(* Title: HOL/Probability/Probability_Mass_Function.thy Author: Johannes Hölzl, TU München Author: Andreas Lochbihler, ETH Zurich *) section ‹ Probability mass function › theory Probability_Mass_Function imports Giry_Monad "HOL-Library.Multiset" begin lemma AE_emeasure_singleton: assumes x: "emeasure M {x} ≠ 0" and ae: "AE x in M. P x" shows "P x" proof - from x have x_M: "{x} ∈ sets M" by (auto intro: emeasure_notin_sets) from ae obtain N where N: "{x∈space M. ¬ P x} ⊆ N" "emeasure M N = 0" "N ∈ sets M" by (auto elim: AE_E) { assume "¬ P x" with x_M[THEN sets.sets_into_space] N have "emeasure M {x} ≤ emeasure M N" by (intro emeasure_mono) auto with x N have False by (auto simp:) } then show "P x" by auto qed lemma AE_measure_singleton: "measure M {x} ≠ 0 ⟹ AE x in M. P x ⟹ P x" by (metis AE_emeasure_singleton measure_def emeasure_empty measure_empty) lemma (in finite_measure) AE_support_countable: assumes [simp]: "sets M = UNIV" shows "(AE x in M. measure M {x} ≠ 0) ⟷ (∃S. countable S ∧ (AE x in M. x ∈ S))" proof assume "∃S. countable S ∧ (AE x in M. x ∈ S)" then obtain S where S[intro]: "countable S" and ae: "AE x in M. x ∈ S" by auto then have "emeasure M (⋃x∈{x∈S. emeasure M {x} ≠ 0}. {x}) = (∫⇧^{+}x. emeasure M {x} * indicator {x∈S. emeasure M {x} ≠ 0} x ∂count_space UNIV)" by (subst emeasure_UN_countable) (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) also have "… = (∫⇧^{+}x. emeasure M {x} * indicator S x ∂count_space UNIV)" by (auto intro!: nn_integral_cong split: split_indicator) also have "… = emeasure M (⋃x∈S. {x})" by (subst emeasure_UN_countable) (auto simp: disjoint_family_on_def nn_integral_restrict_space[symmetric] restrict_count_space) also have "… = emeasure M (space M)" using ae by (intro emeasure_eq_AE) auto finally have "emeasure M {x ∈ space M. x∈S ∧ emeasure M {x} ≠ 0} = emeasure M (space M)" by (simp add: emeasure_single_in_space cong: rev_conj_cong) with finite_measure_compl[of "{x ∈ space M. x∈S ∧ emeasure M {x} ≠ 0}"] have "AE x in M. x ∈ S ∧ emeasure M {x} ≠ 0" by (intro AE_I[OF order_refl]) (auto simp: emeasure_eq_measure measure_nonneg set_diff_eq cong: conj_cong) then show "AE x in M. measure M {x} ≠ 0" by (auto simp: emeasure_eq_measure) qed (auto intro!: exI[of _ "{x. measure M {x} ≠ 0}"] countable_support) subsection ‹ PMF as measure › typedef 'a pmf = "{M :: 'a measure. prob_space M ∧ sets M = UNIV ∧ (AE x in M. measure M {x} ≠ 0)}" morphisms measure_pmf Abs_pmf by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"]) (auto intro!: prob_space_uniform_measure AE_uniform_measureI) declare [[coercion measure_pmf]] lemma prob_space_measure_pmf: "prob_space (measure_pmf p)" using pmf.measure_pmf[of p] by auto interpretation measure_pmf: prob_space "measure_pmf M" for M by (rule prob_space_measure_pmf) interpretation measure_pmf: subprob_space "measure_pmf M" for M by (rule prob_space_imp_subprob_space) unfold_locales lemma subprob_space_measure_pmf: "subprob_space (measure_pmf x)" by unfold_locales locale pmf_as_measure begin setup_lifting type_definition_pmf end context begin interpretation pmf_as_measure . lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV" by transfer blast lemma sets_measure_pmf_count_space[measurable_cong]: "sets (measure_pmf M) = sets (count_space UNIV)" by simp lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV" using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp lemma measure_pmf_UNIV [simp]: "measure (measure_pmf p) UNIV = 1" using measure_pmf.prob_space[of p] by simp lemma measure_pmf_in_subprob_algebra[measurable (raw)]: "measure_pmf x ∈ space (subprob_algebra (count_space UNIV))" by (simp add: space_subprob_algebra subprob_space_measure_pmf) lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV → space N" by (auto simp: measurable_def) lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)" by (intro measurable_cong_sets) simp_all lemma measurable_pair_restrict_pmf2: assumes "countable A" assumes [measurable]: "⋀y. y ∈ A ⟹ (λx. f (x, y)) ∈ measurable M L" shows "f ∈ measurable (M ⨂⇩_{M}restrict_space (measure_pmf N) A) L" (is "f ∈ measurable ?M _") proof - have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" by (simp add: restrict_count_space) show ?thesis by (intro measurable_compose_countable'[where f="λa b. f (fst b, a)" and g=snd and I=A, unfolded prod.collapse] assms) measurable qed lemma measurable_pair_restrict_pmf1: assumes "countable A" assumes [measurable]: "⋀x. x ∈ A ⟹ (λy. f (x, y)) ∈ measurable N L" shows "f ∈ measurable (restrict_space (measure_pmf M) A ⨂⇩_{M}N) L" proof - have [measurable_cong]: "sets (restrict_space (count_space UNIV) A) = sets (count_space A)" by (simp add: restrict_count_space) show ?thesis by (intro measurable_compose_countable'[where f="λa b. f (a, snd b)" and g=fst and I=A, unfolded prod.collapse] assms) measurable qed lift_definition pmf :: "'a pmf ⇒ 'a ⇒ real" is "λM x. measure M {x}" . lift_definition set_pmf :: "'a pmf ⇒ 'a set" is "λM. {x. measure M {x} ≠ 0}" . declare [[coercion set_pmf]] lemma AE_measure_pmf: "AE x in (M::'a pmf). x ∈ M" by transfer simp lemma emeasure_pmf_single_eq_zero_iff: fixes M :: "'a pmf" shows "emeasure M {y} = 0 ⟷ y ∉ M" unfolding set_pmf.rep_eq by (simp add: measure_pmf.emeasure_eq_measure) lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) ⟷ (∀y∈M. P y)" using AE_measure_singleton[of M] AE_measure_pmf[of M] by (auto simp: set_pmf.rep_eq) lemma AE_pmfI: "(⋀y. y ∈ set_pmf M ⟹ P y) ⟹ almost_everywhere (measure_pmf M) P" by(simp add: AE_measure_pmf_iff) lemma countable_set_pmf [simp]: "countable (set_pmf p)" by transfer (metis prob_space.finite_measure finite_measure.countable_support) lemma pmf_positive: "x ∈ set_pmf p ⟹ 0 < pmf p x" by transfer (simp add: less_le) lemma pmf_nonneg[simp]: "0 ≤ pmf p x" by transfer simp lemma pmf_not_neg [simp]: "¬pmf p x < 0" by (simp add: not_less pmf_nonneg) lemma pmf_pos [simp]: "pmf p x ≠ 0 ⟹ pmf p x > 0" using pmf_nonneg[of p x] by linarith lemma pmf_le_1: "pmf p x ≤ 1" by (simp add: pmf.rep_eq) lemma set_pmf_not_empty: "set_pmf M ≠ {}" using AE_measure_pmf[of M] by (intro notI) simp lemma set_pmf_iff: "x ∈ set_pmf M ⟷ pmf M x ≠ 0" by transfer simp lemma pmf_positive_iff: "0 < pmf p x ⟷ x ∈ set_pmf p" unfolding less_le by (simp add: set_pmf_iff) lemma set_pmf_eq: "set_pmf M = {x. pmf M x ≠ 0}" by (auto simp: set_pmf_iff) lemma set_pmf_eq': "set_pmf p = {x. pmf p x > 0}" proof safe fix x assume "x ∈ set_pmf p" hence "pmf p x ≠ 0" by (auto simp: set_pmf_eq) with pmf_nonneg[of p x] show "pmf p x > 0" by simp qed (auto simp: set_pmf_eq) lemma emeasure_pmf_single: fixes M :: "'a pmf" shows "emeasure M {x} = pmf M x" by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) lemma measure_pmf_single: "measure (measure_pmf M) {x} = pmf M x" using emeasure_pmf_single[of M x] by(simp add: measure_pmf.emeasure_eq_measure pmf_nonneg measure_nonneg) lemma emeasure_measure_pmf_finite: "finite S ⟹ emeasure (measure_pmf M) S = (∑s∈S. pmf M s)" by (subst emeasure_eq_sum_singleton) (auto simp: emeasure_pmf_single pmf_nonneg) lemma measure_measure_pmf_finite: "finite S ⟹ measure (measure_pmf M) S = sum (pmf M) S" using emeasure_measure_pmf_finite[of S M] by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg sum_nonneg pmf_nonneg) lemma sum_pmf_eq_1: assumes "finite A" "set_pmf p ⊆ A" shows "(∑x∈A. pmf p x) = 1" proof - have "(∑x∈A. pmf p x) = measure_pmf.prob p A" by (simp add: measure_measure_pmf_finite assms) also from assms have "… = 1" by (subst measure_pmf.prob_eq_1) (auto simp: AE_measure_pmf_iff) finally show ?thesis . qed lemma nn_integral_measure_pmf_support: fixes f :: "'a ⇒ ennreal" assumes f: "finite A" and nn: "⋀x. x ∈ A ⟹ 0 ≤ f x" "⋀x. x ∈ set_pmf M ⟹ x ∉ A ⟹ f x = 0" shows "(∫⇧^{+}x. f x ∂measure_pmf M) = (∑x∈A. f x * pmf M x)" proof - have "(∫⇧^{+}x. f x ∂M) = (∫⇧^{+}x. f x * indicator A x ∂M)" using nn by (intro nn_integral_cong_AE) (auto simp: AE_measure_pmf_iff split: split_indicator) also have "… = (∑x∈A. f x * emeasure M {x})" using assms by (intro nn_integral_indicator_finite) auto finally show ?thesis by (simp add: emeasure_measure_pmf_finite) qed lemma nn_integral_measure_pmf_finite: fixes f :: "'a ⇒ ennreal" assumes f: "finite (set_pmf M)" and nn: "⋀x. x ∈ set_pmf M ⟹ 0 ≤ f x" shows "(∫⇧^{+}x. f x ∂measure_pmf M) = (∑x∈set_pmf M. f x * pmf M x)" using assms by (intro nn_integral_measure_pmf_support) auto lemma integrable_measure_pmf_finite: fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}" shows "finite (set_pmf M) ⟹ integrable M f" by (auto intro!: integrableI_bounded simp: nn_integral_measure_pmf_finite ennreal_mult_less_top) lemma integral_measure_pmf_real: assumes [simp]: "finite A" and "⋀a. a ∈ set_pmf M ⟹ f a ≠ 0 ⟹ a ∈ A" shows "(∫x. f x ∂measure_pmf M) = (∑a∈A. f a * pmf M a)" proof - have "(∫x. f x ∂measure_pmf M) = (∫x. f x * indicator A x ∂measure_pmf M)" using assms(2) by (intro integral_cong_AE) (auto split: split_indicator simp: AE_measure_pmf_iff) also have "… = (∑a∈A. f a * pmf M a)" by (subst integral_indicator_finite_real) (auto simp: measure_def emeasure_measure_pmf_finite pmf_nonneg) finally show ?thesis . qed lemma integrable_pmf: "integrable (count_space X) (pmf M)" proof - have " (∫⇧^{+}x. pmf M x ∂count_space X) = (∫⇧^{+}x. pmf M x ∂count_space (M ∩ X))" by (auto simp add: nn_integral_count_space_indicator set_pmf_iff intro!: nn_integral_cong split: split_indicator) then have "integrable (count_space X) (pmf M) = integrable (count_space (M ∩ X)) (pmf M)" by (simp add: integrable_iff_bounded pmf_nonneg) then show ?thesis by (simp add: pmf.rep_eq measure_pmf.integrable_measure disjoint_family_on_def) qed lemma integral_pmf: "(∫x. pmf M x ∂count_space X) = measure M X" proof - have "(∫x. pmf M x ∂count_space X) = (∫⇧^{+}x. pmf M x ∂count_space X)" by (simp add: pmf_nonneg integrable_pmf nn_integral_eq_integral) also have "… = (∫⇧^{+}x. emeasure M {x} ∂count_space (X ∩ M))" by (auto intro!: nn_integral_cong_AE split: split_indicator simp: pmf.rep_eq measure_pmf.emeasure_eq_measure nn_integral_count_space_indicator AE_count_space set_pmf_iff) also have "… = emeasure M (X ∩ M)" by (rule emeasure_countable_singleton[symmetric]) (auto intro: countable_set_pmf) also have "… = emeasure M X" by (auto intro!: emeasure_eq_AE simp: AE_measure_pmf_iff) finally show ?thesis by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg integral_nonneg pmf_nonneg) qed lemma integral_pmf_restrict: "(f::'a ⇒ 'b::{banach, second_countable_topology}) ∈ borel_measurable (count_space UNIV) ⟹ (∫x. f x ∂measure_pmf M) = (∫x. f x ∂restrict_space M M)" by (auto intro!: integral_cong_AE simp add: integral_restrict_space AE_measure_pmf_iff) lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" proof - have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) then show ?thesis using measure_pmf.emeasure_space_1 by simp qed lemma emeasure_pmf_UNIV [simp]: "emeasure (measure_pmf M) UNIV = 1" using measure_pmf.emeasure_space_1[of M] by simp lemma in_null_sets_measure_pmfI: "A ∩ set_pmf p = {} ⟹ A ∈ null_sets (measure_pmf p)" using emeasure_eq_0_AE[where ?P="λx. x ∈ A" and M="measure_pmf p"] by(auto simp add: null_sets_def AE_measure_pmf_iff) lemma measure_subprob: "measure_pmf M ∈ space (subprob_algebra (count_space UNIV))" by (simp add: space_subprob_algebra subprob_space_measure_pmf) subsection ‹ Monad Interpretation › lemma measurable_measure_pmf[measurable]: "(λx. measure_pmf (M x)) ∈ measurable (count_space UNIV) (subprob_algebra (count_space UNIV))" by (auto simp: space_subprob_algebra intro!: prob_space_imp_subprob_space) unfold_locales lemma bind_measure_pmf_cong: assumes "⋀x. A x ∈ space (subprob_algebra N)" "⋀x. B x ∈ space (subprob_algebra N)" assumes "⋀i. i ∈ set_pmf x ⟹ A i = B i" shows "bind (measure_pmf x) A = bind (measure_pmf x) B" proof (rule measure_eqI) show "sets (measure_pmf x ⤜ A) = sets (measure_pmf x ⤜ B)" using assms by (subst (1 2) sets_bind) (auto simp: space_subprob_algebra) next fix X assume "X ∈ sets (measure_pmf x ⤜ A)" then have X: "X ∈ sets N" using assms by (subst (asm) sets_bind) (auto simp: space_subprob_algebra) show "emeasure (measure_pmf x ⤜ A) X = emeasure (measure_pmf x ⤜ B) X" using assms by (subst (1 2) emeasure_bind[where N=N, OF _ _ X]) (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) qed lift_definition bind_pmf :: "'a pmf ⇒ ('a ⇒ 'b pmf ) ⇒ 'b pmf" is bind proof (clarify, intro conjI) fix f :: "'a measure" and g :: "'a ⇒ 'b measure" assume "prob_space f" then interpret f: prob_space f . assume "sets f = UNIV" and ae_f: "AE x in f. measure f {x} ≠ 0" then have s_f[simp]: "sets f = sets (count_space UNIV)" by simp assume g: "⋀x. prob_space (g x) ∧ sets (g x) = UNIV ∧ (AE y in g x. measure (g x) {y} ≠ 0)" then have g: "⋀x. prob_space (g x)" and s_g[simp]: "⋀x. sets (g x) = sets (count_space UNIV)" and ae_g: "⋀x. AE y in g x. measure (g x) {y} ≠ 0" by auto have [measurable]: "g ∈ measurable f (subprob_algebra (count_space UNIV))" by (auto simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space g) show "prob_space (f ⤜ g)" using g by (intro f.prob_space_bind[where S="count_space UNIV"]) auto then interpret fg: prob_space "f ⤜ g" . show [simp]: "sets (f ⤜ g) = UNIV" using sets_eq_imp_space_eq[OF s_f] by (subst sets_bind[where N="count_space UNIV"]) auto show "AE x in f ⤜ g. measure (f ⤜ g) {x} ≠ 0" apply (simp add: fg.prob_eq_0 AE_bind[where B="count_space UNIV"]) using ae_f apply eventually_elim using ae_g apply eventually_elim apply (auto dest: AE_measure_singleton) done qed adhoc_overloading Monad_Syntax.bind bind_pmf lemma ennreal_pmf_bind: "pmf (bind_pmf N f) i = (∫⇧^{+}x. pmf (f x) i ∂measure_pmf N)" unfolding pmf.rep_eq bind_pmf.rep_eq by (auto simp: measure_pmf.measure_bind[where N="count_space UNIV"] measure_subprob measure_nonneg intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) lemma pmf_bind: "pmf (bind_pmf N f) i = (∫x. pmf (f x) i ∂measure_pmf N)" using ennreal_pmf_bind[of N f i] by (subst (asm) nn_integral_eq_integral) (auto simp: pmf_nonneg pmf_le_1 pmf_nonneg integral_nonneg intro!: nn_integral_eq_integral[symmetric] measure_pmf.integrable_const_bound[where B=1]) lemma bind_pmf_const[simp]: "bind_pmf M (λx. c) = c" by transfer (simp add: bind_const' prob_space_imp_subprob_space) lemma set_bind_pmf[simp]: "set_pmf (bind_pmf M N) = (⋃M∈set_pmf M. set_pmf (N M))" proof - have "set_pmf (bind_pmf M N) = {x. ennreal (pmf (bind_pmf M N) x) ≠ 0}" by (simp add: set_pmf_eq pmf_nonneg) also have "… = (⋃M∈set_pmf M. set_pmf (N M))" unfolding ennreal_pmf_bind by (subst nn_integral_0_iff_AE) (auto simp: AE_measure_pmf_iff pmf_nonneg set_pmf_eq) finally show ?thesis . qed lemma bind_pmf_cong [fundef_cong]: assumes "p = q" shows "(⋀x. x ∈ set_pmf q ⟹ f x = g x) ⟹ bind_pmf p f = bind_pmf q g" unfolding ‹p = q›[symmetric] measure_pmf_inject[symmetric] bind_pmf.rep_eq by (auto simp: AE_measure_pmf_iff Pi_iff space_subprob_algebra subprob_space_measure_pmf sets_bind[where N="count_space UNIV"] emeasure_bind[where N="count_space UNIV"] intro!: nn_integral_cong_AE measure_eqI) lemma bind_pmf_cong_simp: "p = q ⟹ (⋀x. x ∈ set_pmf q =simp=> f x = g x) ⟹ bind_pmf p f = bind_pmf q g" by (simp add: simp_implies_def cong: bind_pmf_cong) lemma measure_pmf_bind: "measure_pmf (bind_pmf M f) = (measure_pmf M ⤜ (λx. measure_pmf (f x)))" by transfer simp lemma nn_integral_bind_pmf[simp]: "(∫⇧^{+}x. f x ∂bind_pmf M N) = (∫⇧^{+}x. ∫⇧^{+}y. f y ∂N x ∂M)" using measurable_measure_pmf[of N] unfolding measure_pmf_bind apply (intro nn_integral_bind[where B="count_space UNIV"]) apply auto done lemma emeasure_bind_pmf[simp]: "emeasure (bind_pmf M N) X = (∫⇧^{+}x. emeasure (N x) X ∂M)" using measurable_measure_pmf[of N] unfolding measure_pmf_bind by (subst emeasure_bind[where N="count_space UNIV"]) auto lift_definition return_pmf :: "'a ⇒ 'a pmf" is "return (count_space UNIV)" by (auto intro!: prob_space_return simp: AE_return measure_return) lemma bind_return_pmf: "bind_pmf (return_pmf x) f = f x" by transfer (auto intro!: prob_space_imp_subprob_space bind_return[where N="count_space UNIV"] simp: space_subprob_algebra) lemma set_return_pmf[simp]: "set_pmf (return_pmf x) = {x}" by transfer (auto simp add: measure_return split: split_indicator) lemma bind_return_pmf': "bind_pmf N return_pmf = N" proof (transfer, clarify) fix N :: "'a measure" assume "sets N = UNIV" then show "N ⤜ return (count_space UNIV) = N" by (subst return_sets_cong[where N=N]) (simp_all add: bind_return') qed lemma bind_assoc_pmf: "bind_pmf (bind_pmf A B) C = bind_pmf A (λx. bind_pmf (B x) C)" by transfer (auto intro!: bind_assoc[where N="count_space UNIV" and R="count_space UNIV"] simp: measurable_def space_subprob_algebra prob_space_imp_subprob_space) definition "map_pmf f M = bind_pmf M (λx. return_pmf (f x))" lemma map_bind_pmf: "map_pmf f (bind_pmf M g) = bind_pmf M (λx. map_pmf f (g x))" by (simp add: map_pmf_def bind_assoc_pmf) lemma bind_map_pmf: "bind_pmf (map_pmf f M) g = bind_pmf M (λx. g (f x))" by (simp add: map_pmf_def bind_assoc_pmf bind_return_pmf) lemma map_pmf_transfer[transfer_rule]: "rel_fun (=) (rel_fun cr_pmf cr_pmf) (λf M. distr M (count_space UNIV) f) map_pmf" proof - have "rel_fun (=) (rel_fun pmf_as_measure.cr_pmf pmf_as_measure.cr_pmf) (λf M. M ⤜ (return (count_space UNIV) o f)) map_pmf" unfolding map_pmf_def[abs_def] comp_def by transfer_prover then show ?thesis by (force simp: rel_fun_def cr_pmf_def bind_return_distr) qed lemma map_pmf_rep_eq: "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f" unfolding map_pmf_def bind_pmf.rep_eq comp_def return_pmf.rep_eq using bind_return_distr[of M f "count_space UNIV"] by (simp add: comp_def) lemma map_pmf_id[simp]: "map_pmf id = id" by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI) lemma map_pmf_ident[simp]: "map_pmf (λx. x) = (λx. x)" using map_pmf_id unfolding id_def . lemma map_pmf_compose: "map_pmf (f ∘ g) = map_pmf f ∘ map_pmf g" by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) lemma map_pmf_comp: "map_pmf f (map_pmf g M) = map_pmf (λx. f (g x)) M" using map_pmf_compose[of f g] by (simp add: comp_def) lemma map_pmf_cong: "p = q ⟹ (⋀x. x ∈ set_pmf q ⟹ f x = g x) ⟹ map_pmf f p = map_pmf g q" unfolding map_pmf_def by (rule bind_pmf_cong) auto lemma pmf_set_map: "set_pmf ∘ map_pmf f = (`) f ∘ set_pmf" by (auto simp add: comp_def fun_eq_iff map_pmf_def) lemma set_map_pmf[simp]: "set_pmf (map_pmf f M) = f`set_pmf M" using pmf_set_map[of f] by (auto simp: comp_def fun_eq_iff) lemma emeasure_map_pmf[simp]: "emeasure (map_pmf f M) X = emeasure M (f -` X)" unfolding map_pmf_rep_eq by (subst emeasure_distr) auto lemma measure_map_pmf[simp]: "measure (map_pmf f M) X = measure M (f -` X)" using emeasure_map_pmf[of f M X] by(simp add: measure_pmf.emeasure_eq_measure measure_nonneg) lemma nn_integral_map_pmf[simp]: "(∫⇧^{+}x. f x ∂map_pmf g M) = (∫⇧^{+}x. f (g x) ∂M)" unfolding map_pmf_rep_eq by (intro nn_integral_distr) auto lemma ennreal_pmf_map: "pmf (map_pmf f p) x = (∫⇧^{+}y. indicator (f -` {x}) y ∂measure_pmf p)" proof (transfer fixing: f x) fix p :: "'b measure" presume "prob_space p" then interpret prob_space p . presume "sets p = UNIV" then show "ennreal (measure (distr p (count_space UNIV) f) {x}) = integral⇧^{N}p (indicator (f -` {x}))" by(simp add: measure_distr measurable_def emeasure_eq_measure) qed simp_all lemma pmf_map: "pmf (map_pmf f p) x = measure p (f -` {x})" proof (transfer fixing: f x) fix p :: "'b measure" presume "prob_space p" then interpret prob_space p . presume "sets p = UNIV" then show "measure (distr p (count_space UNIV) f) {x} = measure p (f -` {x})" by(simp add: measure_distr measurable_def emeasure_eq_measure) qed simp_all lemma nn_integral_pmf: "(∫⇧^{+}x. pmf p x ∂count_space A) = emeasure (measure_pmf p) A" proof - have "(∫⇧^{+}x. pmf p x ∂count_space A) = (∫⇧^{+}x. pmf p x ∂count_space (A ∩ set_pmf p))" by(auto simp add: nn_integral_count_space_indicator indicator_def set_pmf_iff intro: nn_integral_cong) also have "… = emeasure (measure_pmf p) (⋃x∈A ∩ set_pmf p. {x})" by(subst emeasure_UN_countable)(auto simp add: emeasure_pmf_single disjoint_family_on_def) also have "… = emeasure (measure_pmf p) ((⋃x∈A ∩ set_pmf p. {x}) ∪ {x. x ∈ A ∧ x ∉ set_pmf p})" by(rule emeasure_Un_null_set[symmetric])(auto intro: in_null_sets_measure_pmfI) also have "… = emeasure (measure_pmf p) A" by(auto intro: arg_cong2[where f=emeasure]) finally show ?thesis . qed lemma integral_map_pmf[simp]: fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}" shows "integral⇧^{L}(map_pmf g p) f = integral⇧^{L}p (λx. f (g x))" by (simp add: integral_distr map_pmf_rep_eq) lemma pmf_abs_summable [intro]: "pmf p abs_summable_on A" by (rule abs_summable_on_subset[OF _ subset_UNIV]) (auto simp: abs_summable_on_def integrable_iff_bounded nn_integral_pmf) lemma measure_pmf_conv_infsetsum: "measure (measure_pmf p) A = infsetsum (pmf p) A" unfolding infsetsum_def by (simp add: integral_eq_nn_integral nn_integral_pmf measure_def) lemma infsetsum_pmf_eq_1: assumes "set_pmf p ⊆ A" shows "infsetsum (pmf p) A = 1" proof - have "infsetsum (pmf p) A = lebesgue_integral (count_space UNIV) (pmf p)" using assms unfolding infsetsum_altdef set_lebesgue_integral_def by (intro Bochner_Integration.integral_cong) (auto simp: indicator_def set_pmf_eq) also have "… = 1" by (subst integral_eq_nn_integral) (auto simp: nn_integral_pmf) finally show ?thesis . qed lemma map_return_pmf [simp]: "map_pmf f (return_pmf x) = return_pmf (f x)" by transfer (simp add: distr_return) lemma map_pmf_const[simp]: "map_pmf (λ_. c) M = return_pmf c" by transfer (auto simp: prob_space.distr_const) lemma pmf_return [simp]: "pmf (return_pmf x) y = indicator {y} x" by transfer (simp add: measure_return) lemma nn_integral_return_pmf[simp]: "0 ≤ f x ⟹ (∫⇧^{+}x. f x ∂return_pmf x) = f x" unfolding return_pmf.rep_eq by (intro nn_integral_return) auto lemma emeasure_return_pmf[simp]: "emeasure (return_pmf x) X = indicator X x" unfolding return_pmf.rep_eq by (intro emeasure_return) auto lemma measure_return_pmf [simp]: "measure_pmf.prob (return_pmf x) A = indicator A x" proof - have "ennreal (measure_pmf.prob (return_pmf x) A) = emeasure (measure_pmf (return_pmf x)) A" by (simp add: measure_pmf.emeasure_eq_measure) also have "… = ennreal (indicator A x)" by (simp add: ennreal_indicator) finally show ?thesis by simp qed lemma return_pmf_inj[simp]: "return_pmf x = return_pmf y ⟷ x = y" by (metis insertI1 set_return_pmf singletonD) lemma map_pmf_eq_return_pmf_iff: "map_pmf f p = return_pmf x ⟷ (∀y ∈ set_pmf p. f y = x)" proof assume "map_pmf f p = return_pmf x" then have "set_pmf (map_pmf f p) = set_pmf (return_pmf x)" by simp then show "∀y ∈ set_pmf p. f y = x" by auto next assume "∀y ∈ set_pmf p. f y = x" then show "map_pmf f p = return_pmf x" unfolding map_pmf_const[symmetric, of _ p] by (intro map_pmf_cong) auto qed definition "pair_pmf A B = bind_pmf A (λx. bind_pmf B (λy. return_pmf (x, y)))" lemma pmf_pair: "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b" unfolding pair_pmf_def pmf_bind pmf_return apply (subst integral_measure_pmf_real[where A="{b}"]) apply (auto simp: indicator_eq_0_iff) apply (subst integral_measure_pmf_real[where A="{a}"]) apply (auto simp: indicator_eq_0_iff sum_nonneg_eq_0_iff pmf_nonneg) done lemma set_pair_pmf[simp]: "set_pmf (pair_pmf A B) = set_pmf A × set_pmf B" unfolding pair_pmf_def set_bind_pmf set_return_pmf by auto lemma measure_pmf_in_subprob_space[measurable (raw)]: "measure_pmf M ∈ space (subprob_algebra (count_space UNIV))" by (simp add: space_subprob_algebra) intro_locales lemma nn_integral_pair_pmf': "(∫⇧^{+}x. f x ∂pair_pmf A B) = (∫⇧^{+}a. ∫⇧^{+}b. f (a, b) ∂B ∂A)" proof - have "(∫⇧^{+}x. f x ∂pair_pmf A B) = (∫⇧^{+}x. f x * indicator (A × B) x ∂pair_pmf A B)" by (auto simp: AE_measure_pmf_iff intro!: nn_integral_cong_AE) also have "… = (∫⇧^{+}a. ∫⇧^{+}b. f (a, b) * indicator (A × B) (a, b) ∂B ∂A)" by (simp add: pair_pmf_def) also have "… = (∫⇧^{+}a. ∫⇧^{+}b. f (a, b) ∂B ∂A)" by (auto intro!: nn_integral_cong_AE simp: AE_measure_pmf_iff) finally show ?thesis . qed lemma bind_pair_pmf: assumes M[measurable]: "M ∈ measurable (count_space UNIV ⨂⇩_{M}count_space UNIV) (subprob_algebra N)" shows "measure_pmf (pair_pmf A B) ⤜ M = (measure_pmf A ⤜ (λx. measure_pmf B ⤜ (λy. M (x, y))))" (is "?L = ?R") proof (rule measure_eqI) have M'[measurable]: "M ∈ measurable (pair_pmf A B) (subprob_algebra N)" using M[THEN measurable_space] by (simp_all add: space_pair_measure) note measurable_bind[where N="count_space UNIV", measurable] note measure_pmf_in_subprob_space[simp] have sets_eq_N: "sets ?L = N" by (subst sets_bind[OF sets_kernel[OF M']]) auto show "sets ?L = sets ?R" using measurable_space[OF M] by (simp add: sets_eq_N space_pair_measure space_subprob_algebra) fix X assume "X ∈ sets ?L" then have X[measurable]: "X ∈ sets N" unfolding sets_eq_N . then show "emeasure ?L X = emeasure ?R X" apply (simp add: emeasure_bind[OF _ M' X]) apply (simp add: nn_integral_bind[where B="count_space UNIV"] pair_pmf_def measure_pmf_bind[of A] nn_integral_measure_pmf_finite) apply (subst emeasure_bind[OF _ _ X]) apply measurable apply (subst emeasure_bind[OF _ _ X]) apply measurable done qed lemma map_fst_pair_pmf: "map_pmf fst (pair_pmf A B) = A" by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') lemma map_snd_pair_pmf: "map_pmf snd (pair_pmf A B) = B" by (simp add: pair_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') lemma nn_integral_pmf': "inj_on f A ⟹ (∫⇧^{+}x. pmf p (f x) ∂count_space A) = emeasure p (f ` A)" by (subst nn_integral_bij_count_space[where g=f and B="f`A"]) (auto simp: bij_betw_def nn_integral_pmf) lemma pmf_le_0_iff[simp]: "pmf M p ≤ 0 ⟷ pmf M p = 0" using pmf_nonneg[of M p] by arith lemma min_pmf_0[simp]: "min (pmf M p) 0 = 0" "min 0 (pmf M p) = 0" using pmf_nonneg[of M p] by arith+ lemma pmf_eq_0_set_pmf: "pmf M p = 0 ⟷ p ∉ set_pmf M" unfolding set_pmf_iff by simp lemma pmf_map_inj: "inj_on f (set_pmf M) ⟹ x ∈ set_pmf M ⟹ pmf (map_pmf f M) (f x) = pmf M x" by (auto simp: pmf.rep_eq map_pmf_rep_eq measure_distr AE_measure_pmf_iff inj_onD intro!: measure_pmf.finite_measure_eq_AE) lemma pmf_map_inj': "inj f ⟹ pmf (map_pmf f M) (f x) = pmf M x" apply(cases "x ∈ set_pmf M") apply(simp add: pmf_map_inj[OF subset_inj_on]) apply(simp add: pmf_eq_0_set_pmf[symmetric]) apply(auto simp add: pmf_eq_0_set_pmf dest: injD) done lemma pmf_map_outside: "x ∉ f ` set_pmf M ⟹ pmf (map_pmf f M) x = 0" unfolding pmf_eq_0_set_pmf by simp lemma measurable_set_pmf[measurable]: "Measurable.pred (count_space UNIV) (λx. x ∈ set_pmf M)" by simp subsection ‹ PMFs as function › context fixes f :: "'a ⇒ real" assumes nonneg: "⋀x. 0 ≤ f x" assumes prob: "(∫⇧^{+}x. f x ∂count_space UNIV) = 1" begin lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ennreal ∘ f)" proof (intro conjI) have *[simp]: "⋀x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y" by (simp split: split_indicator) show "AE x in density (count_space UNIV) (ennreal ∘ f). measure (density (count_space UNIV) (ennreal ∘ f)) {x} ≠ 0" by (simp add: AE_density nonneg measure_def emeasure_density max_def) show "prob_space (density (count_space UNIV) (ennreal ∘ f))" by standard (simp add: emeasure_density prob) qed simp lemma pmf_embed_pmf: "pmf embed_pmf x = f x" proof transfer have *[simp]: "⋀x y. ennreal (f y) * indicator {x} y = ennreal (f x) * indicator {x} y" by (simp split: split_indicator) fix x show "measure (density (count_space UNIV) (ennreal ∘ f)) {x} = f x" by transfer (simp add: measure_def emeasure_density nonneg max_def) qed lemma set_embed_pmf: "set_pmf embed_pmf = {x. f x ≠ 0}" by(auto simp add: set_pmf_eq pmf_embed_pmf) end lemma embed_pmf_transfer: "rel_fun (eq_onp (λf. (∀x. 0 ≤ f x) ∧ (∫⇧^{+}x. ennreal (f x) ∂count_space UNIV) = 1)) pmf_as_measure.cr_pmf (λf. density (count_space UNIV) (ennreal ∘ f)) embed_pmf" by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer) lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" proof (transfer, elim conjE) fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} ≠ 0" assume "prob_space M" then interpret prob_space M . show "M = density (count_space UNIV) (λx. ennreal (measure M {x}))" proof (rule measure_eqI) fix A :: "'a set" have "(∫⇧^{+}x. ennreal (measure M {x}) * indicator A x ∂count_space UNIV) = (∫⇧^{+}x. emeasure M {x} * indicator (A ∩ {x. measure M {x} ≠ 0}) x ∂count_space UNIV)" by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) also have "… = (∫⇧^{+}x. emeasure M {x} ∂count_space (A ∩ {x. measure M {x} ≠ 0}))" by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) also have "… = emeasure M (⋃x∈(A ∩ {x. measure M {x} ≠ 0}). {x})" by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) (auto simp: disjoint_family_on_def) also have "… = emeasure M A" using ae by (intro emeasure_eq_AE) auto finally show " emeasure M A = emeasure (density (count_space UNIV) (λx. ennreal (measure M {x}))) A" using emeasure_space_1 by (simp add: emeasure_density) qed simp qed lemma td_pmf_embed_pmf: "type_definition pmf embed_pmf {f::'a ⇒ real. (∀x. 0 ≤ f x) ∧ (∫⇧^{+}x. ennreal (f x) ∂count_space UNIV) = 1}" unfolding type_definition_def proof safe fix p :: "'a pmf" have "(∫⇧^{+}x. 1 ∂measure_pmf p) = 1" using measure_pmf.emeasure_space_1[of p] by simp then show *: "(∫⇧^{+}x. ennreal (pmf p x) ∂count_space UNIV) = 1" by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const) show "embed_pmf (pmf p) = p" by (intro measure_pmf_inject[THEN iffD1]) (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def) next fix f :: "'a ⇒ real" assume "∀x. 0 ≤ f x" "(∫⇧^{+}x. f x ∂count_space UNIV) = 1" then show "pmf (embed_pmf f) = f" by (auto intro!: pmf_embed_pmf) qed (rule pmf_nonneg) end lemma nn_integral_measure_pmf: "(∫⇧^{+}x. f x ∂measure_pmf p) = ∫⇧^{+}x. ennreal (pmf p x) * f x ∂count_space UNIV" by(simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg) lemma integral_measure_pmf: fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}" assumes A: "finite A" shows "(⋀a. a ∈ set_pmf M ⟹ f a ≠ 0 ⟹ a ∈ A) ⟹ (LINT x|M. f x) = (∑a∈A. pmf M a *⇩_{R}f a)" unfolding measure_pmf_eq_density apply (simp add: integral_density) apply (subst lebesgue_integral_count_space_finite_support) apply (auto intro!: finite_subset[OF _ ‹finite A›] sum.mono_neutral_left simp: pmf_eq_0_set_pmf) done lemma expectation_return_pmf [simp]: fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}" shows "measure_pmf.expectation (return_pmf x) f = f x" by (subst integral_measure_pmf[of "{x}"]) simp_all lemma pmf_expectation_bind: fixes p :: "'a pmf" and f :: "'a ⇒ 'b pmf" and h :: "'b ⇒ 'c::{banach, second_countable_topology}" assumes "finite A" "⋀x. x ∈ A ⟹ finite (set_pmf (f x))" "set_pmf p ⊆ A" shows "measure_pmf.expectation (p ⤜ f) h = (∑a∈A. pmf p a *⇩_{R}measure_pmf.expectation (f a) h)" proof - have "measure_pmf.expectation (p ⤜ f) h = (∑a∈(⋃x∈A. set_pmf (f x)). pmf (p ⤜ f) a *⇩_{R}h a)" using assms by (intro integral_measure_pmf) auto also have "… = (∑x∈(⋃x∈A. set_pmf (f x)). (∑a∈A. (pmf p a * pmf (f a) x) *⇩_{R}h x))" proof (intro sum.cong refl, goal_cases) case (1 x) thus ?case by (subst pmf_bind, subst integral_measure_pmf[of A]) (insert assms, auto simp: scaleR_sum_left) qed also have "… = (∑j∈A. pmf p j *⇩_{R}(∑i∈(⋃x∈A. set_pmf (f x)). pmf (f j) i *⇩_{R}h i))" by (subst sum.swap) (simp add: scaleR_sum_right) also have "… = (∑j∈A. pmf p j *⇩_{R}measure_pmf.expectation (f j) h)" proof (intro sum.cong refl, goal_cases) case (1 x) thus ?case by (subst integral_measure_pmf[of "(⋃x∈A. set_pmf (f x))"]) (insert assms, auto simp: scaleR_sum_left) qed finally show ?thesis . qed lemma continuous_on_LINT_pmf: ― ‹This is dominated convergence!?› fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::{banach, second_countable_topology}" assumes f: "⋀i. i ∈ set_pmf M ⟹ continuous_on A (f i)" and bnd: "⋀a i. a ∈ A ⟹ i ∈ set_pmf M ⟹ norm (f i a) ≤ B" shows "continuous_on A (λa. LINT i|M. f i a)" proof cases assume "finite M" with f show ?thesis using integral_measure_pmf[OF ‹finite M›] by (subst integral_measure_pmf[OF ‹finite M›]) (auto intro!: continuous_on_sum continuous_on_scaleR continuous_on_const) next assume "infinite M" let ?f = "λi x. pmf (map_pmf (to_nat_on M) M) i *⇩_{R}f (from_nat_into M i) x" show ?thesis proof (rule uniform_limit_theorem) show "∀⇩_{F}n in sequentially. continuous_on A (λa. ∑i<n. ?f i a)" by (intro always_eventually allI continuous_on_sum continuous_on_scaleR continuous_on_const f from_nat_into set_pmf_not_empty) show "uniform_limit A (λn a. ∑i<n. ?f i a) (λa. LINT i|M. f i a) sequentially" proof (subst uniform_limit_cong[where g="λn a. ∑i<n. ?f i a"]) fix a assume "a ∈ A" have 1: "(LINT i|M. f i a) = (LINT i|map_pmf (to_nat_on M) M. f (from_nat_into M i) a)" by (auto intro!: integral_cong_AE AE_pmfI) have 2: "… = (LINT i|count_space UNIV. pmf (map_pmf (to_nat_on M) M) i *⇩_{R}f (from_nat_into M i) a)" by (simp add: measure_pmf_eq_density integral_density) have "(λn. ?f n a) sums (LINT i|M. f i a)" unfolding 1 2 proof (intro sums_integral_count_space_nat) have A: "integrable M (λi. f i a)" using ‹a∈A› by (auto intro!: measure_pmf.integrable_const_bound AE_pmfI bnd) have "integrable (map_pmf (to_nat_on M) M) (λi. f (from_nat_into M i) a)" by (auto simp add: map_pmf_rep_eq integrable_distr_eq intro!: AE_pmfI integrable_cong_AE_imp[OF A]) then show "integrable (count_space UNIV) (λn. ?f n a)" by (simp add: measure_pmf_eq_density integrable_density) qed then show "(LINT i|M. f i a) = (∑ n. ?f n a)" by (simp add: sums_unique) next show "uniform_limit A (λn a. ∑i<n. ?f i a) (λa. (∑ n. ?f n a)) sequentially" proof (rule weierstrass_m_test) fix n a assume "a∈A" then show "norm (?f n a) ≤ pmf (map_pmf (to_nat_on M) M) n * B" using bnd by (auto intro!: mult_mono simp: from_nat_into set_pmf_not_empty) next have "integrable (map_pmf (to_nat_on M) M) (λn. B)" by auto then show "summable (λn. pmf (map_pmf (to_nat_on (set_pmf M)) M) n * B)" by (simp add: measure_pmf_eq_density integrable_density integrable_count_space_nat_iff summable_rabs_cancel) qed qed simp qed simp qed lemma continuous_on_LBINT: fixes f :: "real ⇒ real" assumes f: "⋀b. a ≤ b ⟹ set_integrable lborel {a..b} f" shows "continuous_on UNIV (λb. LBINT x:{a..b}. f x)" proof (subst set_borel_integral_eq_integral) { fix b :: real assume "a ≤ b" from f[OF this] have "continuous_on {a..b} (λb. integral {a..b} f)" by (intro indefinite_integral_continuous_1 set_borel_integral_eq_integral) } note * = this have "continuous_on (⋃b∈{a..}. {a <..< b}) (λb. integral {a..b} f)" proof (intro continuous_on_open_UN) show "b ∈ {a..} ⟹ continuous_on {a<..<b} (λb. integral {a..b} f)" for b using *[of b] by (rule continuous_on_subset) auto qed simp also have "(⋃b∈{a..}. {a <..< b}) = {a <..}" by (auto simp: lt_ex gt_ex less_imp_le) (simp add: Bex_def less_imp_le gt_ex cong: rev_conj_cong) finally have "continuous_on {a+1 ..} (λb. integral {a..b} f)" by (rule continuous_on_subset) auto moreover have "continuous_on {a..a+1} (λb. integral {a..b} f)" by (rule *) simp moreover have "x ≤ a ⟹ {a..x} = (if a = x then {a} else {})" for x by auto then have "continuous_on {..a} (λb. integral {a..b} f)" by (subst continuous_on_cong[OF refl, where g="λx. 0"]) (auto intro!: continuous_on_const) ultimately have "continuous_on ({..a} ∪ {a..a+1} ∪ {a+1 ..}) (λb. integral {a..b} f)" by (intro continuous_on_closed_Un) auto also have "{..a} ∪ {a..a+1} ∪ {a+1 ..} = UNIV" by auto finally show "continuous_on UNIV (λb. integral {a..b} f)" by auto next show "set_integrable lborel {a..b} f" for b using f by (cases "a ≤ b") auto qed locale pmf_as_function begin setup_lifting td_pmf_embed_pmf lemma set_pmf_transfer[transfer_rule]: assumes "bi_total A" shows "rel_fun (pcr_pmf A) (rel_set A) (λf. {x. f x ≠ 0}) set_pmf" using ‹bi_total A› by (auto simp: pcr_pmf_def cr_pmf_def rel_fun_def rel_set_def bi_total_def Bex_def set_pmf_iff) metis+ end context begin interpretation pmf_as_function . lemma pmf_eqI: "(⋀i. pmf M i = pmf N i) ⟹ M = N" by transfer auto lemma pmf_eq_iff: "M = N ⟷ (∀i. pmf M i = pmf N i)" by (auto intro: pmf_eqI) lemma pmf_neq_exists_less: assumes "M ≠ N" shows "∃x. pmf M x < pmf N x" proof (rule ccontr) assume "¬(∃x. pmf M x < pmf N x)" hence ge: "pmf M x ≥ pmf N x" for x by (auto simp: not_less) from assms obtain x where "pmf M x ≠ pmf N x" by (auto simp: pmf_eq_iff) with ge[of x] have gt: "pmf M x > pmf N x" by simp have "1 = measure (measure_pmf M) UNIV" by simp also have "… = measure (measure_pmf N) {x} + measure (measure_pmf N) (UNIV - {x})" by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all also from gt have "measure (measure_pmf N) {x} < measure (measure_pmf M) {x}" by (simp add: measure_pmf_single) also have "measure (measure_pmf N) (UNIV - {x}) ≤ measure (measure_pmf M) (UNIV - {x})" by (subst (1 2) integral_pmf [symmetric]) (intro integral_mono integrable_pmf, simp_all add: ge) also have "measure (measure_pmf M) {x} + … = 1" by (subst measure_pmf.finite_measure_Union [symmetric]) simp_all finally show False by simp_all qed lemma bind_commute_pmf: "bind_pmf A (λx. bind_pmf B (C x)) = bind_pmf B (λy. bind_pmf A (λx. C x y))" unfolding pmf_eq_iff pmf_bind proof fix i interpret B: prob_space "restrict_space B B" by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff) interpret A: prob_space "restrict_space A A" by (intro prob_space_restrict_space measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff) interpret AB: pair_prob_space "restrict_space A A" "restrict_space B B" by unfold_locales have "(∫ x. ∫ y. pmf (C x y) i ∂B ∂A) = (∫ x. (∫ y. pmf (C x y) i ∂restrict_space B B) ∂A)" by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict) also have "… = (∫ x. (∫ y. pmf (C x y) i ∂restrict_space B B) ∂restrict_space A A)" by (intro integral_pmf_restrict B.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 countable_set_pmf borel_measurable_count_space) also have "… = (∫ y. ∫ x. pmf (C x y) i ∂restrict_space A A ∂restrict_space B B)" by (rule AB.Fubini_integral[symmetric]) (auto intro!: AB.integrable_const_bound[where B=1] measurable_pair_restrict_pmf2 simp: pmf_nonneg pmf_le_1 measurable_restrict_space1) also have "… = (∫ y. ∫ x. pmf (C x y) i ∂restrict_space A A ∂B)" by (intro integral_pmf_restrict[symmetric] A.borel_measurable_lebesgue_integral measurable_pair_restrict_pmf2 countable_set_pmf borel_measurable_count_space) also have "… = (∫ y. ∫ x. pmf (C x y) i ∂A ∂B)" by (rule Bochner_Integration.integral_cong) (auto intro!: integral_pmf_restrict[symmetric]) finally show "(∫ x. ∫ y. pmf (C x y) i ∂B ∂A) = (∫ y. ∫ x. pmf (C x y) i ∂A ∂B)" . qed lemma pair_map_pmf1: "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)" proof (safe intro!: pmf_eqI) fix a :: "'a" and b :: "'b" have [simp]: "⋀c d. indicator (apfst f -` {(a, b)}) (c, d) = indicator (f -` {a}) c * (indicator {b} d::ennreal)" by (auto split: split_indicator) have "ennreal (pmf (pair_pmf (map_pmf f A) B) (a, b)) = ennreal (pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b))" unfolding pmf_pair ennreal_pmf_map by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_multc pmf_nonneg emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf) then show "pmf (pair_pmf (map_pmf f A) B) (a, b) = pmf (map_pmf (apfst f) (pair_pmf A B)) (a, b)" by (simp add: pmf_nonneg) qed lemma pair_map_pmf2: "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)" proof (safe intro!: pmf_eqI) fix a :: "'a" and b :: "'b" have [simp]: "⋀c d. indicator (apsnd f -` {(a, b)}) (c, d) = indicator {a} c * (indicator (f -` {b}) d::ennreal)" by (auto split: split_indicator) have "ennreal (pmf (pair_pmf A (map_pmf f B)) (a, b)) = ennreal (pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b))" unfolding pmf_pair ennreal_pmf_map by (simp add: nn_integral_pair_pmf' max_def emeasure_pmf_single nn_integral_cmult nn_integral_multc pmf_nonneg emeasure_map_pmf[symmetric] ennreal_mult del: emeasure_map_pmf) then show "pmf (pair_pmf A (map_pmf f B)) (a, b) = pmf (map_pmf (apsnd f) (pair_pmf A B)) (a, b)" by (simp add: pmf_nonneg) qed lemma map_pair: "map_pmf (λ(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)" by (simp add: pair_map_pmf2 pair_map_pmf1 map_pmf_comp split_beta') end lemma pair_return_pmf1: "pair_pmf (return_pmf x) y = map_pmf (Pair x) y" by(simp add: pair_pmf_def bind_return_pmf map_pmf_def) lemma pair_return_pmf2: "pair_pmf x (return_pmf y) = map_pmf (λx. (x, y)) x" by(simp add: pair_pmf_def bind_return_pmf map_pmf_def) lemma pair_pair_pmf: "pair_pmf (pair_pmf u v) w = map_pmf (λ(x, (y, z)). ((x, y), z)) (pair_pmf u (pair_pmf v w))" by(simp add: pair_pmf_def bind_return_pmf map_pmf_def bind_assoc_pmf) lemma pair_commute_pmf: "pair_pmf x y = map_pmf (λ(x, y). (y, x)) (pair_pmf y x)" unfolding pair_pmf_def by(subst bind_commute_pmf)(simp add: map_pmf_def bind_assoc_pmf bind_return_pmf) lemma set_pmf_subset_singleton: "set_pmf p ⊆ {x} ⟷ p = return_pmf x" proof(intro iffI pmf_eqI) fix i assume x: "set_pmf p ⊆ {x}" hence *: "set_pmf p = {x}" using set_pmf_not_empty[of p] by auto have "ennreal (pmf p x) = ∫⇧^{+}i. indicator {x} i ∂p" by(simp add: emeasure_pmf_single) also have "… = ∫⇧^{+}i. 1 ∂p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * ) also have "… = 1" by simp finally show "pmf p i = pmf (return_pmf x) i" using x by(auto split: split_indicator simp add: pmf_eq_0_set_pmf) qed auto lemma bind_eq_return_pmf: "bind_pmf p f = return_pmf x ⟷ (∀y∈set_pmf p. f y = return_pmf x)" (is "?lhs ⟷ ?rhs") proof(intro iffI strip) fix y assume y: "y ∈ set_pmf p" assume "?lhs" hence "set_pmf (bind_pmf p f) = {x}" by simp hence "(⋃y∈set_pmf p. set_pmf (f y)) = {x}" by simp hence "set_pmf (f y) ⊆ {x}" using y by auto thus "f y = return_pmf x" by(simp add: set_pmf_subset_singleton) next assume *: ?rhs show ?lhs proof(rule pmf_eqI) fix i have "ennreal (pmf (bind_pmf p f) i) = ∫⇧^{+}y. ennreal (pmf (f y) i) ∂p" by (simp add: ennreal_pmf_bind) also have "… = ∫⇧^{+}y. ennreal (pmf (return_pmf x) i) ∂p" by(rule nn_integral_cong_AE)(simp add: AE_measure_pmf_iff * ) also have "… = ennreal (pmf (return_pmf x) i)" by simp finally show "pmf (bind_pmf p f) i = pmf (return_pmf x) i" by (simp add: pmf_nonneg) qed qed lemma pmf_False_conv_True: "pmf p False = 1 - pmf p True" proof - have "pmf p False + pmf p True = measure p {False} + measure p {True}" by(simp add: measure_pmf_single) also have "… = measure p ({False} ∪ {True})" by(subst measure_pmf.finite_measure_Union) simp_all also have "{False} ∪ {True} = space p" by auto finally show ?thesis by simp qed lemma pmf_True_conv_False: "pmf p True = 1 - pmf p False" by(simp add: pmf_False_conv_True) subsection ‹ Conditional Probabilities › lemma measure_pmf_zero_iff: "measure (measure_pmf p) s = 0 ⟷ set_pmf p ∩ s = {}" by (subst measure_pmf.prob_eq_0) (auto simp: AE_measure_pmf_iff) context fixes p :: "'a pmf" and s :: "'a set" assumes not_empty: "set_pmf p ∩ s ≠ {}" begin interpretation pmf_as_measure . lemma emeasure_measure_pmf_not_zero: "emeasure (measure_pmf p) s ≠ 0" proof assume "emeasure (measure_pmf p) s = 0" then have "AE x in measure_pmf p. x ∉ s" by (rule AE_I[rotated]) auto with not_empty show False by (auto simp: AE_measure_pmf_iff) qed lemma measure_measure_pmf_not_zero: "measure (measure_pmf p) s ≠ 0" using emeasure_measure_pmf_not_zero by (simp add: measure_pmf.emeasure_eq_measure measure_nonneg) lift_definition cond_pmf :: "'a pmf" is "uniform_measure (measure_pmf p) s" proof (intro conjI) show "prob_space (uniform_measure (measure_pmf p) s)" by (intro prob_space_uniform_measure) (auto simp: emeasure_measure_pmf_not_zero) show "AE x in uniform_measure (measure_pmf p) s. measure (uniform_measure (measure_pmf p) s) {x} ≠ 0" by (simp add: emeasure_measure_pmf_not_zero measure_measure_pmf_not_zero AE_uniform_measure AE_measure_pmf_iff set_pmf.rep_eq less_top[symmetric]) qed simp lemma pmf_cond: "pmf cond_pmf x = (if x ∈ s then pmf p x / measure p s else 0)" by transfer (simp add: emeasure_measure_pmf_not_zero pmf.rep_eq) lemma set_cond_pmf[simp]: "set_pmf cond_pmf = set_pmf p ∩ s" by (auto simp add: set_pmf_iff pmf_cond measure_measure_pmf_not_zero split: if_split_asm) end lemma measure_pmf_posI: "x ∈ set_pmf p ⟹ x ∈ A ⟹ measure_pmf.prob p A > 0" using measure_measure_pmf_not_zero[of p A] by (subst zero_less_measure_iff) blast lemma cond_map_pmf: assumes "set_pmf p ∩ f -` s ≠ {}" shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))" proof - have *: "set_pmf (map_pmf f p) ∩ s ≠ {}" using assms by auto { fix x have "ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x) = emeasure p (f -` s ∩ f -` {x}) / emeasure p (f -` s)" unfolding ennreal_pmf_map cond_pmf.rep_eq[OF assms] by (simp add: nn_integral_uniform_measure) also have "f -` s ∩ f -` {x} = (if x ∈ s then f -` {x} else {})" by auto also have "emeasure p (if x ∈ s then f -` {x} else {}) / emeasure p (f -` s) = ennreal (pmf (cond_pmf (map_pmf f p) s) x)" using measure_measure_pmf_not_zero[OF *] by (simp add: pmf_cond[OF *] ennreal_pmf_map measure_pmf.emeasure_eq_measure divide_ennreal pmf_nonneg measure_nonneg zero_less_measure_iff pmf_map) finally have "ennreal (pmf (cond_pmf (map_pmf f p) s) x) = ennreal (pmf (map_pmf f (cond_pmf p (f -` s))) x)" by simp } then show ?thesis by (intro pmf_eqI) (simp add: pmf_nonneg) qed lemma bind_cond_pmf_cancel: assumes [simp]: "⋀x. x ∈ set_pmf p ⟹ set_pmf q ∩ {y. R x y} ≠ {}" assumes [simp]: "⋀y. y ∈ set_pmf q ⟹ set_pmf p ∩ {x. R x y} ≠ {}" assumes [simp]: "⋀x y. x ∈ set_pmf p ⟹ y ∈ set_pmf q ⟹ R x y ⟹ measure q {y. R x y} = measure p {x. R x y}" shows "bind_pmf p (λx. cond_pmf q {y. R x y}) = q" proof (rule pmf_eqI) fix i have "ennreal (pmf (bind_pmf p (λx. cond_pmf q {y. R x y})) i) = (∫⇧^{+}x. ennreal (pmf q i / measure p {x. R x i}) * ennreal (indicator {x. R x i} x) ∂p)" by (auto simp add: ennreal_pmf_bind AE_measure_pmf_iff pmf_cond pmf_eq_0_set_pmf pmf_nonneg measure_nonneg intro!: nn_integral_cong_AE) also have "… = (pmf q i * measure p {x. R x i}) / measure p {x. R x i}" by (simp add: pmf_nonneg measure_nonneg zero_ennreal_def[symmetric] ennreal_indicator nn_integral_cmult measure_pmf.emeasure_eq_measure ennreal_mult[symmetric]) also have "… = pmf q i" by (cases "pmf q i = 0") (simp_all add: pmf_eq_0_set_pmf measure_measure_pmf_not_zero pmf_nonneg) finally show "pmf (bind_pmf p (λx. cond_pmf q {y. R x y})) i = pmf q i" by (simp add: pmf_nonneg) qed subsection ‹ Relator › inductive rel_pmf :: "('a ⇒ 'b ⇒ bool) ⇒ 'a pmf ⇒ 'b pmf ⇒ bool" for R p q where "⟦ ⋀x y. (x, y) ∈ set_pmf pq ⟹ R x y; map_pmf fst pq = p; map_pmf snd pq = q ⟧ ⟹ rel_pmf R p q" lemma rel_pmfI: assumes R: "rel_set R (set_pmf p) (set_pmf q)" assumes eq: "⋀x y. x ∈ set_pmf p ⟹ y ∈ set_pmf q ⟹ R x y ⟹ measure p {x. R x y} = measure q {y. R x y}" shows "rel_pmf R p q" proof let ?pq = "bind_pmf p (λx. bind_pmf (cond_pmf q {y. R x y}) (λy. return_pmf (x, y)))" have "⋀x. x ∈ set_pmf p ⟹ set_pmf q ∩ {y. R x y} ≠ {}" using R by (auto simp: rel_set_def) then show "⋀x y. (x, y) ∈ set_pmf ?pq ⟹ R x y" by auto show "map_pmf fst ?pq = p" by (simp add: map_bind_pmf bind_return_pmf') show "map_pmf snd ?pq = q" using R eq apply (simp add: bind_cond_pmf_cancel map_bind_pmf bind_return_pmf') apply (rule bind_cond_pmf_cancel) apply (auto simp: rel_set_def) done qed lemma rel_pmf_imp_rel_set: "rel_pmf R p q ⟹ rel_set R (set_pmf p) (set_pmf q)" by (force simp add: rel_pmf.simps rel_set_def) lemma rel_pmfD_measure: assumes rel_R: "rel_pmf R p q" and R: "⋀a b. R a b ⟹ R a y ⟷ R x b" assumes "x ∈ set_pmf p" "y ∈ set_pmf q" shows "measure p {x. R x y} = measure q {y. R x y}" proof - from rel_R obtain pq where pq: "⋀x y. (x, y) ∈ set_pmf pq ⟹ R x y" and eq: "p = map_pmf fst pq" "q = map_pmf snd pq" by (auto elim: rel_pmf.cases) have "measure p {x. R x y} = measure pq {x. R (fst x) y}" by (simp add: eq map_pmf_rep_eq measure_distr) also have "… = measure pq {y. R x (snd y)}" by (intro measure_pmf.finite_measure_eq_AE) (auto simp: AE_measure_pmf_iff R dest!: pq) also have "… = measure q {y. R x y}" by (simp add: eq map_pmf_rep_eq measure_distr) finally show "measure p {x. R x y} = measure q {y. R x y}" . qed lemma rel_pmf_measureD: assumes "rel_pmf R p q" shows "measure (measure_pmf p) A ≤ measure (measure_pmf q) {y. ∃x∈A. R x y}" (is "?lhs ≤ ?rhs") using assms proof cases fix pq assume R: "⋀x y. (x, y) ∈ set_pmf pq ⟹ R x y" and p[symmetric]: "map_pmf fst pq = p" and q[symmetric]: "map_pmf snd pq = q" have "?lhs = measure (measure_pmf pq) (fst -` A)" by(simp add: p) also have "… ≤ measure (measure_pmf pq) {y. ∃x∈A. R x (snd y)}" by(rule measure_pmf.finite_measure_mono_AE)(auto 4 3 simp add: AE_measure_pmf_iff dest: R) also have "… = ?rhs" by(simp add: q) finally show ?thesis . qed lemma rel_pmf_iff_measure: assumes "symp R" "transp R" shows "rel_pmf R p q ⟷ rel_set R (set_pmf p) (set_pmf q) ∧ (∀x∈set_pmf p. ∀y∈set_pmf q. R x y ⟶ measure p {x. R x y} = measure q {y. R x y})" by (safe intro!: rel_pmf_imp_rel_set rel_pmfI) (auto intro!: rel_pmfD_measure dest: sympD[OF ‹symp R›] transpD[OF ‹transp R›]) lemma quotient_rel_set_disjoint: "equivp R ⟹ C ∈ UNIV // {(x, y). R x y} ⟹ rel_set R A B ⟹ A ∩ C = {} ⟷ B ∩ C = {}" using in_quotient_imp_closed[of UNIV "{(x, y). R x y}" C] by (auto 0 0 simp: equivp_equiv rel_set_def set_eq_iff elim: equivpE) (blast dest: equivp_symp)+ lemma quotientD: "equiv X R ⟹ A ∈ X // R ⟹ x ∈ A ⟹ A = R `` {x}" by (metis Image_singleton_iff equiv_class_eq_iff quotientE) lemma rel_pmf_iff_equivp: assumes "equivp R" shows "rel_pmf R p q ⟷ (∀C∈UNIV // {(x, y). R x y}. measure p C = measure q C)" (is "_ ⟷ (∀C∈_//?R. _)") proof (subst rel_pmf_iff_measure, safe) show "symp R" "transp R" using assms by (auto simp: equivp_reflp_symp_transp) next fix C assume C: "C ∈ UNIV // ?R" and R: "rel_set R (set_pmf p) (set_pmf q)" assume eq: "∀x∈set_pmf p. ∀y∈set_pmf q. R x y ⟶ measure p {x. R x y} = measure q {y. R x y}" show "measure p C = measure q C" proof (cases "p ∩ C = {}") case True then have "q ∩ C = {}" using quotient_rel_set_disjoint[OF assms C R] by simp with True show ?thesis unfolding measure_pmf_zero_iff[symmetric] by simp next case False then have "q ∩ C ≠ {}" using quotient_rel_set_disjoint[OF assms C R] by simp with False obtain x y where in_set: "x ∈ set_pmf p" "y ∈ set_pmf q" and in_C: "x ∈ C" "y ∈ C" by auto then have "R x y" using in_quotient_imp_in_rel[of UNIV ?R C x y] C assms by (simp add: equivp_equiv) with in_set eq have "measure p {x. R x y} = measure q {y. R x y}" by auto moreover have "{y. R x y} = C" using assms ‹x ∈ C› C quotientD[of UNIV ?R C x] by (simp add: equivp_equiv) moreover have "{x. R x y} = C" using assms ‹y ∈ C› C quotientD[of UNIV "?R" C y] sympD[of R] by (auto simp add: equivp_equiv elim: equivpE) ultimately show ?thesis by auto qed next assume eq: "∀C∈UNIV // ?R. measure p C = measure q C" show "rel_set R (set_pmf p) (set_pmf q)" unfolding rel_set_def proof safe fix x assume x: "x ∈ set_pmf p" have "{y. R x y} ∈ UNIV // ?R" by (auto simp: quotient_def) with eq have *: "measure q {y. R x y} = measure p {y. R x y}" by auto have "measure q {y. R x y} ≠ 0" using x assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp) then show "∃y∈set_pmf q. R x y" unfolding measure_pmf_zero_iff by auto next fix y assume y: "y ∈ set_pmf q" have "{x. R x y} ∈ UNIV // ?R" using assms by (auto simp: quotient_def dest: equivp_symp) with eq have *: "measure p {x. R x y} = measure q {x. R x y}" by auto have "measure p {x. R x y} ≠ 0" using y assms unfolding * by (auto simp: measure_pmf_zero_iff set_eq_iff dest: equivp_reflp) then show "∃x∈set_pmf p. R x y" unfolding measure_pmf_zero_iff by auto qed fix x y assume "x ∈ set_pmf p" "y ∈ set_pmf q" "R x y" have "{y. R x y} ∈ UNIV // ?R" "{x. R x y} = {y. R x y}" using assms ‹R x y› by (auto simp: quotient_def dest: equivp_symp equivp_transp) with eq show "measure p {x. R x y} = measure q {y. R x y}" by auto qed bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: rel_pmf proof - show "map_pmf id = id" by (rule map_pmf_id) show "⋀f g. map_pmf (f ∘ g) = map_pmf f ∘ map_pmf g" by (rule map_pmf_compose) show "⋀f g::'a ⇒ 'b. ⋀p. (⋀x. x ∈ set_pmf p ⟹ f x = g x) ⟹ map_pmf f p = map_pmf g p" by (intro map_pmf_cong refl) show "⋀f::'a ⇒ 'b. set_pmf ∘ map_pmf f = (`) f ∘ set_pmf" by (rule pmf_set_map) show "(card_of (set_pmf p), natLeq) ∈ ordLeq" for p :: "'s pmf" proof - have "(card_of (set_pmf p), card_of (UNIV :: nat set)) ∈ ordLeq" by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) (auto intro: countable_set_pmf) also have "(card_of (UNIV :: nat set), natLeq) ∈ ordLeq" by (metis Field_natLeq card_of_least natLeq_Well_order) finally show ?thesis . qed show "⋀R. rel_pmf R = (λx y. ∃z. set_pmf z ⊆ {(x, y). R x y} ∧ map_pmf fst z = x ∧ map_pmf snd z = y)" by (auto simp add: fun_eq_iff rel_pmf.simps) show "rel_pmf R OO rel_pmf S ≤ rel_pmf (R OO S)" for R :: "'a ⇒ 'b ⇒ bool" and S :: "'b ⇒ 'c ⇒ bool" proof - { fix p q r assume pq: "rel_pmf R p q" and qr:"rel_pmf S q r" from pq obtain pq where pq: "⋀x y. (x, y) ∈ set_pmf pq ⟹ R x y" and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto from qr obtain qr where qr: "⋀y z. (y, z) ∈ set_pmf qr ⟹ S y z" and q': "q = map_pmf fst qr" and r: "r = map_pmf snd qr" by cases auto define pr where "pr = bind_pmf pq (λxy. bind_pmf (cond_pmf qr {yz. fst yz = snd xy}) (λyz. return_pmf (fst xy, snd yz)))" have pr_welldefined: "⋀y. y ∈ q ⟹ qr ∩ {yz. fst yz = y} ≠ {}" by (force simp: q') have "rel_pmf (R OO S) p r" proof (rule rel_pmf.intros) fix x z assume "(x, z) ∈ pr" then have "∃y. (x, y) ∈ pq ∧ (y, z) ∈ qr" by (auto simp: q pr_welldefined pr_def split_beta) with pq qr show "(R OO S) x z" by blast next have "map_pmf snd pr = map_pmf snd (bind_pmf q (λy. cond_pmf qr {yz. fst yz = y}))" by (simp add: pr_def q split_beta bind_map_pmf map_pmf_def[symmetric] map_bind_pmf map_pmf_comp) then show "map_pmf snd pr = r" unfolding r q' bind_map_pmf by (subst (asm) bind_cond_pmf_cancel) (auto simp: eq_commute) qed (simp add: pr_def map_bind_pmf split_beta map_pmf_def[symmetric] p map_pmf_comp) } then show ?thesis by(auto simp add: le_fun_def) qed qed (fact natLeq_card_order natLeq_cinfinite)+ lemma map_pmf_idI: "(⋀x. x ∈ set_pmf p ⟹ f x = x) ⟹ map_pmf f p = p" by(simp cong: pmf.map_cong) lemma rel_pmf_conj[simp]: "rel_pmf (λx y. P ∧ Q x y) x y ⟷ P ∧ rel_pmf Q x y" "rel_pmf (λx y. Q x y ∧ P) x y ⟷ P ∧ rel_pmf Q x y" using set_pmf_not_empty by (fastforce simp: pmf.in_rel subset_eq)+ lemma rel_pmf_top[simp]: "rel_pmf top = top" by (auto simp: pmf.in_rel[abs_def] fun_eq_iff map_fst_pair_pmf map_snd_pair_pmf intro: exI[of _ "pair_pmf x y" for x y]) lemma rel_pmf_return_pmf1: "rel_pmf R (return_pmf x) M ⟷ (∀a∈M. R x a)" proof safe fix a assume "a ∈ M" "rel_pmf R (return_pmf x) M" then obtain pq where *: "⋀a b. (a, b) ∈ set_pmf pq ⟹ R a b" and eq: "return_pmf x = map_pmf fst pq" "M = map_pmf snd pq" by (force elim: rel_pmf.cases) moreover have "set_pmf (return_pmf x) = {x}" by simp with ‹a ∈ M› have "(x, a) ∈ pq" by (force simp: eq) with * show "R x a" by auto qed (auto intro!: rel_pmf.intros[where pq="pair_pmf (return_pmf x) M"] simp: map_fst_pair_pmf map_snd_pair_pmf) lemma rel_pmf_return_pmf2: "rel_pmf R M (return_pmf x) ⟷ (∀a∈M. R a x)" by (subst pmf.rel_flip[symmetric]) (simp add: rel_pmf_return_pmf1) lemma rel_return_pmf[simp]: "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2" unfolding rel_pmf_return_pmf2 set_return_pmf by simp lemma rel_pmf_False[simp]: "rel_pmf (λx y. False) x y = False" unfolding pmf.in_rel fun_eq_iff using set_pmf_not_empty by fastforce lemma rel_pmf_rel_prod: "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B') ⟷ rel_pmf R A B ∧ rel_pmf S A' B'" proof safe assume "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" then obtain pq where pq: "⋀a b c d. ((a, c), (b, d)) ∈ set_pmf pq ⟹ R a b ∧ S c d" and eq: "map_pmf fst pq = pair_pmf A A'" "map_pmf snd pq = pair_pmf B B'" by (force elim: rel_pmf.cases) show "rel_pmf R A B" proof (rule rel_pmf.intros) let ?f = "λ(a, b). (fst a, fst b)" have [simp]: "(λx. fst (?f x)) = fst o fst" "(λx. snd (?f x)) = fst o snd" by auto show "map_pmf fst (map_pmf ?f pq) = A" by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) show "map_pmf snd (map_pmf ?f pq) = B" by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_fst_pair_pmf) fix a b assume "(a, b) ∈ set_pmf (map_pmf ?f pq)" then obtain c d where "((a, c), (b, d)) ∈ set_pmf pq" by auto from pq[OF this] show "R a b" .. qed show "rel_pmf S A' B'" proof (rule rel_pmf.intros) let ?f = "λ(a, b). (snd a, snd b)" have [simp]: "(λx. fst (?f x)) = snd o fst" "(λx. snd (?f x)) = snd o snd" by auto show "map_pmf fst (map_pmf ?f pq) = A'" by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) show "map_pmf snd (map_pmf ?f pq) = B'" by (simp add: map_pmf_comp pmf.map_comp[symmetric] eq map_snd_pair_pmf) fix c d assume "(c, d) ∈ set_pmf (map_pmf ?f pq)" then obtain a b where "((a, c), (b, d)) ∈ set_pmf pq" by auto from pq[OF this] show "S c d" .. qed next assume "rel_pmf R A B" "rel_pmf S A' B'" then obtain Rpq Spq where Rpq: "⋀a b. (a, b) ∈ set_pmf Rpq ⟹ R a b" "map_pmf fst Rpq = A" "map_pmf snd Rpq = B" and Spq: "⋀a b. (a, b) ∈ set_pmf Spq ⟹ S a b" "map_pmf fst Spq = A'" "map_pmf snd Spq = B'" by (force elim: rel_pmf.cases) let ?f = "(λ((a, c), (b, d)). ((a, b), (c, d)))" let ?pq = "map_pmf ?f (pair_pmf Rpq Spq)" have [simp]: "(λx. fst (?f x)) = (λ(a, b). (fst a, fst b))" "(λx. snd (?f x)) = (λ(a, b). (snd a, snd b))" by auto show "rel_pmf (rel_prod R S) (pair_pmf A A') (pair_pmf B B')" by (rule rel_pmf.intros[where pq="?pq"]) (auto simp: map_snd_pair_pmf map_fst_pair_pmf map_pmf_comp Rpq Spq map_pair) qed lemma rel_pmf_reflI: assumes "⋀x. x ∈ set_pmf p ⟹ P x x" shows "rel_pmf P p p" by (rule rel_pmf.intros[where pq="map_pmf (λx. (x, x)) p"]) (auto simp add: pmf.map_comp o_def assms) lemma rel_pmf_bij_betw: assumes f: "bij_betw f (set_pmf p) (set_pmf q)" and eq: "⋀x. x ∈ set_pmf p ⟹ pmf p x = pmf q (f x)" shows "rel_pmf (λx y. f x = y) p q" proof(rule rel_pmf.intros) let ?pq = "map_pmf (λx. (x, f x)) p" show "map_pmf fst ?pq = p" by(simp add: pmf.map_comp o_def) have "map_pmf f p = q" proof(rule pmf_eqI) fix i show "pmf (map_pmf f p) i = pmf q i" proof(cases "i ∈ set_pmf q") case True with f obtain j where "i = f j" "j ∈ set_pmf p" by(auto simp add: bij_betw_def image_iff) thus ?thesis using f by(simp add: bij_betw_def pmf_map_inj eq) next case False thus ?thesis by(subst pmf_map_outside)(auto simp add: set_pmf_iff eq[symmetric]) qed qed then show "map_pmf snd ?pq = q" by(simp add: pmf.map_comp o_def) qed auto context begin interpretation pmf_as_measure . definition "join_pmf M = bind_pmf M (λx. x)" lemma bind_eq_join_pmf: "bind_pmf M f = join_pmf (map_pmf f M)" unfolding join_pmf_def bind_map_pmf .. lemma join_eq_bind_pmf: "join_pmf M = bind_pmf M id" by (simp add: join_pmf_def id_def) lemma pmf_join: "pmf (join_pmf N) i = (∫M. pmf M i ∂measure_pmf N)" unfolding join_pmf_def pmf_bind .. lemma ennreal_pmf_join: "ennreal (pmf (join_pmf N) i) = (∫⇧^{+}M. pmf M i ∂measure_pmf N)" unfolding join_pmf_def ennreal_pmf_bind .. lemma set_pmf_join_pmf[simp]: "set_pmf (join_pmf f) = (⋃p∈set_pmf f. set_pmf p)" by (simp add: join_pmf_def) lemma join_return_pmf: "join_pmf (return_pmf M) = M" by (simp add: integral_return pmf_eq_iff pmf_join return_pmf.rep_eq) lemma map_join_pmf: "map_pmf f (join_pmf AA) = join_pmf (map_pmf (map_pmf f) AA)" by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf) lemma join_map_return_pmf: "join_pmf (map_pmf return_pmf A) = A" by (simp add: join_pmf_def map_pmf_def bind_assoc_pmf bind_return_pmf bind_return_pmf') end lemma rel_pmf_joinI: assumes "rel_pmf (rel_pmf P) p q" shows "rel_pmf P (join_pmf p) (join_pmf q)" proof - from assms obtain pq where p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" and P: "⋀x y. (x, y) ∈ set_pmf pq ⟹ rel_pmf P x y" by cases auto from P obtain PQ where PQ: "⋀x y a b. ⟦ (x, y) ∈ set_pmf pq; (a, b) ∈ set_pmf (PQ x y) ⟧ ⟹ P a b" and x: "⋀x y. (x, y) ∈ set_pmf pq ⟹ map_pmf fst (PQ x y) = x" and y: "⋀x y. (x, y) ∈ set_pmf pq ⟹ map_pmf snd (PQ x y) = y" by(metis rel_pmf.simps) let ?r = "bind_pmf pq (λ(x, y). PQ x y)" have "⋀a b. (a, b) ∈ set_pmf ?r ⟹ P a b" by (auto intro: PQ) moreover have "map_pmf fst ?r = join_pmf p" "map_pmf snd ?r = join_pmf q" by (simp_all add: p q x y join_pmf_def map_bind_pmf bind_map_pmf split_def cong: bind_pmf_cong) ultimately show ?thesis .. qed lemma rel_pmf_bindI: assumes pq: "rel_pmf R p q" and fg: "⋀x y. R x y ⟹ rel_pmf P (f x) (g y)" shows "rel_pmf P (bind_pmf p f) (bind_pmf q g)" unfolding bind_eq_join_pmf by (rule rel_pmf_joinI) (auto simp add: pmf.rel_map intro: pmf.rel_mono[THEN le_funD, THEN le_funD, THEN le_boolD, THEN mp, OF _ pq] fg) text ‹ Proof that @{const rel_pmf} preserves orders. Antisymmetry proof follows Thm. 1 in N. Saheb-Djahromi, Cpo's of measures for nondeterminism, Theoretical Computer Science 12(1):19--37, 1980, 🌐‹https://doi.org/10.1016/0304-3975(80)90003-1› › lemma assumes *: "rel_pmf R p q" and refl: "reflp R" and trans: "transp R" shows measure_Ici: "measure p {y. R x y} ≤ measure q {y. R x y}" (is ?thesis1) and measure_Ioi: "measure p {y. R x y ∧ ¬ R y x} ≤ measure q {y. R x y ∧ ¬ R y x}" (is ?thesis2) proof - from * obtain pq where pq: "⋀x y. (x, y) ∈ set_pmf pq ⟹ R x y" and p: "p = map_pmf fst pq" and q: "q = map_pmf snd pq" by cases auto show ?thesis1 ?thesis2 unfolding p q map_pmf_rep_eq using refl trans by(auto 4 3 simp add: measure_distr reflpD AE_measure_pmf_iff intro!: measure_pmf.finite_measure_mono_AE dest!: pq elim: transpE) qed lemma rel_pmf_inf: fixes p q :: "'a pmf" assumes 1: "rel_pmf R p q" assumes 2: "rel_pmf R q p" and refl: "reflp R" and trans: "transp R" shows "rel_pmf (inf R R¯¯) p q" proof (subst rel_pmf_iff_equivp, safe) show "equivp (inf R R¯¯)" using trans refl by (auto simp: equivp_reflp_symp_transp intro: sympI transpI reflpI dest: transpD reflpD) fix C assume "C ∈ UNIV // {(x, y). inf R R¯¯ x y}" then obtain x where C: "C = {y. R x y ∧ R y x}" by (auto elim: quotientE) let ?R = "λx y. R x y ∧ R y x" let ?μR = "λy. measure q {x. ?R x y}" have "measure p {y. ?R x y} = measure p ({y. R x y} - {y. R x y ∧ ¬ R y x})" by(auto intro!: arg_cong[where f="measure p"]) also have "… = measure p {y. R x y} - measure p {y. R x y ∧ ¬ R y x}" by (rule measure_pmf.finite_measure_Diff) auto also have "measure p {y. R x y ∧ ¬ R y x} = measure q {y. R x y ∧ ¬ R y x}" using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ioi) also have "measure p {y. R x y} = measure q {y. R x y}" using 1 2 refl trans by(auto intro!: Orderings.antisym measure_Ici) also have "measure q {y. R x y} - measure q {y. R x y ∧ ¬ R y x} = measure q ({y. R x y} - {y. R x y ∧ ¬ R y x})" by(rule measure_pmf.finite_measure_Diff[symmetric]) auto also have "… = ?μR x" by(auto intro!: arg_cong[where f="measure q"]) finally show "measure p C = measure q C" by (simp add: C conj_commute) qed lemma rel_pmf_antisym: fixes p q :: "'a pmf" assumes 1: "rel_pmf R p q" assumes 2: "rel_pmf R q p" and refl: "reflp R" and trans: "transp R" and antisym: "antisymp R" shows "p = q" proof - from 1 2 refl trans have "rel_pmf (inf R R¯¯) p q" by(rule rel_pmf_inf) also have "inf R R¯¯ = (=)" using refl antisym by (auto intro!: ext simp add: reflpD dest: antisympD) finally show ?thesis unfolding pmf.rel_eq . qed lemma reflp_rel_pmf: "reflp R ⟹ reflp (rel_pmf R)" by (fact pmf.rel_reflp) lemma antisymp_rel_pmf: "⟦ reflp R; transp R; antisymp R ⟧ ⟹ antisymp (rel_pmf R)" by(rule antisympI)(blast intro: rel_pmf_antisym) lemma transp_rel_pmf: assumes "transp R" shows "transp (rel_pmf R)" using assms by (fact pmf.rel_transp) subsection ‹ Distributions › context begin interpretation pmf_as_function . subsubsection ‹ Bernoulli Distribution › lift_definition bernoulli_pmf :: "real ⇒ bool pmf" is "λp b. ((λp. if b then p else 1 - p) ∘ min 1 ∘ max 0) p" by (auto simp: nn_integral_count_space_finite[where A="{False, True}"] UNIV_bool split: split_max split_min) lemma pmf_bernoulli_True[simp]: "0 ≤ p ⟹ p ≤ 1 ⟹ pmf (bernoulli_pmf p) True = p" by transfer simp lemma pmf_bernoulli_False[simp]: "0 ≤ p ⟹ p ≤ 1 ⟹ pmf (bernoulli_pmf p) False = 1 - p" by transfer simp lemma set_pmf_bernoulli[simp]: "0 < p ⟹ p < 1 ⟹ set_pmf (bernoulli_pmf p) = UNIV" by (auto simp add: set_pmf_iff UNIV_bool) lemma nn_integral_bernoulli_pmf[simp]: assumes [simp]: "0 ≤ p" "p ≤ 1" "⋀x. 0 ≤ f x" shows "(∫⇧^{+}x. f x ∂bernoulli_pmf p) = f True * p + f False * (1 - p)" by (subst nn_integral_measure_pmf_support[of UNIV]) (auto simp: UNIV_bool field_simps) lemma integral_bernoulli_pmf[simp]: assumes [simp]: "0 ≤ p" "p ≤ 1" shows "(∫x. f x ∂bernoulli_pmf p) = f True * p + f False * (1 - p)" by (subst integral_measure_pmf[of UNIV]) (auto simp: UNIV_bool) lemma pmf_bernoulli_half [simp]: "pmf (bernoulli_pmf (1 / 2)) x = 1 / 2" by(cases x) simp_all lemma measure_pmf_bernoulli_half: "measure_pmf (bernoulli_pmf (1 / 2)) = uniform_count_measure UNIV" by (rule measure_eqI) (simp_all add: nn_integral_pmf[symmetric] emeasure_uniform_count_measure ennreal_divide_numeral[symmetric] nn_integral_count_space_finite sets_uniform_count_measure divide_ennreal_def mult_ac ennreal_of_nat_eq_real_of_nat) subsubsection ‹ Geometric Distribution › context fixes p :: real assumes p[arith]: "0 < p" "p ≤ 1" begin lift_definition geometric_pmf :: "nat pmf" is "λn. (1 - p)^n * p" proof have "(∑i. ennreal (p * (1 - p) ^ i)) = ennreal (p * (1 / (1 - (1 - p))))" by (intro suminf_ennreal_eq sums_mult geometric_sums) auto then show "(∫⇧^{+}x. ennreal ((1 - p)^x * p) ∂count_space UNIV) = 1" by (simp add: nn_integral_count_space_nat field_simps) qed simp lemma pmf_geometric[simp]: "pmf geometric_pmf n = (1 - p)^n * p" by transfer rule end lemma set_pmf_geometric: "0 < p ⟹ p < 1 ⟹ set_pmf (geometric_pmf p) = UNIV" by (auto simp: set_pmf_iff) subsubsection ‹ Uniform Multiset Distribution › context fixes M :: "'a multiset" assumes M_not_empty: "M ≠ {#}" begin lift_definition pmf_of_multiset :: "'a pmf" is "λx. count M x / size M" proof show "(∫⇧^{+}x. ennreal (real (count M x) / real (size M)) ∂count_space UNIV) = 1" using M_not_empty by (simp add: zero_less_divide_iff nn_integral_count_space nonempty_has_size sum_divide_distrib[symmetric]) (auto simp: size_multiset_overloaded_eq intro!: sum.cong) qed simp lemma pmf_of_multiset[simp]: "pmf pmf_of_multiset x = count M x / size M" by transfer rule lemma set_pmf_of_multiset[simp]: "set_pmf pmf_of_multiset = set_mset M" by (auto simp: set_pmf_iff) end subsubsection ‹ Uniform Distribution › context fixes S :: "'a set" assumes S_not_empty: "S ≠ {}" and S_finite: "finite S" begin lift_definition pmf_of_set :: "'a pmf" is "λx. indicator S x / card S" proof show "(∫⇧^{+}x. ennreal (indicator S x / real (card S)) ∂count_space UNIV) = 1" using S_not_empty S_finite by (subst nn_integral_count_space'[of S]) (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_mult[symmetric]) qed simp lemma pmf_of_set[simp]: "pmf pmf_of_set x = indicator S x / card S" by transfer rule lemma set_pmf_of_set[simp]: "set_pmf pmf_of_set = S" using S_finite S_not_empty by (auto simp: set_pmf_iff) lemma emeasure_pmf_of_set_space[simp]: "emeasure pmf_of_set S = 1" by (rule measure_pmf.emeasure_eq_1_AE) (auto simp: AE_measure_pmf_iff) lemma nn_integral_pmf_of_set: "nn_integral (measure_pmf pmf_of_set) f = sum f S / card S" by (subst nn_integral_measure_pmf_finite) (simp_all add: sum_distrib_right[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def divide_ennreal[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_times_divide) lemma integral_pmf_of_set: "integral⇧^{L}(measure_pmf pmf_of_set) f = sum f S / card S" by (subst integral_measure_pmf[of S]) (auto simp: S_finite sum_divide_distrib) lemma emeasure_pmf_of_set: "emeasure (measure_pmf pmf_of_set) A = card (S ∩ A) / card S" by (subst nn_integral_indicator[symmetric], simp) (simp add: S_finite S_not_empty card_gt_0_iff indicator_def sum.If_cases divide_ennreal ennreal_of_nat_eq_real_of_nat nn_integral_pmf_of_set) lemma measure_pmf_of_set: "measure (measure_pmf pmf_of_set) A = card (S ∩ A) / card S" using emeasure_pmf_of_set[of A] by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure) end lemma pmf_expectation_bind_pmf_of_set: fixes A :: "'a set" and f :: "'a ⇒ 'b pmf" and h :: "'b ⇒ 'c::{banach, second_countable_topology}" assumes "A ≠ {}" "finite A" "⋀x. x ∈ A ⟹ finite (set_pmf (f x))" shows "measure_pmf.expectation (pmf_of_set A ⤜ f) h = (∑a∈A. measure_pmf.expectation (f a) h /⇩_{R}real (card A))" using assms by (subst pmf_expectation_bind[of A]) (auto simp: divide_simps) lemma map_pmf_of_set: assumes "finite A" "A ≠ {}" shows "map_pmf f (pmf_of_set A) = pmf_of_multiset (image_mset f (mset_set A))" (is "?lhs = ?rhs") proof (intro pmf_eqI) fix x from assms have "ennreal (pmf ?lhs x) = ennreal (pmf ?rhs x)" by (subst ennreal_pmf_map) (simp_all add: emeasure_pmf_of_set mset_set_empty_iff count_image_mset Int_commute) thus "pmf ?lhs x = pmf ?rhs x" by simp qed lemma pmf_bind_pmf_of_set: assumes "A ≠ {}" "finite A" shows "pmf (bind_pmf (pmf_of_set A) f) x = (∑xa∈A. pmf (f xa) x) / real_of_nat (card A)" (is "?lhs = ?rhs") proof - from assms have "card A > 0" by auto with assms have "ennreal ?lhs = ennreal ?rhs" by (subst ennreal_pmf_bind) (simp_all add: nn_integral_pmf_of_set max_def pmf_nonneg divide_ennreal [symmetric] sum_nonneg ennreal_of_nat_eq_real_of_nat) thus ?thesis by (subst (asm) ennreal_inj) (auto intro!: sum_nonneg divide_nonneg_nonneg) qed lemma pmf_of_set_singleton: "pmf_of_set {x} = return_pmf x" by(rule pmf_eqI)(simp add: indicator_def) lemma map_pmf_of_set_inj: assumes f: "inj_on f A" and [simp]: "A ≠ {}" "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" (is "?lhs = ?rhs") proof(rule pmf_eqI) fix i show "pmf ?lhs i = pmf ?rhs i" proof(cases "i ∈ f ` A") case True then obtain i' where "i = f i'" "i' ∈ A" by auto thus ?thesis using f by(simp add: card_image pmf_map_inj) next case False hence "pmf ?lhs i = 0" by(simp add: pmf_eq_0_set_pmf set_map_pmf) moreover have "pmf ?rhs i = 0" using False by simp ultimately show ?thesis by simp qed qed lemma map_pmf_of_set_bij_betw: assumes "bij_betw f A B" "A ≠ {}" "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set B" proof - have "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)" by (intro map_pmf_of_set_inj assms bij_betw_imp_inj_on[OF assms(1)]) also from assms have "f ` A = B" by (simp add: bij_betw_def) finally show ?thesis . qed text ‹ Choosing an element uniformly at random from the union of a disjoint family of finite non-empty sets with the same size is the same as first choosing a set from the family uniformly at random and then choosing an element from the chosen set uniformly at random. › lemma pmf_of_set_UN: assumes "finite (UNION A f)" "A ≠ {}" "⋀x. x ∈ A ⟹ f x ≠ {}" "⋀x. x ∈ A ⟹ card (f x) = n" "disjoint_family_on f A" shows "pmf_of_set (UNION A f) = do {x ← pmf_of_set A; pmf_of_set (f x)}" (is "?lhs = ?rhs") proof (intro pmf_eqI) fix x from assms have [simp]: "finite A" using infinite_disjoint_family_imp_infinite_UNION[of A f] by blast from assms have "ereal (pmf (pmf_of_set (UNION A f)) x) = ereal (indicator (⋃x∈A. f x) x / real (card (⋃x∈A. f x)))" by (subst pmf_of_set) auto also from assms have "card (⋃x∈A. f x) = card A * n" by (subst card_UN_disjoint) (auto simp: disjoint_family_on_def) also from assms have "indicator (⋃x∈A. f x) x / real … = indicator (⋃x∈A. f x) x / (n * real (card A))" by (simp add: sum_divide_distrib [symmetric] mult_ac) also from assms have "indicator (⋃x∈A. f x) x = (∑y∈A. indicator (f y) x)" by (intro indicator_UN_disjoint) simp_all also from assms have "ereal ((∑y∈A. indicator (f y) x) / (real n * real (card A))) = ereal (pmf ?rhs x)" by (subst pmf_bind_pmf_of_set) (simp_all add: sum_divide_distrib) finally show "pmf ?lhs x = pmf ?rhs x" by simp qed lemma bernoulli_pmf_half_conv_pmf_of_set: "bernoulli_pmf (1 / 2) = pmf_of_set UNIV" by (rule pmf_eqI) simp_all subsubsection ‹ Poisson Distribution › context fixes rate :: real assumes rate_pos: "0 < rate" begin lift_definition poisson_pmf :: "nat pmf" is "λk. rate ^ k / fact k * exp (-rate)" proof (* by Manuel Eberl *) have summable: "summable (λx::nat. rate ^ x / fact x)" using summable_exp by (simp add: field_simps divide_inverse [symmetric]) have "(∫⇧^{+}(x::nat). rate ^ x / fact x * exp (-rate) ∂count_space UNIV) = exp (-rate) * (∫⇧^{+}(x::nat). rate ^ x / fact x ∂count_space UNIV)" by (simp add: field_simps nn_integral_cmult[symmetric] ennreal_mult'[symmetric]) also from rate_pos have "(∫⇧^{+}(x::nat). rate ^ x / fact x ∂count_space UNIV) = (∑x. rate ^ x / fact x)" by (simp_all add: nn_integral_count_space_nat suminf_ennreal summable ennreal_suminf_neq_top) also have "... = exp rate" unfolding exp_def by (simp add: field_simps divide_inverse [symmetric]) also have "ennreal (exp (-rate)) * ennreal (exp rate) = 1" by (simp add: mult_exp_exp ennreal_mult[symmetric]) finally show "(∫⇧^{+}x. ennreal (rate ^ x / (fact x) * exp (- rate)) ∂count_space UNIV) = 1" . qed (simp add: rate_pos[THEN less_imp_le]) lemma pmf_poisson[simp]: "pmf poisson_pmf k = rate ^ k / fact k * exp (-rate)" by transfer rule lemma set_pmf_poisson[simp]: "set_pmf poisson_pmf = UNIV" using rate_pos by (auto simp: set_pmf_iff) end subsubsection ‹ Binomial Distribution › context fixes n :: nat and p :: real assumes p_nonneg: "0 ≤ p" and p_le_1: "p ≤ 1" begin lift_definition binomial_pmf :: "nat pmf" is "λk. (n choose k) * p^k * (1 - p)^(n - k)" proof have "(∫⇧^{+}k. ennreal (real (n choose k) * p ^ k * (1 - p) ^ (n - k)) ∂count_space UNIV) = ennreal (∑k≤n. real (n choose k) * p ^ k * (1 - p) ^ (n - k))" using p_le_1 p_nonneg by (subst nn_integral_count_space') auto also have "(∑k≤n. real (n choose k) * p ^ k * (1 - p) ^ (n - k)) = (p + (1 - p)) ^ n" by (subst binomial_ring) (simp add: atLeast0AtMost) finally show "(∫⇧^{+}x. ennreal (real (n choose x) * p ^ x * (1 - p) ^ (n - x)) ∂count_space UNIV) = 1" by simp qed (insert p_nonneg p_le_1, simp) lemma pmf_binomial[simp]: "pmf binomial_pmf k = (n choose k) * p^k * (1 - p)^(n - k)" by transfer rule lemma set_pmf_binomial_eq: "set_pmf binomial_pmf = (if p = 0 then {0} else if p = 1 then {n} else {.. n})" using p_nonneg p_le_1 unfolding set_eq_iff set_pmf_iff pmf_binomial by (auto simp: set_pmf_iff) end end lemma set_pmf_binomial_0[simp]: "set_pmf (binomial_pmf n 0) = {0}" by (simp add: set_pmf_binomial_eq) lemma set_pmf_binomial_1[simp]: "set_pmf (binomial_pmf n 1) = {n}" by (simp add: set_pmf_binomial_eq) lemma set_pmf_binomial[simp]: "0 < p ⟹ p < 1 ⟹ set_pmf (binomial_pmf n p) = {..n}" by (simp add: set_pmf_binomial_eq) context includes lifting_syntax begin lemma bind_pmf_parametric [transfer_rule]: "(rel_pmf A ===> (A ===> rel_pmf B) ===> rel_pmf B) bind_pmf bind_pmf" by(blast intro: rel_pmf_bindI dest: rel_funD) lemma return_pmf_parametric [transfer_rule]: "(A ===> rel_pmf A) return_pmf return_pmf" by(rule rel_funI) simp end primrec replicate_pmf :: "nat ⇒ 'a pmf ⇒ 'a list pmf" where "replicate_pmf 0 _ = return_pmf []" | "replicate_pmf (Suc n) p = do {x ← p; xs ← replicate_pmf n p; return_pmf (x#xs)}" lemma replicate_pmf_1: "replicate_pmf 1 p = map_pmf (λx. [x]) p" by (simp add: map_pmf_def bind_return_pmf) lemma set_replicate_pmf: "set_pmf (replicate_pmf n p) = {xs∈lists (set_pmf p). length xs = n}" by (induction n) (auto simp: length_Suc_conv) lemma replicate_pmf_distrib: "replicate_pmf (m + n) p = do {xs ← replicate_pmf m p; ys ← replicate_pmf n p; return_pmf (xs @ ys)}" by (induction m) (simp_all add: bind_return_pmf bind_return_pmf' bind_assoc_pmf) lemma power_diff': assumes "b ≤ a" shows "x ^ (a - b) = (if x = 0 ∧ a = b then 1 else x ^ a / (x::'a::field) ^ b)" proof (cases "x = 0") case True with assms show ?thesis by (cases "a - b") simp_all qed (insert assms, simp_all add: power_diff) lemma binomial_pmf_Suc: assumes "p ∈ {0..1}" shows "binomial_pmf (Suc n) p = do {b ← bernoulli_pmf p; k ← binomial_pmf n p; return_pmf ((if b then 1 else 0) + k)}" (is "_ = ?rhs") proof (intro pmf_eqI) fix k have A: "indicator {Suc a} (Suc b) = indicator {a} b" for a b by (simp add: indicator_def) show "pmf (binomial_pmf (Suc n) p) k = pmf ?rhs k" by (cases k; cases "k > n") (insert assms, auto simp: pmf_bind measure_pmf_single A divide_simps algebra_simps not_less less_eq_Suc_le [symmetric] power_diff') qed lemma binomial_pmf_0: "p ∈ {0..1} ⟹ binomial_pmf 0 p = return_pmf 0" by (rule pmf_eqI) (simp_all add: indicator_def) lemma binomial_pmf_altdef: assumes "p ∈ {0..1}" shows "binomial_pmf n p = map_pmf (length ∘ filter id) (replicate_pmf n (bernoulli_pmf p))" by (induction n) (insert assms, auto simp: binomial_pmf_Suc map_pmf_def bind_return_pmf bind_assoc_pmf bind_return_pmf' binomial_pmf_0 intro!: bind_pmf_cong) subsection ‹PMFs from association lists› definition pmf_of_list ::" ('a × real) list ⇒ 'a pmf" where "pmf_of_list xs = embed_pmf (λx. sum_list (map snd (filter (λz. fst z = x) xs)))" definition pmf_of_list_wf where "pmf_of_list_wf xs ⟷ (∀x∈set (map snd xs) . x ≥ 0) ∧ sum_list (map snd xs) = 1" lemma pmf_of_list_wfI: "(⋀x. x ∈ set (map snd xs) ⟹ x ≥ 0) ⟹ sum_list (map snd xs) = 1 ⟹ pmf_of_list_wf xs" unfolding pmf_of_list_wf_def by simp context begin private lemma pmf_of_list_aux: assumes "⋀x. x ∈ set (map snd xs) ⟹ x ≥ 0" assumes "sum_list (map snd xs) = 1" shows "(∫⇧^{+}x. ennreal (sum_list (map snd [z←xs . fst z = x])) ∂count_space UNIV) = 1" proof - have "(∫⇧^{+}x. ennreal (sum_list (map snd (filter (λz. fst z = x) xs))) ∂count_space UNIV) = (∫⇧^{+}x. ennreal (sum_list (map (λ(x',p). indicator {x'} x * p) xs)) ∂count_space UNIV)" apply (intro nn_integral_cong ennreal_cong, subst sum_list_map_filter') apply (rule arg_cong[where f = sum_list]) apply (auto cong: map_cong) done also have "… = (∑(x',p)←xs. (∫⇧^{+}x. ennreal (indicator {x'} x * p) ∂count_space UNIV))" using assms(1) proof (induction xs) case (Cons x xs) from Cons.prems have "snd x ≥ 0" by simp moreover have "b ≥ 0" if "(a,b) ∈ set xs" for a b using Cons.prems[of b] that by force ultimately have "(∫⇧^{+}y. ennreal (∑(x', p)←x # xs. indicator {x'} y * p) ∂count_space UNIV) = (∫⇧^{+}y. ennreal (indicator {fst x} y * snd x) + ennreal (∑(x', p)←xs. indicator {x'} y * p) ∂count_space UNIV)" by (intro nn_integral_cong, subst ennreal_plus [symmetric]) (auto simp: case_prod_unfold indicator_def intro!: sum_list_nonneg) also have "… = (∫⇧^{+}y. ennreal (indicator {fst x} y * snd x) ∂count_space UNIV) + (∫⇧^{+}y. ennreal (∑(x', p)←xs. indicator {x'} y * p) ∂count_space UNIV)" by (intro nn_integral_add) (force intro!: sum_list_nonneg AE_I2 intro: Cons simp: indicator_def)+ also have "(∫⇧^{+}y. ennreal (∑(x', p)←xs. indicator {x'} y * p) ∂count_space UNIV) = (∑(x', p)←xs. (∫⇧^{+}y. ennreal (indicator {x'} y * p) ∂count_space UNIV))" using Cons(1) by (intro Cons) simp_all finally show ?case by (simp add: case_prod_unfold) qed simp also have "… = (∑(x',p)←xs. ennreal p * (∫⇧^{+}x. indicator {x'} x ∂count_space UNIV))" using assms(1) by (simp cong: map_cong only: case_prod_unfold, subst nn_integral_cmult [symmetric]) (auto intro!: assms(1) simp: max_def times_ereal.simps [symmetric] mult_ac ereal_indicator simp del: times_ereal.simps)+ also from assms have "… = sum_list (map snd xs)" by (simp add: case_prod_unfold sum_list_ennreal) also have "… = 1" using assms(2) by simp finally show ?thesis . qed lemma pmf_pmf_of_list: assumes "pmf_of_list_wf xs" shows "pmf (pmf_of_list xs) x = sum_list (map snd (filter (λz. fst z = x) xs))" using assms pmf_of_list_aux[of xs] unfolding pmf_of_list_def pmf_of_list_wf_def by (subst pmf_embed_pmf) (auto intro!: sum_list_nonneg) end lemma set_pmf_of_list: assumes "pmf_of_list_wf xs" shows "set_pmf (pmf_of_list xs) ⊆ set (map fst xs)" proof clarify fix x assume A: "x ∈ set_pmf (pmf_of_list xs)" show "x ∈ set (map fst xs)" proof (rule ccontr) assume "x ∉ set (map fst xs)" hence "[z←xs . fst z = x] = []" by (auto simp: filter_empty_conv) with A assms show False by (simp add: pmf_pmf_of_list set_pmf_eq) qed qed lemma finite_set_pmf_of_list: assumes "pmf_of_list_wf xs" shows "finite (set_pmf (pmf_of_list xs))" using assms by (rule finite_subset[OF set_pmf_of_list]) simp_all lemma emeasure_Int_set_pmf: "emeasure (measure_pmf p) (A ∩ set_pmf p) = emeasure (measure_pmf p) A" by (rule emeasure_eq_AE) (auto simp: AE_measure_pmf_iff) lemma measure_Int_set_pmf: "measure (measure_pmf p) (A ∩ set_pmf p) = measure (measure_pmf p) A" using emeasure_Int_set_pmf[of p A] by (simp add: Sigma_Algebra.measure_def) lemma measure_prob_cong_0: assumes "⋀x. x ∈ A - B ⟹ pmf p x = 0" assumes "⋀x. x ∈ B - A ⟹ pmf p x = 0" shows "measure (measure_pmf p) A = measure (measure_pmf p) B" proof - have "measure_pmf.prob p A = measure_pmf.prob p (A ∩ set_pmf p)" by (simp add: measure_Int_set_pmf) also have "A ∩ set_pmf p = B ∩ set_pmf p" using assms by (auto simp: set_pmf_eq) also have "measure_pmf.prob p … = measure_pmf.prob p B" by (simp add: measure_Int_set_pmf) finally show ?thesis . qed lemma emeasure_pmf_of_list: assumes "pmf_of_list_wf xs" shows "emeasure (pmf_of_list xs) A = ennreal (sum_list (map snd (filter (λx. fst x ∈ A) xs)))" proof - have "emeasure (pmf_of_list xs) A = nn_integral (measure_pmf (pmf_of_list xs)) (indicator A)" by simp also from assms have "… = (∑x∈set_pmf (pmf_of_list xs) ∩ A. ennreal (sum_list (map snd [z←xs . fst z = x])))" by (subst nn_integral_measure_pmf_finite) (simp_all add: finite_set_pmf_of_list pmf_pmf_of_list Int_def) also from assms have "… = ennreal (∑x∈set_pmf (pmf_of_list xs) ∩ A. sum_list (map snd [z←xs . fst z = x]))" by (subst sum_ennreal) (auto simp: pmf_of_list_wf_def intro!: sum_list_nonneg) also have "… = ennreal (∑x∈set_pmf (pmf_of_list xs) ∩ A. indicator A x * pmf (pmf_of_list xs) x)" (is "_ = ennreal ?S") using assms by (intro ennreal_cong sum.cong) (auto simp: pmf_pmf_of_list) also have "?S = (∑x∈set_pmf (pmf_of_list xs). indicator A x * pmf (pmf_of_list xs) x)" using assms by (intro sum.mono_neutral_left set_pmf_of_list finite_set_pmf_of_list) auto also have "… = (∑x∈set (map fst xs). indicator A x * pmf (pmf_of_list xs) x)" using assms by (intro sum.mono_neutral_left set_pmf_of_list) (auto simp: set_pmf_eq) also have "… = (∑x∈set (map fst xs). indicator A x * sum_list (map snd (filter (λz. fst z = x) xs)))" using assms by (simp add: pmf_pmf_of_list) also have "… = (∑x∈set (map fst xs). sum_list (map snd (filter (λz. fst z = x ∧ x ∈ A) xs)))" by (intro sum.cong) (auto simp: indicator_def) also have "… = (∑x∈set (map fst xs). (∑xa = 0..<length xs. if fst (xs ! xa) = x ∧ x ∈ A then snd (xs ! xa) else 0))" by (intro sum.cong refl, subst sum_list_map_filter', subst sum_list_sum_nth) simp also have "… = (∑xa = 0..<length xs. (∑x∈set (map fst xs). if fst (xs ! xa) = x ∧ x ∈ A then snd (xs ! xa) else 0))" by (rule sum.swap) also have "… = (∑xa = 0..<length xs. if fst (xs ! xa) ∈ A then (∑x∈set (map fst xs). if x = fst (xs ! xa) then snd (xs ! xa) else 0) else 0)" by (auto intro!: sum.cong sum.neutral simp del: sum.delta) also have "… = (∑xa = 0..<length xs. if fst (xs ! xa) ∈ A then snd (xs ! xa) else 0)" by (intro sum.cong refl) (simp_all add: sum.delta) also have "… = sum_list (map snd (filter (λx. fst x ∈ A) xs))" by (subst sum_list_map_filter', subst sum_list_sum_nth) simp_all finally show ?thesis . qed lemma measure_pmf_of_list: assumes "pmf_of_list_wf xs" shows "measure (pmf_of_list xs) A = sum_list (map snd (filter (λx. fst x ∈ A) xs))" using assms unfolding pmf_of_list_wf_def Sigma_Algebra.measure_def by (subst emeasure_pmf_of_list [OF assms], subst enn2real_ennreal) (auto intro!: sum_list_nonneg) (* TODO Move? *) lemma sum_list_nonneg_eq_zero_iff: fixes xs :: "'a :: linordered_ab_group_add list" shows "(⋀x. x ∈ set xs ⟹ x ≥ 0) ⟹ sum_list xs = 0 ⟷ set xs ⊆ {0}" proof (induction xs) case (Cons x xs) from Cons.prems have "sum_list (x#xs) = 0 ⟷ x = 0 ∧ sum_list xs = 0" unfolding sum_list_simps by (subst add_nonneg_eq_0_iff) (auto intro: sum_list_nonneg) with Cons.IH Cons.prems show ?case by simp qed simp_all lemma sum_list_filter_nonzero: "sum_list (filter (λx. x ≠ 0) xs) = sum_list xs" by (induction xs) simp_all (* END MOVE *) lemma set_pmf_of_list_eq: assumes "pmf_of_list_wf xs" "⋀x. x ∈ snd ` set xs ⟹ x > 0" shows "set_pmf (pmf_of_list xs) = fst ` set xs" proof { fix x assume A: "x ∈ fst ` set xs" and B: "x ∉ set_pmf (pmf_of_list xs)" then obtain y where y: "(x, y) ∈ set xs" by auto from B have "sum_list (map snd [z←xs. fst z = x]) = 0" by (simp add: pmf_pmf_of_list[OF assms(1)] set_pmf_eq) moreover from y have "y ∈ snd ` {xa ∈ set xs. fst xa = x}" by force ultimately have "y = 0" using assms(1) by (subst (asm) sum_list_nonneg_eq_zero_iff) (auto simp: pmf_of_list_wf_def) with assms(2) y have False by force } thus "fst ` set xs ⊆ set_pmf (pmf_of_list xs)" by blast qed (insert set_pmf_of_list[OF assms(1)], simp_all) lemma pmf_of_list_remove_zeros: assumes "pmf_of_list_wf xs" defines "xs' ≡ filter (λz. snd z ≠ 0) xs" shows "pmf_of_list_wf xs'" "pmf_of_list xs' = pmf_of_list xs" proof - have "map snd [z←xs . snd z ≠ 0] = filter (λx. x ≠ 0) (map snd xs)" by (induction xs) simp_all with assms(1) show wf: "pmf_of_list_wf xs'" by (auto simp: pmf_of_list_wf_def xs'_def sum_list_filter_nonzero) have "sum_list (map snd [z←xs' . fst z = i]) = sum_list (map snd [z←xs . fst z = i])" for i unfolding xs'_def by (induction xs) simp_all with assms(1) wf show "pmf_of_list xs' = pmf_of_list xs" by (intro pmf_eqI) (simp_all add: pmf_pmf_of_list) qed end