diff --git a/_CoqProject b/_CoqProject index 594eaceef7..d54967a8bb 100644 --- a/_CoqProject +++ b/_CoqProject @@ -36,6 +36,9 @@ theories/derive.v theories/measure.v theories/numfun.v theories/lebesgue_integral.v +theories/kernel.v +theories/prob_lang.v +theories/wip.v theories/summability.v theories/signed.v theories/itv.v diff --git a/theories/kernel.v b/theories/kernel.v new file mode 100644 index 0000000000..279c540567 --- /dev/null +++ b/theories/kernel.v @@ -0,0 +1,1119 @@ +(* mathcomp analysis (c) 2022 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. + +(******************************************************************************) +(* Kernels *) +(* *) +(* 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 *) +(* R.-ker X ~> Y == kernel *) +(* kseries == countable sum of kernels *) +(* 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 *) +(* mset U r == the set probability measures mu such that mu U < r *) +(* pset == the sets mset U r with U measurable and r \in [0,1] *) +(* pprobability == the measurable type generated by pset *) +(* 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. +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. + +Reserved Notation "R .-ker X ~> Y" (at level 42, format "R .-ker 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') + (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 {_ _ _ _ _} _. + +Lemma eq_kernel d d' (T : measurableType d) (T' : measurableType d') + (R : realType) (k1 k2 : R.-ker T ~> T') : + (forall x U, k1 x U = k2 x U) -> k1 = k2. +Proof. +move: k1 k2 => [m1 [[?]]] [m2 [[?]]] /= k12. +have ? : m1 = m2. + by apply/funext => t; apply/eq_measure; apply/funext => U; rewrite k12. +by subst m1; f_equal; f_equal; f_equal; apply/Prop_irrelevance. +Qed. + +Section kseries. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variable k : (R.-ker X ~> Y)^nat. + +Definition kseries : X -> {measure set Y -> \bar R} := + fun x => [the measure _ _ of mseries (k ^~ x) 0]. + +Lemma measurable_fun_kseries (U : set Y) : + measurable U -> + measurable_fun setT (kseries ^~ U). +Proof. +move=> mU. +by apply: ge0_emeasurable_fun_sum => // n; exact/measurable_kernel. +Qed. + +HB.instance Definition _ := + isKernel.Build _ _ _ _ _ kseries measurable_fun_kseries. + +End kseries. + +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[kseries k x]_y (f y) = \sum_(i f0 mf; rewrite /kseries/= ge0_integral_measure_series. +Qed. + +Section measure_fam_uub. +Context d d' (X : measurableType d) (Y : measurableType d') (R : numFieldType). +Variable k : X -> {measure set Y -> \bar R}. + +Definition measure_fam_uub := exists r, forall x, k x [set: Y] < r%:E. + +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. +suff r_gt0 : (0 < r)%R by exists (PosNum r_gt0). +by rewrite -lte_fin; apply: (le_lt_trans _ (kr point)). +Qed. + +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) & + 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 _ _ _ _ R k & Kernel_isSFinite_subdef _ _ X Y R k }. +Notation "R .-sfker X ~> Y" := (sfinite_kernel X Y R). + +Arguments sfinite_subdef {_ _ _ _ _} _. + +Lemma eq_sfkernel d d' (T : measurableType d) (T' : measurableType d') (R : realType) + (k1 k2 : R.-sfker T ~> T') : + (forall x U, k1 x U = k2 x U) -> k1 = k2. +Proof. +move: k1 k2 => [m1 [[?] [?]]] [m2 [[?] [?]]] /= k12. +have ? : m1 = m2. + by apply/funext => t; apply/eq_measure; apply/funext => U; rewrite k12. +by subst m1; f_equal; f_equal; f_equal; apply/Prop_irrelevance. +Qed. + +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 }. + +#[short(type=finite_kernel)] +HB.structure Definition FiniteKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of @SFiniteKernel _ _ _ _ _ k & + SFiniteKernel_isFinite _ _ X Y R k }. +Notation "R .-fker X ~> Y" := (finite_kernel X Y R). + +Arguments measure_uub {_ _ _ _ _} _. + +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 }. + +Section kzero. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Definition kzero : X -> {measure set Y -> \bar R} := fun _ : X => [the measure _ _ of mzero]. + +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 _ _ X Y R kzero measurable_fun_kzero. + +Lemma kzero_uub : measure_fam_uub kzero. +Proof. by exists 1%R => /= t; rewrite /mzero/=. Qed. + +End kzero. + +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 : + exists2 k_ : (R.-ker _ ~> _)^nat, forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> k x U = mseries (k_ ^~ x) 0 U. +Proof. +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_eseriesr _ _ (fun=> 0%E)); last by case. +by rewrite eseries0// adde0. +Qed. + +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. +Context 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. +by exists (fun n => [the _.-fker _ ~> _ of s n]) => x U mU; rewrite /s /= /s; by case: cid2 => ? ? ->. +Qed. + +End sfinite. + +Lemma sfinite_kernel_measure d d' (Z : measurableType d) (X : measurableType d') + (R : realType) (k : R.-sfker Z ~> X) (z : Z) : + sfinite_measure (k z). +Proof. +have [s ks] := sfinite k. +exists (s ^~ z). + move=> n; have [r snr] := measure_uub (s n). + by apply: lty_fin_num_fun; rewrite (lt_le_trans (snr _))// leey. +by move=> U mU; rewrite ks. +Qed. + +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_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 _ _ _ _ _ k & + FiniteKernel_isSubProbability _ _ 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}. + +#[short(type=probability_kernel)] +HB.structure Definition ProbabilityKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) := + {k of @SubProbabilityKernel _ _ _ _ _ k & + SubProbability_isProbability _ _ X Y R k }. +Notation "R .-pker X ~> Y" := (probability_kernel X Y R). + +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 }. + +HB.builders Context d d' (X : measurableType d) (Y : measurableType d') + (R : realType) k of Kernel_isProbability d d' X Y R k. + +Let sprob_kernel : @Kernel_isSubProbability d d' X Y R k. +Proof. +by split; apply: ub_ereal_sup => x [y _ <-{x}]; rewrite prob_kernel. +Qed. + +HB.instance Definition _ := sprob_kernel. + +HB.instance Definition _ := @SubProbability_isProbability.Build _ _ _ _ _ k prob_kernel. + +HB.end. + +Lemma finite_kernel_measure d d' (X : measurableType d) + (Y : measurableType d') (R : realType) (k : R.-fker X ~> Y) (x : X) : + fin_num_fun (k x). +Proof. +have [r k_r] := measure_uub k. +by apply: lty_fin_num_fun; rewrite (@lt_trans _ _ r%:E) ?ltey. +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 + measurability of each measure of the family) + - as a consequence, m2D_bounded holds for all x *) +Section measurable_prod_subset_kernel. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Implicit Types A : set (X * Y). + +Section xsection_kernel. +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. +move=> kD_ub; rewrite measurable_prod_measurableType. +set C := [set A `*` B | A in measurable & B in measurable]. +have CI : setI_closed C. + 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 CXY : C `<=` XY. + move=> _ [A mA [B mB <-]]; split; first exact: measurableM. + rewrite phiM. + apply: emeasurable_funM => //; first exact/measurable_kernel/measurableI. + 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// measureD. +- by rewrite setIidr//; exact: le_xsection. +- exact: measurable_xsection. +- exact: measurable_xsection. +- have [M kM] := kD_ub x. + rewrite (lt_le_trans (kM (xsection A x) _)) ?leey//. + exact: measurable_xsection. +Qed. + +End xsection_kernel. + +End measurable_prod_subset_kernel. + +(* 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. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variable k : R.-fker X ~> Y. +Implicit Types A : set (X * Y). + +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 `<=` 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 k; exists r => B mB. +by rewrite (le_lt_trans _ (hr x)) // /mrestr /= setIT le_measure// inE. +Qed. + +End measurable_fun_xsection_finite_kernel. + +Section measurable_fun_integral_finite_sfinite. +Context 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_ : ({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. +move=> h. +rewrite (_ : (fun x => _) = + (fun x => lim_esup (fun n => \int[l x]_y (k_ n (x, y))%:E))); last first. + apply/funext => x. + transitivity (lim (fun n => \int[l x]_y (k_ n (x, y))%:E)); last first. + rewrite is_cvg_lim_esupE//. + 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; 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; exact/lefP/ndk_. +apply: measurable_fun_lim_esup => n. +rewrite [X in measurable_fun _ X](_ : _ = (fun x => \int[l x]_y + (\sum_(r \in 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 \in 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_fsum//. + - by apply: eq_integral => y _; rewrite -fsumEFin. + - move=> r. + apply/EFin_measurable_fun/measurable_funrM/measurable_fun_prod1 => /=. + rewrite (_ : \1_ _ = mindic R (measurable_sfunP (k_ n) (measurable_set1 r)))//. + exact/measurable_funP. + - by move=> m y _; rewrite nnfun_muleindic_ge0. +apply: emeasurable_fun_fsum => // 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. + 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) (measurable_set1 r)))//. + exact/measurable_funP. + rewrite integral0_eq; last first. + by move=> y _; rewrite preimage_nnfun0// indic0 mule0. + by rewrite integral0_eq ?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. +apply/funext => x; rewrite integral_indic//; last first. + rewrite (_ : (fun x => _) = xsection (k_ n @^-1` [set r]) x). + exact: measurable_xsection. + 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) + (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_] := measure_uub l. +by apply: measurable_fun_xsection_finite_kernel => // /[!inE]. +Qed. + +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). +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 {_ _ _ _ _} k l. +Arguments measurable_fun_integral_finite_kernel {_ _ _ _ _} k l. +Arguments measurable_fun_integral_sfinite_kernel {_ _ _ _ _} k l. + +Section kdirac. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variable f : X -> Y. + +Definition kdirac (mf : measurable_fun setT f) + : X -> {measure set Y -> \bar R} := + fun 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. +move=> mU; apply/EFin_measurable_fun. +by rewrite (_ : (fun x => _) = mindic R mU \o f)//; exact/measurable_funT_comp. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ _ (kdirac mf) + measurable_fun_kdirac. + +Let kdirac_prob x : kdirac mf x setT = 1. +Proof. by rewrite /kdirac/= diracT. Qed. + +HB.instance Definition _ := Kernel_isProbability.Build _ _ _ _ _ + (kdirac mf) kdirac_prob. + +End kdirac. +Arguments kdirac {d d' X Y R f}. + +Section dist_salgebra_instance. +Context d (T : measurableType d) (R : realType). + +Let p0 : probability T R := [the probability _ _ of dirac point]. + +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]. + +Lemma lt0_mset (U : set T) (r : R) : (r < 0)%R -> mset U r = set0. +Proof. +move=> r0; apply/seteqP; split => // x/=. +by apply/negP; rewrite -leNgt (@le_trans _ _ 0)// lee_fin ltW. +Qed. + +Lemma gt1_mset (U : set T) (r : R) : measurable U -> (1 < r)%R -> mset U r = setT. +Proof. +move=> mU r1; apply/seteqP; split => // x/= _. +by rewrite /mset/= (le_lt_trans (probability_le1 _ _)). +Qed. + +Definition pset : set (set (probability T R)) := + [set mset U r | r in `[0%R,1%R] & U in measurable]. + +Definition pprobability : measurableType pset.-sigma := + [the measurableType _ of salgebraType pset]. + +End dist_salgebra_instance. + +Section kprobability. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variable P : X -> pprobability Y R. + +Definition kprobability (mP : measurable_fun setT P) + : X -> {measure set Y -> \bar R} := P. + +Hypothesis mP : measurable_fun setT P. + +Let measurable_fun_kprobability U : measurable U -> + measurable_fun setT (kprobability mP ^~ U). +Proof. +move=> mU. +apply: (measurability (ErealGenInftyO.measurableE R)) => _ /= -[_ [r ->] <-]. +rewrite setTI preimage_itv_infty_o -/(P @^-1` mset U r). +have [r0|r0] := leP 0%R r; last by rewrite lt0_mset// preimage_set0. +have [r1|r1] := leP r 1%R; last by rewrite gt1_mset// preimage_setT. +move: mP => /(_ measurableT (mset U r)); rewrite setTI; apply. +by apply: sub_sigma_algebra; exists r => /=; [rewrite in_itv/= r0|exists U]. +Qed. + +HB.instance Definition _ := + @isKernel.Build _ _ X Y R (kprobability mP) measurable_fun_kprobability. + +Let kprobability_prob x : kprobability mP x setT = 1. +Proof. by rewrite /kprobability/= probability_setT. Qed. + +HB.instance Definition _ := + @Kernel_isProbability.Build _ _ X Y R (kprobability mP) kprobability_prob. + +End kprobability. + +Section kadd. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variables k1 k2 : R.-ker X ~> Y. + +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; 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 _ _ _ _ _ kadd measurable_fun_kadd. +End kadd. + +Section sfkadd. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variables k1 k2 : R.-sfker X ~> Y. + +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 = mseries (k_ ^~ x) 0 U. +Proof. +have [f1 hk1] := sfinite k1; have [f2 hk2] := sfinite k2. +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/= -/(measure_add (k1 x) (k2 x)) measure_addE hk1//= hk2//=. +rewrite /mseries -nneseriesD//; apply: eq_eseriesr => n _ /=. +by rewrite -/(measure_add (f1 n x) (f2 n x)) measure_addE. +Qed. + +HB.instance Definition _ t := + Kernel_isSFinite_subdef.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. +End sfkadd. + +Section fkadd. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variables 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. + +(* TODO: move *) +Section kernel_measurable_preimage. +Context d d' (T : measurableType d) (T' : measurableType d') (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. +have /(_ measurableT [set k]) := measurable_kernel f setT measurableT. +by rewrite setTI; exact. +Qed. + +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` [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. + +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). +Proof. +move=> _ /= B mB; rewrite setTI. +have [/eqP->|/eqP->|/eqP->|/eqP->] := set_bool 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. + +Section mnormalize. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variables (f : X -> {measure set Y -> \bar R}) (P : probability Y R). + +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. +by rewrite /mnormalize; case: ifPn => // _; rewrite measure0 mul0e. +Qed. + +Let mnormalize_ge0 x U : 0 <= mnormalize x U. +Proof. by rewrite /mnormalize; case: ifPn => //; case: ifPn. Qed. + +Let mnormalize_sigma_additive x : semi_sigma_additive (mnormalize x). +Proof. +move=> F mF tF mUF; rewrite /mnormalize/=. +case: ifPn => [_|_]; first exact: measure_semi_sigma_additive. +rewrite (_ : (fun _ => _) = ((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: cvgeMr => //; exact: measure_semi_sigma_additive. +Qed. + +HB.instance Definition _ x := isMeasure.Build _ _ _ (mnormalize x) + (mnormalize0 x) (mnormalize_ge0 x) (@mnormalize_sigma_additive x). + +Let mnormalize1 x : mnormalize x setT = 1. +Proof. +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. + +HB.instance Definition _ x := + Measure_isProbability.Build _ _ _ (mnormalize x) (mnormalize1 x). + +End mnormalize. + +Section knormalize. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Variable f : R.-ker X ~> Y. + +Definition knormalize (P : probability Y R) : X -> {measure set Y -> \bar R} := + fun x => [the measure _ _ of mnormalize f P x]. + +Variable P : probability Y R. + +Let measurable_fun_knormalize U : + measurable U -> measurable_fun setT (knormalize P ^~ U). +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[->//|->]|]; 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]. +- by apply: (@measurable_funS _ _ _ _ setT) => //; 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 _ _ _ _ _ _ [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_funT_comp => /=; first exact: measurable_fun_fine. + by have := measurable_kernel f _ measurableT; exact: measurable_funS. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ R (knormalize P) + measurable_fun_knormalize. + +Let knormalize1 x : knormalize P x setT = 1. +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. +Qed. + +HB.instance Definition _ := + @Kernel_isProbability.Build _ _ _ _ _ (knormalize P) knormalize1. + +End knormalize. + +Section kcomp_def. +Context 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}. + +Definition kcomp x U := \int[l x]_y k (x, y) U. + +End kcomp_def. + +Section kcomp_is_measure. +Context 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. + +Local Notation "l \; k" := (kcomp l k). + +Let kcomp0 x : (l \; k) x set0 = 0. +Proof. +by rewrite /kcomp (eq_integral (cst 0)) ?integral0// => y _; rewrite measure0. +Qed. + +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_nneseries// => n. +by have /measurable_fun_prod1 := measurable_kernel k _ (mU n). +Qed. + +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) : ereal_scope. + +Module KCOMP_FINITE_KERNEL. + +Section kcomp_finite_kernel_kernel. +Context 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 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. + +HB.instance Definition _ := + isKernel.Build _ _ X Z R (l \; k) measurable_fun_kcomp_finite. + +End kcomp_finite_kernel_kernel. + +Section kcomp_finite_kernel_finite. +Context 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 /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 _ measurableT. + exact. + - exact/measurable_fun_cst. + - by move=> y _; exact/ltW/hr. +by rewrite integral_cst//= EFinM lte_pmul2l. +Qed. + +HB.instance Definition _ := + Kernel_isFinite.Build _ _ X Z R (l \; k) mkcomp_finite. + +End kcomp_finite_kernel_finite. +End KCOMP_FINITE_KERNEL. + +Section kcomp_sfinite_kernel. +Context 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 KCOMP_FINITE_KERNEL. + +Lemma mkcomp_sfinite : exists k_ : (R.-fker X ~> Z)^nat, + forall x U, measurable U -> (l \; k) x U = kseries k_ x U. +Proof. +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_nneseries//=; 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_fun_mkcomp_sfinite U : measurable U -> + measurable_fun setT ((l \; k) ^~ U). +Proof. +move=> mU; apply: (measurable_fun_integral_sfinite_kernel (k ^~ U)) => //. +exact/measurable_kernel. +Qed. + +End kcomp_sfinite_kernel. + +Module KCOMP_SFINITE_KERNEL. +Section kcomp_sfinite_kernel. +Context 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). + +#[export] +HB.instance Definition _ := + Kernel_isSFinite.Build _ _ X Z R (l \; k) (mkcomp_sfinite l k). + +End kcomp_sfinite_kernel. +End KCOMP_SFINITE_KERNEL. +HB.export KCOMP_SFINITE_KERNEL. + +Section measurable_fun_preimage_integral. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +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 k_2 : (X * Y -> R)^nat := fun n => k_ n \o snd. + +Let k_2_ge0 n x : (0 <= k_2 n x)%R. Proof. by []. Qed. + +HB.instance Definition _ n := @isNonNegFun.Build _ _ _ (k_2_ge0 n). + +Let mk_2 n : measurable_fun setT (k_2 n). +Proof. by apply: measurable_funT_comp => //; exact: measurable_fun_snd. Qed. + +HB.instance Definition _ n := @isMeasurableFun.Build _ _ _ _ (mk_2 n). + +Let fk_2 n : finite_set (range (k_2 n)). +Proof. +have := @fimfunP _ _ (k_ n). +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 _ _ _ (fk_2 n). + +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 (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 : (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 (comp_preimage _ snd (k_ n)) xsection_preimage_snd. +Qed. + +End measurable_fun_preimage_integral. + +Lemma measurable_fun_integral_kernel + 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 (fun x _ => k0 x). +by apply: (measurable_fun_preimage_integral ndk_ k_k) => n r; exact/ml. +Qed. + +Section integral_kcomp. +Context d d2 d3 (X : measurableType d) (Y : measurableType d2) + (Z : measurableType d3) (R : realType). +Variables (l : R.-sfker X ~> Y) (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z). + +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 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 -fsumEFin//. +rewrite ge0_integral_fsum//; 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. + by move=> z _; rewrite fimfunE -fsumEFin//; over. + rewrite /= ge0_integral_fsum//; 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_fsbigr. + 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_fsum//; last 2 first. + - move=> r; apply: measurable_funeM. + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f (measurable_set1 r)). + by move=> /measurable_fun_prod1; exact. + - move=> n y _. + have := mulemu_ge0 (fun n => f @^-1` [set n]). + by apply; exact: preimage_nnfun0. +apply: eq_fsbigr => r _. +rewrite (integralM_indic _ (fun r => f @^-1` [set r]))//; last first. + exact: preimage_nnfun0. +rewrite /= integral_kcomp_indic; last exact/measurable_sfunP. +have [r0|r0] := leP 0%R r. + rewrite ge0_integralM//; last first. + have := measurable_kernel k (f @^-1` [set r]) (measurable_sfunP f (measurable_set1 r)). + by move/measurable_fun_prod1; exact. + by congr (_ * _); apply: eq_integral => y _; rewrite integral_indic// setIT. +rewrite integral0_eq ?mule0; last first. + by move=> y _; rewrite integral0_eq// => z _; rewrite preimage_nnfun0// indic0. +by rewrite integral0_eq// => y _; rewrite preimage_nnfun0// measure0 mule0. +Qed. + +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))). + by apply/eq_integral => z _; apply/esym/cvg_lim => //=; exact: f_f. +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 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 => //. + + 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. + + 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/lebesgue_integral.v b/theories/lebesgue_integral.v index 2fb442d89e..0d1aa2cd02 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -4029,8 +4029,12 @@ Context 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 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..80847ab9e7 --- /dev/null +++ b/theories/prob_lang.v @@ -0,0 +1,1015 @@ +(* mathcomp analysis (c) 2022 Inria and AIST. License: CeCILL-C. *) +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. +From mathcomp Require Import 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. + +(******************************************************************************) +(* Semantics of a probabilistic programming language using s-finite kernels *) +(* *) +(* 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 *) +(* 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) *) +(* 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 *) +(* behaves as k1 or k2 according to the result *) +(* *) +(* poisson == Poisson distribution function *) +(* exp_density == density function for exponential distribution *) +(* *) +(******************************************************************************) + +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. + +(* TODO: PR *) +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}) : + (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) := + NngNum (onem_nonneg_proof p1). +(* /TODO: PR *) + +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. + +Let bernoulli_setT : bernoulli [set: _] = 1. +Proof. +rewrite /bernoulli/= /measure_add/= /msum 2!big_ord_recr/= big_ord0 add0e/=. +by rewrite /mscale/= !diracT !mule1 -EFinD onem1'. +Qed. + +HB.instance Definition _ := @Measure_isProbability.Build _ _ R bernoulli bernoulli_setT. + +End bernoulli. + +Section mscore. +Context d (T : measurableType d) (R : realType). +Variable f : T -> R. + +Definition mscore t : {measure set _ -> \bar R} := + 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 dirac0 mule0. +by rewrite diracT mule1 (negbTE setT0). +Qed. + +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_funT_comp => //; exact: measurable_funT_comp. +Qed. + +End mscore. + +(* decomposition of score into finite kernels *) +Module SCORE. +Section score. +Context d (T : measurableType d) (R : realType). +Variable f : T -> R. + +Definition k (mf : measurable_fun setT f) i t U := + if i%:R%:E <= mscore f t U < i.+1%:R%:E then + mscore f t U + else + 0. + +Hypothesis mf : measurable_fun setT f. + +Lemma k0 i t : k mf i t (set0 : set unit) = 0 :> \bar R. +Proof. by rewrite /k measure0; case: ifP. Qed. + +Lemma k_ge0 i t B : 0 <= k mf i t B. +Proof. by rewrite /k; case: ifP. Qed. + +Lemma k_sigma_additive i t : semi_sigma_additive (k mf i t). +Proof. +move=> /= F mF tF mUF; rewrite /k /=. +have [F0|UF0] := eqVneq (\bigcup_n F n) set0. + rewrite F0 measure0 (_ : (fun _ => _) = cst 0). + by case: ifPn => _; exact: cvg_cst. + apply/funext => k; rewrite big1// => n _. + by move: F0 => /bigcup0P -> //; rewrite measure0; case: ifPn. +move: (UF0) => /eqP/bigcup0P/existsNP[m /not_implyP[_ /eqP Fm0]]. +rewrite [in X in _ --> X]mscoreE (negbTE UF0). +rewrite -(cvg_shiftn m.+1)/=. +case: ifPn => ir. + rewrite (_ : (fun _ => _) = cst `|(f 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 + _]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). + 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 _]mscoreE (negbTE Fm0) (negbTE ir) add0e. +rewrite big1//= => j jm; 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 eqxx; case: ifP. +Qed. + +HB.instance Definition _ i t := isMeasure.Build _ _ _ + (k mf 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 mf i ^~ U). +Proof. +move=> /= mU; rewrite /k /= (_ : (fun x => _) = + (fun x => if i%:R%:E <= x < i.+1%:R%:E then x else 0) \o (mscore f ^~ U)) //. +apply: measurable_funT_comp => /=; last exact/measurable_fun_mscore. +rewrite (_ : (fun x => _) = (fun x => x * + (\1_(`[i%:R%:E, i.+1%:R%:E [%classic : set _) x)%:E)); last first. + apply/funext => x; case: ifPn => ix; first by rewrite indicE/= mem_set ?mule1. + by rewrite indicE/= memNset ?mule0// /= in_itv/=; exact/negP. +apply: emeasurable_funM => /=; first exact: measurable_fun_id. +apply/EFin_measurable_fun. +by rewrite (_ : \1__ = mindic R (emeasurable_itv `[(i%:R)%:E, (i.+1%:R)%:E[)). +Qed. + +Definition mk i t := [the measure _ _ of k mf i t]. + +HB.instance Definition _ i := + isKernel.Build _ _ _ _ _ (mk i) (measurable_fun_k i). + +Lemma mk_uub i : measure_fam_uub (mk i). +Proof. +exists i.+1%:R => /= t; rewrite /k mscoreE setT_unit. +by case: ifPn => //; case: ifPn => // _ /andP[]. +Qed. + +HB.instance Definition _ i := + Kernel_isFinite.Build _ _ _ _ _ (mk i) (mk_uub i). + +End score. +End SCORE. + +Section kscore. +Context d (T : measurableType d) (R : realType). +Variable f : T -> R. + +Definition kscore (mf : measurable_fun setT f) + : T -> {measure set _ -> \bar R} := + mscore f. + +Variable mf : measurable_fun setT f. + +Let measurable_fun_kscore U : measurable U -> + measurable_fun setT (kscore mf ^~ U). +Proof. by move=> /= _; exact: measurable_fun_mscore. Qed. + +HB.instance Definition _ := isKernel.Build _ _ T _ R + (kscore mf) measurable_fun_kscore. + +Import SCORE. + +Let sfinite_kscore : exists k : (R.-fker T ~> _)^nat, + forall x U, measurable U -> + kscore mf x U = mseries (k ^~ x) 0 U. +Proof. +rewrite /=; exists (fun i => [the R.-fker _ ~> _ of mk mf i]) => /= t U mU. +rewrite /mseries /kscore/= mscoreE; case: ifPn => [/eqP U0|U0]. + by apply/esym/eseries0 => i _; rewrite U0 measure0. +rewrite /mk /= /k /= mscoreE (negbTE U0). +apply/esym/cvg_lim => //. +rewrite -(cvg_shiftn `|floor (fine `|(f t)%:E|)|%N.+1)/=. +rewrite (_ : (fun _ => _) = cst `|(f t)%:E|); first exact: cvg_cst. +apply/funext => n. +pose floor_f := widen_ord (leq_addl n `|floor `|f t| |.+1) + (Ordinal (ltnSn `|floor `|f t| |)). +rewrite big_mkord (bigD1 floor_f)//= ifT; last first. + rewrite lee_fin lte_fin; apply/andP; split. + by rewrite natr_absz (@ger0_norm _ (floor `|f t|)) ?floor_ge0 ?floor_le. + rewrite -addn1 natrD natr_absz. + by rewrite (@ger0_norm _ (floor `|f 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 <= `|f 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 `|f t|)) ?floor_ge0. +- suff : (`|f t| < j%:R)%R by rewrite ltNge => /negbTE ->. + move: jr; rewrite -ltz_nat -(@ltr_int R) (@gez0_abs (floor `|f t|)) ?floor_ge0//. + by rewrite ltr_int -floor_lt_int. +Qed. + +HB.instance Definition _ := + @Kernel_isSFinite.Build _ _ _ _ _ (kscore mf) sfinite_kscore. + +End kscore. + +(* decomposition of ite into s-finite kernels *) +Module ITE. +Section ite. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Section kiteT. +Variable k : R.-ker X ~> Y. + +Definition kiteT : X * bool -> {measure set Y -> \bar R} := + fun xb => if xb.2 then k xb.1 else [the measure _ _ of mzero]. + +Let measurable_fun_kiteT U : measurable U -> measurable_fun setT (kiteT ^~ U). +Proof. +move=> /= mcU; rewrite /kiteT. +rewrite (_ : (fun _ => _) = + (fun x => if x.2 then k x.1 U else mzero U)); last first. + by apply/funext => -[t b]/=; case: ifPn. +apply: (@measurable_fun_if_pair _ _ _ _ (k ^~ U) (fun=> mzero U)). + exact/measurable_kernel. +exact: measurable_fun_cst. +Qed. + +#[export] +HB.instance Definition _ := isKernel.Build _ _ _ _ _ + kiteT measurable_fun_kiteT. +End kiteT. + +Section sfkiteT. +Variable k : R.-sfker X ~> Y. + +Let sfinite_kiteT : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> kiteT k x U = mseries (k_ ^~ x) 0 U. +Proof. +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; first by rewrite hk. +by rewrite /mseries eseries0. +Qed. + +#[export] +HB.instance Definition _ t := @Kernel_isSFinite_subdef.Build _ _ _ _ _ + (kiteT k) sfinite_kiteT. +End sfkiteT. + +Section fkiteT. +Variable 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 _ _ _ _ _ + (kiteT k) kiteT_uub. +End fkiteT. + +Section kiteF. +Variable k : R.-ker X ~> Y. + +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_fun_kiteF U : measurable U -> measurable_fun setT (kiteF ^~ U). +Proof. +move=> /= mcU; rewrite /kiteF. +rewrite (_ : (fun x => _) = + (fun x => if x.2 then mzero U else k x.1 U)); last first. + by apply/funext => -[t b]/=; rewrite if_neg//; case: ifPn. +apply: (@measurable_fun_if_pair _ _ _ _ (fun=> mzero U) (k ^~ U)). + exact: measurable_fun_cst. +exact/measurable_kernel. +Qed. + +#[export] +HB.instance Definition _ := isKernel.Build _ _ _ _ _ + kiteF measurable_fun_kiteF. + +End kiteF. + +Section sfkiteF. +Variable k : R.-sfker X ~> Y. + +Let sfinite_kiteF : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> kiteF k x U = mseries (k_ ^~ x) 0 U. +Proof. +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; first by rewrite hk. +by rewrite /mseries eseries0. +Qed. + +#[export] +HB.instance Definition _ := @Kernel_isSFinite_subdef.Build _ _ _ _ _ + (kiteF k) sfinite_kiteF. + +End sfkiteF. + +Section fkiteF. +Variable k : R.-fker X ~> Y. + +Let kiteF_uub : measure_fam_uub (kiteF k). +Proof. +have /measure_fam_uubP[M hM] := measure_uub k. +by exists M%:num => /= -[]; rewrite /kiteF/= => t; case => //=; rewrite /mzero. +Qed. + +#[export] +HB.instance Definition _ := Kernel_isFinite.Build _ _ _ _ _ + (kiteF k) kiteF_uub. + +End fkiteF. +End ite. +End ITE. + +Section ite. +Context d d' (T : measurableType d) (T' : measurableType d') (R : realType). +Variables (f : T -> bool) (u1 u2 : R.-sfker T ~> T'). + +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. + +Let mite0 t : mite mf t set0 = 0. +Proof. by rewrite /mite; case: ifPn. Qed. + +Let mite_ge0 t U : 0 <= mite mf t U. +Proof. by rewrite /mite; case: ifPn. Qed. + +Let mite_sigma_additive t : semi_sigma_additive (mite mf t). +Proof. +by rewrite /mite; case: ifPn => ft; exact: measure_semi_sigma_additive. +Qed. + +HB.instance Definition _ t := isMeasure.Build _ _ _ (mite mf t) + (mite0 t) (mite_ge0 t) (@mite_sigma_additive t). + +Import ITE. + +(* +Definition kite : R.-sfker T ~> T' := + kdirac mf \; kadd (kiteT u1) (kiteF u2). +*) +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 insn2. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Definition ret (f : X -> Y) (mf : measurable_fun setT f) + : R.-pker X ~> Y := [the R.-pker _ ~> _ of kdirac mf]. + +Definition sample (P : pprobability Y R) : R.-pker X ~> Y := + [the R.-pker _ ~> _ of kprobability (measurable_fun_cst P)]. + +Definition normalize (k : R.-sfker X ~> Y) P : X -> probability Y R := + fun x => [the probability _ _ of mnormalize k P x]. + +Definition ite (f : X -> bool) (mf : measurable_fun setT f) + (k1 k2 : R.-sfker X ~> Y) : 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. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Lemma retE (f : X -> Y) (mf : measurable_fun setT f) x : + ret mf x = \d_(f x) :> (_ -> \bar R). +Proof. by []. Qed. + +Lemma sampleE (P : probability Y R) (x : X) : sample P x = P. +Proof. by []. 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. by rewrite /normalize /= /mnormalize; case: ifPn. Qed. + +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. +apply/eq_measure/funext => U. +rewrite /ite; unlock => /=. +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). +Qed. + +End insn2_lemmas. + +Section insn3. +Context d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). + +Definition letin (l : R.-sfker X ~> Y) (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) + : R.-sfker X ~> Z := + [the R.-sfker X ~> Z of l \; k]. + +End insn3. + +Section insn3_lemmas. +Context d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). + +Lemma letinE (l : R.-sfker X ~> Y) (k : R.-sfker [the measurableType _ of (X * Y)%type] ~> Z) x U : + letin l k x U = \int[l x]_y k (x, y) U. +Proof. by []. Qed. + +End insn3_lemmas. + +(* rewriting laws *) +Section letin_return. +Context d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). + +Lemma letin_kret (k : R.-sfker X ~> Y) + (f : X * Y -> Z) (mf : measurable_fun setT f) x U : + measurable U -> + letin k (ret mf) x U = k x (curry f x @^-1` U). +Proof. +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_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=> mU; rewrite letinE retE integral_dirac ?indicT ?mul1e//. +have /measurable_fun_prod1 := measurable_kernel k _ mU. +exact. +Qed. + +End letin_return. + +Section insn1. +Context d (X : measurableType d) (R : realType). + +Definition score (f : X -> R) (mf : measurable_fun setT f) + : R.-sfker X ~> _ := + [the R.-sfker X ~> _ of kscore mf]. + +End insn1. + +Section hard_constraint. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Definition fail := + letin (score (@measurable_fun_cst _ _ X _ setT (0%R : R))) + (ret (@measurable_fun_cst _ _ _ Y setT point)). + +Lemma failE x U : fail x U = 0. +Proof. by rewrite /fail letinE ge0_integral_mscale//= normr0 mul0e. Qed. + +End hard_constraint. +Arguments fail {d d' X Y R}. + +Module Notations. + +Notation var1of2 := (@measurable_fun_fst _ _ _ _). +Notation var2of2 := (@measurable_fun_snd _ _ _ _). +Notation var1of3 := (measurable_funT_comp (@measurable_fun_fst _ _ _ _) + (@measurable_fun_fst _ _ _ _)). +Notation var2of3 := (measurable_funT_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. +Notation mbool := Datatypes_bool__canonical__measure_Measurable. + +End Notations. + +Section cst_fun. +Context d (T : measurableType d) (R : realType). + +Definition kr (r : R) := @measurable_fun_cst _ _ T _ setT r. +Definition k3 : measurable_fun _ _ := kr 3%:R. +Definition k10 : measurable_fun _ _ := kr 10%:R. +Definition ktt := @measurable_fun_cst _ _ T _ setT tt. + +End cst_fun. +Arguments kr {d T R}. +Arguments k3 {d T R}. +Arguments k10 {d T R}. +Arguments ktt {d T}. + +Section insn1_lemmas. +Import Notations. +Context d (T : measurableType d) (R : realType). + +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//=. +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_funT_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. + +Lemma score_score (f : R -> R) (g : R * unit -> R) + (mf : measurable_fun setT f) + (mg : measurable_fun setT g) : + letin (score mf) (score mg) = + score (measurable_funM mf (measurable_fun_prod2 tt mg)). +Proof. +apply/eq_sfkernel => x U. +rewrite {1}/letin; unlock. +by rewrite kcomp_scoreE/= /mscale/= diracE normrM muleA EFinM. +Qed. + +Import Notations. + +(* hard constraints to express score below 1 *) +Lemma score_fail (r : {nonneg R}) (r1 : (r%:num <= 1)%R) : + score (kr r%:num) = + letin (sample [the probability _ _ of bernoulli r1] : R.-pker T ~> _) + (ite var2of2 (ret ktt) fail). +Proof. +apply/eq_sfkernel => x U. +rewrite letinE/= /sample; unlock. +rewrite integral_measure_add//= ge0_integral_mscale//= ge0_integral_mscale//=. +rewrite integral_dirac//= integral_dirac//= !indicT/= !mul1e. +by rewrite /mscale/= iteE//= iteE//= failE mule0 adde0 ger0_norm. +Qed. + +End insn1_lemmas. + +Section letin_ite. +Context d d2 d3 (T : measurableType d) (T2 : measurableType d2) + (Z : measurableType d3) (R : realType). +Variables (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_iteT : f t -> letin (ite mf k1 k2) u t U = letin k1 u t U. +Proof. +move=> ftT. +rewrite !letinE/=. +apply: eq_measure_integral => V mV _. +by rewrite iteE ftT. +Qed. + +Lemma letin_iteF : ~~ f t -> letin (ite mf k1 k2) u t U = letin k2 u t U. +Proof. +move=> ftF. +rewrite !letinE/=. +apply: eq_measure_integral => V mV _. +by rewrite iteE (negbTE ftF). +Qed. + +End letin_ite. + +Section letinA. +Context d d' d1 d2 d3 (X : measurableType d) (Y : measurableType d') + (T1 : measurableType d1) (T2 : measurableType d2) (T3 : measurableType d3) + (R : realType). +Import Notations. +Variables (t : R.-sfker X ~> T1) + (u : R.-sfker [the measurableType _ of (X * T1)%type] ~> T2) + (v : R.-sfker [the measurableType _ of (X * T2)%type] ~> Y) + (v' : R.-sfker [the measurableType _ of (X * T1 * T2)%type] ~> Y) + (vv' : forall y, v =1 fun xz => v' (xz.1, y, xz.2)). + +Lemma letinA x A : measurable A -> + letin t (letin u v') x A + = + (letin (letin t u) v) x A. +Proof. +move=> mA. +rewrite !letinE. +under eq_integral do rewrite letinE. +rewrite integral_kcomp; [|by []|]. +- apply: eq_integral => y _. + apply: eq_integral => z _. + by rewrite (vv' y). +have /measurable_fun_prod1 := @measurable_kernel _ _ _ _ _ v _ mA. +exact. +Qed. + +End letinA. + +Section letinC. +Context d d1 d' (X : measurableType d) (Y : measurableType d1) + (Z : measurableType d') (R : realType). + +Import Notations. + +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)). + +Definition T z : set X -> \bar R := t z. +Let T0 z : (T z) set0 = 0. Proof. by []. Qed. +Let T_ge0 z x : 0 <= (T z) x. Proof. by []. Qed. +Let T_semi_sigma_additive z : semi_sigma_additive (T z). +Proof. exact: measure_semi_sigma_additive. Qed. +HB.instance Definition _ z := @isMeasure.Build _ R X (T z) (T0 z) (T_ge0 z) + (@T_semi_sigma_additive z). + +Let sfinT z : sfinite_measure (T z). Proof. exact: sfinite_kernel_measure. Qed. +HB.instance Definition _ z := @Measure_isSFinite_subdef.Build _ X R + (T z) (sfinT z). + +Definition U z : set Y -> \bar R := u z. +Let U0 z : (U z) set0 = 0. Proof. by []. Qed. +Let U_ge0 z x : 0 <= (U z) x. Proof. by []. Qed. +Let U_semi_sigma_additive z : semi_sigma_additive (U z). +Proof. exact: measure_semi_sigma_additive. Qed. +HB.instance Definition _ z := @isMeasure.Build _ R Y (U z) (U0 z) (U_ge0 z) + (@U_semi_sigma_additive z). + +Let sfinU z : sfinite_measure (U z). Proof. exact: sfinite_kernel_measure. Qed. +HB.instance Definition _ z := @Measure_isSFinite_subdef.Build _ Y R + (U z) (sfinU z). + +Lemma letinC z A : measurable A -> + letin t + (letin u' + (ret (measurable_fun_pair var2of3 var3of3))) z A = + letin u + (letin t' + (ret (measurable_fun_pair var3of3 var2of3))) 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 + [the {sfinite_measure set X -> \bar R} of T z] + [the {sfinite_measure set Y -> \bar R} of U z] + (fun x => \d_(x.1, x.2) A ))//; last first. + apply/EFin_measurable_fun => /=; rewrite (_ : (fun x => _) = mindic R mA)//. + by apply/funext => -[]. +rewrite /=. +apply: eq_integral => y _. +by rewrite letinE/= -tt'; apply: eq_integral => // x _; rewrite retE. +Qed. + +End letinC. + +(* sample programs *) + +Section constants. +Variable R : realType. +Local Open Scope ring_scope. + +Lemma onem1S n : `1- (1 / n.+1%:R) = (n%:R / n.+1%:R)%:nng%:num :> R. +Proof. +by rewrite /onem/= -{1}(@divrr _ n.+1%:R) ?unitfE// -mulrBl -natr1 addrK. +Qed. + +Lemma p1S n : (1 / n.+1%:R)%:nng%:num <= 1 :> R. +Proof. by rewrite ler_pdivr_mulr//= mul1r ler1n. Qed. + +Lemma p12 : (1 / 2%:R)%:nng%:num <= 1 :> R. Proof. by rewrite p1S. Qed. + +Lemma p14 : (1 / 4%:R)%:nng%:num <= 1 :> R. Proof. by rewrite p1S. Qed. + +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 p12 {R}. +Arguments p14 {R}. +Arguments p27 {R}. + +Section poisson. +Variable R : realType. +Local Open Scope ring_scope. + +(* density function for Poisson *) +Definition poisson k r : R := r ^+ k / k`!%:R^-1 * expR (- r). + +Lemma poisson_ge0 k r : 0 <= r -> 0 <= poisson k r. +Proof. +move=> r0; rewrite /poisson mulr_ge0 ?expR_ge0//. +by rewrite mulr_ge0// exprn_ge0. +Qed. + +Lemma poisson_gt0 k r : 0 < r -> 0 < poisson k.+1 r. +Proof. +move=> r0; rewrite /poisson mulr_gt0 ?expR_gt0//. +by rewrite divr_gt0// ?exprn_gt0// invr_gt0 ltr0n fact_gt0. +Qed. + +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_funT_comp; last exact: measurable_fun_opp. +by apply: continuous_measurable_fun; exact: continuous_expR. +Qed. + +Definition poisson3 := poisson 4 3%:R. (* 0.168 *) +Definition poisson10 := poisson 4 10%:R. (* 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_funT_comp. + by apply: continuous_measurable_fun; exact: continuous_expR. +apply: measurable_funM => /=; first exact: measurable_fun_opp. +exact: measurable_fun_cst. +Qed. + +End exponential. + +Lemma letin_sample_bernoulli d d' (T : measurableType d) + (T' : measurableType d') (R : realType)(r : {nonneg R}) (r1 : (r%:num <= 1)%R) + (u : R.-sfker [the measurableType _ of (T * bool)%type] ~> T') x y : + letin (sample [the probability _ _ of bernoulli r1]) u x y = + r%:num%:E * u (x, true) y + (`1- (r%:num))%:E * u (x, false) y. +Proof. +rewrite letinE/=. +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. +Import Notations. +Context d (T : measurableType d) (R : realType). + +Definition sample_and_return : R.-sfker T ~> _ := + letin + (sample [the probability _ _ of bernoulli p27]) (* T -> B *) + (ret var2of2) (* 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. +Proof. +by rewrite /sample_and_return letin_sample_bernoulli !retE onem27. +Qed. + +End sample_and_return. + +(* trivial example *) +Section sample_and_branch. +Import Notations. +Context d (T : measurableType d) (R : realType). + +(* let x = sample (bernoulli (2/7)) in + let r = case x of {(1, _) => return (k3()), (2, _) => return (k10())} in + return r *) + +Definition sample_and_branch : R.-sfker T ~> mR R := + letin + (sample [the probability _ _ of bernoulli p27]) (* T -> B *) + (ite var2of2 (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. +Proof. +by rewrite /sample_and_branch letin_sample_bernoulli/= !iteE !retE onem27. +Qed. + +End sample_and_branch. + +Section bernoulli_and. +Context d (T : measurableType d) (R : realType). +Import Notations. + +Definition mand (x y : T * mbool * mbool -> mbool) + (t : T * mbool * mbool) : mbool := x t && y t. + +Lemma measurable_fun_mand (x y : T * mbool * mbool -> mbool) : + measurable_fun setT x -> measurable_fun setT y -> + measurable_fun setT (mand x y). +Proof. +move=> /= mx my; apply: (@emeasurable_fun_bool _ _ _ _ true). +rewrite [X in measurable X](_ : _ = + (x @^-1` [set true]) `&` (y @^-1` [set true])); last first. + by rewrite /mand; apply/seteqP; split => z/= /andP. +apply: measurableI. +- by rewrite -[X in measurable X]setTI; exact: mx. +- by rewrite -[X in measurable X]setTI; exact: my. +Qed. + +Definition bernoulli_and : R.-sfker T ~> mbool := + (letin (sample [the probability _ _ of bernoulli p12]) + (letin (sample [the probability _ _ of bernoulli p12]) + (ret (measurable_fun_mand var2of3 var3of3)))). + +Lemma bernoulli_andE t U : + bernoulli_and t U = + sample [the probability _ _ of bernoulli p14] t U. +Proof. +rewrite /bernoulli_and 3!letin_sample_bernoulli/= /mand/= muleDr//= -muleDl//. +rewrite !muleA -addeA -muleDl// -!EFinM !onem1S/= -splitr mulr1. +have -> : (1 / 2 * (1 / 2) = 1 / 4 :> R)%R by rewrite mulf_div mulr1// -natrM. +rewrite /bernoulli/= measure_addE/= /mscale/= -!EFinM; congr( _ + (_ * _)%:E). +have -> : (1 / 2 = 2 / 4 :> R)%R. + by apply/eqP; rewrite eqr_div// ?pnatr_eq0// mul1r -natrM. +by rewrite onem1S// -mulrDl. +Qed. + +End bernoulli_and. + +Section staton_bus. +Import Notations. +Context d (T : measurableType d) (R : realType) (h : R -> R). +Hypothesis mh : measurable_fun setT h. +Definition kstaton_bus : R.-sfker T ~> mbool := + letin (sample [the probability _ _ of bernoulli p27]) + (letin + (letin (ite var2of2 (ret k3) (ret k10)) + (score (measurable_funT_comp mh var3of3))) + (ret var2of3)). + +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 *) +Section staton_bus_poisson. +Import Notations. +Context d (T : measurableType d) (R : realType). +Let poisson4 := @poisson R 4%N. +Let mpoisson4 := @mpoisson R 4%N. + +Definition kstaton_bus_poisson : R.-sfker (mR R) ~> mbool := + kstaton_bus _ mpoisson4. + +Let kstaton_bus_poissonE t U : kstaton_bus_poisson t U = + (2 / 7%:R)%:E * (poisson4 3%:R)%:E * \d_true U + + (5%:R / 7%:R)%:E * (poisson4 10%:R)%:E * \d_false U. +Proof. +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 onem27. + rewrite letin_kret//. + rewrite letin_iteF//. + rewrite letin_retk//. + by rewrite scoreE//= => r r0; exact: poisson_ge0. +Qed. + +(* 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 P (t : R) U : + let N := ((2 / 7%:R) * poisson4 3%:R + + (5%:R / 7%:R) * poisson4 10%:R)%R in + staton_bus mpoisson4 P t U = + ((2 / 7%:R)%:E * (poisson4 3%:R)%:E * \d_true U + + (5%:R / 7%:R)%:E * (poisson4 10%:R)%:E * \d_false U) * N^-1%:E. +Proof. +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. + +(* 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. +Context d (T : measurableType d) (R : realType). +Let exp1560 := @exp_density R (ratr (15%:Q / 60%:Q)). +Let mexp1560 := @mexp_density R (ratr (15%:Q / 60%:Q)). + +(* 15/60 = 0.25 *) + +Definition kstaton_bus_exponential : R.-sfker (mR R) ~> mbool := + kstaton_bus _ mexp1560. + +Let kstaton_bus_exponentialE t U : kstaton_bus_exponential t U = + (2 / 7%:R)%:E * (exp1560 3%:R)%:E * \d_true U + + (5%:R / 7%:R)%:E * (exp1560 10%:R)%:E * \d_false U. +Proof. +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. + +(* 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_exponentialE P (t : R) U : + let N := ((2 / 7%:R) * exp1560 3%:R + + (5%:R / 7%:R) * exp1560 10%:R)%R in + staton_bus mexp1560 P t U = + ((2 / 7%:R)%:E * (exp1560 3%:R)%:E * \d_true U + + (5%:R / 7%:R)%:E * (exp1560 10%:R)%:E * \d_false U) * N^-1%:E. +Proof. +rewrite /staton_bus. +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 new file mode 100644 index 0000000000..897d70c886 --- /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. + +(******************************************************************************) +(* Semantics of a probabilistic programming language using s-finite kernels *) +(* (wip) *) +(******************************************************************************) + +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)%E. + +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_funT_comp => /=. + by apply: continuous_measurable_fun; apply continuous_expR. +apply: measurable_funM; last exact: measurable_fun_cst. +apply: measurable_funT_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. + +Axiom integral_gauss01_density : + (\int[lebesgue_measure]_x (gauss01_density x)%:E = 1%E)%E. + +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 _ := @Measure_isProbability.Build _ _ R mgauss01 mgauss01_setT. + +Definition gauss01 := [the probability _ _ of mgauss01]. + +End gauss. + +Section gauss_lebesgue. +Import Notations. +Context d (T : measurableType d) (R : realType). + +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_funT_comp mf1 var2of2)) + (ret var2of3)). + +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 /=. +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// indicT mul1e 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.