diff --git a/.github/workflows/nix-action-8.19.yml b/.github/workflows/nix-action-8.19.yml index 27378388f..aa6085628 100644 --- a/.github/workflows/nix-action-8.19.yml +++ b/.github/workflows/nix-action-8.19.yml @@ -191,6 +191,10 @@ jobs: name: 'Building/fetching previous CI target: mathcomp-bigenough' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.19" --argstr job "mathcomp-bigenough" + - if: steps.stepCheck.outputs.status == 'built' + name: 'Building/fetching previous CI target: mathcomp-algebra-tactics' + run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.19" --argstr + job "mathcomp-algebra-tactics" - if: steps.stepCheck.outputs.status == 'built' name: 'Building/fetching previous CI target: hierarchy-builder' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.19" --argstr @@ -267,6 +271,10 @@ jobs: name: 'Building/fetching previous CI target: mathcomp-bigenough' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.19" --argstr job "mathcomp-bigenough" + - if: steps.stepCheck.outputs.status == 'built' + name: 'Building/fetching previous CI target: mathcomp-algebra-tactics' + run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.19" --argstr + job "mathcomp-algebra-tactics" - if: steps.stepCheck.outputs.status == 'built' name: 'Building/fetching previous CI target: hierarchy-builder' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.19" --argstr diff --git a/.github/workflows/nix-action-8.20.yml b/.github/workflows/nix-action-8.20.yml index c3f3dd587..5c34e86d4 100644 --- a/.github/workflows/nix-action-8.20.yml +++ b/.github/workflows/nix-action-8.20.yml @@ -191,6 +191,10 @@ jobs: name: 'Building/fetching previous CI target: mathcomp-bigenough' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.20" --argstr job "mathcomp-bigenough" + - if: steps.stepCheck.outputs.status == 'built' + name: 'Building/fetching previous CI target: mathcomp-algebra-tactics' + run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.20" --argstr + job "mathcomp-algebra-tactics" - if: steps.stepCheck.outputs.status == 'built' name: 'Building/fetching previous CI target: hierarchy-builder' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.20" --argstr @@ -267,6 +271,10 @@ jobs: name: 'Building/fetching previous CI target: mathcomp-bigenough' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.20" --argstr job "mathcomp-bigenough" + - if: steps.stepCheck.outputs.status == 'built' + name: 'Building/fetching previous CI target: mathcomp-algebra-tactics' + run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.20" --argstr + job "mathcomp-algebra-tactics" - if: steps.stepCheck.outputs.status == 'built' name: 'Building/fetching previous CI target: hierarchy-builder' run: NIXPKGS_ALLOW_UNFREE=1 nix-build --no-out-link --argstr bundle "8.20" --argstr diff --git a/.github/workflows/nix-action-master.yml b/.github/workflows/nix-action-master.yml index 81711ae1c..d79280162 100644 --- a/.github/workflows/nix-action-master.yml +++ b/.github/workflows/nix-action-master.yml @@ -321,6 +321,7 @@ jobs: needs: - coq - mathcomp-finmap + - hierarchy-builder - mathcomp-bigenough - mathcomp-bigenough - hierarchy-builder diff --git a/.github/workflows/nix-action.yml b/.github/workflows/nix-action.yml new file mode 100644 index 000000000..06697578d --- /dev/null +++ b/.github/workflows/nix-action.yml @@ -0,0 +1,46 @@ +# This file was generated from `meta.yml`, please do not edit manually. +# Follow the instructions on https://github.com/coq-community/templates to regenerate. +name: Nix CI + +on: + push: + branches: + - master + pull_request: + paths: + - .github/workflows/** + pull_request_target: + +jobs: + build: + runs-on: ubuntu-latest + strategy: + matrix: + overrides: + - 'coq = "master"' + fail-fast: false + steps: + - name: Determine which commit to test + run: | + if [[ ${{ github.event_name }} =~ "pull_request" ]]; then + merge_commit=$(git ls-remote ${{ github.event.repository.html_url }} refs/pull/${{ github.event.number }}/merge | cut -f1) + if [ -z "$merge_commit" ]; then + echo "tested_commit=${{ github.event.pull_request.head.sha }}" >> $GITHUB_ENV + else + echo "tested_commit=$merge_commit" >> $GITHUB_ENV + fi + else + echo "tested_commit=${{ github.sha }}" >> $GITHUB_ENV + fi + - uses: cachix/install-nix-action@v14 + with: + nix_path: nixpkgs=channel:nixpkgs-unstable + - uses: cachix/cachix-action@v10 + with: + name: coq-community + extraPullNames: coq, math-comp + - uses: actions/checkout@v2 + with: + ref: ${{ env.tested_ref }} + - run: > + nix-build https://coq.inria.fr/nix/toolbox --argstr job analysis --arg override '{ ${{ matrix.overrides }}; analysis = builtins.filterSource (path: _: baseNameOf path != ".git") ./.; }' diff --git a/.nix/coq-overlays/mathcomp-analysis/default.nix b/.nix/coq-overlays/mathcomp-analysis/default.nix new file mode 100644 index 000000000..3a97b3d7c --- /dev/null +++ b/.nix/coq-overlays/mathcomp-analysis/default.nix @@ -0,0 +1,300 @@ +{ + lib, + mkCoqDerivation, + mathcomp, + mathcomp-finmap, + mathcomp-bigenough, + mathcomp-zify, + mathcomp-algebra-tactics, + hierarchy-builder, + single ? false, + coqPackages, + coq, + version ? null, +}@args: + +let + repo = "analysis"; + owner = "math-comp"; + + release."1.7.0".sha256 = "sha256-GgsMIHqLkWsPm2VyOPeZdOulkN00IoBz++qA6yE9raQ="; + release."1.5.0".sha256 = "sha256-EWogrkr5TC5F9HjQJwO3bl4P8mij8U7thUGJNNI+k88="; + release."1.4.0".sha256 = "sha256-eDggeuEU0fMK7D5FbxvLkbAgpLw5lwL/Rl0eLXAnJeg="; + release."1.2.0".sha256 = "sha256-w6BivDM4dF4Iv4rUTy++2feweNtMAJxgGExPfYGhXxo="; + release."1.1.0".sha256 = "sha256-wl4kZf4mh9zbFfGcqaFEgWRyp0Bj511F505mYodpS6o="; + release."1.0.0".sha256 = "sha256-KiXyaWB4zQ3NuXadq4BSWfoN1cIo1xiLVSN6nW03tC4="; + release."0.7.0".sha256 = "sha256-JwkyetXrFsFHqz8KY3QBpHsrkhmEFnrCGuKztcoen60="; + release."0.6.7".sha256 = "sha256-3i2PBMEwihwgwUmnS0cmrZ8s+aLPFVq/vo0aXMUaUyA="; + release."0.6.6".sha256 = "sha256-tWtv6yeB5/vzwpKZINK9OQ0yQsvD8qu9zVSNHvLMX5Y="; + release."0.6.5".sha256 = "sha256-oJk9/Jl1SWra2aFAXRAVfX7ZUaDfajqdDksYaW8dv8E="; + release."0.6.1".sha256 = "sha256-1VyNXu11/pDMuH4DmFYSUF/qZ4Bo+/Zl3Y0JkyrH/r0="; + release."0.6.0".sha256 = "sha256-0msICcIrK6jbOSiBu0gIVU3RHwoEEvB88CMQqW/06rg="; + release."0.5.3".sha256 = "sha256-1NjFsi5TITF8ZWx1NyppRmi8g6YaoUtTdS9bU/sUe5k="; + release."0.5.2".sha256 = "0yx5p9zyl8jv1vg7rgkyq8dqzkdnkqv969mi62whmhkvxbavgzbw"; + release."0.5.1".sha256 = "1hnzqb1gxf88wgj2n1b0f2xm6sxg9j0735zdsv6j12hlvx5lwk68"; + release."0.3.13".sha256 = "sha256-Yaztew79KWRC933kGFOAUIIoqukaZOdNOdw4XszR1Hg="; + release."0.3.10".sha256 = "sha256-FBH2c8QRibq5Ycw/ieB8mZl0fDiPrYdIzZ6W/A3pIhI="; + release."0.3.9".sha256 = "sha256-uUU9diBwUqBrNRLiDc0kz0CGkwTZCUmigPwLbpDOeg4="; + release."0.3.6".sha256 = "0g2j7b2hca4byz62ssgg90bkbc8wwp7xkb2d3225bbvihi92b4c5"; + release."0.3.4".sha256 = "18mgycjgg829dbr7ps77z6lcj03h3dchjbj5iir0pybxby7gd45c"; + release."0.3.3".sha256 = "1m2mxcngj368vbdb8mlr91hsygl430spl7lgyn9qmn3jykack867"; + release."0.3.1".sha256 = "1iad288yvrjv8ahl9v18vfblgqb1l5z6ax644w49w9hwxs93f2k8"; + release."0.2.3".sha256 = "0p9mr8g1qma6h10qf7014dv98ln90dfkwn76ynagpww7qap8s966"; + + defaultVersion = + let + inherit (lib.versions) range; + in + lib.switch + [ coq.version mathcomp.version ] + [ + { + cases = [ + (range "8.19" "8.20") + (range "2.1.0" "2.2.0") + ]; + out = "1.7.0"; + } + { + cases = [ + (range "8.17" "8.20") + (range "2.0.0" "2.2.0") + ]; + out = "1.1.0"; + } + { + cases = [ + (range "8.17" "8.19") + (range "1.17.0" "1.19.0") + ]; + out = "0.7.0"; + } + { + cases = [ + (range "8.17" "8.18") + (range "1.15.0" "1.18.0") + ]; + out = "0.6.7"; + } + { + cases = [ + (range "8.17" "8.18") + (range "1.15.0" "1.18.0") + ]; + out = "0.6.6"; + } + { + cases = [ + (range "8.14" "8.18") + (range "1.15.0" "1.17.0") + ]; + out = "0.6.5"; + } + { + cases = [ + (range "8.14" "8.18") + (range "1.13.0" "1.16.0") + ]; + out = "0.6.1"; + } + { + cases = [ + (range "8.14" "8.18") + (range "1.13" "1.15") + ]; + out = "0.5.2"; + } + { + cases = [ + (range "8.13" "8.15") + (range "1.13" "1.14") + ]; + out = "0.5.1"; + } + { + cases = [ + (range "8.13" "8.15") + (range "1.12" "1.14") + ]; + out = "0.3.13"; + } + { + cases = [ + (range "8.11" "8.14") + (range "1.12" "1.13") + ]; + out = "0.3.10"; + } + { + cases = [ + (range "8.10" "8.12") + "1.11.0" + ]; + out = "0.3.3"; + } + { + cases = [ + (range "8.10" "8.11") + "1.11.0" + ]; + out = "0.3.1"; + } + { + cases = [ + (range "8.8" "8.11") + (range "1.8" "1.10") + ]; + out = "0.2.3"; + } + ] + null; + + # list of analysis packages sorted by dependency order + packages = { + "classical" = [ ]; + "reals" = [ "classical" ]; + "experimental-reals" = [ "reals" ]; + "analysis" = [ "reals" ]; + "reals-stdlib" = [ "reals" ]; + "analysis-stdlib" = [ + "analysis" + "reals-stdlib" + ]; + }; + + mathcomp_ = + package: + let + classical-deps = [ + mathcomp.algebra + mathcomp-finmap + mathcomp-zify + mathcomp-algebra-tactics + ]; + experimental-reals-deps = [ mathcomp-bigenough ]; + analysis-deps = [ + mathcomp.field + mathcomp-bigenough + ]; + intra-deps = lib.optionals (package != "single") (map mathcomp_ packages.${package}); + pkgpath = lib.switch package [ + { + case = "single"; + out = "."; + } + { + case = "analysis"; + out = "theories"; + } + { + case = "experimental-reals"; + out = "experimental_reals"; + } + { + case = "reals-stdlib"; + out = "reals_stdlib"; + } + { + case = "analysis-stdlib"; + out = "analysis_stdlib"; + } + ] package; + pname = if package == "single" then "mathcomp-analysis-single" else "mathcomp-${package}"; + derivation = mkCoqDerivation ({ + inherit + version + pname + defaultVersion + release + repo + owner + ; + + namePrefix = [ + "coq" + "mathcomp" + ]; + + propagatedBuildInputs = + intra-deps + ++ lib.optionals (lib.elem package [ + "classical" + "single" + ]) classical-deps + ++ lib.optionals (lib.elem package [ + "experimental-reals" + "single" + ]) experimental-reals-deps + ++ lib.optionals (lib.elem package [ + "analysis" + "single" + ]) analysis-deps; + + preBuild = '' + cd ${pkgpath} + ''; + + meta = { + description = "Analysis library compatible with Mathematical Components"; + maintainers = [ lib.maintainers.cohencyril ]; + license = lib.licenses.cecill-c; + }; + + passthru = lib.mapAttrs (package: deps: mathcomp_ package) packages; + }); + # split packages didn't exist before 0.6, so bulding nothing in that case + patched-derivation1 = derivation.overrideAttrs ( + o: + lib.optionalAttrs + ( + o.pname != null + && o.pname != "mathcomp-analysis" + && o.version != null + && o.version != "dev" + && lib.versions.isLt "0.6" o.version + ) + { + preBuild = ""; + buildPhase = "echo doing nothing"; + installPhase = "echo doing nothing"; + } + ); + patched-derivation2 = patched-derivation1.overrideAttrs ( + o: + lib.optionalAttrs ( + o.pname != null + && o.pname == "mathcomp-analysis" + && o.version != null + && o.version != "dev" + && lib.versions.isLt "0.6" o.version + ) { preBuild = ""; } + ); + # only packages classical and analysis existed before 1.7, so bulding nothing in that case + patched-derivation3 = patched-derivation2.overrideAttrs ( + o: + lib.optionalAttrs + ( + o.pname != null + && o.pname != "mathcomp-classical" + && o.pname != "mathcomp-analysis" + && o.version != null + && o.version != "dev" + && lib.versions.isLt "1.7" o.version + ) + { + preBuild = ""; + buildPhase = "echo doing nothing"; + installPhase = "echo doing nothing"; + } + ); + patched-derivation = patched-derivation3.overrideAttrs ( + o: + lib.optionalAttrs (o.version != null && (o.version == "dev" || lib.versions.isGe "0.3.4" o.version)) + { + propagatedBuildInputs = o.propagatedBuildInputs ++ [ hierarchy-builder ]; + } + ); + in + patched-derivation; +in +mathcomp_ (if single then "single" else "analysis") diff --git a/_CoqProject b/_CoqProject index a8b38635c..9cb7d4f39 100644 --- a/_CoqProject +++ b/_CoqProject @@ -93,3 +93,10 @@ theories/pi_irrational.v theories/showcase/summability.v analysis_stdlib/Rstruct_topology.v analysis_stdlib/showcase/uniform_bigO.v +theories/prob_lang.v +theories/prob_lang_wip.v +theories/lang_syntax_util.v +theories/lang_syntax_toy.v +theories/lang_syntax.v +theories/lang_syntax_examples.v +theories/lang_syntax_table_game.v diff --git a/classical/classical_sets.v b/classical/classical_sets.v index 86b9b0ebe..95265327e 100644 --- a/classical/classical_sets.v +++ b/classical/classical_sets.v @@ -550,6 +550,7 @@ Qed. Notation setTP := setTPn (only parsing). Lemma in_set0 (x : T) : (x \in set0) = false. Proof. by rewrite memNset. Qed. + Lemma in_setT (x : T) : x \in setT. Proof. by rewrite mem_set. Qed. Lemma in_setC (x : T) A : (x \in ~` A) = (x \notin A). @@ -1438,9 +1439,15 @@ Implicit Types (A B : set aT) (f : aT -> rT) (Y : set rT). Lemma imageP f A a : A a -> (f @` A) (f a). Proof. by exists a. Qed. +Lemma image_f f A a : a \in A -> f a \in [set f x | x in A]. +Proof. by rewrite !inE; apply/imageP. Qed. + Lemma imageT (f : aT -> rT) (a : aT) : range f (f a). Proof. by apply: imageP. Qed. +Lemma mem_range f a : f a \in range f. +Proof. by rewrite !inE; apply/imageT. Qed. + End base_image_lemmas. #[global] Hint Extern 0 ((?f @` _) (?f _)) => solve [apply: imageP; assumption] : core. @@ -1455,6 +1462,10 @@ Proof. by move=> f_inj; rewrite propeqE; split => [[b Ab /f_inj <-]|/(imageP f)//]. Qed. +Lemma mem_image {f A a} : injective f -> + (f a \in [set f x | x in A]) = (a \in A). +Proof. by move=> /image_inj finj; apply/idP/idP; rewrite !inE finj. Qed. + Lemma image_id A : id @` A = A. Proof. by rewrite eqEsubset; split => a; [case=> /= x Ax <-|exists a]. Qed. @@ -1729,6 +1740,15 @@ Proof. by apply/disj_setPS/disj_setPS; rewrite -some_setI -some_set0 sub_image_someP. Qed. + +Lemma inl_in_set_inr A B (x : A) (Y : set B) : + inl x \in [set inr y | y in Y] = false. +Proof. by apply/negP; rewrite inE/= => -[]. Qed. + +Lemma inr_in_set_inr A B (y : B) (Y : set B) : + inr y \in [set @inr A B y | y in Y] = (y \in Y). +Proof. by apply/idP/idP => [/[!inE][/= [x ? [<-]]]|/[!inE]]//; exists y. Qed. + Section bigop_lemmas. Context {T I : Type}. Implicit Types (A : set T) (i : I) (P : set I) (F G : I -> set T). @@ -2224,6 +2244,9 @@ Notation bigcap_set := bigcap_seq (only parsing). #[deprecated(since="mathcomp-analysis 0.6.4",note="Use bigcap_seq_cond instead")] Notation bigcap_set_cond := bigcap_seq_cond (only parsing). +Lemma in_set1 [T : finType] (x y : T) : (x \in [set y]) = (x \in [set y]%SET). +Proof. by apply/idP/idP; rewrite !inE /= => /eqP. Qed. + Lemma bigcup_pred [T : finType] [U : Type] (P : {pred T}) (f : T -> set U) : \bigcup_(t in [set` P]) f t = \big[setU/set0]_(t in P) f t. Proof. diff --git a/classical/mathcomp_extra.v b/classical/mathcomp_extra.v index 7f5eeb3af..c8637e37f 100644 --- a/classical/mathcomp_extra.v +++ b/classical/mathcomp_extra.v @@ -588,3 +588,9 @@ rewrite mulr_ile1 ?andbT//. by have := xs01 x; rewrite inE xs orbT => /(_ _)/andP[]. by rewrite ih// => e xs; rewrite xs01// in_cons xs orbT. Qed. + +Lemma inr_inj {A B} : injective (@inr A B). +Proof. by move=> ? ? []. Qed. + +Lemma inl_inj {A B} : injective (@inl A B). +Proof. by move=> ? ? []. Qed. diff --git a/coq-mathcomp-analysis.opam b/coq-mathcomp-analysis.opam index a0890eb50..ed4e7d518 100644 --- a/coq-mathcomp-analysis.opam +++ b/coq-mathcomp-analysis.opam @@ -19,6 +19,8 @@ depends: [ "coq-mathcomp-solvable" "coq-mathcomp-field" "coq-mathcomp-bigenough" { (>= "1.0.0") } + "coq-mathcomp-algebra-tactics" { (>= "1.2.3") } + "coq-mathcomp-zify" { (>= "1.5.0") } ] tags: [ diff --git a/reals/constructive_ereal.v b/reals/constructive_ereal.v index a119e963f..b94becb26 100644 --- a/reals/constructive_ereal.v +++ b/reals/constructive_ereal.v @@ -670,6 +670,9 @@ Definition fin_num := [qualify a x : \bar R | (x != -oo) && (x != +oo)]. Fact fin_num_key : pred_key fin_num. Proof. by []. Qed. (*Canonical fin_num_keyd := KeyedQualifier fin_num_key.*) +Lemma fin_numP_EFin x : reflect (exists r, x = r%:E) (x \in fin_num). +Proof. by case: x => [r'||]//=; constructor; [exists r'| case | case ]. Qed. + Lemma fin_numE x : (x \is a fin_num) = (x != -oo) && (x != +oo). Proof. by []. Qed. @@ -2404,6 +2407,11 @@ Qed. Lemma EFin_max : {morph (@EFin R) : r s / Num.max r s >-> maxe r s}. Proof. by move=> a b /=; rewrite -fine_max. Qed. +Lemma EFin_bigmax {I : Type} (s : seq I) (P : I -> bool) (F : I -> R) r : + \big[maxe/r%:E]_(i <- s | P i) (F i)%:E = + (\big[Num.max/r]_(i <- s | P i) F i)%:E. +Proof. by rewrite (big_morph _ EFin_max erefl). Qed. + Lemma fine_min : {in fin_num &, {mono @fine R : x y / mine x y >-> (Num.min x y)%:E}}. Proof. diff --git a/theories/Make b/theories/Make index 35f6699fb..afcc341de 100644 --- a/theories/Make +++ b/theories/Make @@ -59,3 +59,10 @@ kernel.v pi_irrational.v all_analysis.v showcase/summability.v +prob_lang.v +prob_lang_wip.v +lang_syntax_util.v +lang_syntax_toy.v +lang_syntax.v +lang_syntax_examples.v +lang_syntax_table_game.v diff --git a/theories/ftc.v b/theories/ftc.v index bbc908701..08e876b95 100644 --- a/theories/ftc.v +++ b/theories/ftc.v @@ -35,6 +35,24 @@ Notation mu := (@lebesgue_measure R). Local Open Scope ereal_scope. Implicit Types (f : R -> R) (a : itv_bound R). +Let integrable_locally f (A : set R) : measurable A -> + mu.-integrable A (EFin \o f) -> locally_integrable [set: R] (f \_ A). +Proof. +move=> mA intf; split. +- move/integrableP : intf => [mf _]. + by apply/(measurable_restrictT _ _).1 => //; exact/EFin_measurable_fun. +- exact: openT. +- move=> K _ cK. + move/integrableP : intf => [mf]. + rewrite integral_mkcond/=. + under eq_integral do rewrite restrict_EFin restrict_normr. + apply: le_lt_trans. + apply: ge0_subset_integral => //=; first exact: compact_measurable. + apply/EFin_measurable_fun/measurableT_comp/EFin_measurable_fun => //=. + move/(measurable_restrictT _ _).1 : mf => /=. + by rewrite restrict_EFin; exact. +Qed. + Let FTC0 f a : mu.-integrable setT (EFin \o f) -> let F x := (\int[mu]_(t in [set` Interval a (BRight x)]) f t)%R in forall x, a < BRight x -> lebesgue_pt f x -> @@ -679,7 +697,6 @@ Qed. End Rintegration_by_parts. -(* TODO: move to realfun.v? *) Section integration_by_substitution_preliminaries. Context {R : realType}. Notation mu := lebesgue_measure. diff --git a/theories/kernel.v b/theories/kernel.v index 31d8ba818..86fa740c6 100644 --- a/theories/kernel.v +++ b/theories/kernel.v @@ -1,6 +1,7 @@ (* 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 archimedean. From mathcomp Require Import mathcomp_extra boolp classical_sets functions. From mathcomp Require Import cardinality fsbigop reals ereal signed. From mathcomp Require Import topology normedtype sequences esum measure. @@ -9,36 +10,67 @@ From mathcomp Require Import numfun lebesgue_measure lebesgue_integral. (**md**************************************************************************) (* # Kernels *) (* *) -(* This file provides a formation of kernels, s-finite kernels, finite *) -(* kernels, subprobability kernels, and probability kernels. The main *) -(* formalized result is the fact that s-finite kernels are stable by *) -(* composition. *) -(* Reference: *) +(* This file provides a formation of kernels, s-finite kernels, *) +(* sigma-finite kernels, finite transition kernels, finite kernels, *) +(* subprobability kernels, and probability kernels, organized as a hierarchy *) +(* of mathematical structures. The main formalized results are stability by *) +(* composition for s-finite kernels [Lemma 3, Staton, ESOP 2017] and *) +(* subprobability kernels [Theorem 14.26, Klenke 2014]. *) +(* *) +(* References: *) (* - R. Affeldt, C. Cohen, A. Saito. Semantics of probabilistic programs *) (* using s-finite kernels in Coq. CPP 2023 *) +(* - S. Staton. Commutative Semantics for Probabilistic Programming. *) +(* ESOP 2017 *) +(* - A. Klenke. Probability theory: A comprehensive course. 2014 *) (* *) (* ``` *) (* R.-ker X ~> Y == kernel from X to Y where X and Y are of type *) (* measurableType *) (* The HB class is Kernel. *) +(* kseries == countable sum of kernels *) +(* It is declared as an instance of the structure *) +(* Kernel. It will also be shown to be an instance of *) +(* SFiniteKernel if the sum is over s-finite kernels. *) (* measure_fam_uub k == the kernel k is uniformly upper-bounded *) (* R.-sfker X ~> Y == s-finite kernel *) (* The HB class is SFiniteKernel. *) +(* kzero == kernel defined using the mzero measure *) +(* R.-sigmafker X ~> Y == sigma-finite transition kernel *) +(* The HB class is SigmaFiniteTransitionKernel. *) +(* R.-ftker X ~> Y == finite transition kernel *) +(* The HB class is FiniteTransitionKernel. *) (* R.-fker X ~> Y == finite kernel *) (* The HB class is FiniteKernel. *) (* R.-spker X ~> Y == subprobability kernel *) (* The HB class is SubProbabilityKernel. *) (* R.-pker X ~> Y == probability kernel *) (* The HB class is ProbabilityKernel. *) -(* kseries == countable sum of kernels *) -(* It is declared as an instance of the structure *) -(* Kernel. It is also an instance of the structure *) -(* SFiniteKernel if the sum is over s-finite kernels. *) -(* kzero == kernel defined using the mzero measure *) (* kdirac mf == kernel defined by a measurable function *) (* kprobability m == kernel defined by a probability measure *) (* kadd k1 k2 == lifting of the addition of measures to kernels *) -(* l \; k == composition of kernels *) +(* knormalize f P := fun x => mnormalize (f x) P *) +(* kcomp l k == "parameterized" composition for kernels *) +(* l has type X -> {measure set Y -> \bar R} *) +(* k has type X * Y -> {measure set Z -> \bar R} *) +(* kcomp l k as type X -> set Z -> \bar R *) +(* l \; k == same as kcomp l k but equipped with the type *) +(* X -> {measure set Z -> \bar R} *) +(* kfcomp k f == composition of a "kernel" and a function *) +(* k has type X -> {measure set Y -> \bar R}. *) +(* kproduct k1 k2 == "parameterized" product of kernels *) +(* k1 has type T0 -> {measure set T1 -> \bar R}. *) +(* k2 has type T0 * T1 -> {measure set T2 -> \bar R}. *) +(* The result has type T0 -> set (T1 * T2) -> \bar R. *) +(* mkproduct k1 k2 == same as kproduct k1 k2 but equipped with the type *) +(* T0 -> {measure set (T1 * T2) -> \bar R} *) +(* kproduct_snd k1 k2 := kproduct k1 (k2 \o snd) *) +(* k2 has T1 -> {measure set T2 -> \bar R} *) +(* It "ignores" the first argument of k2. *) +(* kcomp_noparam k1 k2 := composition of kernels *) +(* k1 has type T0 -> {measure set T1 -> \bar R}. *) +(* k2 has type T1 -> {measure set T2 -> \bar R}. *) +(* The return type is T0 -> set (T1 * T2) -> \bar R *) (* ``` *) (* *) (******************************************************************************) @@ -57,6 +89,10 @@ 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 .-sigmafker X ~> Y" + (at level 42, format "R .-sigmafker X ~> Y"). +Reserved Notation "R .-ftker X ~> Y" + (at level 42, format "R .-ftker X ~> Y"). Reserved Notation "R .-fker X ~> Y" (at level 42, format "R .-fker X ~> Y"). Reserved Notation "R .-spker X ~> Y" @@ -163,17 +199,29 @@ Qed. End measure_fam_uub. -HB.mixin Record Kernel_isSFinite_subdef d d' +HB.mixin Record isSFiniteKernel_subdef d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : X -> {measure set Y -> \bar R}) := { sfinite_kernel_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 }. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isSFiniteKernel_subdef instead.")] +Notation Kernel_isSFinite_subdef x1 x2 x3 x4 x5 x6 := + (isSFiniteKernel_subdef x1 x2 x3 x4 x5 x6). + +Module Kernel_isSFinite_subdef. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isSFiniteKernel_subdef.Build instead.")] +Notation Build x1 x2 x3 x4 x5 x6 := + (isSFiniteKernel_subdef.Build x1 x2 x3 x4 x5 x6) (only parsing). +End Kernel_isSFinite_subdef. + 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 }. + isSFiniteKernel_subdef _ _ X Y R k }. Notation "R .-sfker X ~> Y" := (SFiniteKernel.type X%type Y R). Arguments sfinite_kernel_subdef {_ _ _ _ _} _. @@ -187,24 +235,6 @@ have ? : m1 = m2. 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%type 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). @@ -222,6 +252,64 @@ Proof. by exists 1%R => /= t; rewrite /mzero/=. Qed. End kzero. +(* interface for sigma-finite kernel *) +HB.mixin Record isSigmaFiniteTransitionKernel + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := { + kernel_sigma_finite : forall x, sigma_finite [set: Y] (k x) }. + +#[short(type=sigma_finite_kernel)] +HB.structure Definition SigmaFiniteTransitionKernel + d d' (X : measurableType d) (Y : measurableType d') (R : realType) := + { k of @Kernel _ _ _ _ _ k & + isSigmaFiniteTransitionKernel _ _ X Y R k }. + +Notation "R .-sigmafker X ~> Y" := (sigma_finite_kernel X%type Y R). + +(* interface for finite transition kernel *) +HB.mixin Record isFiniteTransition + d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> {measure set Y -> \bar R}) := { + kernel_finite_transition : forall x, fin_num_fun (k x) }. + +#[short(type=finite_transition_kernel)] +HB.structure Definition FiniteTransitionKernel + d d' (X : measurableType d) (Y : measurableType d') (R : realType) := + { k of @SFiniteKernel _ _ _ _ _ k & + isFiniteTransition _ _ X Y R k }. + +Notation "R .-ftker X ~> Y" := (finite_transition_kernel X%type Y R). + +HB.mixin Record isMeasureFamUub d d' + (X : measurableType d) (Y : measurableType d') (R : realType) + (k : X -> {measure set Y -> \bar R}) := { + measure_uub : measure_fam_uub k }. + +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isMeasureFamUub instead.")] +Notation SFiniteKernel_isFinite x1 x2 x3 x4 x5 x6 := + (isMeasureFamUub x1 x2 x3 x4 x5 x6). + +Module SFiniteKernel_isFinite. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isMeasureFamUub.Build instead.")] +Notation Build x1 x2 x3 x4 x5 x6 := + (isMeasureFamUub.Build x1 x2 x3 x4 x5 x6) (only parsing). +End SFiniteKernel_isFinite. + +#[short(type=finite_kernel)] +HB.structure Definition FiniteKernel d d' + (X : measurableType d) (Y : measurableType d') (R : realType) := + { k of @FiniteTransitionKernel _ _ _ _ _ k & + isMeasureFamUub _ _ X Y R k }. +Notation "R .-fker X ~> Y" := (finite_kernel X%type 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 }. + HB.builders Context d d' (X : measurableType d) (Y : measurableType d') (R : realType) k of Kernel_isFinite d d' X Y R k. @@ -238,10 +326,32 @@ by rewrite eseries0// adde0. Qed. HB.instance Definition _ := - @Kernel_isSFinite_subdef.Build d d' X Y R k sfinite_finite. + @isSFiniteKernel_subdef.Build d d' X Y R k sfinite_finite. + +Let kernel_sigma_finite x : sigma_finite setT (k x). +Proof. +apply: fin_num_fun_sigma_finite; first by rewrite measure0. +apply: lty_fin_num_fun. +by have [r kr] := measure_uub; rewrite (lt_trans (kr x)) ?ltry. +Qed. HB.instance Definition _ := - @SFiniteKernel_isFinite.Build d d' X Y R k measure_uub. + @isSigmaFiniteTransitionKernel.Build d d' X Y R k kernel_sigma_finite. + +Let finite_transition_finite : forall x, fin_num_fun (k x). +Proof. +move=> x U mU. +have [r kr] := measure_uub. +rewrite ge0_fin_numE//. +rewrite (@le_lt_trans _ _ (k x setT)) ?le_measure ?inE//. +by rewrite (lt_trans (kr x)) ?ltry. +Qed. + +HB.instance Definition _ := + @isFiniteTransition.Build d d' X Y R k finite_transition_finite. + +HB.instance Definition _ := + @isMeasureFamUub.Build d d' X Y R k measure_uub. HB.end. @@ -296,13 +406,13 @@ HB.factory Record Kernel_isSFinite d d' 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. +Lemma sfinite_subdef : isSFiniteKernel_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.instance Definition _ := sfinite_subdef. HB.end. @@ -335,21 +445,47 @@ by rewrite nneseries_esum// fun_true; exact: eq_esum. Qed. HB.instance Definition _ := - Kernel_isSFinite_subdef.Build _ _ _ _ R (kseries k) sfinite_kseries. + isSFiniteKernel_subdef.Build _ _ _ _ R (kseries k) sfinite_kseries. End sfkseries. -HB.mixin Record FiniteKernel_isSubProbability d d' +HB.mixin Record isSubProbabilityKernel 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 [set: X]] <= 1 }. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isSubProbabilityKernel instead.")] +Notation FiniteKernel_isSubProbability x1 x2 x3 x4 x5 x6 := + (isSubProbabilityKernel x1 x2 x3 x4 x5 x6). + +Module FiniteKernel_isSubProbability. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isSubProbabilityKernel.Build instead.")] +Notation Build x1 x2 x3 x4 x5 x6 := + (isSubProbabilityKernel.Build x1 x2 x3 x4 x5 x6) (only parsing). +End FiniteKernel_isSubProbability. + #[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 }. + isSubProbabilityKernel _ _ X Y R k }. Notation "R .-spker X ~> Y" := (sprobability_kernel X%type Y R). +Lemma sprob_kernelP d d' (X : measurableType d) (Y : measurableType d') + (R : realType) (k : X -> set Y -> \bar R) : + (ereal_sup [set k x [set: _] | x in [set: _]] <= 1)%E <-> + forall x, (k x setT <= 1)%E. +Proof. +split => [+ x|k1]; last by apply: ub_ereal_sup => _ /= [z _ <-]; exact: k1. +by apply/le_trans/ereal_sup_ubound => /=; exists x. +Qed. + +Lemma sprob_kernel_le1 d d' (X : measurableType d) + (Y : measurableType d') (R : realType) (k : R.-spker X ~> Y) x : + k x [set: Y] <= 1. +Proof. by apply: (sprob_kernelP (fun x A => k x A)).1; exact: sprob_kernel. Qed. + 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 := { @@ -367,20 +503,32 @@ Qed. HB.instance Definition _ := finite. HB.instance Definition _ := - @FiniteKernel_isSubProbability.Build _ _ _ _ _ k sprob_kernel. + @isSubProbabilityKernel.Build _ _ _ _ _ k sprob_kernel. HB.end. -HB.mixin Record SubProbability_isProbability d d' +HB.mixin Record isProbabilityKernel 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 }. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isProbabilityKernel instead.")] +Notation SubProbability_isProbability x1 x2 x3 x4 x5 x6 := + (isProbabilityKernel x1 x2 x3 x4 x5 x6). + +Module SubProbability_isProbability. +#[deprecated(since="mathcomp-analysis 1.9.0", + note="Use isProbabilityKernel.Build instead.")] +Notation Build x1 x2 x3 x4 x5 x6 := + (isProbabilityKernel.Build x1 x2 x3 x4 x5 x6) (only parsing). +End SubProbability_isProbability. + #[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 }. + isProbabilityKernel _ _ X Y R k }. Notation "R .-pker X ~> Y" := (probability_kernel X%type Y R). HB.factory Record Kernel_isProbability d d' @@ -399,7 +547,7 @@ Qed. HB.instance Definition _ := sprob_kernel. HB.instance Definition _ := - @SubProbability_isProbability.Build _ _ _ _ _ k prob_kernel. + @isProbabilityKernel.Build _ _ _ _ _ k prob_kernel. HB.end. @@ -672,6 +820,7 @@ Qed. HB.instance Definition _ := @isKernel.Build _ _ _ _ _ kadd measurable_fun_kadd. + End kadd. Section sfkadd. @@ -697,7 +846,8 @@ 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. + isSFiniteKernel_subdef.Build _ _ _ _ R (kadd k1 k2) sfinite_kadd. + End sfkadd. Section fkadd. @@ -714,6 +864,7 @@ Qed. HB.instance Definition _ t := Kernel_isFinite.Build _ _ _ _ R (kadd k1 k2) kadd_finite_uub. + End fkadd. Section knormalize. @@ -764,7 +915,6 @@ HB.instance Definition _ (P : probability Y R):= End knormalize. -(* TODO: useful? *) Lemma measurable_fun_mnormalize d d' (X : measurableType d) (Y : measurableType d') (R : realType) (k : R.-ker X ~> Y) : measurable_fun [set: X] (fun x => mnormalize (k x) point : pprobability Y R). @@ -794,16 +944,19 @@ apply: measurable_fun_if => //. + by apply: measurableT_comp => //; exact/measurable_funS/measurable_kernel. Qed. +(* "parameterized composition" of kernels [Lemma 3, Staton ESOP 2017] + return type: X -> set Z -> \bar R *) 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 -> {measure set Z -> \bar R}. -Definition kcomp x U := \int[l x]_y k (x, y) U. +Definition kcomp x (U : set Z) := \int[l x]_y k (x, y) U. End kcomp_def. +(* [Lemma 3, Staton 2017 ESOP] (1/4) *) Section kcomp_is_measure. Context d1 d2 d3 (X : measurableType d1) (Y : measurableType d2) (Z : measurableType d3) (R : realType). @@ -837,6 +990,7 @@ End kcomp_is_measure. Notation "l \; k" := (mkcomp l k) : ereal_scope. +(* [Lemma 3, Staton 2017 ESOP] (2/4) *) Module KCOMP_FINITE_KERNEL. Section kcomp_finite_kernel_kernel. @@ -879,6 +1033,7 @@ HB.instance Definition _ := End kcomp_finite_kernel_finite. End KCOMP_FINITE_KERNEL. +(* [Lemma 3, Staton 2017 ESOP] (3/4) *) Section kcomp_sfinite_kernel. Context d d' d3 (X : measurableType d) (Y : measurableType d') (Z : measurableType d3) (R : realType). @@ -1064,6 +1219,7 @@ rewrite integral0_eq ?mule0; last first. by rewrite integral0_eq// => y _; rewrite preimage_nnfun0// measure0 mule0. Qed. +(* [Lemma 3, Staton 2017 ESOP] (4/4) *) Lemma integral_kcomp x f : (forall z, 0 <= f z) -> measurable_fun [set: Z] f -> \int[kcomp l k x]_z f z = \int[l x]_y (\int[k (x, y)]_z f z). Proof. @@ -1099,3 +1255,538 @@ by apply: eq_integral => z _; apply/cvg_lim => //; exact: cvg_nnsfun_approx. Qed. End integral_kcomp. + +(* [Definition 14.20, Klenke 2014 ]*) +Definition kfcomp d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) + (R : realType) (k : T1 -> {measure set T2 -> \bar R}) + (f : T1 * T2 -> \bar R) : T1 -> \bar R := + fun x => \int[k x]_y f (x, y). + +Section measurable_kfcomp. + +Let finite_measure_sigma_finite d (T : measurableType d) + (R : realType) (mu : {measure set T -> \bar R}) : + fin_num_fun mu -> sigma_finite setT mu. +Proof. +by move=> fmu; apply: fin_num_fun_sigma_finite => //; rewrite measure0. +Qed. + +Let finite_measure_sfinite d (T : measurableType d) + (R : realType) (mu : {measure set T -> \bar R}) : + fin_num_fun mu -> sfinite_measure mu. +Proof. +move=> fmu. +exists (fun n => if n is O then mu else mzero) => [[]//|U mU]. +by rewrite /mseries nneseries_recl// eseries0 ?adde0// => -[|]. +Qed. + +Import HBNNSimple. + +(* [Lemma 14.20, Klenke 2014] *) +Lemma measurable_kfcomp d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) + (R : realType) (k : R.-ftker T1 ~> T2) + (f : T1 * T2 -> \bar R) : measurable_fun [set: T1 * T2] f -> + (forall t, 0 <= f t) -> + measurable_fun [set: T1] (kfcomp k f). +Proof. +move=> mf f0. +rewrite /kfcomp. +pose I (f : T1 * T2 -> \bar R) x := \int[k x]_y f (x, y). +pose g (A1 : set T1) (A2 : set T2) : T1 * T2 -> \bar R := + EFin \o \1_(A1 `*` A2). +have IgE (A1 : set T1) (A2 : set T2) x : measurable A1 -> measurable A2 -> + I (g A1 A2) x = (\1_A1 x)%:E * k x A2. + move=> mA1 mA2. + rewrite /I /g/= integral_indic//; last first. + rewrite [X in measurable X](_ : _ = xsection (A1 `*` A2) x); last first. + by rewrite xsectionE. + by apply: measurable_xsection; exact: measurableX. + have [xA1|xA1] := boolP (x \in A1). + by rewrite indicE xA1 mul1e setIT -[X in _ _ X = _]xsectionE in_xsectionX. + rewrite indicE (negbTE xA1) mul0e setIT. + by rewrite -[X in _ _ X = _]xsectionE notin_xsectionX. +rewrite -[X in measurable_fun _ X]/(I f). +pose f_ := nnsfun_approx measurableT mf. +have If_cvg x : I (EFin \o f_ n) x @[n --> \oo] --> I f x. + pose g' n y := (EFin \o f_ n) (x, y). + rewrite [X in _ --> X](_ : _ = + \int[k x]_y (fun t => limn (g' ^~ t)) y); last first. + apply: eq_integral => y _. + by apply/esym/cvg_lim => //; exact: cvg_nnsfun_approx. + apply: cvg_monotone_convergence => //. + - by move=> n; apply/measurable_EFinP; exact: measurableT_comp. + - by move=> n y _; rewrite /= lee_fin. + - by move=> y _ a b ab/=; rewrite lee_fin; exact/lefP/nd_nnsfun_approx. +apply: (emeasurable_fun_cvg (fun n => I (EFin \o f_ n))). + move=> m. + pose D := [set A | measurable A /\ measurable_fun setT (I (EFin \o \1_A))]. + have setSD_D : setSD_closed D. + move=> B A AB; rewrite /D/= => -[mB mIB] [mA mIA]. + have IE : I (EFin \o \1_(B `\` A)) = I (EFin \o \1_B) \- I (EFin \o \1_A). + apply/funext => x/=. + rewrite /I. + pose kx : {measure set T2 -> \bar R} := k x. + pose H1 := isFinite.Build _ _ _ _ + (@kernel_finite_transition _ _ _ _ R k x). + pose H2 := isSFinite.Build _ _ _ _ (finite_measure_sfinite + (@kernel_finite_transition _ _ _ _ R k x)). + pose H3 := isSigmaFinite.Build _ _ _ _ (finite_measure_sigma_finite + (@kernel_finite_transition _ _ _ _ R k x)). + pose kx' := HB.pack_for (FiniteMeasure.type T2 R) (Measure.sort kx) H1 H2 H3. + have kxE : kx = kx' by exact: eq_measure. + rewrite (*TODO: use RintegralB when merged*) -integralB_EFin//; last 2 first. + - rewrite -/kx kxE; apply: integrable_indic => //. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. + - rewrite -/kx kxE; apply: integrable_indic => //. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. + apply: eq_integral => y _/=. + rewrite setDE indicI indicC/=. + have [/= xyA|/= xyA] := boolP ((x, y) \in _). + rewrite mulr0 EFinN !indicE xyA/= EFinN. + by move/set_mem : xyA => /AB /mem_set ->; rewrite subee. + by rewrite mulr1 EFinN !indicE (negbTE xyA)/= oppr0 adde0. + by split; [exact: measurableD|rewrite IE; exact: emeasurable_funB]. + have lambda_D : lambda_system setT D. + split => //. + - rewrite /D/= indicT/=; split => //. + rewrite [X in measurable_fun _ X](_ : _ = (fun x => k x setT)); last first. + by rewrite /I; apply/funext => x/=; rewrite integral_cst// mul1e. + exact: measurable_kernel. + - suff: D = [set A | (d1, d2).-prod.-measurable A /\ + measurable_fun [set: T1] ((fun C x => k x (xsection C x)) A)]. + by move=> ->; exact: xsection_ndseq_closed. + apply/seteqP; split=> [/= A [mA /= mIA]|/= A [/= mA mIA]]. + split => //. + rewrite [X in measurable_fun _ X](_ : _ = I (EFin \o \1_A))//. + apply/funext => x. + rewrite /I/= integral_indic// ?setIT//. + by rewrite -[X in _ = k x X]/(_ @^-1` _) -xsectionE. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. + split => //. + rewrite /I [X in measurable_fun _ X](_ : _ = (fun x => k x (xsection A x)))//. + apply/funext => x. + rewrite integral_indic//; last first. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. + by rewrite setIT xsectionE. + rewrite /= in lambda_D. + have DE : D = @measurable _ (T1 * T2)%type. + apply/seteqP; split => [/= A []//|]. + rewrite measurable_prod_measurableType. + apply: lambda_system_subset => //. + (* NB: lemma? *) + 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 setXI]. + move=> /= C [A mA [B mB] <-]. + split; first exact: measurableX. + rewrite [X in measurable_fun _ X](_ : _ = + (fun s => (\1_A s)%:E * k s B)); last first. + by apply/funext => s; rewrite IgE. + apply: emeasurable_funM; first exact/measurable_EFinP. + exact: measurable_kernel. + have mI1 (A : set (T1 * T2)) : measurable A -> + measurable_fun setT (I (EFin \o \1_A)). + by rewrite -DE => -[]. + rewrite [X in measurable_fun _ X](_ : _ = I + (EFin \o (fun x => \sum_(y \in range (f_ m)) y * + \1_(f_ m @^-1` [set y]) x))%R); last first. + apply/funext => x/=. + by apply: eq_integral => y _ /=; rewrite fimfunE. + rewrite /I/= [X in measurable_fun _ X](_ : _ = (fun x => + \sum_(y \in range (f_ m)) + (\int[k x]_w2 (y * \1_(f_ m @^-1` [set y]) (x, w2))%:E))); last first. + apply/funext => x. + under eq_integral. + move=> y _; rewrite -fsumEFin//. + over. + rewrite /= ge0_integral_fsum//=. + move=> r. + under eq_fun do rewrite EFinM. + apply: emeasurable_funM => //. + by apply/measurable_EFinP; exact: measurableT_comp. + by move=> r y _; rewrite EFinM nnfun_muleindic_ge0. + apply: emeasurable_fsum => // r. + rewrite [X in measurable_fun _ X](_ : _ = (fun x => + (r%:E * \int[k x]_y (\1_(f_ m @^-1` [set r]) (x, y))%:E))); last first. + apply/funext => x. + under eq_integral do rewrite EFinM. + rewrite integralZl//. + pose kx : {measure set T2 -> \bar R} := k x. + pose H1 := isFinite.Build _ _ _ _ + (@kernel_finite_transition _ _ _ _ R k x). + pose H2 := isSFinite.Build _ _ _ _ (finite_measure_sfinite + (@kernel_finite_transition _ _ _ _ R k x)). + pose H3 := isSigmaFinite.Build _ _ _ _ (finite_measure_sigma_finite + (@kernel_finite_transition _ _ _ _ R k x)). + pose kx' := HB.pack_for (FiniteMeasure.type T2 R) (Measure.sort kx) H1 H2 H3. + have kxE : kx = kx' by apply: eq_measure. + rewrite -/kx kxE; apply: integrable_indic => //. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. + by apply: emeasurable_funM => //; exact: mI1. +by move=> ? _; exact: If_cvg. +Qed. + +End measurable_kfcomp. + +(* [Theorem 14.22, Klenke 2014] + return type : T0 -> set (T1 * T2) -> \bar R *) +Section kproduct_def. +Context d0 d1 d2 (T0 : measurableType d0) (T1 : measurableType d1) + (T2 : measurableType d2) {R : realType}. +Variable k1 : T0 -> {measure set T1 -> \bar R}. +Variable k2 : T0 * T1 -> {measure set T2 -> \bar R}. + +Local Definition intker_indic (k : T0 * T1 -> {measure set T2 -> \bar R}) A := + fun xy => \int[k xy]_z (\1_A (xy.2, z))%:E. + +Definition kproduct x (A : set (T1 * T2)) := + \int[k1 x]_y intker_indic k2 A (x, y). + +End kproduct_def. + +Section intker_indic_lemmas. +Context d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType}. +Variable k : R.-ftker T0 * T1 ~> T2. + +Local Lemma measurable_intker_indic A : measurable A -> + measurable_fun [set: T0 * T1] (intker_indic k A). +Proof. +move=> mA; apply: (@measurable_kfcomp _ _ (T0 * T1)%type T2 R k + (fun abc => (\1_A (abc.1.2, abc.2))%:E)). +- apply/measurable_EFinP => //=; apply: measurableT_comp => //=. + apply/prod_measurable_funP; split => /=. + rewrite [X in measurable_fun _ X](_ : _ = snd \o fst)//. + exact: measurableT_comp. + by rewrite [X in measurable_fun _ X](_ : _ = snd). +- by move=> t; rewrite lee_fin. +Qed. + +Local Lemma intker_indicE A x y : measurable A -> + intker_indic k A (x, y) = k (x, y) (xsection A y). +Proof. +move=> mA; rewrite /intker_indic(*NB:lemma?*) integral_indic//; last first. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. +by rewrite setIT xsectionE. +Qed. + +Local Lemma intker_indic_bigcup x y (A : (set (T1 * T2)) ^nat) : + trivIset [set: nat] A -> (forall n, measurable (A n)) -> + intker_indic k (\bigcup_n A n) (x, y) = + \big[+%R/0%R]_(0 <= i tA mA. +rewrite /intker_indic -(@integral_nneseries _ _ R (k (x, y)) _ measurableT + (fun i z => (\1_(A i) (y, z))%:E))//. +- by apply: eq_integral => z _ /=; rewrite indic_bigcup. +- move=> n; apply/measurable_EFinP => //; apply: measurable_indic. + rewrite -[X in measurable X]/(_ @^-1` _) -xsectionE. + exact: measurable_xsection. +- by move=> n z _; rewrite lee_fin. +Qed. + +End intker_indic_lemmas. + +Section kproduct_snd_def. +Context d0 d1 d2 (T0 : measurableType d0) (T1 : measurableType d1) + (T2 : measurableType d2) {R : realType}. +Variable k1 : T0 -> {measure set T1 -> \bar R}. +Variable k2 : T1 -> {measure set T2 -> \bar R}. + +Definition kproduct_snd : T0 -> set (T1 * T2) -> \bar R := + kproduct k1 (k2 \o snd). + +End kproduct_snd_def. + +(* [Theorem 14.22, Klenke 2014] (1/3) *) +Theorem measurable_kproduct d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-ftker T0 ~> T1) (k2 : R.-ftker (T0 * T1) ~> T2) A : + measurable A -> measurable_fun [set: T0] (kproduct k1 k2 ^~ A). +Proof. +move=> mA; apply: measurable_kfcomp => // t. + exact: measurable_intker_indic. +by apply: integral_ge0 => y _; rewrite lee_fin. +Qed. + +(* [Theorem 14.22, Klenke 2014] (2/3) *) +Theorem semi_sigma_additive_kproduct d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-ftker T0 ~> T1) (k2 : R.-ftker (T0 * T1) ~> T2) x : + semi_sigma_additive (kproduct k1 k2 x). +Proof. +move=> /= A mA tA mbigcup. +rewrite (_ : (fun n => _) = (fun n => + \int[k1 x]_y (\sum_(0 <= i < n) intker_indic k2 (A i) (x, y)))); last first. + apply/funext => n; rewrite ge0_integral_sum//. + by move=> m; exact: measurable_prod2 (measurable_intker_indic k2 _). + by move=> m y _; rewrite intker_indicE. +pose g n y := \sum_(0 <= i < n) intker_indic k2 (A i) (x, y). +rewrite [X in _ --> X](_ : _ = \int[k1 x]_y limn (g ^~ y)); last first. + by apply: eq_integral => y _; rewrite /g intker_indic_bigcup. +apply: (@cvg_monotone_convergence _ _ _ (k1 x) _ measurableT g) => //. +- move=> n; apply: (@emeasurable_sum _ _ R setT _ _ + (fun i t => intker_indic k2 (A i) (x, t))) => m. + exact: measurable_prod2 (measurable_intker_indic k2 (mA m)). +- by move=> n y _; apply: sume_ge0 => m _; rewrite intker_indicE. +- move=> y _ a b ab; apply: lee_sum_nneg_natr => // m _ _. + by rewrite intker_indicE. +Qed. + +Section kproduct_measure. +Context d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-ftker T0 ~> T1) (k2 : R.-ftker (T0 * T1) ~> T2). + +Let kproduct0 x : kproduct k1 k2 x set0 = 0. +Proof. +by apply: integral0_eq => y _; apply: integral0_eq => z _; rewrite indic0. +Qed. + +Let kproduct_ge0 x A : (0 <= kproduct k1 k2 x A)%E. +Proof. +by apply: integral_ge0 => y _; apply: integral_ge0 => z _; rewrite lee_fin. +Qed. + +Let kproduct_additive x : semi_sigma_additive (kproduct k1 k2 x). +Proof. exact: semi_sigma_additive_kproduct. Qed. + +HB.instance Definition _ x := isMeasure.Build + (measure_prod_display (d1, d2)) (T1 * T2)%type R (kproduct k1 k2 x) + (kproduct0 x) (kproduct_ge0 x) (@kproduct_additive x). + +Definition mkproduct := + (kproduct k1 k2 : T0 -> {measure set (T1 * T2) -> \bar R}). + +End kproduct_measure. + +HB.instance Definition _ d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-ftker T0 ~> T1) (k2 : R.-ftker (T0 * T1) ~> T2) := + @isKernel.Build _ _ T0 (T1 * T2)%type R + (mkproduct k1 k2) (measurable_kproduct k1 k2). + +(* [Theorem 14.22, Klenke 2014] (3/3): the composition of finite transition + kernels is sigma-finite *) +Section sigma_finite_kproduct. +Context d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-ftker T0 ~> T1) (k2 : R.-ftker (T0 * T1) ~> T2). + +Theorem sigma_finite_mkproduct x : + sigma_finite [set: T1 * T2] (mkproduct k1 k2 x). +Proof. +pose A n := [set w1 | k2 (x, w1) setT < n%:R%:E]. +have mA n : measurable (A n). + rewrite /A -[X in measurable X]setTI. + apply: emeasurable_fun_infty_o => //. + have := @measurable_kernel _ _ _ _ R k2 _ measurableT. + exact: measurable_prod2. +have bigcupA : \bigcup_n A n = setT. + have fink2 xy : fin_num_fun (k2 xy) by exact: kernel_finite_transition. + apply/seteqP; split => // y _. + have {}fink2 := fink2 (x, y) _ measurableT. + exists (Num.trunc (fine (k2 (x, y) setT))).+1 => //=. + have : (0 <= fine (k2 (x, y) [set: T2]))%R by rewrite fine_ge0. + by move=> /Num.Theory.trunc_itv/andP[_]; rewrite -lte_fin fineK. +have lty n : kproduct k1 k2 x (A n `*` setT) < +oo. + have fink1 : fin_num_fun (k1 x) by exact: kernel_finite_transition. + apply: (@le_lt_trans _ _ (n%:R%:E * k1 x (A n))) => /=. + rewrite [leLHS](_ : _ = \int[k1 x]_(y in A n) k2 (x, y) setT); last first. + rewrite /kproduct [RHS]integral_mkcond; apply: eq_integral => y _. + rewrite intker_indicE//; last exact: measurableX. + rewrite patchE; case: ifPn => [Anx|Anx]. + by rewrite in_xsectionX. + by rewrite notin_xsectionX// measure0. + apply: (@le_trans _ _ (\int[k1 x]_(x in A n) n%:R%:E)). + apply: ge0_le_integral => //. + - apply: measurable_funTS. + have := @measurable_kernel _ _ _ _ _ k2 _ measurableT. + exact: measurable_prod2. + - by move=> y; rewrite /A/= => /ltW. + - by rewrite integral_cst. + by rewrite lte_mul_pinfty// ltey_eq fink1. +exists (fun n => A n `*` setT) => /=. + by rewrite -setX_bigcupl bigcupA setXTT. +by move=> n; split => //; exact: measurableX. +Qed. + +HB.instance Definition _ x := @isSigmaFiniteTransitionKernel.Build d0 + (measure_prod_display (d1, d2)) T0 (T1 * T2)%type R + (mkproduct k1 k2) sigma_finite_mkproduct. + +End sigma_finite_kproduct. + +(* [Definition 14.25, Klenke 2014] + return type: T0 -> set T2 -> \bar R *) +Section kcomp_noparam_def. +Context d0 d1 d2 (T0 : measurableType d0) (T1 : measurableType d1) + (T2 : measurableType d2) {R : realType}. +Variable k1 : T0 -> {measure set T1 -> \bar R}. +Variable k2 : T1 -> {measure set T2 -> \bar R}. + +Definition kcomp_noparam (x : T0) (A : set T2) : \bar R := + \int[k1 x]_y (k2 y A). + +End kcomp_noparam_def. + +(* a parameterized subprobability kernel that ignores its first argument *) +Section spker_snd. +Context d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k : R.-spker T1 ~> T2). + +Local Definition kernel_snd : (T0 * T1)%type -> {measure set T2 -> \bar R} := + k \o snd. + +Let measurable_kernel U : measurable U -> + measurable_fun [set: _] (kernel_snd ^~ U). +Proof. +move=> mU; have /= mk1 := measurable_kernel k _ mU. +move=> _ /= Y mY; have {}mk1 := mk1 measurableT _ mY. +have -> : [set: T0 * T1] `&` (kernel_snd ^~ U) @^-1` Y = + setT `*` ([set: T1] `&` (k ^~ U) @^-1` Y). + by rewrite !setTI setTX. +exact: measurableX. +Qed. + +HB.instance Definition _ := + @isKernel.Build _ _ _ _ _ kernel_snd measurable_kernel. + +Let measure_uub : measure_fam_uub kernel_snd. +Proof. +exists 2%E => /= -[x y]. +rewrite /kernel_snd/= (@le_lt_trans _ _ 1%:E) ?lte1n//. +exact: sprob_kernel_le1. +Qed. + +HB.instance Definition _ := + @Kernel_isFinite.Build _ _ _ _ _ kernel_snd measure_uub. + +Let sprob_kernel : + (ereal_sup [set kernel_snd z [set: _] | z in [set: _]] <= 1)%E. +Proof. +by apply: (sprob_kernelP kernel_snd).2 => -[x y]; exact: sprob_kernel_le1. +Qed. + +HB.instance Definition _ := isSubProbabilityKernel.Build _ _ _ _ _ + kernel_snd sprob_kernel. + +Local Lemma intker_indic_snd (A : set T2) x y : measurable A -> + intker_indic kernel_snd ([set: _] `*` A) (x, y) = k y A. +Proof. +move=> mA; rewrite intker_indicE//=; last exact: measurableX. +by congr (k y _); rewrite xsectionE//= setTX. +Qed. + +End spker_snd. + +(* [Theorem 14.26, Klenke 2014] (1/2) *) +Lemma kcomp_noparamE d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : T0 -> {measure set T1 -> \bar R}) + (k2 : T1 -> {measure set T2 -> \bar R}) x A : + measurable A -> + kcomp_noparam k1 k2 x A = kproduct_snd k1 k2 x (snd @^-1` A). +Proof. +move=> mA; rewrite /kcomp_noparam /kproduct_snd /kproduct. +apply: eq_integral => y _ /=. +by rewrite /intker_indic/= integral_indic// setIT. +Qed. + +Section kcomp_noparam_measure. +Context d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-spker T0 ~> T1) (k2 : R.-spker T1 ~> T2). + +Let kcomp_noparam0 x : kcomp_noparam k1 k2 x set0 = 0. +Proof. by apply: integral0_eq => y _; rewrite measure0. Qed. + +Let kcomp_noparam_ge0 x A : 0 <= kcomp_noparam k1 k2 x A. +Proof. by apply: integral_ge0 => y _; exact: measure_ge0. Qed. + +Let kcomp_noparam_additive x : semi_sigma_additive (kcomp_noparam k1 k2 x). +Proof. +move=> F mF tF mUF. +rewrite kcomp_noparamE// (_ : (fun _ => _) = (fun n => + \sum_(0 <= i < n) kproduct_snd k1 k2 x (snd @^-1` F i))); last first. + by apply/funext => n; apply: eq_bigr => k _; rewrite kcomp_noparamE. +pose F' n := [set: T1] `*` F n. +have kcomp_noparam_bigcup : kcomp_noparam k1 k2 x (\bigcup_n F n) = + kproduct k1 (kernel_snd k2) x (\bigcup_n F' n). + by apply: eq_integral => y _; rewrite -setX_bigcupr intker_indic_snd. +rewrite -kcomp_noparamE// kcomp_noparam_bigcup. +rewrite (_ : (fun n => _) = (fun n => + \sum_(0 <= i < n) kproduct k1 (kernel_snd k2) x (F' i))); last first. + apply/funext => n; apply: eq_bigr => i _. + apply: eq_integral => y _; apply: eq_integral => z _. + by rewrite /F' setTX. +apply: semi_sigma_additive_kproduct => //. +- by move=> i; exact: measurableX. +- apply/trivIsetP => i j _ _ ij. + rewrite /F' -setXI setTI. + move/trivIsetP : tF => /(_ i j Logic.I Logic.I ij) ->. + by rewrite setX0. +- by apply: bigcup_measurable => k _; exact: measurableX. +Qed. + +HB.instance Definition _ x := isMeasure.Build d2 T2 R (kcomp_noparam k1 k2 x) + (kcomp_noparam0 x) (kcomp_noparam_ge0 x) (@kcomp_noparam_additive x). + +Definition mkcomp_noparam := + (kcomp_noparam k1 k2 : T0 -> {measure set T2 -> \bar R}). + +Let measurable_kernel U : + measurable U -> measurable_fun [set: _] (mkcomp_noparam ^~ U). +Proof. +move=> mU. +rewrite [X in measurable_fun _ X](_ : _ = + (kproduct k1 (kernel_snd k2))^~ ([set: T1] `*` U)); last first. + apply/funext => x. + by apply: eq_integral => y _; rewrite intker_indic_snd. +by apply: measurable_kproduct; exact: measurableX. +Qed. + +HB.instance Definition _ := @isKernel.Build _ _ _ _ R + mkcomp_noparam measurable_kernel. + +End kcomp_noparam_measure. + +(* the composition of subprobability kernels is a subprobability kernel *) +Section subprobability_kcomp_noparam. +Context d0 d1 d2 (T0 : measurableType d0) + (T1 : measurableType d1) (T2 : measurableType d2) {R : realType} + (k1 : R.-spker T0 ~> T1) (k2 : R.-spker T1 ~> T2). + +(* [Theorem 14.26, Klenke 2014] (2/2) *) +Lemma sprob_mkcomp_noparam x : (mkcomp_noparam k1 k2 x setT <= 1)%E. +Proof. +rewrite /mkcomp_noparam [leLHS]kcomp_noparamE// preimage_setT. +rewrite /kproduct_snd /kproduct. +rewrite [leLHS](_ : _ = \int[k1 x]_y k2 y setT); last first. + apply: eq_integral => y _. + rewrite /intker_indic integral_indic//; last first. + rewrite [X in measurable X](_ : _ = ysection setT y). + exact: measurable_ysection. + by rewrite ysectionE. + by rewrite setIT. +apply: (@le_trans _ _ (\int[k1 x]__ 1)); last first. + by rewrite integral_cst// mul1e; exact: sprob_kernel_le1. +apply: ge0_le_integral => //; first exact: measurable_kernel. +by move=> y _; exact: sprob_kernel_le1. +Qed. + +Let sprob_kernel : + ereal_sup [set (kcomp_noparam k1 k2) x setT | x in [set: T0]] <= 1. +Proof. by apply/sprob_kernelP => x; exact: sprob_mkcomp_noparam. Qed. + +HB.instance Definition _ := Kernel_isSubProbability.Build + _ _ _ _ R (mkcomp_noparam k1 k2) sprob_kernel. + +End subprobability_kcomp_noparam. diff --git a/theories/lang_syntax.v b/theories/lang_syntax.v new file mode 100644 index 000000000..5ce4e20a6 --- /dev/null +++ b/theories/lang_syntax.v @@ -0,0 +1,2643 @@ +Require Import String. +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval. +From mathcomp Require Import lra. +From mathcomp Require Import mathcomp_extra boolp classical_sets. +From mathcomp Require Import functions cardinality fsbigop. +From mathcomp Require Import signed reals ereal topology normedtype sequences. +From mathcomp Require Import esum measure lebesgue_measure numfun derive realfun. +From mathcomp Require Import lebesgue_integral probability ftc kernel charge. +From mathcomp Require Import prob_lang lang_syntax_util. +From mathcomp Require Import lra. + +(**md**************************************************************************) +(* # Syntax and Evaluation for a Probabilistic Programming Language *) +(* *) +(* Reference: *) +(* - R. Saito, R. Affeldt. Experimenting with an Intrinsically-Typed *) +(* Probabilistic Programming Language in Coq using s-finite kernels in Coq. *) +(* APLAS 2023 *) +(* *) +(* beta distribution specialized to nat *) +(* beta_pdf == probability density function for beta *) +(* beta_prob == beta probability measure *) +(* *) +(* typ == syntax for types of data structures *) +(* measurable_of_typ t == the measurable type corresponding to type t *) +(* It is of type {d & measurableType d} *) +(* mtyp_disp t == the display corresponding to type t *) +(* mtyp t == the measurable type corresponding to type t *) +(* It is of type measurableType (mtyp_disp t) *) +(* measurable_of_seq s == the product space corresponding to the *) +(* list s : seq typ *) +(* It is of type {d & measurableType d} *) +(* acc_typ s n == function that access the nth element of s : seq typ *) +(* It is a function from projT2 (measurable_of_seq s) *) +(* to projT2 (measurable_of_typ (nth Unit s n)) *) +(* ctx == type of context *) +(* := seq (string * type) *) +(* mctx_disp g == the display corresponding to the context g *) +(* mctx g := the measurable type corresponding to the context g *) +(* It is formed of nested pairings of measurable *) +(* spaces. It is of type measurableType (mctx_disp g) *) +(* flag == a flag is either D (deterministic) or *) +(* P (probabilistic) *) +(* exp f g t == syntax of expressions with flag f of type t *) +(* context g *) +(* dval R g t == "deterministic value", i.e., *) +(* function from mctx g to mtyp t *) +(* pval R g t == "probabilistic value", i.e., *) +(* s-finite kernel, from mctx g to mtyp t *) +(* e -D> f ; mf == the evaluation of the deterministic expression e *) +(* leads to the deterministic value f *) +(* (mf is the proof that f is measurable) *) +(* e -P> k == the evaluation of the probabilistic function f *) +(* leads to the probabilistic value k *) +(* execD e == a dependent pair of a function corresponding to the *) +(* evaluation of e and a proof that this function is *) +(* measurable *) +(* execP e == a s-finite kernel corresponding to the evaluation *) +(* of the probabilistic expression e *) +(* *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Import Order.TTheory GRing.Theory Num.Def Num.Theory. +Import numFieldTopology.Exports. + +Reserved Notation "e -D> f ; mf" (at level 40). +Reserved Notation "e -P> k" (at level 40). + +Local Open Scope classical_set_scope. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. + +(* In this module, we use our lemma continuous_FTC2 to compute the value of + * integration of the indicator function over the interval [0, 1]. + * we can use our lemma continuous_FTC2 because it requires continuous + * within [0, 1], which the indicator function satisfies. + * we also shows that the indicator function is not continuous in [0, 1], + * required by previous version of lemma continuous_FTC2. This shows that + * our lemma continuous_FTC2 is + * enough weak to be usable in practice. + *) +Module integral_indicator_function. +Section integral_indicator_function. + +Context {R : realType}. +Notation mu := lebesgue_measure. +Local Open Scope ereal_scope. +Implicit Types (f : R -> R) (a b : R). + +Local Import set_interval. + +Let uni := @indic R R `[0%R, 1%R]%classic. + +Let integrable_uni : mu.-integrable setT (EFin \o uni). +Proof. +apply/integrableP; split. + apply: measurableT_comp => //. + exact: measurable_indic. +apply/abse_integralP => //. + apply: measurableT_comp => //. + exact: measurable_indic. +rewrite -ge0_fin_numE; last exact: abse_ge0. +rewrite abse_fin_num integral_indic// setIT. +by rewrite /= lebesgue_measure_itv ifT. +Qed. + +Let cuni_within : {within `[0%R, 1%R], continuous uni}. +Proof. +apply/continuous_within_itvP => //; split. +- move=> x x01. + apply: (@near_cst_continuous R R 1%R). + near=> z. + rewrite /uni indic_restrict patchE ifT//. + rewrite inE/=. + apply: subset_itv_oo_cc. + near: z. + exact: near_in_itvoo. +- rewrite (_: uni 0 = 1%R); last first. + rewrite /uni indic_restrict patchE ifT//. + by rewrite inE/= boundl_in_itv bnd_simp/=. + apply: cvg_near_cst. + near=> z. + rewrite /uni indic_restrict patchE ifT// inE/= in_itv/=; apply/andP; split => //. + near: z. + exact: nbhs_right_le. +- rewrite (_:uni 1 = 1%R); last first. + rewrite /uni indic_restrict patchE ifT//. + by rewrite inE/= boundr_in_itv bnd_simp/=. + apply: cvg_near_cst. + near=> z. + rewrite /uni indic_restrict patchE ifT// inE/= in_itv/=; apply/andP; split => //. + near: z. + exact: nbhs_left_ge. +Unshelve. all: end_near. Qed. + +Example cuni : ~ {in `[0%R, 1%R], continuous uni}. +Proof. +rewrite -existsNE/=. +exists 0%R. +rewrite not_implyE; split; first by rewrite boundl_in_itv/= bnd_simp. +move/left_right_continuousP. +apply/not_andP; left. +move/(@cvgrPdist_le _ R^o). +apply/existsNP. +exists (2%:R^-1). +rewrite not_implyE; split; first by rewrite invr_gt0. +move=> [e /= e0]. +move/(_ (-(e / 2))%R). +apply/not_implyP; split. + rewrite /= sub0r opprK ger0_norm; last by rewrite divr_ge0// ltW. + rewrite -{1}(add0r e). + exact: (midf_lt e0).2. +apply/not_implyP; split. + rewrite oppr_lt0. + exact: divr_gt0. +apply/negP; rewrite -ltNge. +rewrite /uni !indic_restrict !patchE. +rewrite ifT; last by rewrite inE/= boundl_in_itv/= bnd_simp. +rewrite ifF; last first. + apply: negbTE; apply/negP. + rewrite inE/= in_itv/=. + apply/negP; rewrite negb_and; apply/orP; left. + by rewrite -ltNge oppr_lt0 divr_gt0. +rewrite /point/= {2}/1%R/= subr0. +rewrite ger0_norm//. +rewrite invf_lt1//. +rewrite {1}(_:1%R = 1%:R)//; apply: ltr_nat. +Qed. + +Let dintuni : derivable_oo_continuous_bnd (@id R^o) 0 1. +Proof. +split. +- move=> x _. + exact: derivable_id. +- exact: cvg_at_right_filter. +- exact: cvg_at_left_filter. +Qed. + +Let intuni'uni : {in `]0%R, 1%R[, (@id R^o)^`() =1 uni}. +Proof. +move=> x x01. +rewrite derive1E derive_id. +rewrite /uni indic_restrict patchE ifT// inE/=. +exact: subset_itv_oo_cc. +Qed. + +Lemma intuni1 : (\int[mu]_(x in `[0, 1]) uni x)%R = 1%R. +Proof. +rewrite [RHS](_:1%R = fine (1%:E))//; congr (fine _). +rewrite (continuous_FTC2 ltr01 cuni_within dintuni intuni'uni). +by rewrite sube0. +Qed. + +End integral_indicator_function. +End integral_indicator_function. + +Lemma RintegralZl d {T : measurableType d} {R : realType} + {mu : measure T R} {D : set T} : d.-measurable D -> + forall f : T -> R, + mu.-integrable D (EFin \o f) -> + forall r : R, (\int[mu]_(x in D) (r * f x) = r * \int[mu]_(x in D) f x)%R. +Proof. +move=> mD f intf r. +rewrite /Rintegral. +under eq_integral do rewrite EFinM. +rewrite integralZl// fineM//=. +by apply: integral_fune_fin_num. +Qed. + +(* TODO: naming *) +Lemma cvg_atNP {T : topologicalType} {R : numFieldType} (f : R -> T) (a : R) (l : T) : + f x @[x --> a] --> l <-> (f \o -%R) x @[x --> (- a)%R] --> l. +Proof. +rewrite nbhsN. +have <-// : f x @[x --> a] = fmap [eta f \o -%R] ((- x)%R @[x --> a]). +by apply/seteqP; split=> A; move=> [/= e e0 H]; exists e => //= B /H/=; rewrite opprK. +Qed. + +Lemma derivable_oo_bnd_id {R : numFieldType} (a b : R) : + derivable_oo_continuous_bnd (@id R^o) a b. +Proof. +by split => //; + [exact/cvg_at_right_filter/cvg_id|exact/cvg_at_left_filter/cvg_id]. +Qed. + +Lemma derivable_oo_bndN {R : realFieldType} (f : R -> R^o) a b : + derivable_oo_continuous_bnd f (- a) (- b) -> + derivable_oo_continuous_bnd (f \o -%R) b a. +Proof. +move=> [dF cFa cFb]. +have oppK : (-%R \o -%R) = @id R by apply/funext => x/=; rewrite opprK. +split. +- move=> x xba; apply/derivable1_diffP. + apply/(@differentiable_comp _ _ R^o _ -%R f x) => //. + by apply/derivable1_diffP/dF; rewrite oppr_itvoo 2!opprK. +- by apply/cvg_at_rightNP; rewrite -compA oppK. +- by apply/cvg_at_leftNP; rewrite -compA oppK. +Qed. + +Module increasing_change_of_variables_from_decreasing. +Section lt0. +Context {R : realType}. +Notation mu := lebesgue_measure. +Implicit Types (F G f : R -> R) (a b : R). + +Lemma continuous_withinN f a b : (a < b)%R -> + {within `[(- b)%R, (- a)%R], continuous (f \o -%R)} -> + {within `[a, b], continuous f}. +Proof. +move=> ab cf. +- apply/continuous_within_itvP (* TODO: us [/\ ...] *) => //. +- split; rewrite -ltrN2 in ab. + + move=> x xab. + move/continuous_within_itvP : cf => /(_ ab) [cf _ _]. + rewrite (_ : f = (f \o -%R) \o -%R); last first. + by apply/funext => y; rewrite /= opprK. + apply: continuous_comp; first exact: (@opp_continuous _ R^o). + by apply: cf; rewrite -oppr_itvoo opprK. + + move/continuous_within_itvP : cf => /(_ ab) [_ _ cf]. + apply/cvg_at_rightNP. + by rewrite /= opprK in cf. + + move/continuous_within_itvP : cf => /(_ ab) [_ cf _]. + apply/cvg_at_leftNP. + by rewrite /= opprK in cf. +Qed. + +Lemma oppr_change (f : R -> R) a b : (a < b)%R -> + {within `[a, b], continuous f} -> + \int[mu]_(x in `[a, b]) (f x)%:E = + \int[mu]_(x in `[-%R b, -%R a]) ((f \o -%R) x)%:E. +Proof. +move=> ab cf. +have dN : ((-%R : R -> R^o)^`() = cst (-1) :> (R -> R))%R. (* TODO: lemma? *) + by apply/funext => x/=; rewrite derive1E deriveN// derive_id. +rewrite integration_by_substitution_decreasing//. +- by apply: eq_integral => /= x _; rewrite dN/= opprK mulr1 -compA/= opprK. +- by move=> x y _ _ yx; rewrite ltrN2. +- by move=> y yab; rewrite dN; exact: cvg_cst. +- by rewrite dN; exact: is_cvg_cst. +- by rewrite dN; exact: is_cvg_cst. +- by apply: (@derivable_oo_bndN _ id) => //; exact: derivable_oo_bnd_id. +- apply: continuous_withinN. + + by rewrite ltrN2. + + rewrite -(_ : f = (f \o -%R) \o -%R)//; last first. + by apply/funext => y; rewrite /= opprK. + by rewrite !opprK. +Qed. + +End lt0. +End increasing_change_of_variables_from_decreasing. + +Lemma decreasing_nonincreasing {R : realType} (F : R -> R) (J : interval R) : + {in J &, {homo F : x y /~ (x < y)%R}} -> + {in J &, {homo F : x y /~ (x <= y)%R}}. +Proof. +move=> dF. +move=> x y x01 y01. +by rewrite le_eqVlt => /predU1P[->//|/dF] => /(_ x01 y01)/ltW. +Qed. + +Lemma derive1_onem {R : realType} : (fun x0 : R => (1 - x0)%R : R^o)^`() = (cst (-1)%R). +Proof. +apply/funext => x. +by rewrite derive1E deriveB// derive_id derive_cst sub0r. +Qed. + +Local Close Scope ereal_scope. +Lemma cvg_comp_filter {R : realType} (f g : R -> R) (r l : R) : + continuous f -> + (f \o g) x @[x --> r] --> l -> + f x @[x --> g r] --> l. +Proof. +move=> cf fgrl. +apply/(@cvgrPdist_le _ R^o) => /= e e0. +have e20 : 0 < e / 2 by rewrite divr_gt0. +move/(@cvgrPdist_le _ R^o) : fgrl => /(_ _ e20) fgrl. +have := cf (g r). +move=> /(@cvgrPdist_le _ R^o) => /(_ _ e20)[x x0]H. +exists (minr x (e/2)). + by rewrite lt_min x0. +move=> z. +rewrite /ball_ /= => grze. +rewrite -[X in X - _](subrK (f (g r))). +rewrite -(addrA _ _ (- f z)). +apply: (le_trans (ler_normD _ _)). +rewrite (splitr e) lerD//. + case: fgrl => d /= d0 K. + apply: K. + by rewrite /ball_/= subrr normr0. +apply: H => /=. +by rewrite (lt_le_trans grze)// ge_min lexx. +Qed. +Local Open Scope ereal_scope. + +Section change_of_variables_onem. +Context {R : realType}. +Let mu := (@lebesgue_measure R). + +Lemma onem_change (G : R -> R) (r : R) : + (0 < r <= 1)%R -> + {within `[0%R, r], continuous G} -> + (\int[mu]_(x in `[0%R, r]) (G x)%:E = + \int[mu]_(x in `[(1 - r)%R, 1%R]) (G (1 - x))%:E). +Proof. +move=> r01 cG. +have := @integration_by_substitution_decreasing R (fun x => 1 - x)%R G (1 - r) 1%R. +rewrite opprB subrr addrCA subrr addr0. +move=> ->//. +- apply: eq_integral => x xr. + rewrite !fctE. + by rewrite derive1_onem opprK mulr1. +- rewrite ltrBlDl ltrDr. + by case/andP : r01. +- by move=> x y _ _ xy; rewrite ler_ltB. +- by rewrite derive1_onem; move=> ? ?; apply: cvg_cst. +- by rewrite derive1_onem; exact: is_cvg_cst. +- by rewrite derive1_onem; exact: is_cvg_cst. +- split => /=. + + move=> x xr1. + by apply: derivableB => //. + + apply: cvg_at_right_filter. + rewrite opprB addrCA addrA addrK. + apply: (@cvg_comp_filter _ _ (fun x => 1 - x)%R)=> //=. + move=> x. + apply: (@continuousB _ R^o) => //. + exact: cvg_cst. + under eq_fun do rewrite opprD addrA subrr add0r opprK. + apply: cvg_id. + apply: cvg_at_left_filter. + apply: (@cvgB _ R^o) => //. + exact: cvg_cst. +Qed. + +Lemma Ronem_change (G : R -> R) (r : R) : + (0 < r <= 1)%R -> + {within `[0%R, r], continuous G} -> + (\int[mu]_(x in `[0%R, r]) (G x) = + \int[mu]_(x in `[(1 - r)%R, 1%R]) (G (1 - x)))%R. +Proof. +move=> r01 cG. +rewrite [in LHS]/Rintegral. +by rewrite onem_change. +Qed. + +End change_of_variables_onem. + +Section factD. + +Let factD' n m : (n`! * m`! <= (n + m).+1`!)%N. +Proof. +elim: n m => /= [m|n ih m]. + by rewrite fact0 mul1n add0n factS leq_pmull. +rewrite 2!factS [in X in (_ <= _ * X)%N]addSn -mulnA leq_mul//. +by rewrite ltnS addSnnS leq_addr. +Qed. + +Lemma factD n m : (n`! * m.-1`! <= (n + m)`!)%N. +Proof. +case: m => //= [|m]. + by rewrite fact0 muln1 addn0. +by rewrite addnS factD'. +Qed. + +End factD. + +Lemma leq_prod2 (x y n m : nat) : (n <= x)%N -> (m <= y)%N -> + (\prod_(m <= i < y) i * \prod_(n <= i < x) i <= \prod_(n + m <= i < x + y) i)%N. +Proof. +move=> nx my; rewrite big_addn -addnBA//. +rewrite [in leqRHS]/index_iota -addnBAC// iotaD big_cat/=. +rewrite mulnC leq_mul//. + by apply: leq_prod; move=> i _; rewrite leq_addr. +rewrite subnKC//. +rewrite -[in leqLHS](add0n m) big_addn. +rewrite [in leqRHS](_ : y - m = ((y - m + x) - x))%N; last first. + by rewrite -addnBA// subnn addn0. +rewrite -[X in iota X _](add0n x) big_addn -addnBA// subnn addn0. +by apply: leq_prod => i _; rewrite leq_add2r leq_addr. +Qed. + +Lemma leq_fact2 (x y n m : nat) : (n <= x) %N -> (m <= y)%N -> + (x`! * y`! * ((n + m).+1)`! <= n`! * m`! * ((x + y).+1)`!)%N. +Proof. +move=> nx my. +rewrite (fact_split nx) -!mulnA leq_mul2l; apply/orP; right. +rewrite (fact_split my) mulnCA -!mulnA leq_mul2l; apply/orP; right. +rewrite [leqRHS](_ : _ = (n + m).+1`! * \prod_((n + m).+2 <= i < (x + y).+2) i)%N; last first. + by rewrite -fact_split// ltnS leq_add. +rewrite mulnA mulnC leq_mul2l; apply/orP; right. +do 2 rewrite -addSn -addnS. +exact: leq_prod2. +Qed. + +Lemma bounded_norm_expn_onem {R : realType} (a b : nat) : + [bounded `|x ^+ a * (1 - x) ^+ b|%R : R^o | x in (`[0%R, 1%R]%classic : set R)]. +Proof. +exists 1%R; split; [by rewrite num_real|move=> x x1 /= y]. +rewrite in_itv/= => /andP[y0 y1]. +rewrite ger0_norm// ger0_norm; last first. + by rewrite mulr_ge0 ?exprn_ge0// subr_ge0. +rewrite (le_trans _ (ltW x1))// mulr_ile1 ?exprn_ge0//. +- by rewrite subr_ge0. +- by rewrite exprn_ile1. +- rewrite exprn_ile1 ?subr_ge0//. + by rewrite lerBlDl addrC -lerBlDl subrr. +Qed. + +Lemma measurable_fun_expn_onem {R : realType} a b : + measurable_fun setT (fun x : R => x ^+ a * `1-x ^+ b)%R. +Proof. +apply/measurable_funM => //; apply/measurable_funX => //. +exact: measurable_funB. +Qed. + +Section ubeta_nat_pdf. +Local Open Scope ring_scope. +Context {R : realType}. +Variables a b : nat. + +(* unnormalized pdf *) +(*Definition ubeta_nat_pdf (t : R) := + if (0 <= t <= 1)%R then (t ^+ a.-1 * (`1-t) ^+ b.-1)%R else 0%R. + +Lemma ubeta_nat_pdf_ge0 t : 0 <= ubeta_nat_pdf t. +Proof. +rewrite /ubeta_nat_pdf; case: ifPn => // /andP[t0 t1]. +by rewrite mulr_ge0// exprn_ge0// onem_ge0. +Qed. + +Lemma ubeta_nat_pdf_le1 t : ubeta_nat_pdf t <= 1. +Proof. +rewrite /ubeta_nat_pdf; case: ifPn => // /andP[t0 t1]. +by rewrite mulr_ile1// ?(exprn_ge0,onem_ge0,exprn_ile1,onem_le1). +Qed. + +Lemma measurable_ubeta_nat_pdf : measurable_fun setT ubeta_nat_pdf. +Proof. +rewrite /ubeta_nat_pdf /=; apply: measurable_fun_if => //=; last first. + by rewrite setTI; apply: measurable_funTS; exact: measurable_fun_expn_onem. +by apply: measurable_and => /=; exact: measurable_fun_ler. +Qed. + +Local Notation mu := lebesgue_measure. + +Lemma integral_ubeta_nat_pdf U : + (\int[mu]_(x in U) (ubeta_nat_pdf x)%:E = + \int[mu]_(x in U `&` `[0%R, 1%R]) (ubeta_nat_pdf x)%:E)%E. +Proof. +rewrite [RHS]integral_mkcondr/=; apply: eq_integral => x xU. +rewrite patchE; case: ifPn => //. +rewrite notin_setE/= in_itv/= => /negP. +rewrite negb_and -!ltNge => /orP[x0|x1]. + by rewrite /ubeta_nat_pdf leNgt x0/=. +by rewrite /ubeta_nat_pdf !leNgt x1/= andbF. +Qed. + +Lemma integral_ubeta_nat_pdfT : + (\int[mu]_x (ubeta_nat_pdf x)%:E = + \int[mu]_(x in `[0%R, 1%R]) (ubeta_nat_pdf x)%:E)%E. +Proof. by rewrite integral_ubeta_nat_pdf/= setTI. Qed.*) + +End ubeta_nat_pdf. + +(*Lemma ubeta_nat_pdf11 {R : realType} (x : R) : (0 <= x <= 1)%R -> + ubeta_nat_pdf 1 1 x = 1%R. +Proof. +move=> x01. +by rewrite /ubeta_nat_pdf !expr0 mulr1 x01. +Qed. + +(* normalization constant *) +Definition beta_nat_norm {R : realType} (a b : nat) : R := + fine (\int[@lebesgue_measure R]_x (ubeta_nat_pdf a b x)%:E).*) + +Section beta_nat_Gamma. +Context {R : realType}. +(* +Let mu := @lebesgue_measure R. + +Let B (a b : nat) : \bar R := + \int[mu]_(x in `[0%R, 1%R]%classic) (ubeta_nat_pdf a b x)%:E. +*) +End beta_nat_Gamma. + +(* +Lemma integral_beta_nat_normTE {R : realType} (a b : nat) : + beta_nat_norm a b = + fine (\int[lebesgue_measure]_(t in `[0%R, 1%R]) (t^+a.-1 * (`1-t)^+b.-1)%:E) :> R. +Proof. +rewrite /beta_nat_norm /ubeta_nat_pdf [in RHS]integral_mkcond/=; congr fine. +by apply: eq_integral => /= x _; rewrite patchE mem_setE in_itv/=; case: ifPn. +Qed.*) + +Lemma onemXn_derivable {R : realType} n (x : R) : + derivable (fun y : R^o => `1-y ^+ n : R^o)%R x 1. +Proof. +have := @derivableX R R^o (@onem R) n x 1%R. +rewrite fctE. +apply. +exact: derivableB. +Qed. + +Lemma deriveX_idfun {R : realType} n x : + 'D_1 (@GRing.exp R^o ^~ n.+1) x = n.+1%:R *: (x ^+ n)%R. +Proof. by rewrite exp_derive /GRing.scale/= mulr1. Qed. + +Lemma derive1Mr [R : realFieldType] [f : R^o -> R^o] [x r : R^o] : + derivable f x 1 -> ((fun x => f x * r)^`() x = (r * f^`() x)%R :> R)%R. +Proof. +move=> fx1. +rewrite derive1E (deriveM fx1); last by []. +by rewrite -derive1E derive1_cst scaler0 add0r derive1E. +Qed. + +Lemma derive1Ml [R : realFieldType] [f : R^o -> R^o] [x r : R^o] : + derivable f x 1 -> ((fun x => r * f x)^`() x = (r * f^`() x)%R :> R)%R. +Proof. +under eq_fun do rewrite mulrC. +exact: derive1Mr. +Qed. + +Lemma decreasing_onem {R : numDomainType} : {homo (fun x : R => (1 - x)%R) : x y /~ (x < y)%R}. +Proof. +move=> b a ab. +by rewrite -ltrN2 !opprB ltr_leB. +Qed. + +Lemma continuous_onemXn {R : realType} (n : nat) x : + {for x, continuous (fun y : R => `1-y ^+ n)%R}. +Proof. +apply: (@continuous_comp _ _ _ (@onem R) (fun x => GRing.exp x n)). + by apply: (@cvgB _ R^o); [exact: cvg_cst|exact: cvg_id]. +exact: exprn_continuous. +Qed. + +Local Close Scope ereal_scope. + +(* we define a function to help formalizing the beta distribution *) +Section XMonemX. +Context {R : numDomainType}. + +Definition XMonemX a b := fun x : R => x ^+ a * `1-x ^+ b. + +Lemma XMonemX_ge0 a b x : x \in `[0%R, 1%R] -> 0 <= XMonemX a b x :> R. +Proof. +rewrite in_itv/= => /andP[x0 x1]. +by rewrite /XMonemX mulr_ge0// exprn_ge0// subr_ge0. +Qed. + +Lemma XMonemX0 n x : XMonemX 0 n x = `1-x ^+ n :> R. +Proof. by rewrite /XMonemX/= expr0 mul1r. Qed. + +Lemma XMonemX0' n x : XMonemX n 0 x = x ^+ n :> R. +Proof. by rewrite /XMonemX/= expr0 mulr1. Qed. + +Lemma XMonemX00 x : XMonemX 0 0 x = 1 :> R. +Proof. by rewrite XMonemX0 expr0. Qed. + +Lemma XMonemXC a b x : XMonemX a b (1 - x) = XMonemX b a x :> R. +Proof. +by rewrite /XMonemX [in LHS]/onem opprB addrCA subrr addr0 mulrC. +Qed. + +End XMonemX. + +Lemma continuous_XMonemX {R : realType} a b : + continuous (XMonemX a b : R -> R). +Proof. +by move=> x; apply: cvgM; [exact: exprn_continuous|exact: continuous_onemXn]. +Qed. + +Lemma within_continuous_XMonemX {R : realType} a b (A : set R) : + {within A, continuous (XMonemX a b : R -> R)}. +Proof. +by apply: continuous_in_subspaceT => x _; exact: continuous_XMonemX. +Qed. + +Lemma bounded_XMonemX {R : realType} (a b : nat) : + [bounded XMonemX a b x : R^o | x in (`[0, 1]%classic : set R)]. +Proof. +exists 1%R; split; [by rewrite num_real|move=> x x1 /= y y01]. +rewrite ger0_norm//; last by rewrite XMonemX_ge0. +move: y01; rewrite in_itv/= => /andP[? ?]. +rewrite (le_trans _ (ltW x1))// mulr_ile1 ?exprn_ge0//. +- by rewrite subr_ge0. +- by rewrite exprn_ile1. +- by rewrite exprn_ile1 ?subr_ge0// lerBlDl addrC -lerBlDl subrr. +Qed. + +Lemma measurable_fun_XMonemX {R : realType} a b (A : set R) : + measurable_fun A (XMonemX a b). +Proof. +apply/measurable_funM => //; apply/measurable_funX => //. +exact: measurable_funB. +Qed. + +Local Open Scope ereal_scope. + +Lemma integral_exprn {R : realType} (n : nat) : + fine (\int[lebesgue_measure]_(x in `[0%R, 1%R]) (x ^+ n)%:E) = n.+1%:R^-1 :> R. +Proof. +pose F (x : R) : R^o := (n.+1%:R^-1 * x ^+ n.+1)%R. +have cX m : {in `[0%R, 1%R], continuous (fun x : R => x ^+ m)%R}. + by move=> x x01; exact: exprn_continuous. +have cF0 : {for 0%R, continuous F}. + apply: continuousM; first exact: cvg_cst. + by apply: cX; rewrite /= in_itv/= lexx ler01. +have cF1 : {for 1%R, continuous F}. + apply: continuousM; first exact: cvg_cst. + by apply: cX; rewrite /= in_itv/= lexx ler01. +have dcF : derivable_oo_continuous_bnd F 0 1. + split. + - by move=> x x01; apply: derivableM => //; exact: exprn_derivable. + - apply: continuous_cvg; first exact: mulrl_continuous. + by apply/cvg_at_right_filter/cX; rewrite in_itv/= lexx ler01. + - apply: continuous_cvg; first exact: mulrl_continuous. + by apply/cvg_at_left_filter/cX; rewrite in_itv/= lexx ler01. +have dFE : {in `]0%R, 1%R[, F^`() =1 (fun x : R => x ^+ n : R)%R}. + move=> x x01. + rewrite derive1Ml; last exact: exprn_derivable. + by rewrite derive1E deriveX_idfun mulrA mulVf// mul1r. +rewrite (@continuous_FTC2 _ (fun x : R => x ^+ n)%R F)//. + by rewrite /F/= expr1n expr0n/= mulr1 mulr0 subr0. +by apply: continuous_subspaceT; exact: exprn_continuous. +Qed. + +Lemma derivable_oo_continuous_bnd_onemXnMr {R : realType} (n : nat) (r : R) : + derivable_oo_continuous_bnd (fun x : R => `1-x ^+ n.+1 * r : R^o)%R 0 1. +Proof. +split. +- by move=> x x01; apply: derivableM => //=; exact: onemXn_derivable. +- apply: cvgM; last exact: cvg_cst. + apply: cvg_at_right_filter. + apply: (@cvg_comp _ _ _ (fun x => 1 - x)%R (fun x => GRing.exp x n.+1)%R). + by apply: (@cvgB _ R^o); [exact: cvg_cst|exact: cvg_id]. + exact: exprn_continuous. +- apply: cvg_at_left_filter. + apply: cvgM; last exact: cvg_cst. + apply: (@cvg_comp _ _ _ (fun x => 1 - x)%R (fun x => GRing.exp x n.+1)%R). + by apply: (@cvgB _ R^o); [exact: cvg_cst|exact: cvg_id]. + exact: exprn_continuous. +Qed. + +Lemma derive_onemXn {R : realType} (n : nat) x : + ((fun y : R => `1-y ^+ n.+1 : R^o)^`() x = - n.+1%:R * `1-x ^+ n)%R. +Proof. +rewrite (@derive1_comp _ (@onem R) (fun x => GRing.exp x n.+1))//; last first. + exact: exprn_derivable. +rewrite derive1E deriveX_idfun derive1E deriveB//. +by rewrite -derive1E derive1_cst derive_id sub0r mulrN1 [in RHS]mulNr. +Qed. + +Lemma integral_onemXn {R : realType} (n : nat) : + fine (\int[lebesgue_measure]_(x in `[0%R, 1%R]) (`1-x ^+ n)%:E) = n.+1%:R^-1 :> R. +Proof. +rewrite (@continuous_FTC2 _ _ (fun x : R => ((1 - x) ^+ n.+1 / - n.+1%:R))%R)//=. +- rewrite subrr subr0 expr0n/= mul0r expr1n mul1r sub0r. + by rewrite -invrN -2!mulNrn opprK. +- apply: continuous_in_subspaceT. + by move=> x x01; exact: continuous_onemXn. +- exact: derivable_oo_continuous_bnd_onemXnMr. +- move=> x x01. + rewrite derive1Mr//; last exact: onemXn_derivable. + by rewrite derive_onemXn mulrA mulVf// mul1r. +Qed. + +Lemma integrable_XMonemX {R : realType} (a b : nat) : + lebesgue_measure.-integrable `[0%R, 1%R] (fun x : R => (XMonemX a b x)%:E). +Proof. +apply: continuous_compact_integrable => //. + exact: segment_compact. +apply: continuous_in_subspaceT => x _. +exact: continuous_XMonemX. +Qed. + +Lemma Rintegral_onemXn {R : realType} (n : nat) : + (\int[lebesgue_measure]_(x in `[0%R, 1%R]) (`1-x ^+ n))%R = n.+1%:R^-1 :> R. +Proof. +rewrite /Rintegral. +rewrite (@continuous_FTC2 _ _ (fun x : R => ((1 - x) ^+ n.+1 / - n.+1%:R))%R)//=. +- rewrite subrr subr0 expr0n/= mul0r expr1n mul1r sub0r. + by rewrite -invrN -2!mulNrn opprK. +- apply: continuous_in_subspaceT. +- by move=> x x01; exact: continuous_onemXn. +- exact: derivable_oo_continuous_bnd_onemXnMr. +- move=> x x01. + rewrite derive1Mr//; last exact: onemXn_derivable. + by rewrite derive_onemXn mulrA mulVf// mul1r. +Qed. + +(* TODO: move *) +Lemma normr_onem {R : realType} (x : R) : (0 <= x <= 1 -> `| `1-x | <= 1)%R. +Proof. +move=> /andP[x0 x1]; rewrite ler_norml; apply/andP; split. + by rewrite lerBrDl lerBlDr (le_trans x1)// lerDl. +by rewrite lerBlDr lerDl. +Qed. + +Local Open Scope ereal_scope. + +Local Open Scope ring_scope. + +Section XMonemX01. +Local Open Scope ring_scope. +Context {R : realType}. +Variables a b : nat. + +Definition XMonemX01 := (@XMonemX R a.-1 b.-1) \_ `[0, 1]. + +Lemma XMonemX01_ge0 t : 0 <= XMonemX01 t. +Proof. +rewrite /XMonemX01 patchE ; case: ifPn => //. +rewrite inE/= in_itv/= => /andP[t0 t1]. +by rewrite mulr_ge0// exprn_ge0// onem_ge0. +Qed. + +Lemma XMonemX01_le1 t : XMonemX01 t <= 1. +Proof. +rewrite /XMonemX01 patchE ; case: ifPn => //. +rewrite inE/= in_itv/= => /andP[t0 t1]. +by rewrite mulr_ile1// ?(exprn_ge0,onem_ge0,exprn_ile1,onem_le1). +Qed. + +Lemma measurable_XMonemX01 : measurable_fun [set: R] XMonemX01. +Proof. +rewrite /XMonemX01 /=; apply/(measurable_restrictT _ _).1 => //. +exact: measurable_fun_XMonemX. +Qed. + +Local Notation mu := lebesgue_measure. + +(* TODO: maybe not that useful *) +Lemma integral_XMonemX01 U : + (\int[mu]_(x in U) (XMonemX01 x)%:E = + \int[mu]_(x in U `&` `[0%R, 1%R]) (XMonemX01 x)%:E)%E. +Proof. +rewrite [RHS]integral_mkcondr /=; apply: eq_integral => x xU /=. +by rewrite /XMonemX01/= restrict_EFin -patch_setI setIid. +Qed. + +End XMonemX01. + +Lemma XMonemX_XMonemX01 {R : realType} a b a' b' (x : R) : (0 < a)%N -> (0 < b)%N -> + x \in `[0%R, 1%R]%classic -> + (XMonemX a' b' x * XMonemX01 a b x = XMonemX01 (a + a') (b + b') x :> R)%R. +Proof. +move=> a0 b0 x01; rewrite /XMonemX01 /= !patchE x01. +rewrite mulrCA -mulrA -exprD mulrA -exprD. +congr (_ ^+ _ * _ ^+ _)%R. + by rewrite addnC -!subn1 subDnCA. +by rewrite -!subn1 subDnCA. +Qed. + +Lemma XMonemX01_11 {R : realType} (x : R) : (0 <= x <= 1)%R -> + XMonemX01 1 1 x = 1%R. +Proof. +by move=> x01; rewrite /XMonemX01 patchE mem_setE in_itv/= x01/= XMonemX00. +Qed. + +(* normalization constant *) +Section betafun. +Context {R : realType}. +Notation mu := (@lebesgue_measure R). +Local Open Scope ring_scope. + +Definition betafun (a b : nat) : R := (\int[mu]_x (XMonemX01 a b x))%R. + +Lemma betafun0 (b : nat) : (0 < b)%N -> betafun 0 b = b%:R ^-1:> R. +Proof. +move=> b0. +rewrite -[LHS]Rintegral_mkcond. +under eq_Rintegral do rewrite XMonemX0. +by rewrite Rintegral_onemXn// prednK. +Qed. + +Lemma betafun00 : betafun 0 0 = 1%R :> R. +Proof. +rewrite -[LHS]Rintegral_mkcond. +under eq_Rintegral do rewrite XMonemX00. +rewrite /Rintegral. +rewrite integral_cst/= ?mul1e; last by exact: measurable_itv. +by rewrite lebesgue_measure_itv/= lte_fin ltr01 -EFinB subr0. +Qed. + +Lemma betafun_sym (a b : nat) : betafun a b = betafun b a :> R. +Proof. +rewrite -[LHS]Rintegral_mkcond. +rewrite Ronem_change//=; last 2 first. + by rewrite ltr01 lexx. + apply: continuous_subspaceT. + by move=> x x01; exact: continuous_XMonemX. +rewrite subrr. +rewrite -[RHS]Rintegral_mkcond. +apply: eq_Rintegral => x x01. +by rewrite XMonemXC. +Qed. + +Lemma betafunS (a b : nat) : + (betafun a.+2 b.+1 = a.+1%:R / b.+1%:R * betafun a.+1 b.+2 :> R)%R. +Proof. +rewrite -[LHS]Rintegral_mkcond. +rewrite (@Rintegration_by_parts _ _ + (fun x => `1-x ^+ b.+1 / - b.+1%:R)%R (fun x => a.+1%:R * x ^+ a)%R); last 7 first. + exact: ltr01. + apply/continuous_subspaceT. + move=> x. + apply: cvgM; [exact: cvg_cst|]. + exact: exprn_continuous. + split. + by move=> x x01; exact: exprn_derivable. + by apply: cvg_at_right_filter; exact: exprn_continuous. + by apply: cvg_at_left_filter; exact: exprn_continuous. + by move=> x x01; rewrite derive1E deriveX_idfun. + apply/continuous_subspaceT. + by move=> x x01; exact: continuous_onemXn. + exact: derivable_oo_continuous_bnd_onemXnMr. + move=> x x01. + rewrite derive1Mr; last exact: onemXn_derivable. + by rewrite derive_onemXn mulrA mulVf// mul1r. +rewrite {1}/onem !(expr1n,mul1r,expr0n,subr0,subrr,mul0r,oppr0)/=. +rewrite sub0r. +transitivity (a.+1%:R / b.+1%:R * (\int[lebesgue_measure]_(x in `[0, 1]) (XMonemX a b.+1 x)) : R)%R. + under [in LHS]eq_Rintegral. + move=> x x01. + rewrite mulrA mulrC mulrA (mulrA _ a.+1%:R) -(mulrA (_ * _)%R). + over. + rewrite /=. + rewrite RintegralZl//=; last exact: integrable_XMonemX. + by rewrite -mulNrn -2!mulNr -invrN -mulNrn opprK (mulrC _ a.+1%:R)//=. +by rewrite Rintegral_mkcond. +Qed. + +Lemma betafunSS (a b : nat) : + (betafun a.+1 b.+1 = + a`!%:R / (\prod_(b.+1 <= i < (a + b).+1) i)%:R * betafun 1 (a + b).+1 :> R)%R. +Proof. +elim: a b => [b|a ih b]. + by rewrite fact0 mul1r add0n /index_iota subnn big_nil invr1 mul1r. +rewrite betafunS. +rewrite ih. +rewrite !mulrA. +congr *%R; last by rewrite addSnnS. +rewrite -mulrA. +rewrite mulrCA. +rewrite 2!mulrA. +rewrite -natrM. +rewrite (mulnC a`!). +rewrite -factS. +rewrite -mulrA. +rewrite -invfM. +rewrite big_add1. +rewrite [in RHS]big_nat_recl/=; last by rewrite addSn ltnS leq_addl. +by rewrite -natrM addSnnS. +Qed. + +Lemma betafun1S (n : nat) : (betafun 1 n.+1 = n.+1%:R^-1 :> R)%R. +Proof. +rewrite /betafun -Rintegral_mkcond. +under eq_Rintegral do rewrite XMonemX0. +by rewrite Rintegral_onemXn. +Qed. + +Lemma betafun_fact (a b : nat) : + (betafun a.+1 b.+1 = (a`! * b`!)%:R / (a + b).+1`!%:R :> R)%R. +Proof. +rewrite betafunSS betafun1S. +rewrite natrM -!mulrA; congr *%R. +(* (b+1 b+2 ... b+1 b+a)^-1 / (a+b+1) = b! / (a+b+1)! *) +rewrite factS. +rewrite [in RHS]mulnC. +rewrite natrM. +rewrite invfM. +rewrite mulrA; congr (_ / _). +rewrite -(@invrK _ b`!%:R) -invfM; congr (_^-1). +apply: (@mulfI _ b`!%:R). + by rewrite gt_eqF// ltr0n fact_gt0. +rewrite mulrA divff// ?gt_eqF// ?ltr0n ?fact_gt0//. +rewrite mul1r. +rewrite [in RHS]fact_prod. +rewrite -natrM; congr (_%:R). +rewrite fact_prod -big_cat/=. +by rewrite /index_iota subn1 -iotaD subn1/= subSS addnK addnC. +Qed. + +Lemma betafunE (a b : nat) : betafun a b = + if (a == 0)%N && (0 < b)%N then + b%:R^-1 + else if (b == 0)%N && (0 < a)%N then + a%:R^-1 + else + a.-1`!%:R * b.-1`!%:R / (a + b).-1`!%:R :> R. +Proof. +case: a => [|a]. + rewrite eqxx/=; case: ifPn => [|]. + by case: b => [|b _] //; rewrite betafun0. + rewrite -leqNgt leqn0 => /eqP ->. + by rewrite betafun00 eqxx ltnn/= fact0 mul1r divr1. +case: b => [|b]. + by rewrite betafun_sym betafun0// fact0 addn0/= mulr1 divff. +by rewrite betafun_fact/= natrM// -addnE addnS. +Qed. + +Lemma betafun_gt0 (a b : nat) : (0 < betafun a b :> R)%R. +Proof. +rewrite betafunE. +case: ifPn => [/andP[_ b0]|]; first by rewrite invr_gt0 ltr0n. +rewrite negb_and => /orP[a0|]. + case: ifPn => [/andP[_]|]; first by rewrite invr_gt0// ltr0n. + rewrite negb_and => /orP[b0|]. + by rewrite divr_gt0// ?mulr_gt0 ?ltr0n ?fact_gt0. + by rewrite -leqNgt leqn0 (negbTE a0). +rewrite -leqNgt leqn0 => /eqP ->. +rewrite eqxx/=. +case: ifPn; first by rewrite invr_gt0 ltr0n. +rewrite -leqNgt leqn0 => /eqP ->. +by rewrite fact0 mul1r divr1. +Qed. + +Lemma betafun_ge0 (a b : nat) : (0 <= betafun a b :> R)%R. +Proof. exact/ltW/betafun_gt0. Qed. + +Lemma betafun11 : betafun 1 1 = 1%R :> R. +Proof. by rewrite (betafun1S O) invr1. Qed. + +(* NB: this is not exactly betafun because EFin *) +Definition betafunEFin a b : \bar R := \int[mu]_x (XMonemX01 a b x)%:E. + +(* TODO: rev eq *) +Lemma betafunEFinT a b : + (betafunEFin a b = \int[mu]_(x in `[0%R, 1%R]) (XMonemX01 a b x)%:E)%E. +Proof. by rewrite /betafunEFin integral_XMonemX01/= setTI. Qed. + +Lemma betafunEFin_lty a b : (betafunEFin a b < +oo)%E. +Proof. +have := betafun_gt0 a b; rewrite /betafun /Rintegral /betafunEFin. +by case: (integral _ _ _) => [r _| |//]; rewrite ?ltxx ?ltry. +Qed. + +Lemma betafunEFin_fin_num a b : betafunEFin a b \is a fin_num. +Proof. +rewrite ge0_fin_numE ?betafunEFin_lty//. +by apply: integral_ge0 => //= x _; rewrite lee_fin XMonemX01_ge0. +Qed. + +Lemma betafunEFinE a b : (betafunEFin a b = (betafun a b)%:E :> \bar R)%E. +Proof. by rewrite -[LHS]fineK ?betafunEFin_fin_num. Qed. + +Lemma integrable_XMonemX01 a b : mu.-integrable setT (EFin \o XMonemX01 a b). +Proof. +apply/integrableP; split. + by apply/measurable_EFinP; exact: measurable_XMonemX01. +under eq_integral. + move=> /= x _. + rewrite ger0_norm//; last by rewrite XMonemX01_ge0. + over. +exact: betafunEFin_lty. +Qed. + +End betafun. + +(* normalized pdf for the beta distribution *) +Section beta_pdf. +Local Open Scope ring_scope. +Context {R : realType}. +Variables a b : nat. + +Definition beta_pdf t : R := XMonemX01 a b t / betafun a b. + +Lemma measurable_beta_pdf : measurable_fun setT beta_pdf. +Proof. by apply: measurable_funM => //; exact: measurable_XMonemX01. Qed. + +Lemma beta_pdf_ge0 t : 0 <= beta_pdf t. +Proof. +by rewrite /beta_pdf divr_ge0//; [exact: XMonemX01_ge0|exact: betafun_ge0]. +Qed. + +Lemma beta_pdf_le_betafunV x : beta_pdf x <= (betafun a b)^-1. +Proof. +rewrite /beta_pdf ler_pdivrMr ?betafun_gt0// mulVf ?gt_eqF ?betafun_gt0//. +exact: XMonemX01_le1. +Qed. + +Local Notation mu := lebesgue_measure. + +(* TODO: really useful? *) +Lemma int_beta_pdf01 : + (\int[mu]_(x in `[0%R, 1%R]) (beta_pdf x)%:E = + \int[mu]_x (beta_pdf x)%:E :> \bar R)%E. +Proof. +rewrite /beta_pdf. +under eq_integral do rewrite EFinM. +rewrite /=. +rewrite ge0_integralZr//=; last 3 first. + apply: measurable_funTS => /=; apply/measurable_EFinP => //. + exact: measurable_XMonemX01. + by move=> x _; rewrite lee_fin XMonemX01_ge0. + by rewrite lee_fin invr_ge0// betafun_ge0. +rewrite -betafunEFinT -ge0_integralZr//=. +- by apply/measurableT_comp => //; exact: measurable_XMonemX01. +- by move=> x _; rewrite lee_fin XMonemX01_ge0. +- by rewrite lee_fin invr_ge0// betafun_ge0. +Qed. + +Lemma integrable_beta_pdf : mu.-integrable setT (EFin \o beta_pdf). +Proof. +apply/integrableP; split. + by apply/measurable_EFinP; exact: measurable_beta_pdf. +under eq_integral. + move=> /= x _. + rewrite ger0_norm//; last by rewrite beta_pdf_ge0. + over. +rewrite -int_beta_pdf01. +apply: (@le_lt_trans _ _ (\int[mu]_(x in `[0%R, 1%R]) (betafun a b)^-1%:E)%E). + apply: ge0_le_integral => //=. + - by move=> x _; rewrite lee_fin beta_pdf_ge0. + - by apply/measurable_funTS/measurable_EFinP => /=; exact: measurable_beta_pdf. + - by move=> x _; rewrite lee_fin invr_ge0// betafun_ge0. + - by move=> x _; rewrite lee_fin beta_pdf_le_betafunV. +rewrite integral_cst//= lebesgue_measure_itv//=. +by rewrite lte01 oppr0 adde0 mule1 ltry. +Qed. + +Local Open Scope ring_scope. +Lemma bounded_beta_pdf_01 : + [bounded beta_pdf x : R^o | x in `[0%R, 1%R]%classic : set R]. +Proof. +exists (betafun a b)^-1; split; first by rewrite num_real. +move=> // y y1. +near=> M => /=. +rewrite (le_trans _ (ltW y1))//. +near: M. +move=> M /=. +rewrite in_itv/= => /andP[M0 M1]. +rewrite ler_norml; apply/andP; split. + rewrite lerNl (@le_trans _ _ 0%R)// ?invr_ge0 ?betafun_ge0//. + by rewrite lerNl oppr0 beta_pdf_ge0. +rewrite /beta_pdf ler_pdivrMr ?betafun_gt0//. +by rewrite mulVf ?XMonemX01_le1// gt_eqF// betafun_gt0. +Unshelve. all: by end_near. Qed. +Local Close Scope ring_scope. + +End beta_pdf. + +Section beta. +Local Open Scope ring_scope. +Context {R : realType}. +Variables a b : nat. + +Local Notation mu := (@lebesgue_measure R). + +Let beta_num (U : set (measurableTypeR R)) : \bar R := + \int[mu]_(x in U) (XMonemX01 a b x)%:E. + +Let beta_numT : beta_num setT = betafunEFin a b. +Proof. by rewrite /beta_num/= -/(betafunEFin a b) betafunEFinE. Qed. + +Let beta_num_lty U : measurable U -> (beta_num U < +oo)%E. +Proof. +move=> mU. +apply: (@le_lt_trans _ _ (\int[mu]_(x in U `&` `[0%R, 1%R]%classic) 1)%E); last first. + rewrite integral_cst//= ?mul1e//. + rewrite (le_lt_trans (measureIr _ _ _))//= lebesgue_measure_itv//= lte01//. + by rewrite EFinN sube0 ltry. + exact: measurableI. +rewrite /beta_num integral_XMonemX01 ge0_le_integral//=. +- exact: measurableI. +- by move=> x _; rewrite lee_fin XMonemX01_ge0. +- by apply/measurable_funTS/measurableT_comp => //; exact: measurable_XMonemX01. +- by move=> x _; rewrite lee_fin XMonemX01_le1. +Qed. + +Let beta_num0 : beta_num set0 = 0%:E. +Proof. by rewrite /beta_num integral_set0. Qed. + +Let beta_num_ge0 U : (0 <= beta_num U)%E. +Proof. +by rewrite /beta_num integral_ge0//= => x Ux; rewrite lee_fin XMonemX01_ge0. +Qed. + +Let beta_num_sigma_additive : semi_sigma_additive beta_num. +Proof. +move=> /= F mF tF mUF; rewrite /beta_num; apply: cvg_toP. + apply: ereal_nondecreasing_is_cvgn => m n mn. + apply: lee_sum_nneg_natr => // k _ _; apply: integral_ge0 => /= x Fkx. + by rewrite lee_fin; exact: XMonemX01_ge0. +rewrite ge0_integral_bigcup//=. +- by apply/measurable_funTS/measurableT_comp => //; exact: measurable_XMonemX01. +- by move=> x _; rewrite lee_fin XMonemX01_ge0. +Qed. + +HB.instance Definition _ := isMeasure.Build _ _ _ beta_num + beta_num0 beta_num_ge0 beta_num_sigma_additive. + +Definition beta_prob := + @mscale _ _ _ (invr_nonneg (NngNum (betafun_ge0 a b))) beta_num. + +HB.instance Definition _ := Measure.on beta_prob. + +Let beta_prob_setT : beta_prob setT = 1%:E. +Proof. +rewrite /beta_prob /= /mscale /= beta_numT betafunEFinE//. +by rewrite -EFinM mulVf// gt_eqF// betafun_gt0. +Qed. + +HB.instance Definition _ := + @Measure_isProbability.Build _ _ _ beta_prob beta_prob_setT. + +Lemma beta_prob01 : beta_prob `[0, 1] = 1%:E. +Proof. +rewrite /beta_prob /= /mscale/= /beta_num -betafunEFinT betafunEFinE//. +by rewrite -EFinM mulVf// gt_eqF// betafun_gt0. +Qed. + +Lemma beta_prob_fin_num U : measurable U -> beta_prob U \is a fin_num. +Proof. +move=> mU; rewrite ge0_fin_numE//. +rewrite /beta_prob/= /mscale/= /beta_num lte_mul_pinfty//. + by rewrite lee_fin// invr_ge0 betafun_ge0. +apply: (@le_lt_trans _ _ (betafunEFin a b)). + apply: ge0_subset_integral => //=. + by apply/measurable_EFinP; exact: measurable_XMonemX01. + by move=> x _; rewrite lee_fin XMonemX01_ge0. +by rewrite betafunEFin_lty. +Qed. + +Lemma integral_beta_pdf U : measurable U -> + (\int[mu]_(x in U) (beta_pdf a b x)%:E = beta_prob U :> \bar R)%E. +Proof. +move=> mU. +rewrite /beta_pdf. +under eq_integral do rewrite EFinM/=. +rewrite ge0_integralZr//=. +- by rewrite /beta_prob/= /mscale/= muleC. +- by apply/measurable_funTS/measurableT_comp => //; exact: measurable_XMonemX01. +- by move=> x _; rewrite lee_fin XMonemX01_ge0. +- by rewrite lee_fin invr_ge0// betafun_ge0. +Qed. + +End beta. +Arguments beta_prob {R}. + +Lemma integral_beta_prob_bernoulli_lty {R : realType} a b (f : R -> R) U : + measurable_fun setT f -> + (forall x, x \in `[0%R, 1%R] -> 0 <= f x <= 1)%R -> + (\int[beta_prob a b]_x `|bernoulli (f x) U| < +oo :> \bar R)%E. +Proof. +move=> mf /= f01. +apply: (@le_lt_trans _ _ (\int[beta_prob a b]_x cst 1 x))%E. + apply: ge0_le_integral => //=. + apply: measurableT_comp => //=. + by apply: (measurableT_comp (measurable_bernoulli2 _)). + by move=> x _; rewrite gee0_abs// probability_le1. +by rewrite integral_cst//= mul1e -ge0_fin_numE// beta_prob_fin_num. +Qed. + +Lemma integral_beta_prob_bernoulli_onemX_lty {R : realType} n a b U : + (\int[beta_prob a b]_x `|bernoulli (`1-x ^+ n) U| < +oo :> \bar R)%E. +Proof. +apply: integral_beta_prob_bernoulli_lty => //=. + by apply: measurable_funX => //; exact: measurable_funB. +move=> x; rewrite in_itv/= => /andP[x0 x1]. +rewrite exprn_ge0 ?subr_ge0//= exprn_ile1// ?subr_ge0//. +by rewrite lerBlDl -lerBlDr subrr. +Qed. + +Lemma integral_beta_prob_bernoulli_onem_lty {R : realType} n a b U : + (\int[beta_prob a b]_x `|bernoulli (1 - `1-x ^+ n) U| < +oo :> \bar R)%E. +Proof. +apply: integral_beta_prob_bernoulli_lty => //=. + apply: measurable_funB => //. + by apply: measurable_funX => //; exact: measurable_funB. +move=> x; rewrite in_itv/= => /andP[x0 x1]. +rewrite -lerBlDr opprK add0r. +rewrite andbC lerBlDl -lerBlDr subrr. +rewrite exprn_ge0 ?subr_ge0//= exprn_ile1// ?subr_ge0//. +by rewrite lerBlDl -lerBlDr subrr. +Qed. + +Local Open Scope ring_scope. + +Lemma beta_prob_uniform {R : realType} : beta_prob 1 1 = uniform_prob (@ltr01 R). +Proof. +apply/funext => U. +rewrite /beta_prob /uniform_prob. +rewrite /mscale/= betafun11 invr1 !mul1e. +rewrite integral_XMonemX01 integral_uniform_pdf. +under eq_integral. + move=> /= x. + rewrite inE => -[Ux/=]; rewrite in_itv/= => x10. + rewrite XMonemX01_11//=. + over. +rewrite /=. +under [RHS]eq_integral. + move=> /= x. + rewrite inE => -[Ux/=]; rewrite in_itv/= => x10. + rewrite /uniform_pdf x10 subr0 invr1. + over. +by []. +Qed. + +Lemma beta_prob_integrable {R :realType} a b a' b' : + (beta_prob a b).-integrable `[0, 1] + (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => (XMonemX a' b' x)%:E). +Proof. +apply/integrableP; split. + by apply/measurableT_comp => //; exact: measurable_fun_XMonemX. +apply: (@le_lt_trans _ _ (\int[beta_prob a b]_(x in `[0%R, 1%R]) 1)%E). + apply: ge0_le_integral => //=. + do 2 apply/measurableT_comp => //. + exact: measurable_fun_XMonemX. + move=> x; rewrite in_itv/= => /andP[x0 x1]. + rewrite lee_fin ger0_norm; last first. + by rewrite !mulr_ge0// exprn_ge0// onem_ge0. + by rewrite mulr_ile1// ?exprn_ge0 ?onem_ge0// exprn_ile1// ?onem_ge0// onem_le1. +rewrite integral_cst//= mul1e. +by rewrite -ge0_fin_numE// beta_prob_fin_num. +Qed. + +Lemma beta_prob_integrable_onem {R : realType} a b a' b' : + (beta_prob a b).-integrable `[0, 1] + (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => `1-(XMonemX a' b' x)%:E). +Proof. +apply: (eq_integrable _ (cst 1 \- (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => + (XMonemX a' b' x)%:E))%E) => //. +apply: (@integrableB _ (g_sigma_algebraType R.-ocitv.-measurable)) => //=. + (* TODO: lemma? *) + apply/integrableP; split => //. + rewrite (eq_integral (fun x => (\1_setT x)%:E))/=; last first. + by move=> x _; rewrite /= indicT normr1. + rewrite integral_indic//= setTI /beta_prob /mscale/= lte_mul_pinfty//. + by rewrite lee_fin invr_ge0 betafun_ge0. + have /integrableP[_] := @integrable_XMonemX01 R a b. + under eq_integral. + move=> x _. + rewrite gee0_abs//; last first. + by rewrite lee_fin XMonemX01_ge0. + over. + by rewrite /= -/(betafunEFin a b) /= betafunEFinT. +exact: beta_prob_integrable. +Qed. + +Lemma beta_prob_integrable_dirac {R : realType} a b a' b' (c : bool) U : + (beta_prob a b).-integrable `[0, 1] + (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => (XMonemX a' b' x)%:E * \d_c U)%E. +Proof. +apply: integrableMl => //=; last first. + exists 1; split => // x x1/= _ _; rewrite (le_trans _ (ltW x1))//. + by rewrite ger0_norm// indicE; case: (_ \in _). +exact: beta_prob_integrable. +Qed. + +Lemma beta_prob_integrable_onem_dirac {R : realType} a b a' b' (c : bool) U : + (beta_prob a b).-integrable `[0, 1] + (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => `1-(XMonemX a' b' x)%:E * \d_c U)%E. +Proof. +apply: integrableMl => //=; last first. + exists 1; split => // x x1/= _ _; rewrite (le_trans _ (ltW x1))//. + by rewrite ger0_norm// indicE; case: (_ \in _). +exact: beta_prob_integrable_onem. +Qed. + +Local Close Scope ring_scope. + +Section integral_beta. +Context {R : realType}. +Local Notation mu := lebesgue_measure. + +Lemma beta_prob_dom a b : @beta_prob R a b `<< mu. +Proof. +move=> A mA muA0; rewrite /beta_prob /mscale/=. +apply/eqP; rewrite mule_eq0 eqe invr_eq0 gt_eqF/= ?betafun_gt0//; apply/eqP. +rewrite integral_XMonemX01; apply/eqP; rewrite eq_le; apply/andP; split; last first. + by apply: integral_ge0 => x _; rewrite lee_fin XMonemX01_ge0. +apply: (@le_trans _ _ (\int[mu]_(x in A `&` `[0%R, 1%R]%classic) 1)); last first. + rewrite integral_cst ?mul1e//=; last exact: measurableI. + by rewrite -[leRHS]muA0 measureIl. +apply: ge0_le_integral => //=; first exact: measurableI. +- by move=> x _; rewrite lee_fin XMonemX01_ge0. +- apply/measurable_funTS/measurableT_comp => //. + exact: measurable_XMonemX01. +- by move=> x _; rewrite lee_fin XMonemX01_le1. +Qed. + +Section beta_pdf_Beta. + +Local Open Scope charge_scope. + +(* beta_pdf is almost density function of Beta *) +Lemma beta_pdf_uniq_ae (a b : nat) : + ae_eq mu `[0%R, 1%R]%classic + ('d ((charge_of_finite_measure (@beta_prob R a b))) '/d mu) + (EFin \o (beta_pdf a b)). +Proof. +apply: integral_ae_eq => //. +- apply: (@integrableS _ _ _ _ setT) => //=. + apply: Radon_Nikodym_integrable => //=. + exact: beta_prob_dom. +- apply/measurable_funTS/measurableT_comp => //. + exact: measurable_beta_pdf. +- move=> E E01 mE. + rewrite integral_beta_pdf//. + apply/esym. + rewrite -Radon_Nikodym_integral//=. + exact: beta_prob_dom. +Qed. + +(* need to add lemma about radon-nikodym derivative of + lebesgue_stieltjes measure w.r.t. continuous density function *) +(*Lemma beta_pdf_uniq (a b : nat) : + {in `[0%R, 1%R]%classic, + ('d ((charge_of_finite_measure (@Beta R a b))) '/d mu) =1 + (EFin \o (beta_pdf a b))}. +Proof. Abort.*) + +End beta_pdf_Beta. + +Lemma integral_Beta a b f U : measurable U -> measurable_fun U f -> + \int[beta_prob a b]_(x in U) `|f x| < +oo -> + \int[beta_prob a b]_(x in U) f x = \int[mu]_(x in U) (f x * (beta_pdf a b x)%:E) :> \bar R. +Proof. +move=> mU mf finf. +rewrite -(Radon_Nikodym_change_of_variables (beta_prob_dom a b)) //=; last first. + by apply/integrableP; split. +apply: ae_eq_integral => //. +- apply: emeasurable_funM => //; apply: (measurable_int mu). + apply: (integrableS _ _ (@subsetT _ _)) => //=. + by apply: Radon_Nikodym_integrable; exact: beta_prob_dom. +- apply: emeasurable_funM => //=; apply/measurableT_comp => //=. + by apply/measurable_funTS; exact: measurable_beta_pdf. +- apply: ae_eq_mul2l => /=. + rewrite Radon_NikodymE//=; first exact: beta_prob_dom. + move=> ?. + case: cid => /= h [h1 h2 h3]. +(* uniqueness of Radon-Nikodym derivertive up to equal on non null sets of mu *) + apply: integral_ae_eq => //. + + apply: integrableS h2 => //. (* integrableST? *) + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_beta_pdf. + + by move=> E E01 mE; rewrite -h3//= integral_beta_pdf. +Qed. + +End integral_beta. + +Section beta_prob_bernoulliE. +Context {R : realType}. +Local Notation mu := lebesgue_measure. +Local Open Scope ring_scope. + +Definition div_betafun a b c d : R := betafun (a + c) (b + d) / betafun a b. + +Lemma div_betafun_ge0 a b c d : 0 <= div_betafun a b c d. +Proof. by rewrite /div_betafun divr_ge0// betafun_ge0. Qed. + +Lemma div_betafun_le1 a b c d : (0 < a)%N -> (0 < b)%N -> + div_betafun a b c d <= 1. +Proof. +move=> a0 b0. +rewrite /div_betafun ler_pdivrMr// ?mul1r ?betafun_gt0//. +rewrite !betafunE. +rewrite addn_eq0 (gtn_eqF a0)/=. +rewrite addn_eq0 (gtn_eqF b0)/=. +rewrite ler_pdivrMr ?ltr0n ?fact_gt0//. +rewrite mulrAC. +rewrite ler_pdivlMr ?ltr0n ?fact_gt0//. +rewrite -!natrM ler_nat. +case: a a0 => //. +move=> n. +rewrite addSn. +case: b b0 => //. +move=> m. +rewrite [(n + c).+1.-1]/=. +rewrite [n.+1.-1]/=. +rewrite [m.+1.-1]/=. +rewrite addnS. +rewrite [(_ + m).+1.-1]/=. +rewrite (addSn m d). +rewrite [(m + _).+1.-1]/=. +rewrite (addSn (n + c)). +rewrite [_.+1.-1]/=. +rewrite addSn addnS. +by rewrite leq_fact2// leq_addr. +Qed. + +Definition beta_prob_bernoulli a b c d U : \bar R := + \int[beta_prob a b]_(y in `[0, 1]) bernoulli (XMonemX01 c.+1 d.+1 y) U. + +Lemma beta_prob_bernoulliE a b c d U : (a > 0)%N -> (b > 0)%N -> + beta_prob_bernoulli a b c d U = bernoulli (div_betafun a b c d) U. +Proof. +move=> a0 b0. +rewrite /beta_prob_bernoulli. +under eq_integral => x. + rewrite inE/= in_itv/= => x01. + rewrite bernoulliE/= ?XMonemX01_ge0 ?XMonemX01_le1//. + over. +rewrite /=. +rewrite [in RHS]bernoulliE/= ?div_betafun_ge0 ?div_betafun_le1//=. +under eq_integral => x x01. + rewrite /XMonemX01 patchE x01/=. + over. +rewrite /=. +rewrite integralD//=; last 2 first. + exact: beta_prob_integrable_dirac. + exact: beta_prob_integrable_onem_dirac. +congr (_ + _). + rewrite integralZr//=; last exact: beta_prob_integrable. + congr (_ * _)%E. + rewrite integral_Beta//; last 2 first. + by apply/measurableT_comp => //; exact: measurable_fun_XMonemX. + by have /integrableP[_] := @beta_prob_integrable R a b c d. + rewrite /beta_pdf. + under eq_integral do rewrite EFinM -muleA muleC -muleA. + rewrite /=. + transitivity ((betafun a b)^-1%:E * + \int[mu]_(x in `[0%R, 1%R]) (XMonemX01 (a + c) (b + d) x)%:E : \bar R)%E. + rewrite -integralZl//=; last first. + apply/integrableP; split. + apply/measurable_EFinP/measurable_funTS. + exact: measurable_XMonemX01. + under eq_integral. + move=> x x01. + rewrite gee0_abs; last by rewrite lee_fin XMonemX01_ge0. + over. + by rewrite /= -betafunEFinT betafunEFin_lty. + apply: eq_integral => x x01. + (* TODO: lemma? property of XMonemX? *) + rewrite muleA muleC muleA -(EFinM (x ^+ c)). + rewrite -/(XMonemX c d x) -EFinM mulrA XMonemX_XMonemX01//. + by rewrite -EFinM mulrC. + by rewrite -betafunEFinT betafunEFinE -EFinM mulrC. +under eq_integral do rewrite muleC. +rewrite /=. +rewrite integralZl//=; last exact: beta_prob_integrable_onem. +rewrite muleC; congr (_ * _)%E. +rewrite integral_Beta//=; last 2 first. + apply/measurableT_comp => //=. + by apply/measurable_funB => //; exact: measurable_fun_XMonemX. + by have /integrableP[] := @beta_prob_integrable_onem R a b c d. +rewrite /beta_pdf. +under eq_integral do rewrite EFinM muleA. +rewrite integralZr//=; last first. + apply: integrableMr => //=. + - by apply/measurable_funB => //=; exact: measurable_fun_XMonemX. + - apply/ex_bound => //. + + apply: (@globally_properfilter _ _ 0%R) => //=. + by apply: inferP; rewrite in_itv/= lexx ler01. + + exists 1 => t. + rewrite /= in_itv/= => t01. + apply: normr_onem; apply/andP; split. + by rewrite mulr_ge0// exprn_ge0// ?onem_ge0//; case/andP: t01. + by rewrite mulr_ile1// ?exprn_ge0 ?exprn_ile1// ?onem_ge0 ?onem_le1//; case/andP: t01. + - exact: integrableS (integrable_XMonemX01 _ _). +transitivity ( + (\int[mu]_(x in `[0%R, 1%R]) + ((XMonemX01 a b x)%:E - (XMonemX01 (a + c) (b + d) x)%:E) : \bar R) + * (betafun a b)^-1%:E)%E. + congr (_ * _)%E. + apply: eq_integral => x x01. + rewrite /onem -EFinM mulrBl mul1r EFinB; congr (_ - _)%E. + by rewrite XMonemX_XMonemX01. +rewrite integralB_EFin//=; last 2 first. + exact: integrableS (integrable_XMonemX01 _ _). + exact: integrableS (integrable_XMonemX01 _ _). +rewrite -!betafunEFinT !betafunEFinE. +rewrite -EFinM. +rewrite mulrBl /onem mulfV; last by rewrite gt_eqF// betafun_gt0. +by []. +Qed. + +End beta_prob_bernoulliE. + +Declare Scope lang_scope. +Delimit Scope lang_scope with P. + +Section syntax_of_types. +Import Notations. +Context {R : realType}. + +Inductive typ := +| Unit | Bool | Nat | Real +| Pair : typ -> typ -> typ +| Prob : typ -> typ. + +HB.instance Definition _ := gen_eqMixin typ. + +Fixpoint measurable_of_typ (t : typ) : {d & measurableType d} := + match t with + | Unit => existT _ _ munit + | Bool => existT _ _ mbool + | Nat => existT _ _ (nat : measurableType _) + | Real => existT _ _ + [the measurableType _ of (@measurableTypeR R)] + (* (Real_sort__canonical__measure_Measurable R) *) + | Pair A B => existT _ _ + [the measurableType (projT1 (measurable_of_typ A), + projT1 (measurable_of_typ B)).-prod%mdisp of + (projT2 (measurable_of_typ A) * + projT2 (measurable_of_typ B))%type] + | Prob A => existT _ _ (pprobability (projT2 (measurable_of_typ A)) R) + end. + +Definition mtyp_disp t : measure_display := projT1 (measurable_of_typ t). + +Definition mtyp t : measurableType (mtyp_disp t) := + projT2 (measurable_of_typ t). + +Definition measurable_of_seq (l : seq typ) : {d & measurableType d} := + iter_mprod (List.map measurable_of_typ l). + +End syntax_of_types. +Arguments measurable_of_typ {R}. +Arguments mtyp {R}. +Arguments measurable_of_seq {R}. + +Section accessor_functions. +Context {R : realType}. + +(* NB: almost the same as acc (map (@measurable_of_typ R) s) n l, + modulo commutativity of map and measurable_of_typ *) +Fixpoint acc_typ (s : seq typ) n : + projT2 (@measurable_of_seq R s) -> + projT2 (measurable_of_typ (nth Unit s n)) := + match s return + projT2 (measurable_of_seq s) -> projT2 (measurable_of_typ (nth Unit s n)) + with + | [::] => match n with | 0 => (fun=> tt) | m.+1 => (fun=> tt) end + | a :: l => match n with + | 0 => fst + | m.+1 => fun H => @acc_typ l m H.2 + end + end. + +(*Definition acc_typ : forall (s : seq typ) n, + projT2 (@measurable_of_seq R s) -> + projT2 (@measurable_of_typ R (nth Unit s n)). +fix H 1. +intros s n x. +destruct s as [|s]. + destruct n as [|n]. + exact tt. + exact tt. +destruct n as [|n]. + exact (fst x). +rewrite /=. +apply H. +exact: (snd x). +Show Proof. +Defined.*) + +Lemma measurable_acc_typ (s : seq typ) n : measurable_fun setT (@acc_typ s n). +Proof. +elim: s n => //= h t ih [|m]; first exact: measurable_fst. +by apply: (measurableT_comp (ih _)); exact: measurable_snd. +Qed. + +End accessor_functions. +Arguments acc_typ {R} s n. +Arguments measurable_acc_typ {R} s n. + +Section context. +Variables (R : realType). +Definition ctx := seq (string * typ). + +Definition mctx_disp (g : ctx) := projT1 (@measurable_of_seq R (map snd g)). + +Definition mctx (g : ctx) : measurableType (mctx_disp g) := + projT2 (@measurable_of_seq R (map snd g)). + +End context. +Arguments mctx {R}. + +Section syntax_of_expressions. +Context {R : realType}. + +Inductive flag := D | P. + +Section binop. + +Inductive binop := +| binop_and | binop_or +| binop_add | binop_minus | binop_mult. + +Definition type_of_binop (b : binop) : typ := +match b with +| binop_and => Bool +| binop_or => Bool +| binop_add => Real +| binop_minus => Real +| binop_mult => Real +end. + +Definition fun_of_binop g (b : binop) : (mctx g -> mtyp (type_of_binop b)) -> + (mctx g -> mtyp (type_of_binop b)) -> @mctx R g -> @mtyp R (type_of_binop b) := +match b with +| binop_and => (fun f1 f2 x => f1 x && f2 x : mtyp Bool) +| binop_or => (fun f1 f2 x => f1 x || f2 x : mtyp Bool) +| binop_add => (fun f1 f2 => (f1 \+ f2)%R) +| binop_minus => (fun f1 f2 => (f1 \- f2)%R) +| binop_mult => (fun f1 f2 => (f1 \* f2)%R) +end. + +Definition mfun_of_binop g b + (f1 : @mctx R g -> @mtyp R (type_of_binop b)) (mf1 : measurable_fun setT f1) + (f2 : @mctx R g -> @mtyp R (type_of_binop b)) (mf2 : measurable_fun setT f2) : + measurable_fun [set: @mctx R g] (fun_of_binop f1 f2). +destruct b. +exact: measurable_and mf1 mf2. +exact: measurable_or mf1 mf2. +exact: measurable_funD. +exact: measurable_funB. +exact: measurable_funM. +Defined. + +End binop. + +Section relop. +Inductive relop := +| relop_le | relop_lt | relop_eq . + +Definition fun_of_relop g (r : relop) : (@mctx R g -> @mtyp R Nat) -> + (mctx g -> mtyp Nat) -> @mctx R g -> @mtyp R Bool := +match r with +| relop_le => (fun f1 f2 x => (f1 x <= f2 x)%N) +| relop_lt => (fun f1 f2 x => (f1 x < f2 x)%N) +| relop_eq => (fun f1 f2 x => (f1 x == f2 x)%N) +end. + +Definition mfun_of_relop g r + (f1 : @mctx R g -> @mtyp R Nat) (mf1 : measurable_fun setT f1) + (f2 : @mctx R g -> @mtyp R Nat) (mf2 : measurable_fun setT f2) : + measurable_fun [set: @mctx R g] (fun_of_relop r f1 f2). +destruct r. +exact: measurable_fun_leq. +exact: measurable_fun_ltn. +exact: measurable_fun_eqn. +Defined. + +End relop. + +Inductive exp : flag -> ctx -> typ -> Type := +| exp_unit g : exp D g Unit +| exp_bool g : bool -> exp D g Bool +| exp_nat g : nat -> exp D g Nat +| exp_real g : R -> exp D g Real +| exp_pow g : nat -> exp D g Real -> exp D g Real +| exp_bin (b : binop) g : exp D g (type_of_binop b) -> + exp D g (type_of_binop b) -> exp D g (type_of_binop b) +| exp_rel (r : relop) g : exp D g Nat -> + exp D g Nat -> exp D g Bool +| exp_pair g t1 t2 : exp D g t1 -> exp D g t2 -> exp D g (Pair t1 t2) +| exp_proj1 g t1 t2 : exp D g (Pair t1 t2) -> exp D g t1 +| exp_proj2 g t1 t2 : exp D g (Pair t1 t2) -> exp D g t2 +| exp_var g str t : t = lookup Unit g str -> exp D g t +| exp_bernoulli g : exp D g Real -> exp D g (Prob Bool) +| exp_binomial g (n : nat) : exp D g Real -> exp D g (Prob Nat) +| exp_uniform g (a b : R) (ab : (a < b)%R) : exp D g (Prob Real) +| exp_beta g (a b : nat) (* NB: should be R *) : exp D g (Prob Real) +| exp_poisson g : nat -> exp D g Real -> exp D g Real +| exp_normalize g t : exp P g t -> exp D g (Prob t) +| exp_letin g t1 t2 str : exp P g t1 -> exp P ((str, t1) :: g) t2 -> + exp P g t2 +| exp_sample g t : exp D g (Prob t) -> exp P g t +| exp_score g : exp D g Real -> exp P g Unit +| exp_return g t : exp D g t -> exp P g t +| exp_if z g t : exp D g Bool -> exp z g t -> exp z g t -> exp z g t +| exp_weak z g h t x : exp z (g ++ h) t -> + x.1 \notin dom (g ++ h) -> exp z (g ++ x :: h) t. +Arguments exp_var {g} _ {t}. + +Definition exp_var' (str : string) (t : typ) (g : find str t) := + @exp_var (untag (ctx_of g)) str t (ctx_prf g). +Arguments exp_var' str {t} g. + +Lemma exp_var'E str t (f : find str t) H : + exp_var' str f = exp_var str H :> (@exp _ _ _). +Proof. by rewrite /exp_var'; congr exp_var. Qed. + +End syntax_of_expressions. +Arguments exp {R}. +Arguments exp_unit {R g}. +Arguments exp_bool {R g}. +Arguments exp_nat {R g}. +Arguments exp_real {R g}. +Arguments exp_pow {R g}. +Arguments exp_bin {R} b {g} &. +Arguments exp_rel {R} r {g} &. +Arguments exp_pair {R g} & {t1 t2}. +Arguments exp_var {R g} _ {t} & H. +Arguments exp_bernoulli {R g} &. +Arguments exp_binomial {R g} &. +Arguments exp_uniform {R g} &. +Arguments exp_beta {R g} &. +Arguments exp_poisson {R g}. +Arguments exp_normalize {R g _}. +Arguments exp_letin {R g} & {_ _}. +Arguments exp_sample {R g} & {t}. +Arguments exp_score {R g}. +Arguments exp_return {R g} & {_}. +Arguments exp_if {R z g t} &. +Arguments exp_weak {R} z g h {t} x. +Arguments exp_var' {R} str {t} g &. + +Declare Custom Entry expr. +Notation "[ e ]" := e (e custom expr at level 5) : lang_scope. +Notation "'TT'" := (exp_unit) (in custom expr at level 1) : lang_scope. +Notation "b ':B'" := (@exp_bool _ _ b%bool) + (in custom expr at level 1) : lang_scope. +Notation "n ':N'" := (@exp_nat _ _ n%N) + (in custom expr at level 1) : lang_scope. +Notation "r ':R'" := (@exp_real _ _ r%R) + (in custom expr at level 1, format "r :R") : lang_scope. +Notation "e ^+ n" := (exp_pow n e) + (in custom expr at level 1) : lang_scope. +Notation "e1 && e2" := (exp_bin binop_and e1 e2) + (in custom expr at level 2) : lang_scope. +Notation "e1 || e2" := (exp_bin binop_or e1 e2) + (in custom expr at level 2) : lang_scope. +Notation "e1 + e2" := (exp_bin binop_add e1 e2) + (in custom expr at level 3) : lang_scope. +Notation "e1 - e2" := (exp_bin binop_minus e1 e2) + (in custom expr at level 3) : lang_scope. +Notation "e1 * e2" := (exp_bin binop_mult e1 e2) + (in custom expr at level 2) : lang_scope. +Notation "e1 <= e2" := (exp_rel relop_le e1 e2) + (in custom expr at level 2) : lang_scope. +Notation "e1 == e2" := (exp_rel relop_eq e1 e2) + (in custom expr at level 4) : lang_scope. +Notation "'return' e" := (@exp_return _ _ _ e) + (in custom expr at level 6) : lang_scope. +(*Notation "% str" := (@exp_var _ _ str%string _ erefl) + (in custom expr at level 1, format "% str") : lang_scope.*) +(* Notation "% str H" := (@exp_var _ _ str%string _ H) + (in custom expr at level 1, format "% str H") : lang_scope. *) +Notation "# str" := (@exp_var' _ str%string _ _) + (in custom expr at level 1, format "# str"). +Notation "e :+ str" := (exp_weak _ [::] _ (str, _) e erefl) + (in custom expr at level 1) : lang_scope. +Notation "( e1 , e2 )" := (exp_pair e1 e2) + (in custom expr at level 1) : lang_scope. +Notation "\pi_1 e" := (exp_proj1 e) + (in custom expr at level 1) : lang_scope. +Notation "\pi_2 e" := (exp_proj2 e) + (in custom expr at level 1) : lang_scope. +Notation "'let' x ':=' e 'in' f" := (exp_letin x e f) + (in custom expr at level 5, + x constr, + f custom expr at level 5, + left associativity) : lang_scope. +Notation "{ c }" := c (in custom expr, c constr) : lang_scope. +Notation "x" := x + (in custom expr at level 0, x ident) : lang_scope. +Notation "'Sample' e" := (exp_sample e) + (in custom expr at level 5) : lang_scope. +Notation "'Score' e" := (exp_score e) + (in custom expr at level 5) : lang_scope. +Notation "'Normalize' e" := (exp_normalize e) + (in custom expr at level 0) : lang_scope. +Notation "'if' e1 'then' e2 'else' e3" := (exp_if e1 e2 e3) + (in custom expr at level 6) : lang_scope. + +Section free_vars. +Context {R : realType}. + +Fixpoint free_vars k g t (e : @exp R k g t) : seq string := + match e with + | exp_unit _ => [::] + | exp_bool _ _ => [::] + | exp_nat _ _ => [::] + | exp_real _ _ => [::] + | exp_pow _ _ e => free_vars e + | exp_bin _ _ e1 e2 => free_vars e1 ++ free_vars e2 + | exp_rel _ _ e1 e2 => free_vars e1 ++ free_vars e2 + | exp_pair _ _ _ e1 e2 => free_vars e1 ++ free_vars e2 + | exp_proj1 _ _ _ e => free_vars e + | exp_proj2 _ _ _ e => free_vars e + | exp_var _ x _ _ => [:: x] + | exp_bernoulli _ e => free_vars e + | exp_binomial _ _ e => free_vars e + | exp_uniform _ _ _ _ => [::] + | exp_beta _ _ _ => [::] + | exp_poisson _ _ e => free_vars e + | exp_normalize _ _ e => free_vars e + | exp_letin _ _ _ x e1 e2 => free_vars e1 ++ rem x (free_vars e2) + | exp_sample _ _ _ => [::] + | exp_score _ e => free_vars e + | exp_return _ _ e => free_vars e + | exp_if _ _ _ e1 e2 e3 => free_vars e1 ++ free_vars e2 ++ free_vars e3 + | exp_weak _ _ _ _ x e _ => rem x.1 (free_vars e) + end. + +End free_vars. + +Definition dval R g t := @mctx R g -> @mtyp R t. +Definition pval R g t := R.-sfker @mctx R g ~> @mtyp R t. + + +Section weak. +Context {R : realType}. +Implicit Types (g h : ctx) (x : string * typ). +Local Open Scope ring_scope. + +Fixpoint mctx_strong g h x (f : @mctx R (g ++ x :: h)) : @mctx R (g ++ h) := + match g as g0 return mctx (g0 ++ x :: h) -> mctx (g0 ++ h) with + | [::] => fun f0 : mctx ([::] ++ x :: h) => let (a, b) := f0 in (fun=> id) a b + | a :: t => uncurry (fun a b => (a, @mctx_strong t h x b)) + end f. + +Definition weak g h x t (f : dval R (g ++ h) t) : dval R (g ++ x :: h) t := + f \o @mctx_strong g h x. + +Lemma measurable_fun_mctx_strong g h x : + measurable_fun setT (@mctx_strong g h x). +Proof. +elim: g h x => [h x|x g ih h x0]; first exact: measurable_snd. +apply/prod_measurable_funP; split. +- rewrite [X in measurable_fun _ X](_ : _ = fst)//. + by apply/funext => -[]. +- rewrite [X in measurable_fun _ X](_ : _ = @mctx_strong g h x0 \o snd). + apply: measurableT_comp; last exact: measurable_snd. + exact: ih. + by apply/funext => -[]. +Qed. + +Lemma measurable_weak g h x t (f : dval R (g ++ h) t) : + measurable_fun setT f -> measurable_fun setT (@weak g h x t f). +Proof. +move=> mf; apply: measurableT_comp; first exact: mf. +exact: measurable_fun_mctx_strong. +Qed. + +Definition kweak g h x t (f : pval R (g ++ h) t) + : @mctx R (g ++ x :: h) -> {measure set @mtyp R t -> \bar R} := + f \o @mctx_strong g h x. + +Section kernel_weak. +Context g h x t (f : pval R (g ++ h) t). + +Let mf U : measurable U -> measurable_fun setT (@kweak g h x t f ^~ U). +Proof. +move=> mU. +rewrite (_ : kweak _ ^~ U = f ^~ U \o @mctx_strong g h x)//. +apply: measurableT_comp => //; first exact: measurable_kernel. +exact: measurable_fun_mctx_strong. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ _ (@kweak g h x t f) mf. +End kernel_weak. + +Section sfkernel_weak. +Context g h (x : string * typ) t (f : pval R (g ++ h) t). + +Let sf : exists2 s : (R.-ker @mctx R (g ++ x :: h) ~> @mtyp R t)^nat, + forall n, measure_fam_uub (s n) & + forall z U, measurable U -> (@kweak g h x t f) z U = kseries s z U . +Proof. +have [s hs] := sfinite_kernel f. +exists (fun n => @kweak g h x t (s n)). + by move=> n; have [M hM] := measure_uub (s n); exists M => x0; exact: hM. +by move=> z U mU; by rewrite /kweak/= hs. +Qed. + +HB.instance Definition _ := + isSFiniteKernel_subdef.Build _ _ _ _ _ (@kweak g h x t f) sf. + +End sfkernel_weak. + +Section fkernel_weak. +Context g h x t (f : R.-fker @mctx R (g ++ h) ~> @mtyp R t). + +Let uub : measure_fam_uub (@kweak g h x t f). +Proof. by have [M hM] := measure_uub f; exists M => x0; exact: hM. Qed. + +HB.instance Definition _ := @Kernel_isFinite.Build _ _ _ _ _ + (@kweak g h x t f) uub. +End fkernel_weak. + +End weak. +Arguments weak {R} g h x {t}. +Arguments measurable_weak {R} g h x {t}. +Arguments kweak {R} g h x {t}. + +Section eval. +Context {R : realType}. +Implicit Type (g : ctx) (str : string). +Local Open Scope lang_scope. + +Local Open Scope ring_scope. + +Inductive evalD : forall g t, exp D g t -> + forall f : dval R g t, measurable_fun setT f -> Prop := +| eval_unit g : ([TT] : exp D g _) -D> cst tt ; ktt + +| eval_bool g b : ([b:B] : exp D g _) -D> cst b ; kb b + +| eval_nat g n : ([n:N] : exp D g _) -D> cst n; kn n + +| eval_real g r : ([r:R] : exp D g _) -D> cst r ; kr r + +| eval_pow g n (e : exp D g _) f mf : e -D> f ; mf -> + [e ^+ {n}] -D> (fun x => f x ^+ n) ; (measurable_funX n mf) + +| eval_bin g bop (e1 : exp D g _) f1 mf1 e2 f2 mf2 : + e1 -D> f1 ; mf1 -> e2 -D> f2 ; mf2 -> + exp_bin bop e1 e2 -D> fun_of_binop f1 f2 ; mfun_of_binop mf1 mf2 + +| eval_rel g rop (e1 : exp D g _) f1 mf1 e2 f2 mf2 : + e1 -D> f1 ; mf1 -> e2 -D> f2 ; mf2 -> + exp_rel rop e1 e2 -D> fun_of_relop rop f1 f2 ; mfun_of_relop rop mf1 mf2 + +| eval_pair g t1 (e1 : exp D g t1) f1 mf1 t2 (e2 : exp D g t2) f2 mf2 : + e1 -D> f1 ; mf1 -> e2 -D> f2 ; mf2 -> + [(e1, e2)] -D> fun x => (f1 x, f2 x) ; measurable_fun_prod mf1 mf2 + +| eval_proj1 g t1 t2 (e : exp D g (Pair t1 t2)) f mf : + e -D> f ; mf -> + [\pi_1 e] -D> fst \o f ; measurableT_comp measurable_fst mf + +| eval_proj2 g t1 t2 (e : exp D g (Pair t1 t2)) f mf : + e -D> f ; mf -> + [\pi_2 e] -D> snd \o f ; measurableT_comp measurable_snd mf + +(* | eval_var g str : let i := index str (dom g) in + [% str] -D> acc_typ (map snd g) i ; measurable_acc_typ (map snd g) i *) + +| eval_var g x H : let i := index x (dom g) in + exp_var x H -D> acc_typ (map snd g) i ; measurable_acc_typ (map snd g) i + +| eval_bernoulli g e r mr : + e -D> r ; mr -> (exp_bernoulli e : exp D g _) -D> bernoulli \o r ; + measurableT_comp measurable_bernoulli mr + +| eval_binomial g n e r mr : + e -D> r ; mr -> (exp_binomial n e : exp D g _) -D> binomial_prob n \o r ; + measurableT_comp (measurable_binomial_prob n) mr + +| eval_uniform g (a b : R) (ab : (a < b)%R) : + (exp_uniform a b ab : exp D g _) -D> cst (uniform_prob ab) ; + measurable_cst _ + +| eval_beta g (a b : nat) : + (exp_beta a b : exp D g _) -D> cst (beta_prob a b) ; measurable_cst _ + +| eval_poisson g n (e : exp D g _) f mf : + e -D> f ; mf -> + exp_poisson n e -D> poisson_pdf n \o f ; + measurableT_comp (measurable_poisson_pdf n) mf + +| eval_normalize g t (e : exp P g t) k : + e -P> k -> + [Normalize e] -D> normalize_pt k ; measurable_normalize_pt k + +| evalD_if g t e f mf (e1 : exp D g t) f1 mf1 e2 f2 mf2 : + e -D> f ; mf -> e1 -D> f1 ; mf1 -> e2 -D> f2 ; mf2 -> + [if e then e1 else e2] -D> fun x => if f x then f1 x else f2 x ; + measurable_fun_ifT mf mf1 mf2 + +| evalD_weak g h t e x (H : x.1 \notin dom (g ++ h)) f mf : + e -D> f ; mf -> + (exp_weak _ g h x e H : exp _ _ t) -D> weak g h x f ; + measurable_weak g h x f mf + +where "e -D> v ; mv" := (@evalD _ _ e v mv) + +with evalP : forall g t, exp P g t -> pval R g t -> Prop := + +| eval_letin g t1 t2 str (e1 : exp _ g t1) (e2 : exp _ _ t2) k1 k2 : + e1 -P> k1 -> e2 -P> k2 -> + [let str := e1 in e2] -P> letin' k1 k2 + +| eval_sample g t (e : exp _ _ (Prob t)) + (p : mctx g -> pprobability (mtyp t) R) mp : + e -D> p ; mp -> [Sample e] -P> sample p mp + +| eval_score g (e : exp _ g _) f mf : + e -D> f ; mf -> [Score e] -P> kscore mf + +| eval_return g t (e : exp D g t) f mf : + e -D> f ; mf -> [return e] -P> ret mf + +| evalP_if g t e f mf (e1 : exp P g t) k1 e2 k2 : + e -D> f ; mf -> e1 -P> k1 -> e2 -P> k2 -> + [if e then e1 else e2] -P> ite mf k1 k2 + +| evalP_weak g h t (e : exp P (g ++ h) t) x + (H : x.1 \notin dom (g ++ h)) f : + e -P> f -> + exp_weak _ g h x e H -P> kweak g h x f + +where "e -P> v" := (@evalP _ _ e v). + +End eval. + +Notation "e -D> v ; mv" := (@evalD _ _ _ e v mv) : lang_scope. +Notation "e -P> v" := (@evalP _ _ _ e v) : lang_scope. + +Scheme evalD_mut_ind := Induction for evalD Sort Prop +with evalP_mut_ind := Induction for evalP Sort Prop. + +(* properties of the evaluation relation *) +Section eval_prop. +Variables (R : realType). +Local Open Scope lang_scope. + +Lemma evalD_uniq g t (e : exp D g t) (u v : dval R g t) mu mv : + e -D> u ; mu -> e -D> v ; mv -> u = v. +Proof. +move=> hu. +apply: (@evalD_mut_ind R + (fun g t (e : exp D g t) f mf (h1 : e -D> f; mf) => + forall v mv, e -D> v; mv -> f = v) + (fun g t (e : exp P g t) u (h1 : e -P> u) => + forall v, e -P> v -> u = v)); last exact: hu. +all: (rewrite {g t e u v mu mv hu}). +- move=> g {}v {}mv. + inversion 1; subst g0. + by inj_ex H3. +- move=> g b {}v {}mv. + inversion 1; subst g0 b0. + by inj_ex H3. +- move=> g n {}v {}mv. + inversion 1; subst g0 n0. + by inj_ex H3. +- move=> g r {}v {}mv. + inversion 1; subst g0 r0. + by inj_ex H3. +- move=> g n e f mf ev IH {}v {}mv. + inversion 1; subst g0 n0. + inj_ex H4; subst v. + inj_ex H2; subst e0. + by move: H3 => /IH <-. +- move=> g bop e1 f1 mf1 e2 f2 mf2 ev1 IH1 ev2 IH2 {}v {}mv. + inversion 1; subst g0 bop0. + inj_ex H10; subst v. + inj_ex H5; subst e1. + inj_ex H6; subst e5. + by move: H4 H11 => /IH1 <- /IH2 <-. +- move=> g rop e1 f1 mf1 e2 f2 mf2 ev1 IH1 ev2 IH2 {}v {}mv. + inversion 1; subst g0 rop0. + inj_ex H5; subst v. + inj_ex H1; subst e1. + inj_ex H3; subst e3. + by move: H6 H7 => /IH1 <- /IH2 <-. +- move=> g t1 e1 f1 mf1 t2 e2 f2 mf2 ev1 IH1 ev2 IH2 {}v {}mv. + simple inversion 1 => //; subst g0. + case: H3 => ? ?; subst t0 t3. + inj_ex H4; case: H4 => He1 He2. + inj_ex He1; subst e0. + inj_ex He2; subst e3. + inj_ex H5; subst v. + by move=> /IH1 <- /IH2 <-. +- move=> g t1 t2 e f mf H ih v mv. + inversion 1; subst g0 t3 t0. + inj_ex H11; subst v. + clear H9. + inj_ex H7; subst e1. + by rewrite (ih _ _ H4). +- move=> g t1 t2 e f mf H ih v mv. + inversion 1; subst g0 t3 t0. + inj_ex H11; subst v. + clear H9. + inj_ex H7; subst e1. + by rewrite (ih _ _ H4). +- move=> g str H n {}v {}mv. + inversion 1; subst g0. + inj_ex H9; rewrite -H9. + by inj_ex H10. +- move=> g e r mr ev IH {}v {}mv. + inversion 1; subst g0. + inj_ex H0; subst e0. + inj_ex H3; subst v. + by rewrite (IH _ _ H4). +- move=> g n e f mf ev IH {}v {}mv. + inversion 1; subst g0 n0. + inj_ex H2; subst e0. + inj_ex H4; subst v. + by rewrite (IH _ _ H5). +- move=> g a b ab {}v {}mv. + inversion 1; subst g0 a0 b0. + inj_ex H4; subst v. + by have -> : ab = ab1. +- (* TODO: beta *) move=> g a b {}v {}mv. + inversion 1; subst g0 a0 b0. + by inj_ex H4; subst v. +- move=> g t e k mk ev IH {}v {}mv. + inversion 1; subst g0 t. + inj_ex H2; subst e0. + inj_ex H4; subst v. + by rewrite (IH _ _ H3). +- move=> g t e k ev IH f mf. + inversion 1; subst g0 t0. + inj_ex H2; subst e0. + inj_ex H4; subst f. + inj_ex H5; subst mf. + by rewrite (IH _ H3). +- move=> g t e f mf e1 f1 mf1 e2 f2 mf2 ev ih ev1 ih1 ev2 ih2 v m. + inversion 1; subst g0 t0. + inj_ex H2; subst e0. + inj_ex H6; subst e5. + inj_ex H7; subst e6. + inj_ex H9; subst v. + clear H11. + have ? := ih1 _ _ H12; subst f6. + have ? := ih2 _ _ H13; subst f7. + by rewrite (ih _ _ H5). +- move=> g h t e x H f mf ef ih {}v {}mv. + inversion 1; subst t0 g0 h0 x0. + inj_ex H12; subst e1. + inj_ex H14; subst v. + clear H16. + by rewrite (ih _ _ H5). +- move=> g t1 t2 x e1 e2 k1 k2 ev1 IH1 ev2 IH2 k. + inversion 1; subst g0 t0 t3 x. + inj_ex H7; subst k. + inj_ex H6; subst e5. + inj_ex H5; subst e4. + by rewrite (IH1 _ H4) (IH2 _ H8). +- move=> g t e p mp ev IH k. + inversion 1; subst g0. + inj_ex H5; subst t0. + inj_ex H5; subst e1. + inj_ex H7; subst k. + have ? := IH _ _ H3; subst p1. + by have -> : mp = mp1 by []. +- move=> g e f mf ev IH k. + inversion 1; subst g0. + inj_ex H0; subst e0. + inj_ex H4; subst k. + have ? := IH _ _ H2; subst f1. + by have -> : mf = mf0 by []. +- move=> g t e0 f mf ev IH k. + inversion 1; subst g0 t0. + inj_ex H5; subst e1. + inj_ex H7; subst k. + have ? := IH _ _ H3; subst f1. + by have -> : mf = mf1 by []. +- move=> g t e f mf e1 k1 e2 k2 ev ih ev1 ih1 ev2 ih2 k. + inversion 1; subst g0 t0. + inj_ex H0; subst e0. + inj_ex H1; subst e3. + inj_ex H5; subst k. + inj_ex H2; subst e4. + have ? := ih _ _ H6; subst f1. + have -> : mf = mf0 by []. + by rewrite (ih1 _ H7) (ih2 _ H8). +- move=> g h t e x xgh k ek ih. + inversion 1; subst x0 g0 h0 t0. + inj_ex H13; rewrite -H13. + inj_ex H11; subst e1. + by rewrite (ih _ H4). +Qed. + +Lemma evalP_uniq g t (e : exp P g t) (u v : pval R g t) : + e -P> u -> e -P> v -> u = v. +Proof. +move=> eu. +apply: (@evalP_mut_ind R + (fun g t (e : exp D g t) f mf (h : e -D> f; mf) => + forall v mv, e -D> v; mv -> f = v) + (fun g t (e : exp P g t) u (h : e -P> u) => + forall v, e -P> v -> u = v)); last exact: eu. +all: rewrite {g t e u v eu}. +- move=> g {}v {}mv. + inversion 1; subst g0. + by inj_ex H3. +- move=> g b {}v {}mv. + inversion 1; subst g0 b0. + by inj_ex H3. +- move=> g n {}v {}mv. + inversion 1; subst g0 n0. + by inj_ex H3. +- move=> g r {}v {}mv. + inversion 1; subst g0 r0. + by inj_ex H3. +- move=> g n e f mf ev IH {}v {}mv. + inversion 1; subst g0 n0. + inj_ex H4; subst v. + inj_ex H2; subst e0. + by move: H3 => /IH <-. +- move=> g bop e1 f1 mf1 e2 f2 mf2 ev1 IH1 ev2 IH2 {}v {}mv. + inversion 1; subst g0 bop0. + inj_ex H10; subst v. + inj_ex H5; subst e1. + inj_ex H6; subst e5. + by move: H4 H11 => /IH1 <- /IH2 <-. +- move=> g rop e1 f1 mf1 e2 f2 mf2 ev1 IH1 ev2 IH2 {}v {}mv. + inversion 1; subst g0 rop0. + inj_ex H5; subst v. + inj_ex H1; subst e1. + inj_ex H3; subst e3. + by move: H6 H7 => /IH1 <- /IH2 <-. +- move=> g t1 e1 f1 mf1 t2 e2 f2 mf2 ev1 IH1 ev2 IH2 {}v {}mv. + simple inversion 1 => //; subst g0. + case: H3 => ? ?; subst t0 t3. + inj_ex H4; case: H4 => He1 He2. + inj_ex He1; subst e0. + inj_ex He2; subst e3. + inj_ex H5; subst v. + move=> e1f0 e2f3. + by rewrite (IH1 _ _ e1f0) (IH2 _ _ e2f3). +- move=> g t1 t2 e f mf H ih v mv. + inversion 1; subst g0 t3 t0. + inj_ex H11; subst v. + clear H9. + inj_ex H7; subst e1. + by rewrite (ih _ _ H4). +- move=> g t1 t2 e f mf H ih v mv. + inversion 1; subst g0 t3 t0. + inj_ex H11; subst v. + clear H9. + inj_ex H7; subst e1. + by rewrite (ih _ _ H4). +- move=> g str H n {}v {}mv. + inversion 1; subst g0. + inj_ex H9; rewrite -H9. + by inj_ex H10. +- move=> g e r mr ev IH {}v {}mv. + inversion 1; subst g0. + inj_ex H0; subst e0. + inj_ex H3; subst v. + by rewrite (IH _ _ H4). +- move=> g n e f mf ev IH {}v {}mv. + inversion 1; subst g0 n0. + inj_ex H2; subst e0. + inj_ex H4; subst v. + by rewrite (IH _ _ H5). +- move=> g a b ab {}v {}mv. + inversion 1; subst g0 a0 b0. + inj_ex H4; subst v. + by have -> : ab = ab1. +- (* TODO: beta case*) move=> g a b {}v {}mv. + inversion 1; subst g0 a0 b0. + by inj_ex H4; subst v. +- move=> g n e f mf ev IH {}v {}mv. + inversion 1; subst g0 n0. + inj_ex H2; subst e0. + inj_ex H4; subst v. + inj_ex H5; subst mv. + by rewrite (IH _ _ H3). +- move=> g t e k ev IH {}v {}mv. + inversion 1; subst g0 t0. + inj_ex H2; subst e0. + inj_ex H4; subst v. + inj_ex H5; subst mv. + by rewrite (IH _ H3). +- move=> g t e f mf e1 f1 mf1 e2 f2 mf2 ef ih ef1 ih1 ef2 ih2 {}v {}mv. + inversion 1; subst g0 t0. + inj_ex H2; subst e0. + inj_ex H6; subst e5. + inj_ex H7; subst e6. + inj_ex H9; subst v. + clear H11. + have ? := ih1 _ _ H12; subst f6. + have ? := ih2 _ _ H13; subst f7. + by rewrite (ih _ _ H5). +- move=> g h t e x H f mf ef ih {}v {}mv. + inversion 1; subst x0 g0 h0 t0. + inj_ex H12; subst e1. + inj_ex H14; subst v. + clear H16. + by rewrite (ih _ _ H5). +- move=> g t1 t2 x e1 e2 k1 k2 ev1 IH1 ev2 IH2 k. + inversion 1; subst g0 x t3 t0. + inj_ex H7; subst k. + inj_ex H5; subst e4. + inj_ex H6; subst e5. + by rewrite (IH1 _ H4) (IH2 _ H8). +- move=> g t e p mp ep IH v. + inversion 1; subst g0 t0. + inj_ex H7; subst v. + inj_ex H5; subst e1. + have ? := IH _ _ H3; subst p1. + by have -> : mp = mp1 by []. +- move=> g e f mf ev IH k. + inversion 1; subst g0. + inj_ex H0; subst e0. + inj_ex H4; subst k. + have ? := IH _ _ H2; subst f1. + by have -> : mf = mf0 by []. +- move=> g t e f mf ev IH k. + inversion 1; subst g0 t0. + inj_ex H7; subst k. + inj_ex H5; subst e1. + have ? := IH _ _ H3; subst f1. + by have -> : mf = mf1 by []. +- move=> g t e f mf e1 k1 e2 k2 ev ih ev1 ih1 ev2 ih2 k. + inversion 1; subst g0 t0. + inj_ex H0; subst e0. + inj_ex H1; subst e3. + inj_ex H5; subst k. + inj_ex H2; subst e4. + have ? := ih _ _ H6; subst f1. + have -> : mf0 = mf by []. + by rewrite (ih1 _ H7) (ih2 _ H8). +- move=> g h t e x xgh k ek ih. + inversion 1; subst x0 g0 h0 t0. + inj_ex H13; rewrite -H13. + inj_ex H11; subst e1. + by rewrite (ih _ H4). +Qed. + +Lemma eval_total z g t (e : @exp R z g t) : + (match z with + | D => fun e => exists f mf, e -D> f ; mf + | P => fun e => exists k, e -P> k + end) e. +Proof. +elim: e. +all: rewrite {z g t}. +- by do 2 eexists; exact: eval_unit. +- by do 2 eexists; exact: eval_bool. +- by do 2 eexists; exact: eval_nat. +- by do 2 eexists; exact: eval_real. +- move=> g n e [f [mf H]]. + by exists (fun x => (f x ^+ n)%R); eexists; exact: eval_pow. +- move=> b g e1 [f1 [mf1 H1]] e2 [f2 [mf2 H2]]. + by exists (fun_of_binop f1 f2); eexists; exact: eval_bin. +- move=> r g e1 [f1 [mf1 H1]] e2 [f2 [mf2 H2]]. + by exists (fun_of_relop r f1 f2); eexists; exact: eval_rel. +- move=> g t1 t2 e1 [f1 [mf1 H1]] e2 [f2 [mf2 H2]]. + by exists (fun x => (f1 x, f2 x)); eexists; exact: eval_pair. +- move=> g t1 t2 e [f [mf H]]. + by exists (fst \o f); eexists; exact: eval_proj1. +- move=> g t1 t2 e [f [mf H]]. + by exists (snd \o f); eexists; exact: eval_proj2. +- by move=> g x t tE; subst t; eexists; eexists; exact: eval_var. +- move=> g e [p [mp H]]. + exists ((bernoulli : R -> pprobability bool R) \o p). + by eexists; exact: eval_bernoulli. +- move=> g n e [p [mp H]]. + exists ((binomial_prob n : R -> pprobability nat R) \o p). + by eexists; exact: (eval_binomial n). +- by eexists; eexists; exact: eval_uniform. +- by eexists; eexists; exact: eval_beta. +- move=> g h e [f [mf H]]. + by exists (poisson_pdf h \o f); eexists; exact: eval_poisson. +- move=> g t e [k ek]. + by exists (normalize_pt k); eexists; exact: eval_normalize. +- move=> g t1 t2 x e1 [k1 ev1] e2 [k2 ev2]. + by exists (letin' k1 k2); exact: eval_letin. +- move=> g t e [f [/= mf ef]]. + by eexists; exact: (@eval_sample _ _ _ _ _ mf). +- move=> g e [f [mf f_mf]]. + by exists (kscore mf); exact: eval_score. +- by move=> g t e [f [mf f_mf]]; exists (ret mf); exact: eval_return. +- case. + + move=> g t e1 [f [mf H1]] e2 [f2 [mf2 H2]] e3 [f3 [mf3 H3]]. + by exists (fun g => if f g then f2 g else f3 g), + (measurable_fun_ifT mf mf2 mf3); exact: evalD_if. + + move=> g t e1 [f [mf H1]] e2 [k2 H2] e3 [k3 H3]. + by exists (ite mf k2 k3); exact: evalP_if. +- case=> [g h t x e [f [mf ef]] xgh|g h st x e [k ek] xgh]. + + by exists (weak _ _ _ f), (measurable_weak _ _ _ _ mf); exact/evalD_weak. + + by exists (kweak _ _ _ k); exact: evalP_weak. +Qed. + +Lemma evalD_total g t (e : @exp R D g t) : exists f mf, e -D> f ; mf. +Proof. exact: (eval_total e). Qed. + +Lemma evalP_total g t (e : @exp R P g t) : exists k, e -P> k. +Proof. exact: (eval_total e). Qed. + +End eval_prop. + +Section execution_functions. +Local Open Scope lang_scope. +Context {R : realType}. +Implicit Type g : ctx. + +Definition execD g t (e : exp D g t) : + {f : dval R g t & measurable_fun setT f} := + let: exist _ H := cid (evalD_total e) in + existT _ _ (projT1 (cid H)). + +Lemma eq_execD g t (p1 p2 : @exp R D g t) : + projT1 (execD p1) = projT1 (execD p2) -> execD p1 = execD p2. +Proof. +rewrite /execD /=. +case: cid => /= f1 [mf1 ev1]. +case: cid => /= f2 [mf2 ev2] f12. +subst f2. +have ? : mf1 = mf2 by []. +subst mf2. +congr existT. +rewrite /sval. +case: cid => mf1' ev1'. +have ? : mf1 = mf1' by []. +subst mf1'. +case: cid => mf2' ev2'. +have ? : mf1 = mf2' by []. +by subst mf2'. +Qed. + +Definition execP g t (e : exp P g t) : pval R g t := + projT1 (cid (evalP_total e)). + +Lemma execD_evalD g t e x mx: + @execD g t e = existT _ x mx <-> e -D> x ; mx. +Proof. +rewrite /execD; split. + case: cid => x' [mx' H] [?]; subst x'. + have ? : mx = mx' by []. + by subst mx'. +case: cid => f' [mf' f'mf']/=. +move/evalD_uniq => /(_ _ _ f'mf') => ?; subst f'. +by case: cid => //= ? ?; congr existT. +Qed. + +Lemma evalD_execD g t (e : exp D g t) : + e -D> projT1 (execD e); projT2 (execD e). +Proof. +by rewrite /execD; case: cid => // x [mx xmx]/=; case: cid. +Qed. + +Lemma execP_evalP g t (e : exp P g t) x : + execP e = x <-> e -P> x. +Proof. +rewrite /execP; split; first by move=> <-; case: cid. +case: cid => // x0 Hx0. +by move/evalP_uniq => /(_ _ Hx0) ?; subst x. +Qed. + +Lemma evalP_execP g t (e : exp P g t) : e -P> execP e. +Proof. by rewrite /execP; case: cid. Qed. + +Lemma execD_unit g : @execD g _ [TT] = existT _ (cst tt) ktt. +Proof. exact/execD_evalD/eval_unit. Qed. + +Lemma execD_bool g b : @execD g _ [b:B] = existT _ (cst b) (kb b). +Proof. exact/execD_evalD/eval_bool. Qed. + +Lemma execD_nat g n : @execD g _ [n:N] = existT _ (cst n) (kn n). +Proof. exact/execD_evalD/eval_nat. Qed. + +Lemma execD_real g r : @execD g _ [r:R] = existT _ (cst r) (kr r). +Proof. exact/execD_evalD/eval_real. Qed. + +Local Open Scope ring_scope. +Lemma execD_pow g (e : exp D g _) n : + let f := projT1 (execD e) in let mf := projT2 (execD e) in + execD (exp_pow n e) = + @existT _ _ (fun x => f x ^+ n) (measurable_funX n mf). +Proof. +by move=> f mf; apply/execD_evalD/eval_pow/evalD_execD. +Qed. + +Lemma execD_bin g bop (e1 : exp D g _) (e2 : exp D g _) : + let f1 := projT1 (execD e1) in let f2 := projT1 (execD e2) in + let mf1 := projT2 (execD e1) in let mf2 := projT2 (execD e2) in + execD (exp_bin bop e1 e2) = + @existT _ _ (fun_of_binop f1 f2) (mfun_of_binop mf1 mf2). +Proof. +by move=> f1 f2 mf1 mf2; apply/execD_evalD/eval_bin; exact/evalD_execD. +Qed. + +Lemma execD_rel g rop (e1 : exp D g _) (e2 : exp D g _) : + let f1 := projT1 (execD e1) in let f2 := projT1 (execD e2) in + let mf1 := projT2 (execD e1) in let mf2 := projT2 (execD e2) in + execD (exp_rel rop e1 e2) = + @existT _ _ (fun_of_relop rop f1 f2) (mfun_of_relop rop mf1 mf2). +Proof. +by move=> f1 f2 mf1 mf2; apply/execD_evalD/eval_rel; exact: evalD_execD. +Qed. + +Lemma execD_pair g t1 t2 (e1 : exp D g t1) (e2 : exp D g t2) : + let f1 := projT1 (execD e1) in let f2 := projT1 (execD e2) in + let mf1 := projT2 (execD e1) in let mf2 := projT2 (execD e2) in + execD [(e1, e2)] = + @existT _ _ (fun z => (f1 z, f2 z)) + (@measurable_fun_prod _ _ _ (mctx g) (mtyp t1) (mtyp t2) + f1 f2 mf1 mf2). +Proof. +by move=> f1 f2 mf1 mf2; apply/execD_evalD/eval_pair; exact: evalD_execD. +Qed. + +Lemma execD_proj1 g t1 t2 (e : exp D g (Pair t1 t2)) : + let f := projT1 (execD e) in + let mf := projT2 (execD e) in + execD [\pi_1 e] = @existT _ _ (fst \o f) + (measurableT_comp measurable_fst mf). +Proof. +by move=> f mf; apply/execD_evalD/eval_proj1; exact: evalD_execD. +Qed. + +Lemma execD_proj2 g t1 t2 (e : exp D g (Pair t1 t2)) : + let f := projT1 (execD e) in let mf := projT2 (execD e) in + execD [\pi_2 e] = @existT _ _ (snd \o f) + (measurableT_comp measurable_snd mf). +Proof. +by move=> f mf; apply/execD_evalD/eval_proj2; exact: evalD_execD. +Qed. + +Lemma execD_var_erefl g str : let i := index str (dom g) in + @execD g _ (exp_var str erefl) = existT _ (acc_typ (map snd g) i) + (measurable_acc_typ (map snd g) i). +Proof. by move=> i; apply/execD_evalD; exact: eval_var. Qed. + +Lemma execD_var g x (H : nth Unit (map snd g) (index x (dom g)) = lookup Unit g x) : + let i := index x (dom g) in + @execD g _ (exp_var x H) = existT _ (acc_typ (map snd g) i) + (measurable_acc_typ (map snd g) i). +Proof. by move=> i; apply/execD_evalD; exact: eval_var. Qed. + +Lemma execD_bernoulli g e : + @execD g _ (exp_bernoulli e) = + existT _ ((bernoulli : R -> pprobability bool R) \o projT1 (execD e)) + (measurableT_comp measurable_bernoulli (projT2 (execD e))). +Proof. exact/execD_evalD/eval_bernoulli/evalD_execD. Qed. + +Lemma execD_binomial g n e : + @execD g _ (exp_binomial n e) = + existT _ ((binomial_prob n : R -> pprobability nat R) \o projT1 (execD e)) + (measurableT_comp (measurable_binomial_prob n) (projT2 (execD e))). +Proof. exact/execD_evalD/eval_binomial/evalD_execD. Qed. + +Lemma execD_uniform g a b ab0 : + @execD g _ (exp_uniform a b ab0) = + existT _ (cst [the probability _ _ of uniform_prob ab0]) (measurable_cst _). +Proof. exact/execD_evalD/eval_uniform. Qed. + +Lemma execD_beta g a b : + @execD g _ (exp_beta a b) = + existT _ (cst [the probability _ _ of beta_prob a b]) (measurable_cst _). +Proof. exact/execD_evalD/eval_beta. Qed. + +Lemma execD_normalize_pt g t (e : exp P g t) : + @execD g _ [Normalize e] = + existT _ (normalize_pt (execP e) : _ -> pprobability _ _) + (measurable_normalize_pt (execP e)). +Proof. exact/execD_evalD/eval_normalize/evalP_execP. Qed. + +Lemma execD_poisson g n (e : exp D g Real) : + execD (exp_poisson n e) = + existT _ (poisson_pdf n \o projT1 (execD e)) + (measurableT_comp (measurable_poisson_pdf n) (projT2 (execD e))). +Proof. exact/execD_evalD/eval_poisson/evalD_execD. Qed. + +Lemma execP_if g st e1 e2 e3 : + @execP g st [if e1 then e2 else e3] = + ite (projT2 (execD e1)) (execP e2) (execP e3). +Proof. +by apply/execP_evalP/evalP_if; [apply: evalD_execD| exact: evalP_execP..]. +Qed. + +Lemma execP_letin g x t1 t2 (e1 : exp P g t1) (e2 : exp P ((x, t1) :: g) t2) : + execP [let x := e1 in e2] = letin' (execP e1) (execP e2) :> (R.-sfker _ ~> _). +Proof. by apply/execP_evalP/eval_letin; exact: evalP_execP. Qed. + +Lemma execP_sample g t (e : @exp R D g (Prob t)) : + let x := execD e in + execP [Sample e] = sample (projT1 x) (projT2 x). +Proof. exact/execP_evalP/eval_sample/evalD_execD. Qed. + +Lemma execP_score g (e : exp D g Real) : + execP [Score e] = score (projT2 (execD e)). +Proof. exact/execP_evalP/eval_score/evalD_execD. Qed. + +Lemma execP_return g t (e : exp D g t) : + execP [return e] = ret (projT2 (execD e)). +Proof. exact/execP_evalP/eval_return/evalD_execD. Qed. + +Lemma execP_weak g h x t (e : exp P (g ++ h) t) + (xl : x.1 \notin dom (g ++ h)) : + execP (exp_weak P g h _ e xl) = kweak _ _ _ (execP e). +Proof. exact/execP_evalP/evalP_weak/evalP_execP. Qed. + +End execution_functions. +Arguments execD_var_erefl {R g} str. +Arguments execP_weak {R} g h x {t} e. +Arguments exp_var'E {R} str. + +Local Open Scope lang_scope. +Lemma congr_letinl {R : realType} g t1 t2 str (e1 e2 : @exp _ _ g t1) + (e : @exp _ _ (_ :: g) t2) x U : + (forall y V, execP e1 y V = execP e2 y V) -> + measurable U -> + @execP R g t2 [let str := e1 in e] x U = + @execP R g t2 [let str := e2 in e] x U. +Proof. by move=> + mU; move/eq_sfkernel => He; rewrite !execP_letin He. Qed. + +Lemma congr_letinr {R : realType} g t1 t2 str (e : @exp _ _ _ t1) + (e1 e2 : @exp _ _ (_ :: g) t2) x U : + (forall y V, execP e1 (y, x) V = execP e2 (y, x) V) -> + @execP R g t2 [let str := e in e1] x U = @execP R g t2 [let str := e in e2] x U. +Proof. +by move=> He; rewrite !execP_letin !letin'E; apply: eq_integral => ? _; exact: He. +Qed. + +Lemma congr_normalize {R : realType} g t (e1 e2 : @exp R _ g t) : + (forall x U, execP e1 x U = execP e2 x U) -> + execD [Normalize e1] = execD [Normalize e2]. +Proof. +move=> He; apply: eq_execD. +rewrite !execD_normalize_pt /=. +f_equal. +apply: eq_kernel => y V. +exact: He. +Qed. +Local Close Scope lang_scope. diff --git a/theories/lang_syntax_examples.v b/theories/lang_syntax_examples.v new file mode 100644 index 000000000..41efab0ee --- /dev/null +++ b/theories/lang_syntax_examples.v @@ -0,0 +1,975 @@ +From Coq Require Import String. +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval. +From mathcomp.classical Require Import mathcomp_extra boolp. +From mathcomp Require Import classical_sets. +From mathcomp.classical Require Import functions cardinality fsbigop. +From mathcomp Require Import signed reals ereal topology normedtype sequences. +From mathcomp Require Import esum measure lebesgue_measure numfun. +From mathcomp Require Import lebesgue_integral probability kernel prob_lang. +From mathcomp Require Import lang_syntax_util lang_syntax. +From mathcomp Require Import ring lra. + +(**md**************************************************************************) +(* # Examples using the Probabilistic Programming Language of lang_syntax.v *) +(* *) +(* sample_pair1213 := normalize ( *) +(* let x := sample (bernoulli 1/2) in *) +(* let y := sample (bernoulli 1/3) in *) +(* return (x, y)) *) +(* *) +(* sample_and1213 := normalize ( *) +(* let x := sample (bernoulli 1/2) in *) +(* let y := sample (bernoulli 1/3) in *) +(* return (x && y)) *) +(* *) +(* bernoulli13_score := normalize ( *) +(* let x := sample (bernoulli 1/3) in *) +(* let _ := if x then score (1/3) else score (2/3) in *) +(* return x) *) +(* *) +(* sample_binomial3 := *) +(* let x := sample (binomial 3 1/2) in *) +(* return x *) +(* *) +(* hard_constraint := let x := Score {0}:R in return TT *) +(* *) +(* guard := *) +(* let p := sample (bernoulli (1 / 3)) in *) +(* let _ := if p then return TT else score 0 in *) +(* return p *) +(* *) +(* more examples about uniform, beta, and bernoulli distributions *) +(* *) +(* associativity of let-in expressions *) +(* *) +(* staton_bus_syntax == example from [Staton, ESOP 2017] *) +(* *) +(* staton_busA_syntax == same as staton_bus_syntax module associativity of *) +(* let-in expression *) +(* *) +(* commutativity of let-in expressions *) +(* *) +(******************************************************************************) + +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. + +Local Open Scope lang_scope. + +Local Close Scope lang_scope. + +(* simple tests to check bidirectional hints *) +Module bidi_tests. +Section bidi_tests. +Local Open Scope lang_scope. +Import Notations. +Context (R : realType). + +Definition bidi_test1 x : @exp R P [::] _ := [ + let x := return {1}:R in + return #x]. + +Definition bidi_test2 (a b : string) + (a := "a") (b := "b") + (* (ba : infer (b != a)) *) + : @exp R P [::] _ := [ + let a := return {1}:R in + let b := return {true}:B in + (* let c := return {3}:R in + let d := return {4}:R in *) + return (#a, #b)]. + +Definition bidi_test3 (a b c d : string) + (ba : infer (b != a)) (ca : infer (c != a)) + (cb : infer (c != b)) (ab : infer (a != b)) + (ac : infer (a != c)) (bc : infer (b != c)) : @exp R P [::] _ := [ + let a := return {1}:R in + let b := return {2}:R in + let c := return {3}:R in + (* let d := return {4}:R in *) + return (#b, #a)]. + +Definition bidi_test4 (a b c d : string) + (ba : infer (b != a)) (ca : infer (c != a)) + (cb : infer (c != b)) (ab : infer (a != b)) + (ac : infer (a != c)) (bc : infer (b != c)) : @exp R P [::] _ := [ + let a := return {1}:R in + let b := return {2}:R in + let c := return {3}:R in + (* let d := return {4}:R in *) + return {exp_poisson O [#c(*{exp_var c erefl}*)]}]. + +End bidi_tests. +End bidi_tests. + +Section trivial_example. +Local Open Scope lang_scope. +Import Notations. +Context {R : realType}. + +Lemma exec_normalize_return g x r : + projT1 (@execD _ g _ [Normalize return r:R]) x = + @dirac _ (measurableTypeR R) r _ :> probability _ R. + (* NB: \d_r notation? *) +Proof. +by rewrite execD_normalize_pt execP_return execD_real//=; exact: normalize_kdirac. +Qed. + +End trivial_example. + +Section sample_pair. +Local Open Scope lang_scope. +Local Open Scope ring_scope. +Import Notations. +Context {R : realType}. + +Definition sample_pair1213' : @exp R _ [::] _ := + [let "x" := Sample {exp_bernoulli [{1 / 2}:R]} in + let "y" := Sample {exp_bernoulli [{1 / 3}:R]} in + return (#{"x"}, #{"y"})]. + +Definition sample_pair1213 : exp _ [::] _ := [Normalize {sample_pair1213'}]. + +Lemma exec_sample_pair1213' (A : set (bool * bool)) : + @execP R [::] _ sample_pair1213' tt A = + ((1 / 2)%:E * + ((1 / 3)%:E * \d_(true, true) A + + (1 - 1 / 3)%:E * \d_(true, false) A) + + (1 - 1 / 2)%:E * + ((1 / 3)%:E * \d_(false, true) A + + (1 - 1 / 3)%:E * \d_(false, false) A))%E. +Proof. +rewrite !execP_letin !execP_sample !execD_bernoulli !execP_return /=. +rewrite execD_pair !exp_var'E (execD_var_erefl "x") (execD_var_erefl "y") /=. +rewrite !execD_real//=. +do 2 (rewrite letin'E/= integral_bernoulli//=; last lra). +by rewrite letin'E/= integral_bernoulli//=; lra. +Qed. + +Lemma exec_sample_pair1213'_TandT : + @execP R [::] _ sample_pair1213' tt [set (true, true)] = (1 / 6)%:E. +Proof. +rewrite exec_sample_pair1213' !diracE mem_set//; do 3 rewrite memNset//=. +by rewrite /= !mule0 mule1 !add0e mule0 adde0; congr (_%:E); lra. +Qed. + +Lemma exec_sample_pair1213'_TandT' : + @execP R [::] _ sample_pair1213' tt [set p | p.1 && p.2] = (1 / 6)%:E. +Proof. +rewrite exec_sample_pair1213' !diracE mem_set//; do 3 rewrite memNset//=. +by rewrite /= !mule0 mule1 !add0e mule0 adde0; congr (_%:E); lra. +Qed. + +Lemma exec_sample_pair1213'_TandF : + @execP R [::] _ sample_pair1213' tt [set (true, false)] = (1 / 3)%:E. +Proof. +rewrite exec_sample_pair1213' !diracE memNset// mem_set//; do 2 rewrite memNset//. +by rewrite /= !mule0 mule1 !add0e mule0 adde0; congr (_%:E); lra. +Qed. + +Lemma exec_sample_pair1213_TorT : + (projT1 (execD sample_pair1213)) tt [set p | p.1 || p.2] = (2 / 3)%:E. +Proof. +rewrite execD_normalize_pt normalizeE/= exec_sample_pair1213'. +rewrite !diracE; do 4 rewrite mem_set//=. +rewrite eqe ifF; last by apply/negbTE/negP => /orP[/eqP|//]; lra. +rewrite exec_sample_pair1213' !diracE; do 3 rewrite mem_set//; rewrite memNset//=. +by rewrite !mule1; congr (_%:E); field. +Qed. + +End sample_pair. + +Section sample_and. +Local Open Scope lang_scope. +Local Open Scope ring_scope. +Import Notations. +Context {R : realType}. + +Definition sample_and1213' : @exp R _ [::] _ := + [let "x" := Sample {exp_bernoulli [{1 / 2}:R]} in + let "y" := Sample {exp_bernoulli [{1 / 3}:R]} in + return #{"x"} && #{"y"}]. + +Lemma exec_sample_and1213' (A : set bool) : + @execP R [::] _ sample_and1213' tt A = ((1 / 6)%:E * \d_true A + + (1 - 1 / 6)%:E * \d_false A)%E. +Proof. +rewrite !execP_letin !execP_sample/= !execD_bernoulli execP_return /=. +rewrite !(@execD_bin _ _ binop_and) !exp_var'E (execD_var_erefl "x"). +rewrite (execD_var_erefl "y") /= !letin'E/= !execD_real/=. +rewrite integral_bernoulli//=; last lra. +rewrite !letin'E/= integral_bernoulli//=; last lra. +rewrite integral_bernoulli//=; last lra. +rewrite /onem. +rewrite muleDr// -addeA; congr (_ + _)%E. + by rewrite !muleA; congr (_%:E); congr (_ * _); field. +rewrite -muleDl// !muleA -muleDl//. +by congr (_%:E); congr (_ * _); field. +Qed. + +Definition sample_and121212 : @exp R _ [::] _ := + [let "x" := Sample {exp_bernoulli [{1 / 2}:R]} in + let "y" := Sample {exp_bernoulli [{1 / 2}:R]} in + let "z" := Sample {exp_bernoulli [{1 / 2}:R]} in + return #{"x"} && #{"y"} && #{"z"}]. + +Lemma exec_sample_and121212 t U : + execP sample_and121212 t U = ((1 / 8)%:E * \d_true U + + (1 - 1 / 8)%:E * \d_false U)%E. +Proof. +rewrite !execP_letin !execP_sample !execD_bernoulli !execP_return /=. +rewrite !(@execD_bin _ _ binop_and) !exp_var'E (execD_var_erefl "x"). +rewrite (execD_var_erefl "y") (execD_var_erefl "z") /= !execD_real/=. +do 3 (rewrite !letin'E/= integral_bernoulli//=; last lra). +do 2 (rewrite integral_bernoulli//=; last lra). +rewrite !letin'E/= integral_bernoulli//=; last lra. +rewrite !muleDr// -!addeA; congr (_ + _)%E. + by rewrite !muleA; congr *%E; congr EFin; field. +rewrite !muleA -!muleDl//; congr *%E; congr EFin. +by rewrite /onem; field. +Qed. + +End sample_and. + +Section sample_score. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Import Notations. +Context {R : realType}. + +Definition bernoulli13_score := [Normalize + let "x" := Sample {@exp_bernoulli R [::] [{1 / 3}:R]} in + let "_" := if #{"x"} then Score {1 / 3}:R else Score {2 / 3}:R in + return #{"x"}]. + +Lemma exec_bernoulli13_score : + execD bernoulli13_score = execD (exp_bernoulli [{1 / 5}:R]). +Proof. +apply: eq_execD. +rewrite execD_bernoulli/= /bernoulli13_score execD_normalize_pt 2!execP_letin. +rewrite execP_sample/= execD_bernoulli/= execP_if /= exp_var'E. +rewrite (execD_var_erefl "x")/= !execP_return/= 2!execP_score !execD_real/=. +apply: funext=> g; apply: eq_probability => U. +rewrite normalizeE !letin'E/=. +under eq_integral. + move=> x _. + rewrite !letin'E. + under eq_integral do rewrite retE /=. + over. +rewrite /=. +rewrite integral_bernoulli//=; [|lra|by move=> b; rewrite integral_ge0]. +rewrite iteE/= !ge0_integral_mscale//=. +rewrite ger0_norm//. +rewrite !integral_indic//= !iteE/= /mscale/=. +rewrite setTI !diracT !mule1. +rewrite ger0_norm//. +rewrite -EFinD/= eqe ifF; last first. + by apply/negbTE/negP => /orP[/eqP|//]; rewrite /onem; lra. +rewrite integral_bernoulli//=; last lra. +rewrite !letin'E/= !iteE/=. +rewrite !ge0_integral_mscale//=. +rewrite ger0_norm//. +rewrite !integral_dirac//= !diracT !mul1e ger0_norm//. +rewrite exp_var'E (execD_var_erefl "x")/=. +rewrite !indicT/= !mulr1. +rewrite bernoulliE//=; last lra. +by rewrite muleDl//; congr (_ + _)%E; + rewrite -!EFinM; congr (_%:E); + rewrite !indicE /onem /=; case: (_ \in _); field. +Qed. + +Definition bernoulli12_score := [Normalize + let "x" := Sample {@exp_bernoulli R [::] [{1 / 2}:R]} in + let "r" := if #{"x"} then Score {1 / 3}:R else Score {2 / 3}:R in + return #{"x"}]. + +Lemma exec_bernoulli12_score : + execD bernoulli12_score = execD (exp_bernoulli [{1 / 3}:R]). +Proof. +apply: eq_execD. +rewrite execD_bernoulli/= /bernoulli12_score execD_normalize_pt 2!execP_letin. +rewrite execP_sample/= execD_bernoulli/= execP_if /= exp_var'E. +rewrite (execD_var_erefl "x")/= !execP_return/= 2!execP_score !execD_real/=. +apply: funext=> g; apply: eq_probability => U. +rewrite normalizeE !letin'E/=. +under eq_integral. + move=> x _. + rewrite !letin'E. + under eq_integral do rewrite retE /=. + over. +rewrite /= integral_bernoulli//=; [|lra|by move=> b; rewrite integral_ge0]. +rewrite iteE/= !ge0_integral_mscale//=. +rewrite ger0_norm//. +rewrite !integral_indic//= !iteE/= /mscale/=. +rewrite setTI !diracT !mule1. +rewrite ger0_norm//. +rewrite -EFinD/= eqe ifF; last first. + apply/negbTE/negP => /orP[/eqP|//]. + by rewrite /onem; lra. +rewrite integral_bernoulli//=; last lra. +rewrite !letin'E/= !iteE/=. +rewrite !ge0_integral_mscale//=. +rewrite ger0_norm//. +rewrite !integral_dirac//= !diracT !mul1e ger0_norm//. +rewrite exp_var'E (execD_var_erefl "x")/=. +rewrite bernoulliE//=; last lra. +rewrite !mul1r. +rewrite muleDl//; congr (_ + _)%E; + rewrite -!EFinM; + congr (_%:E); + by rewrite !indicT !indicE /onem /=; case: (_ \in _); field. +Qed. + +(* https://dl.acm.org/doi/pdf/10.1145/2933575.2935313 (Sect. 4) *) +Definition bernoulli14_score := [Normalize + let "x" := Sample {@exp_bernoulli R [::] [{1 / 4}:R]} in + let "r" := if #{"x"} then Score {5}:R else Score {2}:R in + return #{"x"}]. + +Lemma exec_bernoulli14_score : + execD bernoulli14_score = execD (exp_bernoulli [{5%:R / 11%:R}:R]). +Proof. +apply: eq_execD. +rewrite execD_bernoulli/= execD_normalize_pt 2!execP_letin. +rewrite execP_sample/= execD_bernoulli/= execP_if /= !exp_var'E. +rewrite !execP_return/= 2!execP_score !execD_real/=. +rewrite !(execD_var_erefl "x")/=. +apply: funext=> g; apply: eq_probability => U. +rewrite normalizeE !letin'E/=. +under eq_integral. + move=> x _. + rewrite !letin'E. + under eq_integral do rewrite retE /=. + over. +rewrite /= integral_bernoulli//=; [|lra|by move=> b; rewrite integral_ge0]. +rewrite iteE/= !ge0_integral_mscale//=. +rewrite ger0_norm//. +rewrite !integral_cst//= !diracT !(mule1,mul1e). +rewrite !indicT/= !mule1. +rewrite !iteE/= /mscale/=. +rewrite ger0_norm//. +rewrite !diracT/= !mul1r. +rewrite -EFinD/= eqe ifF; last first. + apply/negbTE/negP => /orP[/eqP|//]. + by rewrite /onem; lra. +rewrite integral_bernoulli//=; last lra. +rewrite !letin'E/= !iteE/=. +rewrite !ge0_integral_mscale//=. +rewrite ger0_norm//. +rewrite !integral_dirac//= !diracT !mul1e ger0_norm//. +rewrite bernoulliE//=; last lra. +rewrite !indicT. +rewrite muleDl//; congr (_ + _)%E; + rewrite -!EFinM; + congr (_%:E); + by rewrite !indicE /onem /=; case: (_ \in _); field. +Qed. + +End sample_score. + +Section sample_binomial. +Context {R : realType}. +Open Scope lang_scope. +Open Scope ring_scope. + +Definition sample_binomial3 : @exp R _ [::] _ := + [let "x" := Sample {exp_binomial 3 [{1 / 2}:R]} in + return #{"x"}]. + +Lemma exec_sample_binomial3 t U : measurable U -> + execP sample_binomial3 t U = ((1 / 8)%:E * \d_0%N U + + (3 / 8)%:E * \d_1%N U + + (3 / 8)%:E * \d_2%N U + + (1 / 8)%:E * \d_3%N U)%E. +Proof. +move=> mU; rewrite /sample_binomial3 execP_letin execP_sample execP_return. +rewrite exp_var'E (execD_var_erefl "x") !execD_binomial/= execD_real//=. +rewrite letin'E/= /= integral_binomial//=; [lra|move=> _]. +rewrite !big_ord_recl big_ord0/=. +rewrite /bump. +rewrite !binS/= !bin0 bin1 bin2 bin_small// addn0. +rewrite expr0 mulr1 mul1r subn0. +rewrite -2!addeA !mul1r. +congr _%:E. +rewrite !indicE /onem !addrA addr0 expr1/=. +by congr (_ + _ + _ + _); congr (_ * _); field. +Qed. + +End sample_binomial. + +Section hard_constraint. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Import Notations. +Context {R : realType} {str : string}. + +Definition hard_constraint g : @exp R _ g _ := + [let str := Score {0}:R in return TT]. + +Lemma exec_hard_constraint g mg U : + execP (hard_constraint g) mg U = fail' (false, tt) U. +Proof. +rewrite execP_letin execP_score execD_real execP_return execD_unit/=. +rewrite letin'E integral_indic//= /mscale/= normr0 mul0e. +by rewrite /fail' letin'E/= ge0_integral_mscale//= normr0 mul0e. +Qed. + +Lemma exec_score_fail (r : R) (r01 : (0 <= r <= 1)%R) : + execP (g := [::]) [Score {r}:R] = + execP [let str := Sample {exp_bernoulli [{r}:R]} in + if #str then return TT else {hard_constraint _}]. +Proof. +move: r01 => /andP[r0 r1]//. +rewrite execP_score execD_real /= score_fail' ?r0 ?r1//. +rewrite execP_letin execP_sample/= execD_bernoulli execP_if execP_return. +rewrite execD_unit/= exp_var'E /=. + exact/ctx_prf_head (* TODO *). +move=> h. +apply: eq_sfkernel=> /= -[] U. +rewrite [LHS]letin'E/= [RHS]letin'E/=. +rewrite execD_real/=. +apply: eq_integral => b _. +rewrite 2!iteE//=. +case: b => //=. +- suff : projT1 (@execD R _ _ (exp_var str h)) (true, tt) = true by move=> ->. + set g := [:: (str, Bool)]. + have /= := @execD_var R [:: (str, Bool)] str. + by rewrite eqxx => /(_ h) ->. +- have -> : projT1 (@execD R _ _ (exp_var str h)) (false, tt) = false. + set g := [:: (str, Bool)]. + have /= := @execD_var R [:: (str, Bool)] str. + by rewrite eqxx /= => /(_ h) ->. + by rewrite (@exec_hard_constraint [:: (str, Bool)]). +Qed. + +End hard_constraint. + +Section test_uniform. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Context (R : realType). + +Definition uniform_syntax : @exp R _ [::] _ := + [let "p" := Sample {exp_uniform 0 1 (@ltr01 R)} in + return #{"p"}]. + +Lemma exec_uniform_syntax t U : measurable U -> + execP uniform_syntax t U = uniform_prob (@ltr01 R) U. +Proof. +move=> mU. +rewrite /uniform_syntax execP_letin execP_sample execP_return !execD_uniform. +rewrite exp_var'E (execD_var_erefl "p")/=. +rewrite letin'E /=. +rewrite integral_uniform//=; last exact: measurable_fun_dirac. +rewrite subr0 invr1 mul1e. +rewrite {1}/uniform_prob. +rewrite integral_mkcond//=. +rewrite [in RHS]integral_mkcond//=. +apply: eq_integral => x _. +rewrite !patchE. +case: ifPn => //; case: ifPn => //. +- move=> xU. + rewrite inE/= in_itv/= => x01. + by rewrite /uniform_pdf x01 diracE xU subr0 invr1. +- by rewrite diracE => /negbTE ->. +- move=> xU. + rewrite notin_setE/= in_itv/= => /negP/negbTE x01. + by rewrite /uniform_pdf x01. +Qed. + +End test_uniform. + +Section guard. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Context (R : realType). + +Definition guard : @exp R _ [::] _ := [ + let "p" := Sample {exp_bernoulli [{1 / 3}:R]} in + let "_" := if #{"p"} then return TT else Score {0}:R in + return #{"p"} +]. + +Lemma exec_guard t U : execP guard t U = ((1 / 3)%:E * \d_true U)%E. +Proof. +rewrite /guard 2!execP_letin execP_sample execD_bernoulli execD_real. +rewrite execP_if/= !execP_return !exp_var'E !(execD_var_erefl "p") execD_unit. +rewrite execP_score execD_real/=. +rewrite letin'E/= integral_bernoulli//=; last lra. +rewrite !letin'E !iteE/= integral_dirac// ge0_integral_mscale//=. +by rewrite normr0 mul0e !mule0 !adde0 !diracT !mul1e. +Qed. + +End guard. + +Section test_binomial. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Context (R : realType). + +Definition binomial_le : @exp R _ [::] Bool := + [let "a2" := Sample {exp_binomial 3 [{1 / 2}:R]} in + return {1}:N <= #{"a2"}]. + +Lemma exec_binomial_le t U : + execP binomial_le t U = ((7 / 8)%:E * \d_true U + + (1 / 8)%:E * \d_false U)%E. +Proof. +rewrite /binomial_le execP_letin execP_sample execP_return execD_rel execD_nat. +rewrite exp_var'E (execD_var_erefl "a2") execD_binomial/= !execD_real/=. +rewrite letin'E//= integral_binomial//=; [lra|move=> _]. +rewrite !big_ord_recl big_ord0//=. +rewrite /bump. +rewrite !binS/= !bin0 bin1 bin2 bin_small// addn0. +rewrite addeC adde0. +congr (_ + _)%:E. + rewrite !indicE !(mul0n,add0n,lt0n,mul1r)/=. + rewrite -!mulrDl; congr (_ * _). + rewrite /onem. + lra. +rewrite !expr0 ltnn indicE/= !(mul1r,mul1e) /onem. +lra. +Qed. + +Definition binomial_guard : @exp R _ [::] Nat := + [let "a1" := Sample {exp_binomial 3 [{1 / 2}:R]} in + let "_" := if #{"a1"} == {1}:N then return TT else Score {0}:R in + return #{"a1"}]. + +Lemma exec_binomial_guard t U : + execP binomial_guard t U = ((3 / 8)%:E * \d_1%N U)%E. +Proof. +rewrite /binomial_guard !execP_letin execP_sample execP_return execP_if. +rewrite !exp_var'E execD_rel !(execD_var_erefl "a1") execP_return. +rewrite execD_unit execD_binomial execD_nat execP_score !execD_real. +rewrite !letin'E//=. +rewrite integral_binomial//=; [lra|move=> _]. +rewrite !big_ord_recl big_ord0. +rewrite /bump/=. +rewrite !binS/= !bin0 bin1 bin2 bin_small//. +rewrite !letin'E//= !iteE/=. +rewrite !ge0_integral_mscale//=. +rewrite !integral_dirac//= !diracE/=. +rewrite /bump/=. +rewrite !(normr0,mul0e,mule0,add0e,add0n,mul1e,adde0). +rewrite mem_set//=. +rewrite /onem mul1e. +congr (_%:E * _)%E. +lra. +Qed. + +End test_binomial. + +Section beta_bernoulli_bernoulli. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Context (R : realType). +Local Notation mu := (@lebesgue_measure R). + +(* TODO: move? *) +Lemma integrable_bernoulli_XMonemX01 a b U + (mu : {measure set (g_sigma_algebraType R.-ocitv.-measurable) -> \bar R}) : + measurable U -> (mu `[0%R, 1%R]%classic < +oo)%E -> + mu.-integrable `[0, 1] (fun x => bernoulli (XMonemX01 a b x) U). +Proof. +move=> mU mu01oo. +apply/integrableP; split. + apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. + by apply: measurable_funTS; exact: measurable_XMonemX01. +apply: (@le_lt_trans _ _ (\int[mu]_(x in `[0%R, 1%R]) cst 1 x)%E). + apply: ge0_le_integral => //=. + apply/measurable_funTS/measurableT_comp => //=. + apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. + exact: measurable_XMonemX01. + by move=> x _; rewrite gee0_abs// probability_le1. +by rewrite integral_cst//= mul1e. +Qed. + +Let measurable_bernoulli_XMonemX01 U : + measurable_fun setT (fun x : R => bernoulli (XMonemX01 2 1 x) U). +Proof. +apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. +exact: measurable_XMonemX01. +Qed. + +Lemma beta_bernoulli_bernoulli U : measurable U -> + @execP R [::] _ [let "p" := Sample {exp_beta 6 4} in + Sample {exp_bernoulli [#{"p"}]}] tt U = + @execP R [::] _ [Sample {exp_bernoulli [{3 / 5}:R]}] tt U. +Proof. +move=> mU. +rewrite execP_letin !execP_sample execD_beta !execD_bernoulli/=. +rewrite !execD_real/= exp_var'E (execD_var_erefl "p")/=. +transitivity (beta_prob_bernoulli 6 4 1 0 U : \bar R). + rewrite /beta_prob_bernoulli !letin'E/=. + rewrite integral_Beta//=; last 2 first. + exact: measurable_bernoulli2. + exact: integral_beta_prob_bernoulli_lty. + rewrite integral_Beta//=; last 2 first. + by apply: measurable_funTS => /=; exact: measurable_bernoulli_XMonemX01. + rewrite integral_Beta//=. + + suff: mu.-integrable `[0%R, 1%R] + (fun x => bernoulli (XMonemX01 2 1 x) U * (beta_pdf 6 4 x)%:E)%E. + move=> /integrableP[_]. + under eq_integral. + move=> x _. + rewrite gee0_abs//; last first. + by rewrite mule_ge0// lee_fin beta_pdf_ge0. + over. + move=> ?. + by under eq_integral do rewrite gee0_abs//. + + apply: integrableMl => //=. + * apply: integrable_bernoulli_XMonemX01 => //=. + by rewrite lebesgue_measure_itv//= lte01 EFinN sube0 ltry. + * by apply: measurable_funTS; exact: measurable_beta_pdf. + * exact: bounded_beta_pdf_01. + + apply/measurableT_comp => //; apply: measurable_funTS => /=. + exact: measurable_bernoulli_XMonemX01. + + under eq_integral do rewrite gee0_abs//=. + have : (beta_prob 6 4 `[0%R, 1%R] < +oo :> \bar R)%E. + by rewrite -ge0_fin_numE// beta_prob_fin_num. + by move=> /(@integrable_bernoulli_XMonemX01 2 1 _ (beta_prob 6 4) mU) /integrableP[]. + rewrite [RHS]integral_mkcond. + apply: eq_integral => x _ /=. + rewrite patchE. + case: ifPn => x01. + by rewrite /XMonemX01 patchE x01 XMonemX0' expr1. + by rewrite /beta_pdf /XMonemX01 patchE (negbTE x01) mul0r mule0. +rewrite beta_prob_bernoulliE// !bernoulliE//=; last 2 first. + lra. + by rewrite div_betafun_ge0 div_betafun_le1. +by congr (_ * _ + _ * _)%:E; + rewrite /div_betafun/= /onem !betafunE/= !factE/=; field. +Qed. + +End beta_bernoulli_bernoulli. + +Section letinA. +Local Open Scope lang_scope. +Variable R : realType. + +Lemma letinA g x y t1 t2 t3 (xyg : x \notin dom ((y, t2) :: g)) + (e1 : @exp R P g t1) + (e2 : exp P [:: (x, t1) & g] t2) + (e3 : exp P [:: (y, t2) & g] t3) : + forall U, measurable U -> + execP [let x := e1 in + let y := e2 in + {@exp_weak _ _ [:: (y, t2)] _ _ (x, t1) e3 xyg}] ^~ U = + execP [let y := + let x := e1 in e2 in + e3] ^~ U. +Proof. +move=> U mU; apply/funext=> z1. +rewrite !execP_letin. +rewrite (execP_weak [:: (y, t2)]). +apply: letin'A => //= z2 z3. +rewrite /kweak /mctx_strong /=. +by destruct z3. +Qed. + +Example letinA12 : forall U, measurable U -> + @execP R [::] _ [let "y" := return {1}:R in + let "x" := return {2}:R in + return #{"x"}] ^~ U = + @execP R [::] _ [let "x" := + let "y" := return {1}:R in return {2}:R in + return #{"x"}] ^~ U. +Proof. +move=> U mU. +rewrite !execP_letin !execP_return !execD_real. +apply: funext=> x. +rewrite !exp_var'E /= !(execD_var_erefl "x")/=. +exact: letin'A. +Qed. + +End letinA. + +Section staton_bus. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Import Notations. +Context {R : realType}. + +Definition staton_bus_syntax0 : @exp R _ [::] _ := + [let "x" := Sample {exp_bernoulli [{2 / 7}:R]} in + let "r" := if #{"x"} then return {3}:R else return {10}:R in + let "_" := Score {exp_poisson 4 [#{"r"}]} in + return #{"x"}]. + +Definition staton_bus_syntax := [Normalize {staton_bus_syntax0}]. + +Let sample_bern : R.-sfker munit ~> mbool := + sample _ (measurableT_comp measurable_bernoulli (measurable_cst (2 / 7 : R)%R)). + +Let ite_3_10 : R.-sfker mbool * munit ~> measurableTypeR R := + ite macc0of2 (@ret _ _ _ (measurableTypeR R) R _ (kr 3)) (@ret _ _ _ (measurableTypeR R) R _ (kr 10)). + +Let score_poisson4 : R.-sfker measurableTypeR R * (mbool * munit) ~> munit := + score (measurableT_comp (measurable_poisson_pdf 4) (@macc0of2 _ _ (measurableTypeR R) _)). + +Let kstaton_bus' := + letin' sample_bern + (letin' ite_3_10 + (letin' score_poisson4 (ret macc2of4'))). + +Lemma eval_staton_bus0 : staton_bus_syntax0 -P> kstaton_bus'. +Proof. +apply: eval_letin. + by apply: eval_sample; apply: eval_bernoulli; exact: eval_real. +apply: eval_letin. + apply/evalP_if; [|exact/eval_return/eval_real..]. + rewrite exp_var'E. + by apply/execD_evalD; rewrite (execD_var_erefl "x")/=; congr existT. +apply: eval_letin. + apply/eval_score/eval_poisson. + rewrite exp_var'E. + by apply/execD_evalD; rewrite (execD_var_erefl "r")/=; congr existT. +apply/eval_return/execD_evalD. +by rewrite exp_var'E (execD_var_erefl "x")/=; congr existT. +Qed. + +Lemma exec_staton_bus0' : execP staton_bus_syntax0 = kstaton_bus'. +Proof. +rewrite 3!execP_letin execP_sample/= execD_bernoulli/= !execD_real. +rewrite /kstaton_bus'; congr letin'. +rewrite !execP_if !execP_return !execD_real/=. +rewrite exp_var'E (execD_var_erefl "x")/=. +have -> : measurable_acc_typ [:: Bool] 0 = macc0of2 by []. +congr letin'. +rewrite execP_score execD_poisson/=. +rewrite exp_var'E (execD_var_erefl "r")/=. +have -> : measurable_acc_typ [:: Real; Bool] 0 = macc0of2 by []. +congr letin'. +by rewrite exp_var'E (execD_var_erefl "x") /=; congr ret. +Qed. + +Lemma exec_staton_bus : execD staton_bus_syntax = + existT _ (normalize_pt kstaton_bus') (measurable_normalize_pt _). +Proof. by rewrite execD_normalize_pt exec_staton_bus0'. Qed. + +Let poisson4 := @poisson_pdf R 4%N. + +Let staton_bus_probability U := + ((2 / 7)%:E * (poisson4 3)%:E * \d_true U + + (5 / 7)%:E * (poisson4 10)%:E * \d_false U)%E. + +Lemma exec_staton_bus0 (U : set bool) : + execP staton_bus_syntax0 tt U = staton_bus_probability U. +Proof. +rewrite exec_staton_bus0' /staton_bus_probability /kstaton_bus'. +rewrite /sample_bern. +rewrite letin'E/=. +rewrite integral_bernoulli//=; last lra. +rewrite -!muleA; congr (_ * _ + _ * _)%E. +- rewrite letin'_iteT//. + rewrite letin'_retk//. + rewrite letin'_kret//. + rewrite /score_poisson4. + by rewrite /score/= /mscale/= ger0_norm//= poisson_pdf_ge0. +- by rewrite onem27. +- rewrite letin'_iteF//. + rewrite letin'_retk//. + rewrite letin'_kret//. + rewrite /score_poisson4. + by rewrite /score/= /mscale/= ger0_norm//= poisson_pdf_ge0. +Qed. + +End staton_bus. + +(* same as staton_bus module associativity of letin *) +Section staton_busA. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Import Notations. +Context {R : realType}. + +Definition staton_busA_syntax0 : @exp R _ [::] _ := + [let "x" := Sample {exp_bernoulli [{2 / 7}:R]} in + let "_" := + let "r" := if #{"x"} then return {3}:R else return {10}:R in + Score {exp_poisson 4 [#{"r"}]} in + return #{"x"}]. + +Definition staton_busA_syntax : exp _ [::] _ := + [Normalize {staton_busA_syntax0}]. + +Let sample_bern : R.-sfker munit ~> mbool := + sample _ (measurableT_comp measurable_bernoulli (measurable_cst (2 / 7 : R)%R)). + +Let ite_3_10 : R.-sfker mbool * munit ~> measurableTypeR R := + ite macc0of2 (@ret _ _ _ (measurableTypeR R) R _ (kr 3)) (@ret _ _ _ (measurableTypeR R) R _ (kr 10)). + +Let score_poisson4 : R.-sfker measurableTypeR R * (mbool * munit) ~> munit := + score (measurableT_comp (measurable_poisson_pdf 4) (@macc0of3' _ _ _ (measurableTypeR R) _ _)). + +(* same as kstaton_bus _ (measurable_poisson 4) but expressed with letin' + instead of letin *) +Let kstaton_busA' := + letin' sample_bern + (letin' + (letin' ite_3_10 + score_poisson4) + (ret macc1of3')). + +Lemma eval_staton_busA0 : staton_busA_syntax0 -P> kstaton_busA'. +Proof. +apply: eval_letin. + by apply: eval_sample; apply: eval_bernoulli; exact: eval_real. +apply: eval_letin. + apply: eval_letin. + apply/evalP_if; [|exact/eval_return/eval_real..]. + rewrite exp_var'E. + by apply/execD_evalD; rewrite (execD_var_erefl "x")/=; congr existT. + apply/eval_score/eval_poisson. + rewrite exp_var'E. + by apply/execD_evalD; rewrite (execD_var_erefl "r")/=; congr existT. +apply/eval_return. +by apply/execD_evalD; rewrite exp_var'E (execD_var_erefl "x")/=; congr existT. +Qed. + +Lemma exec_staton_busA0' : execP staton_busA_syntax0 = kstaton_busA'. +Proof. +rewrite 3!execP_letin execP_sample/= execD_bernoulli execD_real. +rewrite /kstaton_busA'; congr letin'. +rewrite !execP_if !execP_return !execD_real/=. +rewrite exp_var'E (execD_var_erefl "x")/=. +have -> : measurable_acc_typ [:: Bool] 0 = macc0of2 by []. +congr letin'. + rewrite execP_score execD_poisson/=. + rewrite exp_var'E (execD_var_erefl "r")/=. + by have -> : measurable_acc_typ [:: Real; Bool] 0 = macc0of3' by []. +by rewrite exp_var'E (execD_var_erefl "x") /=; congr ret. +Qed. + +Lemma exec_statonA_bus : execD staton_busA_syntax = + existT _ (normalize_pt kstaton_busA') (measurable_normalize_pt _). +Proof. by rewrite execD_normalize_pt exec_staton_busA0'. Qed. + +(* equivalence between staton_bus and staton_busA *) +Lemma staton_bus_staton_busA : + execP staton_bus_syntax0 = @execP R _ _ staton_busA_syntax0. +Proof. +rewrite /staton_bus_syntax0 /staton_busA_syntax0. +rewrite execP_letin. +rewrite [in RHS]execP_letin. +congr (letin' _). +set e1 := exp_if _ _ _. +set e2 := exp_score _. +set e3 := (exp_return _ in RHS). +pose f := @found _ Unit "x" Bool [::]. +have r_f : "r" \notin [seq i.1 | i <- ("_", Unit) :: untag (ctx_of f)] by []. +have H := @letinA _ _ _ _ _ _ + (lookup Unit (("_", Unit) :: untag (ctx_of f)) "x") + r_f e1 e2 e3. +apply/eq_sfkernel => /= x U. +have mU : + (@mtyp_disp R (lookup Unit (("_", Unit) :: untag (ctx_of f)) "x")).-measurable U. + by []. +move: H => /(_ U mU) /(congr1 (fun f => f x)) <-. +set e3' := exp_return _. +set e3_weak := exp_weak _ _ _ _. +rewrite !execP_letin. +suff: execP e3' = execP (e3_weak e3 r_f) by move=> <-. +rewrite execP_return/= exp_var'E (execD_var_erefl "x") /= /e3_weak. +rewrite (@execP_weak R [:: ("_", Unit)] _ ("r", Real) _ e3 r_f). +rewrite execP_return exp_var'E/= (execD_var_erefl "x") //=. +by apply/eq_sfkernel => /= -[[] [a [b []]]] U0. +Qed. + +Let poisson4 := @poisson_pdf R 4%N. + +Lemma exec_staton_busA0 U : execP staton_busA_syntax0 tt U = + ((2 / 7%:R)%:E * (poisson4 3%:R)%:E * \d_true U + + (5%:R / 7%:R)%:E * (poisson4 10%:R)%:E * \d_false U)%E. +Proof. by rewrite -staton_bus_staton_busA exec_staton_bus0. Qed. + +End staton_busA. + +Section letinC. +Local Open Scope lang_scope. +Variable (R : realType). + +Let weak_head g {t1 t2} x (e : @exp R P g t2) (xg : x \notin dom g) := + exp_weak P [::] _ (x, t1) e xg. + +Lemma letinC g t1 t2 (e1 : @exp R P g t1) (e2 : exp P g t2) + (x y : string) + (xy : infer (x != y)) (yx : infer (y != x)) + (xg : x \notin dom g) (yg : y \notin dom g) : + forall U, measurable U -> + execP [ + let x := e1 in + let y := {weak_head e2 xg} in + return (#x, #y)] ^~ U = + execP [ + let y := e2 in + let x := {weak_head e1 yg} in + return (#x, #y)] ^~ U. +Proof. +move=> U mU; apply/funext => z. +rewrite 4!execP_letin. +rewrite 2!(execP_weak [::] g). +rewrite 2!execP_return/=. +rewrite 2!execD_pair/=. +rewrite !exp_var'E. +- exact/(ctx_prf_tail _ yx)/ctx_prf_head. +- exact/ctx_prf_head. +- exact/ctx_prf_head. +- exact/(ctx_prf_tail _ xy)/ctx_prf_head. +- move=> h1 h2 h3 h4. + set g1 := [:: (y, t2), (x, t1) & g]. + set g2 := [:: (x, t1), (y, t2) & g]. + have /= := @execD_var R g1 x. + rewrite (negbTE yx) eqxx => /(_ h4) ->. + have /= := @execD_var R g2 x. + rewrite (negbTE yx) eqxx => /(_ h2) ->. + have /= := @execD_var R g1 y. + rewrite eqxx => /(_ h3) ->. + have /= := @execD_var R g2 y. + rewrite (negbTE xy) eqxx => /(_ h1) -> /=. + have -> : measurable_acc_typ [:: t2, t1 & map snd g] 0 = macc0of3' by []. + have -> : measurable_acc_typ [:: t2, t1 & map snd g] 1 = macc1of3' by []. + rewrite (letin'C _ _ (execP e2) + [the R.-sfker _ ~> _ of @kweak _ [::] _ (y, t2) _ (execP e1)]); + [ |by [] | by [] |by []]. + have -> : measurable_acc_typ [:: t1, t2 & map snd g] 0 = macc0of3' by []. + by have -> : measurable_acc_typ [:: t1, t2 & map snd g] 1 = macc1of3' by []. +Qed. + +Example letinC_ground_variables g t1 t2 (e1 : @exp R P g t1) (e2 : exp P g t2) + (x := "x") (y := "y") + (xg : x \notin dom g) (yg : y \notin dom g) : + forall U, measurable U -> + execP [ + let x := e1 in + let y := {exp_weak _ [::] _ (x, t1) e2 xg} in + return (#x, #y)] ^~ U = + execP [ + let y := e2 in + let x := {exp_weak _ [::] _ (y, t2) e1 yg} in + return (#x, #y)] ^~ U. +Proof. by move=> U mU; rewrite letinC. Qed. + +Example letinC_ground (g := [:: ("a", Unit); ("b", Bool)]) t1 t2 + (e1 : @exp R P g t1) + (e2 : exp P g t2) : + forall U, measurable U -> + execP [let "x" := e1 in + let "y" := e2 :+ {"x"} in + return (#{"x"}, #{"y"})] ^~ U = + execP [let "y" := e2 in + let "x" := e1 :+ {"y"} in + return (#{"x"}, #{"y"})] ^~ U. +Proof. move=> U mU; exact: letinC. Qed. + +End letinC. diff --git a/theories/lang_syntax_table_game.v b/theories/lang_syntax_table_game.v new file mode 100644 index 000000000..dbd2c1a37 --- /dev/null +++ b/theories/lang_syntax_table_game.v @@ -0,0 +1,717 @@ +Require Import String. +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval. +From mathcomp Require Import ring lra. +From mathcomp Require Import mathcomp_extra boolp classical_sets. +From mathcomp Require Import functions cardinality fsbigop. +From mathcomp Require Import signed reals ereal topology normedtype sequences. +From mathcomp Require Import esum measure charge lebesgue_measure numfun. +From mathcomp Require Import lebesgue_integral probability kernel prob_lang. +From mathcomp Require Import lang_syntax_util lang_syntax lang_syntax_examples. + +(**md**************************************************************************) +(* # Eddy's table game example *) +(* *) +(* ref: *) +(* - Chung-chieh Shan, Equational reasoning for probabilistic programming, *) +(* POPL TutorialFest 2018 *) +(* https://homes.luddy.indiana.edu/ccshan/rational/equational-handout.pdf *) +(* - Sean R Eddy, What is Bayesian statistics?, Nature Biotechnology 22(9), *) +(* 1177--1178 (2004) *) +(* *) +(******************************************************************************) + +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. +Lemma letin'_sample_uniform {R : realType} d d' (T : measurableType d) + (T' : measurableType d') (a b : R) (ab : (a < b)%R) + (u : R.-sfker [the measurableType _ of (_ * T)%type] ~> T') x y : + measurable y -> + letin' (sample_cst (uniform_prob ab)) u x y = + (b - a)^-1%:E * \int[lebesgue_measure]_(x0 in `[a, b]) u (x0, x) y. +Proof. +move=> my; rewrite letin'E/=. +rewrite integral_uniform//=. +move => _ /= Y mY /=. +have /= := measurable_kernel u _ my measurableT _ mY. +move/measurable_ysection => /(_ x) /=. +set A := (X in measurable X). +set B := (X in _ -> measurable X). +suff : A = B by move=> ->. +by rewrite {}/A {}/B !setTI ysectionE. +Qed. + +Local Open Scope lang_scope. +Lemma execP_letin_uniform {R : realType} + g t str (s0 s1 : exp P ((str, Real) :: g) t) : + (forall (p : R) x U, (0 <= p <= 1)%R -> + execP s0 (p, x) U = execP s1 (p, x) U) -> + forall x U, measurable U -> + execP [let str := Sample {@exp_uniform _ g 0 1 (@ltr01 R)} in {s0}] x U = + execP [let str := Sample {@exp_uniform _ g 0 1 (@ltr01 R)} in {s1}] x U. +Proof. +move=> s01 x U mU. +rewrite !execP_letin execP_sample execD_uniform/=. +rewrite !letin'_sample_uniform//. +congr *%E. +apply: eq_integral => p p01. +apply: s01. +by rewrite inE in p01. +Qed. +Local Close Scope lang_scope. +Local Close Scope ereal_scope. + +Section bounded. +Local Open Scope ring_scope. +Local Open Scope lang_scope. +Local Open Scope ereal_scope. +Context {R : realType}. + +Lemma bounded_id_01 : [bounded x0 : R^o | x0 in `[0%R, 1%R]%classic : set R]. +Proof. +exists 1%R; split => // y y1. +near=> M => /=. +rewrite (le_trans _ (ltW y1))//. +near: M. +move=> M /=. +rewrite in_itv/= => /andP[M0 M1]. +by rewrite ler_norml M1 andbT (le_trans _ M0). +Unshelve. all: by end_near. Qed. + +Lemma bounded_onem_01 : [bounded (`1- x : R^o) | x in `[0%R, 1%R]%classic : set R]. +Proof. +exists 1%R; split => // y y1. +near=> M => /=. +rewrite (le_trans _ (ltW y1))//. +near: M. +move=> M /=. +rewrite in_itv/= => /andP[M0 M1]. +rewrite ler_norml (@le_trans _ _ 0%R)//=. + by rewrite lerBlDr addrC -lerBlDr subrr. +by rewrite onem_ge0. +Unshelve. all: by end_near. Qed. + +Lemma bounded_cst_01 (x : R^o) : [bounded x | _ in `[0%R, 1%R]%classic : set R]. +Proof. +exists `|x|%R; split. + by rewrite num_real. +move=> y y1/= z. +rewrite in_itv/= => /andP[z0 z1]. +by rewrite (le_trans _ (ltW y1)). +Qed. + +Lemma bounded_norm (f : R -> R) : + [bounded f x : R^o | x in (`[0%R, 1%R]%classic : set R)] <-> + [bounded `|f x|%R : R^o | x in (`[0%R, 1%R]%classic : set R)]. +Proof. +split. + move=> [M [Mreal HM]]. + exists `|M|%R; split; first by rewrite normr_real. + move=> r Mr x/= x01. + by rewrite ger0_norm// HM// (le_lt_trans _ Mr)// ler_norm. +move=> [M [Mreal HM]]. +exists `|M|%R; split; first by rewrite normr_real. +move=> r Mr x/= x01. +rewrite -[leLHS]ger0_norm// HM//. +by rewrite (le_lt_trans _ Mr)// ler_norm. +Qed. + +Lemma boundedMl k (f : R -> R) : + [bounded f x : R^o | x in (`[0%R, 1%R]%classic : set R)] -> + [bounded (k * f x)%R : R^o | x in (`[0%R, 1%R]%classic : set R)]. +Proof. +move=> [M [Mreal HM]]. +exists `|k * M|%R; split; first by rewrite normr_real. +move=> r kMr x/= x01. +rewrite normrM. +have [->|k0] := eqVneq k 0%R. + by rewrite normr0 mul0r (le_trans _ (ltW kMr)). +rewrite -ler_pdivlMl ?normr_gt0//. +apply: HM => //. +rewrite ltr_pdivlMl ?normr_gt0//. +rewrite (le_lt_trans _ kMr)//. +by rewrite normrM ler_pM2l ?normr_gt0// ler_norm. +Qed. + +End bounded. + +Lemma measurable_bernoulli_expn {R : realType} U n : + measurable_fun [set: g_sigma_algebraType R.-ocitv.-measurable] + (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => bernoulli (`1-x ^+ n) U). +Proof. +apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. +by apply: measurable_funX => //=; exact: measurable_funB. +Qed. + +Lemma integrable_bernoulli_beta_pdf {R : realType} U : measurable U -> + (@lebesgue_measure R).-integrable [set: g_sigma_algebraType R.-ocitv.-measurable] + (fun x => bernoulli (1 - `1-x ^+ 3) U * (beta_pdf 6 4 x)%:E)%E. +Proof. +move=> mU. +have ? : measurable_fun [set: g_sigma_algebraType R.-ocitv.-measurable] + (fun x => bernoulli (1 - `1-x ^+ 3) U). + apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. + apply: measurable_funB => //; apply: measurable_funX => //. + exact: measurable_funB. +apply/integrableP; split => /=. + apply: emeasurable_funM => //=. + by apply/measurable_EFinP; exact: measurable_beta_pdf. +apply: (@le_lt_trans _ _ (\int[lebesgue_measure]_(x in `[0%R, 1%R]) (beta_pdf 6 4 x)%:E))%E. + rewrite [leRHS]integral_mkcond /=. + apply: ge0_le_integral => //=. + - apply: measurableT_comp => //; apply: emeasurable_funM => //. + by apply/measurable_EFinP; exact: measurable_beta_pdf. + - move=> x _ /=; rewrite patchE; case: ifPn => // _. + by rewrite lee_fin beta_pdf_ge0. + - apply/(measurable_restrict _ _ _) => //. + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_beta_pdf. + - move=> x _. + rewrite patchE; case: ifPn => x01. + rewrite gee0_abs//. + rewrite gee_pMl// ?probability_le1//. + by rewrite ge0_fin_numE// (le_lt_trans (probability_le1 _ _))// ltry. + by rewrite lee_fin beta_pdf_ge0. + by rewrite mule_ge0// lee_fin beta_pdf_ge0. + by rewrite /beta_pdf /XMonemX01 patchE (negbTE x01) mul0r mule0 abse0. +apply: (@le_lt_trans _ _ + (\int[lebesgue_measure]_(x in `[0%R, 1%R]) (betafun 6 4)^-1%:E)%E); last first. + by rewrite integral_cst//= lebesgue_measure_itv/= lte01 EFinN sube0 mule1 ltry. +apply: ge0_le_integral => //=. +- by move=> ? _; rewrite lee_fin beta_pdf_ge0. +- by apply/measurable_funTS/measurableT_comp => //; exact: measurable_beta_pdf. +- by move=> ? _; rewrite lee_fin invr_ge0// betafun_ge0. +- by move=> x _; rewrite lee_fin beta_pdf_le_betafunV. +Qed. + +Section game_programs. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. +Local Open Scope lang_scope. +Context (R : realType). +Local Notation mu := lebesgue_measure. + +Definition guard {g} str n : @exp R P [:: (str, _) ; g] _ := + [if #{str} == {n}:N then return TT else Score {0}:R]. + +Definition prog0 : @exp R _ [::] _ := + [Normalize + let "p" := Sample {exp_uniform 0 1 (@ltr01 R)} in + let "x" := Sample {exp_binomial 8 [#{"p"}]} in + let "_" := {guard "x" 5} in + let "y" := Sample {exp_binomial 3 [#{"p"}]} in + return {1}:N <= #{"y"}]. + +Definition tail1 : @exp R _ [:: ("_", Unit); ("x", Nat) ; ("p", Real)] _ := + [Sample {exp_bernoulli [{1}:R - {[{1}:R - #{"p"}]} ^+ {3}]}]. + +Definition tail2 : @exp R _ [:: ("_", Unit); ("p", Real)] _ := + [Sample {exp_bernoulli [{1}:R - {[{1}:R - #{"p"}]} ^+ {3}]}]. + +Definition tail3 : @exp R _ [:: ("p", Real); ("_", Unit)] _ := + [Sample {exp_bernoulli [{1}:R - {[{1}:R - #{"p"}]} ^+ {3}]}]. + +Definition prog1 : @exp R _ [::] _ := + [Normalize + let "p" := Sample {exp_uniform 0 1 (@ltr01 R)} in + let "x" := Sample {exp_binomial 8 [#{"p"}]} in + let "_" := {guard "x" 5} in + {tail1}]. + +Definition prog2 : @exp R _ [::] _ := + [Normalize + let "p" := Sample {exp_uniform 0 1 (@ltr01 R)} in + let "_" := + Score {[{56}:R * #{"p"} ^+ {5} * {[{1}:R - #{"p"}]} ^+ {3}]} in + {tail2}]. + +Definition prog2' : @exp R _ [::] _ := + [Normalize + let "p" := Sample {exp_beta 1 1} in + let "_" := Score + {[{56}:R * #{"p"} ^+ {5} * {[{1}:R - #{"p"}]} ^+ {3}]} in + {tail2}]. + +Definition prog3 : @exp R _ [::] _ := + [Normalize + let "_" := Score {1 / 9}:R in + let "p" := Sample {exp_beta 6 4} in + {tail3}]. + +Definition prog4 : @exp R _ [::] _ := + [Normalize + let "_" := Score {1 / 9}:R in + Sample {exp_bernoulli [{10 / 11}:R]}]. + +Definition prog5 : @exp R _ [::] _ := + [Normalize Sample {exp_bernoulli [{10 / 11}:R]}]. + +End game_programs. +Arguments tail1 {R}. +Arguments tail2 {R}. +Arguments guard {R g}. + +Section from_prog0_to_prog1. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. +Local Open Scope lang_scope. +Context (R : realType). +Local Notation mu := lebesgue_measure. + +Let prog01_subproof + (x : mctx (untag (ctx_of (recurse Unit (recurse Nat (found "p" Real [::])))))) + U : (0 <= x.2.2.1 <= 1)%R -> + execP [let "y" := Sample {exp_binomial 3 [#{"p"}]} in + return {1}:N <= #{"y"}] x U = + execP (@tail1 R) x U. +Proof. +move=> x01. +rewrite /tail1. +(* reduce lhs *) +rewrite execP_letin execP_sample execD_binomial/= execP_return/= execD_rel/=. +rewrite exp_var'E (execD_var_erefl "p")/=. +rewrite exp_var'E (execD_var_erefl "y")/=. +rewrite execD_nat/=. +rewrite [LHS]letin'E/=. +(* reduce rhs *) +rewrite execP_sample/= execD_bernoulli/= (@execD_bin _ _ binop_minus)/=. +rewrite execD_real/= execD_pow/= (@execD_bin _ _ binop_minus)/= execD_real/=. +rewrite (execD_var_erefl "p")/=. +exact/integral_binomial_prob. +Qed. + +Lemma prog01 : execD (@prog0 R) = execD (@prog1 R). +Proof. +rewrite /prog0 /prog1. +apply: congr_normalize => y A. +apply: execP_letin_uniform => // p [] B p01. +apply: congr_letinr => a1 V0. +apply: congr_letinr => -[] V1. +exact: prog01_subproof. +Qed. + +End from_prog0_to_prog1. + +Section from_prog1_to_prog2. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. +Local Open Scope lang_scope. +Context (R : realType). +Local Notation mu := lebesgue_measure. + +Let prog12_subproof (y : @mctx R [::]) (V : set (@mtyp R Bool)) + (p : R) + (x : projT2 (existT measurableType default_measure_display unit)) + (U : set (mtyp Bool)) + (p0 : (0 <= p)%R) + (p1 : (p <= 1)%R) : + \int[binomial_prob 8 p]_y0 + execP [let "_" := {guard "x" 5} in {tail1}] + (y0, (p, x)) U = + \int[mscale (NngNum (normr_ge0 (56 * XMonemX 5 3 p))) \d_tt]_y0 + execP tail2 (y0, (p, x)) U. +Proof. +rewrite integral_binomial//=. +rewrite (bigD1 (inord 5))//=. +rewrite big1 ?adde0; last first. + move=> i i5. + rewrite execP_letin/= execP_if/= execD_rel/=. + rewrite exp_var'E/= (execD_var_erefl "x")/=. + rewrite execD_nat/= execP_score/= execD_real/= execP_return/=. + rewrite letin'E iteE/=. + move: i => [[|[|[|[|[|[|[|[|[|//]]]]]]]]]]//= Hi in i5 *; + rewrite ?ge0_integral_mscale//= ?execD_real/= ?normr0 ?(mul0e,mule0)//. + by rewrite -val_eqE/= inordK in i5. +(* reduce lhs *) +rewrite -[(p ^+ _ * _ ^+ _)%R]/(XMonemX _ _ p). +rewrite execP_letin/= execP_if/= execD_rel/=. +rewrite exp_var'E/= (execD_var_erefl "x")/=. +rewrite execD_nat/= execP_score/= execD_real/= execP_return/=. +rewrite letin'E iteE/=. +rewrite inordK// eqxx. +rewrite integral_dirac//= execD_unit/= diracE mem_set// mul1e. +(* reduce rhs *) +rewrite ge0_integral_mscale//=. +rewrite integral_dirac//= diracE mem_set// mul1e. +rewrite ger0_norm ?mulr_ge0 ?subr_ge0//. +rewrite mulr_natl. +(* same score *) +congr *%E. +(* the tails are the same module the shape of the environment *) +rewrite /tail1 /tail2 !execP_sample/=. +rewrite !execD_bernoulli/=. +rewrite !(@execD_bin _ _ binop_minus)/=. +rewrite !execD_pow/=. +rewrite !execD_real/=. +rewrite !(@execD_bin _ _ binop_minus)/=. +by rewrite !execD_real/= !exp_var'E/= !(execD_var_erefl "p")/=. +Qed. + +Lemma prog12 : execD (@prog1 R) = execD (@prog2 R). +Proof. +apply: congr_normalize => y V. +apply: execP_letin_uniform => // p x U /andP[p0 p1]. +(* reduce the lhs *) +rewrite execP_letin execP_sample execD_binomial/=. +rewrite letin'E/=. +rewrite [in LHS]exp_var'E/= (execD_var_erefl "p")/=. +(* reduce the rhs *) +rewrite [in RHS]execP_letin execP_score/=. +rewrite letin'E/=. +do 2 rewrite (@execD_bin _ _ binop_mult)/=/=. +rewrite [in RHS]exp_var'E/=. +rewrite execD_pow/=. +rewrite (execD_var_erefl "p")/=. +rewrite execD_pow/=. +rewrite (@execD_bin _ _ binop_minus)/=/=. +rewrite 2!execD_real/=. +rewrite (execD_var_erefl "p")/=. +rewrite -(mulrA 56). +exact: prog12_subproof. +Qed. + +End from_prog1_to_prog2. + +Local Open Scope ereal_scope. + +Lemma measurable_bernoulli_onemXn {R : realType} U : + measurable_fun [set: g_sigma_algebraType R.-ocitv.-measurable] + (fun x => bernoulli (1 - `1-x ^+ 3) U). +Proof. +apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. +apply: measurable_funB => //. +by apply: measurable_funX; exact: measurable_funB. +Qed. + +Lemma bounded_norm_XnonemXn {R : realType} : + [bounded normr (56 * XMonemX 5 3 x)%R : R^o | x in `[0%R, 1%R] : set R]. +Proof. exact/(bounded_norm _).1/boundedMl/bounded_XMonemX. Qed. + +Lemma integrable_bernoulli_XMonemX {R : realType} U : + (beta_prob 1 1).-integrable [set: R] + (fun x => bernoulli (1 - `1-x ^+ 3) U * (normr (56 * XMonemX 5 3 x))%:E). +Proof. +apply/integrableP; split. + apply: emeasurable_funM; first exact: measurable_bernoulli_onemXn. + apply/measurable_EFinP => //; apply: measurableT_comp => //. + by apply: measurable_funM => //; exact: measurable_fun_XMonemX. +rewrite beta_prob_uniform integral_uniform//=. + rewrite subr0 invr1 mul1e. + suff : lebesgue_measure.-integrable `[0%R, 1%R] + (fun y : R => bernoulli (1 - `1-y ^+ 3) U * (normr (56 * XMonemX 5 3 y))%:E). + by move=> /integrableP[]. + apply: integrableMl => //=. + - apply/integrableP; split. + by apply: measurable_funTS; exact: measurable_bernoulli_onemXn. + have := @integral_beta_prob_bernoulli_onem_lty R 3 1%N 1%N U. + rewrite beta_prob_uniform integral_uniform//=; last first. + by apply: measurableT_comp => //=; exact: measurable_bernoulli_onemXn. + by rewrite subr0 invr1 mul1e. + - apply: @measurableT_comp => //=; apply: measurable_funM => //. + exact: measurable_fun_XMonemX. + exact: bounded_norm_XnonemXn. +apply: @measurableT_comp => //; apply: emeasurable_funM => //=. + exact: measurable_bernoulli_onemXn. +do 2 apply: @measurableT_comp => //=. +by apply: measurable_funM => //; exact: measurable_fun_XMonemX. +Qed. + +Section from_prog2_to_prog3. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. +Local Open Scope lang_scope. +Context (R : realType). +Local Notation mu := lebesgue_measure. + +Lemma prog22' : execD (@prog2 R) = execD (@prog2' R). +Proof. +apply: congr_normalize => // x U. +apply: congr_letinl => // y V. +rewrite !execP_sample execD_uniform/= execD_beta/=. +by rewrite beta_prob_uniform. +Qed. + +Lemma prog23 : execD (@prog2' R) = execD (@prog3 R). +Proof. +apply: congr_normalize => x U. +(* reduce the LHS *) +rewrite 2![in LHS]execP_letin. +rewrite ![in LHS]execP_sample. +rewrite [in LHS]execP_score. +rewrite [in LHS]execD_beta/=. +rewrite [in LHS]execD_bernoulli. +rewrite 2![in LHS](@execD_bin _ _ binop_mult)/=. +rewrite 2![in LHS]execD_pow/=. +rewrite 2![in LHS](@execD_bin _ _ binop_minus)/=. +rewrite 3![in LHS]execD_real. +rewrite [in LHS]exp_var'E [in LHS](execD_var_erefl "p")/=. +rewrite [in LHS]execD_pow/=. +rewrite [in LHS](@execD_bin _ _ binop_minus)/=. +rewrite [in LHS]execD_real. +rewrite [in LHS]exp_var'E [in LHS](execD_var_erefl "p")/=. +(* reduce the RHS *) +rewrite [in RHS]execP_letin. +rewrite [in RHS]execP_score. +rewrite [in RHS]execP_letin/=. +rewrite [in RHS]execP_sample/=. +rewrite [in RHS]execD_beta/=. +rewrite [in RHS]execP_sample/=. +rewrite [in RHS]execD_bernoulli/=. +rewrite [in RHS]execD_real/=. +rewrite [in RHS](@execD_bin _ _ binop_minus)/=. +rewrite [in RHS]execD_real/=. +rewrite [in RHS]execD_pow/=. +rewrite [in RHS](@execD_bin _ _ binop_minus)/=. +rewrite [in RHS]exp_var'E [in RHS](execD_var_erefl "p")/=. +rewrite [in RHS]execD_real/=. +rewrite [LHS]letin'E/=. +under eq_integral => y _. + rewrite letin'E/=. + rewrite integral_cst//= /mscale/= diracT mule1 -mulrA -/(XMonemX _ _ _). + over. +rewrite [RHS]letin'E/=. +under [in RHS]eq_integral => y _. + rewrite letin'E/=. + over. +rewrite /=. +rewrite [RHS]ge0_integral_mscale//=; last first. + by move=> _ _; rewrite integral_ge0. +rewrite integral_Beta//=; last 2 first. + - apply: emeasurable_funM => //=. + exact: measurable_bernoulli_onemXn. + apply/measurable_EFinP; apply: measurableT_comp => //. + by apply: measurable_funM => //; exact: measurable_fun_XMonemX. + - by have /integrableP[] := @integrable_bernoulli_XMonemX R U. +rewrite ger0_norm// integral_dirac// diracT mul1e. +rewrite integral_Beta/=; [|by []|exact: measurable_bernoulli_onemXn + |exact: integral_beta_prob_bernoulli_onem_lty]. +rewrite -integralZl//=; last exact: integrable_bernoulli_beta_pdf. +apply: eq_integral => y _. +rewrite [in RHS]muleCA -[in LHS]muleA; congr *%E. +rewrite /beta_pdf /XMonemX01 2!patchE; case: ifPn => [y01|_]; last first. + by rewrite !mul0r 2!mule0. +rewrite ger0_norm; last first. + by rewrite mulr_ge0// XMonemX_ge0//; rewrite inE in y01. +rewrite [X in _ = _ * X]EFinM [in RHS]muleCA. +rewrite /= XMonemX00 mul1r [in LHS](mulrC 56) [in LHS]EFinM -[in LHS]muleA; congr *%E. +by rewrite !betafunE/= !factE/= -EFinM; congr EFin; lra. + +Qed. + +End from_prog2_to_prog3. + +Local Open Scope ereal_scope. +(* TODO: move? *) +Lemma int_beta_prob01 {R : realType} (f : R -> R) a b U : + measurable_fun [set: R] f -> + (forall x, x \in `[0%R, 1%R] -> 0 <= f x <= 1)%R -> + \int[beta_prob a b]_y bernoulli (f y) U = + \int[beta_prob a b]_(y in `[0%R, 1%R] : set R) bernoulli (f y) U. +Proof. +move=> mf f01. +rewrite [LHS]integral_Beta//=; last 2 first. + apply: measurable_funTS. + by apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. + exact: integral_beta_prob_bernoulli_lty. +rewrite [RHS]integral_Beta//; last 2 first. + apply/measurable_funTS => //=. + by apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. + apply: (le_lt_trans _ (lang_syntax.integral_beta_prob_bernoulli_lty a b U mf f01)). + apply: ge0_subset_integral => //=; apply: measurableT_comp => //=. + by apply: (measurableT_comp (measurable_bernoulli2 _)) => //=. +rewrite [RHS]integral_mkcond/=; apply: eq_integral => x _ /=. +rewrite !patchE; case: ifPn => // x01. +by rewrite /beta_pdf /XMonemX01 patchE (negbTE x01) mul0r mule0. +Qed. + +Lemma expr_onem_01 {R : realType} (x : R) : x \in `[0%R, 1%R] -> + (0 <= `1-x ^+ 3 <= 1)%R. +Proof. +rewrite in_itv/= => /andP[x0 x1]. +rewrite exprn_ge0 ?subr_ge0//= exprn_ile1// ?subr_ge0//. +by rewrite lerBlDl -lerBlDr subrr. +Qed. + +Lemma int_beta_prob_bernoulli {R : realType} (U : set (@mtyp R Bool)) : + \int[beta_prob 6 4]_y bernoulli (`1-y ^+ 3) U = bernoulli (1 / 11) U :> \bar R. +Proof. +rewrite int_beta_prob01//; last 2 first. + by apply: measurable_funX => //; exact: measurable_funB. + exact: expr_onem_01. +have := @beta_prob_bernoulliE R 6 4 0 3 U isT isT. +rewrite /beta_prob_bernoulli. +under eq_integral. + move=> x x0. + rewrite /XMonemX01 patchE x0 XMonemX0. + over. +rewrite /= => ->; congr bernoulli. +by rewrite /div_betafun addn0 !betafunE/= !factE/=; field. +Qed. + +Lemma dirac_bool {R : realType} (U : set bool) : + \d_false U + \d_true U = (\sum_(x \in U) (1%E : \bar R))%R. +Proof. +have [| | |] := set_bool U => /eqP ->; rewrite !diracE. +- by rewrite memNset// mem_set//= fsbig_set1 add0e. +- by rewrite mem_set// memNset//= fsbig_set1 adde0. +- by rewrite !in_set0 fsbig_set0 adde0. +- rewrite !in_setT setT_bool fsbigU0//=; last by move=> x [->]. + by rewrite !fsbig_set1. +Qed. + +Lemma int_beta_prob_bernoulli_onem {R : realType} (U : set (@mtyp R Bool)) : + \int[beta_prob 6 4]_y bernoulli (`1-(`1-y ^+ 3)) U = bernoulli (10 / 11) U :> \bar R. +Proof. +transitivity (\d_false U + \d_true U - bernoulli (1 / 11) U : \bar R)%E; last first. + rewrite /bernoulli ifT; last lra. + rewrite ifT; last lra. + apply/eqP; rewrite sube_eq//; last first. + rewrite ge0_adde_def// inE. + by apply/sume_ge0 => //= b _; rewrite lee_fin bernoulli_pmf_ge0//; lra. + by apply/sume_ge0 => //= b _; rewrite lee_fin bernoulli_pmf_ge0//; lra. + rewrite -fsbig_split//=. + under eq_fsbigr. + move=> /= x _. + rewrite -EFinD /bernoulli_pmf [X in X%:E](_ : _ = 1%R); last first. + case: x => //; lra. + over. + by rewrite /= dirac_bool. +rewrite -int_beta_prob_bernoulli. +apply/esym/eqP; rewrite sube_eq//; last first. + by rewrite ge0_adde_def// inE; exact: integral_ge0. +rewrite int_beta_prob01; last 2 first. + apply: measurable_funB => //; apply: measurable_funX => //. + exact: measurable_funB. + move=> x x01. + by rewrite subr_ge0 andbC lerBlDr -lerBlDl subrr expr_onem_01. +rewrite [X in _ == _ + X]int_beta_prob01; last 2 first. + by apply: measurable_funX => //; exact: measurable_funB. + exact: expr_onem_01. +rewrite -ge0_integralD//=; last 2 first. + apply: (@measurableT_comp _ _ _ _ _ _ (bernoulli ^~ U)) => /=. + exact: measurable_bernoulli2. + apply: measurable_funB => //=; apply: measurable_funX => //=. + exact: measurable_funB. + apply: (@measurableT_comp _ _ _ _ _ _ (bernoulli ^~ U)) => /=. + exact: measurable_bernoulli2. + by apply: measurable_funX => //=; exact: measurable_funB. +apply/eqP; transitivity + (\int[beta_prob 6 4]_(x in `[0%R, 1%R]) (\d_false U + \d_true U) : \bar R). + by rewrite integral_cst//= beta_prob01 mule1 EFinD. +apply: eq_integral => /= x x01. +rewrite /bernoulli subr_ge0 lerBlDr -lerBlDl subrr andbC. +rewrite (_ : (_ <= _ <= _)%R); last first. + by apply: expr_onem_01; rewrite inE in x01. +rewrite -fsbig_split//=. +under eq_fsbigr. + move=> /= y yU. + rewrite -EFinD /bernoulli_pmf. + rewrite [X in X%:E](_ : _ = 1%R); last first. + by case: ifPn => _; rewrite subrK. + over. +by rewrite /= dirac_bool. +Qed. + +Local Close Scope ereal_scope. + +Section from_prog3_to_prog4. +Local Open Scope ereal_scope. +Local Open Scope lang_scope. +Context (R : realType). + +(* NB: not used *) +Lemma prog34' U : + @execP R [::] _ [let "p" := Sample {exp_beta 6 4} in + Sample {exp_bernoulli [{[{1}:R - #{"p"}]} ^+ {3%N}]}] tt U = + @execP R [::] _ [Sample {exp_bernoulli [{1 / 11}:R]}] tt U. +Proof. +(* reduce the lhs *) +rewrite execP_letin. +rewrite execP_sample execD_beta/=. +rewrite execP_sample/= execD_bernoulli/=. +rewrite execD_pow/= (@execD_bin _ _ binop_minus) execD_real/=. +rewrite exp_var'E (execD_var_erefl "p")/=. +(* reduce the rhs *) +rewrite execP_sample execD_bernoulli/= execD_real. +(* semantics of lhs *) +rewrite letin'E/=. +exact: int_beta_prob_bernoulli. +Qed. + +Lemma prog34 l u U : + @execP R l _ [let "p" := Sample {exp_beta 6 4} in + Sample {exp_bernoulli [{1}:R - {[{1}:R - #{"p"}]} ^+ {3%N}]}] u U = + @execP R l _ [Sample {exp_bernoulli [{10 / 11}:R]}] u U. +Proof. +(* reduce the lhs *) +rewrite execP_letin. +rewrite execP_sample execD_beta/=. +rewrite execP_sample/= execD_bernoulli/=. +rewrite (@execD_bin _ _ binop_minus)/=. +rewrite execD_pow/= (@execD_bin _ _ binop_minus) execD_real/=. +rewrite exp_var'E (execD_var_erefl "p")/=. +(* reduce the rhs *) +rewrite execP_sample execD_bernoulli/= execD_real. +(* semantics of lhs *) +rewrite letin'E/=. +exact: int_beta_prob_bernoulli_onem. +Qed. + +End from_prog3_to_prog4. + +Section from_prog4_to_prog5. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. +Local Open Scope lang_scope. +Context (R : realType). +Local Notation mu := lebesgue_measure. + +Lemma normalize_score_bernoulli g p q (p0 : (0 < p)%R) (q01 : (0 <= q <= 1)%R) : + @execD R g _ [Normalize let "_" := Score {p}:R in + Sample {exp_bernoulli [{q}:R]}] = + execD [Normalize Sample {exp_bernoulli [{q}:R]}]. +Proof. +apply: eq_execD. +rewrite !execD_normalize_pt/= !execP_letin !execP_score. +rewrite !execP_sample !execD_bernoulli !execD_real/=. +apply: funext=> x. +apply: eq_probability=> /= U. +rewrite !normalizeE/=. +rewrite !bernoulliE//=; [|lra..]. +rewrite !diracT !mule1 -EFinD add_onemK onee_eq0/=. +rewrite !letin'E. +under eq_integral. + move=> A _ /=. + rewrite !bernoulliE//=; [|lra..]. + rewrite !diracT !mule1 -EFinD add_onemK. + over. +rewrite !ge0_integral_mscale//= (ger0_norm (ltW p0))//. +rewrite integral_dirac// !diracT !indicT /= !mule1 !mulr1. +rewrite add_onemK invr1 mule1. +rewrite gt_eqF ?lte_fin//=. +rewrite integral_dirac//= diracT mul1e. +by rewrite muleAC -EFinM divff ?gt_eqF// mul1e bernoulliE. +Qed. + +Lemma prog45 : execD (@prog4 R) = execD (@prog5 R). +Proof. by rewrite normalize_score_bernoulli//; lra. Qed. + +End from_prog4_to_prog5. + +Lemma from_prog0_to_prog5 {R : realType} : execD (@prog0 R) = execD (@prog5 R). +Proof. +rewrite prog01 prog12 prog22' prog23. +rewrite -prog45. +apply: congr_normalize => y V. +apply: congr_letinr => x U. +by rewrite -prog34. +Qed. diff --git a/theories/lang_syntax_toy.v b/theories/lang_syntax_toy.v new file mode 100644 index 000000000..ae6463adb --- /dev/null +++ b/theories/lang_syntax_toy.v @@ -0,0 +1,550 @@ +From Coq Require Import String Classical. +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg. +From mathcomp Require Import mathcomp_extra boolp. +From mathcomp Require Import signed reals topology normedtype. +From mathcomp Require Import lang_syntax_util. + +(******************************************************************************) +(* Intrinsically-typed concrete syntax for a toy language *) +(* *) +(* The main module provided by this file is "lang_intrinsic_tysc" which *) +(* provides an example of intrinsically-typed concrete syntax for a toy *) +(* language (a simplification of the syntax/evaluation formalized in *) +(* lang_syntax.v). Other modules provide even more simplified language for *) +(* pedagogical purposes. *) +(* *) +(* lang_extrinsic == non-intrinsic definition of expression *) +(* lang_intrinsic_ty == intrinsically-typed syntax *) +(* lang_intrinsic_sc == intrinsically-scoped syntax *) +(* lang_intrinsic_tysc == intrinsically-typed/scoped syntax *) +(* *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Set Printing Implicit Defensive. + +Import numFieldTopology.Exports. + +Local Open Scope classical_set_scope. +Local Open Scope ring_scope. +Local Open Scope ereal_scope. + +Section type. +Variables (R : realType). + +Inductive typ := Real | Unit. + +HB.instance Definition _ := gen_eqMixin typ. + +Definition iter_pair (l : list Type) : Type := + foldr (fun x y => (x * y)%type) unit l. + +Definition Type_of_typ (t : typ) : Type := + match t with + | Real => R + | Unit => unit + end. + +Definition ctx := seq (string * typ). + +Definition Type_of_ctx (g : ctx) := iter_pair (map (Type_of_typ \o snd) g). + +Goal Type_of_ctx [:: ("x", Real); ("y", Real)] = (R * (R * unit))%type. +Proof. by []. Qed. + +End type. + +Module lang_extrinsic. +Section lang_extrinsic. +Variable R : realType. +Implicit Types str : string. + +Inductive exp : Type := +| exp_unit : exp +| exp_real : R -> exp +| exp_var (g : ctx) t str : t = lookup Unit g str -> exp +| exp_add : exp -> exp -> exp +| exp_letin str : exp -> exp -> exp. +Arguments exp_var {g t}. + +Fail Example letin_once : exp := + exp_letin "x" (exp_real 1) (exp_var "x" erefl). +Example letin_once : exp := + exp_letin "x" (exp_real 1) (@exp_var [:: ("x", Real)] Real "x" erefl). + +End lang_extrinsic. +End lang_extrinsic. + +Module lang_intrinsic_ty. +Section lang_intrinsic_ty. +Variable R : realType. +Implicit Types str : string. + +Inductive exp : typ -> Type := +| exp_unit : exp Unit +| exp_real : R -> exp Real +| exp_var g t str : t = lookup Unit g str -> exp t +| exp_add : exp Real -> exp Real -> exp Real +| exp_letin t u : string -> exp t -> exp u -> exp u. +Arguments exp_var {g t}. + +Fail Example letin_once : exp Real := + exp_letin "x" (exp_real 1) (exp_var "x" erefl). +Example letin_once : exp Real := + exp_letin "x" (exp_real 1) (@exp_var [:: ("x", Real)] _ "x" erefl). + +End lang_intrinsic_ty. +End lang_intrinsic_ty. + +Module lang_intrinsic_sc. +Section lang_intrinsic_sc. +Variable R : realType. +Implicit Types str : string. + +Inductive exp : ctx -> Type := +| exp_unit g : exp g +| exp_real g : R -> exp g +| exp_var g t str : t = lookup Unit g str -> exp g +| exp_add g : exp g -> exp g -> exp g +| exp_letin g t str : exp g -> exp ((str, t) :: g) -> exp g. +Arguments exp_real {g}. +Arguments exp_var {g t}. +Arguments exp_letin {g t}. + +Declare Custom Entry expr. + +Notation "[ e ]" := e (e custom expr at level 5). +Notation "{ x }" := x (in custom expr, x constr). +Notation "x ':R'" := (exp_real x) (in custom expr at level 1). +Notation "x" := x (in custom expr at level 0, x ident). +Notation "$ x" := (exp_var x erefl) (in custom expr at level 1). +Notation "x + y" := (exp_add x y) + (in custom expr at level 2, left associativity). +Notation "'let' x ':=' e1 'in' e2" := (exp_letin x e1 e2) + (in custom expr at level 3, x constr, + e1 custom expr at level 2, e2 custom expr at level 3, + left associativity). + +Fail Example letin_once : exp [::] := + [let "x" := {1%R}:R in ${"x"}]. +Example letin_once : exp [::] := + [let "x" := {1%R}:R in {@exp_var [:: ("x", Real)] _ "x" erefl}]. + +Fixpoint acc (g : ctx) (i : nat) : + Type_of_ctx R g -> @Type_of_typ R (nth Unit (map snd g) i) := + match g return Type_of_ctx R g -> Type_of_typ R (nth Unit (map snd g) i) with + | [::] => match i with | O => id | j.+1 => id end + | _ :: _ => match i with + | O => fst + | j.+1 => fun H => acc j H.2 + end + end. +Arguments acc : clear implicits. + +Inductive eval : forall g (t : typ), exp g -> (Type_of_ctx R g -> Type_of_typ R t) -> Prop := +| eval_real g c : @eval g Real [c:R] (fun=> c) +| eval_plus g (e1 e2 : exp g) (v1 v2 : R) : + @eval g Real e1 (fun=> v1) -> + @eval g Real e2 (fun=> v2) -> + @eval g Real [e1 + e2] (fun=> v1 + v2) +| eval_var (g : ctx) str i : + i = index str (map fst g) -> eval [$ str] (acc g i). + +Goal @eval [::] Real [{1}:R] (fun=> 1). +Proof. exact: eval_real. Qed. +Goal @eval [::] Real [{1}:R + {2}:R] (fun=> 3). +Proof. exact/eval_plus/eval_real/eval_real. Qed. +Goal @eval [:: ("x", Real)] _ [$ {"x"}] (acc [:: ("x", Real)] 0). +Proof. exact: eval_var. Qed. + +End lang_intrinsic_sc. +End lang_intrinsic_sc. + +Module lang_intrinsic_tysc. +Section lang_intrinsic_tysc. +Variable R : realType. +Implicit Types str : string. + +Inductive typ := Real | Unit | Pair : typ -> typ -> typ. + +HB.instance Definition _ := gen_eqMixin typ. + +Fixpoint mtyp (t : typ) : Type := + match t with + | Real => R + | Unit => unit + | Pair t1 t2 => (mtyp t1 * mtyp t2) + end. + +Definition ctx := seq (string * typ). + +Definition Type_of_ctx (g : ctx) := iter_pair (map (mtyp \o snd) g). + +Goal Type_of_ctx [:: ("x", Real); ("y", Real)] = (R * (R * unit))%type. +Proof. by []. Qed. + +Inductive exp : ctx -> typ -> Type := +| exp_unit g : exp g Unit +| exp_real g : R -> exp g Real +| exp_var g t str : t = lookup Unit g str -> exp g t +| exp_add g : exp g Real -> exp g Real -> exp g Real +| exp_pair g t1 t2 : exp g t1 -> exp g t2 -> exp g (Pair t1 t2) +| exp_letin g t1 t2 x : exp g t1 -> exp ((x, t1) :: g) t2 -> exp g t2. + +Definition exp_var' str (t : typ) (g : find str t) := + @exp_var (untag (ctx_of g)) t str (ctx_prf g). + +Section no_bidirectional_hints. + +Arguments exp_unit {g}. +Arguments exp_real {g}. +Arguments exp_var {g t}. +Arguments exp_add {g}. +Arguments exp_pair {g t1 t2}. +Arguments exp_letin {g t1 t2}. +Arguments exp_var' str {t} g. + +Fail Example letin_add : exp [::] _ := + exp_letin "x" (exp_real 1) + (exp_letin "y" (exp_real 2) + (exp_add (exp_var "x" erefl) + (exp_var "y" erefl))). +Example letin_add : exp [::] _ := + exp_letin "x" (exp_real 1) + (exp_letin "y" (exp_real 2) + (exp_add (@exp_var [:: ("y", Real); ("x", Real)] _ "x" erefl) + (exp_var "y" erefl))). +Reset letin_add. + +Declare Custom Entry expr. + +Notation "[ e ]" := e (e custom expr at level 5). +Notation "{ x }" := x (in custom expr, x constr). +Notation "x ':R'" := (exp_real x) (in custom expr at level 1). +Notation "x" := x (in custom expr at level 0, x ident). +Notation "$ x" := (exp_var x erefl) (in custom expr at level 1). +Notation "# x" := (exp_var' x%string _) (in custom expr at level 1). +Notation "e1 + e2" := (exp_add e1 e2) + (in custom expr at level 2, + (* e1 custom expr at level 1, e2 custom expr at level 2, *) + left associativity). +Notation "( e1 , e2 )" := (exp_pair e1 e2) + (in custom expr at level 1). +Notation "'let' x ':=' e1 'in' e2" := (exp_letin x e1 e2) + (in custom expr at level 3, x constr, + e1 custom expr at level 2, e2 custom expr at level 3, + left associativity). + +Fail Definition let3_add_erefl (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + $a + $b]. +(* The term "[$ a]" has type "exp ?g2 (lookup Unit ?g2 a)" while it is expected to have type "exp ?g2 Real". *) + +Definition let3_pair_erefl (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + ($a, $b)]. + +Fail Definition let3_add (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + #a + #b]. +(* The term "[# a + # b]" has type + "exp (untag (ctx_of (recurse (str':=b) Real ?f))) Real" +while it is expected to have type "exp ((c, Real) :: ?g1) ?u1" +(cannot unify "(b, Real)" and "(c, Real)"). *) + +Fail Definition let3_pair (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + (#a, #b)]. +(* The term "[# a + # b]" has type "exp (untag (ctx_of (recurse (str':=b) Real ?f))) Real" while it is expected to have type + "exp ((c, Real) :: ?g1) ?u1" (cannot unify "(b, Real)" and "(c, Real)"). *) + +End no_bidirectional_hints. + +Section with_bidirectional_hints. + +Arguments exp_unit {g}. +Arguments exp_real {g}. +Arguments exp_var {g t}. +Arguments exp_add {g} &. +Arguments exp_pair {g} & {t1 t2}. +Arguments exp_letin {g} & {t1 t2}. +Arguments exp_var' str {t} g. + +Declare Custom Entry expr. + +Notation "[ e ]" := e (e custom expr at level 5). +Notation "{ x }" := x (in custom expr, x constr). +Notation "x ':R'" := (exp_real x) (in custom expr at level 1). +Notation "x" := x (in custom expr at level 0, x ident). +Notation "$ x" := (exp_var x%string erefl) (in custom expr at level 1). +Notation "# x" := (exp_var' x%string _) (in custom expr at level 1). +Notation "e1 + e2" := (exp_add e1 e2) + (in custom expr at level 2, + left associativity). +Notation "( e1 , e2 )" := (exp_pair e1 e2) + (in custom expr at level 1). +Notation "'let' x ':=' e1 'in' e2" := (exp_letin x e1 e2) + (in custom expr at level 3, x constr, + e1 custom expr at level 2, e2 custom expr at level 3, + left associativity). + +Fail Definition let2_add_erefl_bidi (a b : string) + (ba : infer (b != a)) (ab : infer (a != b)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + $a + $b]. + +Definition let2_add_erefl_bidi (a b : string) + (ba : infer (b != a)) (ab : infer (a != b)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + #a + #b]. + +Fail Definition let3_add_erefl_bidi (a b c d : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + $a + $b]. +(* The term "[$ a]" has type "exp [:: (c, Real); (b, Real); (a, Real)] (lookup Unit [:: (c, Real); (b, Real); (a, Real)] a)" +while it is expected to have type "exp [:: (c, Real); (b, Real); (a, Real)] Real" +(cannot unify "lookup Unit [:: (c, Real); (b, Real); (a, Real)] a" and "Real"). *) + +Definition let3_pair_erefl_bidi (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + ($a, $b)]. + +Check let3_pair_erefl_bidi. +(* exp [::] (Pair (lookup Unit [:: (c, Real); (b, Real); (a, Real)] a) (lookup Unit [:: (c, Real); (b, Real); (a, Real)] b)) *) + +Definition let3_add_bidi (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + #a + #b]. + +Definition let3_pair_bidi (a b c : string) + (ba : infer (b != a)) (ca : infer (c != a)) (cb : infer (c != b)) + (ab : infer (a != b)) (ac : infer (a != c)) (bc : infer (b != c)) + : exp [::] _ := [ + let a := {1}:R in + let b := {2}:R in + let c := {3}:R in + (#a , #b)]. + +Check let3_pair_bidi. +(* exp [::] (Pair Real Real) *) + +Example e0 : exp [::] _ := exp_real 1. +Example letin1 : exp [::] _ := + exp_letin "x" (exp_real 1) (exp_var "x" erefl). +Example letin2 : exp [::] _ := + exp_letin "x" (exp_real 1) + (exp_letin "y" (exp_real 2) + (exp_var "x" erefl)). + +Example letin_add : exp [::] _ := + exp_letin "x" (exp_real 1) + (exp_letin "y" (exp_real 2) + (exp_add (exp_var "x" erefl) + (exp_var "y" erefl))). +Reset letin_add. +Fail Example letin_add (x y : string) + (xy : infer (x != y)) (yx : infer (y != x)) : exp [::] _ := + exp_letin x (exp_real 1) + (exp_letin y (exp_real 2) + (exp_add (exp_var x erefl) (exp_var y erefl))). +Example letin_add (x y : string) + (xy : infer (x != y)) (yx : infer (y != x)) : exp [::] _ := + exp_letin x (exp_real 1) + (exp_letin y (exp_real 2) + (exp_add (exp_var' x _) (exp_var' y _))). +Reset letin_add. + +Example letin_add_custom : exp [::] _ := + [let "x" := {1}:R in + let "y" := {2}:R in + #{"x"} + #{"y"}]. + +Section eval. + +Fixpoint acc (g : ctx) (i : nat) : + Type_of_ctx g -> mtyp (nth Unit (map snd g) i) := + match g return Type_of_ctx g -> mtyp (nth Unit (map snd g) i) with + | [::] => match i with | O => id | j.+1 => id end + | _ :: _ => match i with + | O => fst + | j.+1 => fun H => acc j H.2 + end + end. +Arguments acc : clear implicits. + +Reserved Notation "e '-e->' v" (at level 40). + +Inductive eval : forall g t, exp g t -> (Type_of_ctx g -> mtyp t) -> Prop := +| eval_tt g : (exp_unit : exp g _) -e-> (fun=> tt) +| eval_real g c : (exp_real c : exp g _) -e-> (fun=> c) +| eval_plus g (e1 e2 : exp g Real) v1 v2 : + e1 -e-> v1 -> + e2 -e-> v2 -> + [e1 + e2] -e-> fun x => v1 x + v2 x +| eval_var g str : + let i := index str (map fst g) in + exp_var str erefl -e-> acc g i +| eval_pair g t1 t2 e1 e2 v1 v2 : + e1 -e-> v1 -> + e2 -e-> v2 -> + @exp_pair g t1 t2 e1 e2 -e-> fun x => (v1 x, v2 x) +| eval_letin g t t' str (e1 : exp g t) (e2 : exp ((str, t) :: g) t') v1 v2 : + e1 -e-> v1 -> + e2 -e-> v2 -> + exp_letin str e1 e2 -e-> (fun a => v2 (v1 a, a)) +where "e '-e->' v" := (@eval _ _ e v). + +Lemma eval_uniq g t (e : exp g t) u v : + e -e-> u -> e -e-> v -> u = v. +Proof. +move=> hu. +apply: (@eval_ind + (fun g t (e : exp g t) (u : Type_of_ctx g -> mtyp t) => + forall v, e -e-> v -> u = v)); last exact: hu. +all: (rewrite {g t e u v hu}). +- move=> g v. + inversion 1. + by inj_ex H3. +- move=> g c v. + inversion 1. + by inj_ex H3. +- move=> g e1 e2 v1 v2 ev1 IH1 ev2 IH2 v. + inversion 1. + inj_ex H0; inj_ex H1; subst. + inj_ex H5; subst. + by rewrite (IH1 _ H3) (IH2 _ H4). +- move=> g x i v. + inversion 1. + by inj_ex H6; subst. +- move=> g t1 t2 e1 e2 v1 v2 ev1 IH1 ev2 IH2 v. + inversion 1. + inj_ex H3; inj_ex H4; subst. + inj_ex H5; subst. + by rewrite (IH1 _ H6) (IH2 _ H7). +- move=> g t t' x0 e0 e1 v1 v2 ev1 IH1 ev2 IH2 v. + inversion 1. + inj_ex H5; subst. + inj_ex H6; subst. + inj_ex H7; subst. + by rewrite (IH1 _ H4) (IH2 _ H8). +Qed. + +Lemma eval_total g t (e : exp g t) : exists v, e -e-> v. +Proof. +elim: e. +- by eexists; exact: eval_tt. +- by eexists; exact: eval_real. +- move=> {}g {}t x e; subst t. + by eexists; exact: eval_var. +- move=> {}g e1 [v1] IH1 e2 [v2] IH2. + by eexists; exact: (eval_plus IH1 IH2). +- move=> {}g t1 t2 e1 [v1] IH1 e2 [v2] IH2. + by eexists; exact: (eval_pair IH1 IH2). +- move=> {}g {}t u x e1 [v1] IH1 e2 [v2] IH2. + by eexists; exact: (eval_letin IH1 IH2). +Qed. + +Definition exec g t (e : exp g t) : Type_of_ctx g -> mtyp t := + proj1_sig (cid (@eval_total g t e)). + +Lemma exec_eval g t (e : exp g t) v : exec e = v <-> e -e-> v. +Proof. +split. + by move=> <-; rewrite /exec; case: cid. +move=> ev. +by rewrite /exec; case: cid => f H/=; apply: eval_uniq; eauto. +Qed. + +Lemma eval_exec g t (e : exp g t) : e -e-> exec e. +Proof. by rewrite /exec; case: cid. Qed. + +Lemma exec_real g r : @exec g Real (exp_real r) = (fun=> r). +Proof. exact/exec_eval/eval_real. Qed. + +Lemma exec_var g str t H : + exec (@exp_var _ t str H) = + eq_rect _ (fun a => Type_of_ctx g -> mtyp a) + (acc g (index str (map fst g))) + _ (esym H). +Proof. +subst t. +rewrite {1}/exec. +case: cid => f H. +inversion H; subst g0 str0. +by inj_ex H6; subst f. +Qed. + +Lemma exp_var'E str t (f : find str t) H : exp_var' str f = exp_var str H. +Proof. by rewrite /exp_var'; congr exp_var. Qed. + +Lemma exec_letin g x t1 t2 (e1 : exp g t1) (e2 : exp ((x, t1) :: g) t2) : + exec [let x := e1 in e2] = (fun a => (exec e2) ((exec e1) a, a)). +Proof. by apply/exec_eval/eval_letin; exact: eval_exec. Qed. + +Goal ([{1}:R] : exp [::] _) -e-> (fun=> 1). +Proof. exact: eval_real. Qed. +Goal @eval [::] _ [{1}:R + {2}:R] (fun=> 3). +Proof. exact/eval_plus/eval_real/eval_real. Qed. +Goal @eval [:: ("x", Real)] _ (exp_var "x" erefl) (@acc [:: ("x", Real)] 0). +Proof. exact: eval_var. Qed. +Goal @eval [::] _ [let "x" := {1}:R in #{"x"}] (fun=> 1). +Proof. +apply/exec_eval; rewrite exec_letin/=. +apply/funext => t/=. +by rewrite exp_var'E exec_real/= exec_var/=. +Qed. + +Goal exec (g := [::]) [let "x" := {1}:R in #{"x"}] = (fun=> 1). +Proof. +rewrite exec_letin//=. +apply/funext => x. +by rewrite exp_var'E exec_var/= exec_real. +Qed. + +End eval. + +End with_bidirectional_hints. + +End lang_intrinsic_tysc. +End lang_intrinsic_tysc. diff --git a/theories/lang_syntax_util.v b/theories/lang_syntax_util.v new file mode 100644 index 000000000..253c1d8ff --- /dev/null +++ b/theories/lang_syntax_util.v @@ -0,0 +1,79 @@ +From Coq Require Import String. +From HB Require Import structures. +Require Import Classical_Prop. (* NB: to compile with Coq 8.17 *) +From mathcomp Require Import all_ssreflect. +From mathcomp Require Import signed. + +(******************************************************************************) +(* Shared by lang_syntax_*.v files *) +(******************************************************************************) + +HB.instance Definition _ := hasDecEq.Build string eqb_spec. + +Ltac inj_ex H := revert H; + match goal with + | |- existT ?P ?l (existT ?Q ?t (existT ?R ?u (existT ?T ?v ?v1))) = + existT ?P ?l (existT ?Q ?t (existT ?R ?u (existT ?T ?v ?v2))) -> _ => + (intro H; do 4 apply Classical_Prop.EqdepTheory.inj_pair2 in H) + | |- existT ?P ?l (existT ?Q ?t (existT ?R ?u ?v1)) = + existT ?P ?l (existT ?Q ?t (existT ?R ?u ?v2)) -> _ => + (intro H; do 3 apply Classical_Prop.EqdepTheory.inj_pair2 in H) + | |- existT ?P ?l (existT ?Q ?t ?v1) = + existT ?P ?l (existT ?Q ?t ?v2) -> _ => + (intro H; do 2 apply Classical_Prop.EqdepTheory.inj_pair2 in H) + | |- existT ?P ?l (existT ?Q ?t ?v1) = + existT ?P ?l (existT ?Q ?t' ?v2) -> _ => + (intro H; do 2 apply Classical_Prop.EqdepTheory.inj_pair2 in H) + | |- existT ?P ?l ?v1 = + existT ?P ?l ?v2 -> _ => + (intro H; apply Classical_Prop.EqdepTheory.inj_pair2 in H) + | |- existT ?P ?l ?v1 = + existT ?P ?l' ?v2 -> _ => + (intro H; apply Classical_Prop.EqdepTheory.inj_pair2 in H) +end. + +Set Implicit Arguments. +Unset Strict Implicit. +Set Printing Implicit Defensive. + +Section tagged_context. +Context {T : eqType} {t0 : T}. +Let ctx := seq (string * T). +Implicit Types (str : string) (g : ctx) (t : T). + +Definition dom g := map fst g. + +Definition lookup g str := nth t0 (map snd g) (index str (dom g)). + +Structure tagged_ctx := Tag {untag : ctx}. + +Structure find str t := Find { + ctx_of : tagged_ctx ; + #[canonical=no] ctx_prf : t = lookup (untag ctx_of) str}. + +Lemma ctx_prf_head str t g : t = lookup ((str, t) :: g) str. +Proof. by rewrite /lookup /= !eqxx. Qed. + +Lemma ctx_prf_tail str t g str' t' : + str' != str -> + t = lookup g str -> + t = lookup ((str', t') :: g) str. +Proof. +move=> str'str tg /=; rewrite /lookup/=. +by case: ifPn => //=; rewrite (negbTE str'str). +Qed. + +Definition recurse_tag g := Tag g. +Canonical found_tag g := recurse_tag g. + +Canonical found str t g : find str t := + @Find str t (found_tag ((str, t) :: g)) + (@ctx_prf_head str t g). + +Canonical recurse str t str' t' {H : infer (str' != str)} + (g : find str t) : find str t := + @Find str t (recurse_tag ((str', t') :: untag (ctx_of g))) + (@ctx_prf_tail str t (untag (ctx_of g)) str' t' H (ctx_prf g)). + +End tagged_context. +Arguments lookup {T} t0 g str. diff --git a/theories/lebesgue_integral.v b/theories/lebesgue_integral.v index 284f18536..d02164a02 100644 --- a/theories/lebesgue_integral.v +++ b/theories/lebesgue_integral.v @@ -5950,7 +5950,7 @@ Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). Variables (m1 : {sfinite_measure set X -> \bar R}). Variables (m2 : {sfinite_measure set Y -> \bar R}). Variables (f : X * Y -> \bar R) (f0 : forall xy, 0 <= f xy). -Hypothesis mf : measurable_fun setT f. +Hypothesis mf : measurable_fun [set: X * Y] f. Lemma sfinite_Fubini : \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). diff --git a/theories/lebesgue_measure.v b/theories/lebesgue_measure.v index 2a1e1993c..5ebbbcdfb 100644 --- a/theories/lebesgue_measure.v +++ b/theories/lebesgue_measure.v @@ -1450,6 +1450,7 @@ Lemma measurable_EFinP d (T : measurableType d) (R : realType) (D : set T) Proof. split=> [mf mD A mA|]; last by move=> mg; exact: measurableT_comp. rewrite [X in measurable X](_ : _ = D `&` (EFin \o g) @^-1` (EFin @` A)). + (* TODO: use measurable_image_EFin? *) by apply: mf => //; exists A => //; exists set0; [constructor|rewrite setU0]. congr (_ `&` _);rewrite eqEsubset; split=> [|? []/= _ /[swap] -[->//]]. by move=> ? ?; exact: preimage_image. diff --git a/theories/measure.v b/theories/measure.v index cc1637134..6b8898697 100644 --- a/theories/measure.v +++ b/theories/measure.v @@ -1507,6 +1507,12 @@ by move=> fg mf mD A mA; rewrite [X in measurable X](_ : _ = D `&` f @^-1` A); [exact: mf|exact/esym/eq_preimage]. Qed. +Lemma measurable_fun_eqP D (f g : T1 -> T2) : + {in D, f =1 g} -> measurable_fun D f <-> measurable_fun D g. +Proof. +by move=> eq_fg; split; apply/eq_measurable_fun => // ? ?; rewrite eq_fg. +Qed. + Lemma measurable_cst D (r : T2) : measurable_fun D (cst r : T1 -> _). Proof. by move=> mD /= Y mY; rewrite preimage_cst; case: ifPn; rewrite ?setIT ?setI0. @@ -1574,6 +1580,8 @@ End measurable_fun. solve [apply: measurable_id] : core. Arguments eq_measurable_fun {d1 d2 T1 T2 D} f {g}. #[deprecated(since="mathcomp-analysis 0.6.2", note="renamed `eq_measurable_fun`")] +Arguments measurable_fun_eqP {d1 d2 T1 T2 D} f {g}. +#[deprecated(since="mathcomp-analysis 0.6.2", note="renamed `eq_measurable_fun`")] Notation measurable_fun_ext := eq_measurable_fun (only parsing). #[deprecated(since="mathcomp-analysis 0.6.3", note="renamed `measurable_id`")] Notation measurable_fun_id := measurable_id (only parsing). @@ -3603,6 +3611,13 @@ HB.instance Definition _ := End mnormalize. +Lemma mnormalize_id d (T : measurableType d) (R : realType) + (P P' : probability T R) : mnormalize P P' = P. +Proof. +apply/funext => x; rewrite /mnormalize/= probability_setT. +by rewrite onee_eq0/= invr1 mule1. +Qed. + Section pdirac. Context d (T : measurableType d) (R : realType). @@ -4206,7 +4221,7 @@ Lemma le_outer_measure : {homo mu : A B / A `<=` B >-> A <= B}. Proof. move=> A B AB; pose B_ k := if k is 0%N then B else set0. have -> : mu B = \sum_(n ?; rewrite outer_measure_ge0. + rewrite nneseries_recl//; last by move=> ? ?; rewrite outer_measure_ge0. rewrite eseries_cond/= eseries0 ?adde0// => -[|]//= k _ _. by rewrite outer_measure0. apply: subset_outer_measure_sigma_subadditive => //. @@ -5252,8 +5267,10 @@ End partial_measurable_fun. #[global] Hint Extern 0 (measurable_fun _ (pair^~ _)) => solve [apply: measurable_pair2] : core. +(* [Lemma 14.13, Klenke 2014] *) Section measurable_section. -Context d1 d2 (T1 : measurableType d1) (T2 : measurableType d2). +Context d1 d2 d3 (T1 : measurableType d1) (T2 : measurableType d2) + (T3 : measurableType d3). Lemma measurable_xsection (A : set (T1 * T2)) (x : T1) : measurable A -> measurable (xsection A x). @@ -5271,6 +5288,14 @@ have mi : measurable_fun setT i by exact: measurable_pair2. by rewrite ysectionE -[X in measurable X]setTI; exact: mi. Qed. +Lemma measurable_prod1 (f : T1 * T2 -> T3) (y : T2) : + measurable_fun setT f -> measurable_fun setT (fun x => f (x, y)). +Proof. by move=> mf; exact: measurableT_comp. Qed. + +Lemma measurable_prod2 (f : T1 * T2 -> T3) (x : T1) : + measurable_fun setT f -> measurable_fun setT (fun y => f (x, y)). +Proof. by move=> mf; exact: measurableT_comp. Qed. + End measurable_section. Section absolute_continuity. diff --git a/theories/numfun.v b/theories/numfun.v index acb61398b..353cab228 100644 --- a/theories/numfun.v +++ b/theories/numfun.v @@ -311,6 +311,9 @@ Proof. by apply/funext=> x; rewrite indicE in_set0. Qed. Lemma indicI A B : \1_(A `&` B) = \1_A \* \1_B :> (_ -> R). Proof. by apply/funext=> u/=; rewrite !indicE in_setI -natrM mulnb. Qed. +Lemma indicC A : \1_(~` A) = (fun x => (~~ (x \in A))%:R) :> (_ -> R). +Proof. by apply/funext=> u/=; rewrite indicE in_setC. Qed. + Lemma image_indic D A : \1_D @` A = (if A `\` D != set0 then [set 0] else set0) `|` (if A `&` D != set0 then [set 1 : R] else set0). @@ -363,6 +366,25 @@ Qed. End indic_lemmas. +Lemma indic_bigcup T {R : realType} (A : (set T)^nat) (t : T) : + trivIset [set: nat] A -> + (\1_(\bigcup_n A n) t)%:E = (\sum_(0 <= n \bar R. +Proof. +move=> tA. +have [At|At] := eqVneq (\1_(\bigcup_n A n) t) (1%R : R). +- move: (At) => /eqP; rewrite pnatr_eq1 eqb1 => /asboolP[i _] Ait. + rewrite At (@nneseriesD1 _ _ i)//; last by move=> j; rewrite lee_fin. + rewrite indicE mem_set// eseries0 ?adde0// => j _/= ji. + rewrite indicE memNset// => Ajt. + move/trivIsetP : tA => /(_ j i Logic.I Logic.I ji). + by apply/eqP/set0P; exists t. +- have {}At : \1_(\bigcup_n A n) t = 0%R :> R. + by apply/eqP; move: At; rewrite pnatr_eq1 eqb1 pnatr_eq0 eqb0. + move: (At) => /eqP; rewrite pnatr_eq0 eqb0 notin_setE => AtC. + rewrite At eseries0// => j _ _. + by rewrite indicE memNset// => Ajt; apply: AtC; exists j. +Qed. + Lemma patch_indic T {R : numFieldType} (f : T -> R) (D : set T) : f \_ D = (f \* \1_D)%R. Proof. diff --git a/theories/prob_lang.v b/theories/prob_lang.v new file mode 100644 index 000000000..56cc11e21 --- /dev/null +++ b/theories/prob_lang.v @@ -0,0 +1,2183 @@ +(* 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 archimedean. +From mathcomp Require Import lra. +From mathcomp Require Import mathcomp_extra boolp classical_sets. +From mathcomp Require Import functions cardinality fsbigop. +From mathcomp Require Import reals ereal signed topology normedtype sequences. +From mathcomp Require Import esum measure lebesgue_measure numfun. +From mathcomp Require Import lebesgue_integral probability exp kernel. +From mathcomp Require Import ring lra. + +(**md**************************************************************************) +(* # Semantics of a probabilistic programming language using s-finite kernels *) +(* *) +(* Reference: *) +(* - R. Affeldt, C. Cohen, A. Saito. Semantics of probabilistic programs *) +(* using s-finite kernels in Coq. CPP 2023 *) +(* - S. Staton. Commutative Semantics for Probabilistic Programming. *) +(* ESOP 2017 *) +(* *) +(* ``` *) +(* poisson_pdf == Poisson pdf *) +(* exponential_pdf == exponential distribution pdf *) +(* measurable_sum X Y == the type X + Y, as a measurable type *) +(* *) +(* mscore f t := mscale `|f t| \d_tt *) +(* kscore f := fun=> mscore f *) +(* This is an s-finite kernel. *) +(* kite k1 k2 mf := kdirac mf \; kadd (kiteT k1) (kiteF k2). *) +(* k1 has type R.-sfker T ~> T'. *) +(* k2 has type R.-sfker T ~> T'. *) +(* mf is a proof that f : T -> bool is measurable. *) +(* KITE.kiteT k1 is k1 \o fst if f returne true *) +(* and zero otherwise. *) +(* KITE.kiteF k2 is k2 \o fst if f returne false *) +(* and zero otherwise. *) +(* *) +(* ret mf == access the context with f and return the result *) +(* mf is a proof that f is measurable. *) +(* This is a probability kernel. *) +(* sample mP == sample according to the probability measure P *) +(* mP is a proof that P is a measurable function. *) +(* sample_cst P == same as sample with a constant probability measure *) +(* normalize k P == normalize the kernel k into a probability kernel *) +(* P is a default probability in case normalization *) +(* is not possible. *) +(* normalize_pt k := normalize k point *) +(* ite mf k1 k2 == access the context with the boolean function f and *) +(* behaves as k1 or k2 according to the result *) +(* letin l k == execute l, augment the context, and execute k *) +(* fail := let _ := score 0 in ret point *) +(* 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) *) +(* acc0of2, acc1of2, etc. == accessor function *) +(* case_nat t u_ == case analysis on natural numbers *) +(* t has type R.-sfker T ~> nat *) +(* u_ has type nat -> R.-sfker T ~> T' *) +(* CASE_SUM.case_sum g k1 k2 == case analysis on a sum type *) +(* g has type R.-sfker X ~> (A + B). *) +(* k1 has type A -> R.-sfker X ~> Y. *) +(* k2 has type B -> R.-sfker X ~> Y. *) +(* kcounting == the counting measure as a kernel *) +(* iterate k mu == iteration *) +(* k has type R.-sfker G * A ~> (A + B). *) +(* mu is a proof that u : G -> A is measurable. *) +(* flift_neq == an s-finite kernel to test that two expressions *) +(* are different *) +(* ``` *) +(* *) +(* Examples: Staton's bus, von Neumann's trick, etc. *) +(* *) +(* ``` *) +(* mkswap k == given a kernel k : (Y * X) ~> Z, *) +(* returns a kernel of type (X * Y) ~> Z *) +(* letin' := mkcomp \o mkswap *) +(* ``` *) +(******************************************************************************) + +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. + +Lemma eq_probability R d (Y : measurableType d) (m1 m2 : probability Y R) : + (m1 =1 m2 :> (set Y -> \bar R)) -> m1 = m2. +Proof. +move: m1 m2 => [m1 +] [m2 +] /= m1m2. +move/funext : m1m2 => <- -[[c11 c12] [m01] [sf1] [sig1] [fin1] [sub1] [p1]] + [[c21 c22] [m02] [sf2] [sig2] [fin2] [sub2] [p2]]. +have ? : c11 = c21 by []. +subst c21. +have ? : c12 = c22 by []. +subst c22. +have ? : m01 = m02 by []. +subst m02. +have ? : sf1 = sf2 by []. +subst sf2. +have ? : sig1 = sig2 by []. +subst sig2. +have ? : fin1 = fin2 by []. +subst fin2. +have ? : sub1 = sub2 by []. +subst sub2. +have ? : p1 = p2 by []. +subst p2. +by f_equal. +Qed. + +(* NB: to be PRed to probability.v *) +Section poisson_pdf. +Variable R : realType. +Local Open Scope ring_scope. + +(* density function for Poisson *) +Definition poisson_pdf k r : R := + if r > 0 then r ^+ k / k`!%:R^-1 * expR (- r) else 1%:R. + +Lemma poisson_pdf_ge0 k r : 0 <= poisson_pdf k r. +Proof. +rewrite /poisson_pdf; case: ifPn => r0//. +by rewrite mulr_ge0 ?expR_ge0// mulr_ge0// exprn_ge0 ?ltW. +Qed. + +Lemma poisson_pdf_gt0 k r : 0 < r -> 0 < poisson_pdf k.+1 r. +Proof. +move=> r0; rewrite /poisson_pdf r0 mulr_gt0 ?expR_gt0//. +by rewrite divr_gt0// ?exprn_gt0// invr_gt0 ltr0n fact_gt0. +Qed. + +Lemma measurable_poisson_pdf k : measurable_fun setT (poisson_pdf k). +Proof. +rewrite /poisson_pdf; apply: measurable_fun_if => //. + exact: measurable_fun_ltr. +by apply: measurable_funM => /=; + [exact: measurable_funM|exact: measurableT_comp]. +Qed. + +Definition poisson3 := poisson_pdf 4 3%:R. (* 0.168 *) +Definition poisson10 := poisson_pdf 4 10%:R. (* 0.019 *) + +End poisson_pdf. + +Section exponential_pdf. +Variable R : realType. +Local Open Scope ring_scope. + +(* density function for exponential *) +Definition exponential_pdf x r : R := r * expR (- r * x). + +Lemma exponential_pdf_gt0 x r : 0 < r -> 0 < exponential_pdf x r. +Proof. by move=> r0; rewrite /exponential_pdf mulr_gt0// expR_gt0. Qed. + +Lemma exponential_pdf_ge0 x r : 0 <= r -> 0 <= exponential_pdf x r. +Proof. by move=> r0; rewrite /exponential_pdf mulr_ge0// expR_ge0. Qed. + +Lemma measurable_exponential_pdf x : measurable_fun setT (exponential_pdf x). +Proof. +apply: measurable_funM => //=; apply: measurableT_comp => //. +exact: measurable_funM. +Qed. + +End exponential_pdf. + +(* X + Y is a measurableType if X and Y are *) +HB.instance Definition _ (X Y : pointedType) := + isPointed.Build (X + Y)%type (@inl X Y point). + +Section measurable_sum. +Context d d' (X : measurableType d) (Y : measurableType d'). + +Definition measurable_sum : set (set (X + Y)) := setT. + +Let sum0 : measurable_sum set0. Proof. by []. Qed. + +Let sumC A : measurable_sum A -> measurable_sum (~` A). Proof. by []. Qed. + +Let sumU (F : (set (X + Y))^nat) : (forall i, measurable_sum (F i)) -> + measurable_sum (\bigcup_i F i). +Proof. by []. Qed. + +HB.instance Definition _ := @isMeasurable.Build default_measure_display + (X + Y)%type measurable_sum sum0 sumC sumU. + +End measurable_sum. + +Lemma measurable_fun_sum dA dB d' (A : measurableType dA) (B : measurableType dB) + (Y : measurableType d') (f : A -> Y) (g : B -> Y) : + measurable_fun setT f -> measurable_fun setT g -> + measurable_fun setT (fun tb : A + B => + match tb with inl a => f a | inr b => g b end). +Proof. +move=> mx my/= _ Z mZ /=; rewrite setTI /=. +rewrite (_ : _ @^-1` Z = inl @` (f @^-1` Z) `|` inr @` (g @^-1` Z)). + exact: measurableU. +apply/seteqP; split. + by move=> [a Zxa|b Zxb]/=; [left; exists a|right; exists b]. +by move=> z [/= [a Zxa <-//=]|]/= [b Zyb <-//=]. +Qed. + +(* TODO: measurable_fun_if_pair -> measurable_fun_if_pair_bool? *) +Lemma measurable_fun_if_pair_nat d d' (X : measurableType d) + (Y : measurableType d') (f g : X -> Y) (n : nat) : + measurable_fun setT f -> measurable_fun setT g -> + measurable_fun setT (fun xn => if xn.2 == n then f xn.1 else g xn.1). +Proof. +move=> mx my; apply: measurable_fun_ifT => //=. +- have h : measurable_fun [set: nat] (fun t => t == n) by []. + exact: (measurableT_comp h). +- exact: measurableT_comp. +- exact: measurableT_comp. +Qed. + +Module Notations. +Notation munit := Datatypes_unit__canonical__measure_Measurable. +Notation mbool := Datatypes_bool__canonical__measure_Measurable. +Notation mnat := Datatypes_nat__canonical__measure_Measurable. +End Notations. + +Lemma invr_nonneg_proof (R : numDomainType) (p : {nonneg R}) : + (0 <= (p%:num)^-1)%R. +Proof. by rewrite invr_ge0. Qed. + +(* TODO: move *) +Definition invr_nonneg (R : numDomainType) (p : {nonneg R}) := + NngNum (invr_nonneg_proof p). + +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_pdivrMr//= 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_pdivrMr// mul1r ler_nat. Qed.*) + +End constants. +Arguments p12 {R}. +Arguments p14 {R}. +(*Arguments p27 {R}.*) +Arguments p1S {R}. + +Section mscore. +Context d (T : measurableType d) (R : realType). +Variable f : T -> R. + +Definition mscore t : {measure set unit -> \bar R} := + let p := NngNum (normr_ge0 (f t)) in mscale p \d_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_cst. +by apply: measurableT_comp => //; exact: measurableT_comp. +Qed. + +End mscore. + +(* decomposition of score into finite kernels [Section 3.2, Staton ESOP 2017] *) +Module SCORE. +Section score. +Context d (T : measurableType d) (R : realType). +Variable f : T -> R. + +Definition k (mf : measurable_fun [set: T] 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: measurableT_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 => //=; apply/measurable_EFinP. +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 ?ge_floor. + rewrite -addn1 natrD natr_absz. + by rewrite (@ger0_norm _ (floor `|f t|)) ?floor_ge0// intrD1 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 [Section 3.2, Staton ESOP 2017] *) +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 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. +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_kernel 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 _ := @isSFiniteKernel_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 _ := 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 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_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_kernel 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 _ := @isSFiniteKernel_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'). + +(* NB: not used? *) +Definition mite (mf : measurable_fun setT f) : T -> set T' -> \bar R := + fun t => if f t then u1 t else u2 t. + +Hypothesis mf : measurable_fun [set: T] 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). + +End ite. + +Section insn2. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Definition ret (f : X -> Y) (mf : measurable_fun [set: X] f) + : R.-pker X ~> Y := kdirac mf. + +Definition sample (P : X -> pprobability Y R) (mP : measurable_fun [set: X] P) + : R.-pker X ~> Y := + kprobability mP. + +Definition sample_cst (P : pprobability Y R) : R.-pker X ~> Y := + sample (measurable_cst P). + +Definition normalize (k : R.-ker X ~> Y) P : X -> probability Y R := + knormalize k P. + +Definition normalize_pt (k : R.-ker X ~> Y) : X -> probability Y R := + normalize k point. + +Lemma measurable_normalize_pt (f : R.-ker X ~> Y) : + measurable_fun [set: X] (normalize_pt f : X -> pprobability Y R). +Proof. +apply: (@measurability _ _ _ _ _ _ + (@pset _ _ _ : set (set (pprobability Y R)))) => //. +move=> _ -[_ [r r01] [Ys mYs <-]] <-. +apply: emeasurable_fun_infty_o => //. +exact: (measurable_kernel (knormalize f point) Ys). +Qed. + +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_cst {d d' X Y R}. +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 sample_cstE (P : probability Y R) (x : X) : sample_cst P x = P. +Proof. by []. Qed. + +Lemma sampleE (P : X -> pprobability Y R) (mP : measurable_fun setT P) (x : X) : sample P mP x = P x. +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 diracT 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. + +Lemma normalize_kdirac (R : realType) + d (T : measurableType d) d' (T' : measurableType d') (x : T) (r : T') P : + normalize (kdirac (measurable_cst r)) P x = \d_r :> probability T' R. +Proof. +apply: eq_probability => U. +rewrite normalizeE /= diracE in_setT/=. +by rewrite onee_eq0/= indicE in_setT/= -div1r divr1 mule1. +Qed. + +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 (X * Y) ~> 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 [set: X * Y] 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// -[X in measurable X]setTI. +exact: (measurableT_comp mf). +Qed. + +Lemma letin_retk (f : X -> Y) + (mf : measurable_fun [set: X] f) (k : R.-sfker X * Y ~> Z) x U : + measurable U -> + letin (ret mf) k x U = k (x, f x) U. +Proof. +move=> mU; rewrite letinE retE integral_dirac ?diracT ?mul1e//. +exact: (measurableT_comp (measurable_kernel k _ mU)). +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 : R.-sfker X ~> Y := + letin (score (measurable_cst (0%R : R))) + (ret (measurable_cst 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}. + +Section cst_fun. +Context d (T : measurableType d) (R : realType). + +Definition kr (r : R) := @measurable_cst _ _ T _ setT r. +Definition k3 : measurable_fun _ _ := kr 3%:R. +Definition k10 : measurable_fun _ _ := kr 10%:R. +Definition ktt := @measurable_cst _ _ T _ setT tt. +Definition kb (b : bool) := @measurable_cst _ _ T _ setT b. +Definition kn (n : nat) := @measurable_cst _ _ T _ setT n. + +End cst_fun. +Arguments kr {d T R}. +Arguments k3 {d T R}. +Arguments k10 {d T R}. +Arguments ktt {d T}. +Arguments kb {d T}. +Arguments kn {d T}. + +Section iter_mprod. +Local Open Scope type_scope. + +Fixpoint iter_mprod (l : seq {d & measurableType d}) : {d & measurableType d} := + match l with + | [::] => existT measurableType _ unit + | h :: t => let t' := iter_mprod t in + existT _ _ [the measurableType _ of projT2 h * projT2 t'] + end. + +End iter_mprod. + +Section acc. +Import Notations. +Context {R : realType}. + +Fixpoint acc (l : seq {d & measurableType d}) k : + projT2 (iter_mprod l) -> projT2 (nth (existT _ _ munit) l k) := + match l with + | [::] => match k with O => id | _ => id end + | _ :: _ => match k with + | O => fst + | m.+1 => fun x => acc m x.2 + end + end. + +Lemma measurable_acc (l : seq {d & measurableType d}) n : + measurable_fun setT (@acc l n). +Proof. +by elim: l n => //= h t ih [|m] //; exact: (measurableT_comp (ih _)). +Qed. +End acc. +Arguments acc : clear implicits. +Arguments measurable_acc : clear implicits. + +Section rpair_pairA. +Context d0 d1 d2 (T0 : measurableType d0) (T1 : measurableType d1) + (T2 : measurableType d2). + +Definition rpair d (T : measurableType d) t : T0 -> T0 * T := + fun x => (x, t). + +Lemma mrpair d (T : measurableType d) t : measurable_fun setT (@rpair _ T t). +Proof. exact: measurable_fun_prod. Qed. + +Definition pairA : T0 * T1 * T2 -> T0 * (T1 * T2) := + fun x => (x.1.1, (x.1.2, x.2)). + +Definition mpairA : measurable_fun [set: (T0 * T1) * T2] pairA. +Proof. +apply: measurable_fun_prod => /=; first exact: measurableT_comp. +by apply: measurable_fun_prod => //=; exact: measurableT_comp. +Qed. + +Definition pairAi : T0 * (T1 * T2) -> T0 * T1 * T2 := + fun x => (x.1, x.2.1, x.2.2). + +Definition mpairAi : measurable_fun setT pairAi. +Proof. +apply: measurable_fun_prod => //=; last exact: measurableT_comp. +apply: measurable_fun_prod => //=; exact: measurableT_comp. +Qed. + +End rpair_pairA. +Arguments rpair {d0 T0 d} T. +#[global] Hint Extern 0 (measurable_fun _ (rpair _ _)) => + solve [apply: mrpair] : core. +Arguments pairA {d0 d1 d2 T0 T1 T2}. +#[global] Hint Extern 0 (measurable_fun _ pairA) => + solve [apply: mpairA] : core. +Arguments pairAi {d0 d1 d2 T0 T1 T2}. +#[global] Hint Extern 0 (measurable_fun _ pairAi) => + solve [apply: mpairAi] : core. + +Section rpair_pairA_comp. +Import Notations. +Context d0 d1 d2 d3 (T0 : measurableType d0) (T1 : measurableType d1) + (T2 : measurableType d2) (T3 : measurableType d3) (R : realType). + +Definition pairAr d (T : measurableType d) t : T0 * T1 -> T0 * (T1 * T) := + pairA \o rpair T t. +Arguments pairAr {d} T. + +Lemma mpairAr d (T : measurableType d) t : measurable_fun setT (pairAr T t). +Proof. exact: measurableT_comp. Qed. + +Definition pairAAr : T0 * T1 * T2 -> T0 * (T1 * (T2 * unit)) := + pairA \o pairA \o rpair unit tt. + +Lemma mpairAAr : measurable_fun setT pairAAr. +Proof. by do 2 apply: measurableT_comp => //. Qed. + +Definition pairAAAr : T0 * T1 * T2 * T3 -> T0 * (T1 * (T2 * (T3 * unit))) := + pairA \o pairA \o pairA \o rpair unit tt. + +Lemma mpairAAAr : measurable_fun setT pairAAAr. +Proof. by do 3 apply: measurableT_comp => //. Qed. + +Definition pairAArAi : T0 * (T1 * T2) -> T0 * (T1 * (T2 * unit)) := + pairAAr \o pairAi. + +Lemma mpairAArAi : measurable_fun setT pairAArAi. +Proof. by apply: measurableT_comp => //=; exact: mpairAAr. Qed. + +Definition pairAAArAAi : T3 * (T0 * (T1 * T2)) -> T3 * (T0 * (T1 * (T2 * unit))) := + pairA \o pairA \o pairA \o rpair unit tt \o pairAi \o pairAi. + +Lemma mpairAAARAAAi : measurable_fun setT pairAAArAAi. +Proof. by do 5 apply: measurableT_comp => //=. Qed. + +End rpair_pairA_comp. +Arguments pairAr {d0 d1 T0 T1 d} T. +Arguments pairAAr {d0 d1 d2 T0 T1 T2}. +Arguments pairAAAr {d0 d1 d2 d3 T0 T1 T2 T3}. +Arguments pairAArAi {d0 d1 d2 T0 T1 T2}. +Arguments pairAAArAAi {d0 d1 d2 d3 T0 T1 T2 T3}. + +Section accessor_functions. +Import Notations. +Context d0 d1 d2 d3 (T0 : measurableType d0) (T1 : measurableType d1) + (T2 : measurableType d2) (T3 : measurableType d3) (R : realType). + +Let T01 : seq {d & measurableType d} := [:: existT _ _ T0; existT _ _ T1]. + +Definition acc0of2 : T0 * T1 -> T0 := + acc T01 0 \o pairAr unit tt. + +Lemma macc0of2 : measurable_fun setT acc0of2. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T01 0)|exact: mpairAr]. +Qed. + +Definition acc1of2 : T0 * T1 -> T1 := + acc T01 1 \o pairAr unit tt. + +Lemma macc1of2 : measurable_fun setT acc1of2. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T01 1)|exact: mpairAr]. +Qed. + +Let T02 := [:: existT _ _ T0; existT _ _ T1; existT _ _ T2]. + +Definition acc1of3 : T0 * T1 * T2 -> T1 := + acc T02 1 \o pairAAr. + +Lemma macc1of3 : measurable_fun setT acc1of3. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T02 1)|exact: mpairAAr]. +Qed. + +Definition acc2of3 : T0 * T1 * T2 -> T2 := + acc T02 2 \o pairAAr. + +Lemma macc2of3 : measurable_fun setT acc2of3. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T02 2)|exact: mpairAAr]. +Qed. + +Definition acc0of3' : T0 * (T1 * T2) -> T0 := + acc T02 0 \o pairAArAi. + +Lemma macc0of3' : measurable_fun setT acc0of3'. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T02 0)|exact: mpairAArAi]. +Qed. + +Definition acc1of3' : T0 * (T1 * T2) -> T1 := + acc T02 1 \o pairAArAi. + +Lemma macc1of3' : measurable_fun setT acc1of3'. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T02 1)|exact: mpairAArAi]. +Qed. + +Definition acc2of3' : T0 * (T1 * T2) -> T2 := + acc T02 2 \o pairAArAi. + +Lemma macc2of3' : measurable_fun setT acc2of3'. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T02 2)|exact: mpairAArAi]. +Qed. + +Let T03 := [:: existT _ _ T0; existT _ _ T1; existT _ d2 T2; existT _ d3 T3]. + +Definition acc1of4 : T0 * T1 * T2 * T3 -> T1 := + acc T03 1 \o pairAAAr. + +Lemma macc1of4 : measurable_fun setT acc1of4. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T03 1)|exact: mpairAAAr]. +Qed. + +Definition acc2of4' : T0 * (T1 * (T2 * T3)) -> T2 := + acc T03 2 \o pairAAArAAi. + +Lemma macc2of4' : measurable_fun setT acc2of4'. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T03 2)|exact: mpairAAARAAAi]. +Qed. + +Definition acc3of4 : T0 * T1 * T2 * T3 -> T3 := + acc T03 3 \o pairAAAr. + +Lemma macc3of4 : measurable_fun setT acc3of4. +Proof. +by apply: measurableT_comp; [exact: (measurable_acc T03 3)|exact: mpairAAAr]. +Qed. + +End accessor_functions. +Arguments macc0of2 {d0 d1 _ _}. +Arguments macc1of2 {d0 d1 _ _}. +Arguments macc0of3' {d0 d1 d2 _ _ _}. +Arguments macc1of3 {d0 d1 d2 _ _ _}. +Arguments macc1of3' {d0 d1 d2 _ _ _}. +Arguments macc2of3 {d0 d1 d2 _ _ _}. +Arguments macc2of3' {d0 d1 d2 _ _ _}. +Arguments macc1of4 {d0 d1 d2 d3 _ _ _ _}. +Arguments macc2of4' {d0 d1 d2 d3 _ _ _ _}. +Arguments macc3of4 {d0 d1 d2 d3 _ _ _ _}. + +Module CASE_NAT. +Section case_nat. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Section case_nat_ker. +Variable k : R.-ker X ~> Y. + +Definition case_nat_ m (xn : X * nat) : {measure set Y -> \bar R} := + if xn.2 == m then k xn.1 else mzero. + +Let measurable_fun_case_nat_ m U : measurable U -> + measurable_fun setT (case_nat_ m ^~ U). +Proof. +move=> mU/=; rewrite /case_nat_ (_ : (fun _ => _) = + (fun x => if x.2 == m then k x.1 U else mzero U)) /=; last first. + by apply/funext => -[t b]/=; case: ifPn. +apply: (@measurable_fun_if_pair_nat _ _ _ _ (k ^~ U) (fun=> mzero U)) => //. +exact/measurable_kernel. +Qed. + +#[export] +HB.instance Definition _ m := isKernel.Build _ _ _ _ _ + (case_nat_ m) (measurable_fun_case_nat_ m). +End case_nat_ker. + +Section sfcase_nat. +Variable k : R.-sfker X ~> Y. + +Let sfcase_nat_ m : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> case_nat_ k m x U = mseries (k_ ^~ x) 0 U. +Proof. +have [k_ hk /=] := sfinite_kernel k. +exists (fun n => case_nat_ (k_ n) m) => /=. + move=> n; have /measure_fam_uubP[r k_r] := measure_uub (k_ n). + exists r%:num => /= -[x [|n']]; rewrite /case_nat_//= /mzero//. + by case: ifPn => //= ?; rewrite /mzero. + by case: ifPn => // ?; rewrite /= /mzero. +move=> [x b] U mU; rewrite /case_nat_; case: ifPn => hb; first by rewrite hk. +by rewrite /mseries eseries0. +Qed. + +#[export] +HB.instance Definition _ m := @isSFiniteKernel_subdef.Build _ _ _ _ _ + (case_nat_ k m) (sfcase_nat_ m). +End sfcase_nat. + +Section fkcase_nat. +Variable k : R.-fker X ~> Y. + +Let case_nat_uub n : measure_fam_uub (case_nat_ k n). +Proof. +have /measure_fam_uubP[M hM] := measure_uub k. +exists M%:num => /= -[]; rewrite /case_nat_ => t [|n']/=. + by case: ifPn => //= ?; rewrite /mzero. +by case: ifPn => //= ?; rewrite /mzero. +Qed. + +#[export] +HB.instance Definition _ n := Kernel_isFinite.Build _ _ _ _ _ + (case_nat_ k n) (case_nat_uub n). +End fkcase_nat. + +End case_nat. +End CASE_NAT. + +Import CASE_NAT. + +Section case_nat. +Context d d' (T : measurableType d) (T' : measurableType d') (R : realType). + +Import CASE_NAT. + +Definition case_nat (t : R.-sfker T ~> nat) (u_ : (R.-sfker T ~> T')^nat) + : R.-sfker T ~> T' := + t \; kseries (fun n => case_nat_ (u_ n) n). + +End case_nat. + +Definition measure_sum_display : + measure_display * measure_display -> measure_display. +Proof. exact. Qed. + +Definition g_sigma_imageU d1 d2 + (T1 : measurableType d1) (T2 : measurableType d2) (T : Type) + (f1 : T1 -> T) (f2 : T2 -> T) := + <>. + +Section sum_salgebra_instance. +Context d1 d2 (T1 : measurableType d1) (T2 : measurableType d2). +Let f1 : T1 -> T1 + T2 := @inl T1 T2. +Let f2 : T2 -> T1 + T2 := @inr T1 T2. + +Lemma sum_salgebra_set0 : g_sigma_imageU f1 f2 (set0 : set (T1 + T2)). +Proof. exact: sigma_algebra0. Qed. + +Lemma sum_salgebra_setC A : g_sigma_imageU f1 f2 A -> + g_sigma_imageU f1 f2 (~` A). +Proof. exact: sigma_algebraC. Qed. + +Lemma sum_salgebra_bigcup (F : _^nat) : (forall i, g_sigma_imageU f1 f2 (F i)) -> + g_sigma_imageU f1 f2 (\bigcup_i (F i)). +Proof. exact: sigma_algebra_bigcup. Qed. + +HB.instance Definition sum_salgebra_mixin := + @isMeasurable.Build (measure_sum_display (d1, d2)) + (T1 + T2)%type (g_sigma_imageU f1 f2) + sum_salgebra_set0 sum_salgebra_setC sum_salgebra_bigcup. + +End sum_salgebra_instance. +Reserved Notation "p .-sum" (at level 1, format "p .-sum"). +Reserved Notation "p .-sum.-measurable" + (at level 2, format "p .-sum.-measurable"). +Notation "p .-sum" := (measure_sum_display p) : measure_display_scope. +Notation "p .-sum.-measurable" := + ((p.-sum).-measurable : set (set (_ + _))) : + classical_set_scope. + +#[short(type="measurableCountType")] +HB.structure Definition MeasurableCountable d := + {T of Measurable d T & Countable T }. + +#[short(type="measurableFinType")] +HB.structure Definition MeasurableFinite d := + {T of Measurable d T & Finite T }. + +Definition measurableTypeUnit := unit. + +HB.instance Definition _ := Pointed.on measurableTypeUnit. +HB.instance Definition _ := Finite.on measurableTypeUnit. +HB.instance Definition _ := Measurable.on measurableTypeUnit. + +Definition measurableTypeBool := bool. + +HB.instance Definition _ := Pointed.on measurableTypeBool. +HB.instance Definition _ := Finite.on measurableTypeBool. +HB.instance Definition _ := Measurable.on measurableTypeBool. + +Module CASE_SUM. + +Section case_sum'. + +Section kcase_sum'. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Context dA (A : measurableCountType dA) dB (B : measurableCountType dB). +Variables (k1 : A -> R.-sfker X ~> Y) (k2 : B -> R.-sfker X ~> Y). + +Definition case_sum' : X * (A + B) -> {measure set Y -> \bar R} := + fun xab => match xab with + | (x, inl a) => k1 a x + | (x, inr b) => k2 b x + end. + +Let measurable_fun_case_sum' U : measurable U -> + measurable_fun setT (case_sum' ^~ U). +Proof. +rewrite /= => mU. +apply: (measurability (ErealGenInftyO.measurableE R)) => //. +move=> /= _ [_ [x ->] <-]; apply: measurableI => //. +rewrite /case_sum'/= (_ : _ @^-1` _ = + (\bigcup_a ([set x1 | k1 a x1 U < x%:E] `*` inl @` [set a])) `|` + (\bigcup_b ([set x1 | k2 b x1 U < x%:E] `*` inr @` [set b]))); last first. + apply/seteqP; split. + - move=> z/=; rewrite in_itv/=. + move: z => [z [a|b]]/= ?. + + by left; exists a => //; split => //=; exists a. + + by right; exists b => //; split => //=; exists b. + - move=> z/=; rewrite in_itv/=. + move: z => [z [a|b]]/= [|]. + + by case => a' _ /= [] /[swap] [] [_ ->] [->]. + + by case => b' _ /= [] b'x [_ ->]. + + by case => b' _ /= [] b'x [_ ->]. + + by case => b' _ /= [] /[swap] [] [_ ->] [->]. +apply: measurableU. +- pose h1 a := [set xub : X * (A + B) | k1 a xub.1 U < x%:E]. + apply: countable_bigcupT_measurable; first exact: countableP. + move=> a; apply: measurableX => //. + rewrite [X in measurable X](_ : _ = ysection (h1 a) (inl a)). + + apply: measurable_ysection. + rewrite -[X in measurable X]setTI. + apply: emeasurable_fun_infty_o => //= => _ /= C mC; rewrite setTI. + have : measurable_fun setT (fun x => k1 a x U) by exact/measurable_kernel. + move=> /(_ measurableT _ mC); rewrite setTI => H. + rewrite [X in measurable X](_ : _ = ((fun x => k1 a x U) @^-1` C) `*` setT)//. + exact: measurableX. + by apply/seteqP; split => [z//=| z/= []]. + + by rewrite ysectionE. +- pose h2 a := [set xub : X * (A + B)| k2 a xub.1 U < x%:E]. + apply: countable_bigcupT_measurable; first exact: countableP. + move=> b; apply: measurableX => //. + rewrite [X in measurable X](_ : _ = ysection (h2 b) (inr b))//. + + apply: measurable_ysection. + rewrite -[X in measurable X]setTI. + apply: emeasurable_fun_infty_o => //= _ /= C mC; rewrite setTI. + have : measurable_fun setT (fun x => k2 b x U) by exact/measurable_kernel. + move=> /(_ measurableT _ mC); rewrite setTI => H. + rewrite [X in measurable X](_ : _ = ((fun x => k2 b x U) @^-1` C) `*` setT)//. + exact: measurableX. + by apply/seteqP; split => [z //=|z/= []]. + + by rewrite ysectionE. +Qed. + +#[export] +HB.instance Definition _ := isKernel.Build _ _ _ _ _ + case_sum' measurable_fun_case_sum'. +End kcase_sum'. + +Section sfkcase_sum'. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Context dA (A : measurableFinType dA) dB (B : measurableFinType dB). +Variables (k1 : A -> R.-sfker X ~> Y) (k2 : B-> R.-sfker X ~> Y). + +Let sfinite_case_sum' : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> case_sum' k1 k2 x U = mseries (k_ ^~ x) 0 U. +Proof. +rewrite /=. +set f1 : A -> (R.-fker _ ~> _)^nat := + fun ab : A => sval (cid (sfinite_kernel (k1 ab))). +set Hf1 := fun ab : A => svalP (cid (sfinite_kernel (k1 ab))). +rewrite /= in Hf1. +set f2 : B -> (R.-fker _ ~> _)^nat := + fun ab : B => sval (cid (sfinite_kernel (k2 ab))). +set Hf2 := fun ab : B => svalP (cid (sfinite_kernel (k2 ab))). +rewrite /= in Hf2. +exists (fun n => case_sum' (f1 ^~ n) (f2 ^~ n)). + move=> n /=. + pose f1' a := sval (cid (measure_uub (f1 a n))). + pose f2' b := sval (cid (measure_uub (f2 b n))). + red. + exists (maxr (\big[Order.max/0%R]_a f1' a) (\big[Order.max/0%R]_b (f2' b)))%R. + move=> /= [x [a|b]]. + - have [bnd Hbnd] := measure_uub (f1 a n). + rewrite EFin_max lt_max; apply/orP; left. + rewrite /case_sum' -EFin_bigmax. + apply: lt_le_trans; last exact: le_bigmax_cond. + by rewrite /f1'; case: cid => /=. + - have [bnd Hbnd] := measure_uub (f2 b n). + rewrite EFin_max lt_max; apply/orP; right. + rewrite /case_sum' -EFin_bigmax. + apply: lt_le_trans; last exact: le_bigmax_cond. + by rewrite /f2'; case: cid => /=C. +move=> [x [a|b]] U mU/=-. +- by rewrite (Hf1 a x _ mU). +- by rewrite (Hf2 b x _ mU). +Qed. + +#[export] +HB.instance Definition _ := @isSFiniteKernel_subdef.Build _ _ _ _ _ + (case_sum' k1 k2) (sfinite_case_sum'). +End sfkcase_sum'. + +End case_sum'. + +Section case_sum. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). +Context dA (A : measurableFinType dA) dB (B : measurableFinType dB). + +Definition case_sum (f : R.-sfker X ~> (A + B)%type) + (k1 : A -> R.-sfker X ~> Y) (k2 : B -> R.-sfker X ~> Y) : R.-sfker X ~> Y := + f \; case_sum' k1 k2. + +End case_sum. + +End CASE_SUM. + +(* counting measure as a kernel *) +Section kcounting. +Context d (G : measurableType d) (R : realType). + +Definition kcounting : G -> {measure set nat -> \bar R} := fun=> counting. + +Let mkcounting U : measurable U -> measurable_fun setT (kcounting ^~ U). +Proof. by []. Qed. + +HB.instance Definition _ := isKernel.Build _ _ _ _ _ kcounting mkcounting. + +Let sfkcounting : exists2 k_ : (R.-ker _ ~> _)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> kcounting x U = mseries (k_ ^~ x) 0 U. +Proof. +exists (fun n => [the R.-fker _ ~> _ of + @kdirac _ _ G nat R _ (@measurable_cst _ _ _ _ setT n)]). + by move=> n /=; exact: measure_uub. +by move=> g U mU; rewrite /kcounting/= counting_dirac. +Qed. + +HB.instance Definition _ := + isSFiniteKernel_subdef.Build _ _ _ _ R kcounting sfkcounting. + +End kcounting. + +(* formalization of the iterate construct [Section 4.2, Staton ESOP 2017] *) +Section iterate. +Context d {G : measurableType d} {R : realType}. +Context dA (A : measurableFinType dA) dB (B : measurableFinType dB). + +Import CASE_SUM. + +(* formalization of iterate^n + Gamma |-p iterate^n t from x = u : B *) +Variables (t : R.-sfker (G * A) ~> (A + B)%type) + (u : G -> A) (mu : measurable_fun setT u). + +Fixpoint iterate_ n : R.-sfker G ~> B := + match n with + | 0%N => case_sum (letin (ret mu) t) + (fun u' => fail) + (fun v => ret (measurable_cst v)) + | m.+1 => case_sum (letin (ret mu) t) + (fun u' => iterate_ m) + (fun v => fail) + end. + +(* formalization of iterate + Gamma, x : A |-p t : A + B Gamma |-d u : A +----------------------------------------------- + Gamma |-p iterate t from x = u : B *) +Definition iterate : R.-sfker G ~> B := case_nat (kcounting R) iterate_. + +End iterate. + +Section iterate_unit. + +Let unit := measurableTypeUnit. +Let bool := measurableTypeBool. +Context d {G : measurableType d} {R : realType}. +Context dB (B : measurableFinType dB). + +Section iterate_elim. +Variables (t : R.-sfker (G * unit) ~> (unit + B)%type) + (u : G -> unit) (mu : measurable_fun setT u). +Variables (r : R) (tlE : forall gamma, t (gamma, tt) [set inl tt] = r%:E). + +Variables (gamma : G) (X : set B) (q : R). +Hypothesis trE : t (gamma, tt) [set inr x | x in X] = q%:E. + +Let q_ge0 : (0 <= q)%R. Proof. by rewrite -lee_fin -trE measure_ge0. Qed. +Let r_ge0 : (0 <= r)%R. +Proof. by rewrite -lee_fin -(tlE gamma) measure_ge0. Qed. + +Lemma iterate_E n : iterate_ t mu n gamma X = (geometric q r n)%:E. +Proof. +elim: n => [|n IHn] //=; + rewrite /kcomp; rewrite integral_kcomp//=; + rewrite /= integral_dirac//= ?diracT ?mul1e ?expr0 ?exprS ?mulr1. + rewrite (eq_integral (EFin \o \1_[set inr x | x in X]))//=; last first. + move=> [a' _|b _]//=; last first. + by rewrite diracE indicE/= (mem_image inr_inj). + rewrite /kcomp/= indicE /= ge0_integral_mscale//= normr0 mul0e. + by rewrite [_ \in _](introF idP)// inE /= => -[]. + by rewrite ?unitE integral_indic//= setIT. +pose g : unit + B -> R^o := (geometric q r n \o* \1_[set inl tt])%R. +rewrite (eq_integral (EFin \o g))//=; last first. + move=> [[] _|b _]//=. + by rewrite /g/= indicE//= in_set1 inE eqxx mul1r. + rewrite /kcomp/= ge0_integral_mscale//= normr0 mul0e. + by rewrite /g /= indicE//= in_set1 inE mul0r. +rewrite /g /=; under eq_integral do rewrite EFinM. +rewrite integralZr//=; last first. + apply/integrableP; split=> //. + under eq_integral => x. + rewrite gee0_abs//=; last first. + by rewrite indicE lee_fin natr_ge0//. + over. + by rewrite /= integral_indic// setIT [u gamma]unitE tlE ltey. +by rewrite integral_indic//= [u gamma]unitE setIT tlE -EFinM mulrCA. +Qed. + +Hypothesis r_lt1 : (r < 1)%R. + +Lemma iterateE : iterate t mu gamma X = (q / (1 - r))%:E. +Proof. +rewrite /= /kcomp/= /case_nat_/= /mseries. +under eq_integral => n _. + under (@congr_lim _ _ _ \o @eq_fun _ _ _ _) => k. + under eq_bigr do rewrite fun_if/= (fun_if (@^~ _))/mzero eq_sym. + rewrite -big_mkcond/= big_nat1_eq. + over. +over. +rewrite /= (eq_integral (EFin \o geometric q r))//=; last first. + move=> k _; apply/lim_near_cst => //; rewrite iterate_E ?r_ge0 ?r_lt1//. + by near do rewrite ifT//. +have cvgg: series (geometric q r) x @[x --> \oo] --> (q / (1 - r))%R. + by apply/cvg_geometric_series; rewrite ger0_norm ?r_lt1//. +have limgg := cvg_lim (@Rhausdorff R) cvgg. +have sumgE : \big[+%R/0%R]_(0 <= k x; rewrite inE trueE. +rewrite -(@nneseries_esum _ _ predT)//=. +under eq_eseriesr do rewrite ger0_norm// ?geometric_ge0//. +by rewrite sumgE ltey. +Unshelve. all: end_near. Qed. + +End iterate_elim. + +Import CASE_SUM. + +Variables (t : R.-pker (G * unit) ~> (unit + B)%type) + (u : G -> unit) (mu : measurable_fun setT u). +Variables (r : R) (r_lt1 : (r < 1)%R). +Hypothesis (tlE : forall gamma, t (gamma, tt) [set inl tt] = r%:E). + +Let trE gamma X : t (gamma, tt) [set inr x | x in X] \in fin_num. +Proof. +apply/fin_numPlt; rewrite (@lt_le_trans _ _ 0)//=. +rewrite (@le_lt_trans _ _ 1)//= ?ltey//. +rewrite -( @prob_kernel _ _ _ _ _ t (gamma, tt) ). +by apply/le_measure => //=; rewrite inE//=. +Qed. + +Lemma iterate_normalize p : + iterate t mu = knormalize (case_sum (letin (ret mu) t) + (fun u' => fail) + (fun v => ret (measurable_cst v))) p. +Proof. +apply/eq_sfkernel => gamma U. +have /fin_numP_EFin[q trE'] := trE gamma U. +rewrite (iterateE mu tlE trE')//; symmetry. +rewrite /= /mnormalize/= (fun_if (@^~ U))/=. +set m := kcomp _ _ _. +have mE V : m V = t (gamma, tt) [set inr x | x in V]. + rewrite /m/= /kcomp/= integral_kcomp//= integral_dirac//= diracT mul1e. + rewrite (eq_integral (EFin \o \1_[set inr x | x in V])). + by rewrite integral_indic ?setIT ?unitE. + move=> [x|x] xV /=; rewrite indicE. + rewrite ?inl_in_set_inr /kcomp/=. + by rewrite ge0_integral_mscale//= ?normr0 mul0e. + by rewrite inr_in_set_inr// indicE. +rewrite !mE trE'. +suff -> : t (gamma, tt) (range inr) = 1 - t (gamma, tt) [set inl tt]. + by rewrite tlE -EFinB/= orbF eqe subr_eq0 eq_sym lt_eqF. +rewrite -( @prob_kernel _ _ _ _ _ t (gamma, tt) ). +have -> : [set: unit + B] = [set inl tt] `|` (range inr). + symmetry; apply/eq_set => -[[]|b]//=; apply/propT; first by left. + by right; exists b. +rewrite measureU//=; first by rewrite addeAC subee ?add0e// ?tlE//. +by apply/eq_set => -[[]|b]//=; apply/propF; case=> []// _ []. +Qed. + +End iterate_unit. + +Section lift_neq. +Context {R : realType} d (G : measurableType d). +Variables (f : G -> bool) (g : G -> bool). + +Definition flift_neq : G -> bool := fun x' => f x' != g x'. + +Hypotheses (mf : measurable_fun setT f) (mg : measurable_fun setT g). + +(* see also emeasurable_fun_neq *) +Lemma measurable_fun_flift_neq : measurable_fun setT flift_neq. +Proof. +apply: (@measurable_fun_bool _ _ _ _ true). +rewrite setTI. +rewrite /flift_neq /= (_ : _ @^-1` _ = ([set x | f x] `&` [set x | ~~ g x]) `|` + ([set x | ~~ f x] `&` [set x | g x])). + apply: measurableU; apply: measurableI. + - by rewrite -[X in measurable X]setTI; exact: mf. + - rewrite [X in measurable X](_ : _ = ~` [set x | g x]); last first. + by apply/seteqP; split => x /= /negP. + by apply: measurableC; rewrite -[X in measurable X]setTI; exact: mg. + - rewrite [X in measurable X](_ : _ = ~` [set x | f x]); last first. + by apply/seteqP; split => x /= /negP. + by apply: measurableC; rewrite -[X in measurable X]setTI; exact: mf. + - by rewrite -[X in measurable X]setTI; exact: mg. +by apply/seteqP; split => x /=; move: (f x) (g x) => [|] [|]//=; intuition. +Qed. + +Definition lift_neq : R.-sfker G ~> bool := ret measurable_fun_flift_neq. + +End lift_neq. + +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// diracT 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 (measurableT_comp mf (@macc1of2 _ _ _ _)) + (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 [set: R] f) + (mg : measurable_fun [set: R * unit] g) : + letin (score mf) (score mg) = + score (measurable_funM mf (measurableT_comp mg (measurable_pair2 tt))). +Proof. +apply/eq_sfkernel => x U. +rewrite {1}/letin; unlock. +by rewrite kcomp_scoreE/= /mscale/= diracE normrM muleA EFinM. +Qed. + +(* hard constraints to express score below 1 *) +Lemma score_fail (r : R) : (0 <= r <= 1)%R -> + score (kr r) = + letin (sample_cst (bernoulli r) : R.-pker T ~> _) + (ite (@macc1of2 _ _ _ _) (ret ktt) fail). +Proof. +move=> /andP[r0 r1]; apply/eq_sfkernel => x U. +rewrite letinE/= /sample; unlock. +by rewrite /mscale/= ger0_norm// integral_bernoulli ?r0//= 2!iteE//= failE mule0 adde0. +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. + +(* associativity of let [Section 4.2, Staton ESOP 2017] *) +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 (X * T1) ~> T2) + (v : R.-sfker (X * T2) ~> Y) + (v' : R.-sfker (X * T1 * T2) ~> 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). +- exact: (measurableT_comp (measurable_kernel v _ mA)). +Qed. + +End letinA. + +(* commutativity of let [Section 4.2, Staton ESOP 2017] *) +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 _ X R (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 := @isSFinite.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 _ Y R (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 := @isSFinite.Build _ Y R (U z) (sfinU z). + +Lemma letinC z A : measurable A -> + letin t + (letin u' + (ret (measurable_fun_prod macc1of3 macc2of3))) z A = + letin u + (letin t' + (ret (measurable_fun_prod macc2of3 macc1of3))) 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/measurable_EFinP => /=; 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. + +(* examples *) + +Lemma letin_sample_bernoulli d d' (T : measurableType d) + (T' : measurableType d') (R : realType) (r : R) + (u : R.-sfker [the measurableType _ of (T * bool)%type] ~> T') x y : + (0 <= r <= 1)%R -> + letin (sample_cst (bernoulli r)) u x y = + r%:E * u (x, true) y + (`1- r)%:E * u (x, false) y. +Proof. by move=> r01; rewrite letinE/= integral_bernoulli. Qed. + +Section sample_and_return. +Import Notations. +Context d (T : measurableType d) (R : realType). + +Definition sample_and_return : R.-sfker T ~> _ := + letin + (sample_cst (bernoulli (2 / 7))) (* T -> B *) + (ret macc1of2) (* 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. +rewrite /sample_and_return letin_sample_bernoulli; last lra. +by rewrite !retE onem27. +Qed. + +End sample_and_return. + +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 ~> _ := + letin + (sample_cst (bernoulli (2 / 7))) (* T -> B *) + (ite macc1of2 (ret (@k3 _ _ R)) (ret k10)). + +Lemma sample_and_branchE t U : sample_and_branch t U = + (2 / 7)%:E * \d_(3 : R) U + (5 / 7)%:E * \d_(10 : R) U. +Proof. +rewrite /sample_and_branch letin_sample_bernoulli/=; last lra. +by rewrite !iteE/= onem27. +Qed. + +End sample_and_branch. + +Section bernoulli_and. +Context d (T : measurableType d) (R : realType). +Import Notations. + +Definition bernoulli_and : R.-sfker T ~> mbool := + (letin (sample_cst (bernoulli (1 / 2))) + (letin (sample_cst (bernoulli (1 / 2))) + (ret (measurable_and macc1of3 macc2of3)))). + +Lemma bernoulli_andE t U : + bernoulli_and t U = sample_cst (bernoulli (1 / 4)) t U. +Proof. +rewrite /bernoulli_and. +rewrite letin_sample_bernoulli; last lra. +rewrite (letin_sample_bernoulli (r := 1 / 2)); last lra. +rewrite (letin_sample_bernoulli (r := 1 / 2)); last lra. +rewrite muleDr//= -muleDl//. +rewrite !muleA -addeA -muleDl// -!EFinM !onem1S/= -splitr mulr1. +have -> : (1 / 2 * (1 / 2) = 1 / 4%:R :> R)%R by rewrite mulf_div mulr1// -natrM. +rewrite [in RHS](_ : 1 / 4 = (1 / 4)%:nng%:num)%R//. +rewrite bernoulliE/=; last lra. +rewrite -!EFinM; congr( _ + (_ * _)%:E). +by rewrite /onem; lra. +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_cst (bernoulli (2 / 7))) + (letin + (letin (ite macc1of2 (ret k3) (ret k10)) + (score (measurableT_comp mh macc2of3))) + (ret macc1of3)). + +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_pdf R 4%N. +Let mpoisson4 := @measurable_poisson_pdf R 4%N. + +Definition kstaton_bus_poisson : R.-sfker R ~> mbool := + kstaton_bus _ mpoisson4. + +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 /kstaton_bus_poisson /kstaton_bus. +rewrite letin_sample_bernoulli; last lra. +rewrite -!muleA; congr (_ * _ + _ * _). +- rewrite letin_kret//. + rewrite letin_iteT//. + rewrite letin_retk//. + by rewrite scoreE//= => r r0; exact: poisson_pdf_ge0. +- by rewrite onem27. + rewrite letin_kret//. + rewrite letin_iteF//. + rewrite letin_retk//. + by rewrite scoreE//= => r r0; exact: poisson_pdf_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) * 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 normalizeE !kstaton_bus_poissonE !diracT !mule1 ifF //. +apply/negbTE; rewrite gt_eqF// lte_fin. +by rewrite addr_gt0// mulr_gt0//= ?divr_gt0// ?ltr0n// poisson_pdf_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 := @exponential_pdf R (ratr (15%:Q / 60%:Q)). +Let mexp1560 := @measurable_exponential_pdf R (ratr (15%:Q / 60%:Q)). + +(* 15/60 = 0.25 *) + +Definition kstaton_bus_exponential : R.-sfker R ~> mbool := + kstaton_bus _ mexp1560. + +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. +rewrite /kstaton_bus. +rewrite letin_sample_bernoulli; last lra. +rewrite -!muleA; congr (_ * _ + _ * _). +- rewrite letin_kret//. + rewrite letin_iteT//. + rewrite letin_retk//. + rewrite scoreE//= => r r0; exact: exponential_pdf_ge0. +- by rewrite onem27. + rewrite letin_kret//. + rewrite letin_iteF//. + rewrite letin_retk//. + by rewrite scoreE//= => r r0; exact: exponential_pdf_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) * 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. +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// exponential_pdf_gt0 ?ltr0n. +Qed. + +End staton_bus_exponential. + + +Section von_neumann_trick. +Context d {T : measurableType d} {R : realType}. + +Definition minltt {d1 d2} {T1 : measurableType d1} {T2 : measurableType d2} := + @measurable_cst _ _ T1 _ setT (@inl _ T2 tt). + +Definition finrb d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) : + T1 * bool -> T2 + bool := fun t1b => inr t1b.2. + +Lemma minrb {d1 d2} {T1 : measurableType d1} {T2 : measurableType d2} : + measurable_fun setT (@finrb _ _ T1 T2). +Proof. exact: measurableT_comp. Qed. + +Variable (D : pprobability bool R). (* biased coin *) +Let unit := measurableTypeUnit. +Let bool := measurableTypeBool. + +Definition trick : R.-sfker (T * unit) ~> (unit + bool)%type := + letin (sample_cst D) + (letin (sample_cst D) + (letin (lift_neq macc1of3 macc2of3) + (ite macc3of4 + (letin (ret macc1of4) (ret minrb)) + (ret minltt)))). + +HB.instance Definition _ := SFiniteKernel.on trick. +HB.instance Definition _ x := Measure.on (trick x). + +Definition kvon_neumann_trick : _ -> _ := + (@iterate _ _ R _ unit _ bool trick _ ktt). +Definition von_neumann_trick x : _ -> _ := kvon_neumann_trick x. + +HB.instance Definition _ := SFiniteKernel.on kvon_neumann_trick. +HB.instance Definition _ x := Measure.on (von_neumann_trick x). + +Section von_neumann_trick_proof. + +Let p : R := fine (D [set true]). +Let q : R := p * (1 - p). +Let r : R := p ^+ 2 + (1 - p) ^+ 2. + +Let Dtrue : D [set true] = p%:E. +Proof. by rewrite fineK//= fin_num_measure. Qed. + +Lemma trickE gamma X : trick gamma X = + (r *+ (inl tt \in X) + + q *+ ((inr true \in X) + (inr false \in X)))%:E. +Proof. +have Dbernoulli : D =1 bernoulli p by exact/eq_bernoulli/Dtrue. +have p_itv01 : (0 <= p <= 1)%R. + by rewrite -2!lee_fin -Dtrue?measure_ge0 ?probability_le1. +pose eqbern := eq_measure_integral _ (fun x _ _ => Dbernoulli x). +rewrite /trick/= /kcomp. +do 2?rewrite ?eqbern ?integral_bernoulli//= /kcomp/=. +rewrite !integral_dirac ?diracT//= ?mul1e. +rewrite !iteE//= ?diracE/= /kcomp/=. +rewrite !integral_dirac /acc1of4/= ?diracT ?diracE ?mul1e//. +rewrite /finrb /acc1of4/= -?(EFinB, EFinN, EFinM, EFinD) /q /r /onem. +by congr (_)%:E; do 3!move: (_ \in _) => ? /=; ring. +Qed. + +Lemma trick_prob_kernelT gamma : trick gamma setT = 1. +Proof. +by rewrite trickE !mem_setT mulr2n mulr1n /r /q; congr (_)%:E; ring. +Qed. + +HB.instance Definition _ gamma := Measure_isProbability.Build _ _ _ + (trick gamma) (trick_prob_kernelT gamma). +HB.instance Definition _ := Kernel_isProbability.Build _ _ _ _ _ + trick trick_prob_kernelT. + +Hypothesis D_nontrivial : 0 < D [set true] < 1. + +Let p_gt0 : (0 < p)%R. +Proof. by rewrite -lte_fin -Dtrue; case/andP : D_nontrivial. Qed. + +Let p_lt1 : (p < 1)%R. +Proof. by rewrite -lte_fin -Dtrue; case/andP : D_nontrivial. Qed. + +Let p'_gt0 : (0 < 1 - p)%R. Proof. by rewrite subr_gt0. Qed. + +Let r_lt1 : (r < 1)%R. +Proof. +rewrite /r -subr_gt0 [ltRHS](_ : _ = 2 * p * (1 - p))%R; last by ring. +by rewrite !mulr_gt0. +Qed. + +Lemma von_neumann_trick_prob_kernel gamma b : + kvon_neumann_trick gamma [set b] = 2^-1%:E. +Proof. +rewrite [LHS](@iterateE _ _ _ _ _ _ _ _ r _ _ _ q)//=. +- rewrite /r /q; congr (_)%:E. + suff: (1 - ((p ^+ 2)%R + ((1 - p) ^+ 2)%R)%E)%R != 0%R by move=> *; field. + rewrite [X in X != _](_ : _ = 2 * (p * (1 - p)))%R; last by ring. + by rewrite mulf_eq0 ?pnatr_eq0/= mulf_neq0// gt_eqF ?p_gt0 ?p'_gt0. +- by move=> gamma'; rewrite trickE//= ?in_set1 ?inE//= addr0. +- rewrite trickE/= ?inl_in_set_inr ?inr_in_set_inr// add0r !in_set1 !inE. + by case: b. +Qed. + +Lemma von_neumann_trick_prob_kernelT gamma : + von_neumann_trick gamma [set: bool] = 1. +Proof. +rewrite setT_bool measureU//=; last by rewrite disjoints_subset => -[]. +rewrite !von_neumann_trick_prob_kernel -EFinD. +by have := splitr (1 : R); rewrite mul1r => <-. +Qed. + +HB.instance Definition _ gamma := Measure.on (von_neumann_trick gamma). +HB.instance Definition _ gamma := Measure_isProbability.Build _ _ _ + (von_neumann_trick gamma) (von_neumann_trick_prob_kernelT gamma). +HB.instance Definition _ := Kernel_isProbability.Build _ _ _ _ _ + kvon_neumann_trick von_neumann_trick_prob_kernelT. + +Theorem von_neumann_trickP gamma : von_neumann_trick gamma =1 bernoulli 2^-1. +Proof. by apply: eq_bernoulli; rewrite von_neumann_trick_prob_kernel. Qed. + +End von_neumann_trick_proof. + +End von_neumann_trick. + +(**md + letin' variants +*) + +Section mswap. +Context d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable k : R.-ker Y * X ~> Z. + +Definition mswap xy U := k (swap xy) U. + +Let mswap0 xy : mswap xy set0 = 0. +Proof. done. Qed. + +Let mswap_ge0 x U : 0 <= mswap x U. +Proof. done. Qed. + +Let mswap_sigma_additive x : semi_sigma_additive (mswap x). +Proof. exact: measure_semi_sigma_additive. Qed. + +HB.instance Definition _ x := isMeasure.Build _ _ R + (mswap x) (mswap0 x) (mswap_ge0 x) (@mswap_sigma_additive x). + +Definition mkswap : _ -> {measure set Z -> \bar R} := + fun x => mswap x. + +Let measurable_fun_kswap U : + measurable U -> measurable_fun setT (mkswap ^~ U). +Proof. +move=> mU. +rewrite [X in measurable_fun _ X](_ : _ = k ^~ U \o @swap _ _)//. +apply measurableT_comp => //=; first exact: measurable_kernel. +exact: measurable_swap. +Qed. + +HB.instance Definition _ := isKernel.Build _ _ + (X * Y)%type Z R mkswap measurable_fun_kswap. + +End mswap. + +Section mswap_sfinite_kernel. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). +Variable k : R.-sfker Y * X ~> Z. + +Let mkswap_sfinite : + exists2 k_ : (R.-ker X * Y ~> Z)^nat, + forall n, measure_fam_uub (k_ n) & + forall x U, measurable U -> mkswap k x U = kseries k_ x U. +Proof. +have [k_ /= kE] := sfinite_kernel k. +exists (fun n => mkswap (k_ n)). + move=> n. + have /measure_fam_uubP[M hM] := measure_uub (k_ n). + by exists M%:num => x/=; exact: hM. +move=> xy U mU. +by rewrite /mswap/= kE. +Qed. + +HB.instance Definition _ := + isSFiniteKernel_subdef.Build _ _ _ Z R (mkswap k) mkswap_sfinite. + +End mswap_sfinite_kernel. + +Section kswap_finite_kernel_finite. +Context d d' d3 (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType) + (k : R.-fker Y * X ~> Z). + +Let mkswap_finite : measure_fam_uub (mkswap k). +Proof. +have /measure_fam_uubP[r hr] := measure_uub k. +apply/measure_fam_uubP; exists (PosNum [gt0 of r%:num%R]) => x /=. +exact: hr. +Qed. + +HB.instance Definition _ := + Kernel_isFinite.Build _ _ _ Z R (mkswap k) mkswap_finite. + +End kswap_finite_kernel_finite. + +Reserved Notation "f .; g" (at level 60, right associativity, + format "f .; '/ ' g"). + +Notation "l .; k" := (mkcomp l (mkswap k)) : ereal_scope. + +Section letin'. +Variables (d d' d3 : _) (X : measurableType d) (Y : measurableType d') + (Z : measurableType d3) (R : realType). + +Definition letin' (l : R.-sfker X ~> Y) (k : R.-sfker Y * X ~> Z) := + locked [the R.-sfker X ~> Z of l .; k]. + +Lemma letin'E (l : R.-sfker X ~> Y) (k : R.-sfker Y * X ~> Z) x U : + letin' l k x U = \int[l x]_y k (y, x) U. +Proof. by rewrite /letin'; unlock. Qed. + +Lemma letin'_letin (l : R.-sfker X ~> Y) (k : R.-sfker Y * X ~> Z) : + letin' l k = letin l (mkswap k). +Proof. by rewrite /letin'; unlock. Qed. + +End letin'. + +Section letin'C. +Import Notations. +Context d d1 d' (X : measurableType d) (Y : measurableType d1) + (Z : measurableType d') (R : realType). +Variables (t : R.-sfker Z ~> X) + (u' : R.-sfker X * Z ~> Y) + (u : R.-sfker Z ~> Y) + (t' : R.-sfker Y * Z ~> X) + (tt' : forall y, t =1 fun z => t' (y, z)) + (uu' : forall x, u =1 fun z => u' (x, z)). + +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 _ X R (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 := @isSFinite.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 _ Y R (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 := @isSFinite.Build _ Y R + (U' z) (sfinU z). + +Lemma letin'C z A : measurable A -> + letin' t + (letin' u' + (ret (measurable_fun_prod macc1of3' macc0of3'))) z A = + letin' u + (letin' t' + (ret (measurable_fun_prod macc0of3' macc1of3'))) z A. +Proof. +move=> mA. +rewrite !letin'E. +under eq_integral. + move=> x _. + rewrite letin'E -uu'. + under eq_integral do rewrite retE /=. + over. +rewrite (sfinite_Fubini (T' z) (U' z) (fun x => \d_(x.1, x.2) A ))//; last first. + apply/measurable_EFinP => /=; rewrite (_ : (fun x => _) = mindic R mA)//. + by apply/funext => -[]. +rewrite /=. +apply: eq_integral => y _. +by rewrite letin'E/= -tt'; apply: eq_integral => // x _; rewrite retE. +Qed. + +End letin'C. +Arguments letin'C {d d1 d' X Y Z R} _ _ _ _. + +Section letin'A. +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 T1 * X ~> T2) + (v : R.-sfker T2 * X ~> Y) + (v' : R.-sfker T2 * (T1 * X) ~> Y) + (vv' : forall y, v =1 fun xz => v' (xz.1, (y, xz.2))). + +Lemma letin'A x A : measurable A -> + letin' t (letin' u v') x A + = + (letin' (letin' t u) v) x A. +Proof. +move=> mA. +rewrite !letin'E. +under eq_integral do rewrite letin'E. +rewrite letin'_letin/=. +rewrite integral_kcomp; [|by []|]. + apply: eq_integral => z _. + apply: eq_integral => y _. + by rewrite (vv' z). +exact: measurableT_comp (@measurable_kernel _ _ _ _ _ v _ mA) _. +Qed. + +End letin'A. + +Lemma letin'_sample_bernoulli d d' (T : measurableType d) + (T' : measurableType d') (R : realType) (r : R) (r01 : (0 <= r <= 1)%R) + (u : R.-sfker bool * T ~> T') x y : + letin' (sample_cst (bernoulli r)) u x y = + r%:E * u (true, x) y + (`1- r)%:E * u (false, x) y. +Proof. by rewrite letin'_letin letin_sample_bernoulli. Qed. + +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 : Y * X -> 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 letin'E. +under eq_integral do rewrite retE. +rewrite integral_indic ?setIT// -[X in measurable X]setTI. +exact: (measurableT_comp mf). +Qed. + +Lemma letin'_retk (f : X -> Y) (mf : measurable_fun setT f) + (k : R.-sfker Y * X ~> Z) x U : + measurable U -> letin' (ret mf) k x U = k (f x, x) U. +Proof. +move=> mU; rewrite letin'E retE integral_dirac ?diracT ?mul1e//. +exact: (measurableT_comp (measurable_kernel k _ mU)). +Qed. + +End letin'_return. + +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 Z * T ~> 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. by move=> ftT; rewrite !letin'_letin letin_iteT. Qed. + +Lemma letin'_iteF : ~~ f t -> letin' (ite mf k1 k2) u t U = letin' k2 u t U. +Proof. by move=> ftF; rewrite !letin'_letin letin_iteF. Qed. + +End letin'_ite. + +Section hard_constraint'. +Context d d' (X : measurableType d) (Y : measurableType d') (R : realType). + +Definition fail' : R.-sfker X ~> Y := + letin' (score (measurable_cst (0%R : R))) + (ret (measurable_cst point)). + +Lemma fail'E x U : fail' x U = 0. +Proof. by rewrite /fail' letin'_letin failE. Qed. + +End hard_constraint'. +Arguments fail' {d d' X Y R}. + +Lemma score_fail' d (X : measurableType d) {R : realType} + (r : R) (r01 : (0 <= r <= 1)%R) : + score (kr r) = + letin' (sample_cst (bernoulli r) : R.-pker X ~> _) + (ite macc0of2 (ret ktt) fail'). +Proof. +move: r01 => /andP[r0 r1]; apply/eq_sfkernel => x U. +rewrite letin'E/= /sample; unlock. +rewrite integral_bernoulli ?r0//=. +by rewrite /mscale/= iteE//= iteE//= fail'E mule0 adde0 ger0_norm. +Qed. diff --git a/theories/prob_lang_wip.v b/theories/prob_lang_wip.v new file mode 100644 index 000000000..e084df971 --- /dev/null +++ b/theories/prob_lang_wip.v @@ -0,0 +1,327 @@ +From HB Require Import structures. +From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. +From mathcomp Require Import rat. +From mathcomp Require Import mathcomp_extra boolp classical_sets. +From mathcomp Require Import functions cardinality fsbigop. +From mathcomp Require Import signed reals ereal topology normedtype sequences. +From mathcomp Require Import esum measure lebesgue_measure numfun. +From mathcomp Require Import lebesgue_integral exp kernel trigo prob_lang. +From mathcomp Require Import realfun charge. + +(******************************************************************************) +(* Semantics of a probabilistic programming language using s-finite kernels *) +(* (wip about the definition of Lebesgue measure) *) +(******************************************************************************) + +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_pdf. +Context {R : realType}. +Local Open Scope ring_scope. + +Definition gauss_pdf m s x : R := + (s * sqrtr (pi *+ 2))^-1 * expR (- ((x - m) / s) ^+ 2 / 2%:R). + +Lemma gauss_pdf_ge0 m s x : 0 <= s -> 0 <= gauss_pdf m s x. +Proof. by move=> s0; rewrite mulr_ge0 ?expR_ge0// invr_ge0 mulr_ge0. Qed. + +Lemma gauss_pdf_gt0 m s x : 0 < s -> 0 < gauss_pdf m s x. +Proof. +move=> s0; rewrite mulr_gt0 ?expR_gt0// invr_gt0 mulr_gt0//. +by rewrite sqrtr_gt0 pmulrn_rgt0// pi_gt0. +Qed. + +Lemma measurable_gauss_pdf m s : measurable_fun setT (gauss_pdf m s). +Proof. +apply: measurable_funM => //=; apply: measurableT_comp => //=. +apply: measurable_funM => //=; apply: measurableT_comp => //=. +apply: measurableT_comp (exprn_measurable _) _ => /=. +by apply: measurable_funM => //=; exact: measurable_funD. +Qed. + +Definition gauss_pdf01 : R -> R := gauss_pdf 0 1. + +Lemma gauss_pdf01E x : + gauss_pdf01 x = (sqrtr (pi *+ 2))^-1 * expR (- (x ^+ 2) / 2%:R). +Proof. by rewrite /gauss_pdf01 /gauss_pdf mul1r subr0 divr1. Qed. + +Lemma gauss_pdf01_ub x : gauss_pdf01 x <= (Num.sqrt (pi *+ 2))^-1. +Proof. +rewrite -[leRHS]mulr1. +rewrite /gauss_pdf01 /gauss_pdf; last first. +rewrite mul1r subr0 ler_pM2l ?invr_gt0// ?sqrtr_gt0; last by rewrite mulrn_wgt0// pi_gt0. +by rewrite -[leRHS]expR0 ler_expR mulNr oppr_le0 mulr_ge0// sqr_ge0. +Qed. + +Lemma continuous_gauss_pdf1 x : {for x, continuous gauss_pdf01}. +Proof. +apply: continuousM => //=; first exact: cvg_cst. +apply: continuous_comp => /=; last exact: continuous_expR. +apply: continuousM => //=; last exact: cvg_cst. +apply: continuous_comp => //=; last exact: (@continuousN _ R^o). +apply: (@continuous_comp _ _ _ _ (fun x : R => x ^+ 2)%R); last exact: exprn_continuous. +apply: continuousM => //=; last exact: cvg_cst. +by apply: (@continuousD _ R^o) => //=; exact: cvg_cst. +Qed. + +End gauss_pdf. + +Definition gauss01 {R : realType} + of \int[@lebesgue_measure R]_x (gauss_pdf01 x)%:E = 1%E : set _ -> \bar R := + fun V => (\int[lebesgue_measure]_(x in V) (gauss_pdf01 x)%:E)%E. + +Section gauss. +Variable R : realType. +Local Open Scope ring_scope. + +Hypothesis integral_gauss_pdf01 : + (\int[@lebesgue_measure R]_x (gauss_pdf01 x)%:E = 1%E)%E. + +Local Notation gauss01 := (gauss01 integral_gauss_pdf01). + +Let gauss010 : gauss01 set0 = 0%E. +Proof. by rewrite /gauss01 integral_set0. Qed. + +Let gauss01_ge0 A : (0 <= gauss01 A)%E. +Proof. +by rewrite /gauss01 integral_ge0//= => x _; rewrite lee_fin gauss_pdf_ge0. +Qed. + +Let gauss01_sigma_additive : semi_sigma_additive gauss01. +Proof. +move=> /= F mF tF mUF. +rewrite /gauss01/= integral_bigcup//=; last first. + apply/integrableP; split. + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_gauss_pdf. + rewrite (_ : (fun x => _) = EFin \o gauss_pdf01); last first. + by apply/funext => x; rewrite gee0_abs// lee_fin gauss_pdf_ge0. + apply: le_lt_trans. + apply: (@ge0_subset_integral _ _ _ _ _ setT) => //=. + by apply/measurable_EFinP; exact: measurable_gauss_pdf. + by move=> ? _; rewrite lee_fin gauss_pdf_ge0. + by rewrite integral_gauss_pdf01 // ltey. +apply: is_cvg_ereal_nneg_natsum_cond => n _ _. +by apply: integral_ge0 => /= x ?; rewrite lee_fin gauss_pdf_ge0. +Qed. + +HB.instance Definition _ := isMeasure.Build _ _ _ + gauss01 gauss010 gauss01_ge0 gauss01_sigma_additive. + +Let gauss01_setT : gauss01 [set: _] = 1%E. +Proof. by rewrite /gauss01 integral_gauss_pdf01. Qed. + +HB.instance Definition _ := @Measure_isProbability.Build _ _ R gauss01 gauss01_setT. + +End gauss. + +Section gauss_lebesgue. +Context d (T : measurableType d) (R : realType). +Notation mu := (@lebesgue_measure R). +Hypothesis integral_gauss_pdf01 : \int[mu]_x (gauss_pdf01 x)%:E = 1%E. + +Lemma gauss01_dom : gauss01 integral_gauss_pdf01 `<< mu. +Proof. +move=> A mA muA0; rewrite /gauss01. +apply/eqP; rewrite eq_le; apply/andP; split; last first. + by apply: integral_ge0 => x _; rewrite lee_fin gauss_pdf_ge0. +apply: (@le_trans _ _ (\int[mu]_(x in A) (Num.sqrt (pi *+ 2))^-1%:E))%E; last first. + by rewrite integral_cst//= muA0 mule0. +apply: ge0_le_integral => //=. +- by move=> x _; rewrite lee_fin gauss_pdf_ge0. +- apply/measurable_funTS/measurableT_comp => //. + exact: measurable_gauss_pdf. +- by move=> x _; rewrite lee_fin gauss_pdf01_ub. +Qed. + +Let f1 (x : g_sigma_algebraType (R.-ocitv.-measurable)) := (gauss_pdf01 x) ^-1. + +Lemma measurable_fun_f1 : measurable_fun setT f1. +Proof. +apply: continuous_measurable_fun => x. +apply: (@continuousV _ _ gauss_pdf01). + by rewrite gt_eqF// gauss_pdf_gt0. +exact: continuous_gauss_pdf1. +Qed. + +Lemma integrable_f1 U : measurable U -> + (gauss01 integral_gauss_pdf01).-integrable U (fun x : g_sigma_algebraType (R.-ocitv.-measurable) => (f1 x)%:E). +Proof. +Admitted. + +Lemma integral_mgauss01 : forall U, measurable U -> + \int[gauss01 integral_gauss_pdf01]_(y in U) (f1 y)%:E = + \int[mu]_(x0 in U) (gauss_pdf01 x0 * f1 x0)%:E. +Proof. +move=> U mU. +under [in RHS]eq_integral do rewrite EFinM/= muleC. +rewrite -(Radon_Nikodym_change_of_variables gauss01_dom _ (integrable_f1 mU))//=. +apply: ae_eq_integral => //=. +- apply: emeasurable_funM => //. + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_fun_f1. + apply: (measurable_int mu). + apply: (integrableS _ _ (@subsetT _ _)) => //=. + apply: Radon_Nikodym_integrable => //=. + exact: gauss01_dom. +- apply: emeasurable_funM => //. + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_fun_f1. + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_gauss_pdf. +- apply: ae_eq_mul2l => /=. + rewrite Radon_NikodymE//=. + exact: gauss01_dom. + move=> gauss01_dom'. + case: cid => //= h [h1 h2 h3]. + apply: integral_ae_eq => //=. + + exact: integrableS h2. + + apply/measurable_funTS/measurableT_comp => //. + exact: measurable_gauss_pdf. + + by move=> E EU mE; rewrite -(h3 _ mE). +Qed. + +(*Hypothesis integral_mgauss01 : forall U, measurable U -> + \int[gauss01 integral_gauss_pdf01]_(y in U) (f1 y)%:E = + \int[mu]_(x0 in U) (gauss_pdf01 x0 * f1 x0)%:E.*) + +Let mf1 : measurable_fun setT f1. +Proof. +apply: (measurable_comp (F := [set r : R | r != 0%R])) => //. +- exact: open_measurable. +- by move=> /= r [t _ <-]; rewrite gt_eqF// gauss_pdf_gt0. +- apply: open_continuous_measurable_fun => //. + by apply/in_setP => x /= x0; exact: inv_continuous. +- exact: measurable_gauss_pdf. +Qed. + +Definition staton_lebesgue : R.-sfker T ~> _ := + letin (sample_cst (gauss01 integral_gauss_pdf01 : pprobability _ _)) + (letin + (score (measurableT_comp mf1 macc1of2)) + (ret macc1of3)). + +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[gauss01 integral_gauss_pdf01]_(y in U) (f1 y)%:E). + rewrite -[in RHS](setTI U) integral_mkcondr/=. + apply: eq_integral => //= r _. + rewrite letinE/= ge0_integral_mscale//= ger0_norm//; last first. + by rewrite invr_ge0// gauss_pdf_ge0. + rewrite integral_dirac// diracT mul1e/= diracE epatch_indic/=. + by rewrite indicE. +rewrite integral_mgauss01//. +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_pdf_gt0. +by rewrite integral_indic//= setIid. +Qed. + +End gauss_lebesgue. + +(* assuming x > 0 *) +Definition Gamma {R : realType} (x : R) : \bar R := + \int[lebesgue_measure]_(t in `[0%R, +oo[%classic) (expR (- t) * powR t (x - 1))%:E. + +Definition Rfact {R : realType} (x : R) := Gamma (x + 1)%R. + +Section poisson. +Variable R : realType. +Local Open Scope ring_scope. +Notation mu := (@lebesgue_measure R). +Hypothesis integral_poisson_density : forall k, + (\int[mu]_x (@poisson_pdf R k x)%:E = 1%E)%E. + +(* density function for poisson *) +Definition poisson1 := @poisson_pdf R 1%N. + +Lemma poisson1_ge0 (x : R) : 0 <= poisson1 x. +Proof. exact: poisson_pdf_ge0. Qed. + +Definition mpoisson1 (V : set R) : \bar R := + (\int[lebesgue_measure]_(x in V) (poisson1 x)%:E)%E. + +Lemma measurable_fun_poisson1 : measurable_fun setT poisson1. +Proof. exact: measurable_poisson_pdf. Qed. + +Let mpoisson10 : mpoisson1 set0 = 0%E. +Proof. by rewrite /mpoisson1 integral_set0. Qed. + +Lemma mpoisson1_ge0 A : (0 <= mpoisson1 A)%E. +Proof. +apply: integral_ge0 => x Ax. +by rewrite lee_fin poisson1_ge0. +Qed. + +Let mpoisson1_sigma_additive : semi_sigma_additive mpoisson1. +Proof. +move=> /= F mF tF mUF. +rewrite /mpoisson1/= integral_bigcup//=; last first. + apply/integrableP; split. + apply/measurable_EFinP. + exact: measurable_funS (measurable_poisson_pdf _). + rewrite (_ : (fun x => _) = (EFin \o poisson1)); last first. + by apply/funext => x; rewrite gee0_abs// lee_fin poisson1_ge0//. + apply: le_lt_trans. + apply: (@ge0_subset_integral _ _ _ _ _ setT) => //=. + by apply/measurable_EFinP; exact: measurable_poisson_pdf. + by move=> ? _; rewrite lee_fin poisson1_ge0//. + by rewrite /= integral_poisson_density// ltry. +apply: is_cvg_ereal_nneg_natsum_cond => n _ _. +by apply: integral_ge0 => /= x ?; rewrite lee_fin poisson1_ge0. +Qed. + +HB.instance Definition _ := isMeasure.Build _ _ _ + mpoisson1 mpoisson10 mpoisson1_ge0 mpoisson1_sigma_additive. + +Let mpoisson1_setT : mpoisson1 [set: _] = 1%E. +Proof. exact: integral_poisson_density. Qed. + +HB.instance Definition _ := @Measure_isProbability.Build _ _ R + mpoisson1 mpoisson1_setT. + +Definition poisson' := [the probability _ _ of mpoisson1]. + +End poisson. + +(* Staton's definition of the counting measure + Staton ESOP 2017, Sect. 4.2, equation (13) *) +Section staton_counting. +Context d (X : measurableType d). +Variable R : realType. +Notation mu := (@lebesgue_measure R). +Import Notations. +Hypothesis integral_poisson_density : forall k, + (\int[mu]_x (@poisson_pdf R k x)%:E = 1%E)%E. + +Let f1 x := (poisson1 (x : R)) ^-1. + +Let mf1 : measurable_fun setT f1. +rewrite /f1 /poisson1 /poisson_pdf. +apply: (measurable_comp (F := [set r : R | r != 0%R])) => //. +- exact: open_measurable. +- move=> /= r [t ? <-]. + by case: ifPn => // t0; rewrite gt_eqF ?mulr_gt0 ?expR_gt0//= invrK ltr0n. +- apply: open_continuous_measurable_fun => //. + by apply/in_setP => x /= x0; exact: inv_continuous. +- exact: measurable_poisson_pdf. +Qed. + +Definition staton_counting : R.-sfker X ~> _ := + letin (sample_cst (@poisson' R integral_poisson_density : pprobability _ _)) + (letin + (score (measurableT_comp mf1 macc1of2)) + (ret macc1of3)). + +End staton_counting. diff --git a/theories/probability.v b/theories/probability.v index 1ff17ee46..2909baf6f 100644 --- a/theories/probability.v +++ b/theories/probability.v @@ -937,6 +937,23 @@ Qed. HB.instance Definition _ := @Measure_isProbability.Build _ _ R bernoulli bernoulli_setT. +Lemma eq_bernoulli (P : probability bool R) : + P [set true] = p%:E -> P =1 bernoulli. +Proof. +move=> Ptrue sb; rewrite /bernoulli /bernoulli_pmf. +have Pfalse: P [set false] = (1 - p%:E)%E. + rewrite -Ptrue -(@probability_setT _ _ _ P) setT_bool measureU//; last first. + by rewrite disjoints_subset => -[]//. + by rewrite addeAC subee ?add0e//= Ptrue. +have: (0 <= p%:E <= 1)%E by rewrite -Ptrue measure_ge0 probability_le1. +rewrite !lee_fin => ->. +have eq_sb := etrans (bigcup_imset1 (_ : set bool) id) (image_id _). +rewrite -[in LHS](eq_sb sb)/= measure_fin_bigcup//; last 2 first. +- exact: finite_finset. +- by move=> [] [] _ _ [[]]//= []. +- by apply: eq_fsbigr => /= -[]. +Qed. + End bernoulli. Section bernoulli_measure. @@ -966,6 +983,17 @@ Qed. End bernoulli_measure. Arguments bernoulli {R}. +Lemma eq_bernoulliV2 {R : realType} (P : probability bool R) : + P [set true] = P [set false] -> P =1 bernoulli 2^-1. +Proof. +move=> Ptrue_eq_false; apply/eq_bernoulli. +have : P [set: bool] = 1%E := probability_setT. +rewrite setT_bool measureU//=; last first. + by rewrite disjoints_subset => -[]//. +rewrite Ptrue_eq_false -mule2n; move/esym/eqP. +by rewrite -mule_natl -eqe_pdivrMl// mule1 => /eqP<-. +Qed. + Section integral_bernoulli. Context {R : realType}. Variables (p : R) (p01 : (0 <= p <= 1)%R). diff --git a/theories/sequences.v b/theories/sequences.v index 285fb6255..ac185a2d6 100644 --- a/theories/sequences.v +++ b/theories/sequences.v @@ -1004,6 +1004,10 @@ rewrite exprDn (bigD1 (inord 1)) ?inordK// subn1 expr1 bin1 lerDl sumr_ge0//. by move=> i; rewrite ?(mulrn_wge0, mulr_ge0, exprn_ge0, subr_ge0)// ltW. Unshelve. all: by end_near. Qed. +Lemma geometric_ge0 (R : numFieldType) (a z : R) n : + 0 <= a -> 0 <= z -> geometric a z n >= 0. +Proof. by move=> *; rewrite mulr_ge0// exprn_ge0. Qed. + Lemma geometric_seriesE (R : numFieldType) (a z : R) : z != 1 -> series (geometric a z) = [sequence a * (1 - z ^+ n) / (1 - z)]_n. Proof. @@ -1921,7 +1925,7 @@ rewrite -lim_shift_cst; last by rewrite (@lt_le_trans _ _ 0)// f0// leq_addr. Unshelve. all: by end_near. Qed. Lemma nneseries_split_cond (R : realType) (f : nat -> \bar R) N n (P : pred nat) : - (forall k, P k -> 0 <= f k)%E -> + (forall k, P k -> 0 <= f k) -> \sum_(N <= k k Nk. by case: ifPn => //; exact: NPf. Qed. +Lemma nneseriesD1 {R : realType} (f : nat -> \bar R) n (P : pred nat) : + (forall k, P k -> 0 <= f k) -> P n -> + \sum_(0 <= k f0 Pn. +rewrite (@nneseries_split_cond _ f 0%N n.+1 P)// add0n big_mkcond/=. +rewrite big_nat_recr//= Pn -big_mkcond/= -addrA addrCA; congr +%E. +rewrite [RHS]eseries_mkcondr. +rewrite [in RHS](@nneseries_split_cond _ _ _ n.+1 P)//; last first. + by move=> k Pk; case: ifPn => // _; exact: f0. +rewrite add0n [X in _ = X + _]big_mkcond/= big_nat_recr//= Pn eqxx/= adde0. +rewrite -big_mkcond//=; congr +%E. + rewrite big_seq_cond [RHS]big_seq_cond; apply: eq_bigr => /= i. + by rewrite mem_index_iota leq0n/= => /andP[ij Pi]; rewrite lt_eqF. +rewrite eseries_cond [RHS]eseries_cond; apply: eq_eseriesr => i /andP[Pi ji]. +by rewrite gt_eqF. +Qed. + End nneseries_split. Arguments nneseries_split {R f} _ _. Arguments nneseries_split_cond {R f} _ _ _. +Arguments nneseriesD1 {R f} n {P}. -Lemma nneseries_recl (R : realType) (f : nat -> \bar R) : - (forall k, 0 <= f k) -> \sum_(k \bar R) : + (forall k, P k -> 0 <= f k) -> P 0%N -> + \sum_(0 <= k f0; rewrite [LHS](nneseries_split _ 1)// add0n. -by rewrite /index_iota subn0/= big_cons big_nil addr0. +move=> F0 P0; rewrite (nneseriesD1 0%N)//; congr +%E. +by rewrite [RHS]eseries_cond; apply: eq_eseriesl => n; rewrite lt0n. Qed. Lemma nneseries_tail_cvg (R : realType) (f : (\bar R)^nat) P :