src/HOL/Probability/Probability_Mass_Function.thy
 author hoelzl Mon Oct 06 16:27:07 2014 +0200 (2014-10-06) changeset 58587 5484f6079bcd child 58606 9c66f7c541fb permissions -rw-r--r--
add type for probability mass functions, i.e. discrete probability distribution
 hoelzl@58587 ` 1` ```theory Probability_Mass_Function ``` hoelzl@58587 ` 2` ``` imports Probability_Measure ``` hoelzl@58587 ` 3` ```begin ``` hoelzl@58587 ` 4` hoelzl@58587 ` 5` ```lemma sets_Pair: "{x} \ sets M1 \ {y} \ sets M2 \ {(x, y)} \ sets (M1 \\<^sub>M M2)" ``` hoelzl@58587 ` 6` ``` using pair_measureI[of "{x}" M1 "{y}" M2] by simp ``` hoelzl@58587 ` 7` hoelzl@58587 ` 8` ```lemma finite_subset_card: ``` hoelzl@58587 ` 9` ``` assumes X: "infinite X" shows "\A\X. finite A \ card A = n" ``` hoelzl@58587 ` 10` ```proof (induct n) ``` hoelzl@58587 ` 11` ``` case (Suc n) then guess A .. note A = this ``` hoelzl@58587 ` 12` ``` with X obtain x where "x \ X" "x \ A" ``` hoelzl@58587 ` 13` ``` by (metis subset_antisym subset_eq) ``` hoelzl@58587 ` 14` ``` with A show ?case ``` hoelzl@58587 ` 15` ``` by (intro exI[of _ "insert x A"]) auto ``` hoelzl@58587 ` 16` ```qed (simp cong: conj_cong) ``` hoelzl@58587 ` 17` hoelzl@58587 ` 18` ```lemma (in prob_space) countable_support: ``` hoelzl@58587 ` 19` ``` "countable {x. measure M {x} \ 0}" ``` hoelzl@58587 ` 20` ```proof - ``` hoelzl@58587 ` 21` ``` let ?m = "\x. measure M {x}" ``` hoelzl@58587 ` 22` ``` have *: "{x. ?m x \ 0} = (\n. {x. inverse (real (Suc n)) < ?m x})" ``` hoelzl@58587 ` 23` ``` by (auto intro!: measure_nonneg reals_Archimedean order_le_neq_trans) ``` hoelzl@58587 ` 24` ``` have **: "\n. finite {x. inverse (Suc n) < ?m x}" ``` hoelzl@58587 ` 25` ``` proof (rule ccontr) ``` hoelzl@58587 ` 26` ``` fix n assume "infinite {x. inverse (Suc n) < ?m x}" (is "infinite ?X") ``` hoelzl@58587 ` 27` ``` then obtain X where "finite X" "card X = Suc (Suc n)" "X \ ?X" ``` hoelzl@58587 ` 28` ``` by (metis finite_subset_card) ``` hoelzl@58587 ` 29` ``` from this(3) have *: "\x. x \ X \ 1 / Suc n \ ?m x" ``` hoelzl@58587 ` 30` ``` by (auto simp: inverse_eq_divide) ``` hoelzl@58587 ` 31` ``` { fix x assume "x \ X" ``` hoelzl@58587 ` 32` ``` from *[OF this] have "?m x \ 0" by auto ``` hoelzl@58587 ` 33` ``` then have "{x} \ sets M" by (auto dest: measure_notin_sets) } ``` hoelzl@58587 ` 34` ``` note singleton_sets = this ``` hoelzl@58587 ` 35` ``` have "1 < (\x\X. 1 / Suc n)" ``` hoelzl@58587 ` 36` ``` by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc) ``` hoelzl@58587 ` 37` ``` also have "\ \ (\x\X. ?m x)" ``` hoelzl@58587 ` 38` ``` by (rule setsum_mono) fact ``` hoelzl@58587 ` 39` ``` also have "\ = measure M (\x\X. {x})" ``` hoelzl@58587 ` 40` ``` using singleton_sets `finite X` ``` hoelzl@58587 ` 41` ``` by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def) ``` hoelzl@58587 ` 42` ``` finally show False ``` hoelzl@58587 ` 43` ``` using prob_le_1[of "\x\X. {x}"] by arith ``` hoelzl@58587 ` 44` ``` qed ``` hoelzl@58587 ` 45` ``` show ?thesis ``` hoelzl@58587 ` 46` ``` unfolding * by (intro countable_UN countableI_type countable_finite[OF **]) ``` hoelzl@58587 ` 47` ```qed ``` hoelzl@58587 ` 48` hoelzl@58587 ` 49` ```lemma measure_count_space: "measure (count_space A) X = (if X \ A then card X else 0)" ``` hoelzl@58587 ` 50` ``` unfolding measure_def ``` hoelzl@58587 ` 51` ``` by (cases "finite X") (simp_all add: emeasure_notin_sets) ``` hoelzl@58587 ` 52` hoelzl@58587 ` 53` ```typedef 'a pmf = "{M :: 'a measure. prob_space M \ sets M = UNIV \ (AE x in M. measure M {x} \ 0)}" ``` hoelzl@58587 ` 54` ``` morphisms measure_pmf Abs_pmf ``` hoelzl@58587 ` 55` ``` apply (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"]) ``` hoelzl@58587 ` 56` ``` apply (auto intro!: prob_space_uniform_measure simp: measure_count_space) ``` hoelzl@58587 ` 57` ``` apply (subst uniform_measure_def) ``` hoelzl@58587 ` 58` ``` apply (simp add: AE_density AE_count_space split: split_indicator) ``` hoelzl@58587 ` 59` ``` done ``` hoelzl@58587 ` 60` hoelzl@58587 ` 61` ```declare [[coercion measure_pmf]] ``` hoelzl@58587 ` 62` hoelzl@58587 ` 63` ```lemma prob_space_measure_pmf: "prob_space (measure_pmf p)" ``` hoelzl@58587 ` 64` ``` using pmf.measure_pmf[of p] by auto ``` hoelzl@58587 ` 65` hoelzl@58587 ` 66` ```interpretation measure_pmf!: prob_space "measure_pmf M" for M ``` hoelzl@58587 ` 67` ``` by (rule prob_space_measure_pmf) ``` hoelzl@58587 ` 68` hoelzl@58587 ` 69` ```locale pmf_as_measure ``` hoelzl@58587 ` 70` ```begin ``` hoelzl@58587 ` 71` hoelzl@58587 ` 72` ```setup_lifting type_definition_pmf ``` hoelzl@58587 ` 73` hoelzl@58587 ` 74` ```end ``` hoelzl@58587 ` 75` hoelzl@58587 ` 76` ```context ``` hoelzl@58587 ` 77` ```begin ``` hoelzl@58587 ` 78` hoelzl@58587 ` 79` ```interpretation pmf_as_measure . ``` hoelzl@58587 ` 80` hoelzl@58587 ` 81` ```lift_definition pmf :: "'a pmf \ 'a \ real" is "\M x. measure M {x}" . ``` hoelzl@58587 ` 82` hoelzl@58587 ` 83` ```lift_definition set_pmf :: "'a pmf \ 'a set" is "\M. {x. measure M {x} \ 0}" . ``` hoelzl@58587 ` 84` hoelzl@58587 ` 85` ```lift_definition map_pmf :: "('a \ 'b) \ 'a pmf \ 'b pmf" is ``` hoelzl@58587 ` 86` ``` "\f M. distr M (count_space UNIV) f" ``` hoelzl@58587 ` 87` ```proof safe ``` hoelzl@58587 ` 88` ``` fix M and f :: "'a \ 'b" ``` hoelzl@58587 ` 89` ``` let ?D = "distr M (count_space UNIV) f" ``` hoelzl@58587 ` 90` ``` assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \ 0" ``` hoelzl@58587 ` 91` ``` interpret prob_space M by fact ``` hoelzl@58587 ` 92` ``` from ae have "AE x in M. measure M (f -` {f x}) \ 0" ``` hoelzl@58587 ` 93` ``` proof eventually_elim ``` hoelzl@58587 ` 94` ``` fix x ``` hoelzl@58587 ` 95` ``` have "measure M {x} \ measure M (f -` {f x})" ``` hoelzl@58587 ` 96` ``` by (intro finite_measure_mono) auto ``` hoelzl@58587 ` 97` ``` then show "measure M {x} \ 0 \ measure M (f -` {f x}) \ 0" ``` hoelzl@58587 ` 98` ``` using measure_nonneg[of M "{x}"] by auto ``` hoelzl@58587 ` 99` ``` qed ``` hoelzl@58587 ` 100` ``` then show "AE x in ?D. measure ?D {x} \ 0" ``` hoelzl@58587 ` 101` ``` by (simp add: AE_distr_iff measure_distr measurable_def) ``` hoelzl@58587 ` 102` ```qed (auto simp: measurable_def prob_space.prob_space_distr) ``` hoelzl@58587 ` 103` hoelzl@58587 ` 104` ```declare [[coercion set_pmf]] ``` hoelzl@58587 ` 105` hoelzl@58587 ` 106` ```lemma countable_set_pmf: "countable (set_pmf p)" ``` hoelzl@58587 ` 107` ``` by transfer (metis prob_space.countable_support) ``` hoelzl@58587 ` 108` hoelzl@58587 ` 109` ```lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV" ``` hoelzl@58587 ` 110` ``` by transfer metis ``` hoelzl@58587 ` 111` hoelzl@58587 ` 112` ```lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV" ``` hoelzl@58587 ` 113` ``` using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp ``` hoelzl@58587 ` 114` hoelzl@58587 ` 115` ```lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \ space N" ``` hoelzl@58587 ` 116` ``` by (auto simp: measurable_def) ``` hoelzl@58587 ` 117` hoelzl@58587 ` 118` ```lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)" ``` hoelzl@58587 ` 119` ``` by (intro measurable_cong_sets) simp_all ``` hoelzl@58587 ` 120` hoelzl@58587 ` 121` ```lemma pmf_positive: "x \ set_pmf p \ 0 < pmf p x" ``` hoelzl@58587 ` 122` ``` by transfer (simp add: less_le measure_nonneg) ``` hoelzl@58587 ` 123` hoelzl@58587 ` 124` ```lemma pmf_nonneg: "0 \ pmf p x" ``` hoelzl@58587 ` 125` ``` by transfer (simp add: measure_nonneg) ``` hoelzl@58587 ` 126` hoelzl@58587 ` 127` ```lemma emeasure_pmf_single: ``` hoelzl@58587 ` 128` ``` fixes M :: "'a pmf" ``` hoelzl@58587 ` 129` ``` shows "emeasure M {x} = pmf M x" ``` hoelzl@58587 ` 130` ``` by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) ``` hoelzl@58587 ` 131` hoelzl@58587 ` 132` ```lemma AE_measure_pmf: "AE x in (M::'a pmf). x \ M" ``` hoelzl@58587 ` 133` ``` by transfer simp ``` hoelzl@58587 ` 134` hoelzl@58587 ` 135` ```lemma emeasure_pmf_single_eq_zero_iff: ``` hoelzl@58587 ` 136` ``` fixes M :: "'a pmf" ``` hoelzl@58587 ` 137` ``` shows "emeasure M {y} = 0 \ y \ M" ``` hoelzl@58587 ` 138` ``` by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure]) ``` hoelzl@58587 ` 139` hoelzl@58587 ` 140` ```lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \ (\y\M. P y)" ``` hoelzl@58587 ` 141` ```proof - ``` hoelzl@58587 ` 142` ``` { fix y assume y: "y \ M" and P: "AE x in M. P x" "\ P y" ``` hoelzl@58587 ` 143` ``` with P have "AE x in M. x \ y" ``` hoelzl@58587 ` 144` ``` by auto ``` hoelzl@58587 ` 145` ``` with y have False ``` hoelzl@58587 ` 146` ``` by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) } ``` hoelzl@58587 ` 147` ``` then show ?thesis ``` hoelzl@58587 ` 148` ``` using AE_measure_pmf[of M] by auto ``` hoelzl@58587 ` 149` ```qed ``` hoelzl@58587 ` 150` hoelzl@58587 ` 151` ```lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)" ``` hoelzl@58587 ` 152` ```proof (transfer, elim conjE) ``` hoelzl@58587 ` 153` ``` fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \ 0" ``` hoelzl@58587 ` 154` ``` assume "prob_space M" then interpret prob_space M . ``` hoelzl@58587 ` 155` ``` show "M = density (count_space UNIV) (\x. ereal (measure M {x}))" ``` hoelzl@58587 ` 156` ``` proof (rule measure_eqI) ``` hoelzl@58587 ` 157` ``` fix A :: "'a set" ``` hoelzl@58587 ` 158` ``` have "(\\<^sup>+ x. ereal (measure M {x}) * indicator A x \count_space UNIV) = ``` hoelzl@58587 ` 159` ``` (\\<^sup>+ x. emeasure M {x} * indicator (A \ {x. measure M {x} \ 0}) x \count_space UNIV)" ``` hoelzl@58587 ` 160` ``` by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator) ``` hoelzl@58587 ` 161` ``` also have "\ = (\\<^sup>+ x. emeasure M {x} \count_space (A \ {x. measure M {x} \ 0}))" ``` hoelzl@58587 ` 162` ``` by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space) ``` hoelzl@58587 ` 163` ``` also have "\ = emeasure M (\x\(A \ {x. measure M {x} \ 0}). {x})" ``` hoelzl@58587 ` 164` ``` by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support) ``` hoelzl@58587 ` 165` ``` (auto simp: disjoint_family_on_def) ``` hoelzl@58587 ` 166` ``` also have "\ = emeasure M A" ``` hoelzl@58587 ` 167` ``` using ae by (intro emeasure_eq_AE) auto ``` hoelzl@58587 ` 168` ``` finally show " emeasure M A = emeasure (density (count_space UNIV) (\x. ereal (measure M {x}))) A" ``` hoelzl@58587 ` 169` ``` using emeasure_space_1 by (simp add: emeasure_density) ``` hoelzl@58587 ` 170` ``` qed simp ``` hoelzl@58587 ` 171` ```qed ``` hoelzl@58587 ` 172` hoelzl@58587 ` 173` ```lemma set_pmf_not_empty: "set_pmf M \ {}" ``` hoelzl@58587 ` 174` ``` using AE_measure_pmf[of M] by (intro notI) simp ``` hoelzl@58587 ` 175` hoelzl@58587 ` 176` ```lemma set_pmf_iff: "x \ set_pmf M \ pmf M x \ 0" ``` hoelzl@58587 ` 177` ``` by transfer simp ``` hoelzl@58587 ` 178` hoelzl@58587 ` 179` ```lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1" ``` hoelzl@58587 ` 180` ```proof - ``` hoelzl@58587 ` 181` ``` have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)" ``` hoelzl@58587 ` 182` ``` by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf) ``` hoelzl@58587 ` 183` ``` then show ?thesis ``` hoelzl@58587 ` 184` ``` using measure_pmf.emeasure_space_1 by simp ``` hoelzl@58587 ` 185` ```qed ``` hoelzl@58587 ` 186` hoelzl@58587 ` 187` ```lemma map_pmf_id[simp]: "map_pmf id = id" ``` hoelzl@58587 ` 188` ``` by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI) ``` hoelzl@58587 ` 189` hoelzl@58587 ` 190` ```lemma map_pmf_compose: "map_pmf (f \ g) = map_pmf f \ map_pmf g" ``` hoelzl@58587 ` 191` ``` by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) ``` hoelzl@58587 ` 192` hoelzl@58587 ` 193` ```lemma map_pmf_cong: ``` hoelzl@58587 ` 194` ``` assumes "p = q" ``` hoelzl@58587 ` 195` ``` shows "(\x. x \ set_pmf q \ f x = g x) \ map_pmf f p = map_pmf g q" ``` hoelzl@58587 ` 196` ``` unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq ``` hoelzl@58587 ` 197` ``` by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI) ``` hoelzl@58587 ` 198` hoelzl@58587 ` 199` ```lemma pmf_set_map: ``` hoelzl@58587 ` 200` ``` fixes f :: "'a \ 'b" ``` hoelzl@58587 ` 201` ``` shows "set_pmf \ map_pmf f = op ` f \ set_pmf" ``` hoelzl@58587 ` 202` ```proof (rule, transfer, clarsimp simp add: measure_distr measurable_def) ``` hoelzl@58587 ` 203` ``` fix f :: "'a \ 'b" and M :: "'a measure" ``` hoelzl@58587 ` 204` ``` assume "prob_space M" and ae: "AE x in M. measure M {x} \ 0" and [simp]: "sets M = UNIV" ``` hoelzl@58587 ` 205` ``` interpret prob_space M by fact ``` hoelzl@58587 ` 206` ``` show "{x. measure M (f -` {x}) \ 0} = f ` {x. measure M {x} \ 0}" ``` hoelzl@58587 ` 207` ``` proof safe ``` hoelzl@58587 ` 208` ``` fix x assume "measure M (f -` {x}) \ 0" ``` hoelzl@58587 ` 209` ``` moreover have "measure M (f -` {x}) = measure M {y. f y = x \ measure M {y} \ 0}" ``` hoelzl@58587 ` 210` ``` using ae by (intro finite_measure_eq_AE) auto ``` hoelzl@58587 ` 211` ``` ultimately have "{y. f y = x \ measure M {y} \ 0} \ {}" ``` hoelzl@58587 ` 212` ``` by (metis measure_empty) ``` hoelzl@58587 ` 213` ``` then show "x \ f ` {x. measure M {x} \ 0}" ``` hoelzl@58587 ` 214` ``` by auto ``` hoelzl@58587 ` 215` ``` next ``` hoelzl@58587 ` 216` ``` fix x assume "measure M {x} \ 0" ``` hoelzl@58587 ` 217` ``` then have "0 < measure M {x}" ``` hoelzl@58587 ` 218` ``` using measure_nonneg[of M "{x}"] by auto ``` hoelzl@58587 ` 219` ``` also have "measure M {x} \ measure M (f -` {f x})" ``` hoelzl@58587 ` 220` ``` by (intro finite_measure_mono) auto ``` hoelzl@58587 ` 221` ``` finally show "measure M (f -` {f x}) = 0 \ False" ``` hoelzl@58587 ` 222` ``` by simp ``` hoelzl@58587 ` 223` ``` qed ``` hoelzl@58587 ` 224` ```qed ``` hoelzl@58587 ` 225` hoelzl@58587 ` 226` ```context ``` hoelzl@58587 ` 227` ``` fixes f :: "'a \ real" ``` hoelzl@58587 ` 228` ``` assumes nonneg: "\x. 0 \ f x" ``` hoelzl@58587 ` 229` ``` assumes prob: "(\\<^sup>+x. f x \count_space UNIV) = 1" ``` hoelzl@58587 ` 230` ```begin ``` hoelzl@58587 ` 231` hoelzl@58587 ` 232` ```lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \ f)" ``` hoelzl@58587 ` 233` ```proof (intro conjI) ``` hoelzl@58587 ` 234` ``` have *[simp]: "\x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y" ``` hoelzl@58587 ` 235` ``` by (simp split: split_indicator) ``` hoelzl@58587 ` 236` ``` show "AE x in density (count_space UNIV) (ereal \ f). ``` hoelzl@58587 ` 237` ``` measure (density (count_space UNIV) (ereal \ f)) {x} \ 0" ``` hoelzl@58587 ` 238` ``` by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator) ``` hoelzl@58587 ` 239` ``` show "prob_space (density (count_space UNIV) (ereal \ f))" ``` hoelzl@58587 ` 240` ``` by default (simp add: emeasure_density prob) ``` hoelzl@58587 ` 241` ```qed simp ``` hoelzl@58587 ` 242` hoelzl@58587 ` 243` ```lemma pmf_embed_pmf: "pmf embed_pmf x = f x" ``` hoelzl@58587 ` 244` ```proof transfer ``` hoelzl@58587 ` 245` ``` have *[simp]: "\x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y" ``` hoelzl@58587 ` 246` ``` by (simp split: split_indicator) ``` hoelzl@58587 ` 247` ``` fix x show "measure (density (count_space UNIV) (ereal \ f)) {x} = f x" ``` hoelzl@58587 ` 248` ``` by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg) ``` hoelzl@58587 ` 249` ```qed ``` hoelzl@58587 ` 250` hoelzl@58587 ` 251` ```end ``` hoelzl@58587 ` 252` hoelzl@58587 ` 253` ```lemma embed_pmf_transfer: ``` hoelzl@58587 ` 254` ``` "rel_fun (eq_onp (\f::'a \ real. (\x. 0 \ f x) \ (\\<^sup>+x. ereal (f x) \count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\f. density (count_space UNIV) (ereal \ f)) embed_pmf" ``` hoelzl@58587 ` 255` ``` by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer) ``` hoelzl@58587 ` 256` hoelzl@58587 ` 257` ```lemma td_pmf_embed_pmf: ``` hoelzl@58587 ` 258` ``` "type_definition pmf embed_pmf {f::'a \ real. (\x. 0 \ f x) \ (\\<^sup>+x. ereal (f x) \count_space UNIV) = 1}" ``` hoelzl@58587 ` 259` ``` unfolding type_definition_def ``` hoelzl@58587 ` 260` ```proof safe ``` hoelzl@58587 ` 261` ``` fix p :: "'a pmf" ``` hoelzl@58587 ` 262` ``` have "(\\<^sup>+ x. 1 \measure_pmf p) = 1" ``` hoelzl@58587 ` 263` ``` using measure_pmf.emeasure_space_1[of p] by simp ``` hoelzl@58587 ` 264` ``` then show *: "(\\<^sup>+ x. ereal (pmf p x) \count_space UNIV) = 1" ``` hoelzl@58587 ` 265` ``` by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const) ``` hoelzl@58587 ` 266` hoelzl@58587 ` 267` ``` show "embed_pmf (pmf p) = p" ``` hoelzl@58587 ` 268` ``` by (intro measure_pmf_inject[THEN iffD1]) ``` hoelzl@58587 ` 269` ``` (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def) ``` hoelzl@58587 ` 270` ```next ``` hoelzl@58587 ` 271` ``` fix f :: "'a \ real" assume "\x. 0 \ f x" "(\\<^sup>+x. f x \count_space UNIV) = 1" ``` hoelzl@58587 ` 272` ``` then show "pmf (embed_pmf f) = f" ``` hoelzl@58587 ` 273` ``` by (auto intro!: pmf_embed_pmf) ``` hoelzl@58587 ` 274` ```qed (rule pmf_nonneg) ``` hoelzl@58587 ` 275` hoelzl@58587 ` 276` ```end ``` hoelzl@58587 ` 277` hoelzl@58587 ` 278` ```locale pmf_as_function ``` hoelzl@58587 ` 279` ```begin ``` hoelzl@58587 ` 280` hoelzl@58587 ` 281` ```setup_lifting td_pmf_embed_pmf ``` hoelzl@58587 ` 282` hoelzl@58587 ` 283` ```end ``` hoelzl@58587 ` 284` hoelzl@58587 ` 285` ```(* ``` hoelzl@58587 ` 286` hoelzl@58587 ` 287` ```definition ``` hoelzl@58587 ` 288` ``` "rel_pmf P d1 d2 \ (\p3. (\(x, y) \ set_pmf p3. P x y) \ map_pmf fst p3 = d1 \ map_pmf snd p3 = d2)" ``` hoelzl@58587 ` 289` hoelzl@58587 ` 290` ```lift_definition pmf_join :: "real \ 'a pmf \ 'a pmf \ 'a pmf" is ``` hoelzl@58587 ` 291` ``` "\p M1 M2. density (count_space UNIV) (\x. p * measure M1 {x} + (1 - p) * measure M2 {x})" ``` hoelzl@58587 ` 292` ```sorry ``` hoelzl@58587 ` 293` hoelzl@58587 ` 294` ```lift_definition pmf_single :: "'a \ 'a pmf" is ``` hoelzl@58587 ` 295` ``` "\x. uniform_measure (count_space UNIV) {x}" ``` hoelzl@58587 ` 296` ```sorry ``` hoelzl@58587 ` 297` hoelzl@58587 ` 298` ```bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: pmf_rel ``` hoelzl@58587 ` 299` ```proof - ``` hoelzl@58587 ` 300` ``` show "map_pmf id = id" by (rule map_pmf_id) ``` hoelzl@58587 ` 301` ``` show "\f g. map_pmf (f \ g) = map_pmf f \ map_pmf g" by (rule map_pmf_compose) ``` hoelzl@58587 ` 302` ``` show "\f g::'a \ 'b. \p. (\x. x \ set_pmf p \ f x = g x) \ map_pmf f p = map_pmf g p" ``` hoelzl@58587 ` 303` ``` by (intro map_pmg_cong refl) ``` hoelzl@58587 ` 304` hoelzl@58587 ` 305` ``` show "\f::'a \ 'b. set_pmf \ map_pmf f = op ` f \ set_pmf" ``` hoelzl@58587 ` 306` ``` by (rule pmf_set_map) ``` hoelzl@58587 ` 307` hoelzl@58587 ` 308` ``` { fix p :: "'s pmf" ``` hoelzl@58587 ` 309` ``` have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \ ordLeq" ``` hoelzl@58587 ` 310` ``` by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"]) ``` hoelzl@58587 ` 311` ``` (auto intro: countable_set_pmf inj_on_to_nat_on) ``` hoelzl@58587 ` 312` ``` also have "(card_of (UNIV :: nat set), natLeq) \ ordLeq" ``` hoelzl@58587 ` 313` ``` by (metis Field_natLeq card_of_least natLeq_Well_order) ``` hoelzl@58587 ` 314` ``` finally show "(card_of (set_pmf p), natLeq) \ ordLeq" . } ``` hoelzl@58587 ` 315` hoelzl@58587 ` 316` ``` show "\R. pmf_rel R = ``` hoelzl@58587 ` 317` ``` (BNF_Util.Grp {x. set_pmf x \ {(x, y). R x y}} (map_pmf fst))\\ OO ``` hoelzl@58587 ` 318` ``` BNF_Util.Grp {x. set_pmf x \ {(x, y). R x y}} (map_pmf snd)" ``` hoelzl@58587 ` 319` ``` by (auto simp add: fun_eq_iff pmf_rel_def BNF_Util.Grp_def OO_def) ``` hoelzl@58587 ` 320` hoelzl@58587 ` 321` ``` { let ?f = "map_pmf fst" and ?s = "map_pmf snd" ``` hoelzl@58587 ` 322` ``` fix R :: "'a \ 'b \ bool" and A assume "\x y. (x, y) \ set_pmf A \ R x y" ``` hoelzl@58587 ` 323` ``` fix S :: "'b \ 'c \ bool" and B assume "\y z. (y, z) \ set_pmf B \ S y z" ``` hoelzl@58587 ` 324` ``` assume "?f B = ?s A" ``` hoelzl@58587 ` 325` ``` have "\C. (\(x, z)\set_pmf C. \y. R x y \ S y z) \ ?f C = ?f A \ ?s C = ?s B" ``` hoelzl@58587 ` 326` ``` sorry } ``` hoelzl@58587 ` 327` ```oops ``` hoelzl@58587 ` 328` ``` then show "\R::'a \ 'b \ bool. \S::'b \ 'c \ bool. pmf_rel R OO pmf_rel S \ pmf_rel (R OO S)" ``` hoelzl@58587 ` 329` ``` by (auto simp add: subset_eq pmf_rel_def fun_eq_iff OO_def Ball_def) ``` hoelzl@58587 ` 330` ```qed (fact natLeq_card_order natLeq_cinfinite)+ ``` hoelzl@58587 ` 331` hoelzl@58587 ` 332` ```notepad ``` hoelzl@58587 ` 333` ```begin ``` hoelzl@58587 ` 334` ``` fix x y :: "nat \ real" ``` hoelzl@58587 ` 335` ``` def IJz \ "rec_nat ((0, 0), \_. 0) (\n ((I, J), z). ``` hoelzl@58587 ` 336` ``` let a = x I - (\ji b then I + 1 else I, if b \ a then J + 1 else J), z((I, J) := min a b)))" ``` hoelzl@58587 ` 338` ``` def I == "fst \ fst \ IJz" def J == "snd \ fst \ IJz" def z == "snd \ IJz" ``` hoelzl@58587 ` 339` ``` let ?a = "\n. x (I n) - (\jp. z 0 p = 0" "I 0 = 0" "J 0 = 0" ``` hoelzl@58587 ` 341` ``` by (simp_all add: I_def J_def z_def IJz_def) ``` hoelzl@58587 ` 342` ``` have z_Suc[simp]: "\n. z (Suc n) = (z n)((I n, J n) := min (?a n) (?b n))" ``` hoelzl@58587 ` 343` ``` by (simp add: z_def I_def J_def IJz_def Let_def split_beta) ``` hoelzl@58587 ` 344` ``` have I_Suc[simp]: "\n. I (Suc n) = (if ?a n \ ?b n then I n + 1 else I n)" ``` hoelzl@58587 ` 345` ``` by (simp add: z_def I_def J_def IJz_def Let_def split_beta) ``` hoelzl@58587 ` 346` ``` have J_Suc[simp]: "\n. J (Suc n) = (if ?b n \ ?a n then J n + 1 else J n)" ``` hoelzl@58587 ` 347` ``` by (simp add: z_def I_def J_def IJz_def Let_def split_beta) ``` hoelzl@58587 ` 348` ``` ``` hoelzl@58587 ` 349` ``` { fix N have "\p. z N p \ 0 \ \nj. z n (i, j)) = x i" ``` hoelzl@58587 ` 354` ``` oops ``` hoelzl@58587 ` 355` ```*) ``` hoelzl@58587 ` 356` hoelzl@58587 ` 357` ```end ``` hoelzl@58587 ` 358`