src/HOL/Probability/Probability_Mass_Function.thy
author hoelzl
Tue, 07 Oct 2014 10:34:24 +0200
changeset 58606 9c66f7c541fb
parent 58587 5484f6079bcd
child 58730 b3fd0628f849
permissions -rw-r--r--
add Giry monad
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
58606
9c66f7c541fb add Giry monad
hoelzl
parents: 58587
diff changeset
     1
(*  Title:      HOL/Probability/Probability_Mass_Function.thy
9c66f7c541fb add Giry monad
hoelzl
parents: 58587
diff changeset
     2
    Author:     Johannes Hölzl, TU München *)
9c66f7c541fb add Giry monad
hoelzl
parents: 58587
diff changeset
     3
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
     4
theory Probability_Mass_Function
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
     5
  imports Probability_Measure
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
     6
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
     7
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
     8
lemma (in prob_space) countable_support:
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
     9
  "countable {x. measure M {x} \<noteq> 0}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    10
proof -
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    11
  let ?m = "\<lambda>x. measure M {x}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    12
  have *: "{x. ?m x \<noteq> 0} = (\<Union>n. {x. inverse (real (Suc n)) < ?m x})"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    13
    by (auto intro!: measure_nonneg reals_Archimedean order_le_neq_trans)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    14
  have **: "\<And>n. finite {x. inverse (Suc n) < ?m x}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    15
  proof (rule ccontr)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    16
    fix n assume "infinite {x. inverse (Suc n) < ?m x}" (is "infinite ?X")
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    17
    then obtain X where "finite X" "card X = Suc (Suc n)" "X \<subseteq> ?X"
58606
9c66f7c541fb add Giry monad
hoelzl
parents: 58587
diff changeset
    18
      by (metis infinite_arbitrarily_large)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    19
    from this(3) have *: "\<And>x. x \<in> X \<Longrightarrow> 1 / Suc n \<le> ?m x" 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    20
      by (auto simp: inverse_eq_divide)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    21
    { fix x assume "x \<in> X"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    22
      from *[OF this] have "?m x \<noteq> 0" by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    23
      then have "{x} \<in> sets M" by (auto dest: measure_notin_sets) }
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    24
    note singleton_sets = this
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    25
    have "1 < (\<Sum>x\<in>X. 1 / Suc n)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    26
      by (simp add: `card X = Suc (Suc n)` real_eq_of_nat[symmetric] real_of_nat_Suc)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    27
    also have "\<dots> \<le> (\<Sum>x\<in>X. ?m x)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    28
      by (rule setsum_mono) fact
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    29
    also have "\<dots> = measure M (\<Union>x\<in>X. {x})"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    30
      using singleton_sets `finite X`
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    31
      by (intro finite_measure_finite_Union[symmetric]) (auto simp: disjoint_family_on_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    32
    finally show False
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    33
      using prob_le_1[of "\<Union>x\<in>X. {x}"] by arith
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    34
  qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    35
  show ?thesis
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    36
    unfolding * by (intro countable_UN countableI_type countable_finite[OF **])
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    37
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    38
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    39
typedef 'a pmf = "{M :: 'a measure. prob_space M \<and> sets M = UNIV \<and> (AE x in M. measure M {x} \<noteq> 0)}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    40
  morphisms measure_pmf Abs_pmf
58606
9c66f7c541fb add Giry monad
hoelzl
parents: 58587
diff changeset
    41
  by (intro exI[of _ "uniform_measure (count_space UNIV) {undefined}"])
9c66f7c541fb add Giry monad
hoelzl
parents: 58587
diff changeset
    42
     (auto intro!: prob_space_uniform_measure AE_uniform_measureI)
58587
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    43
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    44
declare [[coercion measure_pmf]]
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    45
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    46
lemma prob_space_measure_pmf: "prob_space (measure_pmf p)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    47
  using pmf.measure_pmf[of p] by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    48
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    49
interpretation measure_pmf!: prob_space "measure_pmf M" for M
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    50
  by (rule prob_space_measure_pmf)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    51
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    52
locale pmf_as_measure
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    53
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    54
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    55
setup_lifting type_definition_pmf
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    56
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    57
end
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    58
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    59
context
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    60
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    61
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    62
interpretation pmf_as_measure .
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    63
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    64
lift_definition pmf :: "'a pmf \<Rightarrow> 'a \<Rightarrow> real" is "\<lambda>M x. measure M {x}" .
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    65
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    66
lift_definition set_pmf :: "'a pmf \<Rightarrow> 'a set" is "\<lambda>M. {x. measure M {x} \<noteq> 0}" .
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    67
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    68
lift_definition map_pmf :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pmf \<Rightarrow> 'b pmf" is
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    69
  "\<lambda>f M. distr M (count_space UNIV) f"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    70
proof safe
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    71
  fix M and f :: "'a \<Rightarrow> 'b"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    72
  let ?D = "distr M (count_space UNIV) f"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    73
  assume "prob_space M" and [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    74
  interpret prob_space M by fact
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    75
  from ae have "AE x in M. measure M (f -` {f x}) \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    76
  proof eventually_elim
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    77
    fix x
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    78
    have "measure M {x} \<le> measure M (f -` {f x})"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    79
      by (intro finite_measure_mono) auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    80
    then show "measure M {x} \<noteq> 0 \<Longrightarrow> measure M (f -` {f x}) \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    81
      using measure_nonneg[of M "{x}"] by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    82
  qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    83
  then show "AE x in ?D. measure ?D {x} \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    84
    by (simp add: AE_distr_iff measure_distr measurable_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    85
qed (auto simp: measurable_def prob_space.prob_space_distr)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    86
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    87
declare [[coercion set_pmf]]
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    88
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    89
lemma countable_set_pmf: "countable (set_pmf p)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    90
  by transfer (metis prob_space.countable_support)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    91
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    92
lemma sets_measure_pmf[simp]: "sets (measure_pmf p) = UNIV"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    93
  by transfer metis
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    94
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    95
lemma space_measure_pmf[simp]: "space (measure_pmf p) = UNIV"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    96
  using sets_eq_imp_space_eq[of "measure_pmf p" "count_space UNIV"] by simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    97
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    98
lemma measurable_pmf_measure1[simp]: "measurable (M :: 'a pmf) N = UNIV \<rightarrow> space N"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
    99
  by (auto simp: measurable_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   100
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   101
lemma measurable_pmf_measure2[simp]: "measurable N (M :: 'a pmf) = measurable N (count_space UNIV)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   102
  by (intro measurable_cong_sets) simp_all
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   103
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   104
lemma pmf_positive: "x \<in> set_pmf p \<Longrightarrow> 0 < pmf p x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   105
  by transfer (simp add: less_le measure_nonneg)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   106
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   107
lemma pmf_nonneg: "0 \<le> pmf p x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   108
  by transfer (simp add: measure_nonneg)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   109
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   110
lemma emeasure_pmf_single:
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   111
  fixes M :: "'a pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   112
  shows "emeasure M {x} = pmf M x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   113
  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   114
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   115
lemma AE_measure_pmf: "AE x in (M::'a pmf). x \<in> M"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   116
  by transfer simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   117
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   118
lemma emeasure_pmf_single_eq_zero_iff:
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   119
  fixes M :: "'a pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   120
  shows "emeasure M {y} = 0 \<longleftrightarrow> y \<notin> M"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   121
  by transfer (simp add: finite_measure.emeasure_eq_measure[OF prob_space.finite_measure])
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   122
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   123
lemma AE_measure_pmf_iff: "(AE x in measure_pmf M. P x) \<longleftrightarrow> (\<forall>y\<in>M. P y)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   124
proof -
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   125
  { fix y assume y: "y \<in> M" and P: "AE x in M. P x" "\<not> P y"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   126
    with P have "AE x in M. x \<noteq> y"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   127
      by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   128
    with y have False
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   129
      by (simp add: emeasure_pmf_single_eq_zero_iff AE_iff_measurable[OF _ refl]) }
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   130
  then show ?thesis
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   131
    using AE_measure_pmf[of M] by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   132
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   133
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   134
lemma measure_pmf_eq_density: "measure_pmf p = density (count_space UNIV) (pmf p)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   135
proof (transfer, elim conjE)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   136
  fix M :: "'a measure" assume [simp]: "sets M = UNIV" and ae: "AE x in M. measure M {x} \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   137
  assume "prob_space M" then interpret prob_space M .
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   138
  show "M = density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   139
  proof (rule measure_eqI)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   140
    fix A :: "'a set"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   141
    have "(\<integral>\<^sup>+ x. ereal (measure M {x}) * indicator A x \<partial>count_space UNIV) = 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   142
      (\<integral>\<^sup>+ x. emeasure M {x} * indicator (A \<inter> {x. measure M {x} \<noteq> 0}) x \<partial>count_space UNIV)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   143
      by (auto intro!: nn_integral_cong simp: emeasure_eq_measure split: split_indicator)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   144
    also have "\<dots> = (\<integral>\<^sup>+ x. emeasure M {x} \<partial>count_space (A \<inter> {x. measure M {x} \<noteq> 0}))"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   145
      by (subst nn_integral_restrict_space[symmetric]) (auto simp: restrict_count_space)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   146
    also have "\<dots> = emeasure M (\<Union>x\<in>(A \<inter> {x. measure M {x} \<noteq> 0}). {x})"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   147
      by (intro emeasure_UN_countable[symmetric] countable_Int2 countable_support)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   148
         (auto simp: disjoint_family_on_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   149
    also have "\<dots> = emeasure M A"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   150
      using ae by (intro emeasure_eq_AE) auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   151
    finally show " emeasure M A = emeasure (density (count_space UNIV) (\<lambda>x. ereal (measure M {x}))) A"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   152
      using emeasure_space_1 by (simp add: emeasure_density)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   153
  qed simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   154
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   155
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   156
lemma set_pmf_not_empty: "set_pmf M \<noteq> {}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   157
  using AE_measure_pmf[of M] by (intro notI) simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   158
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   159
lemma set_pmf_iff: "x \<in> set_pmf M \<longleftrightarrow> pmf M x \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   160
  by transfer simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   161
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   162
lemma emeasure_pmf: "emeasure (M::'a pmf) M = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   163
proof -
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   164
  have "emeasure (M::'a pmf) M = emeasure (M::'a pmf) (space M)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   165
    by (intro emeasure_eq_AE) (simp_all add: AE_measure_pmf)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   166
  then show ?thesis
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   167
    using measure_pmf.emeasure_space_1 by simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   168
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   169
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   170
lemma map_pmf_id[simp]: "map_pmf id = id"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   171
  by (rule, transfer) (auto simp: emeasure_distr measurable_def intro!: measure_eqI)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   172
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   173
lemma map_pmf_compose: "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   174
  by (rule, transfer) (simp add: distr_distr[symmetric, where N="count_space UNIV"] measurable_def) 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   175
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   176
lemma map_pmf_cong:
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   177
  assumes "p = q"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   178
  shows "(\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g q"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   179
  unfolding `p = q`[symmetric] measure_pmf_inject[symmetric] map_pmf.rep_eq
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   180
  by (auto simp add: emeasure_distr AE_measure_pmf_iff intro!: emeasure_eq_AE measure_eqI)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   181
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   182
lemma pmf_set_map: 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   183
  fixes f :: "'a \<Rightarrow> 'b"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   184
  shows "set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   185
proof (rule, transfer, clarsimp simp add: measure_distr measurable_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   186
  fix f :: "'a \<Rightarrow> 'b" and M :: "'a measure"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   187
  assume "prob_space M" and ae: "AE x in M. measure M {x} \<noteq> 0" and [simp]: "sets M = UNIV"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   188
  interpret prob_space M by fact
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   189
  show "{x. measure M (f -` {x}) \<noteq> 0} = f ` {x. measure M {x} \<noteq> 0}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   190
  proof safe
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   191
    fix x assume "measure M (f -` {x}) \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   192
    moreover have "measure M (f -` {x}) = measure M {y. f y = x \<and> measure M {y} \<noteq> 0}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   193
      using ae by (intro finite_measure_eq_AE) auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   194
    ultimately have "{y. f y = x \<and> measure M {y} \<noteq> 0} \<noteq> {}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   195
      by (metis measure_empty)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   196
    then show "x \<in> f ` {x. measure M {x} \<noteq> 0}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   197
      by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   198
  next
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   199
    fix x assume "measure M {x} \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   200
    then have "0 < measure M {x}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   201
      using measure_nonneg[of M "{x}"] by auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   202
    also have "measure M {x} \<le> measure M (f -` {f x})"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   203
      by (intro finite_measure_mono) auto
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   204
    finally show "measure M (f -` {f x}) = 0 \<Longrightarrow> False"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   205
      by simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   206
  qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   207
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   208
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   209
context
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   210
  fixes f :: "'a \<Rightarrow> real"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   211
  assumes nonneg: "\<And>x. 0 \<le> f x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   212
  assumes prob: "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   213
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   214
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   215
lift_definition embed_pmf :: "'a pmf" is "density (count_space UNIV) (ereal \<circ> f)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   216
proof (intro conjI)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   217
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   218
    by (simp split: split_indicator)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   219
  show "AE x in density (count_space UNIV) (ereal \<circ> f).
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   220
    measure (density (count_space UNIV) (ereal \<circ> f)) {x} \<noteq> 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   221
    by (simp add: AE_density nonneg emeasure_density measure_def nn_integral_cmult_indicator)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   222
  show "prob_space (density (count_space UNIV) (ereal \<circ> f))"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   223
    by default (simp add: emeasure_density prob)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   224
qed simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   225
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   226
lemma pmf_embed_pmf: "pmf embed_pmf x = f x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   227
proof transfer
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   228
  have *[simp]: "\<And>x y. ereal (f y) * indicator {x} y = ereal (f x) * indicator {x} y"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   229
    by (simp split: split_indicator)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   230
  fix x show "measure (density (count_space UNIV) (ereal \<circ> f)) {x} = f x"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   231
    by transfer (simp add: measure_def emeasure_density nn_integral_cmult_indicator nonneg)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   232
qed
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   233
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   234
end
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   235
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   236
lemma embed_pmf_transfer:
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   237
  "rel_fun (eq_onp (\<lambda>f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1)) pmf_as_measure.cr_pmf (\<lambda>f. density (count_space UNIV) (ereal \<circ> f)) embed_pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   238
  by (auto simp: rel_fun_def eq_onp_def embed_pmf.transfer)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   239
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   240
lemma td_pmf_embed_pmf:
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   241
  "type_definition pmf embed_pmf {f::'a \<Rightarrow> real. (\<forall>x. 0 \<le> f x) \<and> (\<integral>\<^sup>+x. ereal (f x) \<partial>count_space UNIV) = 1}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   242
  unfolding type_definition_def
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   243
proof safe
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   244
  fix p :: "'a pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   245
  have "(\<integral>\<^sup>+ x. 1 \<partial>measure_pmf p) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   246
    using measure_pmf.emeasure_space_1[of p] by simp
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   247
  then show *: "(\<integral>\<^sup>+ x. ereal (pmf p x) \<partial>count_space UNIV) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   248
    by (simp add: measure_pmf_eq_density nn_integral_density pmf_nonneg del: nn_integral_const)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   249
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   250
  show "embed_pmf (pmf p) = p"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   251
    by (intro measure_pmf_inject[THEN iffD1])
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   252
       (simp add: * embed_pmf.rep_eq pmf_nonneg measure_pmf_eq_density[of p] comp_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   253
next
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   254
  fix f :: "'a \<Rightarrow> real" assume "\<forall>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>count_space UNIV) = 1"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   255
  then show "pmf (embed_pmf f) = f"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   256
    by (auto intro!: pmf_embed_pmf)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   257
qed (rule pmf_nonneg)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   258
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   259
end
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   260
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   261
locale pmf_as_function
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   262
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   263
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   264
setup_lifting td_pmf_embed_pmf
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   265
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   266
end 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   267
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   268
(*
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   269
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   270
definition
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   271
  "rel_pmf P d1 d2 \<longleftrightarrow> (\<exists>p3. (\<forall>(x, y) \<in> set_pmf p3. P x y) \<and> map_pmf fst p3 = d1 \<and> map_pmf snd p3 = d2)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   272
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   273
lift_definition pmf_join :: "real \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf \<Rightarrow> 'a pmf" is
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   274
  "\<lambda>p M1 M2. density (count_space UNIV) (\<lambda>x. p * measure M1 {x} + (1 - p) * measure M2 {x})"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   275
sorry
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   276
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   277
lift_definition pmf_single :: "'a \<Rightarrow> 'a pmf" is
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   278
  "\<lambda>x. uniform_measure (count_space UNIV) {x}"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   279
sorry
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   280
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   281
bnf pmf: "'a pmf" map: map_pmf sets: set_pmf bd : "natLeq" rel: pmf_rel
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   282
proof -
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   283
  show "map_pmf id = id" by (rule map_pmf_id)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   284
  show "\<And>f g. map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g" by (rule map_pmf_compose) 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   285
  show "\<And>f g::'a \<Rightarrow> 'b. \<And>p. (\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x) \<Longrightarrow> map_pmf f p = map_pmf g p"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   286
    by (intro map_pmg_cong refl)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   287
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   288
  show "\<And>f::'a \<Rightarrow> 'b. set_pmf \<circ> map_pmf f = op ` f \<circ> set_pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   289
    by (rule pmf_set_map)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   290
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   291
  { fix p :: "'s pmf"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   292
    have "(card_of (set_pmf p), card_of (UNIV :: nat set)) \<in> ordLeq"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   293
      by (rule card_of_ordLeqI[where f="to_nat_on (set_pmf p)"])
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   294
         (auto intro: countable_set_pmf inj_on_to_nat_on)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   295
    also have "(card_of (UNIV :: nat set), natLeq) \<in> ordLeq"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   296
      by (metis Field_natLeq card_of_least natLeq_Well_order)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   297
    finally show "(card_of (set_pmf p), natLeq) \<in> ordLeq" . }
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   298
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   299
  show "\<And>R. pmf_rel R =
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   300
         (BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   301
         BNF_Util.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   302
     by (auto simp add: fun_eq_iff pmf_rel_def BNF_Util.Grp_def OO_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   303
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   304
  { let ?f = "map_pmf fst" and ?s = "map_pmf snd"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   305
    fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and A assume "\<And>x y. (x, y) \<in> set_pmf A \<Longrightarrow> R x y"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   306
    fix S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" and B assume "\<And>y z. (y, z) \<in> set_pmf B \<Longrightarrow> S y z"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   307
    assume "?f B = ?s A"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   308
    have "\<exists>C. (\<forall>(x, z)\<in>set_pmf C. \<exists>y. R x y \<and> S y z) \<and> ?f C = ?f A \<and> ?s C = ?s B"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   309
      sorry }
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   310
oops
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   311
  then show "\<And>R::'a \<Rightarrow> 'b \<Rightarrow> bool. \<And>S::'b \<Rightarrow> 'c \<Rightarrow> bool. pmf_rel R OO pmf_rel S \<le> pmf_rel (R OO S)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   312
      by (auto simp add: subset_eq pmf_rel_def fun_eq_iff OO_def Ball_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   313
qed (fact natLeq_card_order natLeq_cinfinite)+
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   314
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   315
notepad
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   316
begin
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   317
  fix x y :: "nat \<Rightarrow> real"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   318
  def IJz \<equiv> "rec_nat ((0, 0), \<lambda>_. 0) (\<lambda>n ((I, J), z).
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   319
    let a = x I - (\<Sum>j<J. z (I, j)) ; b = y J - (\<Sum>i<I. z (i, J)) in
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   320
      ((if a \<le> b then I + 1 else I, if b \<le> a then J + 1 else J), z((I, J) := min a b)))"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   321
  def I == "fst \<circ> fst \<circ> IJz" def J == "snd \<circ> fst \<circ> IJz" def z == "snd \<circ> IJz"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   322
  let ?a = "\<lambda>n. x (I n) - (\<Sum>j<J n. z n (I n, j))" and ?b = "\<lambda>n. y (J n) - (\<Sum>i<I n. z n (i, J n))"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   323
  have IJz_0[simp]: "\<And>p. z 0 p = 0" "I 0 = 0" "J 0 = 0"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   324
    by (simp_all add: I_def J_def z_def IJz_def)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   325
  have z_Suc[simp]: "\<And>n. z (Suc n) = (z n)((I n, J n) := min (?a n) (?b n))"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   326
    by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   327
  have I_Suc[simp]: "\<And>n. I (Suc n) = (if ?a n \<le> ?b n then I n + 1 else I n)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   328
    by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   329
  have J_Suc[simp]: "\<And>n. J (Suc n) = (if ?b n \<le> ?a n then J n + 1 else J n)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   330
    by (simp add: z_def I_def J_def IJz_def Let_def split_beta)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   331
  
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   332
  { fix N have "\<And>p. z N p \<noteq> 0 \<Longrightarrow> \<exists>n<N. p = (I n, J n)"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   333
      by (induct N) (auto simp add: less_Suc_eq split: split_if_asm) }
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   334
  
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   335
  { fix i n assume "i < I n"
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   336
    then have "(\<Sum>j. z n (i, j)) = x i" 
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   337
    oops
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   338
*)
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   339
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   340
end
5484f6079bcd add type for probability mass functions, i.e. discrete probability distribution
hoelzl
parents:
diff changeset
   341