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Datei: fubini.prf Sprache: Lisp

Original von: PVS^©

(fubini
(integrable_is_integrable1 0
  (integrable_is_integrable1-1 nil 3453179746
   ("" (skosimp)
    (("" (typepred "f!1")
      (("" (lemma "integrable_pm_def" ("f0" "f!1"))
        (("" (flatten)
          (("" (hide -2)
            (("" (split)
              (("1" (flatten)
                (("1" (lemma "fubini_tonelli_1" ("h" "minus(f!1)"))
                  (("1" (lemma "fubini_tonelli_1" ("h" "plus(f!1)"))
                    (("1" (assert)
                      (("1"
                        (lemma "integrable1_diff"
                         ("g1" "plus(f!1)" "h1" "minus(f!1)"))
                        (("1"
                          (case-replace
                           "(-[[T1, T2]])(plus(f!1), minus(f!1))=f!1")
                          (("1" (hide-all-but 1)
                            (("1" (apply-extensionality :hide? t)
                              (("1" (grind) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (expand "nn_measurable?")
                      (("2"
                        (use "integrable_is_measurable[[T1,T2],sigma_times(S1,S2),m_times(mu,nu)]")
                        (("2" (assert)
                          (("2" (skosimp) (("2" (assert) nil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (expand "nn_measurable?")
                    (("2"
                      (use "integrable_is_measurable[[T1,T2],sigma_times(S1,S2),m_times(mu,nu)]")
                      (("2" (assert)
                        (("2" (skosimp) (("2" (assert) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (propax) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((integrable nonempty-type-eq-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (nu formal-const-decl "sigma_finite_measure[T2, S2]" fubini nil)
    (mu formal-const-decl "sigma_finite_measure[T1, S1]" fubini nil)
    (m_times const-decl
     "sigma_finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_measure 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)
    (S2 formal-const-decl "sigma_algebra[T2]" fubini nil)
    (S1 formal-const-decl "sigma_algebra[T1]" fubini nil)
    (sigma_times const-decl "sigma_algebra[[T1, T2]]" product_sigma_def
     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)
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    (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)
    (T2 formal-type-decl nil fubini nil)
    (T1 formal-type-decl nil fubini nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (minus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (nn_measurable nonempty-type-eq-decl nil measure_space nil)
    (nn_measurable? const-decl "bool" measure_space nil)
    (fubini_tonelli_1 formula-decl nil fubini_tonelli nil)
    (integrable_minus application-judgement "integrable" fubini nil)
    (minus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (integrable_is_measurable judgement-tcc nil integral nil)
    (- const-decl "[T -> real]" real_fun_ops "reals/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (integrable_diff application-judgement "integrable" fubini nil)
    (diff_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (integrable1 nonempty-type-eq-decl nil product_integral_def nil)
    (integrable1? const-decl "bool" product_integral_def nil)
    (integrable1_diff judgement-tcc nil product_integral_def nil)
    (plus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (integrable_plus application-judgement "integrable" fubini nil)
    (plus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_pm_def formula-decl nil integral nil))
   shostak))
(integrable_is_integrable2 0
  (integrable_is_integrable2-1 nil 3453180306
   ("" (skosimp)
    (("" (typepred "f!1")
      (("" (lemma "integrable_pm_def" ("f0" "f!1"))
        (("" (flatten)
          (("" (hide -2)
            (("" (split)
              (("1" (flatten)
                (("1" (lemma "fubini_tonelli_2" ("h" "minus(f!1)"))
                  (("1" (lemma "fubini_tonelli_2" ("h" "plus(f!1)"))
                    (("1" (assert)
                      (("1"
                        (lemma "integrable2_diff"
                         ("g2" "plus(f!1)" "h2" "minus(f!1)"))
                        (("1"
                          (case-replace
                           "(-[[T1, T2]])(plus(f!1), minus(f!1))=f!1")
                          (("1" (apply-extensionality :hide? t)
                            (("1" (hide-all-but 1)
                              (("1" (grind) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (expand "nn_measurable?")
                      (("2"
                        (use "integrable_is_measurable[[T1,T2],sigma_times(S1,S2),m_times(mu,nu)]")
                        (("2" (assert)
                          (("2" (hide-all-but 1)
                            (("2" (grind) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (expand "nn_measurable?")
                    (("2"
                      (use "integrable_is_measurable[[T1,T2],sigma_times(S1,S2),m_times(mu,nu)]")
                      (("2" (assert)
                        (("2" (hide-all-but 1) (("2" (grind) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (propax) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((integrable nonempty-type-eq-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (nu formal-const-decl "sigma_finite_measure[T2, S2]" fubini nil)
    (mu formal-const-decl "sigma_finite_measure[T1, S1]" fubini nil)
    (m_times const-decl
     "sigma_finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_measure 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)
    (S2 formal-const-decl "sigma_algebra[T2]" fubini nil)
    (S1 formal-const-decl "sigma_algebra[T1]" fubini nil)
    (sigma_times const-decl "sigma_algebra[[T1, T2]]" product_sigma_def
     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)
    (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)
    (T2 formal-type-decl nil fubini nil)
    (T1 formal-type-decl nil fubini nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (minus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (nn_measurable nonempty-type-eq-decl nil measure_space nil)
    (nn_measurable? const-decl "bool" measure_space nil)
    (fubini_tonelli_2 formula-decl nil fubini_tonelli nil)
    (integrable_minus application-judgement "integrable" fubini nil)
    (minus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_is_measurable judgement-tcc nil integral nil)
    (- const-decl "[T -> real]" real_fun_ops "reals/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (integrable_diff application-judgement "integrable" fubini nil)
    (diff_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (integrable2 nonempty-type-eq-decl nil product_integral_def nil)
    (integrable2? const-decl "bool" product_integral_def nil)
    (integrable2_diff judgement-tcc nil product_integral_def nil)
    (plus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (integrable_plus application-judgement "integrable" fubini nil)
    (plus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_pm_def formula-decl nil integral nil))
   shostak))
(fubini1_TCC1 0
  (fubini1_TCC1-1 nil 3459090851
   ("" (skosimp)
    (("" (lemma "integrable_is_integrable1" ("f" "f!1"))
      (("" (propax) nil nil)) nil))
    nil)
   ((integrable nonempty-type-eq-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (nu formal-const-decl "sigma_finite_measure[T2, S2]" fubini nil)
    (mu formal-const-decl "sigma_finite_measure[T1, S1]" fubini nil)
    (m_times const-decl
     "sigma_finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_measure 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)
    (S2 formal-const-decl "sigma_algebra[T2]" fubini nil)
    (S1 formal-const-decl "sigma_algebra[T1]" fubini nil)
    (sigma_times const-decl "sigma_algebra[[T1, T2]]" product_sigma_def
     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)
    (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)
    (T2 formal-type-decl nil fubini nil)
    (T1 formal-type-decl nil fubini nil)
    (integrable_is_integrable1 formula-decl nil fubini nil))
   nil))
(fubini1 0
  (fubini1-1 nil 3431174025
   ("" (skosimp)
    (("" (rewrite "integral_pm")
      (("" (lemma "integral_pm" ("f" "integral1(f!1)"))
        (("" (replace -1)
          (("" (hide -1)
            (("" (lemma "fubini_tonelli_3" ("g" "plus(f!1)"))
              (("1" (lemma "fubini_tonelli_3" ("g" "minus(f!1)"))
                (("1" (replace -1)
                  (("1" (replace -2)
                    (("1" (hide -1 -2)
                      (("1" (rewrite "integral_diff" 1 :dir rl)
                        (("1" (rewrite "integral_diff" 1 :dir rl)
                          (("1"
                            (case-replace
                             "plus(integral1(f!1)) - minus(integral1(f!1))=integral1(f!1)")
                            (("1" (hide -1)
                              (("1"
                                (lemma
                                 "integral1_diff"
                                 ("g1" "plus(f!1)" "h1" "minus(f!1)"))
                                (("1"
                                  (case-replace
                                   "plus(f!1) - minus(f!1)=f!1")
                                  (("1"
                                    (hide -1)
                                    (("1"
                                      (lemma
                                       "integral_ae_eq"
                                       ("f"
                                        "product_integral_def[T1, T2, S1, S2, mu, nu].integral1(f!1)"
                                        "h"
                                        "product_integral_def[T1, T2, S1, S2, mu, nu].integral1(plus(f!1)) - product_integral_def[T1, T2, S1, S2, mu, nu].integral1(minus(f!1))"))
                                      (("1" (assert) nil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but 1)
                                    (("2"
                                      (apply-extensionality :hide? t)
                                      (("2" (grind) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (name-replace "DRL" "integral1(f!1)")
                              (("2"
                                (hide 2)
                                (("2"
                                  (apply-extensionality :hide? t)
                                  (("2" (grind) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2"
                  (lemma "nn_integrable_is_nn_integrable"
                   ("f" "minus[[T1, T2]](f!1)"))
                  (("2" (assert)
                    (("2" (hide-all-but 1) (("2" (grind) nil nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2"
                (lemma "nn_integrable_is_nn_integrable"
                 ("f" "plus[[T1, T2]](f!1)"))
                (("2" (assert)
                  (("2" (hide-all-but 1) (("2" (grind) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((integral_pm formula-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (integrable nonempty-type-eq-decl nil integral nil)
    (T1 formal-type-decl nil fubini nil)
    (T2 formal-type-decl nil fubini 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 nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_times const-decl "sigma_algebra[[T1, T2]]" product_sigma_def
     nil)
    (S1 formal-const-decl "sigma_algebra[T1]" fubini nil)
    (S2 formal-const-decl "sigma_algebra[T2]" fubini 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)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (m_times const-decl
     "sigma_finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_measure nil)
    (mu formal-const-decl "sigma_finite_measure[T1, S1]" fubini nil)
    (nu formal-const-decl "sigma_finite_measure[T2, S2]" fubini nil)
    (integrable_minus application-judgement "integrable" fubini nil)
    (minus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_plus application-judgement "integrable" fubini nil)
    (plus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (fubini_tonelli_3 formula-decl nil fubini_tonelli nil)
    (nn_integrable? const-decl "bool" nn_integral nil)
    (nn_integrable nonempty-type-eq-decl nil nn_integral nil)
    (plus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (nn_integrable_is_nn_integrable formula-decl nil integral nil)
    (integrable_minus application-judgement "integrable" fubini nil)
    (minus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_plus application-judgement "integrable" fubini nil)
    (plus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_diff application-judgement "integrable" fubini nil)
    (diff_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integral_ae_eq formula-decl nil integral nil)
    (measurable_function? const-decl "bool" measure_space_def nil)
    (measurable_function nonempty-type-eq-decl nil measure_space_def
     nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (integral1_diff formula-decl nil product_integral_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (- const-decl "[T -> real]" real_fun_ops "reals/")
    (diff_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_diff application-judgement "integrable" fubini nil)
    (integral_diff formula-decl nil integral nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (minus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (integral1 const-decl "integrable[T1, S1, mu]" product_integral_def
     nil)
    (integrable1 nonempty-type-eq-decl nil product_integral_def nil)
    (integrable1? const-decl "bool" product_integral_def nil))
   shostak))
(fubini2_TCC1 0
  (fubini2_TCC1-1 nil 3459090851
   ("" (skosimp)
    (("" (lemma "integrable_is_integrable2" ("f" "f!1"))
      (("" (propax) nil nil)) nil))
    nil)
   ((integrable nonempty-type-eq-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (nu formal-const-decl "sigma_finite_measure[T2, S2]" fubini nil)
    (mu formal-const-decl "sigma_finite_measure[T1, S1]" fubini nil)
    (m_times const-decl
     "sigma_finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_measure 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)
    (S2 formal-const-decl "sigma_algebra[T2]" fubini nil)
    (S1 formal-const-decl "sigma_algebra[T1]" fubini nil)
    (sigma_times const-decl "sigma_algebra[[T1, T2]]" product_sigma_def
     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)
    (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)
    (T2 formal-type-decl nil fubini nil)
    (T1 formal-type-decl nil fubini nil)
    (integrable_is_integrable2 formula-decl nil fubini nil))
   nil))
(fubini2 0
  (fubini2-1 nil 3431180694
   ("" (skosimp)
    (("" (rewrite "integral_pm")
      (("" (lemma "integral_pm" ("f" "integral2(f!1)"))
        (("" (replace -1)
          (("" (hide -1)
            (("" (lemma "fubini_tonelli_4" ("g" "plus(f!1)"))
              (("1" (lemma "fubini_tonelli_4" ("g" "minus(f!1)"))
                (("1" (replace -1)
                  (("1" (replace -2)
                    (("1" (hide -1 -2)
                      (("1" (rewrite "integral_diff" 1 :dir rl)
                        (("1" (rewrite "integral_diff" 1 :dir rl)
                          (("1"
                            (case-replace
                             "plus(integral2(f!1)) - minus(integral2(f!1))=integral2(f!1)")
                            (("1" (hide -1)
                              (("1"
                                (lemma
                                 "integral2_diff"
                                 ("g2" "plus(f!1)" "h2" "minus(f!1)"))
                                (("1"
                                  (case-replace
                                   "plus(f!1) - minus(f!1)=f!1")
                                  (("1"
                                    (hide -1)
                                    (("1"
                                      (lemma
                                       "integral_ae_eq"
                                       ("f"
                                        "product_integral_def[T1, T2, S1, S2, mu, nu].integral2(f!1)"
                                        "h"
                                        "product_integral_def[T1, T2, S1, S2, mu, nu].integral2(plus(f!1)) - product_integral_def[T1, T2, S1, S2, mu, nu].integral2(minus(f!1))"))
                                      (("1" (assert) nil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but 1)
                                    (("2"
                                      (apply-extensionality :hide? t)
                                      (("2" (grind) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (name-replace "DRL" "integral2(f!1)")
                              (("2"
                                (hide 2)
                                (("2"
                                  (apply-extensionality :hide? t)
                                  (("2" (grind) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2"
                  (lemma "nn_integrable_is_nn_integrable"
                   ("f" "minus[[T1, T2]](f!1)"))
                  (("2" (assert)
                    (("2" (hide-all-but 1) (("2" (grind) nil nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2"
                (lemma "nn_integrable_is_nn_integrable"
                 ("f" "plus[[T1, T2]](f!1)"))
                (("2" (assert)
                  (("2" (hide-all-but 1) (("2" (grind) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((integral_pm formula-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (integrable nonempty-type-eq-decl nil integral nil)
    (T1 formal-type-decl nil fubini nil)
    (T2 formal-type-decl nil fubini 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 nil)
    (sigma_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (sigma_times const-decl "sigma_algebra[[T1, T2]]" product_sigma_def
     nil)
    (S1 formal-const-decl "sigma_algebra[T1]" fubini nil)
    (S2 formal-const-decl "sigma_algebra[T2]" fubini 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)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (m_times const-decl
     "sigma_finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_measure nil)
    (mu formal-const-decl "sigma_finite_measure[T1, S1]" fubini nil)
    (nu formal-const-decl "sigma_finite_measure[T2, S2]" fubini nil)
    (integrable_minus application-judgement "integrable" fubini nil)
    (minus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_plus application-judgement "integrable" fubini nil)
    (plus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (fubini_tonelli_4 formula-decl nil fubini_tonelli nil)
    (nn_integrable? const-decl "bool" nn_integral nil)
    (nn_integrable nonempty-type-eq-decl nil nn_integral nil)
    (plus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (nn_integrable_is_nn_integrable formula-decl nil integral nil)
    (integrable_minus application-judgement "integrable" fubini nil)
    (minus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_plus application-judgement "integrable" fubini nil)
    (plus_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_diff application-judgement "integrable" fubini nil)
    (diff_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integral_ae_eq formula-decl nil integral nil)
    (measurable_function? const-decl "bool" measure_space_def nil)
    (measurable_function nonempty-type-eq-decl nil measure_space_def
     nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (integral2_diff formula-decl nil product_integral_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (- const-decl "[T -> real]" real_fun_ops "reals/")
    (diff_measurable application-judgement "measurable_function[T, S]"
     fubini nil)
    (integrable_diff application-judgement "integrable" fubini nil)
    (integral_diff formula-decl nil integral nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (minus const-decl "[T -> nnreal]" real_fun_ops_aux "reals/")
    (integral2 const-decl "integrable[T2, S2, nu]" product_integral_def
     nil)
    (integrable2 nonempty-type-eq-decl nil product_integral_def nil)
    (integrable2? const-decl "bool" product_integral_def nil))
   shostak)))

¤ Dauer der Verarbeitung: 0.8 Sekunden (vorverarbeitet) ¤

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