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Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei:   Sprache: Lisp

Untersuchung PVS©

(finite_measure
 (fm_emptyset 0
  (fm_emptyset-1 nil 3321844720
   ("" (typepred "mu")
    (("" (expand "finite_measure?") (("" (flatten) nil nil)) nil)) nil)
   ((sigma_algebra_IUnion_rew application-judgement "(S)"
     finite_measure 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-type-decl nil finite_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 nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "sigma_algebra" finite_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)
    (finite_measure? const-decl "bool" generalized_measure_def nil)
    (finite_measure nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (mu formal-const-decl "finite_measure" finite_measure nil))
   shostak))
 (fm_convergence 0
  (fm_convergence-1 nil 3321844756
   ("" (skosimp)
    (("" (typepred "mu")
      (("" (expand "finite_measure?")
        (("" (flatten)
          (("" (inst - "X!1")
            (("" (expand "disjoint_indexed_measurable?")
              (("" (propax) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((mu formal-const-decl "finite_measure" finite_measure nil)
    (finite_measure nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (finite_measure? const-decl "bool" generalized_measure_def 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" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil finite_measure nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (X!1 skolem-const-decl "[nat -> (S)]" finite_measure nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def 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)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     finite_measure nil))
   shostak))
 (fm_disjointunion 0
  (fm_disjointunion-1 nil 3321844999
   ("" (skosimp)
    ((""
      (name "AA"
            "lambda (n:nat): IF n=0 THEN A!1 ELSIF n=1 THEN B!1 ELSE emptyset[T] ENDIF")
      (("" (lemma "fm_convergence" ("X" "AA"))
        (("" (split -1)
          (("1" (hide -2)
            (("1" (lemma "zero_tail_series" ("a" "mu o AA" "n" "1"))
              (("1" (split)
                (("1" (rewrite "limit_equiv")
                  (("1" (rewrite "limit_equiv")
                    (("1" (flatten)
                      (("1" (replace -2)
                        (("1" (hide-all-but (-4 1))
                          (("1" (expand "o ")
                            (("1" (expand "series")
                              (("1"
                                (case-replace
                                 "IUnion(AA)=union(A!1, B!1)")
                                (("1"
                                  (expand "sigma")
                                  (("1"
                                    (expand "sigma")
                                    (("1"
                                      (expand "sigma")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (expand "AA")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide-all-but 1)
                                  (("2"
                                    (apply-extensionality :hide? t)
                                    (("2"
                                      (expand "union")
                                      (("2"
                                        (expand "IUnion")
                                        (("2"
                                          (expand "AA")
                                          (("2"
                                            (expand "member")
                                            (("2"
                                              (case-replace "A!1(x!1)")
                                              (("1"
                                                (inst + "0")
                                                nil
                                                nil)
                                               ("2"
                                                (case-replace
                                                 "B!1(x!1)")
                                                (("1"
                                                  (inst + "1")
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (expand
                                                       "emptyset")
                                                      (("2"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide-all-but 1)
                  (("2" (skosimp)
                    (("2" (expand "AA")
                      (("2" (expand "o")
                        (("2" (typepred "mu")
                          (("2" (expand "finite_measure?")
                            (("2" (flatten)
                              (("2"
                                (replace -1)
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide -1 2)
            (("2" (expand "AA")
              (("2" (expand "disjoint?")
                (("2" (skosimp)
                  (("2" (case-replace "i!1=0")
                    (("1" (assert)
                      (("1" (case-replace "j!1=1")
                        (("1" (expand "disjoint?")
                          (("1" (propax) nil nil)) nil)
                         ("2" (expand "disjoint?")
                          (("2" (assert)
                            (("2" (expand "intersection")
                              (("2"
                                (expand "empty?")
                                (("2"
                                  (expand "member")
                                  (("2"
                                    (skosimp)
                                    (("2"
                                      (expand "emptyset")
                                      (("2" (propax) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (case-replace "i!1=1")
                      (("1" (assert)
                        (("1" (case-replace "j!1=0")
                          (("1" (expand "disjoint?")
                            (("1" (expand "intersection")
                              (("1"
                                (expand "empty?")
                                (("1"
                                  (expand "member")
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (inst - "x!1")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (assert)
                            (("2" (expand "disjoint?")
                              (("2"
                                (expand "intersection")
                                (("2"
                                  (expand "empty?")
                                  (("2"
                                    (expand "member")
                                    (("2"
                                      (expand "emptyset")
                                      (("2" (propax) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (assert)
                        (("2" (expand "disjoint?")
                          (("2" (expand "emptyset")
                            (("2" (expand "intersection")
                              (("2"
                                (expand "empty?")
                                (("2"
                                  (expand "member")
                                  (("2" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((emptyset const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil finite_measure nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (subset_algebra_emptyset name-judgement "(S)" finite_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)
    (zero_tail_series formula-decl nil series_aux "series/")
    (sequence type-eq-decl nil sequences nil)
    (nnreal type-eq-decl nil real_types nil)
    (O const-decl "T3" function_props nil)
    (finite_measure? const-decl "bool" generalized_measure_def nil)
    (finite_measure nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (mu formal-const-decl "finite_measure" finite_measure nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (series const-decl "sequence[real]" series "series/")
    (limit_equiv formula-decl nil convergence_ops "analysis/")
    (member const-decl "bool" sets nil)
    (sigma def-decl "real" sigma "reals/")
    (AA skolem-const-decl "[nat -> (S)]" finite_measure nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (subset_algebra_union application-judgement "(S)" finite_measure
     nil)
    (union const-decl "set" sets nil)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     finite_measure nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (finite_intersection2 application-judgement "finite_set[T]"
     countable_setofsets "sets_aux/")
    (countable_intersection1 application-judgement "countable_set[T]"
     countable_setofsets "sets_aux/")
    (subset_algebra_intersection application-judgement "(S)"
     finite_measure nil)
    (intersection const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil)
    (finite_intersection1 application-judgement "finite_set[T]"
     countable_setofsets "sets_aux/")
    (countable_intersection2 application-judgement "countable_set[T]"
     countable_setofsets "sets_aux/")
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (disjoint? const-decl "bool" sets nil)
    (fm_convergence formula-decl nil finite_measure nil))
   shostak))
 (fm_complement 0
  (fm_complement-1 nil 3321856987
   ("" (skosimp)
    (("" (lemma "fm_disjointunion" ("A" "A!1" "B" "complement(A!1)"))
      (("1" (split -1)
        (("1" (rewrite "union_complement") (("1" (assertnil nil))
          nil)
         ("2" (hide 2)
          (("2" (expand "complement")
            (("2" (expand "disjoint?")
              (("2" (expand "intersection")
                (("2" (expand "empty?")
                  (("2" (expand "member") (("2" (skosimp*) nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (lemma "sigma_algebra_complement" ("x" "A!1"))
        (("2" (expand "member") (("2" (propax) nil nil)) nil)) nil))
      nil))
    nil)
   ((complement const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (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-type-decl nil finite_measure nil)
    (fm_disjointunion formula-decl nil finite_measure nil)
    (subset_algebra_complement application-judgement "(S)"
     finite_measure nil)
    (disjoint? const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (union_complement formula-decl nil sets_lemmas_aux nil)
    (subset_algebra_fullset name-judgement "(S)" finite_measure nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   shostak))
 (fm_union 0
  (fm_union-1 nil 3321853771
   ("" (skosimp)
    (("" (rewrite "union_difference")
      (("" (lemma "difference_disjoint" ("a" "A!1" "b" "B!1"))
        ((""
          (lemma "fm_disjointunion"
           ("A" "A!1" "B" "difference(B!1, A!1)"))
          (("1" (assert)
            (("1" (replace -1)
              (("1" (hide -1)
                (("1" (assert)
                  (("1" (rewrite "intersection_commutative")
                    (("1"
                      (case "B!1 = union(difference(B!1, A!1),intersection(B!1, A!1))")
                      (("1"
                        (case "disjoint?(difference(B!1, A!1), intersection(B!1, A!1))")
                        (("1"
                          (lemma "fm_disjointunion"
                           ("A" "difference(B!1, A!1)" "B"
                            "intersection(B!1, A!1)"))
                          (("1" (assertnil nil)
                           ("2"
                            (lemma "sigma_algebra_intersection"
                             ("x" "B!1" "y" "A!1"))
                            (("2" (expand "member")
                              (("2" (propax) nil nil)) nil))
                            nil))
                          nil)
                         ("2" (hide -1 -2 2)
                          (("2" (expand "disjoint?")
                            (("2" (expand "intersection")
                              (("2"
                                (expand "difference")
                                (("2"
                                  (expand "empty?")
                                  (("2"
                                    (expand "member")
                                    (("2" (skosimp*) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (hide-all-but 1)
                        (("2" (apply-extensionality :hide? t)
                          (("2" (expand "intersection")
                            (("2" (expand "difference")
                              (("2"
                                (expand "union")
                                (("2"
                                  (expand "member")
                                  (("2"
                                    (case-replace "B!1(x!1)")
                                    (("1"
                                      (assert)
                                      (("1" (flatten) nil nil))
                                      nil)
                                     ("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide-all-but 1)
            (("2"
              (lemma "sigma_algebra_difference" ("x" "B!1" "y" "A!1"))
              (("2" (expand "member") (("2" (propax) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((union_difference formula-decl nil sets_lemmas nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "sigma_algebra" finite_measure nil)
    (T formal-type-decl nil finite_measure nil)
    (subset_algebra_difference application-judgement "(S)"
     finite_measure nil)
    (subset_algebra_union application-judgement "(S)" finite_measure
     nil)
    (fm_disjointunion formula-decl nil finite_measure nil)
    (difference const-decl "set" sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (union const-decl "set" sets nil)
    (intersection const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (disjoint? const-decl "bool" sets nil)
    (intersection_commutative formula-decl nil sets_lemmas 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)"
     finite_measure nil)
    (difference_disjoint formula-decl nil sets_lemmas nil))
   shostak))
 (fm_intersection 0
  (fm_intersection-1 nil 3321854440
   ("" (skosimp)
    (("" (lemma "fm_union" ("A" "A!1" "B" "B!1"))
      (("" (assertnil nil)) nil))
    nil)
   ((S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (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-type-decl nil finite_measure nil)
    (fm_union formula-decl nil finite_measure nil)
    (subset_algebra_union application-judgement "(S)" finite_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)"
     finite_measure nil))
   shostak))
 (fm_difference 0
  (fm_difference-1 nil 3321854478
   ("" (skosimp)
    (("" (lemma "disjoint_diff_inter" ("a" "A!1" "b" "B!1"))
      (("" (lemma "disjoint_diff_inter" ("a" "B!1" "b" "A!1"))
        ((""
          (lemma "fm_disjointunion"
           ("A" "difference(A!1, B!1)" "B" "intersection(A!1, B!1)"))
          (("1"
            (lemma "fm_disjointunion"
             ("A" "difference(B!1, A!1)" "B" "intersection(B!1, A!1)"))
            (("1" (assert)
              (("1" (rewrite "union_diff_inter")
                (("1" (rewrite "union_diff_inter")
                  (("1" (rewrite "intersection_commutative")
                    (("1" (assertnil nil)) nil))
                  nil))
                nil))
              nil)
             ("2"
              (lemma "sigma_algebra_intersection"
               ("x" "B!1" "y" "A!1"))
              (("2" (expand "member") (("2" (propax) nil nil)) nil))
              nil)
             ("3"
              (lemma "sigma_algebra_difference" ("x" "B!1" "y" "A!1"))
              (("3" (expand "member") (("3" (propax) nil nil)) nil))
              nil))
            nil)
           ("2"
            (lemma "sigma_algebra_intersection" ("y" "B!1" "x" "A!1"))
            (("2" (expand "member") (("2" (propax) nil nil)) nil)) nil)
           ("3"
            (lemma "sigma_algebra_difference" ("y" "B!1" "x" "A!1"))
            (("3" (expand "member") (("3" (propax) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil finite_measure nil)
    (S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (disjoint_diff_inter formula-decl nil sets_lemmas_aux nil)
    (intersection const-decl "set" sets nil)
    (difference const-decl "set" sets nil)
    (fm_disjointunion formula-decl nil finite_measure nil)
    (subset_algebra_intersection application-judgement "(S)"
     finite_measure nil)
    (subset_algebra_difference application-judgement "(S)"
     finite_measure 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)
    (intersection_commutative formula-decl nil sets_lemmas nil)
    (union_diff_inter formula-decl nil sets_lemmas_aux nil))
   shostak))
 (fm_subset 0
  (fm_subset-1 nil 3321855702
   ("" (skosimp)
    (("" (lemma "fm_difference" ("A" "A!1" "B" "B!1"))
      (("" (lemma "difference_subset2" ("a" "A!1" "b" "B!1"))
        (("" (assert)
          (("" (replace -1)
            (("" (rewrite "fm_emptyset") (("" (assertnil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (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-type-decl nil finite_measure nil)
    (fm_difference formula-decl nil finite_measure nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas_aux nil)
    (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)"
     finite_measure nil)
    (fm_emptyset formula-decl nil finite_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/")
    (subset_algebra_emptyset name-judgement "(S)" finite_measure nil)
    (difference_subset2 formula-decl nil sets_lemmas nil)
    (set type-eq-decl nil sets nil))
   shostak))
 (fm_subset_le 0
  (fm_subset_le-1 nil 3321855786
   ("" (skosimp)
    (("" (lemma "fm_subset" ("A" "A!1" "B" "B!1"))
      (("" (assert)
        (("" (typepred "mu(difference(B!1, A!1))")
          (("1" (assertnil nil)
           ("2"
            (lemma "sigma_algebra_difference" ("x" "B!1" "y" "A!1"))
            (("2" (expand "member") (("2" (propax) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (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-type-decl nil finite_measure nil)
    (fm_subset formula-decl nil finite_measure nil)
    (difference const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (mu formal-const-decl "finite_measure" finite_measure nil)
    (finite_measure nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (finite_measure? const-decl "bool" generalized_measure_def 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)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (subset_algebra_difference application-judgement "(S)"
     finite_measure 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 nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (fm_monotone 0
  (fm_monotone-1 nil 3455335004
   ("" (skosimp)
    (("" (lemma "fm_subset_le" ("A" "A!1" "B" "B!1"))
      (("" (assertnil nil)) nil))
    nil)
   ((S formal-const-decl "sigma_algebra" finite_measure nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_algebra? const-decl "bool" subset_algebra_def nil)
    (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-type-decl nil finite_measure nil)
    (fm_subset_le formula-decl nil finite_measure 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 nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (fm_IUnion 0
  (fm_IUnion-1 nil 3321857494
   ("" (skosimp)
    ((""
      (name "DS"
            "LAMBDA (i:nat): IF i = 0 THEN X!1(i) ELSE difference(X!1(i),X!1(i-1)) ENDIF")
      (("1" (case "disjoint?(DS)")
        (("1" (case-replace "IUnion(X!1) = IUnion(DS)")
          (("1" (lemma "fm_convergence" ("X" "DS"))
            (("1" (assert)
              (("1" (expand "o")
                (("1"
                  (case-replace
                   "series(LAMBDA (x: nat): mu(DS(x))) = LAMBDA (x: nat): mu(X!1(x))")
                  (("1" (hide-all-but (-3 -5 1))
                    (("1" (expand "series")
                      (("1"
                        (rewrite "extensionality_postulate" 1 :dir rl)
                        (("1" (induct "x_1")
                          (("1" (expand "sigma")
                            (("1" (expand "DS")
                              (("1"
                                (expand "sigma")
                                (("1" (propax) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (skosimp)
                            (("2" (expand "sigma" 1)
                              (("2"
                                (replace -1 1)
                                (("2"
                                  (expand "DS")
                                  (("2"
                                    (rewrite "fm_difference")
                                    (("2"
                                      (expand "increasing?")
                                      (("2"
                                        (inst - "j!1" "1+j!1")
                                        (("2"
                                          (assert)
                                          (("2"
                                            (lemma
                                             "difference_subset2"
                                             ("a"
                                              "X!1(j!1)"
                                              "b"
                                              "X!1(1 + j!1)"))
                                            (("2"
                                              (assert)
                                              (("2"
                                                (replace -1)
                                                (("2"
                                                  (rewrite
                                                   "fm_emptyset")
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide -2 2)
            (("2" (apply-extensionality :hide? t)
              (("2" (expand "IUnion")
                (("2" (expand "DS")
                  (("2" (hide -1)
                    (("2" (case-replace "EXISTS (i: nat): X!1(i)(x!1)")
                      (("1" (skosimp)
                        (("1"
                          (case "forall (n:nat): X!1(n) = IUnion(lambda (i:nat): IF i <= n THEN DS(i) ELSE emptyset[T] ENDIF)")
                          (("1" (inst - "i!1")
                            (("1" (replace -1 -2)
                              (("1"
                                (hide -1)
                                (("1"
                                  (expand "IUnion")
                                  (("1"
                                    (skosimp*)
                                    (("1"
                                      (expand "emptyset")
                                      (("1"
                                        (case-replace "i!2 <= i!1")
                                        (("1"
                                          (expand "DS")
                                          (("1"
                                            (inst + "i!2")
                                            nil
                                            nil))
                                          nil)
                                         ("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide -1 2)
                            (("2" (induct "n")
                              (("1"
                                (expand "IUnion")
                                (("1"
                                  (apply-extensionality :hide? t)
                                  (("1"
                                    (case-replace "X!1(0)(x!2)")
                                    (("1"
                                      (inst + "0")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (expand "DS")
                                          (("1" (propax) nil nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (assert)
                                      (("2"
                                        (expand "emptyset")
                                        (("2"
                                          (skosimp*)
                                          (("2"
                                            (assert)
                                            (("2"
                                              (expand "<=")
                                              (("2"
                                                (case-replace "i!2=0")
                                                (("1"
                                                  (expand "DS")
                                                  (("1"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (skosimp*)
                                (("2"
                                  (apply-extensionality :hide? t)
                                  (("2"
                                    (expand "IUnion")
                                    (("2"
                                      (expand "emptyset")
                                      (("2"
                                        (expand "increasing?")
                                        (("2"
                                          (inst - "j!1" "j!1+1")
                                          (("2"
                                            (rewrite
                                             "extensionality_postulate"
                                             -1
                                             :dir
                                             rl)
                                            (("2"
                                              (inst - "x!2")
                                              (("2"
                                                (case-replace
                                                 "X!1(j!1)(x!2)")
                                                (("1"
                                                  (expand "subset?")
                                                  (("1"
                                                    (inst - "x!2")
                                                    (("1"
                                                      (expand "member")
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (assert)
                                                          (("1"
                                                            (skosimp*)
                                                            (("1"
                                                              (inst
                                                               +
                                                               "i!2")
                                                              (("1"
                                                                (case-replace
                                                                 "i!2 <= j!1")
                                                                (("1"
                                                                  (assert)
                                                                  nil
                                                                  nil)
                                                                 ("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  (("2"
                                                    (case-replace
                                                     "X!1(1 + j!1)(x!2)")
                                                    (("1"
                                                      (inst + "1+j!1")
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (expand "DS")
                                                          (("1"
                                                            (expand
                                                             "difference")
                                                            (("1"
                                                              (expand
                                                               "member")
                                                              (("1"
                                                                (propax)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (replace 1 3)
                                                      (("2"
                                                        (assert)
                                                        (("2"
                                                          (skosimp)
                                                          (("2"
                                                            (case-replace
                                                             "i!2 <= 1 + j!1")
                                                            (("1"
                                                              (expand
                                                               "<="
                                                               -1)
                                                              (("1"
                                                                (split
                                                                 -1)
                                                                (("1"
                                                                  (inst
                                                                   +
                                                                   "i!2")
                                                                  (("1"
                                                                    (assert)
                                                                    nil
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (replace
                                                                   -1)
                                                                  (("2"
                                                                    (expand
                                                                     "DS")
                                                                    (("2"
                                                                      (expand
                                                                       "difference")
                                                                      (("2"
                                                                        (expand
                                                                         "member")
                                                                        (("2"
                                                                          (propax)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (replace 1 2)
                        (("2" (assert)
                          (("2" (skosimp*)
                            (("2" (inst + "i!1")
                              (("2"
                                (case-replace "i!1=0")
                                (("2"
                                  (assert)
                                  (("2"
                                    (expand "difference")
                                    (("2"
                                      (expand "member")
                                      (("2" (propax) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (hide -1 2)
          (("2" (expand "disjoint?")
            (("2" (skosimp*)
              (("2" (expand "increasing?")
                (("2"
                  (case "forall (i,j:nat): i < j => disjoint?(DS(i), DS(j))")
                  (("1" (lemma "trich_lt" ("x" "i!1" "y" "j!1"))
                    (("1" (split -1)
                      (("1" (inst - "i!1" "j!1")
                        (("1" (assertnil nil)) nil)
                       ("2" (propax) nil nil)
                       ("3" (inst - "j!1" "i!1")
                        (("3" (assert)
                          (("3" (rewrite "disjoint_commutative"nil
                            nil))
                          nil))
                        nil))
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    (set type-eq-decl nil sets nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
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    (setofsets type-eq-decl nil sets nil)
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     finite_measure nil)
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    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
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    (subset_algebra_intersection application-judgement "(S)"
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    (IUnion const-decl "set[T]" indexed_sets nil)
    (sigma_algebra_IUnion_rew application-judgement "(S)"
     finite_measure nil)
    (mu formal-const-decl "finite_measure" finite_measure nil)
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    (nat_induction formula-decl nil naturalnumbers nil)
    (DS skolem-const-decl "[nat -> (S)]" finite_measure nil)
    (increasing? const-decl "bool" fun_preds_partial "structures/")
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas_aux nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (fm_emptyset formula-decl nil finite_measure nil)
    (difference_subset2 formula-decl nil sets_lemmas nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (fm_difference formula-decl nil finite_measure nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (extensionality_postulate formula-decl nil functions nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (sigma def-decl "real" sigma "reals/")
    (O const-decl "T3" function_props nil)
    (fm_convergence formula-decl nil finite_measure nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (emptyset const-decl "set" sets nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (FALSE const-decl "bool" booleans nil)
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
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   shostak))
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--> --------------------

--> maximum size reached

--> --------------------

¤ Dauer der Verarbeitung: 0.77 Sekunden  (vorverarbeitet)  ¤





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