R.-sfker T ~> _ := - (letin (sample (bernoulli p27)) - (letin - (letin (ite var2_of2 - (ret R k3) - (ret R k10)) - (score (measurable_fun_comp (@mpoisson R 4) var3_of3))) - (ret R var2_of3))). +Definition kstaton_bus_poisson : R.-sfker (mR R) ~> mbool := + kstaton_bus _ mpoisson4. -(* true -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) -(* false -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) - -Let staton_bus'E t U : staton_bus' t U = - (2 / 7%:R)%:E * (poisson 3%:R 4)%:E * \d_true U + - (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U. +Let kstaton_bus_poissonE t U : kstaton_bus_poisson t U = + (2 / 7)%:E * (poisson4 3)%:E * \d_true U + + (5 / 7)%:E * (poisson4 10)%:E * \d_false U. Proof. -rewrite /staton_bus'. +rewrite /kstaton_bus. rewrite letin_sample_bernoulli. rewrite -!muleA; congr (_ * _ + _ * _). - rewrite letin_kret//. rewrite letin_iteT//. rewrite letin_retk//. - by rewrite scoreE// => r r0; exact: poisson_ge0. + by rewrite scoreE//= => r r0; exact: poisson_ge0. - by rewrite onem27. rewrite letin_kret//. rewrite letin_iteF//. rewrite letin_retk//. - by rewrite scoreE// => r r0; exact: poisson_ge0. + by rewrite scoreE//= => r r0; exact: poisson_ge0. Qed. -Definition staton_bus : R.-pker T ~> mbool := pnormalize staton_bus' P. +(* true -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) +(* false -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) -Lemma staton_busE t U : - let N := ((2 / 7%:R) * poisson 3%:R 4 + - (5%:R / 7%:R) * poisson 10%:R 4)%R in - staton_bus t U = - ((2 / 7%:R)%:E * (poisson 3%:R 4)%:E * \d_true U + - (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. +Lemma staton_busE P (t : R) U : + let N := ((2 / 7) * poisson4 3 + + (5 / 7) * poisson4 10)%R in + staton_bus mpoisson4 P t U = + ((2 / 7)%:E * (poisson4 3)%:E * \d_true U + + (5 / 7)%:E * (poisson4 10)%:E * \d_false U) * N^-1%:E. Proof. rewrite /staton_bus. -rewrite pnormalizeE /=. -rewrite !staton_bus'E. +rewrite normalizeE /=. +rewrite !kstaton_bus_poissonE. rewrite diracE mem_set// mule1. rewrite diracE mem_set// mule1. rewrite ifF //. apply/negbTE. -by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= poisson_gt0. +by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// poisson_gt0// ltr0n. Qed. -Definition dstaton_bus (t : T) : probability mbool R := dnormalize t staton_bus' P. +End staton_bus_poisson. -Lemma dstaton_busE t U : - let N := ((2 / 7%:R) * poisson 3%:R 4 + - (5%:R / 7%:R) * poisson 10%:R 4)%R in - dstaton_bus t U = - ((2 / 7%:R)%:E * (poisson 3%:R 4)%:E * \d_true U + - (5%:R / 7%:R)%:E * (poisson 10%:R 4)%:E * \d_false U) * N^-1%:E. -Proof. -rewrite /staton_bus. -rewrite dnormalizeE /=. -rewrite !staton_bus'E. -rewrite diracE mem_set// mule1. -rewrite diracE mem_set// mule1. -rewrite ifF //. -apply/negbTE. -by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= poisson_gt0. -Qed. - -End staton_bus. - -(* TODO: move *) -Section measurable_fun_pair. -Variables (d d' d3 : _) (X : measurableType d) - (Y : measurableType d') (Z : measurableType d3). - -Lemma measurable_fun_pair (f : X -> Y) (g : X -> Z) : - measurable_fun setT f -> - measurable_fun setT g -> - measurable_fun setT (fun x => (f x, g x)). -Proof. -by move=> mf mg; apply/prod_measurable_funP. -Qed. +(* let x = sample (bernoulli (2/7)) in + let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in + let _ = score (r e^-(15/60 r)) in + return x *) +Section staton_bus_exponential. +Import Notations. +Variables (R : realType) (d : _) (T : measurableType d). +Let exp1560 := @exp_density R (ratr (15%:Q / 60%:Q)). +Let mexp1560 := @mexp_density R (ratr (15%:Q / 60%:Q)). -End measurable_fun_pair. +(* 15/60 = 0.25 *) -(* TODO: move *) -Lemma finite_kernel_measure (d d' : _) (X : measurableType d) - (Y : measurableType d') (R : realType) (k : R.-fker X ~> Y) (x : X) : - finite_measure (k x). -Proof. -have [r k_r] := measure_uub k. -by rewrite /finite_measure (@lt_trans _ _ r%:E) ?ltey. -Qed. +Definition kstaton_bus_exponential : R.-sfker (mR R) ~> mbool := + kstaton_bus _ mexp1560. -Lemma sfinite_kernel_measure (d d' : _) (X : measurableType d) - (Y : measurableType d') (R : realType) (k : R.-sfker X ~> Y) (x : X) : - sfinite_measure (k x). +Let kstaton_bus_exponentialE t U : kstaton_bus_exponential t U = + (2 / 7)%:E * (exp1560 3)%:E * \d_true U + + (5 / 7)%:E * (exp1560 10)%:E * \d_false U. Proof. -have [k_ k_E] := sfinite k. -exists (fun n => k_ n x); split; last by move=> A mA; rewrite k_E. -by move=> n; exact: finite_kernel_measure. +rewrite /kstaton_bus. +rewrite letin_sample_bernoulli. +rewrite -!muleA; congr (_ * _ + _ * _). +- rewrite letin_kret//. + rewrite letin_iteT//. + rewrite letin_retk//. + rewrite scoreE//= => r r0; exact: exp_density_ge0. +- by rewrite onem27. + rewrite letin_kret//. + rewrite letin_iteF//. + rewrite letin_retk//. + by rewrite scoreE//= => r r0; exact: exp_density_ge0. Qed. -Section letinC. -Variables (d d1 : _) (X : measurableType d) (Y : measurableType d1). -Variables (R : realType) (d' : _) (Z : measurableType d'). - -Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var3_of3 := (@measurable_fun_snd _ _ _ _). - -Variables (t : R.-sfker Z ~> X) - (t' : R.-sfker [the measurableType _ of (Z * Y)%type] ~> X) - (tt' : forall y, t =1 fun z => t' (z, y)) - (u : R.-sfker Z ~> Y) - (u' : R.-sfker [the measurableType _ of (Z * X)%type] ~> Y) - (uu' : forall x, u =1 fun z => u' (z, x)). +(* true -> 5/7 * 0.019 = 5/7 * 10^4 e^-10 / 4! *) +(* false -> 2/7 * 0.168 = 2/7 * 3^4 e^-3 / 4! *) -Lemma letinC z A : measurable A -> - letin t - (letin u' - (ret R (measurable_fun_pair var2_of3 var3_of3))) z A = - letin u - (letin t' - (ret R (measurable_fun_pair var3_of3 var2_of3))) z A. +Lemma staton_bus_exponentialE P (t : R) U : + let N := ((2 / 7) * exp1560 3 + + (5 / 7) * exp1560 10)%R in + staton_bus mexp1560 P t U = + ((2 / 7)%:E * (exp1560 3)%:E * \d_true U + + (5 / 7)%:E * (exp1560 10)%:E * \d_false U) * N^-1%:E. Proof. -move=> mA. -rewrite !letinE. -under eq_integral. - move=> x _. - rewrite letinE/= -uu'. - under eq_integral do rewrite retE /=. - over. -rewrite (sfinite_fubini _ _ (fun x => \d_(x.1, x.2) A ))//; last 3 first. - exact: sfinite_kernel_measure. - exact: sfinite_kernel_measure. - apply/EFin_measurable_fun => /=; rewrite (_ : (fun x => _) = mindic R mA)//. - by apply/funext => -[]. -apply eq_integral => y _. -by rewrite letinE/= -tt'; apply eq_integral => // x _; rewrite retE. +rewrite /staton_bus. +rewrite normalizeE /=. +rewrite !kstaton_bus_exponentialE. +rewrite diracE mem_set// mule1. +rewrite diracE mem_set// mule1. +rewrite ifF //. +apply/negbTE. +by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// exp_density_gt0 ?ltr0n. Qed. -End letinC. - -Section dist_salgebra_instance. -Variables (d : measure_display) (T : measurableType d) (R : realType). -Variables p0 : probability T R. - -Definition prob_pointed := Pointed.Class - (Choice.Class gen_eqMixin (Choice.Class gen_eqMixin gen_choiceMixin)) p0. - -Canonical probability_eqType := EqType (probability T R) prob_pointed. -Canonical probability_choiceType := ChoiceType (probability T R) prob_pointed. -Canonical probability_ptType := PointedType (probability T R) prob_pointed. - -Definition mset (U : set T) (r : R) := [set mu : probability T R | mu U < r%:E]. - -Definition pset : set (set (probability T R)) := - [set mset U r | r in `[0%R,1%R]%classic & U in @measurable d T]. - -Definition sset := [the measurableType pset.-sigma of salgebraType pset]. - -End dist_salgebra_instance. +End staton_bus_exponential. From 1a63831eeeb4d30a5d8e3e606a0b783b9bbabea8 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 14 Sep 2022 12:22:46 +0900 Subject: [PATCH 39/42] wip (gauss) --- _CoqProject | 1 + theories/kernel.v | 96 ++++++++++++++-------------- theories/prob_lang.v | 21 +++--- theories/wip.v | 149 +++++++++++++++++++++++++++++++++++++++++++ 4 files changed, 207 insertions(+), 60 deletions(-) create mode 100644 theories/wip.v diff --git a/_CoqProject b/_CoqProject index b5a4ba0fef..600698fb52 100644 --- a/_CoqProject +++ b/_CoqProject @@ -34,6 +34,7 @@ theories/numfun.v theories/lebesgue_integral.v theories/kernel.v theories/prob_lang.v +theories/wip.v theories/summability.v theories/functions.v theories/signed.v diff --git a/theories/kernel.v b/theories/kernel.v index 08083a9df4..cecc473867 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -313,9 +313,8 @@ Definition finite_measure d (T : measurableType d) (R : realType) Definition sfinite_measure d (T : measurableType d) (R : realType) (mu : set T -> \bar R) := - exists mu_ : {measure set T -> \bar R}^nat, - (forall n, finite_measure (mu_ n)) /\ - (forall U, measurable U -> mu U = mseries mu_ 0 U). + exists2 mu_ : {measure set T -> \bar R}^nat, + forall n, finite_measure (mu_ n) & forall U, measurable U -> mu U = mseries mu_ 0 U. Lemma finite_measure_sigma_finite d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}) : @@ -337,8 +336,8 @@ Variable (mf : measurable_fun setT f). Lemma sfinite_fubini : \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). Proof. -have [m1_ [fm1 m1E]] := sfm1. -have [m2_ [fm2 m2E]] := sfm2. +have [m1_ fm1 m1E] := sfm1. +have [m2_ fm2 m2E] := sfm2. rewrite [LHS](eq_measure_integral [the measure _ _ of mseries m1_ 0]); last first. by move=> A mA _; rewrite m1E. transitivity (\int[[the measure _ _ of mseries m1_ 0]]_x @@ -417,7 +416,7 @@ Lemma measurable_fun_kseries (U : set Y) : measurable_fun setT (kseries ^~ U). Proof. move=> mU; rewrite /kseries /= /mseries. -by apply: ge0_emeasurable_fun_sum => // n; apply/measurable_kernel. +by apply: ge0_emeasurable_fun_sum => // n; exact/measurable_kernel. Qed. HB.instance Definition _ := @@ -579,7 +578,7 @@ Lemma sfinite_kernel_measure (d d' : _) (X : measurableType d) sfinite_measure (k x). Proof. have [k_ k_E] := sfinite k. -exists (fun n => k_ n x); split; last by move=> A mA; rewrite k_E. +exists (fun n => k_ n x); last by move=> A mA; rewrite k_E. by move=> n; exact: finite_kernel_measure. Qed. @@ -589,48 +588,51 @@ Qed. measurability of each measure of the family) - as a consequence, m2D_bounded holds for all x *) Section measurable_prod_subset_kernel. -Variables (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Implicit Types A : set (T1 * T2). +Implicit Types A : set (X * Y). Section xsection_kernel. -Variable (m2 : R.-ker T1 ~> T2) (D : set T2) (mD : measurable D). -Let m2D x := mrestr (m2 x) mD. -HB.instance Definition _ x := Measure.on (m2D x). -Let phi A := fun x => m2D x (xsection A x). -Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. - -Lemma measurable_prod_subset_xsection_kernel - (m2D_bounded : forall x, exists M, forall X, measurable X -> (m2D x X < M%:E)%E) : - measurable `<=` B. +Variable (k : R.-ker X ~> Y) (D : set Y) (mD : measurable D). +Let kD x := mrestr (k x) mD. +HB.instance Definition _ x := Measure.on (kD x). +Let phi A := fun x => kD x (xsection A x). +Let XY := [set A | measurable A /\ measurable_fun setT (phi A)]. + +Let phiM (A : set X) B : phi (A `*` B) = (fun x => kD x B * (\1_A x)%:E). +Proof. +rewrite funeqE => x; rewrite indicE /phi/=. +have [xA|xA] := boolP (x \in A); first by rewrite mule1 in_xsectionM. +by rewrite mule0 notin_xsectionM// set0I measure0. +Qed. + +Lemma measurable_prod_subset_xsection_kernel : + (forall x, exists M, forall X, measurable X -> kD x X < M%:E) -> + measurable `<=` XY. Proof. -rewrite measurable_prod_measurableType. -set C := [set A1 `*` A2 | A1 in measurable & A2 in measurable]. +move=> kD_ub; rewrite measurable_prod_measurableType. +set C := [set A `*` B | A in measurable & B in measurable]. have CI : setI_closed C. - move=> X Y [X1 mX1 [X2 mX2 <-{X}]] [Y1 mY1 [Y2 mY2 <-{Y}]]. + move=> _ _ [X1 mX1 [X2 mX2 <-]] [Y1 mY1 [Y2 mY2 <-]]. exists (X1 `&` Y1); first exact: measurableI. by exists (X2 `&` Y2); [exact: measurableI|rewrite setMI]. have CT : C setT by exists setT => //; exists setT => //; rewrite setMTT. -have CB : C `<=` B. - move=> X [X1 mX1 [X2 mX2 <-{X}]]; split; first exact: measurableM. - have -> : phi (X1 `*` X2) = (fun x => m2D x X2 * (\1_X1 x)%:E)%E. - rewrite funeqE => x; rewrite indicE /phi /m2/= /mrestr. - have [xX1|xX1] := boolP (x \in X1); first by rewrite mule1 in_xsectionM. - by rewrite mule0 notin_xsectionM// set0I measure0. +have CXY : C `<=` XY. + move=> _ [A mA [B mB <-]]; split; first exact: measurableM. + rewrite phiM. apply: emeasurable_funM => //; first exact/measurable_kernel/measurableI. - apply/EFin_measurable_fun. - by rewrite (_ : \1_ _ = mindic R mX1). -suff monoB : monotone_class setT B by exact: monotone_class_subset. -split => //; [exact: CB| |exact: xsection_ndseq_closed]. -move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. -suff : phi (X `\` Y) = (fun x => phi X x - phi Y x)%E. + by apply/EFin_measurable_fun; rewrite (_ : \1_ _ = mindic R mA). +suff monoB : monotone_class setT XY by exact: monotone_class_subset. +split => //; [exact: CXY| |exact: xsection_ndseq_closed]. +move=> A B BA [mA mphiA] [mB mphiB]; split; first exact: measurableD. +suff : phi (A `\` B) = (fun x => phi A x - phi B x). by move=> ->; exact: emeasurable_funB. -rewrite funeqE => x; rewrite /phi/= xsectionD// /m2D measureD. +rewrite funeqE => x; rewrite /phi/= xsectionD// measureD. - by rewrite setIidr//; exact: le_xsection. - exact: measurable_xsection. - exact: measurable_xsection. -- move: (m2D_bounded x) => [M m2M]. - rewrite (lt_le_trans (m2M (xsection X x) _))// ?leey//. +- have [M kM] := kD_ub x. + rewrite (lt_le_trans (kM (xsection A x) _)) ?leey//. exact: measurable_xsection. Qed. @@ -642,26 +644,24 @@ End measurable_prod_subset_kernel. the difference is that this section uses a finite kernel m2 instead of a sigma-finite measure m2 *) Section measurable_fun_xsection_finite_kernel. -Variables (d1 d2 : _) (T1 : measurableType d1) (T2 : measurableType d2) +Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Variable m2 : R.-fker T1 ~> T2. -Implicit Types A : set (T1 * T2). +Variable k : R.-fker X ~> Y. +Implicit Types A : set (X * Y). -Let phi A := fun x => m2 x (xsection A x). -Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. +Let phi A := fun x => k x (xsection A x). +Let XY := [set A | measurable A /\ measurable_fun setT (phi A)]. Lemma measurable_fun_xsection_finite_kernel A : A \in measurable -> measurable_fun setT (phi A). Proof. -move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. -move=> /= X mX; rewrite /B/=; split => //; rewrite /phi. -rewrite -(_ : (fun x => mrestr (m2 x) measurableT (xsection X x)) = - (fun x => m2 x (xsection X x)))//; last first. +move: A; suff : measurable `<=` XY by move=> + A; rewrite inE => /[apply] -[]. +move=> /= A mA; rewrite /XY/=; split => //; rewrite (_ : phi _ = + (fun x => mrestr (k x) measurableT (xsection A x))); last first. by apply/funext => x//=; rewrite /mrestr setIT. apply measurable_prod_subset_xsection_kernel => // x. -have [r hr] := measure_uub m2; exists r => Y mY. -rewrite (le_lt_trans _ (hr x)) // /mrestr /= setIT. -by apply: le_measure => //; rewrite inE. +have [r hr] := measure_uub k; exists r => B mB. +by rewrite (le_lt_trans _ (hr x)) // /mrestr /= setIT le_measure// inE. Qed. End measurable_fun_xsection_finite_kernel. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index 3a6579a012..2bc2e68b11 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -4,7 +4,6 @@ From mathcomp Require Import rat. Require Import mathcomp_extra boolp classical_sets signed functions cardinality. Require Import reals ereal topology normedtype sequences esum measure. Require Import lebesgue_measure fsbigop numfun lebesgue_integral exp kernel. -Require Import exp. (******************************************************************************) (* Semantics of a programming language PPL using s-finite kernels *) @@ -32,7 +31,7 @@ Require Import exp. Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. -Import Order.TTheory GRing.Theory Num.Def Num.Theory. +Import Order.TTheory GRing.Theory Num.Def Num.ExtraDef Num.Theory. Import numFieldTopology.Exports. Local Open Scope classical_set_scope. @@ -84,14 +83,14 @@ Variables (d : _) (T : measurableType d). Variables (R : realType) (f : T -> R). Definition mscore t : {measure set _ -> \bar R} := - let p := NngNum (@normr_ge0 _ _ (`| f t |)%R) in + let p := NngNum (normr_ge0 (f t)) in [the measure _ _ of mscale p [the measure _ _ of dirac tt]]. Lemma mscoreE t U : mscore t U = if U == set0 then 0 else `| (f t)%:E |. Proof. rewrite /mscore/= /mscale/=; have [->|->] := set_unit U. by rewrite eqxx diracE in_set0 mule0. -by rewrite diracE in_setT mule1 (negbTE (setT0 _)) normr_id. +by rewrite diracE in_setT mule1 (negbTE (setT0 _)). Qed. Lemma measurable_fun_mscore U : measurable_fun setT f -> @@ -566,7 +565,7 @@ Let kcomp_scoreE d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) f (mf : measurable_fun setT f) r U : (score mf \; g) r U = `|f r|%:E * g (r, tt) U. Proof. -rewrite /= /kcomp /kscore /= ge0_integral_mscale//= normr_id. +rewrite /= /kcomp /kscore /= ge0_integral_mscale//=. by rewrite integral_dirac// indicE in_setT mul1e. Qed. @@ -576,9 +575,7 @@ Lemma scoreE d' (T' : measurableType d') (x : T * T') (U : set T') (f : R -> R) score (measurable_fun_comp mf var2_of2) (x, r) (curry (snd \o fst) x @^-1` U) = (f r)%:E * \d_x.2 U. -Proof. -by rewrite /score/= /mscale/= normr_id ger0_norm// f0. -Qed. +Proof. by rewrite /score/= /mscale/= ger0_norm// f0. Qed. Lemma score_score (f : R -> R) (g : R * unit -> R) (mf : measurable_fun setT f) @@ -588,7 +585,7 @@ Lemma score_score (f : R -> R) (g : R * unit -> R) Proof. rewrite {1}/letin. unlock. -by rewrite kcomp_scoreE/= /mscale/= diracE !normr_id normrM muleA EFinM. +by rewrite kcomp_scoreE/= /mscale/= diracE normrM muleA EFinM. Qed. End insn1_lemmas. @@ -804,13 +801,13 @@ End sample_and_branch. Section staton_bus. Import Notations. -Variables (R : realType) (d : _) (T : measurableType d) (density : R -> R). -Hypothesis mdensity : measurable_fun setT density. +Variables (R : realType) (d : _) (T : measurableType d) (h : R -> R). +Hypothesis mh : measurable_fun setT h. Definition kstaton_bus : R.-sfker T ~> mbool := letin (sample (bernoulli p27)) (letin (letin (ite var2_of2 (ret k3) (ret k10)) - (score (measurable_fun_comp mdensity var3_of3))) + (score (measurable_fun_comp mh var3_of3))) (ret var2_of3)). Definition staton_bus := normalize kstaton_bus. diff --git a/theories/wip.v b/theories/wip.v new file mode 100644 index 0000000000..334697b692 --- /dev/null +++ b/theories/wip.v @@ -0,0 +1,149 @@ +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. +From mathcomp Require Import rat. +Require Import mathcomp_extra boolp classical_sets signed functions cardinality. +Require Import reals ereal topology normedtype sequences esum measure. +Require Import lebesgue_measure fsbigop numfun lebesgue_integral exp kernel. +Require Import trigo prob_lang. + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. +Import Order.TTheory GRing.Theory Num.Def Num.ExtraDef Num.Theory. +Import numFieldTopology.Exports. + +Local Open Scope classical_set_scope. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. + +Section gauss. +Variable R : realType. +Local Open Scope ring_scope. + +(* density function for gauss *) +Definition gauss_density m s x : R := + (s * sqrtr (pi *+ 2))^-1 * expR (- ((x - m) / s) ^+ 2 / 2%:R). + +Lemma gauss_density_ge0 m s x : 0 <= s -> 0 <= gauss_density m s x. +Proof. by move=> s0; rewrite mulr_ge0 ?expR_ge0// invr_ge0 mulr_ge0. Qed. + +Lemma gauss_density_gt0 m s x : 0 < s -> 0 < gauss_density m s x. +Proof. +move=> s0; rewrite mulr_gt0 ?expR_gt0// invr_gt0 mulr_gt0//. +by rewrite sqrtr_gt0 pmulrn_rgt0// pi_gt0. +Qed. + +Definition gauss01_density : R -> R := gauss_density 0 1. + +Lemma gauss01_densityE x : + gauss01_density x = (sqrtr (pi *+ 2))^-1 * expR (- (x ^+ 2) / 2%:R). +Proof. by rewrite /gauss01_density /gauss_density mul1r subr0 divr1. Qed. + +Definition mgauss01 (V : set R) := + \int[lebesgue_measure]_(x in V) (gauss01_density x)%:E. + +Lemma integral_gauss01_density : + \int[lebesgue_measure]_x (gauss01_density x)%:E = 1%E. +Proof. +Admitted. + +Lemma measurable_fun_gauss_density m s : + measurable_fun setT (gauss_density m s). +Proof. +apply: measurable_funM; first exact: measurable_fun_cst. +apply: measurable_fun_comp => /=. + by apply: continuous_measurable_fun; apply continuous_expR. +apply: measurable_funM; last exact: measurable_fun_cst. +apply: measurable_fun_comp => /=; first exact: measurable_fun_opp. +apply: measurable_fun_exprn. +apply: measurable_funM => /=; last exact: measurable_fun_cst. +apply: measurable_funD => //; first exact: measurable_fun_id. +exact: measurable_fun_cst. +Qed. + +Let mgauss010 : mgauss01 set0 = 0%E. +Proof. by rewrite /mgauss01 integral_set0. Qed. + +Let mgauss01_ge0 A : (0 <= mgauss01 A)%E. +Proof. +by rewrite /mgauss01 integral_ge0//= => x _; rewrite lee_fin gauss_density_ge0. +Qed. + +Let mgauss01_sigma_additive : semi_sigma_additive mgauss01. +Proof. +move=> /= F mF tF mUF. +rewrite /mgauss01/= integral_bigcup//=; last first. + split. + apply/EFin_measurable_fun. + exact: measurable_funS (measurable_fun_gauss_density 0 1). + rewrite (_ : (fun x => _) = (EFin \o gauss01_density)); last first. + by apply/funext => x; rewrite gee0_abs// lee_fin gauss_density_ge0. + apply: le_lt_trans. + apply: (@subset_integral _ _ _ _ _ setT) => //=. + apply/EFin_measurable_fun. + exact: measurable_fun_gauss_density. + by move=> ? _; rewrite lee_fin gauss_density_ge0. + by rewrite integral_gauss01_density// ltey. +apply: is_cvg_ereal_nneg_natsum_cond => n _ _. +by apply: integral_ge0 => /= x ?; rewrite lee_fin gauss_density_ge0. +Qed. + +HB.instance Definition _ := isMeasure.Build _ _ _ + mgauss01 mgauss010 mgauss01_ge0 mgauss01_sigma_additive. + +Let mgauss01_setT : mgauss01 [set: _] = 1%E. +Proof. by rewrite /mgauss01 integral_gauss01_density. Qed. + +HB.instance Definition _ := @isProbability.Build _ _ R mgauss01 mgauss01_setT. + +Definition gauss01 := [the probability _ _ of mgauss01]. + +End gauss. + +Section gauss_lebesgue. +Import Notations. +Variables (R : realType) (d : _) (T : measurableType d). + +Let f1 (x : R) := (gauss01_density x) ^-1. + +Let mf1 : measurable_fun setT f1. +Proof. +apply: (measurable_fun_comp' (F := [set r : R | r != 0%R])) => //. +- exact: open_measurable. +- by move=> /= r [t _ <-]; rewrite gt_eqF// gauss_density_gt0. +- apply: open_continuous_measurable_fun => //. + by apply/in_setP => x /= x0; exact: inv_continuous. +- exact: measurable_fun_gauss_density. +Qed. + +Variable mu : {measure set mR R -> \bar R}. + +Definition staton_lebesgue : R.-sfker T ~> _ := + letin (sample (@gauss01 R)) + (letin + (score (measurable_fun_comp mf1 var2_of2)) + (ret var2_of3)). + +Lemma staton_lebesgueE x U : measurable U -> + staton_lebesgue x U = lebesgue_measure U. +Proof. +move=> mU; rewrite [in LHS]/staton_lebesgue/=. +rewrite [in LHS]letinE. +rewrite [in LHS]/sample. +unlock. +rewrite [in LHS]/=. +transitivity (\int[@mgauss01 R]_(y in U) (f1 y)%:E). + rewrite -[in RHS](setTI U) integral_setI_indic//=. + apply: eq_integral => /= r _. + rewrite letinE/= ge0_integral_mscale//= ger0_norm//; last first. + by rewrite invr_ge0// gauss_density_ge0. + by rewrite integral_dirac// indicE in_setT mul1e retE/= diracE indicE. +transitivity (\int[lebesgue_measure]_(x in U) (gauss01_density x * f1 x)%:E). + admit. +transitivity (\int[lebesgue_measure]_(x in U) (\1_U x)%:E). + apply: eq_integral => /= y yU. + by rewrite /f1 divrr ?indicE ?yU// unitfE gt_eqF// gauss_density_gt0. +by rewrite integral_indic//= setIid. +Abort. + +End gauss_lebesgue. From 7439b0aad6768b7c6ed59502780a589b7c24dd7e Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Fri, 16 Sep 2022 22:34:18 +0900 Subject: [PATCH 40/42] linearize hierarchy --- theories/kernel.v | 275 +++++++++++++++++++++++++++---------------- theories/prob_lang.v | 112 ++++++++++++------ 2 files changed, 250 insertions(+), 137 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index cecc473867..a94d7db6ea 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -208,27 +208,27 @@ Qed. End measurable_fun_comp. -Lemma measurable_fun_if (d d' : _) (T : measurableType d) - (T' : measurableType d') (x y : T -> T') D (md : measurable D) - (f : T -> bool) (mf : measurable_fun setT f) : +Lemma measurable_fun_if (d d' : _) (X : measurableType d) + (Y : measurableType d') (x y : X -> Y) D (md : measurable D) + (f : X -> bool) (mf : measurable_fun setT f) : measurable_fun (D `&` (f @^-1` [set true])) x -> measurable_fun (D `&` (f @^-1` [set false])) y -> measurable_fun D (fun t => if f t then x t else y t). Proof. -move=> mx my /= _ Y mY. +move=> mx my /= _ B mB. have mDf : measurable (D `&` [set b | f b]). apply: measurableI => //. rewrite [X in measurable X](_ : _ = f @^-1` [set true])//. by have := mf measurableT [set true]; rewrite setTI; exact. -have := mx mDf Y mY. +have := mx mDf _ mB. have mDNf : measurable (D `&` f @^-1` [set false]). apply: measurableI => //. by have := mf measurableT [set false]; rewrite setTI; exact. -have := my mDNf Y mY. -move=> yY xY. -rewrite (_ : _ @^-1` Y = - ((f @^-1` [set true]) `&` (x @^-1` Y) `&` (f @^-1` [set true])) `|` - ((f @^-1` [set false]) `&` (y @^-1` Y) `&` (f @^-1` [set false]))); last first. +have := my mDNf _ mB. +move=> yB xB. +rewrite (_ : _ @^-1` B = + ((f @^-1` [set true]) `&` (x @^-1` B) `&` (f @^-1` [set true])) `|` + ((f @^-1` [set false]) `&` (y @^-1` B) `&` (f @^-1` [set false]))); last first. apply/seteqP; split=> [t /=| t]. by case: ifPn => ft; [left|right]. by move=> /= [|]; case: ifPn => ft; case=> -[]. @@ -239,8 +239,8 @@ rewrite setIUr; apply: measurableU. by apply: measurableI => //; rewrite setIA. Qed. -Lemma measurable_fun_ifT (d d' : _) (T : measurableType d) - (T' : measurableType d') (x y : T -> T') (f : T -> bool) +Lemma measurable_fun_ifT (d d' : _) (X : measurableType d) + (Y : measurableType d') (x y : X -> Y) (f : X -> bool) (mf : measurable_fun setT f) : measurable_fun setT x -> measurable_fun setT y -> measurable_fun setT (fun t => if f t then x t else y t). @@ -249,15 +249,15 @@ by move=> mx my; apply: measurable_fun_if => //; [exact: measurable_funS mx|exact: measurable_funS my]. Qed. -Lemma measurable_fun_if_pair (d d' : _) (T : measurableType d) - (T' : measurableType d') (x y : T -> T') : +Lemma measurable_fun_if_pair (d d' : _) (X : measurableType d) + (Y : measurableType d') (x y : X -> Y) : measurable_fun setT x -> measurable_fun setT y -> measurable_fun setT (fun tb => if tb.2 then x tb.1 else y tb.1). Proof. move=> mx my. -have {}mx : measurable_fun [set: T * bool] (x \o fst). +have {}mx : measurable_fun [set: X * bool] (x \o fst). by apply: measurable_fun_comp => //; exact: measurable_fun_fst. -have {}my : measurable_fun [set: T * bool] (y \o fst). +have {}my : measurable_fun [set: X * bool] (y \o fst). by apply: measurable_fun_comp => //; exact: measurable_fun_fst. by apply: measurable_fun_ifT => //=; exact: measurable_fun_snd. Qed. @@ -450,7 +450,22 @@ Qed. End measure_fam_uub. -HB.mixin Record isFiniteFam +HB.mixin Record Kernel_isSFinite_subdef + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := { + sfinite_subdef : exists2 s : (R.-ker X ~> Y)^nat, forall n, measure_fam_uub (s n) & + forall x U, measurable U -> k x U = kseries s x U }. + +#[short(type=sfinite_kernel)] +HB.structure Definition SFiniteKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of Kernel_isSFinite_subdef _ _ X Y R k & isKernel d d' X Y R k }. +Notation "R .-sfker X ~> Y" := (sfinite_kernel X Y R). + +Arguments sfinite_subdef {_ _ _ _ _} _. + +HB.mixin Record SFiniteKernel_isFinite d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { measure_uub : measure_fam_uub k }. @@ -459,71 +474,128 @@ HB.mixin Record isFiniteFam HB.structure Definition FiniteKernel d d' (X : measurableType d) (Y : measurableType d') (R : realType) := - {k of isFiniteFam _ _ X Y R k & isKernel _ _ X Y R k}. + {k of SFiniteKernel_isFinite _ _ X Y R k & @SFiniteKernel _ _ X Y R k }. Notation "R .-fker X ~> Y" := (finite_kernel X Y R). Arguments measure_uub {_ _ _ _ _} _. -Section kernel_from_mzero. -Variables (d : _) (T : measurableType d) (R : realType). -Variables (d' : _) (T' : measurableType d'). +HB.factory Record Kernel_isFinite d d' (X : measurableType d) + (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) of isKernel _ _ _ _ _ k := { + measure_uub : measure_fam_uub k }. -Definition kernel_from_mzero : T' -> {measure set T -> \bar R} := - fun _ : T' => [the measure _ _ of mzero]. +Section kzero. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variable R : realType. + +Definition kzero : X -> {measure set Y -> \bar R} := + fun _ : X => [the measure _ _ of mzero]. -Lemma kernel_from_mzeroP : forall U, measurable U -> - measurable_fun setT (kernel_from_mzero ^~ U). -Proof. by move=> U mU/=; exact: measurable_fun_cst. Qed. +Let measurable_fun_kzero U : measurable U -> + measurable_fun setT (kzero ^~ U). +Proof. by move=> ?/=; exact: measurable_fun_cst. Qed. HB.instance Definition _ := - @isKernel.Build _ _ T' T R kernel_from_mzero - kernel_from_mzeroP. + @isKernel.Build _ _ X Y R kzero measurable_fun_kzero. -Let kernel_from_mzero_uub : measure_fam_uub kernel_from_mzero. -Proof. by exists 1%R => /= t; rewrite /mzero/=. Qed. +(*Let kernel_from_mzero_sfinite0 : exists2 s : (R.-ker T' ~> T)^nat, forall n, measure_fam_uub (s n) & + forall x U, measurable U -> kernel_from_mzero x U = kseries s x U. +Proof. +exists (fun=> [the _.-ker _ ~> _ of kernel_from_mzero]). + move=> _. + by exists 1%R => y; rewrite /= /mzero. +by move=> t U mU/=; rewrite /mseries nneseries0. +Qed. HB.instance Definition _ := - @isFiniteFam.Build _ _ _ T R kernel_from_mzero - kernel_from_mzero_uub. + @isSFinite0.Build _ _ _ T R kernel_from_mzero + kernel_from_mzero_sfinite0.*) -End kernel_from_mzero. - -HB.mixin Record isSFinite - d d' (X : measurableType d) (Y : measurableType d') - (R : realType) (k : X -> {measure set Y -> \bar R}) := { - sfinite : exists s : (R.-fker X ~> Y)^nat, - forall x U, measurable U -> k x U = kseries s x U }. +Lemma kzero_uub : measure_fam_uub kzero. +Proof. by exists 1%R => /= t; rewrite /mzero/=. Qed. -#[short(type=sfinite_kernel)] -HB.structure Definition SFiniteKernel - d d' (X : measurableType d) (Y : measurableType d') - (R : realType) := - {k of isSFinite _ _ X Y R k & isKernel _ _ X Y _ k}. -Notation "R .-sfker X ~> Y" := (sfinite_kernel X Y R). +(*HB.instance Definition _ := + @SFiniteKernel_isFinite.Build _ _ _ T R kernel_from_mzero + kernel_from_mzero_uub.*) -Arguments sfinite {_ _ _ _ _} _. +End kzero. -(* a finite kernel is always an s-finite kernel *) -Section finite_is_sfinite. -Variables (d d' : _) (X : measurableType d) (T : measurableType d'). -Variables (R : realType) (k : R.-fker T ~> X). +HB.builders Context d d' (X : measurableType d) (Y : measurableType d') + (R : realType) k of Kernel_isFinite d d' X Y R k. Lemma sfinite_finite : - exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> - k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. + exists2 k_ : (R.-ker _ ~> _)^nat, forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -exists (fun n => if n is O then k else - [the finite_kernel _ _ _ of @kernel_from_mzero _ X R _ T]). -move=> t U mU/=. -rewrite /mseries. +exists (fun n => if n is O then [the _.-ker _ ~> _ of k] else + [the _.-ker _ ~> _ of @kzero _ _ X Y R]). + by case => [|_]; [exact: measure_uub|exact: kzero_uub]. +move=> t U mU/=; rewrite /mseries. rewrite (nneseries_split 1%N)// big_ord_recl/= big_ord0 adde0. rewrite ereal_series (@eq_nneseries _ _ (fun=> 0%E)); last by case. by rewrite nneseries0// adde0. Qed. -End finite_is_sfinite. +HB.instance Definition _ := @Kernel_isSFinite_subdef.Build d d' X Y R k sfinite_finite. + +HB.instance Definition _ := @SFiniteKernel_isFinite.Build d d' X Y R k measure_uub. + +HB.end. + +Section sfinite. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-sfker X ~> Y). + +Let s : (X -> {measure set Y -> \bar R})^nat := + let: exist2 x _ _ := cid2 (sfinite_subdef k) in x. + +Let ms n : @isKernel d d' X Y R (s n). +Proof. +split; rewrite /s; case: cid2 => /= s' s'_uub kE. +exact: measurable_kernel. +Qed. + +HB.instance Definition _ n := ms n. + +Let s_uub n : measure_fam_uub (s n). +Proof. by rewrite /s; case: cid2. Qed. + +HB.instance Definition _ n := + @Kernel_isFinite.Build d d' X Y R (s n) (s_uub n). + +Lemma sfinite : exists s : (R.-fker X ~> Y)^nat, + forall x U, measurable U -> k x U = kseries s x U. +Proof. +exists (fun n => [the _.-fker _ ~> _ of s n]) => x U mU. +by rewrite /s /= /s; by case: cid2 => ? ? ->. +Qed. + +End sfinite. -HB.mixin Record isProbabilityFam +HB.instance Definition _ (d d' : _) (X : measurableType d) + (Y : measurableType d') (R : realType) := + @Kernel_isFinite.Build _ _ _ _ R (@kzero _ _ X Y R) + (@kzero_uub _ _ X Y R). + +HB.factory Record Kernel_isSFinite d d' (X : measurableType d) + (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) + of isKernel _ _ _ _ _ k := { + sfinite : exists s : (R.-fker X ~> Y)^nat, + forall x U, measurable U -> k x U = kseries s x U }. + +HB.builders Context d d' (X : measurableType d) (Y : measurableType d') + (R : realType) k of Kernel_isSFinite d d' X Y R k. + +Lemma sfinite_subdef : Kernel_isSFinite_subdef d d' X Y R k. +Proof. +split; have [s sE] := sfinite; exists s => //. +by move=> n; exact: measure_uub. +Qed. + +HB.instance Definition _ := (*@isSFinite0.Build d d' X Y R k*) sfinite_subdef. + +HB.end. + +HB.mixin Record FiniteKernel_isProbability d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { prob_kernel : forall x, k x [set: Y] = 1}. @@ -532,36 +604,28 @@ HB.mixin Record isProbabilityFam HB.structure Definition ProbabilityKernel (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType) := - {k of isProbabilityFam _ _ X Y R k & isKernel _ _ X Y R k & - isFiniteFam _ _ X Y R k & isSFinite _ _ X Y R k}. + {k of FiniteKernel_isProbability _ _ X Y R k & + @FiniteKernel _ _ X Y R k}. Notation "R .-pker X ~> Y" := (probability_kernel X Y R). -HB.factory Record isProbabilityKernel +HB.factory Record Kernel_isProbability d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := - { prob_kernel' : forall x, k x setT = 1 }. + { prob_kernel : forall x, k x setT = 1 }. HB.builders Context d d' (X : measurableType d) (Y : measurableType d') - (R : realType) k of isProbabilityKernel d d' X Y R k. + (R : realType) k of Kernel_isProbability d d' X Y R k. -Let is_finite_kernel : measure_fam_uub k. +Let finite : @Kernel_isFinite d d' X Y R k. Proof. +split. exists 2%R => /= ?. -by rewrite (@le_lt_trans _ _ 1%:E) ?lte_fin ?ltr1n// prob_kernel'. +by rewrite (@le_lt_trans _ _ 1%:E) ?lte_fin ?ltr1n// prob_kernel. Qed. -HB.instance Definition _ := @isFiniteFam.Build _ _ _ _ _ _ is_finite_kernel. +HB.instance Definition _ := finite. -Lemma is_sfinite_kernel : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> - k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. -Proof. exact: sfinite_finite. Qed. - -HB.instance Definition _ := @isSFinite.Build _ _ _ _ _ _ is_sfinite_kernel. - -Lemma is_probability_kernel : forall x, k x setT = 1. - exact/prob_kernel'. Qed. - -HB.instance Definition _ := @isProbabilityFam.Build _ _ _ _ _ _ is_probability_kernel. +HB.instance Definition _ := @FiniteKernel_isProbability.Build _ _ _ _ _ k prob_kernel. HB.end. @@ -579,7 +643,8 @@ Lemma sfinite_kernel_measure (d d' : _) (X : measurableType d) Proof. have [k_ k_E] := sfinite k. exists (fun n => k_ n x); last by move=> A mA; rewrite k_E. -by move=> n; exact: finite_kernel_measure. +move=> n; rewrite /finite_measure. +exact: finite_kernel_measure. Qed. (* see measurable_prod_subset in lebesgue_integral.v; @@ -646,7 +711,7 @@ End measurable_prod_subset_kernel. Section measurable_fun_xsection_finite_kernel. Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Variable k : R.-fker X ~> Y. +Variables (k : R.-fker X ~> Y). Implicit Types A : set (X * Y). Let phi A := fun x => k x (xsection A x). @@ -783,7 +848,7 @@ Let kprobability_prob x : kprobability x setT = 1. Proof. by rewrite /kprobability/= probability_setT. Qed. HB.instance Definition _ := - @isProbabilityKernel.Build _ _ X Y R kprobability kprobability_prob. + @Kernel_isProbability.Build _ _ X Y R kprobability kprobability_prob. End kprobability. @@ -810,7 +875,7 @@ HB.instance Definition _ := isKernel.Build _ _ _ _ _ (kdirac mf) Let kdirac_prob x : kdirac mf x setT = 1. Proof. by rewrite /kdirac/= diracE in_setT. Qed. -HB.instance Definition _ := isProbabilityKernel.Build _ _ _ _ _ +HB.instance Definition _ := Kernel_isProbability.Build _ _ _ _ _ (kdirac mf) kdirac_prob. End kdirac. @@ -836,33 +901,23 @@ HB.instance Definition _ := @isKernel.Build _ _ _ _ _ kadd measurable_fun_kadd. End kadd. -Section fkadd. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k1 k2 : R.-fker X ~> Y). - -Let kadd_finite_uub : measure_fam_uub (kadd k1 k2). -Proof. -have [f1 hk1] := measure_uub k1; have [f2 hk2] := measure_uub k2. -exists (f1 + f2)%R => x; rewrite /kadd /=. -rewrite -/(measure_add (k1 x) (k2 x)). -by rewrite measure_addE EFinD; exact: lte_add. -Qed. - -HB.instance Definition _ t := - isFiniteFam.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. -End fkadd. - Section sfkadd. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variables (R : realType) (k1 k2 : R.-sfker X ~> Y). -Let sfinite_kadd : exists k_ : (R.-fker _ ~> _)^nat, +Let sfinite_kadd : exists2 k_ : (R.-ker _ ~> _)^nat, forall n, measure_fam_uub (k_ n) & forall x U, measurable U -> kadd k1 k2 x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. have [f1 hk1] := sfinite k1. have [f2 hk2] := sfinite k2. -exists (fun n => [the finite_kernel _ _ _ of kadd (f1 n) (f2 n)]) => x U mU. +exists (fun n => [the _.-ker _ ~> _ of kadd (f1 n) (f2 n)]). + move=> n. + have [r1 f1r1] := measure_uub (f1 n). + have [r2 f2r2] := measure_uub (f2 n). + exists (r1 + r2)%R => x/=. + by rewrite /msum !big_ord_recr/= big_ord0 add0e EFinD lte_add. +move=> x U mU. rewrite /kadd/=. rewrite -/(measure_add (k1 x) (k2 x)) measure_addE. rewrite /mseries. @@ -873,9 +928,25 @@ by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE. Qed. HB.instance Definition _ t := - isSFinite.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. + Kernel_isSFinite_subdef.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. End sfkadd. +Section fkadd. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k1 k2 : R.-fker X ~> Y). + +Let kadd_finite_uub : measure_fam_uub (kadd k1 k2). +Proof. +have [f1 hk1] := measure_uub k1; have [f2 hk2] := measure_uub k2. +exists (f1 + f2)%R => x; rewrite /kadd /=. +rewrite -/(measure_add (k1 x) (k2 x)). +by rewrite measure_addE EFinD; exact: lte_add. +Qed. + +HB.instance Definition _ t := + Kernel_isFinite.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. +End fkadd. + Section kernel_measurable_preimage. Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). Variable R : realType. @@ -1021,7 +1092,7 @@ by rewrite -EFinM divrr// ?lte_fin ?ltr1n// ?unitfE fine_eq0. Qed. HB.instance Definition _ := - @isProbabilityKernel.Build _ _ _ _ _ (knormalize P) knormalize1. + @Kernel_isProbability.Build _ _ _ _ _ (knormalize P) knormalize1. End knormalize. @@ -1112,7 +1183,7 @@ by rewrite integral_cst//= EFinM lte_pmul2l. Qed. HB.instance Definition _ := - isFiniteFam.Build _ _ X Z R (l \; k) mkcomp_finite. + Kernel_isFinite.Build _ _ X Z R (l \; k) mkcomp_finite. End kcomp_finite_kernel_finite. End KCOMP_FINITE_KERNEL. @@ -1174,7 +1245,7 @@ HB.instance Definition _ := #[export] HB.instance Definition _ := - isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). + Kernel_isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). End kcomp_sfinite_kernel. End KCOMP_SFINITE_KERNEL. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index 2bc2e68b11..bc0e9b5350 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -193,7 +193,7 @@ rewrite (_ : [set tt] == set0 = false); last first. by case: ifPn => // /andP[]. Qed. -HB.instance Definition _ i := @isFiniteFam.Build _ _ _ _ R (mk i) (mk_uub i). +HB.instance Definition _ i := @Kernel_isFinite.Build _ _ _ _ R (mk i) (mk_uub i). End score. End SCORE. @@ -245,7 +245,7 @@ move: jk; rewrite neq_ltn/= => /orP[|] jr. by rewrite -floor_lt_int. Qed. -HB.instance Definition _ := @isSFinite.Build _ _ _ _ _ (kscore mr) sfinite_kscore. +HB.instance Definition _ := @Kernel_isSFinite.Build _ _ _ _ _ (kscore mr) sfinite_kscore. End kscore. @@ -269,28 +269,29 @@ apply: (@measurable_fun_if_pair _ _ _ _ (k ^~ U) (fun=> mzero U)). exact: measurable_fun_cst. Qed. +#[export] HB.instance Definition _ := isKernel.Build _ _ _ _ R kiteT measurable_fun_kiteT. End kiteT. -Section fkiteT. +Section sfkiteT. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k : R.-fker X ~> Y). +Variables (R : realType) (k : R.-sfker X ~> Y). -Let kiteT_uub : measure_fam_uub (kiteT k). +Let sfinite_kiteT : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> + kiteT k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -have /measure_fam_uubP[M hM] := measure_uub k. -exists M%:num => /= -[]; rewrite /kiteT => t [|]/=; first exact: hM. -by rewrite /= /mzero. +have [k_ hk /=] := sfinite k. +exists (fun n => [the _.-ker _ ~> _ of kiteT (k_ n)]) => /=. + move=> n; have /measure_fam_uubP[r k_r] := measure_uub (k_ n). + by exists r%:num => /= -[x []]; rewrite /kiteT//= /mzero//. +move=> [x b] U mU; rewrite /kiteT; case: ifPn => hb. + by rewrite hk. +by rewrite /mseries nneseries0. Qed. -HB.instance Definition _ t := isFiniteFam.Build _ _ _ _ R (kiteT k) kiteT_uub. -End fkiteT. - -Section sfkiteT. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k : R.-sfker X ~> Y). - -Let sfinite_kiteT : exists k_ : (R.-fker _ ~> _)^nat, +(*Let sfinite_kiteT : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> kiteT k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. @@ -302,12 +303,29 @@ rewrite /kiteT; case: ifPn => hb. by rewrite /kiteT hb. rewrite /= /mseries nneseries0// => n _. by rewrite /kiteT (negbTE hb). -Qed. - -HB.instance Definition _ t := @isSFinite.Build _ _ _ _ _ (kiteT k) sfinite_kiteT. +Qed.*) +(* NB: we could also want to use Kernel_isSFinite *) +#[export] +HB.instance Definition _ t := @Kernel_isSFinite_subdef.Build _ _ _ _ _ + (kiteT k) sfinite_kiteT. End sfkiteT. +Section fkiteT. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-fker X ~> Y). + +Let kiteT_uub : measure_fam_uub (kiteT k). +Proof. +have /measure_fam_uubP[M hM] := measure_uub k. +exists M%:num => /= -[]; rewrite /kiteT => t [|]/=; first exact: hM. +by rewrite /= /mzero. +Qed. + +#[export] +HB.instance Definition _ t := Kernel_isFinite.Build _ _ _ _ R (kiteT k) kiteT_uub. +End fkiteT. + Section kiteF. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). Variables (R : realType) (k : R.-ker X ~> Y). @@ -326,30 +344,30 @@ apply: (@measurable_fun_if_pair _ _ _ _ (fun=> mzero U) (k ^~ U)). exact/measurable_kernel. Qed. +#[export] HB.instance Definition _ := isKernel.Build _ _ _ _ R kiteF measurable_fun_kiteF. End kiteF. -Section fkiteF. +Section sfkiteF. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k : R.-fker X ~> Y). +Variables (R : realType) (k : R.-sfker X ~> Y). -Let kiteF_uub : measure_fam_uub (kiteF k). +Let sfinite_kiteF : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> + kiteF k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. -have /measure_fam_uubP[M hM] := measure_uub k. -exists M%:num => /= -[]; rewrite /kiteF/= => t. -by case => //=; rewrite /mzero. +have [k_ hk /=] := sfinite k. +exists (fun n => [the _.-ker _ ~> _ of kiteF (k_ n)]) => /=. + move=> n; have /measure_fam_uubP[r k_r] := measure_uub (k_ n). + by exists r%:num => /= -[x []]; rewrite /kiteF//= /mzero//. +move=> [x b] U mU; rewrite /kiteF; case: ifPn => hb. + by rewrite hk. +by rewrite /mseries nneseries0. Qed. -HB.instance Definition _ := isFiniteFam.Build _ _ _ _ R (kiteF k) kiteF_uub. - -End fkiteF. - -Section sfkiteF. -Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). -Variables (R : realType) (k : R.-sfker X ~> Y). - -Let sfinite_kiteF : exists k_ : (R.-fker _ ~> _)^nat, +(*Let sfinite_kiteF : exists k_ : (R.-fker _ ~> _)^nat, forall x U, measurable U -> kiteF k x U = [the measure _ _ of mseries (k_ ^~ x) 0] U. Proof. @@ -359,11 +377,35 @@ rewrite /= /kiteF /=; case: ifPn => hb. by rewrite /mseries hk//= /mseries/=. by rewrite /= /mseries nneseries0. Qed. +*) -HB.instance Definition _ := @isSFinite.Build _ _ _ _ _ (kiteF k) sfinite_kiteF. +#[export] +HB.instance Definition _ := @Kernel_isSFinite_subdef.Build _ _ _ _ _ + (kiteF k) sfinite_kiteF. End sfkiteF. + +Section fkiteF. +Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). +Variables (R : realType) (k : R.-fker X ~> Y). + +Let kiteF_uub : measure_fam_uub (kiteF k). +Proof. +have /measure_fam_uubP[M hM] := measure_uub k. +exists M%:num => /= -[]; rewrite /kiteF/= => t. +by case => //=; rewrite /mzero. +Qed. + +#[export] +HB.instance Definition _ := Kernel_isFinite.Build _ _ _ _ R (kiteF k) kiteF_uub. + +End fkiteF. + +(*Module Exports. +HB.reexport. +End Exports.*) End ITE. +(*Export ITE.Exports.*) Section ite. Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). From 7e2fdee2f9403758a5c49bcd180e0d56e5e61bd3 Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Sun, 18 Sep 2022 08:19:51 +0900 Subject: [PATCH 41/42] subprob kernel --- theories/kernel.v | 60 +++++++++++++++++++++++++++++++++++++---------- 1 file changed, 47 insertions(+), 13 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index a94d7db6ea..008887b582 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -15,8 +15,9 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (* sfinite_measure mu == the measure mu is s-finite *) (* R.-ker X ~> Y == kernel *) (* kseries == countable sum of kernels *) -(* R.-fker X ~> Y == finite kernel *) (* R.-sfker X ~> Y == s-finite kernel *) +(* R.-fker X ~> Y == finite kernel *) +(* R.-spker X ~> Y == subprobability kernel *) (* R.-pker X ~> Y == probability kernel *) (* kprobability m == kernel defined by a probability measure *) (* kdirac mf == kernel defined by a measurable function *) @@ -387,8 +388,9 @@ End sfinite_fubini. Arguments sfinite_fubini {d d' X Y R m1} _ {m2} _ f. Reserved Notation "R .-ker X ~> Y" (at level 42, format "R .-ker X ~> Y"). -Reserved Notation "R .-fker X ~> Y" (at level 42, format "R .-fker X ~> Y"). Reserved Notation "R .-sfker X ~> Y" (at level 42, format "R .-sfker X ~> Y"). +Reserved Notation "R .-fker X ~> Y" (at level 42, format "R .-fker X ~> Y"). +Reserved Notation "R .-spker X ~> Y" (at level 42, format "R .-spker X ~> Y"). Reserved Notation "R .-pker X ~> Y" (at level 42, format "R .-pker X ~> Y"). HB.mixin Record isKernel d d' (X : measurableType d) (Y : measurableType d') @@ -415,7 +417,7 @@ Lemma measurable_fun_kseries (U : set Y) : measurable U -> measurable_fun setT (kseries ^~ U). Proof. -move=> mU; rewrite /kseries /= /mseries. +move=> mU. by apply: ge0_emeasurable_fun_sum => // n; exact/measurable_kernel. Qed. @@ -453,7 +455,8 @@ End measure_fam_uub. HB.mixin Record Kernel_isSFinite_subdef d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { - sfinite_subdef : exists2 s : (R.-ker X ~> Y)^nat, forall n, measure_fam_uub (s n) & + sfinite_subdef : exists2 s : (R.-ker X ~> Y)^nat, + forall n, measure_fam_uub (s n) & forall x U, measurable U -> k x U = kseries s x U }. #[short(type=sfinite_kernel)] @@ -595,7 +598,40 @@ HB.instance Definition _ := (*@isSFinite0.Build d d' X Y R k*) sfinite_subdef. HB.end. -HB.mixin Record FiniteKernel_isProbability +HB.mixin Record FiniteKernel_isSubProbability + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := + { sprob_kernel : ereal_sup [set k x [set: Y] | x in setT] <= 1}. + +#[short(type=sprobability_kernel)] +HB.structure Definition SubProbabilityKernel + (d d' : _) (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of FiniteKernel_isSubProbability _ _ X Y R k & + @FiniteKernel _ _ X Y R k}. +Notation "R .-spker X ~> Y" := (sprobability_kernel X Y R). + +HB.factory Record Kernel_isSubProbability + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) of isKernel _ _ X Y R k := + { sprob_kernel : ereal_sup [set k x [set: Y] | x in setT] <= 1}. + +HB.builders Context d d' (X : measurableType d) (Y : measurableType d') + (R : realType) k of Kernel_isSubProbability d d' X Y R k. + +Let finite : @Kernel_isFinite d d' X Y R k. +Proof. +split; exists 2%R => /= ?; rewrite (@le_lt_trans _ _ 1%:E) ?lte_fin ?ltr1n//. +by rewrite (le_trans _ sprob_kernel)//; exact: ereal_sup_ub. +Qed. + +HB.instance Definition _ := finite. + +HB.instance Definition _ := @FiniteKernel_isSubProbability.Build _ _ _ _ _ k sprob_kernel. + +HB.end. + +HB.mixin Record SubProbability_isProbability d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { prob_kernel : forall x, k x [set: Y] = 1}. @@ -604,8 +640,8 @@ HB.mixin Record FiniteKernel_isProbability HB.structure Definition ProbabilityKernel (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType) := - {k of FiniteKernel_isProbability _ _ X Y R k & - @FiniteKernel _ _ X Y R k}. + {k of SubProbability_isProbability _ _ X Y R k & + @SubProbabilityKernel _ _ X Y R k}. Notation "R .-pker X ~> Y" := (probability_kernel X Y R). HB.factory Record Kernel_isProbability @@ -616,16 +652,14 @@ HB.factory Record Kernel_isProbability HB.builders Context d d' (X : measurableType d) (Y : measurableType d') (R : realType) k of Kernel_isProbability d d' X Y R k. -Let finite : @Kernel_isFinite d d' X Y R k. +Let sprob_kernel : @Kernel_isSubProbability d d' X Y R k. Proof. -split. -exists 2%R => /= ?. -by rewrite (@le_lt_trans _ _ 1%:E) ?lte_fin ?ltr1n// prob_kernel. +by split; apply: ub_ereal_sup => x [y _ <-{x}]; rewrite prob_kernel. Qed. -HB.instance Definition _ := finite. +HB.instance Definition _ := sprob_kernel. -HB.instance Definition _ := @FiniteKernel_isProbability.Build _ _ _ _ _ k prob_kernel. +HB.instance Definition _ := @SubProbability_isProbability.Build _ _ _ _ _ k prob_kernel. HB.end. From 536597fd9988e080b916258d413f34248005eeeb Mon Sep 17 00:00:00 2001 From: Reynald Affeldt Date: Wed, 21 Sep 2022 09:36:09 +0900 Subject: [PATCH 42/42] cleaning --- theories/kernel.v | 314 +++++++++++++++++++------------------------ theories/prob_lang.v | 129 ++++++++---------- theories/wip.v | 19 +-- 3 files changed, 207 insertions(+), 255 deletions(-) diff --git a/theories/kernel.v b/theories/kernel.v index 008887b582..d84168a0d0 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -10,6 +10,8 @@ Require Import lebesgue_measure fsbigop numfun lebesgue_integral. (* *) (* This file provides a formation of kernels and extends the theory of *) (* measures with, e.g., Tonelli-Fubini's theorem for s-finite measures. *) +(* The main result is the fact that s-finite kernels are stable by *) +(* composition. *) (* *) (* finite_measure mu == the measure mu is finite *) (* sfinite_measure mu == the measure mu is s-finite *) @@ -58,7 +60,7 @@ Qed. End probability_lemmas. (* /PR 516 in progress *) -(* TODO: PR *) +(* TODO: PR*) Lemma setT0 (T : pointedType) : setT != set0 :> set T. Proof. by apply/eqP => /seteqP[] /(_ point) /(_ Logic.I). Qed. @@ -93,6 +95,10 @@ have -> : B = set0. by apply/or4P; rewrite eqxx/= !orbT. Qed. +Lemma xsection_preimage_snd (X Y Z : Type) (f : Y -> Z) (A : set Z) (x : X) : + xsection ((f \o snd) @^-1` A) x = f @^-1` A. +Proof. by apply/seteqP; split; move=> y/=; rewrite /xsection/= inE. Qed. + Canonical unit_pointedType := PointedType unit tt. Section discrete_measurable_unit. @@ -269,28 +275,22 @@ apply: continuous_measurable_fun. by have := (@opp_continuous R [the normedModType R of R^o]). Qed. -Section integralM_0ifneg. -Local Open Scope ereal_scope. -Variables (d : _) (T : measurableType d) (R : realType). -Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). - -Lemma integralM_0ifneg (f : R -> T -> \bar R) (k : R) - (f0 : forall r t, D t -> 0 <= f r t) : - ((k < 0)%R -> f k = cst 0%E) -> measurable_fun setT (f k) -> - \int[m]_(x in D) (k%:E * (f k) x) = k%:E * \int[m]_(x in D) ((f k) x). +Lemma integral_eq0 d (T : measurableType d) (R : realType) + (mu : {measure set T -> \bar R}) (D : set T) f : + (forall x, D x -> f x = 0) -> \int[mu]_(x in D) f x = 0. Proof. -move=> fk0 mfk; have [k0|k0] := ltP k 0%R. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. - by move=> x _; rewrite fk0// mule0. - rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. - by rewrite fk0// indic0. -rewrite ge0_integralM//. -- by apply/(@measurable_funS _ _ _ _ setT) => //. -- by move=> y Dy; rewrite f0. +move=> f0; under eq_integral. + by move=> x /[1!inE] /f0 ->; over. +by rewrite integral0. Qed. -End integralM_0ifneg. -Arguments integralM_0ifneg {d T R} m {D} mD f. +Lemma dirac0 d (T : measurableType d) (R : realFieldType) (a : T) : + \d_a set0 = 0%E :> \bar R. +Proof. by rewrite /dirac indic0. Qed. + +Lemma diracT d (T : measurableType d) (R : realFieldType) (a : T) : + \d_a setT = 1%E :> \bar R. +Proof. by rewrite /dirac indicT. Qed. Section fubini_tonelli. Local Open Scope ereal_scope. @@ -765,14 +765,13 @@ Qed. End measurable_fun_xsection_finite_kernel. -(* pollard? *) Section measurable_fun_integral_finite_sfinite. Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). +Variable k : X * Y -> \bar R. Lemma measurable_fun_xsection_integral (l : X -> {measure set Y -> \bar R}) - (k : X * Y -> \bar R) (k_ : ({nnsfun [the measurableType _ of (X * Y)%type] >-> R})^nat) (ndk_ : nondecreasing_seq (k_ : (X * Y -> R)^nat)) (k_k : forall z, EFin \o (k_ ^~ z) --> k z) : @@ -799,11 +798,11 @@ rewrite (_ : (fun x => _) = - by move=> y _ m n mn; rewrite lee_fin; exact/lefP/ndk_. apply: measurable_fun_elim_sup => n. rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y - (\sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR 743 is merged*) + (\sum_(r <- fset_set (range (k_ n)))(*TODO: upd when PR 743 is merged*) r * \1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. by apply/funext => x; apply: eq_integral => y _; rewrite fimfunE. rewrite [X in measurable_fun _ X](_ : _ = (fun x => - \sum_(r <- fset_set (range (k_ n)))(*TODO: upd when the PR 743 is merged*) + \sum_(r <- fset_set (range (k_ n)))(*TODO: upd when PR 743 is merged*) (\int[l x]_y (r * \1_(k_ n @^-1` [set r]) (x, y))%:E))); last first. apply/funext => x; rewrite -ge0_integral_sum//. - by apply: eq_integral => y _; rewrite sumEFin. @@ -815,14 +814,15 @@ rewrite [X in measurable_fun _ X](_ : _ = (fun x => apply emeasurable_fun_sum => r. rewrite [X in measurable_fun _ X](_ : _ = (fun x => r%:E * \int[l x]_y (\1_(k_ n @^-1` [set r]) (x, y))%:E)); last first. - apply/funext => x. - under eq_integral do rewrite EFinM. - rewrite (integralM_0ifneg _ _ (fun k y => (\1_(k_ n @^-1` [set r]) (x, y))%:E))//. - - by move=> _ y _; rewrite lee_fin. - - by move=> r0; apply/funext => y; rewrite preimage_nnfun0// indicE in_set0. - - apply/EFin_measurable_fun/measurable_fun_prod1 => /=. + apply/funext => x; under eq_integral do rewrite EFinM. + have [r0|r0] := leP 0%R r. + rewrite ge0_integralM//; last by move=> y _; rewrite lee_fin. + apply/EFin_measurable_fun/measurable_fun_prod1 => /=. rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) r))//. exact/measurable_funP. + rewrite integral_eq0; last first. + by move=> y _; rewrite preimage_nnfun0// indic0 mule0. + by rewrite integral_eq0 ?mule0// => y _; rewrite preimage_nnfun0// indic0. apply/measurable_funeM. rewrite (_ : (fun x => _) = (fun x => l x (xsection (k_ n @^-1` [set r]) x))). exact/h. @@ -834,9 +834,8 @@ congr (l x _); apply/funext => y1/=; rewrite /xsection/= inE. by apply/propext; tauto. Qed. -Lemma measurable_fun_integral_finite_kernel - (l : R.-fker X ~> Y) - (k : X * Y -> \bar R) (k0 : forall z, 0 <= k z) (mk : measurable_fun setT k) : +Lemma measurable_fun_integral_finite_kernel (l : R.-fker X ~> Y) + (k0 : forall z, 0 <= k z) (mk : measurable_fun setT k) : measurable_fun setT (fun x => \int[l x]_y k (x, y)). Proof. have [k_ [ndk_ k_k]] := approximation measurableT mk (fun x _ => k0 x). @@ -845,9 +844,8 @@ have [l_ hl_] := measure_uub l. by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. Qed. -Lemma measurable_fun_integral_sfinite_kernel - (l : R.-sfker X ~> Y) - (k : X * Y -> \bar R) (k0 : forall t, 0 <= k t) (mk : measurable_fun setT k) : +Lemma measurable_fun_integral_sfinite_kernel (l : R.-sfker X ~> Y) + (k0 : forall t, 0 <= k t) (mk : measurable_fun setT k) : measurable_fun setT (fun x => \int[l x]_y k (x, y)). Proof. have [k_ [ndk_ k_k]] := approximation measurableT mk (fun xy _ => k0 xy). @@ -861,9 +859,9 @@ by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. Qed. End measurable_fun_integral_finite_sfinite. -Arguments measurable_fun_xsection_integral {_ _ _ _ _} l k. -Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} l k. -Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} l k. +Arguments measurable_fun_xsection_integral {_ _ _ _ _} k l. +Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} k l. +Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} k l. Section kprobability. Variables (d d' : _) (X : measurableType d) (Y : measurableType d'). @@ -907,7 +905,7 @@ HB.instance Definition _ := isKernel.Build _ _ _ _ _ (kdirac mf) measurable_fun_kdirac. Let kdirac_prob x : kdirac mf x setT = 1. -Proof. by rewrite /kdirac/= diracE in_setT. Qed. +Proof. by rewrite /kdirac/= diracT. Qed. HB.instance Definition _ := Kernel_isProbability.Build _ _ _ _ _ (kdirac mf) kdirac_prob. @@ -981,6 +979,7 @@ HB.instance Definition _ t := Kernel_isFinite.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. End fkadd. +(* TODO: move *) Section kernel_measurable_preimage. Variables (d d' : _) (T : measurableType d) (T' : measurableType d'). Variable R : realType. @@ -1005,6 +1004,7 @@ Qed. End kernel_measurable_preimage. +(* TODO: move *) Lemma measurable_fun_eq_cst (d d' : _) (T : measurableType d) (T' : measurableType d') (R : realType) (f : R.-ker T ~> T') k : measurable_fun setT (fun t => f t setT == k). @@ -1080,36 +1080,29 @@ Proof. move=> mU; rewrite /knormalize/= /mnormalize /=. rewrite (_ : (fun _ => _) = (fun x => if f x setT == 0 then P U else if f x setT == +oo then P U - else f x U * ((fine (f x setT))^-1)%:E)); last first. - apply/funext => x; case: ifPn => [/orP[->//|->]|]. - by case: ifPn. + else f x U * (fine (f x setT))^-1%:E)); last first. + apply/funext => x; case: ifPn => [/orP[->//|->]|]; first by case: ifPn. by rewrite negb_or=> /andP[/negbTE -> /negbTE ->]. +apply: measurable_fun_if => //; + [exact: measurable_fun_eq_cst|exact: measurable_fun_cst|]. apply: measurable_fun_if => //. +- rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]). + exact: measurable_neq_cst. + by apply/seteqP; split => [x /negbT//|x /negbTE]. - exact: measurable_fun_eq_cst. - exact: measurable_fun_cst. -- apply: measurable_fun_if => //. - + rewrite setTI [X in measurable X](_ : _ = [set t | f t setT != 0]); last first. - by apply/seteqP; split => [x /negbT//|x /negbTE]. - exact: measurable_neq_cst. - + exact: measurable_fun_eq_cst. - + exact: measurable_fun_cst. - + apply: emeasurable_funM. - by have := measurable_kernel f U mU; exact: measurable_funS. - apply/EFin_measurable_fun. - apply: (measurable_fun_comp' (F := [set r : R | r != 0%R])) => //. - * exact: open_measurable. - * move=> /= r [t] [] [_ H1] H2 H3. - apply/eqP => H4; subst r. - move/eqP : H4. - rewrite fine_eq0 ?H1//. - rewrite ge0_fin_numE//. - by rewrite lt_neqAle leey H2. - * apply: open_continuous_measurable_fun => //. - apply/in_setP => x /= x0. - by apply: inv_continuous. - * apply: measurable_fun_comp => /=. - exact: measurable_fun_fine. - by have := measurable_kernel f _ measurableT; exact: measurable_funS. +- apply: emeasurable_funM. + by have := measurable_kernel f U mU; exact: measurable_funS. + apply/EFin_measurable_fun. + apply: (@measurable_fun_comp' _ _ _ _ _ _ [set r : R | r != 0%R]) => //. + + exact: open_measurable. + + move=> /= r [t] [] [_ ft0] ftoo ftr; apply/eqP => r0. + move: (ftr); rewrite r0 => /eqP; rewrite fine_eq0 ?ft0//. + by rewrite ge0_fin_numE// lt_neqAle leey ftoo. + + apply: open_continuous_measurable_fun => //; apply/in_setP => x /= x0. + exact: inv_continuous. + + apply: measurable_fun_comp => /=; first exact: measurable_fun_fine. + by have := measurable_kernel f _ measurableT; exact: measurable_funS. Qed. HB.instance Definition _ := isKernel.Build _ _ _ _ R (knormalize P) @@ -1158,13 +1151,13 @@ Let kcomp_ge0 x U : 0 <= (l \; k) x U. Proof. exact: integral_ge0. Qed. Let kcomp_sigma_additive x : semi_sigma_additive ((l \; k) x). Proof. move=> U mU tU mUU; rewrite [X in _ --> X](_ : _ = - \int[l x]_y (\sum_(n V _. by apply/esym/cvg_lim => //; exact/measure_semi_sigma_additive. apply/cvg_closeP; split. by apply: is_cvg_nneseries => n _; exact: integral_ge0. rewrite closeE// integral_sum// => n. -by have /measurable_fun_prod1 := measurable_kernel k (U n) (mU n). +by have /measurable_fun_prod1 := measurable_kernel k _ (mU n). Qed. HB.instance Definition _ x := isMeasure.Build _ R _ @@ -1187,7 +1180,7 @@ Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') Lemma measurable_fun_kcomp_finite U : measurable U -> measurable_fun setT ((l \; k) ^~ U). Proof. -move=> mU; apply: (measurable_fun_integral_finite_kernel _ (k ^~ U)) => //=. +move=> mU; apply: (measurable_fun_integral_finite_kernel (k ^~ U)) => //=. exact/measurable_kernel. Qed. @@ -1209,7 +1202,7 @@ have /measure_fam_uubP[s hs] := measure_uub l. apply/measure_fam_uubP; exists (PosNum [gt0 of (r%:num * s%:num)%R]) => x /=. apply: (@le_lt_trans _ _ (\int[l x]__ r%:num%:E)). apply: ge0_le_integral => //. - - have /measurable_fun_prod1 := measurable_kernel k setT measurableT. + - have /measurable_fun_prod1 := measurable_kernel k _ measurableT. exact. - exact/measurable_fun_cst. - by move=> y _; exact/ltW/hr. @@ -1261,7 +1254,7 @@ Qed. Lemma measurable_fun_mkcomp_sfinite U : measurable U -> measurable_fun setT ((l \; k) ^~ U). Proof. -move=> mU; apply: (measurable_fun_integral_sfinite_kernel _ (k ^~ U)) => //. +move=> mU; apply: (measurable_fun_integral_sfinite_kernel (k ^~ U)) => //. exact/measurable_kernel. Qed. @@ -1285,119 +1278,97 @@ End kcomp_sfinite_kernel. End KCOMP_SFINITE_KERNEL. HB.export KCOMP_SFINITE_KERNEL. -(* pollard? *) -Section measurable_fun_integral_kernel'. +Section measurable_fun_preimage_integral. Variables (d d' : _) (X : measurableType d) (Y : measurableType d') (R : realType). -Variables (l : X -> {measure set Y -> \bar R}) - (k : Y -> \bar R) +Variables (k : Y -> \bar R) (k_ : ({nnsfun Y >-> R}) ^nat) (ndk_ : nondecreasing_seq (k_ : (Y -> R)^nat)) (k_k : forall z, setT z -> EFin \o (k_ ^~ z) --> k z). -Let p : (X * Y -> R)^nat := fun n xy => k_ n xy.2. +Let k_2 : (X * Y -> R)^nat := fun n => k_ n \o snd. -Let p_ge0 n x : (0 <= p n x)%R. Proof. by []. Qed. +Let k_2_ge0 n x : (0 <= k_2 n x)%R. Proof. by []. Qed. -HB.instance Definition _ n := @IsNonNegFun.Build _ R (p n) (p_ge0 n). +HB.instance Definition _ n := @IsNonNegFun.Build _ _ _ (k_2_ge0 n). -Let mp n : measurable_fun setT (p n). -Proof. -rewrite /p => _ /= B mB; rewrite setTI. -have mk_n : measurable_fun setT (k_ n) by []. -rewrite (_ : _ @^-1` _ = setT `*` (k_ n @^-1` B)); last first. - by apply/seteqP; split => xy /=; tauto. -apply: measurableM => //. -have := mk_n measurableT _ mB. -by rewrite setTI. -Qed. +Let mk_2 n : measurable_fun setT (k_2 n). +Proof. by apply: measurable_fun_comp => //; exact: measurable_fun_snd. Qed. -HB.instance Definition _ n := @IsMeasurableFun.Build _ _ R (p n) (mp n). +HB.instance Definition _ n := @IsMeasurableFun.Build _ _ _ _ (mk_2 n). -Let fp n : finite_set (range (p n)). +Let fk_2 n : finite_set (range (k_2 n)). Proof. have := @fimfunP _ _ (k_ n). -suff : range (k_ n) = range (p n) by move=> <-. +suff : range (k_ n) = range (k_2 n) by move=> <-. by apply/seteqP; split => r [y ?] <-; [exists (point, y)|exists y.2]. Qed. -HB.instance Definition _ n := @FiniteImage.Build _ _ (p n) (fp n). +HB.instance Definition _ n := @FiniteImage.Build _ _ _ (fk_2 n). -Lemma measurable_fun_preimage_integral : - (forall n r, measurable_fun setT (fun x => l x (k_ n @^-1` [set r]))) -> +Lemma measurable_fun_preimage_integral (l : X -> {measure set Y -> \bar R}) : + (forall n r, measurable_fun setT (l ^~ (k_ n @^-1` [set r]))) -> measurable_fun setT (fun x => \int[l x]_z k z). Proof. -move=> h. -apply: (measurable_fun_xsection_integral l (fun xy => k xy.2) - (fun n => [the {nnsfun _ >-> _} of p n])) => /=. -- by rewrite /p => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_. +move=> h; apply: (measurable_fun_xsection_integral (k \o snd) l + (fun n => [the {nnsfun _ >-> _} of k_2 n])) => /=. +- by rewrite /k_2 => m n mn; apply/lefP => -[x y] /=; exact/lefP/ndk_. - by move=> [x y]; exact: k_k. - move=> n r _ /= B mB. - have := h n r measurableT B mB. - rewrite !setTI. - suff : ((fun x => l x (k_ n @^-1` [set r])) @^-1` B) = - ((fun x => l x (xsection (p n @^-1` [set r]) x)) @^-1` B) by move=> ->. - apply/seteqP; split => x/=. - suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. - by apply/seteqP; split; move=> y/=; - rewrite /xsection/= /p /preimage/= inE/=. - suff : (k_ n @^-1` [set r]) = (xsection (p n @^-1` [set r]) x) by move=> ->. - by apply/seteqP; split; move=> y/=; rewrite /xsection/= /p /preimage/= inE/=. + have := h n r measurableT B mB; rewrite !setTI. + suff : (l ^~ (k_ n @^-1` [set r])) @^-1` B = + (fun x => l x (xsection (k_2 n @^-1` [set r]) x)) @^-1` B by move=> ->. + by apply/seteqP; split => x/=; rewrite xsection_preimage_snd. Qed. -End measurable_fun_integral_kernel'. +End measurable_fun_preimage_integral. Lemma measurable_fun_integral_kernel - (d d' d3 : _) (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) - (l : R.-ker [the measurableType _ of (X * Y)%type] ~> Z) c - (k : Z -> \bar R) (k0 : forall z, True -> 0 <= k z) (mk : measurable_fun setT k) : - measurable_fun setT (fun y => \int[l (c, y)]_z k z). + d d' (X : measurableType d) (Y : measurableType d') (R : realType) + (l : X -> {measure set Y -> \bar R}) + (ml : forall U, measurable U -> measurable_fun setT (l ^~ U)) + (* NB: l is really just a kernel *) + (k : Y -> \bar R) (k0 : forall z, 0 <= k z) (mk : measurable_fun setT k) : + measurable_fun setT (fun x => \int[l x]_y k y). Proof. -have [k_ [ndk_ k_k]] := approximation measurableT mk k0. -apply: (measurable_fun_preimage_integral ndk_ k_k) => n r. -have := measurable_kernel l (k_ n @^-1` [set r]) (measurable_sfunP (k_ n) r). -by move=> /measurable_fun_prod1; exact. +have [k_ [ndk_ k_k]] := approximation measurableT mk (fun x _ => k0 x). +by apply: (measurable_fun_preimage_integral ndk_ k_k) => n r; exact/ml. Qed. Section integral_kcomp. +Variables (d d2 d3 : _) (X : measurableType d) (Y : measurableType d2) + (Z : measurableType d3) (R : realType). +Variable l : R.-sfker X ~> Y. +Variables k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z. -Let integral_kcomp_indic d d' d3 (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) - x (E : set _) (mE : measurable E) : +Let integral_kcomp_indic x E (mE : measurable E) : \int[(l \; k) x]_z (\1_E z)%:E = \int[l x]_y (\int[k (x, y)]_z (\1_E z)%:E). Proof. rewrite integral_indic//= /kcomp. by apply eq_integral => y _; rewrite integral_indic. Qed. -Let integral_kcomp_nnsfun d d' d3 (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) - x (f : {nnsfun Z >-> R}) : +Let integral_kcomp_nnsfun x (f : {nnsfun Z >-> R}) : \int[(l \; k) x]_z (f z)%:E = \int[l x]_y (\int[k (x, y)]_z (f z)%:E). Proof. under [in LHS]eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM. - have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. + - move=> r; apply/EFin_measurable_fun/measurable_funrM. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + - by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. under [in RHS]eq_integral. move=> y _. under eq_integral. - move=> z _. - rewrite fimfunE -sumEFin. - over. + by move=> z _; rewrite fimfunE -sumEFin; over. rewrite /= ge0_integral_sum//; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM. - have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. - by rewrite (_ : \1__ = mindic R fr). - by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. + - move=> r; apply/EFin_measurable_fun/measurable_funrM. + have fr : measurable (f @^-1` [set r]) by exact/measurable_sfunP. + by rewrite (_ : \1__ = mindic R fr). + - by move=> r z _; rewrite EFinM nnfun_muleindic_ge0. under eq_bigr. move=> r _. - rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. + rewrite (integralM_indic _ (fun r => f @^-1` [set r]))//; last first. by move=> r0; rewrite preimage_nnfun0. rewrite integral_indic// setIT. over. @@ -1407,64 +1378,55 @@ rewrite /= ge0_integral_sum//; last 2 first. have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). by move=> /measurable_fun_prod1; exact. - move=> n y _. - have := @mulemu_ge0 _ _ _ (k (x, y)) n (fun n => f @^-1` [set n]). + have := mulemu_ge0 (fun n => f @^-1` [set n]). by apply; exact: preimage_nnfun0. apply eq_bigr => r _. -rewrite (@integralM_indic _ _ _ _ _ _ (fun r => f @^-1` [set r]))//; last first. +rewrite (integralM_indic _ (fun r => f @^-1` [set r]))//; last first. exact: preimage_nnfun0. rewrite /= integral_kcomp_indic; last exact/measurable_sfunP. -rewrite (@integralM_0ifneg _ _ _ _ _ _ (fun r t => k (x, t) (f @^-1` [set r])))//; last 2 first. - move=> r0. - apply/funext => y. - by rewrite preimage_nnfun0// measure0. - have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). - by move/measurable_fun_prod1; exact. -congr (_ * _); apply eq_integral => y _. -by rewrite integral_indic// setIT. +have [r0|r0] := leP 0%R r. + rewrite ge0_integralM//; last first. + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f r). + by move/measurable_fun_prod1; exact. + by congr (_ * _); apply eq_integral => y _; rewrite integral_indic// setIT. +rewrite integral_eq0 ?mule0; last first. + by move=> y _; rewrite integral_eq0// => z _; rewrite preimage_nnfun0// indic0. +by rewrite integral_eq0// => y _; rewrite preimage_nnfun0// measure0 mule0. Qed. -Lemma integral_kcomp d d' d3 (X : measurableType d) (Y : measurableType d') - (Z : measurableType d3) (R : realType) (l : R.-sfker X ~> Y) - (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) - x f : (forall z, 0 <= f z) -> measurable_fun setT f -> +Lemma integral_kcomp x f : (forall z, 0 <= f z) -> measurable_fun setT f -> \int[(l \; k) x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). Proof. move=> f0 mf. have [f_ [ndf_ f_f]] := approximation measurableT mf (fun z _ => f0 z). -transitivity (\int[(l \; k) x]_z (lim (EFin \o (f_^~ z)))). - apply/eq_integral => z _. - apply/esym/cvg_lim => //=. - exact: f_f. +transitivity (\int[(l \; k) x]_z (lim (EFin \o f_^~ z))). + by apply/eq_integral => z _; apply/esym/cvg_lim => //=; exact: f_f. rewrite monotone_convergence//; last 3 first. - by move=> n; apply/EFin_measurable_fun. + by move=> n; exact/EFin_measurable_fun. by move=> n z _; rewrite lee_fin. by move=> z _ a b /ndf_ /lefP ab; rewrite lee_fin. -rewrite (_ : (fun _ => _) = (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first. +rewrite (_ : (fun _ => _) = + (fun n => \int[l x]_y (\int[k (x, y)]_z (f_ n z)%:E)))//; last first. by apply/funext => n; rewrite integral_kcomp_nnsfun. transitivity (\int[l x]_y lim (fun n => \int[k (x, y)]_z (f_ n z)%:E)). rewrite -monotone_convergence//; last 3 first. - move=> n. - apply: measurable_fun_integral_kernel => //. - - by move=> z; rewrite lee_fin. - - by apply/EFin_measurable_fun. - - move=> n y _. - by apply integral_ge0 => // z _; rewrite lee_fin. - - move=> y _ a b ab. - apply: ge0_le_integral => //. + - move=> n; apply: measurable_fun_integral_kernel => //. + + move=> U mU; have := measurable_kernel k _ mU. + by move=> /measurable_fun_prod1; exact. + + by move=> z; rewrite lee_fin. + + exact/EFin_measurable_fun. + - by move=> n y _; apply integral_ge0 => // z _; rewrite lee_fin. + - move=> y _ a b ab; apply: ge0_le_integral => //. + by move=> z _; rewrite lee_fin. + exact/EFin_measurable_fun. + by move=> z _; rewrite lee_fin. + exact/EFin_measurable_fun. - + move: ab => /ndf_ /lefP ab z _. - by rewrite lee_fin. -apply eq_integral => y _. -rewrite -monotone_convergence//; last 3 first. - move=> n; exact/EFin_measurable_fun. - by move=> n z _; rewrite lee_fin. - by move=> z _ a b /ndf_ /lefP; rewrite lee_fin. -apply eq_integral => z _. -apply/cvg_lim => //. -exact: f_f. + + by move: ab => /ndf_ /lefP ab z _; rewrite lee_fin. +apply eq_integral => y _; rewrite -monotone_convergence//; last 3 first. + - by move=> n; exact/EFin_measurable_fun. + - by move=> n z _; rewrite lee_fin. + - by move=> z _ a b /ndf_ /lefP; rewrite lee_fin. +by apply eq_integral => z _; apply/cvg_lim => //; exact: f_f. Qed. End integral_kcomp. diff --git a/theories/prob_lang.v b/theories/prob_lang.v index bc0e9b5350..ccae7124bb 100644 --- a/theories/prob_lang.v +++ b/theories/prob_lang.v @@ -42,11 +42,12 @@ Local Open Scope ereal_scope. Lemma onem1' (R : numDomainType) (p : R) : (p + `1- p = 1)%R. Proof. by rewrite /onem addrCA subrr addr0. Qed. -Lemma onem_nonneg_proof (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) : - (0 <= `1-(p%:num))%R. +Lemma onem_nonneg_proof (R : numDomainType) (p : {nonneg R}) : + (p%:num <= 1 -> 0 <= `1-(p%:num))%R. Proof. by rewrite /onem/= subr_ge0. Qed. -Definition onem_nonneg (R : numDomainType) (p : {nonneg R}) (p1 : (p%:num <= 1)%R) := +Definition onem_nonneg (R : numDomainType) (p : {nonneg R}) + (p1 : (p%:num <= 1)%R) := NngNum (onem_nonneg_proof p1). Lemma expR_ge0 (R : realType) (x : R) : (0 <= expR x)%R. @@ -69,10 +70,11 @@ Local Close Scope ring_scope. Let mbernoulli_setT : mbernoulli [set: _] = 1. Proof. rewrite /mbernoulli/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. -by rewrite /mscale/= !diracE !in_setT !mule1 -EFinD onem1'. +by rewrite /mscale/= !diracT !mule1 -EFinD onem1'. Qed. -HB.instance Definition _ := @isProbability.Build _ _ R mbernoulli mbernoulli_setT. +HB.instance Definition _ := + @isProbability.Build _ _ R mbernoulli mbernoulli_setT. Definition bernoulli := [the probability _ _ of mbernoulli]. @@ -89,8 +91,8 @@ Definition mscore t : {measure set _ -> \bar R} := Lemma mscoreE t U : mscore t U = if U == set0 then 0 else `| (f t)%:E |. Proof. rewrite /mscore/= /mscale/=; have [->|->] := set_unit U. - by rewrite eqxx diracE in_set0 mule0. -by rewrite diracE in_setT mule1 (negbTE (setT0 _)). + by rewrite eqxx dirac0 mule0. +by rewrite diracT mule1 (negbTE (setT0 _)). Qed. Lemma measurable_fun_mscore U : measurable_fun setT f -> @@ -164,18 +166,19 @@ HB.instance Definition _ i t := isMeasure.Build _ _ _ Lemma measurable_fun_k i U : measurable U -> measurable_fun setT (k mr i ^~ U). Proof. move=> /= mU; rewrite /k /=. -rewrite (_ : (fun x : T => _) = (fun x => if (i%:R)%:E <= x < (i.+1%:R)%:E then x else 0) \o - (mscore r ^~ U)) //. +rewrite (_ : (fun x : T => _) = + (fun x => if i%:R%:E <= x < i.+1%:R%:E then x else 0) \o (mscore r ^~ U)) //. apply: measurable_fun_comp => /=; last exact/measurable_fun_mscore. -pose A : _ -> \bar R := (fun x => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E). +pose A : _ -> \bar R := + fun x => x * (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set (\bar R)) x)%:E. rewrite (_ : (fun x => _) = A); last first. apply/funext => x; rewrite /A; case: ifPn => ix. - by rewrite indicE/= mem_set ?mule1//. + by rewrite indicE/= mem_set ?mule1. by rewrite indicE/= memNset ?mule0// /= in_itv/=; exact/negP. rewrite {}/A. apply emeasurable_funM => /=; first exact: measurable_fun_id. apply/EFin_measurable_fun. -have mi : measurable (`[(i%:R)%:E, (i.+1%:R)%:E[%classic : set (\bar R)). +have mi : measurable (`[i%:R%:E, i.+1%:R%:E[%classic : set (\bar R)). exact: emeasurable_itv. by rewrite (_ : \1__ = mindic R mi). Qed. @@ -188,12 +191,11 @@ HB.instance Definition _ i := Lemma mk_uub (i : nat) : measure_fam_uub (mk i). Proof. exists i.+1%:R => /= t; rewrite /k mscoreE setT_unit. -rewrite (_ : [set tt] == set0 = false); last first. - by apply/eqP => /seteqP[] /(_ tt) /(_ erefl). -by case: ifPn => // /andP[]. +by case: ifPn => //; case: ifPn => // _ /andP[]. Qed. -HB.instance Definition _ i := @Kernel_isFinite.Build _ _ _ _ R (mk i) (mk_uub i). +HB.instance Definition _ i := + @Kernel_isFinite.Build _ _ _ _ R (mk i) (mk_uub i). End score. End SCORE. @@ -207,7 +209,8 @@ Definition kscore (mr : measurable_fun setT r) Variable (mr : measurable_fun setT r). -Let measurable_fun_kscore U : measurable U -> measurable_fun setT (kscore mr ^~ U). +Let measurable_fun_kscore U : measurable U -> + measurable_fun setT (kscore mr ^~ U). Proof. by move=> /= _; exact: measurable_fun_mscore. Qed. HB.instance Definition _ := isKernel.Build _ _ T _ R @@ -228,11 +231,13 @@ apply/esym/cvg_lim => //. rewrite -(cvg_shiftn `|floor (fine `|(r t)%:E|)|%N.+1)/=. rewrite (_ : (fun _ => _) = cst `|(r t)%:E|); first exact: cvg_cst. apply/funext => n. -pose floor_r := widen_ord (leq_addl n `|floor `|r t| |.+1) (Ordinal (ltnSn `|floor `|r t| |)). +pose floor_r := widen_ord (leq_addl n `|floor `|r t| |.+1) + (Ordinal (ltnSn `|floor `|r t| |)). rewrite big_mkord (bigD1 floor_r)//= ifT; last first. rewrite lee_fin lte_fin; apply/andP; split. by rewrite natr_absz (@ger0_norm _ (floor `|r t|)) ?floor_ge0 ?floor_le. - by rewrite -addn1 natrD natr_absz (@ger0_norm _ (floor `|r t|)) ?floor_ge0 ?lt_succ_floor. + rewrite -addn1 natrD natr_absz. + by rewrite (@ger0_norm _ (floor `|r t|)) ?floor_ge0 ?lt_succ_floor. rewrite big1 ?adde0//= => j jk. rewrite ifF// lte_fin lee_fin. move: jk; rewrite neq_ltn/= => /orP[|] jr. @@ -241,11 +246,12 @@ move: jk; rewrite neq_ltn/= => /orP[|] jr. move: jr; rewrite -lez_nat => /le_trans; apply. by rewrite -[leRHS](@ger0_norm _ (floor `|r t|)) ?floor_ge0. - suff : (`|r t| < j%:R)%R by rewrite ltNge => /negbTE ->. - move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|r t|)) ?floor_ge0// ltr_int. - by rewrite -floor_lt_int. + move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|r t|)) ?floor_ge0//. + by rewrite ltr_int -floor_lt_int. Qed. -HB.instance Definition _ := @Kernel_isSFinite.Build _ _ _ _ _ (kscore mr) sfinite_kscore. +HB.instance Definition _ := + @Kernel_isSFinite.Build _ _ _ _ _ (kscore mr) sfinite_kscore. End kscore. @@ -507,12 +513,9 @@ Lemma iteE (f : X -> bool) (mf : measurable_fun setT f) Proof. apply/eq_measure/funext => U. rewrite /ite; unlock => /=. -rewrite /kcomp/=. -rewrite integral_dirac//=. -rewrite indicT. -rewrite mul1e. -rewrite -/(measure_add (ITE.kiteT k1 (x, f x)) - (ITE.kiteF k2 (x, f x))). +rewrite /kcomp/= integral_dirac//=. +rewrite indicT mul1e. +rewrite -/(measure_add (ITE.kiteT k1 (x, f x)) (ITE.kiteF k2 (x, f x))). rewrite measure_addE. rewrite /ITE.kiteT /ITE.kiteF/=. by case: ifPn => fx /=; rewrite /mzero ?(adde0,add0e). @@ -566,8 +569,7 @@ Lemma letin_retk x U : measurable U -> letin (ret mf) k x U = k (x, f x) U. Proof. -move=> mU; rewrite letinE retE integral_dirac//. - by rewrite indicE mem_set// mul1e. +move=> mU; rewrite letinE retE integral_dirac ?indicT ?mul1e//. have /measurable_fun_prod1 := measurable_kernel k _ mU. exact. Qed. @@ -584,13 +586,13 @@ End insn1. Module Notations. -Notation var1_of2 := (@measurable_fun_fst _ _ _ _). -Notation var2_of2 := (@measurable_fun_snd _ _ _ _). -Notation var1_of3 := (measurable_fun_comp (@measurable_fun_fst _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var3_of3 := (@measurable_fun_snd _ _ _ _). +Notation var1of2 := (@measurable_fun_fst _ _ _ _). +Notation var2of2 := (@measurable_fun_snd _ _ _ _). +Notation var1of3 := (measurable_fun_comp (@measurable_fun_fst _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var2of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var3of3 := (@measurable_fun_snd _ _ _ _). Notation mR := Real_sort__canonical__measure_Measurable. Notation munit := Datatypes_unit__canonical__measure_Measurable. @@ -608,13 +610,13 @@ Let kcomp_scoreE d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) (score mf \; g) r U = `|f r|%:E * g (r, tt) U. Proof. rewrite /= /kcomp /kscore /= ge0_integral_mscale//=. -by rewrite integral_dirac// indicE in_setT mul1e. +by rewrite integral_dirac// indicT mul1e. Qed. Lemma scoreE d' (T' : measurableType d') (x : T * T') (U : set T') (f : R -> R) (r : R) (r0 : (0 <= r)%R) (f0 : (forall r, 0 <= r -> 0 <= f r)%R) (mf : measurable_fun setT f) : - score (measurable_fun_comp mf var2_of2) + score (measurable_fun_comp mf var2of2) (x, r) (curry (snd \o fst) x @^-1` U) = (f r)%:E * \d_x.2 U. Proof. by rewrite /score/= /mscale/= ger0_norm// f0. Qed. @@ -625,8 +627,7 @@ Lemma score_score (f : R -> R) (g : R * unit -> R) letin (score mf) (score mg) x U = score (measurable_funM mf (measurable_fun_prod2 tt mg)) x U. Proof. -rewrite {1}/letin. -unlock. +rewrite {1}/letin; unlock. by rewrite kcomp_scoreE/= /mscale/= diracE normrM muleA EFinM. Qed. @@ -661,9 +662,7 @@ Section letinC. Variables (d d1 : _) (X : measurableType d) (Y : measurableType d1). Variables (R : realType) (d' : _) (Z : measurableType d'). -Notation var2_of3 := (measurable_fun_comp (@measurable_fun_snd _ _ _ _) - (@measurable_fun_fst _ _ _ _)). -Notation var3_of3 := (@measurable_fun_snd _ _ _ _). +Import Notations. Variables (t : R.-sfker Z ~> X) (t' : R.-sfker [the measurableType _ of (Z * Y)%type] ~> X) @@ -675,10 +674,10 @@ Variables (t : R.-sfker Z ~> X) Lemma letinC z A : measurable A -> letin t (letin u' - (ret (measurable_fun_pair var2_of3 var3_of3))) z A = + (ret (measurable_fun_pair var2of3 var3of3))) z A = letin u (letin t' - (ret (measurable_fun_pair var3_of3 var2_of3))) z A. + (ret (measurable_fun_pair var3of3 var2of3))) z A. Proof. move=> mA. rewrite !letinE. @@ -788,11 +787,8 @@ Lemma letin_sample_bernoulli (R : realType) (d d' : _) (T : measurableType d) r%:num%:E * u (x, true) y + (`1- (r%:num : R))%:E * u (x, false) y. Proof. rewrite letinE/= sampleE. -rewrite ge0_integral_measure_sum//. -rewrite 2!big_ord_recl/= big_ord0 adde0/=. -rewrite !ge0_integral_mscale//=. -rewrite !integral_dirac//=. -by rewrite indicE in_setT mul1e indicE in_setT mul1e. +rewrite ge0_integral_measure_sum// 2!big_ord_recl/= big_ord0 adde0/=. +by rewrite !ge0_integral_mscale//= !integral_dirac//= indicT 2!mul1e. Qed. Section sample_and_return. @@ -802,7 +798,7 @@ Variables (R : realType) (d : _) (T : measurableType d). Definition sample_and_return : R.-sfker T ~> _ := letin (sample (bernoulli p27)) (* T -> B *) - (ret var2_of2) (* T * B -> B *). + (ret var2of2) (* T * B -> B *). Lemma sample_and_returnE t U : sample_and_return t U = (2 / 7)%:E * \d_true U + (5 / 7)%:E * \d_false U. @@ -828,7 +824,7 @@ Definition sample_and_branch : R.-sfker T ~> mR R := letin (sample (bernoulli p27)) (* T -> B *) - (ite var2_of2 (ret k3) (ret k10)). + (ite var2of2 (ret k3) (ret k10)). Lemma sample_and_branchE t U : sample_and_branch t U = (2 / 7)%:E * \d_(3 : R) U + @@ -848,9 +844,9 @@ Hypothesis mh : measurable_fun setT h. Definition kstaton_bus : R.-sfker T ~> mbool := letin (sample (bernoulli p27)) (letin - (letin (ite var2_of2 (ret k3) (ret k10)) - (score (measurable_fun_comp mh var3_of3))) - (ret var2_of3)). + (letin (ite var2of2 (ret k3) (ret k10)) + (score (measurable_fun_comp mh var3of3))) + (ret var2of3)). Definition staton_bus := normalize kstaton_bus. @@ -897,14 +893,9 @@ Lemma staton_busE P (t : R) U : ((2 / 7)%:E * (poisson4 3)%:E * \d_true U + (5 / 7)%:E * (poisson4 10)%:E * \d_false U) * N^-1%:E. Proof. -rewrite /staton_bus. -rewrite normalizeE /=. -rewrite !kstaton_bus_poissonE. -rewrite diracE mem_set// mule1. -rewrite diracE mem_set// mule1. -rewrite ifF //. -apply/negbTE. -by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// poisson_gt0// ltr0n. +rewrite /staton_bus normalizeE /= !kstaton_bus_poissonE !diracT !mule1 ifF //. +apply/negbTE; rewrite gt_eqF// lte_fin. +by rewrite addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// poisson_gt0// ltr0n. Qed. End staton_bus_poisson. @@ -953,13 +944,9 @@ Lemma staton_bus_exponentialE P (t : R) U : (5 / 7)%:E * (exp1560 10)%:E * \d_false U) * N^-1%:E. Proof. rewrite /staton_bus. -rewrite normalizeE /=. -rewrite !kstaton_bus_exponentialE. -rewrite diracE mem_set// mule1. -rewrite diracE mem_set// mule1. -rewrite ifF //. -apply/negbTE. -by rewrite gt_eqF// lte_fin addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// exp_density_gt0 ?ltr0n. +rewrite normalizeE /= !kstaton_bus_exponentialE !diracT !mule1 ifF //. +apply/negbTE; rewrite gt_eqF// lte_fin. +by rewrite addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// exp_density_gt0 ?ltr0n. Qed. End staton_bus_exponential. diff --git a/theories/wip.v b/theories/wip.v index 334697b692..c31538faac 100644 --- a/theories/wip.v +++ b/theories/wip.v @@ -6,6 +6,11 @@ Require Import reals ereal topology normedtype sequences esum measure. Require Import lebesgue_measure fsbigop numfun lebesgue_integral exp kernel. Require Import trigo prob_lang. +(******************************************************************************) +(* Semantics of a programming language PPL using s-finite kernels (wip) *) +(* *) +(******************************************************************************) + Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. @@ -42,11 +47,6 @@ Proof. by rewrite /gauss01_density /gauss_density mul1r subr0 divr1. Qed. Definition mgauss01 (V : set R) := \int[lebesgue_measure]_(x in V) (gauss01_density x)%:E. -Lemma integral_gauss01_density : - \int[lebesgue_measure]_x (gauss01_density x)%:E = 1%E. -Proof. -Admitted. - Lemma measurable_fun_gauss_density m s : measurable_fun setT (gauss_density m s). Proof. @@ -69,6 +69,9 @@ Proof. by rewrite /mgauss01 integral_ge0//= => x _; rewrite lee_fin gauss_density_ge0. Qed. +Axiom integral_gauss01_density : + \int[lebesgue_measure]_x (gauss01_density x)%:E = 1%E. + Let mgauss01_sigma_additive : semi_sigma_additive mgauss01. Proof. move=> /= F mF tF mUF. @@ -121,8 +124,8 @@ Variable mu : {measure set mR R -> \bar R}. Definition staton_lebesgue : R.-sfker T ~> _ := letin (sample (@gauss01 R)) (letin - (score (measurable_fun_comp mf1 var2_of2)) - (ret var2_of3)). + (score (measurable_fun_comp mf1 var2of2)) + (ret var2of3)). Lemma staton_lebesgueE x U : measurable U -> staton_lebesgue x U = lebesgue_measure U. @@ -137,7 +140,7 @@ transitivity (\int[@mgauss01 R]_(y in U) (f1 y)%:E). apply: eq_integral => /= r _. rewrite letinE/= ge0_integral_mscale//= ger0_norm//; last first. by rewrite invr_ge0// gauss_density_ge0. - by rewrite integral_dirac// indicE in_setT mul1e retE/= diracE indicE. + by rewrite integral_dirac// indicT mul1e retE/= diracE indicE. transitivity (\int[lebesgue_measure]_(x in U) (gauss01_density x * f1 x)%:E). admit. transitivity (\int[lebesgue_measure]_(x in U) (\1_U x)%:E).