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Datei: probability_measure.prf   Sprache: Lisp

Original von: PVS©

(probability_measure
 (trivial_probability_measure_TCC1 0
  (trivial_probability_measure_TCC1-1 nil 3424438960
   ("" (skosimp)
    (("" (expand "nonempty?")
      (("" (expand "empty?")
        (("" (expand "member")
          (("" (expand "fullset") (("" (propax) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((nonempty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (fullset const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil))
   nil))
 (probability_measure_TCC1 0
  (probability_measure_TCC1-1 nil 3424438960
   ("" (expand "probability_measure?")
    (("" (expand "finite_measure?")
      (("" (split)
        (("1" (expand "probability_measure_full?")
          (("1" (expand "trivial_probability_measure")
            (("1" (lift-if) (("1" (assertnil nil)) nil)) nil))
          nil)
         ("2" (expand "trivial_probability_measure")
          (("2" (expand "emptyset")
            (("2" (expand "member") (("2" (propax) nil nil)) nil))
            nil))
          nil)
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                              (("1"
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                                (("1"
                                  (lemma
                                   "single_nz_series"
                                   ("a"
                                    "LAMBDA (x: nat):
                           IF X!1(x)(CC) THEN 1 ELSE 0 ENDIF"
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                     ("2" (replace 1 2)
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                           IF X!1(x)(choose(fullset[T])) THEN 1
                           ELSE 0
                           ENDIF)=(lambda (x:nat): 0)")
                        (("1" (hide-all-but 2)
                          (("1" (expand "convergence")
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    nil)
   ((sigma_algebra_IUnion_rew application-judgement "(S)"
     probability_measure nil)
    (finite_measure? const-decl "bool" generalized_measure_def
     "measure_integration/")
    (O const-decl "T3" function_props nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def "measure_integration/")
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def "measure_integration/")
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
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    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
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    (rational_pred const-decl "[real -> boolean]" rationals nil)
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    (number_field_pred const-decl "[number -> boolean]" number_fields
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    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (sigma_zero formula-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
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    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (series const-decl "sequence[real]" series "series/")
    (sigma_nat application-judgement "nat" sigma_nat "reals/")
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (single_nz_series formula-decl nil series_aux "series/")
    (sequence type-eq-decl nil sequences nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
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    (intersection const-decl "set" sets nil)
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    (fullset const-decl "set" sets nil)
    (emptyset const-decl "set" sets nil)
    (probability_measure_full? const-decl "bool" probability_measure
     nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     probability_measure nil)
    (subset_algebra_fullset name-judgement "(S)" probability_measure
     nil)
    (member const-decl "bool" sets nil)
    (trivial_probability_measure const-decl "probability"
     probability_measure nil)
    (probability_measure? const-decl "bool" probability_measure nil))
   nil))
 (probability_measure_is_finite_measure 0
  (probability_measure_is_finite_measure-1 nil 3424499703
   ("" (judgement-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
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    (nnreal type-eq-decl nil real_types nil)
    (<= const-decl "bool" reals nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure_full? const-decl "bool" probability_measure
     nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     probability_measure nil)
    (subset_algebra_fullset name-judgement "(S)" probability_measure
     nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (measurable_emptyset name-judgement "measurable_set[T, S]"
     probability_measure nil)
    (subset_algebra_emptyset name-judgement "(S)" probability_measure
     nil)
    (series const-decl "sequence[real]" series "series/")
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (finite_measure? const-decl "bool" generalized_measure_def
     "measure_integration/")
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     probability_measure nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/"))
   nil))
 (random_variable_TCC1 0
  (random_variable_TCC1-1 nil 3424499077
   ("" (lemma "constant_is_measurable")
    (("" (inst - "(LAMBDA (x: T): 0)")
      (("" (hide 2)
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    (constant? const-decl "bool" const_fun_def "structures/")
    (const_fun const-decl "[S -> T]" const_fun_def "structures/")
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    (T formal-nonempty-type-decl nil probability_measure nil)
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    (bool nonempty-type-eq-decl nil booleans nil)
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    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
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     "measure_integration/")
    (S formal-const-decl "sigma_algebra" probability_measure nil))
   nil))
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        nil))
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   ((probability_measure nonempty-type-eq-decl nil probability_measure
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    (<= const-decl "bool" reals nil)
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     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
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    (T formal-nonempty-type-decl nil probability_measure nil)
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    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
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     nil))
   shostak))
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  (P_convergence-1 nil 3424499739
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                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
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    (>= const-decl "bool" reals nil)
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     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
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    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     probability_measure nil)
    (finite_measure? const-decl "bool" generalized_measure_def
     "measure_integration/")
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def "measure_integration/")
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def "measure_integration/")
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (subset_algebra_emptyset name-judgement "(S)" probability_measure
     nil)
    (measurable_emptyset name-judgement "measurable_set[T, S]"
     probability_measure nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (O const-decl "T3" function_props nil))
   shostak))
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    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     probability_measure nil)
    (finite_measure? const-decl "bool" generalized_measure_def
     "measure_integration/"))
   shostak))
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  (P_disjointunion-1 nil 3424499882
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          (("" (flatten)
            ((""
              (lemma "fm_disjointunion[T,S,P!1]" ("A" "A!1" "B" "B!1"))
              (("1" (assertnil nil) ("2" (propax) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (subset_algebra_union application-judgement "(S)"
     probability_measure nil)
    (fm_disjointunion formula-decl nil finite_measure
     "measure_integration/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (<= const-decl "bool" reals nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability_measure nonempty-type-eq-decl nil probability_measure
     nil))
   shostak))
 (P_complement 0
  (P_complement-1 nil 3424583049
   ("" (skosimp)
    ((""
      (lemma "P_disjointunion"
       ("P" "P!1" "A" "A!1" "B" "complement(A!1)"))
      (("" (case-replace "union(A!1, complement(A!1))=fullset[T]")
        (("1" (rewrite "P_fullset")
          (("1" (split -2)
            (("1" (assertnil nil)
             ("2" (hide-all-but 1) (("2" (grind) nil nil)) nil))
            nil))
          nil)
         ("2" (hide-all-but 1)
          (("2" (apply-extensionality :hide? t) (("2" (grind) nil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (complement const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_disjointunion formula-decl nil probability_measure nil)
    (subset_algebra_complement application-judgement "(S)"
     probability_measure nil)
    (P_fullset formula-decl nil probability_measure nil)
    (disjoint? const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (member const-decl "bool" sets nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (subset_algebra_fullset name-judgement "(S)" probability_measure
     nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     probability_measure nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (subset_algebra_union application-judgement "(S)"
     probability_measure nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (union const-decl "set" sets nil)
    (fullset const-decl "set" sets nil))
   shostak))
 (P_union 0
  (P_union-1 nil 3424956361
   ("" (skosimp)
    ((""
      (lemma "P_disjointunion"
       ("A" "A!1" "B" "difference(B!1,A!1)" "P" "P!1"))
      ((""
        (lemma "P_disjointunion"
         ("A" "B!1" "B" "difference(A!1,B!1)" "P" "P!1"))
        (("" (rewrite "difference_disjoint")
          (("" (rewrite "difference_disjoint")
            ((""
              (case-replace
               "union(B!1, difference(A!1, B!1))=union(A!1,B!1)")
              (("1" (hide -1)
                (("1" (replace -1)
                  (("1" (assert)
                    (("1" (hide -1)
                      (("1" (hide -1)
                        (("1" (rewrite "difference_intersection")
                          (("1"
                            (lemma "P_disjointunion"
                             ("P" "P!1" "A"
                              "intersection(A!1, complement(B!1))" "B"
                              "intersection(A!1, B!1)"))
                            (("1"
                              (case-replace
                               "union(intersection(A!1, complement(B!1)),
                 intersection(A!1, B!1))=A!1")
                              (("1"
                                (split -2)
                                (("1" (assertnil nil)
                                 ("2"
                                  (hide-all-but 1)
                                  (("2" (grind) nil nil))
                                  nil))
                                nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (apply-extensionality :hide? t)
                                  (("2" (grind) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (apply-extensionality :hide? t)
                  (("2" (grind) nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (difference const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_disjointunion formula-decl nil probability_measure nil)
    (subset_algebra_difference application-judgement "(S)"
     probability_measure nil)
    (difference_disjoint formula-decl nil sets_lemmas nil)
    (union const-decl "set" sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (subset_algebra_union application-judgement "(S)"
     probability_measure nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (difference_intersection formula-decl nil sets_lemmas nil)
    (subset_algebra_complement application-judgement "(S)"
     probability_measure nil)
    (disjoint? const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (complement const-decl "set" sets nil)
    (intersection const-decl "set" sets nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil))
   shostak))
 (P_intersection 0
  (P_intersection-1 nil 3424693483
   ("" (skosimp)
    (("" (lemma "P_union" ("A" "A!1" "B" "B!1" "P" "P!1"))
      (("" (assertnil nil)) nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_union formula-decl nil probability_measure nil)
    (subset_algebra_union application-judgement "(S)"
     probability_measure nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil))
   shostak))
 (P_difference 0
  (P_difference-1 nil 3424693520
   ("" (skosimp)
    (("" (lemma "fm_difference[T,S,P!1]" ("A" "A!1" "B" "B!1"))
      (("1" (propax) nil nil) ("2" (postpone) nil nil)) nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (fm_difference formula-decl nil finite_measure
     "measure_integration/"))
   shostak))
 (P_subset 0
  (P_subset-1 nil 3424957252
   ("" (skosimp)
    (("" (lemma "P_difference" ("P" "P!1" "A" "A!1" "B" "B!1"))
      (("" (lemma "difference_subset2" ("a" "A!1" "b" "B!1"))
        (("" (assert)
          (("" (replace -1)
            (("" (rewrite "P_emptyset") (("" (assertnil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_difference formula-decl nil probability_measure nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas_aux "measure_integration/")
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (subset_algebra_difference application-judgement "(S)"
     probability_measure nil)
    (P_emptyset formula-decl nil probability_measure nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (measurable_emptyset name-judgement "measurable_set[T, S]"
     probability_measure nil)
    (subset_algebra_emptyset name-judgement "(S)" probability_measure
     nil)
    (set type-eq-decl nil sets nil)
    (difference_subset2 formula-decl nil sets_lemmas nil))
   shostak))
 (P_subset_le 0
  (P_subset_le-1 nil 3424957345
   ("" (skosimp)
    (("" (lemma "P_subset" ("A" "A!1" "B" "B!1" "P" "P!1"))
      (("" (assert)
        (("" (typepred "P!1(difference(B!1, A!1))")
          (("" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_subset formula-decl nil probability_measure nil)
    (subset_algebra_difference application-judgement "(S)"
     probability_measure nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (difference const-decl "set" sets nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas_aux "measure_integration/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (P_IUnion 0
  (P_IUnion-1 nil 3424958483
   ("" (skosimp)
    (("" (lemma "fm_IUnion[T,S,P!1]" ("X" "X!1"))
      (("" (assert)
        (("" (expand "convergence")
          (("" (expand "o ") (("" (propax) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (fm_IUnion formula-decl nil finite_measure "measure_integration/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (O const-decl "T3" function_props nil)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     probability_measure nil))
   shostak))
 (P_IIntersection 0
  (P_IIntersection-1 nil 3424958571
   ("" (skosimp)
    (("" (lemma "fm_IIntersection[T,S,P!1]" ("X" "X!1"))
      (("" (expand "o ") (("" (assertnil nil)) nil)) nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (fm_IIntersection formula-decl nil finite_measure
     "measure_integration/")
    (sigma_algebra_IIntersection_rew application-judgement "(S)"
     probability_measure nil)
    (O const-decl "T3" function_props nil))
   shostak))
 (P0_intersection 0
  (P0_intersection-1 nil 3424957493
   ("" (skosimp)
    ((""
      (lemma "P_subset_le"
       ("P" "P!1" "A" "intersection(A!1,B!1)" "B" "A!1"))
      (("" (rewrite "intersection_subset1") (("" (assertnil nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (intersection const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_subset_le formula-decl nil probability_measure nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (intersection_subset1 formula-decl nil sets_lemmas nil))
   shostak))
 (P0_union 0
  (P0_union-1 nil 3424957599
   ("" (skosimp)
    (("" (rewrite "P_union")
      (("" (lemma "P0_intersection" ("P" "P!1" "A" "A!1" "B" "B!1"))
        (("" (assertnil nil)) nil))
      nil))
    nil)
   ((P_union formula-decl nil probability_measure nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (<= const-decl "bool" reals nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (P0_intersection formula-decl nil probability_measure nil))
   shostak))
 (P1_intersection 0
  (P1_intersection-1 nil 3424957828
   ("" (skosimp)
    ((""
      (lemma "P_disjointunion"
       ("P" "P!1" "A" "intersection(B!1, A!1)" "B"
        "intersection(B!1, complement(A!1))"))
      (("" (split -1)
        (("1"
          (case-replace "union(intersection(B!1, A!1),
                intersection(B!1, complement(A!1)))=B!1")
          (("1" (hide -1)
            (("1"
              (lemma "P0_intersection"
               ("P" "P!1" "B" "B!1" "A" "complement(A!1)"))
              (("1" (rewrite "P_complement")
                (("1" (replace -3)
                  (("1" (assert)
                    (("1" (rewrite "intersection_commutative")
                      (("1" (replace -1)
                        (("1" (hide -1 -3)
                          (("1" (rewrite "intersection_commutative")
                            (("1" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide-all-but 1)
            (("2" (apply-extensionality :hide? t)
              (("2" (grind) nil nil)) nil))
            nil))
          nil)
         ("2" (hide-all-but 1) (("2" (grind) nil nil)) nil))
        nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (complement const-decl "set" sets nil)
    (intersection const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P_disjointunion formula-decl nil probability_measure nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil)
    (subset_algebra_complement application-judgement "(S)"
     probability_measure nil)
    (disjoint? const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (union const-decl "set" sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (subset_algebra_union application-judgement "(S)"
     probability_measure nil)
    (P0_intersection formula-decl nil probability_measure nil)
    (intersection_commutative formula-decl nil sets_lemmas nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (odd_minus_odd_is_even application-judgement "even_int" integers
     nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (P_complement formula-decl nil probability_measure nil)
    (member const-decl "bool" sets nil))
   shostak))
 (P1_union 0
  (P1_union-1 nil 3424958254
   ("" (skosimp)
    (("" (lemma "P1_intersection" ("P" "P!1" "A" "A!1" "B" "B!1"))
      (("" (assert)
        (("" (rewrite "P_union") (("" (assertnil nil)) nil)) nil))
      nil))
    nil)
   ((probability_measure nonempty-type-eq-decl nil probability_measure
     nil)
    (probability_measure? const-decl "bool" probability_measure nil)
    (probability nonempty-type-eq-decl nil probability_measure nil)
    (<= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (P1_intersection formula-decl nil probability_measure nil)
    (P_union formula-decl nil probability_measure nil)
    (subset_algebra_intersection application-judgement "(S)"
     probability_measure nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (subset_algebra_union application-judgement "(S)"
     probability_measure nil))
   shostak))
 (measure_to_df_TCC1 0
  (measure_to_df_TCC1-1 nil 3425001461
   ("" (skosimp)
    (("" (typepred "Y!1")
      (("" (rewrite "measurable_le") (("" (inst - "x!1"nil nil))
        nil))
      nil))
    nil)
   ((random_variable nonempty-type-eq-decl nil probability_measure nil)
    (measurable_function? const-decl "bool" measure_space_def
     "measure_integration/")
    (S formal-const-decl "sigma_algebra" probability_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def
     "measure_integration/")
    (sigma_algebra? const-decl "bool" subset_algebra_def
     "measure_integration/")
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (T formal-nonempty-type-decl nil probability_measure nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (measurable_le formula-decl nil measure_space_def
     "measure_integration/"))
   nil)))


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