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Datei: sigma_finite_measure_props.prf   Sprache: Lisp

Original von: PVS©

(sigma_finite_measure_props
 (sfm_integrable_TCC1 0
  (sfm_integrable_TCC1-1 nil 3459603923
   ("" (expand "measurable_set?") (("" (propax) nil nil)) nil)
   ((measurable_set? const-decl "bool" measure_space_def nil)) nil))
 (sfm_integrable_TCC2 0
  (sfm_integrable_TCC2-1 nil 3459603923 ("" (skosimp*) nil nilnil
   nil))
 (sfm_integrable 0
  (sfm_integrable-1 nil 3459603934
   ("" (skosimp)
    ((""
      (lemma "contraction_integrable_def[T, S, mu]"
       ("h" "h!1" "A" "A_of(mu)"))
      (("" (rewrite "A_of_def1")
        (("" (typepred "h!1")
          (("" (expand "nn_measurable?")
            (("" (flatten)
              (("" (expand ">=" -3)
                (("" (replace -2)
                  (("" (replace -3 1 rl) (("" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((mu formal-const-decl "sigma_finite_measure"
     sigma_finite_measure_props nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (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" sigma_finite_measure_props
     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 sigma_finite_measure_props nil)
    (nn_measurable nonempty-type-eq-decl nil measure_space nil)
    (nn_measurable? const-decl "bool" measure_space nil)
    (measurable_function nonempty-type-eq-decl nil measure_space_def
     nil)
    (measurable_function? const-decl "bool" measure_space_def nil)
    (A_of const-decl "disjoint_indexed_measurable" measure_def 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)
    (contraction_integrable_def formula-decl nil
     measure_contraction_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     sigma_finite_measure_props nil)
    (subset_algebra_fullset name-judgement "(S)"
     sigma_finite_measure_props nil)
    (A_of_def1 formula-decl nil measure_def nil))
   shostak))
 (sfm_integral_TCC1 0
  (sfm_integral_TCC1-1 nil 3459603923
   ("" (skosimp)
    (("" (typepred "f!1")
      (("" (rewrite "contraction_integrable")
        (("1" (rewrite "indefinite_integrable[T, S, mu]")
          (("1" (expand "measurable_set?") (("1" (propax) nil nil))
            nil))
          nil)
         ("2" (expand "measurable_set?") (("2" (propax) nil nil)) nil))
        nil))
      nil))
    nil)
   ((nn_integrable nonempty-type-eq-decl nil nn_integral nil)
    (nn_integrable? const-decl "bool" nn_integral nil)
    (mu formal-const-decl "sigma_finite_measure"
     sigma_finite_measure_props nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (S formal-const-decl "sigma_algebra" sigma_finite_measure_props
     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)
    (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)
    (T formal-type-decl nil sigma_finite_measure_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (integrable nonempty-type-eq-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (indefinite_integrable formula-decl nil integral nil)
    (prod_measurable application-judgement "measurable_function[T, S]"
     sigma_finite_measure_props nil)
    (phi_is_simple application-judgement "simple"
     sigma_finite_measure_props nil)
    (measure_type nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (measurable_function nonempty-type-eq-decl nil measure_space_def
     nil)
    (measurable_function? const-decl "bool" measure_space_def nil)
    (A_of const-decl "disjoint_indexed_measurable" measure_def 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)
    (measurable_set nonempty-type-eq-decl nil measure_space_def nil)
    (measurable_set? const-decl "bool" measure_space_def nil)
    (set type-eq-decl nil sets nil)
    (contraction_integrable formula-decl nil measure_contraction nil))
   nil))
 (sfm_integral 0
  (sfm_integral-1 nil 3459604066
   ("" (skosimp)
    ((""
      (lemma "indefinite_countably_additive[T, S, mu]"
       ("f" "f!1" "DX" "A_of(mu)"))
      (("" (rewrite "A_of_def1" -1)
        (("" (rewrite "indefinite_fullset[T, S, mu]")
          ((""
            (case-replace "(LAMBDA n:
                            integral[T, S, contraction(mu, A_of(mu)(n))]
                                (f!1))=LAMBDA n: integral[T,S,mu](A_of(mu)(n), f!1)")
            (("1" (apply-extensionality :hide? t)
              (("1" (hide 2)
                (("1" (rewrite "contraction_integral")
                  (("1" (expand "integral" 1 2)
                    (("1" (propax) nil nil)) nil))
                  nil))
                nil)
               ("2" (hide-all-but 1)
                (("2" (typepred "f!1")
                  (("2" (skosimp)
                    (("2"
                      (lemma "nn_integrable_is_integrable[T, S, mu]")
                      (("2" (inst - "f!1")
                        (("2" (expand "integrable?" 1)
                          (("2"
                            (rewrite "indefinite_integrable[T, S, mu]")
                            (("2" (expand "measurable_set?")
                              (("2" (propax) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (expand "measurable_set?") (("3" (propax) nil nil))
                nil)
               ("4" (skosimp)
                (("4" (hide-all-but 1)
                  (("4" (typepred "f!1")
                    (("4"
                      (lemma "nn_integrable_is_integrable[T, S, mu]")
                      (("4" (inst - "f!1")
                        (("4" (rewrite "contraction_integrable")
                          (("1"
                            (rewrite "indefinite_integrable[T, S, mu]")
                            (("1" (expand "measurable_set?")
                              (("1" (propax) nil nil)) nil))
                            nil)
                           ("2" (expand "measurable_set?")
                            (("2" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but 1)
              (("2" (typepred "f!1")
                (("2" (skosimp)
                  (("2" (expand "integrable?")
                    (("2" (rewrite "indefinite_integrable[T, S, mu]")
                      (("2" (expand "measurable_set?")
                        (("2" (propax) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("3" (expand "measurable_set?") (("3" (propax) nil nil))
              nil)
             ("4" (hide-all-but 1)
              (("4" (typepred "f!1")
                (("4" (skosimp)
                  (("4" (lemma "nn_integrable_is_integrable[T, S, mu]")
                    (("4" (inst - "f!1")
                      (("4" (rewrite "contraction_integrable" 1)
                        (("1"
                          (rewrite "indefinite_integrable[T, S, mu]")
                          (("1" (expand "measurable_set?")
                            (("1" (propax) nil nil)) nil))
                          nil)
                         ("2" (expand "measurable_set?")
                          (("2" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((mu formal-const-decl "sigma_finite_measure"
     sigma_finite_measure_props nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (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" sigma_finite_measure_props
     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 sigma_finite_measure_props nil)
    (nn_integrable nonempty-type-eq-decl nil nn_integral nil)
    (nn_integrable? const-decl "bool" nn_integral nil)
    (integrable nonempty-type-eq-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (A_of const-decl "disjoint_indexed_measurable" measure_def 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)
    (indefinite_countably_additive formula-decl nil indefinite_integral
     nil)
    (indefinite_fullset formula-decl nil indefinite_integral nil)
    (f!1 skolem-const-decl "nn_integrable[T, S, mu]"
     sigma_finite_measure_props nil)
    (contraction_integral formula-decl nil measure_contraction nil)
    (phi_is_simple application-judgement "simple"
     sigma_finite_measure_props nil)
    (prod_measurable application-judgement "measurable_function[T, S]"
     sigma_finite_measure_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nn_integrable_is_integrable judgement-tcc nil integral nil)
    (indefinite_integrable formula-decl nil integral nil)
    (measurable_function nonempty-type-eq-decl nil measure_space_def
     nil)
    (measurable_function? const-decl "bool" measure_space_def nil)
    (contraction_integrable formula-decl nil measure_contraction nil)
    (integral const-decl "real" indefinite_integral nil)
    (integrable? const-decl "bool" indefinite_integral nil)
    (integral const-decl "real" integral nil)
    (contraction const-decl "measure_type" measure_contraction nil)
    (measurable_set nonempty-type-eq-decl nil measure_space_def nil)
    (measurable_set? const-decl "bool" measure_space_def nil)
    (set type-eq-decl nil sets nil)
    (measure_type nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     sigma_finite_measure_props nil)
    (subset_algebra_fullset name-judgement "(S)"
     sigma_finite_measure_props nil)
    (A_of_def1 formula-decl nil measure_def nil))
   shostak))
 (sfm_component_eq_TCC1 0
  (sfm_component_eq_TCC1-1 nil 3459609819
   ("" (expand "measurable_set?")
    (("" (skosimp) (("" (rewrite "A_of_def2"nil nil)) nil)) nil)
   ((S formal-const-decl "sigma_algebra" sigma_finite_measure_props
     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 sigma_finite_measure_props 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)
    (mu formal-const-decl "sigma_finite_measure"
     sigma_finite_measure_props nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (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)
    (A_of_def2 formula-decl nil measure_def nil)
    (measurable_set? const-decl "bool" measure_space_def nil))
   nil))
 (sfm_component_eq 0
  (sfm_component_eq-1 nil 3459609938
   ("" (skosimp)
    (("" (expand "fm_contraction")
      (("" (expand "contraction")
        (("" (expand "to_measure")
          (("" (expand "x_eq")
            (("" (lemma "A_of_def2" ("n" "n!1" "mu" "mu"))
              ((""
                (lemma "m_monotone[T,S,mu]"
                 ("a" "intersection(A_of(mu)(n!1), A!1)" "b"
                  "A_of(mu)(n!1)"))
                (("1" (assert)
                  (("1" (expand "x_le")
                    (("1" (expand "subset?")
                      (("1" (expand "intersection")
                        (("1" (expand "member")
                          (("1" (skosimp) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (expand "measurable_set?")
                  (("2" (propax) nil nil)) nil)
                 ("3" (expand "measurable_set?")
                  (("3" (propax) nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((fm_contraction const-decl "finite_measure" measure_contraction
     nil)
    (to_measure const-decl "measure_type" generalized_measure_def nil)
    (S formal-const-decl "sigma_algebra" sigma_finite_measure_props
     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 sigma_finite_measure_props 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)
    (mu formal-const-decl "sigma_finite_measure"
     sigma_finite_measure_props nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (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)
    (A_of_def2 formula-decl nil measure_def nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (subset_algebra_intersection application-judgement "(S)"
     sigma_finite_measure_props nil)
    (m_monotone formula-decl nil measure_props nil)
    (set type-eq-decl nil sets nil)
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