Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/measure_integration/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 1 MB image not shown  

Quelle  finite_fubini_tonelli.prf

  Sprache: Lisp
 

(finite_fubini_tonelli
 (mu_TCC1 0
  (mu_TCC1-1 nil 3458551591
   ("" (typepred "S1")
    (("" (expand "sigma_algebra?")
      (("" (expand "subset_algebra_empty?")
        (("" (flatten)
          (("" (expand "member") (("" (propax) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((member const-decl "bool" sets nil)
    (subset_algebra_empty? const-decl "bool" subset_algebra_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T1 formal-type-decl nil finite_fubini_tonelli 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)
    (S1 formal-const-decl "sigma_algebra[T1]" finite_fubini_tonelli
     nil))
   nil))
 (nu_TCC1 0
  (nu_TCC1-1 nil 3458551591
   ("" (typepred "S2")
    (("" (expand "sigma_algebra?")
      (("" (expand "subset_algebra_empty?")
        (("" (flatten)
          (("" (expand "member") (("" (propax) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((member const-decl "bool" sets nil)
    (subset_algebra_empty? const-decl "bool" subset_algebra_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T2 formal-type-decl nil finite_fubini_tonelli 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)
    (S2 formal-const-decl "sigma_algebra[T2]" finite_fubini_tonelli
     nil))
   nil))
 (finite_fubini_tonelli_1 0
  (finite_fubini_tonelli_1-1 nil 3458552933
   ("" (skosimp)
    (("" (split)
      (("1" (flatten)
        (("1" (typepred "h!1")
          (("1" (expand "nn_measurable?")
            (("1" (flatten)
              (("1"
                (lemma "nn_integrable_is_nn_integrable" ("f" "h!1"))
                (("1" (split)
                  (("1" (hide -4)
                    (("1" (lemma "nn_integral_def" ("f" "h!1"))
                      (("1" (skolem - "HH")
                        (("1" (flatten)
                          (("1" (typepred "HH")
                            (("1" (case "FORALL (n: nat): isf?(HH(n))")
                              (("1"
                                (case
                                 "FORALL (n: nat,z:[T1,T2]): HH(n)(z)>=0")
                                (("1"
                                  (name
                                   "FF"
                                   "lambda (n:nat): lambda (x:T1): isf_integral(lambda (y:T2): HH(n)(x,y))")
                                  (("1"
                                    (case
                                     "FORALL (n: nat): integrable?[T1, S1, to_measure(mu)](FF(n))")
                                    (("1"
                                      (case
                                       "forall (n:nat): integral[T1,S1,to_measure(mu)](FF(n))=integral[[T1,T2],sigma_times(S1,S2),to_measure(fm_times(mu,nu))](HH(n))")
                                      (("1"
                                        (case-replace
                                         "(isf_integral o HH)=integral o FF")
                                        (("1"
                                          (case
                                           "forall (n:nat,x:T1): FF(n)(x)>=0")
                                          (("1"
                                            (case
                                             "pointwise_increasing?(FF)")
                                            (("1"
                                              (case
                                               "bounded?(integral o FF)")
                                              (("1"
                                                (lemma
                                                 "monotone_convergence[T1, S1, to_measure(mu)]"
                                                 ("F" "FF"))
                                                (("1"
                                                  (split -1)
                                                  (("1"
                                                    (flatten)
                                                    (("1"
                                                      (hide -1)
                                                      (("1"
                                                        (split -1)
                                                        (("1"
                                                          (skolem
                                                           -
                                                           "F")
                                                          (("1"
                                                            (inst
                                                             -
                                                             "F")
                                                            (("1"
                                                              (assert)
                                                              (("1"
                                                                (typepred
                                                                 "F")
                                                                (("1"
                                                                  (expand
                                                                   "integrable1?")
                                                                  (("1"
                                                                    (expand
                                                                     "ae_convergence?")
                                                                    (("1"
                                                                      (expand
                                                                       "fullset")
                                                                      (("1"
                                                                        (expand
                                                                         "ae_convergence_in?")
                                                                        (("1"
                                                                          (expand
                                                                           "ae_in?")
                                                                          (("1"
                                                                            (skosimp)
                                                                            (("1"
                                                                              (expand
                                                                               "member")
                                                                              (("1"
                                                                                (typepred
                                                                                 "E!1")
                                                                                (("1"
                                                                                  (expand
                                                                                   "negligible_set?")
                                                                                  (("1"
                                                                                    (skolem
                                                                                     -
                                                                                     "N1")
                                                                                    (("1"
                                                                                      (flatten)
                                                                                      (("1"
                                                                                        (inst
                                                                                         +
                                                                                         "N1"
                                                                                         "F")
                                                                                        (("1"
                                                                                          (skosimp)
                                                                                          (("1"
                                                                                            (expand
                                                                                             "subset?")
                                                                                            (("1"
                                                                                              (inst
                                                                                               -
                                                                                               "x!1")
                                                                                              (("1"
                                                                                                (expand
                                                                                                 "member")
                                                                                                (("1"
                                                                                                  (inst
                                                                                                   -
                                                                                                   "x!1")
                                                                                                  (("1"
                                                                                                    (assert)
                                                                                                    (("1"
                                                                                                      (lemma
                                                                                                       "monotone_convergence_scaf[T2,S2,to_measure(nu)]"
                                                                                                       ("f"
                                                                                                        "LAMBDA (y: T2): h!1(x!1, y)"
                                                                                                        "F"
                                                                                                        "lambda (n:nat): lambda (y:T2): HH(n)(x!1,y)"))
                                                                                                      (("1"
                                                                                                        (split
                                                                                                         -1)
                                                                                                        (("1"
                                                                                                          (flatten)
                                                                                                          (("1"
                                                                                                            (replace
                                                                                                             -1)
                                                                                                            (("1"
                                                                                                              (name-replace
                                                                                                               "LHS"
                                                                                                               "integral(LAMBDA (y: T2): h!1(x!1, y))")
                                                                                                              (("1"
                                                                                                                (case-replace
                                                                                                                 "(integral o
                        (LAMBDA (n: nat): LAMBDA (y: T2): HH(n)(x!1, y)))=LAMBDA (n:nat): FF(n)(x!1)")
                                                                                                                (("1"
                                                                                                                  (expand
                                                                                                                   "converges_upto?")
                                                                                                                  (("1"
                                                                                                                    (flatten)
                                                                                                                    (("1"
                                                                                                                      (hide-all-but
                                                                                                                       (-3
                                                                                                                        -7
                                                                                                                        3))
                                                                                                                      (("1"
                                                                                                                        (lemma
                                                                                                                         "hausdorff_convergence.unique_limit"
                                                                                                                         ("u"
                                                                                                                          "LAMBDA (n: nat): FF(n)(x!1)"
                                                                                                                          "l1"
                                                                                                                          "LHS"
                                                                                                                          "l2"
                                                                                                                          "F(x!1)"))
                                                                                                                        (("1"
                                                                                                                          (assert)
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("2"
                                                                                                                  (apply-extensionality
                                                                                                                   :hide?
                                                                                                                   t)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "FF"
                                                                                                                     1)
                                                                                                                    (("1"
                                                                                                                      (expand
                                                                                                                       "o"
                                                                                                                       1)
                                                                                                                      (("1"
                                                                                                                        (inst
                                                                                                                         -15
                                                                                                                         "x!2")
                                                                                                                        (("1"
                                                                                                                          (lemma
                                                                                                                           "isf_x_section"
                                                                                                                           ("x"
                                                                                                                            "x!1"
                                                                                                                            "i"
                                                                                                                            "HH(x!2)"))
                                                                                                                          (("1"
                                                                                                                            (rewrite
                                                                                                                             "isf_integral"
                                                                                                                             1)
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (skosimp)
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       -15
                                                                                                                       "n!1")
                                                                                                                      (("2"
                                                                                                                        (lemma
                                                                                                                         "isf_x_section"
                                                                                                                         ("x"
                                                                                                                          "x!1"
                                                                                                                          "i"
                                                                                                                          "HH(n!1)"))
                                                                                                                        (("2"
                                                                                                                          (lemma
                                                                                                                           "isf_is_integrable[T2, S2, to_measure(nu)]")
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             -
                                                                                                                             "LAMBDA (y: T2): HH(n!1)(x!1, y)")
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (hide-all-but
                                                                                                           (1
                                                                                                            -14
                                                                                                            -15))
                                                                                                          (("2"
                                                                                                            (expand
                                                                                                             "increasing_nn_isf?")
                                                                                                            (("2"
                                                                                                              (expand
                                                                                                               "pointwise_converges_upto?")
                                                                                                              (("2"
                                                                                                                (split)
                                                                                                                (("1"
                                                                                                                  (expand
                                                                                                                   "pointwise_convergence?")
                                                                                                                  (("1"
                                                                                                                    (skolem
                                                                                                                     +
                                                                                                                     "y!1")
                                                                                                                    (("1"
                                                                                                                      (inst
                                                                                                                       -
                                                                                                                       "(x!1,y!1)")
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("2"
                                                                                                                  (expand
                                                                                                                   "pointwise_increasing?")
                                                                                                                  (("2"
                                                                                                                    (skolem
                                                                                                                     +
                                                                                                                     "y!1")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       -
                                                                                                                       "(x!1,y!1)")
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("3"
                                                                                                          (replace
                                                                                                           -8
                                                                                                           -5
                                                                                                           rl)
                                                                                                          (("3"
                                                                                                            (hide-all-but
                                                                                                             (-7
                                                                                                              1
                                                                                                              -13
                                                                                                              -12
                                                                                                              -8
                                                                                                              -3
                                                                                                              -6))
                                                                                                            (("3"
                                                                                                              (expand
                                                                                                               "FF")
                                                                                                              (("3"
                                                                                                                (expand
                                                                                                                 "o ")
                                                                                                                (("3"
                                                                                                                  (expand
                                                                                                                   "pointwise_increasing?")
                                                                                                                  (("3"
                                                                                                                    (inst
                                                                                                                     -2
                                                                                                                     "x!1")
                                                                                                                    (("3"
                                                                                                                      (hide
                                                                                                                       -4)
                                                                                                                      (("3"
                                                                                                                        (expand
                                                                                                                         "bounded?")
                                                                                                                        (("3"
                                                                                                                          (split)
                                                                                                                          (("1"
                                                                                                                            (expand
                                                                                                                             "bounded_above?")
                                                                                                                            (("1"
                                                                                                                              (inst
                                                                                                                               +
                                                                                                                               "F(x!1)")
                                                                                                                              (("1"
                                                                                                                                (skolem
                                                                                                                                 +
                                                                                                                                 "n!1")
                                                                                                                                (("1"
                                                                                                                                  (inst-cp
                                                                                                                                   -5
                                                                                                                                   "n!1")
                                                                                                                                  (("1"
                                                                                                                                    (lemma
                                                                                                                                     "isf_x_section"
                                                                                                                                     ("i"
                                                                                                                                      "HH(n!1)"
                                                                                                                                      "x"
                                                                                                                                      "x!1"))
                                                                                                                                    (("1"
                                                                                                                                      (rewrite
                                                                                                                                       "isf_integral")
                                                                                                                                      (("1"
                                                                                                                                        (rewrite
                                                                                                                                         "metric_convergence_def")
                                                                                                                                        (("1"
                                                                                                                                          (expand
                                                                                                                                           "metric_converges_to")
                                                                                                                                          (("1"
                                                                                                                                            (inst
                                                                                                                                             -
                                                                                                                                             "isf_integral(LAMBDA (y: T2): HH(n!1)(x!1, y))-F(x!1)")
                                                                                                                                            (("1"
                                                                                                                                              (skosimp)
                                                                                                                                              (("1"
                                                                                                                                                (inst
                                                                                                                                                 -
                                                                                                                                                 "n!1+n!2")
                                                                                                                                                (("1"
                                                                                                                                                  (assert)
                                                                                                                                                  (("1"
                                                                                                                                                    (expand
                                                                                                                                                     "ball")
                                                                                                                                                    (("1"
                                                                                                                                                      (expand
                                                                                                                                                       "increasing?")
                                                                                                                                                      (("1"
                                                                                                                                                        (inst
                                                                                                                                                         -3
                                                                                                                                                         "n!1"
                                                                                                                                                         "n!1+n!2")
                                                                                                                                                        (("1"
                                                                                                                                                          (assert)
                                                                                                                                                          nil
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil)
                                                                                                                                             ("2"
                                                                                                                                              (assert)
                                                                                                                                              nil
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil)
                                                                                                                                         ("2"
                                                                                                                                          (skosimp)
                                                                                                                                          (("2"
                                                                                                                                            (rewrite
                                                                                                                                             "isf_x_section")
                                                                                                                                            nil
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil)
                                                                                                                           ("2"
                                                                                                                            (expand
                                                                                                                             "bounded_below?")
                                                                                                                            (("2"
                                                                                                                              (inst
                                                                                                                               +
                                                                                                                               "0")
                                                                                                                              (("2"
                                                                                                                                (skolem
                                                                                                                                 +
                                                                                                                                 "n!1")
                                                                                                                                (("2"
                                                                                                                                  (inst
                                                                                                                                   -5
                                                                                                                                   "n!1")
                                                                                                                                  (("2"
                                                                                                                                    (lemma
                                                                                                                                     "isf_x_section"
                                                                                                                                     ("i"
                                                                                                                                      "HH(n!1)"
                                                                                                                                      "x"
                                                                                                                                      "x!1"))
                                                                                                                                    (("2"
                                                                                                                                      (rewrite
                                                                                                                                       "isf_integral"
                                                                                                                                       1)
                                                                                                                                      (("2"
                                                                                                                                        (inst
                                                                                                                                         -4
                                                                                                                                         "n!1"
                                                                                                                                         "x!1")
                                                                                                                                        (("2"
                                                                                                                                          (assert)
                                                                                                                                          nil
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (skosimp)
                                                                                                        (("2"
                                                                                                          (inst
                                                                                                           -13
                                                                                                           "n!1")
                                                                                                          (("2"
                                                                                                            (lemma
                                                                                                             "isf_x_section"
                                                                                                             ("i"
                                                                                                              "HH(n!1)"
                                                                                                              "x"
                                                                                                              "x!1"))
                                                                                                            (("2"
                                                                                                              (lemma
                                                                                                               "isf_is_integrable[T2, S2, to_measure(nu)]")
                                                                                                              (("2"
                                                                                                                (inst
                                                                                                                 -
                                                                                                                 "LAMBDA (y: T2): HH(n!1)(x!1, y)")
                                                                                                                nil
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  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
                                                           (-2 1 -7))
                                                          (("2"
                                                            (expand
                                                             "bounded?")
                                                            (("2"
                                                              (skosimp)
                                                              (("2"
                                                                (split)
                                                                (("1"
                                                                  (expand
                                                                   "bounded_above?")
                                                                  (("1"
                                                                    (inst
                                                                     +
                                                                     "c!1")
                                                                    (("1"
                                                                      (skolem
                                                                       +
                                                                       "n!1")
                                                                      (("1"
                                                                        (inst
                                                                         -
                                                                         "n!1")
                                                                        (("1"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (expand
                                                                   "bounded_below?")
                                                                  (("2"
                                                                    (inst
                                                                     +
                                                                     "-c!1")
                                                                    (("2"
                                                                      (skolem
                                                                       +
                                                                       "n!1")
                                                                      (("2"
                                                                        (inst
                                                                         -
                                                                         "n!1")
                                                                        (("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but
                                                     (1 -2))
                                                    (("2"
                                                      (expand
                                                       "ae_increasing?")
                                                      (("2"
                                                        (inst
                                                         +
                                                         "emptyset")
                                                        (("2"
                                                          (skosimp)
                                                          (("2"
                                                            (expand
                                                             "pointwise_increasing?")
                                                            (("2"
                                                              (inst
                                                               -
                                                               "x!1")
                                                              (("2"
                                                                (expand
                                                                 "increasing?")
                                                                (("2"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (replace -3 * rl)
                                                (("2"
                                                  (hide-all-but
                                                   (1
                                                    -11
                                                    -12
                                                    -9
                                                    -8
                                                    -7))
                                                  (("2"
                                                    (expand "bounded?")
                                                    (("2"
                                                      (inst
                                                       +
                                                       "nn_integral(h!1)")
                                                      (("2"
                                                        (skolem
                                                         +
                                                         "n!1")
                                                        (("2"
                                                          (lemma
                                                           "isf_integral_pos"
                                                           ("i"
                                                            "HH(n!1)"))
                                                          (("2"
                                                            (split)
                                                            (("1"
                                                              (expand
                                                               "abs"
                                                               1)
                                                              (("1"
                                                                (assert)
                                                                (("1"
                                                                  (expand
                                                                   "o"
                                                                   1)
                                                                  (("1"
                                                                    (rewrite
                                                                     "metric_convergence_def")
                                                                    (("1"
                                                                      (expand
                                                                       "metric_converges_to")
                                                                      (("1"
                                                                        (inst
                                                                         -5
                                                                         "isf_integral(HH(n!1))-nn_integral(h!1)")
                                                                        (("1"
                                                                          (skosimp)
                                                                          (("1"
                                                                            (inst
                                                                             -5
                                                                             "n!1+n!2")
                                                                            (("1"
                                                                              (assert)
                                                                              (("1"
                                                                                (expand
                                                                                 "ball")
                                                                                (("1"
                                                                                  (expand
                                                                                   "increasing_nn_isf?")
                                                                                  (("1"
                                                                                    (expand
                                                                                     "pointwise_increasing?")
                                                                                    (("1"
                                                                                      (inst-cp
                                                                                       -3
                                                                                       "n!1")
                                                                                      (("1"
                                                                                        (inst
                                                                                         -3
                                                                                         "n!1+n!2")
                                                                                        (("1"
                                                                                          (case
                                                                                           "isf_integral(HH(n!1)) <= isf_integral(HH(n!1+n!2))")
                                                                                          (("1"
                                                                                            (expand
                                                                                             "o ")
                                                                                            (("1"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil))
                                                                                            nil)
                                                                                           ("2"
                                                                                            (assert)
                                                                                            (("2"
                                                                                              (hide
                                                                                               2
                                                                                               -6)
                                                                                              (("2"
                                                                                                (lemma
                                                                                                 "isf_integral_le"
                                                                                                 ("i1"
                                                                                                  "HH(n!1)"
                                                                                                  "i2"
                                                                                                  "HH(n!1 + n!2)"))
                                                                                                (("2"
                                                                                                  (assert)
                                                                                                  (("2"
                                                                                                    (skolem
                                                                                                     +
                                                                                                     "z!1")
                                                                                                    (("2"
                                                                                                      (inst
                                                                                                       -5
                                                                                                       "z!1")
                                                                                                      (("2"
                                                                                                        (expand
                                                                                                         "increasing?")
                                                                                                        (("2"
                                                                                                          (inst
                                                                                                           -5
                                                                                                           "n!1"
                                                                                                           "n!1+n!2")
                                                                                                          (("2"
                                                                                                            (assert)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (skolem
                                                               +
                                                               "z!1")
                                                              (("2"
                                                                (inst
                                                                 -
                                                                 "n!1"
                                                                 "z!1")
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (expand
                                               "increasing_nn_isf?")
                                              (("2"
                                                (hide-all-but
                                                 (1 -8 -7))
                                                (("2"
                                                  (expand "FF")
                                                  (("2"
                                                    (expand
                                                     "pointwise_increasing?")
                                                    (("2"
                                                      (skosimp)
                                                      (("2"
                                                        (expand
                                                         "increasing?")
                                                        (("2"
                                                          (skolem
                                                           +
                                                           ("i!1"
                                                            "j!1"))
                                                          (("2"
                                                            (flatten)
                                                            (("2"
                                                              (inst-cp
                                                               -
                                                               "j!1")
                                                              (("2"
                                                                (inst
                                                                 -
                                                                 "i!1")
                                                                (("2"
                                                                  (lemma
                                                                   "isf_x_section"
                                                                   ("x"
                                                                    "x!1"
                                                                    "i"
                                                                    "HH(j!1)"))
                                                                  (("2"
                                                                    (lemma
                                                                     "isf_x_section"
                                                                     ("x"
                                                                      "x!1"
                                                                      "i"
                                                                      "HH(i!1)"))
                                                                    (("2"
                                                                      (lemma
                                                                       "isf_integral_le"
                                                                       ("i1"
                                                                        "LAMBDA (y: T2): HH(i!1)(x!1, y)"
                                                                        "i2"
                                                                        "LAMBDA (y: T2): HH(j!1)(x!1, y)"))
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (skolem
                                                                           +
                                                                           "y!1")
                                                                          (("1"
                                                                            (inst
                                                                             -6
                                                                             "(x!1,y!1)")
                                                                            (("1"
                                                                              (inst
                                                                               -
                                                                               "i!1"
                                                                               "j!1")
                                                                              (("1"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil)
                                                                       ("2"
                                                                        (propax)
                                                                        nil
                                                                        nil)
                                                                       ("3"
                                                                        (propax)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (skosimp)
                                            (("2"
                                              (expand "FF" 1)
                                              (("2"
                                                (hide-all-but
                                                 (-5 -6 1))
                                                (("2"
                                                  (inst - "n!1")
                                                  (("2"
                                                    (lemma
                                                     "isf_x_section"
                                                     ("x"
                                                      "x!1"
                                                      "i"
                                                      "HH(n!1)"))
                                                    (("2"
                                                      (lemma
                                                       "isf_integral_pos"
                                                       ("i"
                                                        "LAMBDA (y: T2): HH(n!1)(x!1, y)"))
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (skolem
                                                           +
                                                           "y!1")
                                                          (("1"
                                                            (inst
                                                             -
                                                             "n!1"
                                                             "(x!1,y!1)")
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (apply-extensionality
                                           :hide?
                                           t)
                                          (("2"
                                            (expand "o")
                                            (("2"
                                              (inst - "x!1")
                                              (("2"
                                                (replace -1)
                                                (("2"
                                                  (inst -5 "x!1")
                                                  (("2"
                                                    (rewrite
                                                     "isf_integral")
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (skosimp)
                                        (("2"
                                          (inst - "n!1")
                                          (("2"
                                            (expand "FF")
                                            (("2"
                                              (inst -4 "n!1")
                                              (("2"
                                                (rewrite
                                                 "isf_integral[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]"
                                                 1)
                                                (("2"
                                                  (lemma
                                                   "isf_fubini_tonelli_3"
                                                   ("i" "HH(n!1)"))
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("3" (propax) nil nil))
                                      nil)
                                     ("2"
                                      (skosimp)
                                      (("2"
                                        (expand "FF")
                                        (("2"
                                          (inst -3 "n!1")
                                          (("2"
                                            (rewrite "isf_integral_x")
                                            (("2"
                                              (case-replace
                                               "(LAMBDA (x: T1, y: T2): HH(n!1)(x, y))=HH(n!1)")
                                              (("2"
                                                (apply-extensionality
                                                 :hide?
                                                 t)
                                                nil
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (skosimp)
                                    (("2"
                                      (inst -2 "n!1")
                                      (("2"
                                        (rewrite "isf_x_section")
                                        nil
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (skosimp)
                                  (("2"
                                    (typepred "HH(n!1)")
                                    (("2"
                                      (expand "nn_isf?")
                                      (("2" (inst - "z!1"nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (skosimp)
                                (("2"
                                  (typepred "HH(n!1)")
                                  (("2" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (propax) nil nil))
                      nil))
                    nil)
                   ("2" (skosimp)
                    (("2" (inst - "x!1") (("2" (assertnil nil)) nil))
                    nil))
                  nil)
                 ("2" (propax) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (flatten)
        (("2" (expand "integrable1?")
          (("2" (skolem - ("N!1" "f!1"))
            (("2" (expand "member")
              (("2" (typepred "N!1")
                (("2" (name "N" "complement(N!1)")
                  (("2"
                    (name "NN"
                          "cross_product(complement(N!1),fullset[T2])")
                    (("2"
                      (lemma "sigma_algebra_complement[T1,S1]"
                       ("x" "N!1"))
                      (("1" (expand "member")
                        (("1" (replace -3)
                          (("1"
                            (lemma "cross_product_is_sigma_times"
                             ("Y" "fullset[T2]" "X" "N"))
                            (("1" (name "H" "*[[T1,T2]](phi(NN),h!1)")
                              (("1"
                                (name "F" "*[T1](phi(N),f!1)")
                                (("1"
                                  (case
                                   "forall (x:T1): integrable?(LAMBDA (y: T2): H(x, y))")
                                  (("1"
                                    (case
                                     "forall (x:T1): integral(LAMBDA (y: T2): H(x, y))=F(x)")
                                    (("1"
                                      (hide -10)
                                      (("1"
                                        (case "integrable?(F)")
                                        (("1"
                                          (case
                                           "forall (x:T1): F(x)>=0")
                                          (("1"
                                            (case
                                             "forall (z:[T1,T2]): H(z)>=0")
                                            (("1"
                                              (case
                                               "nn_integrable?(H)")
                                              (("1"
                                                (lemma
                                                 "integral_ae_eq"
                                                 ("f" "H" "h" "h!1"))
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (hide 2 -1)
                                                    (("1"
                                                      (expand "ae_eq?")
                                                      (("1"
                                                        (expand
                                                         "restrict")
                                                        (("1"
                                                          (expand
                                                           "pointwise_ae?")
                                                          (("1"
                                                            (expand
                                                             "ae?")
                                                            (("1"
                                                              (expand
                                                               "fullset"
                                                               1)
                                                              (("1"
                                                                (expand
                                                                 "H")
                                                                (("1"
                                                                  (expand
                                                                   "*")
                                                                  (("1"
                                                                    (expand
                                                                     "ae_in?")
                                                                    (("1"
                                                                      (expand
                                                                       "member")
                                                                      (("1"
                                                                        (inst
                                                                         +
                                                                         "cross_product(N!1,fullset[T2])")
                                                                        (("1"
                                                                          (skolem
                                                                           +
                                                                           "y!1")
                                                                          (("1"
                                                                            (flatten)
                                                                            (("1"
                                                                              (expand
                                                                               "NN")
                                                                              (("1"
                                                                                (expand
                                                                                 "fullset")
                                                                                (("1"
                                                                                  (expand
                                                                                   "complement")
                                                                                  (("1"
                                                                                    (expand
                                                                                     "cross_product")
                                                                                    (("1"
                                                                                      (expand
                                                                                       "phi")
                                                                                      (("1"
                                                                                        (expand
                                                                                         "member")
                                                                                        (("1"
                                                                                          (assert)
                                                                                          nil
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (expand
                                                                           "negligible_set?")
                                                                          (("2"
                                                                            (inst
                                                                             +
                                                                             "cross_product[T1, T2](N!1, fullset[T2])")
                                                                            (("2"
                                                                              (split)
                                                                              (("1"
                                                                                (hide-all-but
                                                                                 (-12
                                                                                  1))
                                                                                (("1"
                                                                                  (expand
                                                                                   "null_set?")
                                                                                  (("1"
                                                                                    (expand
                                                                                     "mu_fin?")
                                                                                    (("1"
                                                                                      (expand
                                                                                       "to_measure")
                                                                                      (("1"
                                                                                        (expand
                                                                                         "mu")
                                                                                        (("1"
                                                                                          (expand
                                                                                           "to_measure")
                                                                                          (("1"
                                                                                            (flatten)
                                                                                            (("1"
                                                                                              (expand
                                                                                               "measurable_set?")
                                                                                              (("1"
                                                                                                (rewrite
                                                                                                 "cross_product_is_sigma_times")
                                                                                                (("1"
                                                                                                  (expand
                                                                                                   "fm_times")
                                                                                                  (("1"
                                                                                                    (lemma
                                                                                                     "rectangle_measure1"
                                                                                                     ("M"
                                                                                                      "cross_product[T1, T2](N!1,fullset[T2])"
                                                                                                      "Y"
                                                                                                      "fullset[T2]"
                                                                                                      "X"
                                                                                                      "N!1"
                                                                                                      "mu"
                                                                                                      "mu"
                                                                                                      "nu"
                                                                                                      "nu"))
                                                                                                    (("1"
                                                                                                      (assert)
                                                                                                      nil
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (rewrite
                                                                                                       "cross_product_is_sigma_times")
                                                                                                      nil
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (expand
                                                                                 "subset?")
                                                                                (("2"
                                                                                  (skosimp)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (typepred "h!1")
                                                  (("2"
                                                    (expand
                                                     "nn_measurable?")
                                                    (("2"
                                                      (flatten)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("3"
                                                  (lemma
                                                   "nn_integrable_is_integrable[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]")
                                                  (("3"
                                                    (inst - "H")
                                                    nil
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide 2)
                                                (("2"
                                                  (case
                                                   "nn_measurable?(H)")
                                                  (("1"
                                                    (hide
                                                     (-7
                                                      -8
                                                      -9
                                                      -10
                                                      -11
                                                      -12
                                                      -13))
                                                    (("1"
                                                      (lemma
                                                       "nn_integrable_is_nn_integrable"
                                                       ("f" "F"))
                                                      (("1"
                                                        (replace -4)
                                                        (("1"
                                                          (lemma
                                                           "nn_measurable_def"
                                                           ("f" "H"))
                                                          (("1"
                                                            (split -1)
                                                            (("1"
                                                              (flatten)
                                                              (("1"
                                                                (hide
                                                                 -2)
                                                                (("1"
                                                                  (split
                                                                   -1)
                                                                  (("1"
                                                                    (skolem
                                                                     -
                                                                     "HH")
                                                                    (("1"
                                                                      (typepred
                                                                       "HH")
                                                                      (("1"
                                                                        (hide
                                                                         -3
                                                                         -4)
                                                                        (("1"
                                                                          (case
                                                                           "forall (n:nat): isf?(HH(n))")
                                                                          (("1"
                                                                            (case
                                                                             "forall (n:nat,z:[T1, T2]): HH(n)(z) >= 0")
                                                                            (("1"
                                                                              (name
                                                                               "FF"
                                                                               "lambda (n:nat): lambda (x:T1): isf_integral(LAMBDA (y: T2): HH(n)(x, y))")
                                                                              (("1"
                                                                                (case
                                                                                 "forall (n:nat): integrable?(FF(n))")
                                                                                (("1"
                                                                                  (case
                                                                                   "pointwise_increasing?(FF)")
                                                                                  (("1"
                                                                                    (case
                                                                                     "pointwise_converges_upto?(FF, F)")
                                                                                    (("1"
                                                                                      (case
                                                                                       "bounded?(integral o HH)")
                                                                                      (("1"
                                                                                        (lemma
                                                                                         "monotone_convergence_scaf[[T1, T2], sigma_times(S1, S2),
       to_measure(fm_times(mu, nu))]"
                                                                                         ("F"
                                                                                          "HH"
                                                                                          "f"
                                                                                          "H"))
                                                                                        (("1"
                                                                                          (assert)
                                                                                          (("1"
                                                                                            (split
                                                                                             -1)
                                                                                            (("1"
                                                                                              (flatten)
                                                                                              (("1"
                                                                                                (lemma
                                                                                                 "nn_integrable_is_nn_integrable"
                                                                                                 ("f"
                                                                                                  "H"))
                                                                                                (("1"
                                                                                                  (replace
                                                                                                   -13)
                                                                                                  (("1"
                                                                                                    (propax)
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("2"
                                                                                              (hide-all-but
                                                                                               (-1
                                                                                                1))
                                                                                              (("2"
                                                                                                (expand
                                                                                                 "bounded?")
                                                                                                (("2"
                                                                                                  (skosimp)
                                                                                                  (("2"
                                                                                                    (split)
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "bounded_above?")
                                                                                                      (("1"
                                                                                                        (inst
                                                                                                         +
                                                                                                         "c!1")
                                                                                                        (("1"
                                                                                                          (skosimp)
                                                                                                          (("1"
                                                                                                            (inst
                                                                                                             -
                                                                                                             "x!1")
                                                                                                            (("1"
                                                                                                              (assert)
                                                                                                              nil
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (expand
                                                                                                       "bounded_below?")
                                                                                                      (("2"
                                                                                                        (inst
                                                                                                         +
                                                                                                         "-c!1")
                                                                                                        (("2"
                                                                                                          (skosimp)
                                                                                                          (("2"
                                                                                                            (inst
                                                                                                             -
                                                                                                             "x!1")
                                                                                                            (("2"
                                                                                                              (assert)
                                                                                                              nil
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (case-replace
                                                                                         "integral o HH=isf_integral o HH")
                                                                                        (("1"
                                                                                          (expand
                                                                                           "bounded?")
                                                                                          (("1"
                                                                                            (inst
                                                                                             +
                                                                                             "integral(F)")
                                                                                            (("1"
                                                                                              (skolem
                                                                                               +
                                                                                               "n!1")
                                                                                              (("1"
                                                                                                (expand
                                                                                                 "o ")
                                                                                                (("1"
                                                                                                  (expand
                                                                                                   "abs")
                                                                                                  (("1"
                                                                                                    (lemma
                                                                                                     "isf_integral_pos"
                                                                                                     ("i"
                                                                                                      "HH(n!1)"))
                                                                                                    (("1"
                                                                                                      (inst
                                                                                                       -7
                                                                                                       "n!1"
                                                                                                       "_")
                                                                                                      (("1"
                                                                                                        (replace
                                                                                                         -7)
                                                                                                        (("1"
                                                                                                          (assert)
                                                                                                          (("1"
                                                                                                            (hide
                                                                                                             -1
                                                                                                             2)
                                                                                                            (("1"
                                                                                                              (rewrite
                                                                                                               "isf_fubini_tonelli_3")
                                                                                                              (("1"
                                                                                                                (lemma
                                                                                                                 "integral_ae_le"
                                                                                                                 ("f1"
                                                                                                                  "FF(n!1)"
                                                                                                                  "f2"
                                                                                                                  "F"))
                                                                                                                (("1"
                                                                                                                  (split
                                                                                                                   -1)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "FF")
                                                                                                                    (("1"
                                                                                                                      (propax)
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (hide-all-but
                                                                                                                     (-2
                                                                                                                      -3
                                                                                                                      1))
                                                                                                                    (("2"
                                                                                                                      (expand
                                                                                                                       "pointwise_converges_upto?")
                                                                                                                      (("2"
                                                                                                                        (expand
                                                                                                                         "pointwise_increasing?")
                                                                                                                        (("2"
                                                                                                                          (expand
                                                                                                                           "pointwise_convergence?")
                                                                                                                          (("2"
                                                                                                                            (expand
                                                                                                                             "ae_le?")
                                                                                                                            (("2"
                                                                                                                              (expand
                                                                                                                               "pointwise_ae?")
                                                                                                                              (("2"
                                                                                                                                (expand
                                                                                                                                 "ae?")
                                                                                                                                (("2"
                                                                                                                                  (expand
                                                                                                                                   "fullset")
                                                                                                                                  (("2"
                                                                                                                                    (expand
                                                                                                                                     "ae_in?")
                                                                                                                                    (("2"
                                                                                                                                      (inst
                                                                                                                                       +
                                                                                                                                       "emptyset")
                                                                                                                                      (("2"
                                                                                                                                        (skosimp)
                                                                                                                                        (("2"
                                                                                                                                          (hide
                                                                                                                                           1)
                                                                                                                                          (("2"
                                                                                                                                            (inst
                                                                                                                                             -
                                                                                                                                             "x!1")
                                                                                                                                            (("2"
                                                                                                                                              (inst
                                                                                                                                               -
                                                                                                                                               "x!1")
                                                                                                                                              (("2"
                                                                                                                                                (rewrite
                                                                                                                                                 "metric_convergence_def")
                                                                                                                                                (("2"
                                                                                                                                                  (expand
                                                                                                                                                   "metric_converges_to")
                                                                                                                                                  (("2"
                                                                                                                                                    (assert)
                                                                                                                                                    (("2"
                                                                                                                                                      (inst
                                                                                                                                                       -
                                                                                                                                                       "FF(n!1)(x!1)-F(x!1)")
                                                                                                                                                      (("2"
                                                                                                                                                        (skosimp)
                                                                                                                                                        (("2"
                                                                                                                                                          (expand
                                                                                                                                                           "increasing?")
                                                                                                                                                          (("2"
                                                                                                                                                            (inst
                                                                                                                                                             -
                                                                                                                                                             "n!1+n!2")
                                                                                                                                                            (("2"
                                                                                                                                                              (inst
                                                                                                                                                               -
                                                                                                                                                               "n!1"
                                                                                                                                                               "n!1+n!2")
                                                                                                                                                              (("2"
                                                                                                                                                                (expand
                                                                                                                                                                 "ball")
                                                                                                                                                                (("2"
                                                                                                                                                                  (assert)
                                                                                                                                                                  nil
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("2"
                                                                                                                  (inst
                                                                                                                   -4
                                                                                                                   "n!1")
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (inst
                                                                                                       -7
                                                                                                       "n!1")
                                                                                                      nil
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("2"
                                                                                              (lemma
                                                                                               "integral_nonneg"
                                                                                               ("f"
                                                                                                "F"))
                                                                                              (("2"
                                                                                                (assert)
                                                                                                nil
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil)
                                                                                         ("2"
                                                                                          (hide
                                                                                           -4
                                                                                           2
                                                                                           3)
                                                                                          (("2"
                                                                                            (apply-extensionality
                                                                                             :hide?
                                                                                             t)
                                                                                            (("2"
                                                                                              (expand
                                                                                               "o ")
                                                                                              (("2"
                                                                                                (inst
                                                                                                 -5
                                                                                                 "x!1")
                                                                                                (("2"
                                                                                                  (rewrite
                                                                                                   "isf_integral")
                                                                                                  nil
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("3"
                                                                                        (skolem
                                                                                         +
                                                                                         "n!1")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -6
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (lemma
                                                                                             "isf_is_integrable[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]")
                                                                                            (("3"
                                                                                              (inst
                                                                                               -
                                                                                               "HH(n!1)")
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil)
                                                                                     ("2"
                                                                                      (expand
                                                                                       "pointwise_converges_upto?")
                                                                                      (("2"
                                                                                        (assert)
                                                                                        (("2"
                                                                                          (hide
                                                                                           2)
                                                                                          (("2"
                                                                                            (expand
                                                                                             "pointwise_convergence?")
                                                                                            (("2"
                                                                                              (skosimp)
                                                                                              (("2"
                                                                                                (inst
                                                                                                 -12
                                                                                                 "x!1")
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -13
                                                                                                   "x!1")
                                                                                                  (("2"
                                                                                                    (replace
                                                                                                     -12
                                                                                                     1
                                                                                                     rl)
                                                                                                    (("2"
                                                                                                      (expand
                                                                                                       "FF"
                                                                                                       1)
                                                                                                      (("2"
                                                                                                        (lemma
                                                                                                         "monotone_convergence_scaf[T2,S2,to_measure(nu)]"
                                                                                                         ("f"
                                                                                                          "LAMBDA (y: T2): H(x!1, y)"
                                                                                                          "F"
                                                                                                          "lambda (n:nat): lambda (y:T2): HH(n)(x!1, y)"))
                                                                                                        (("1"
                                                                                                          (split
                                                                                                           -1)
                                                                                                          (("1"
                                                                                                            (flatten)
                                                                                                            (("1"
                                                                                                              (expand
                                                                                                               "converges_upto?")
                                                                                                              (("1"
                                                                                                                (flatten)
                                                                                                                (("1"
                                                                                                                  (expand
                                                                                                                   "o")
                                                                                                                  (("1"
                                                                                                                    (case-replace
                                                                                                                     "(LAMBDA (x_1: nat):
                     integral(LAMBDA (y: T2): HH(x_1)(x!1, y)))=LAMBDA (n:nat): isf_integral(LAMBDA (y: T2): HH(n)(x!1, y))")
                                                                                                                    (("1"
                                                                                                                      (apply-extensionality
                                                                                                                       :hide?
                                                                                                                       t)
                                                                                                                      (("1"
                                                                                                                        (inst
                                                                                                                         -8
                                                                                                                         "x!2")
                                                                                                                        (("1"
                                                                                                                          (lemma
                                                                                                                           "isf_x_section"
                                                                                                                           ("x"
                                                                                                                            "x!1"
                                                                                                                            "i"
                                                                                                                            "HH(x!2)"))
                                                                                                                          (("1"
                                                                                                                            (rewrite
                                                                                                                             "isf_integral"
                                                                                                                             1)
                                                                                                                            nil
                                                                                                                            nil)
                                                                                                                           ("2"
                                                                                                                            (propax)
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("2"
                                                                                                                        (skosimp)
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -8
                                                                                                                           "n!1")
                                                                                                                          (("2"
                                                                                                                            (rewrite
                                                                                                                             "isf_x_section")
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("3"
                                                                                                                        (skolem
                                                                                                                         +
                                                                                                                         "n!1")
                                                                                                                        (("3"
                                                                                                                          (inst
                                                                                                                           -8
                                                                                                                           "n!1")
                                                                                                                          (("3"
                                                                                                                            (lemma
                                                                                                                             "isf_x_section"
                                                                                                                             ("x"
                                                                                                                              "x!1"
                                                                                                                              "i"
                                                                                                                              "HH(n!1)"))
                                                                                                                            (("1"
                                                                                                                              (lemma
                                                                                                                               "isf_is_integrable[T2, S2, to_measure(nu)]")
                                                                                                                              (("1"
                                                                                                                                (inst
                                                                                                                                 -
                                                                                                                                 "LAMBDA (y: T2): HH(n!1)(x!1, y)")
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil)
                                                                                                                             ("2"
                                                                                                                              (propax)
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil)
                                                                                                                     ("2"
                                                                                                                      (skosimp)
                                                                                                                      (("2"
                                                                                                                        (inst
                                                                                                                         -8
                                                                                                                         "n!1")
                                                                                                                        (("2"
                                                                                                                          (rewrite
                                                                                                                           "isf_x_section")
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (hide
                                                                                                             2)
                                                                                                            (("2"
                                                                                                              (expand
                                                                                                               "pointwise_converges_upto?")
                                                                                                              (("2"
                                                                                                                (split
                                                                                                                 1)
                                                                                                                (("1"
                                                                                                                  (expand
                                                                                                                   "pointwise_convergence?")
                                                                                                                  (("1"
                                                                                                                    (skolem
                                                                                                                     +
                                                                                                                     "y!1")
                                                                                                                    (("1"
                                                                                                                      (inst
                                                                                                                       -7
                                                                                                                       "(x!1,y!1)")
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("2"
                                                                                                                  (expand
                                                                                                                   "pointwise_increasing?")
                                                                                                                  (("2"
                                                                                                                    (skolem
                                                                                                                     +
                                                                                                                     "y!1")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       -8
                                                                                                                       "(x!1,y!1)")
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("3"
                                                                                                            (hide
                                                                                                             2)
                                                                                                            (("3"
                                                                                                              (expand
                                                                                                               "bounded?")
                                                                                                              (("3"
                                                                                                                (expand
                                                                                                                 "o")
                                                                                                                (("3"
                                                                                                                  (split
                                                                                                                   1)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "bounded_above?")
                                                                                                                    (("1"
                                                                                                                      (inst
                                                                                                                       +
                                                                                                                       "F(x!1)")
                                                                                                                      (("1"
                                                                                                                        (skolem
                                                                                                                         +
                                                                                                                         "n!1")
                                                                                                                        (("1"
                                                                                                                          (lemma
                                                                                                                           "integral_ae_le[T2,S2,to_measure(nu)]"
                                                                                                                           ("f1"
                                                                                                                            "LAMBDA (y: T2): HH(n!1)(x!1, y)"
                                                                                                                            "f2"
                                                                                                                            "LAMBDA (y: T2): H(x!1,y)"))
                                                                                                                          (("1"
                                                                                                                            (split
                                                                                                                             -1)
                                                                                                                            (("1"
                                                                                                                              (assert)
                                                                                                                              nil
                                                                                                                              nil)
                                                                                                                             ("2"
                                                                                                                              (hide-all-but
                                                                                                                               (1
                                                                                                                                -6
                                                                                                                                -7))
                                                                                                                              (("2"
                                                                                                                                (expand
                                                                                                                                 "increasing_nn_simple?")
                                                                                                                                (("2"
                                                                                                                                  (flatten)
                                                                                                                                  (("2"
                                                                                                                                    (hide
                                                                                                                                     -1)
                                                                                                                                    (("2"
                                                                                                                                      (expand
                                                                                                                                       "pointwise_increasing?")
                                                                                                                                      (("2"
                                                                                                                                        (expand
                                                                                                                                         "ae_le?")
                                                                                                                                        (("2"
                                                                                                                                          (expand
                                                                                                                                           "pointwise_ae?")
                                                                                                                                          (("2"
                                                                                                                                            (expand
                                                                                                                                             "ae?")
                                                                                                                                            (("2"
                                                                                                                                              (expand
                                                                                                                                               "fullset")
                                                                                                                                              (("2"
                                                                                                                                                (expand
                                                                                                                                                 "ae_in?")
                                                                                                                                                (("2"
                                                                                                                                                  (inst
                                                                                                                                                   +
                                                                                                                                                   "emptyset")
                                                                                                                                                  (("2"
                                                                                                                                                    (skolem
                                                                                                                                                     +
                                                                                                                                                     "y!1")
                                                                                                                                                    (("2"
                                                                                                                                                      (expand
                                                                                                                                                       "member")
                                                                                                                                                      (("2"
                                                                                                                                                        (flatten)
                                                                                                                                                        (("2"
                                                                                                                                                          (hide
                                                                                                                                                           1)
                                                                                                                                                          (("2"
                                                                                                                                                            (inst
                                                                                                                                                             -
                                                                                                                                                             "(x!1,y!1)")
                                                                                                                                                            (("2"
                                                                                                                                                              (inst
                                                                                                                                                               -
                                                                                                                                                               "(x!1,y!1)")
                                                                                                                                                              (("2"
                                                                                                                                                                (rewrite
                                                                                                                                                                 "metric_convergence_def")
                                                                                                                                                                (("2"
                                                                                                                                                                  (expand
                                                                                                                                                                   "metric_converges_to")
                                                                                                                                                                  (("2"
                                                                                                                                                                    (inst
                                                                                                                                                                     -
                                                                                                                                                                     "HH(n!1)(x!1, y!1)-H(x!1, y!1)")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (skosimp)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (inst
                                                                                                                                                                         -
                                                                                                                                                                         "n!1+n!2")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (expand
                                                                                                                                                                           "ball")
                                                                                                                                                                          (("1"
                                                                                                                                                                            (expand
                                                                                                                                                                             "increasing?")
                                                                                                                                                                            (("1"
                                                                                                                                                                              (inst
                                                                                                                                                                               -
                                                                                                                                                                               "n!1"
                                                                                                                                                                               "n!1+n!2")
                                                                                                                                                                              (("1"
                                                                                                                                                                                (assert)
                                                                                                                                                                                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))
                                                                                                                              nil))
                                                                                                                            nil)
                                                                                                                           ("2"
                                                                                                                            (propax)
                                                                                                                            nil
                                                                                                                            nil)
                                                                                                                           ("3"
                                                                                                                            (lemma
                                                                                                                             "isf_x_section"
                                                                                                                             ("x"
                                                                                                                              "x!1"
                                                                                                                              "i"
                                                                                                                              "HH(n!1)"))
                                                                                                                            (("1"
                                                                                                                              (lemma
                                                                                                                               "isf_is_integrable[T2, S2, to_measure(nu)]")
                                                                                                                              (("1"
                                                                                                                                (inst
                                                                                                                                 -
                                                                                                                                 "LAMBDA (y: T2): HH(n!1)(x!1, y)")
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil)
                                                                                                                             ("2"
                                                                                                                              (inst
                                                                                                                               -5
                                                                                                                               "n!1")
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (expand
                                                                                                                     "bounded_below?")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       +
                                                                                                                       "0")
                                                                                                                      (("2"
                                                                                                                        (skolem
                                                                                                                         +
                                                                                                                         "n!1")
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -5
                                                                                                                           "n!1")
                                                                                                                          (("2"
                                                                                                                            (lemma
                                                                                                                             "isf_x_section"
                                                                                                                             ("x"
                                                                                                                              "x!1"
                                                                                                                              "i"
                                                                                                                              "HH(n!1)"))
                                                                                                                            (("1"
                                                                                                                              (rewrite
                                                                                                                               "isf_integral"
                                                                                                                               1)
                                                                                                                              (("1"
                                                                                                                                (lemma
                                                                                                                                 "isf_integral_pos"
                                                                                                                                 ("i"
                                                                                                                                  "LAMBDA (y: T2): HH(n!1)(x!1, y)"))
                                                                                                                                (("1"
                                                                                                                                  (assert)
                                                                                                                                  (("1"
                                                                                                                                    (skolem
                                                                                                                                     +
                                                                                                                                     "y!1")
                                                                                                                                    (("1"
                                                                                                                                      (inst
                                                                                                                                       -5
                                                                                                                                       "n!1"
                                                                                                                                       "(x!1,y!1)")
                                                                                                                                      nil
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("2"
                                                                                                                                  (propax)
                                                                                                                                  nil
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil)
                                                                                                                             ("2"
                                                                                                                              (propax)
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (skosimp)
                                                                                                          (("2"
                                                                                                            (inst
                                                                                                             -5
                                                                                                             "n!1")
                                                                                                            (("2"
                                                                                                              (lemma
                                                                                                               "isf_x_section"
                                                                                                               ("x"
                                                                                                                "x!1"
                                                                                                                "i"
                                                                                                                "HH(n!1)"))
                                                                                                              (("1"
                                                                                                                (lemma
                                                                                                                 "isf_is_integrable[T2, S2, to_measure(nu)]")
                                                                                                                (("1"
                                                                                                                  (inst
                                                                                                                   -
                                                                                                                   "LAMBDA (y: T2): HH(n!1)(x!1, y)")
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil)
                                                                                                               ("2"
                                                                                                                (propax)
                                                                                                                nil
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (expand
                                                                                     "increasing_nn_simple?")
                                                                                    (("2"
                                                                                      (flatten)
                                                                                      (("2"
                                                                                        (expand
                                                                                         "pointwise_increasing?")
                                                                                        (("2"
                                                                                          (skosimp)
                                                                                          (("2"
                                                                                            (expand
                                                                                             "FF")
                                                                                            (("2"
                                                                                              (hide-all-but
                                                                                               (-4
                                                                                                -6
                                                                                                1))
                                                                                              (("2"
                                                                                                (expand
                                                                                                 "increasing?")
                                                                                                (("2"
                                                                                                  (skolem
                                                                                                   +
                                                                                                   ("i!1"
                                                                                                    "j!1"))
                                                                                                  (("2"
                                                                                                    (flatten)
                                                                                                    (("2"
                                                                                                      (inst-cp
                                                                                                       -
                                                                                                       "j!1")
                                                                                                      (("2"
                                                                                                        (inst
                                                                                                         -
                                                                                                         "i!1")
                                                                                                        (("2"
                                                                                                          (lemma
                                                                                                           "isf_x_section"
                                                                                                           ("x"
                                                                                                            "x!1"
                                                                                                            "i"
                                                                                                            "HH(j!1)"))
                                                                                                          (("1"
                                                                                                            (lemma
                                                                                                             "isf_x_section"
                                                                                                             ("x"
                                                                                                              "x!1"
                                                                                                              "i"
                                                                                                              "HH(i!1)"))
                                                                                                            (("1"
                                                                                                              (lemma
                                                                                                               "isf_integral_le"
                                                                                                               ("i1"
                                                                                                                "LAMBDA (y: T2): HH(i!1)(x!1, y)"
                                                                                                                "i2"
                                                                                                                "LAMBDA (y: T2): HH(j!1)(x!1, y)"))
                                                                                                              (("1"
                                                                                                                (assert)
                                                                                                                (("1"
                                                                                                                  (skolem
                                                                                                                   +
                                                                                                                   "y!1")
                                                                                                                  (("1"
                                                                                                                    (inst
                                                                                                                     -6
                                                                                                                     "(x!1,y!1)")
                                                                                                                    (("1"
                                                                                                                      (inst
                                                                                                                       -6
                                                                                                                       "i!1"
                                                                                                                       "j!1")
                                                                                                                      (("1"
                                                                                                                        (assert)
                                                                                                                        nil
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil)
                                                                                                               ("2"
                                                                                                                (propax)
                                                                                                                nil
                                                                                                                nil)
                                                                                                               ("3"
                                                                                                                (propax)
                                                                                                                nil
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (propax)
                                                                                                              nil
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (propax)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (skosimp)
                                                                                  (("2"
                                                                                    (hide
                                                                                     2
                                                                                     -1)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "FF")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -2
                                                                                         "n!1")
                                                                                        (("2"
                                                                                          (lemma
                                                                                           "isf_integral_x"
                                                                                           ("i"
                                                                                            "HH(n!1)"))
                                                                                          (("1"
                                                                                            (propax)
                                                                                            nil
                                                                                            nil)
                                                                                           ("2"
                                                                                            (propax)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (skosimp)
                                                                                (("2"
                                                                                  (inst
                                                                                   -2
                                                                                   "n!1")
                                                                                  (("2"
                                                                                    (rewrite
                                                                                     "isf_x_section")
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("2"
                                                                              (skosimp)
                                                                              (("2"
                                                                                (expand
                                                                                 "increasing_nn_simple?")
                                                                                (("2"
                                                                                  (expand
                                                                                   "nn_simple?")
                                                                                  (("2"
                                                                                    (flatten)
                                                                                    (("2"
                                                                                      (inst
                                                                                       -2
                                                                                       "n!1")
                                                                                      (("2"
                                                                                        (flatten)
                                                                                        (("2"
                                                                                          (inst
                                                                                           -2
                                                                                           "z!1")
                                                                                          (("2"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (skosimp)
                                                                            (("2"
                                                                              (expand
                                                                               "increasing_nn_simple?")
                                                                              (("2"
                                                                                (flatten)
                                                                                (("2"
                                                                                  (inst
                                                                                   -
                                                                                   "n!1")
                                                                                  (("2"
                                                                                    (expand
                                                                                     "nn_simple?")
                                                                                    (("2"
                                                                                      (flatten)
                                                                                      (("2"
                                                                                        (expand
                                                                                         "isf?")
                                                                                        (("2"
                                                                                          (expand
                                                                                           "mu_fin?")
                                                                                          (("2"
                                                                                            (expand
                                                                                             "to_measure")
                                                                                            (("2"
                                                                                              (propax)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (expand
                                                                     "nn_measurable?")
                                                                    (("2"
                                                                      (flatten)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (skosimp)
                                                              (("2"
                                                                (inst
                                                                 -
                                                                 "x!1")
                                                                (("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide -3 -4 -5 2)
                                                    (("2"
                                                      (typepred "h!1")
                                                      (("2"
                                                        (expand
                                                         "nn_measurable?")
                                                        (("2"
                                                          (flatten)
                                                          (("2"
                                                            (split)
                                                            (("1"
                                                              (expand
                                                               "H")
                                                              (("1"
                                                                (rewrite
                                                                 "prod_measurable")
                                                                (("1"
                                                                  (lemma
                                                                   "phi_is_simple[[T1, T2], sigma_times(S1, S2)]"
                                                                   ("X"
                                                                    "NN"))
                                                                  (("1"
                                                                    (expand
                                                                     "simple?")
                                                                    (("1"
                                                                      (propax)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (skosimp)
                                                              (("2"
                                                                (inst
                                                                 -3
                                                                 "x!1")
                                                                (("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3" (propax) nil nil))
                                              nil)
                                             ("2"
                                              (skosimp)
                                              (("2"
                                                (expand "H")
                                                (("2"
                                                  (typepred "h!1")
                                                  (("2"
                                                    (expand
                                                     "nn_measurable?")
                                                    (("2"
                                                      (flatten)
                                                      (("2"
                                                        (inst - "z!1")
                                                        (("2"
                                                          (expand "*")
                                                          (("2"
                                                            (expand
                                                             "phi")
                                                            (("2"
                                                              (lift-if
                                                               1)
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (skosimp)
                                            (("2"
                                              (inst - "x!1")
                                              (("2"
                                                (inst - "x!1")
                                                (("2"
                                                  (lemma
                                                   "integral_nonneg"
                                                   ("f"
                                                    "LAMBDA (y: T2): H(x!1, y)"))
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (skolem + "y!1")
                                                      (("1"
                                                        (expand "H")
                                                        (("1"
                                                          (expand "*")
                                                          (("1"
                                                            (expand
                                                             "phi")
                                                            (("1"
                                                              (lift-if)
                                                              (("1"
                                                                (typepred
                                                                 "h!1")
                                                                (("1"
                                                                  (expand
                                                                   "nn_measurable?")
                                                                  (("1"
                                                                    (flatten)
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "(x!1,y!1)")
                                                                      (("1"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (typepred "f!1")
                                          (("2"
                                            (expand "F")
                                            (("2"
                                              (rewrite
                                               "indefinite_integrable")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide 2)
                                      (("2"
                                        (skosimp)
                                        (("2"
                                          (inst - "x!1")
                                          (("2"
                                            (inst -9 "x!1")
                                            (("2"
                                              (expand "H")
                                              (("2"
                                                (expand "F")
                                                (("2"
                                                  (expand "*")
                                                  (("2"
                                                    (expand "phi")
                                                    (("2"
                                                      (expand "NN")
                                                      (("2"
                                                        (expand
                                                         "cross_product")
                                                        (("2"
                                                          (expand
                                                           "fullset")
                                                          (("2"
                                                            (expand
                                                             "complement")
                                                            (("2"
                                                              (expand
                                                               "member")
                                                              (("2"
                                                                (expand
                                                                 "N")
                                                                (("2"
                                                                  (expand
                                                                   "complement")
                                                                  (("2"
                                                                    (expand
                                                                     "member")
                                                                    (("2"
                                                                      (lift-if
                                                                       1)
                                                                      (("2"
                                                                        (case-replace
                                                                         "N!1(x!1)")
                                                                        (("1"
                                                                          (rewrite
                                                                           "integral_zero")
                                                                          (("1"
                                                                            (assert)
                                                                            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)
                                     ("3" (propax) nil nil))
                                    nil)
                                   ("2"
                                    (skosimp)
                                    (("2"
                                      (expand "H")
                                      (("2"
                                        (expand "NN")
                                        (("2"
                                          (expand "cross_product")
                                          (("2"
                                            (expand "fullset")
                                            (("2"
                                              (expand "complement")
                                              (("2"
                                                (expand "phi")
                                                (("2"
                                                  (expand "member")
                                                  (("2"
                                                    (expand "*")
                                                    (("2"
                                                      (inst -8 "x!1")
                                                      (("2"
                                                        (case-replace
                                                         "N!1(x!1)")
                                                        (("1"
                                                          (rewrite
                                                           "integrable_zero")
                                                          nil
                                                          nil)
                                                         ("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (propax) nil nil))
                            nil))
                          nil))
                        nil)
                       ("2" (expand "null_set?")
                        (("2" (expand "measurable_set?")
                          (("2" (flatten) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nn_measurable nonempty-type-eq-decl nil measure_space nil)
    (nn_measurable? const-decl "bool" measure_space nil)
    (S2 formal-const-decl "sigma_algebra[T2]" finite_fubini_tonelli
     nil)
    (S1 formal-const-decl "sigma_algebra[T1]" finite_fubini_tonelli
     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 finite_fubini_tonelli nil)
    (T1 formal-type-decl nil finite_fubini_tonelli nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (nn_integral_def formula-decl nil nn_integral nil)
    (nn_integrable? const-decl "bool" nn_integral nil)
    (nn_integrable nonempty-type-eq-decl nil nn_integral nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (isf_integral const-decl "real" isf nil)
    (isf_integral_x formula-decl nil finite_fubini_scaf nil)
    (integral const-decl "real" integral nil)
    (bounded? const-decl "bool" sup_norm nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (integrable1? const-decl "bool" product_integral_def nil)
    (fullset const-decl "set" sets nil)
    (ae_in? const-decl "bool" measure_theory nil)
    (member const-decl "bool" sets nil)
    (TRUE const-decl "bool" booleans nil)
    (monotone_convergence_scaf formula-decl nil
     integral_convergence_scaf nil)
    (metric_converges_to const-decl "bool" metric_space_def
     "metric_space/")
    (increasing? const-decl "bool" real_fun_preds "reals/")
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (ball const-decl "set[T]" metric_space_def "metric_space/")
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (F skolem-const-decl "integrable[T1, S1, to_measure(mu)]"
     finite_fubini_tonelli nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_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)
    (metric_convergence_def formula-decl nil metric_space
     "metric_space/")
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (bounded? const-decl "bool" real_fun_preds "reals/")
    (pointwise_converges_upto? const-decl "bool" pointwise_convergence
     nil)
    (pointwise_convergence? const-decl "bool" pointwise_convergence
     nil)
    (x!1 skolem-const-decl "T1" finite_fubini_tonelli nil)
    (HH skolem-const-decl "increasing_nn_isf
    [[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]"
     finite_fubini_tonelli nil)
    (isf_x_section formula-decl nil finite_fubini_scaf nil)
    (isf_integral formula-decl nil integral nil)
    (FF skolem-const-decl "[nat -> [T1 -> real]]" finite_fubini_tonelli
     nil)
    (isf_is_integrable judgement-tcc nil integral nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (converges_upto? const-decl "bool" convergence_aux "metric_space/")
    (metric_induced_topology const-decl "setofsets[T]" metric_space_def
     "metric_space/")
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (unique_limit formula-decl nil hausdorff_convergence "topology/")
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (subset? const-decl "bool" sets nil)
    (null_set nonempty-type-eq-decl nil measure_theory nil)
    (N1 skolem-const-decl "set[T1]" finite_fubini_tonelli nil)
    (null_set? const-decl "bool" measure_theory nil)
    (negligible nonempty-type-eq-decl nil measure_theory nil)
    (negligible_set? const-decl "bool" measure_theory nil)
    (set type-eq-decl nil sets nil)
    (ae_convergence_in? const-decl "bool" measure_theory nil)
    (ae_convergence? const-decl "bool" measure_theory nil)
    (ae_increasing? const-decl "bool" measure_theory nil)
    (emptyset const-decl "set" sets 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/")
    (finite_emptyset name-judgement "finite_set[T]" sigma_countable
     "sigma_set/")
    (subset_algebra_emptyset name-judgement "(S)" finite_fubini_tonelli
     nil)
    (null_emptyset name-judgement "null_set" finite_fubini_tonelli nil)
    (monotone_convergence formula-decl nil integral_convergence nil)
    (nn_integral const-decl "nnreal" nn_integral nil)
    (isf_integral_pos formula-decl nil isf nil)
    (<= const-decl "bool" reals nil)
    (isf_integral_le formula-decl nil isf nil)
    (h!1 skolem-const-decl
     "nn_measurable[[T1, T2], sigma_times(S1, S2)]"
     finite_fubini_tonelli nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (pointwise_increasing? const-decl "bool" pointwise_convergence nil)
    (O const-decl "T3" function_props nil)
    (isf_fubini_tonelli_3 formula-decl nil finite_fubini_scaf nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (isf? const-decl "bool" isf nil)
    (isf nonempty-type-eq-decl nil isf nil)
    (nn_isf? const-decl "bool" nn_integral nil)
    (nn_isf nonempty-type-eq-decl nil nn_integral nil)
    (sequence type-eq-decl nil sequences nil)
    (increasing_nn_isf? const-decl "bool" nn_integral nil)
    (increasing_nn_isf nonempty-type-eq-decl nil nn_integral nil)
    (nn_integrable_is_nn_integrable formula-decl nil integral nil)
    (integrable? const-decl "bool" integral nil)
    (integrable nonempty-type-eq-decl nil integral 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)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (measure? const-decl "bool" generalized_measure_def nil)
    (measure_type nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (to_measure const-decl "measure_type" generalized_measure_def nil)
    (fm_times const-decl
     "finite_measure[[T1, T2], sigma_times(S1, S2)]"
     product_finite_measure nil)
    (mu formal-const-decl "finite_measure[T1, S1]"
     finite_fubini_tonelli nil)
    (nu formal-const-decl "finite_measure[T2, S2]"
     finite_fubini_tonelli nil)
    (subset_algebra_complement application-judgement "(S)"
     sigma_algebra nil)
    (measurable_complement application-judgement "measurable_set"
     finite_fubini_tonelli nil)
    (complement const-decl "set" sets nil)
    (sigma_algebra_complement formula-decl nil sigma_algebra nil)
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (phi const-decl "nat" measure_space nil)
    (integral_zero formula-decl nil integral nil)
    (N skolem-const-decl "(null_set?)" finite_fubini_tonelli nil)
    (measurable_set nonempty-type-eq-decl nil measure_space_def nil)
    (indefinite_integrable formula-decl nil integral nil)
    (nn_integrable_is_integrable judgement-tcc nil integral nil)
    (ae_eq? const-decl "bool" measure_theory nil)
    (pointwise_ae? const-decl "bool" measure_theory nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (measurable_set? const-decl "bool" measure_space_def nil)
    (rectangle_measure1 formula-decl nil product_finite_measure nil)
    (mu const-decl "nnreal" measure_props nil)
    (mu_fin? const-decl "bool" measure_props nil)
    (NN skolem-const-decl "set[[T1, T2]]" finite_fubini_tonelli nil)
    (N!1 skolem-const-decl "null_set[T1, S1, to_measure(mu)]"
     finite_fubini_tonelli nil)
    (H skolem-const-decl "[[T1, T2] -> real]" finite_fubini_tonelli
     nil)
    (ae? const-decl "bool" measure_theory nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     finite_fubini_tonelli nil)
    (subset_algebra_fullset name-judgement "(S)" finite_fubini_tonelli
     nil)
    (restrict const-decl "R" restrict nil)
    (measurable_function nonempty-type-eq-decl nil measure_space_def
     nil)
    (measurable_function? const-decl "bool" measure_space_def nil)
    (integral_ae_eq formula-decl nil integral nil)
    (nn_measurable_def formula-decl nil measure_space nil)
    (increasing_nn_simple? const-decl "bool" measure_space nil)
    (increasing_nn_simple nonempty-type-eq-decl nil measure_space nil)
    (nn_simple? const-decl "bool" measure_space nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (x!1 skolem-const-decl "T1" finite_fubini_tonelli nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (null_emptyset name-judgement "null_set" finite_fubini_tonelli nil)
    (subset_algebra_emptyset name-judgement "(S)" finite_fubini_tonelli
     nil)
    (y!1 skolem-const-decl "({x | TRUE})" finite_fubini_tonelli nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (integral_nonneg formula-decl nil integral nil)
    (FF skolem-const-decl "[nat -> [T1 -> real]]" finite_fubini_tonelli
     nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     finite_fubini_tonelli nil)
    (subset_algebra_fullset name-judgement "(S)" finite_fubini_tonelli
     nil)
    (ae_le? const-decl "bool" measure_theory nil)
    (integral_ae_le formula-decl nil integral nil)
    (F skolem-const-decl "[T1 -> real]" finite_fubini_tonelli nil)
    (HH skolem-const-decl
     "increasing_nn_simple[[T1, T2], sigma_times(S1, S2)]"
     finite_fubini_tonelli nil)
    (n!1 skolem-const-decl "nat" finite_fubini_tonelli nil)
    (phi_is_simple judgement-tcc nil measure_space nil)
    (simple? const-decl "bool" measure_space nil)
    (prod_measurable judgement-tcc nil measure_space nil)
    (integrable_zero formula-decl nil integral nil)
    (cross_product_is_sigma_times formula-decl nil product_sigma nil)
    (measurable_fullset name-judgement "measurable_set[T, S]"
     finite_fubini_tonelli nil)
    (subset_algebra_fullset name-judgement "(S)" finite_fubini_tonelli
     nil)
    (cross_product const-decl "set[[T1, T2]]" cross_product
     "topology/"))
   shostak))
 (finite_fubini_tonelli_2 0
  (finite_fubini_tonelli_2-1 nil 3459084018
   ("" (skosimp)
    (("" (split)
      (("1" (flatten)
        (("1" (typepred "h!1")
          (("1" (expand "nn_measurable?")
            (("1" (flatten)
              (("1"
                (lemma "nn_integrable_is_nn_integrable" ("f" "h!1"))
                (("1" (split)
                  (("1" (hide -4)
                    (("1" (lemma "nn_integral_def" ("f" "h!1"))
                      (("1" (skolem - "HH")
                        (("1" (flatten)
                          (("1" (typepred "HH")
                            (("1" (case "FORALL (n: nat): isf?(HH(n))")
                              (("1"
                                (case
                                 "FORALL (n: nat,z:[T1,T2]): HH(n)(z)>=0")
                                (("1"
                                  (name
                                   "GG"
                                   "lambda (n:nat): lambda (y:T2): isf_integral(lambda (x:T1): HH(n)(x,y))")
                                  (("1"
                                    (case
                                     "FORALL (n: nat): integrable?[T2, S2, to_measure(nu)](GG(n))")
                                    (("1"
                                      (case
                                       "forall (n:nat): integral[T2,S2,to_measure(nu)](GG(n))=integral[[T1,T2],sigma_times(S1,S2),to_measure(fm_times(mu,nu))](HH(n))")
                                      (("1"
                                        (case-replace
                                         "(isf_integral o HH)=integral o GG")
                                        (("1"
                                          (hide -1)
                                          (("1"
                                            (case
                                             "forall (n:nat,y:T2): GG(n)(y)>=0")
                                            (("1"
                                              (case
                                               "pointwise_increasing?(GG)")
                                              (("1"
                                                (case
                                                 "bounded?(integral o GG)")
                                                (("1"
                                                  (lemma
                                                   "monotone_convergence[T2, S2, to_measure(nu)]"
                                                   ("F" "GG"))
                                                  (("1"
                                                    (split -1)
                                                    (("1"
                                                      (flatten)
                                                      (("1"
                                                        (hide -1)
                                                        (("1"
                                                          (split -1)
                                                          (("1"
                                                            (skolem
                                                             -
                                                             "G")
                                                            (("1"
                                                              (inst
                                                               -
                                                               "G")
                                                              (("1"
                                                                (assert)
                                                                (("1"
                                                                  (typepred
                                                                   "G")
                                                                  (("1"
                                                                    (expand
                                                                     "integrable2?")
                                                                    (("1"
                                                                      (expand
                                                                       "ae_convergence?")
                                                                      (("1"
                                                                        (expand
                                                                         "fullset")
                                                                        (("1"
                                                                          (expand
                                                                           "ae_convergence_in?")
                                                                          (("1"
                                                                            (expand
                                                                             "ae_in?")
                                                                            (("1"
                                                                              (skosimp)
                                                                              (("1"
                                                                                (typepred
                                                                                 "E!1")
                                                                                (("1"
                                                                                  (expand
                                                                                   "negligible_set?")
                                                                                  (("1"
                                                                                    (skolem
                                                                                     -
                                                                                     "N2")
                                                                                    (("1"
                                                                                      (flatten)
                                                                                      (("1"
                                                                                        (inst
                                                                                         +
                                                                                         "N2"
                                                                                         "G")
                                                                                        (("1"
                                                                                          (skosimp)
                                                                                          (("1"
                                                                                            (expand
                                                                                             "subset?")
                                                                                            (("1"
                                                                                              (expand
                                                                                               "member")
                                                                                              (("1"
                                                                                                (inst
                                                                                                 -
                                                                                                 "y!1")
                                                                                                (("1"
                                                                                                  (assert)
                                                                                                  (("1"
                                                                                                    (inst
                                                                                                     -3
                                                                                                     "y!1")
                                                                                                    (("1"
                                                                                                      (assert)
                                                                                                      (("1"
                                                                                                        (lemma
                                                                                                         "monotone_convergence_scaf[T1,S1,to_measure(mu)]"
                                                                                                         ("f"
                                                                                                          "LAMBDA (x: T1): h!1(x, y!1)"
                                                                                                          "F"
                                                                                                          "lambda (n:nat): lambda (x:T1): HH(n)(x,y!1)"))
                                                                                                        (("1"
                                                                                                          (split
                                                                                                           -1)
                                                                                                          (("1"
                                                                                                            (flatten)
                                                                                                            (("1"
                                                                                                              (replace
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (name-replace
                                                                                                                 "LHS"
                                                                                                                 "integral(LAMBDA (x: T1): h!1(x, y!1))")
                                                                                                                (("1"
                                                                                                                  (case-replace
                                                                                                                   "(integral o
                        (LAMBDA (n: nat): LAMBDA (x: T1): HH(n)(x, y!1)))=LAMBDA (n:nat): GG(n)(y!1)")
                                                                                                                  (("1"
                                                                                                                    (hide
                                                                                                                     -1)
                                                                                                                    (("1"
                                                                                                                      (expand
                                                                                                                       "converges_upto?")
                                                                                                                      (("1"
                                                                                                                        (flatten)
                                                                                                                        (("1"
                                                                                                                          (hide-all-but
                                                                                                                           (-2
                                                                                                                            -6
                                                                                                                            3))
                                                                                                                          (("1"
                                                                                                                            (lemma
                                                                                                                             "hausdorff_convergence.unique_limit"
                                                                                                                             ("u"
                                                                                                                              "LAMBDA (n: nat): GG(n)(y!1)"
                                                                                                                              "l1"
                                                                                                                              "LHS"
                                                                                                                              "l2"
                                                                                                                              "G(y!1)"))
                                                                                                                            (("1"
                                                                                                                              (assert)
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (hide
                                                                                                                     4)
                                                                                                                    (("2"
                                                                                                                      (apply-extensionality
                                                                                                                       :hide?
                                                                                                                       t)
                                                                                                                      (("1"
                                                                                                                        (expand
                                                                                                                         "GG"
                                                                                                                         1)
                                                                                                                        (("1"
                                                                                                                          (expand
                                                                                                                           "o"
                                                                                                                           1)
                                                                                                                          (("1"
                                                                                                                            (inst
                                                                                                                             -14
                                                                                                                             "x!1")
                                                                                                                            (("1"
                                                                                                                              (lemma
                                                                                                                               "isf_y_section"
                                                                                                                               ("y"
                                                                                                                                "y!1"
                                                                                                                                "i"
                                                                                                                                "HH(x!1)"))
                                                                                                                              (("1"
                                                                                                                                (rewrite
                                                                                                                                 "isf_integral")
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("2"
                                                                                                                        (skosimp)
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -14
                                                                                                                           "n!1")
                                                                                                                          (("2"
                                                                                                                            (lemma
                                                                                                                             "isf_y_section"
                                                                                                                             ("y"
                                                                                                                              "y!1"
                                                                                                                              "i"
                                                                                                                              "HH(n!1)"))
                                                                                                                            (("2"
                                                                                                                              (lemma
                                                                                                                               "isf_is_integrable[T1, S1, to_measure(mu)]")
                                                                                                                              (("2"
                                                                                                                                (inst
                                                                                                                                 -
                                                                                                                                 "LAMBDA (x: T1): HH(n!1)(x, y!1)")
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (hide-all-but
                                                                                                             (1
                                                                                                              -14
                                                                                                              -13))
                                                                                                            (("2"
                                                                                                              (expand
                                                                                                               "increasing_nn_isf?")
                                                                                                              (("2"
                                                                                                                (expand
                                                                                                                 "pointwise_converges_upto?")
                                                                                                                (("2"
                                                                                                                  (split)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "pointwise_convergence?")
                                                                                                                    (("1"
                                                                                                                      (skosimp)
                                                                                                                      (("1"
                                                                                                                        (inst
                                                                                                                         -
                                                                                                                         "(x!1,y!1)")
                                                                                                                        nil
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (expand
                                                                                                                     "pointwise_increasing?")
                                                                                                                    (("2"
                                                                                                                      (skosimp)
                                                                                                                      (("2"
                                                                                                                        (inst
                                                                                                                         -
                                                                                                                         "(x!1,y!1)")
                                                                                                                        nil
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("3"
                                                                                                            (hide-all-but
                                                                                                             (-3
                                                                                                              -6
                                                                                                              1
                                                                                                              -11
                                                                                                              -12
                                                                                                              -7))
                                                                                                            (("3"
                                                                                                              (expand
                                                                                                               "bounded?")
                                                                                                              (("3"
                                                                                                                (expand
                                                                                                                 "o")
                                                                                                                (("3"
                                                                                                                  (split)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "bounded_above?")
                                                                                                                    (("1"
                                                                                                                      (inst
                                                                                                                       +
                                                                                                                       "G(y!1)")
                                                                                                                      (("1"
                                                                                                                        (skolem
                                                                                                                         +
                                                                                                                         "n!1")
                                                                                                                        (("1"
                                                                                                                          (expand
                                                                                                                           "GG")
                                                                                                                          (("1"
                                                                                                                            (expand
                                                                                                                             "pointwise_increasing?")
                                                                                                                            (("1"
                                                                                                                              (inst
                                                                                                                               -
                                                                                                                               "y!1")
                                                                                                                              (("1"
                                                                                                                                (rewrite
                                                                                                                                 "metric_convergence_def")
                                                                                                                                (("1"
                                                                                                                                  (expand
                                                                                                                                   "metric_converges_to")
                                                                                                                                  (("1"
                                                                                                                                    (inst-cp
                                                                                                                                     -5
                                                                                                                                     "n!1")
                                                                                                                                    (("1"
                                                                                                                                      (lemma
                                                                                                                                       "isf_y_section"
                                                                                                                                       ("y"
                                                                                                                                        "y!1"
                                                                                                                                        "i"
                                                                                                                                        "HH(n!1)"))
                                                                                                                                      (("1"
                                                                                                                                        (rewrite
                                                                                                                                         "isf_integral"
                                                                                                                                         1)
                                                                                                                                        (("1"
                                                                                                                                          (inst
                                                                                                                                           -
                                                                                                                                           "isf_integral(LAMBDA (x: T1): HH(n!1)(x, y!1))-G(y!1)")
                                                                                                                                          (("1"
                                                                                                                                            (skosimp)
                                                                                                                                            (("1"
                                                                                                                                              (inst
                                                                                                                                               -
                                                                                                                                               "n!1+n!2")
                                                                                                                                              (("1"
                                                                                                                                                (assert)
                                                                                                                                                (("1"
                                                                                                                                                  (expand
                                                                                                                                                   "ball")
                                                                                                                                                  (("1"
                                                                                                                                                    (expand
                                                                                                                                                     "increasing?")
                                                                                                                                                    (("1"
                                                                                                                                                      (inst
                                                                                                                                                       -3
                                                                                                                                                       "n!1"
                                                                                                                                                       "n!1+n!2")
                                                                                                                                                      (("1"
                                                                                                                                                        (assert)
                                                                                                                                                        nil
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil)
                                                                                                                                           ("2"
                                                                                                                                            (assert)
                                                                                                                                            nil
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("2"
                                                                                                                                  (skosimp)
                                                                                                                                  (("2"
                                                                                                                                    (rewrite
                                                                                                                                     "isf_y_section")
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (expand
                                                                                                                     "bounded_below?")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       +
                                                                                                                       "0")
                                                                                                                      (("2"
                                                                                                                        (skolem
                                                                                                                         +
                                                                                                                         "n!1")
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -5
                                                                                                                           "n!1")
                                                                                                                          (("2"
                                                                                                                            (lemma
                                                                                                                             "isf_y_section"
                                                                                                                             ("y"
                                                                                                                              "y!1"
                                                                                                                              "i"
                                                                                                                              "HH(n!1)"))
                                                                                                                            (("2"
                                                                                                                              (rewrite
                                                                                                                               "isf_integral")
                                                                                                                              (("2"
                                                                                                                                (lemma
                                                                                                                                 "isf_integral_pos"
                                                                                                                                 ("i"
                                                                                                                                  "LAMBDA (x: T1): HH(n!1)(x, y!1)"))
                                                                                                                                (("1"
                                                                                                                                  (assert)
                                                                                                                                  (("1"
                                                                                                                                    (skosimp)
                                                                                                                                    (("1"
                                                                                                                                      (assert)
                                                                                                                                      (("1"
                                                                                                                                        (inst
                                                                                                                                         -5
                                                                                                                                         "n!1"
                                                                                                                                         "(x!1,y!1)")
                                                                                                                                        nil
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("2"
                                                                                                                                  (propax)
                                                                                                                                  nil
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (skosimp)
                                                                                                          (("2"
                                                                                                            (inst
                                                                                                             -12
                                                                                                             "n!1")
                                                                                                            (("2"
                                                                                                              (lemma
                                                                                                               "isf_y_section"
                                                                                                               ("y"
                                                                                                                "y!1"
                                                                                                                "i"
                                                                                                                "HH(n!1)"))
                                                                                                              (("2"
                                                                                                                (lemma
                                                                                                                 "isf_is_integrable[T1, S1, to_measure(mu)]")
                                                                                                                (("2"
                                                                                                                  (inst
                                                                                                                   -
                                                                                                                   "LAMBDA (x: T1): HH(n!1)(x, y!1)")
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (hide
                                                             -1
                                                             -5
                                                             2)
                                                            (("2"
                                                              (expand
                                                               "bounded?")
                                                              (("2"
                                                                (skosimp)
                                                                (("2"
                                                                  (split
                                                                   1)
                                                                  (("1"
                                                                    (expand
                                                                     "bounded_above?")
                                                                    (("1"
                                                                      (inst
                                                                       +
                                                                       "c!1")
                                                                      (("1"
                                                                        (skolem
                                                                         +
                                                                         "n!1")
                                                                        (("1"
                                                                          (inst
                                                                           -
                                                                           "n!1")
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (expand
                                                                     "bounded_below?")
                                                                    (("2"
                                                                      (inst
                                                                       +
                                                                       "-c!1")
                                                                      (("2"
                                                                        (skolem
                                                                         +
                                                                         "n!1")
                                                                        (("2"
                                                                          (inst
                                                                           -
                                                                           "n!1")
                                                                          (("2"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide-all-but
                                                       (-2 1))
                                                      (("2"
                                                        (expand
                                                         "ae_increasing?")
                                                        (("2"
                                                          (expand
                                                           "pointwise_increasing?")
                                                          (("2"
                                                            (expand
                                                             "increasing?")
                                                            (("2"
                                                              (inst
                                                               +
                                                               "emptyset")
                                                              (("2"
                                                                (skolem
                                                                 +
                                                                 "y!1")
                                                                (("2"
                                                                  (inst
                                                                   -
                                                                   "y!1")
                                                                  (("2"
                                                                    (flatten)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide -5 2)
                                                  (("2"
                                                    (expand "bounded?")
                                                    (("2"
                                                      (inst
                                                       +
                                                       "nn_integral(h!1)")
                                                      (("2"
                                                        (skolem
                                                         +
                                                         "n!1")
                                                        (("2"
                                                          (expand "o ")
                                                          (("2"
                                                            (lemma
                                                             "integral_nonneg"
                                                             ("f"
                                                              "GG(n!1)"))
                                                            (("2"
                                                              (inst-cp
                                                               -3
                                                               "n!1"
                                                               "_")
                                                              (("2"
                                                                (replace
                                                                 -4)
                                                                (("2"
                                                                  (expand
                                                                   "abs")
                                                                  (("2"
                                                                    (assert)
                                                                    (("2"
                                                                      (rewrite
                                                                       "metric_convergence_def")
                                                                      (("2"
                                                                        (expand
                                                                         "metric_converges_to")
                                                                        (("2"
                                                                          (inst
                                                                           -11
                                                                           "integral(GG(n!1))-nn_integral(h!1)")
                                                                          (("2"
                                                                            (skosimp)
                                                                            (("2"
                                                                              (inst
                                                                               -11
                                                                               "n!1+n!2")
                                                                              (("2"
                                                                                (assert)
                                                                                (("2"
                                                                                  (expand
                                                                                   "ball")
                                                                                  (("2"
                                                                                    (expand
                                                                                     "pointwise_increasing?")
                                                                                    (("2"
                                                                                      (inst-cp
                                                                                       -6
                                                                                       "n!1")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -6
                                                                                         "n!1+n!2")
                                                                                        (("2"
                                                                                          (case
                                                                                           "integral(GG(n!1)) <= integral(GG(n!1 + n!2))")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil)
                                                                                           ("2"
                                                                                            (hide
                                                                                             2)
                                                                                            (("2"
                                                                                              (lemma
                                                                                               "integral_ae_le"
                                                                                               ("f1"
                                                                                                "GG(n!1)"
                                                                                                "f2"
                                                                                                "GG(n!1 + n!2)"))
                                                                                              (("2"
                                                                                                (assert)
                                                                                                (("2"
                                                                                                  (hide
                                                                                                   2)
                                                                                                  (("2"
                                                                                                    (expand
                                                                                                     "ae_le?")
                                                                                                    (("2"
                                                                                                      (expand
                                                                                                       "pointwise_ae?")
                                                                                                      (("2"
                                                                                                        (expand
                                                                                                         "ae?")
                                                                                                        (("2"
                                                                                                          (expand
                                                                                                           "fullset")
                                                                                                          (("2"
                                                                                                            (expand
                                                                                                             "ae_in?")
                                                                                                            (("2"
                                                                                                              (inst
                                                                                                               +
                                                                                                               "emptyset")
                                                                                                              (("2"
                                                                                                                (skolem
                                                                                                                 +
                                                                                                                 "y!1")
                                                                                                                (("2"
                                                                                                                  (expand
                                                                                                                   "member")
                                                                                                                  (("2"
                                                                                                                    (flatten)
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       -
                                                                                                                       "y!1")
                                                                                                                      (("2"
                                                                                                                        (expand
                                                                                                                         "increasing?")
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -
                                                                                                                           "n!1"
                                                                                                                           "n!1+n!2")
                                                                                                                          (("2"
                                                                                                                            (assert)
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (expand
                                                 "pointwise_increasing?")
                                                (("2"
                                                  (skolem + "y!1")
                                                  (("2"
                                                    (expand
                                                     "increasing_nn_isf?")
                                                    (("2"
                                                      (expand
                                                       "pointwise_increasing?")
                                                      (("2"
                                                        (expand
                                                         "increasing?")
                                                        (("2"
                                                          (skolem
                                                           +
                                                           ("i!1"
                                                            "j!1"))
                                                          (("2"
                                                            (flatten)
                                                            (("2"
                                                              (hide
                                                               -3
                                                               -4
                                                               2
                                                               -5)
                                                              (("2"
                                                                (expand
                                                                 "GG")
                                                                (("2"
                                                                  (inst-cp
                                                                   -4
                                                                   "i!1")
                                                                  (("2"
                                                                    (inst
                                                                     -4
                                                                     "j!1")
                                                                    (("2"
                                                                      (lemma
                                                                       "isf_y_section"
                                                                       ("i"
                                                                        "HH(i!1)"
                                                                        "y"
                                                                        "y!1"))
                                                                      (("2"
                                                                        (lemma
                                                                         "isf_y_section"
                                                                         ("i"
                                                                          "HH(j!1)"
                                                                          "y"
                                                                          "y!1"))
                                                                        (("2"
                                                                          (lemma
                                                                           "isf_integral_le"
                                                                           ("i1"
                                                                            "LAMBDA (x: T1): HH(i!1)(x, y!1)"
                                                                            "i2"
                                                                            "LAMBDA (x: T1): HH(j!1)(x, y!1)"))
                                                                          (("1"
                                                                            (assert)
                                                                            (("1"
                                                                              (skosimp)
                                                                              (("1"
                                                                                (inst
                                                                                 -8
                                                                                 "(x!1,y!1)")
                                                                                (("1"
                                                                                  (inst
                                                                                   -8
                                                                                   "i!1"
                                                                                   "j!1")
                                                                                  (("1"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (propax)
                                                                            nil
                                                                            nil)
                                                                           ("3"
                                                                            (propax)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide 2 -1 -2 -3)
                                              (("2"
                                                (skosimp)
                                                (("2"
                                                  (expand "GG")
                                                  (("2"
                                                    (inst - "n!1" "_")
                                                    (("2"
                                                      (inst -2 "n!1")
                                                      (("2"
                                                        (lemma
                                                         "isf_y_section"
                                                         ("y"
                                                          "y!1"
                                                          "i"
                                                          "HH(n!1)"))
                                                        (("2"
                                                          (lemma
                                                           "isf_integral_pos"
                                                           ("i"
                                                            "LAMBDA (x: T1): HH(n!1)(x, y!1)"))
                                                          (("1"
                                                            (assert)
                                                            (("1"
                                                              (skosimp)
                                                              (("1"
                                                                (inst
                                                                 -
                                                                 "(x!1,y!1)")
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (apply-extensionality
                                             :hide?
                                             t)
                                            (("2"
                                              (expand "o ")
                                              (("2"
                                                (inst - "x!1")
                                                (("2"
                                                  (replace -1)
                                                  (("2"
                                                    (inst -5 "x!1")
                                                    (("2"
                                                      (rewrite
                                                       "isf_integral[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]"
                                                       +)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (skosimp)
                                        (("2"
                                          (inst - "n!1")
                                          (("2"
                                            (inst -4 "n!1")
                                            (("2"
                                              (hide -2 2)
                                              (("2"
                                                (rewrite
                                                 "isf_integral[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]")
                                                (("2"
                                                  (rewrite
                                                   "isf_fubini_tonelli_4"
                                                   +)
                                                  (("2"
                                                    (expand "GG")
                                                    (("2"
                                                      (propax)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("3" (propax) nil nil))
                                      nil)
                                     ("2"
                                      (hide -1 2)
                                      (("2"
                                        (skosimp)
                                        (("2"
                                          (expand "GG")
                                          (("2"
                                            (rewrite "isf_integral_y")
                                            (("2"
                                              (case-replace
                                               "(LAMBDA (x: T1, y: T2): HH(n!1)(x, y))=HH(n!1)")
                                              (("1" (assertnil nil)
                                               ("2"
                                                (apply-extensionality
                                                 :hide?
                                                 t)
                                                nil
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (skosimp)
                                    (("2"
                                      (rewrite "isf_y_section")
                                      nil
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (skosimp)
                                  (("2"
                                    (typepred "HH(n!1)")
                                    (("2"
                                      (expand "nn_isf?")
                                      (("2" (inst - "z!1"nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (skosimp)
                                (("2"
                                  (typepred "HH(n!1)")
                                  (("2" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (propax) nil nil))
                      nil))
                    nil)
                   ("2" (skosimp)
                    (("2" (inst - "x!1") (("2" (assertnil nil)) nil))
                    nil))
                  nil)
                 ("2" (propax) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (flatten)
        (("2" (expand "integrable2?")
          (("2" (skolem - ("N!1" "g!1"))
            (("2" (expand "member")
              (("2" (typepred "N!1")
                (("2" (name "N" "complement(N!1)")
                  (("2"
                    (name "NN"
                          "cross_product(fullset[T1],complement(N!1))")
                    (("2"
                      (lemma "sigma_algebra_complement[T2,S2]"
                       ("x" "N!1"))
                      (("1" (expand "member")
                        (("1" (replace -3)
                          (("1"
                            (lemma "cross_product_is_sigma_times"
                             ("X" "fullset[T1]" "Y" "N"))
                            (("1" (replace -3)
                              (("1"
                                (name "H" "*[[T1,T2]](phi(NN),h!1)")
                                (("1"
                                  (name "G" "*[T2](phi(N),g!1)")
                                  (("1"
                                    (case
                                     "forall (y:T2): integrable?(LAMBDA (x: T1): H(x, y))")
                                    (("1"
                                      (case
                                       "forall (y:T2): integral(LAMBDA (x: T1): H(x, y))=G(y)")
                                      (("1"
                                        (hide -10)
                                        (("1"
                                          (case "integrable?(G)")
                                          (("1"
                                            (case
                                             "forall (y:T2): G(y)>=0")
                                            (("1"
                                              (case
                                               "forall (z:[T1,T2]): H(z)>=0")
                                              (("1"
                                                (case
                                                 "nn_integrable?(H)")
                                                (("1"
                                                  (lemma
                                                   "integral_ae_eq"
                                                   ("f" "H" "h" "h!1"))
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (hide 2 -1)
                                                      (("1"
                                                        (expand
                                                         "ae_eq?")
                                                        (("1"
                                                          (expand
                                                           "pointwise_ae?")
                                                          (("1"
                                                            (expand
                                                             "restrict")
                                                            (("1"
                                                              (expand
                                                               "ae?")
                                                              (("1"
                                                                (expand
                                                                 "fullset"
                                                                 1)
                                                                (("1"
                                                                  (expand
                                                                   "H")
                                                                  (("1"
                                                                    (expand
                                                                     "*")
                                                                    (("1"
                                                                      (expand
                                                                       "phi")
                                                                      (("1"
                                                                        (expand
                                                                         "ae_in?")
                                                                        (("1"
                                                                          (expand
                                                                           "member")
                                                                          (("1"
                                                                            (inst
                                                                             +
                                                                             "cross_product(fullset[T1], N!1)")
                                                                            (("1"
                                                                              (skolem
                                                                               +
                                                                               "z!1")
                                                                              (("1"
                                                                                (expand
                                                                                 "NN")
                                                                                (("1"
                                                                                  (expand
                                                                                   "fullset")
                                                                                  (("1"
                                                                                    (expand
                                                                                     "cross_product")
                                                                                    (("1"
                                                                                      (expand
                                                                                       "complement")
                                                                                      (("1"
                                                                                        (expand
                                                                                         "member")
                                                                                        (("1"
                                                                                          (flatten)
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("2"
                                                                              (expand
                                                                               "negligible_set?")
                                                                              (("2"
                                                                                (inst
                                                                                 +
                                                                                 "cross_product[T1, T2](fullset[T1], N!1)")
                                                                                (("2"
                                                                                  (split)
                                                                                  (("1"
                                                                                    (hide-all-but
                                                                                     (-12
                                                                                      1))
                                                                                    (("1"
                                                                                      (expand
                                                                                       "null_set?")
                                                                                      (("1"
                                                                                        (flatten)
                                                                                        (("1"
                                                                                          (expand
                                                                                           "mu_fin?")
                                                                                          (("1"
                                                                                            (expand
                                                                                             "mu")
                                                                                            (("1"
                                                                                              (expand
                                                                                               "to_measure")
                                                                                              (("1"
                                                                                                (lemma
                                                                                                 "cross_product_is_sigma_times"
                                                                                                 ("X"
                                                                                                  "fullset[T1]"
                                                                                                  "Y"
                                                                                                  "N!1"))
                                                                                                (("1"
                                                                                                  (expand
                                                                                                   "measurable_set?")
                                                                                                  (("1"
                                                                                                    (assert)
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "fm_times")
                                                                                                      (("1"
                                                                                                        (lemma
                                                                                                         "rectangle_measure1"
                                                                                                         ("M"
                                                                                                          "cross_product[T1, T2](fullset[T1], N!1)"
                                                                                                          "X"
                                                                                                          "fullset[T1]"
                                                                                                          "Y"
                                                                                                          "N!1"
                                                                                                          "mu"
                                                                                                          "mu"
                                                                                                          "nu"
                                                                                                          "nu"))
                                                                                                        (("1"
                                                                                                          (assert)
                                                                                                          nil
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (expand
                                                                                     "subset?")
                                                                                    (("2"
                                                                                      (skosimp)
                                                                                      nil
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (typepred "h!1")
                                                    (("2"
                                                      (expand
                                                       "nn_measurable?")
                                                      (("2"
                                                        (flatten)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("3"
                                                    (lemma
                                                     "nn_integrable_is_integrable[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]")
                                                    (("3"
                                                      (inst - "H")
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide 2)
                                                  (("2"
                                                    (case
                                                     "nn_measurable?(H)")
                                                    (("1"
                                                      (hide
                                                       (-7
                                                        -8
                                                        -9
                                                        -10
                                                        -11
                                                        -12
                                                        -13))
                                                      (("1"
                                                        (lemma
                                                         "nn_integrable_is_nn_integrable"
                                                         ("f" "G"))
                                                        (("1"
                                                          (replace -4)
                                                          (("1"
                                                            (lemma
                                                             "nn_measurable_def"
                                                             ("f" "H"))
                                                            (("1"
                                                              (split)
                                                              (("1"
                                                                (flatten)
                                                                (("1"
                                                                  (split
                                                                   -1)
                                                                  (("1"
                                                                    (skolem
                                                                     -
                                                                     "HH")
                                                                    (("1"
                                                                      (typepred
                                                                       "HH")
                                                                      (("1"
                                                                        (hide
                                                                         -3
                                                                         -4)
                                                                        (("1"
                                                                          (case
                                                                           "forall (n:nat): isf?(HH(n))")
                                                                          (("1"
                                                                            (case
                                                                             "forall (n:nat,z:[T1, T2]): HH(n)(z) >= 0")
                                                                            (("1"
                                                                              (name
                                                                               "GG"
                                                                               "lambda (n:nat): lambda (y:T2): isf_integral(LAMBDA (x: T1): HH(n)(x, y))")
                                                                              (("1"
                                                                                (case
                                                                                 "forall (n:nat): integrable?(GG(n))")
                                                                                (("1"
                                                                                  (case
                                                                                   "pointwise_increasing?(GG)")
                                                                                  (("1"
                                                                                    (case
                                                                                     "pointwise_converges_upto?(GG, G)")
                                                                                    (("1"
                                                                                      (case
                                                                                       "bounded?(integral o HH)")
                                                                                      (("1"
                                                                                        (lemma
                                                                                         "monotone_convergence_scaf[[T1, T2], sigma_times(S1, S2),
       to_measure(fm_times(mu, nu))]"
                                                                                         ("F"
                                                                                          "HH"
                                                                                          "f"
                                                                                          "H"))
                                                                                        (("1"
                                                                                          (assert)
                                                                                          (("1"
                                                                                            (split
                                                                                             -1)
                                                                                            (("1"
                                                                                              (flatten)
                                                                                              (("1"
                                                                                                (lemma
                                                                                                 "nn_integrable_is_nn_integrable"
                                                                                                 ("f"
                                                                                                  "H"))
                                                                                                (("1"
                                                                                                  (replace
                                                                                                   -14)
                                                                                                  (("1"
                                                                                                    (propax)
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil)
                                                                                             ("2"
                                                                                              (hide-all-but
                                                                                               (-1
                                                                                                1))
                                                                                              (("2"
                                                                                                (expand
                                                                                                 "o ")
                                                                                                (("2"
                                                                                                  (assert)
                                                                                                  (("2"
                                                                                                    (expand
                                                                                                     "bounded?")
                                                                                                    (("2"
                                                                                                      (skosimp)
                                                                                                      (("2"
                                                                                                        (split)
                                                                                                        (("1"
                                                                                                          (expand
                                                                                                           "bounded_above?")
                                                                                                          (("1"
                                                                                                            (inst
                                                                                                             +
                                                                                                             "c!1")
                                                                                                            (("1"
                                                                                                              (skolem
                                                                                                               +
                                                                                                               "n!1")
                                                                                                              (("1"
                                                                                                                (inst
                                                                                                                 -
                                                                                                                 "n!1")
                                                                                                                (("1"
                                                                                                                  (assert)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (expand
                                                                                                           "bounded_below?")
                                                                                                          (("2"
                                                                                                            (inst
                                                                                                             +
                                                                                                             "-c!1")
                                                                                                            (("2"
                                                                                                              (skolem
                                                                                                               +
                                                                                                               "n!1")
                                                                                                              (("2"
                                                                                                                (inst
                                                                                                                 -
                                                                                                                 "n!1")
                                                                                                                (("2"
                                                                                                                  (assert)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide
                                                                                         2)
                                                                                        (("2"
                                                                                          (case-replace
                                                                                           "integral o HH=isf_integral o HH")
                                                                                          (("1"
                                                                                            (hide
                                                                                             -1)
                                                                                            (("1"
                                                                                              (expand
                                                                                               "bounded?")
                                                                                              (("1"
                                                                                                (inst
                                                                                                 +
                                                                                                 "integral(G)")
                                                                                                (("1"
                                                                                                  (skolem
                                                                                                   +
                                                                                                   "n!1")
                                                                                                  (("1"
                                                                                                    (expand
                                                                                                     "o"
                                                                                                     1)
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "abs"
                                                                                                       1)
                                                                                                      (("1"
                                                                                                        (inst-cp
                                                                                                         -6
                                                                                                         "n!1")
                                                                                                        (("1"
                                                                                                          (lemma
                                                                                                           "isf_integral_pos"
                                                                                                           ("i"
                                                                                                            "HH(n!1)"))
                                                                                                          (("1"
                                                                                                            (inst
                                                                                                             -6
                                                                                                             "n!1"
                                                                                                             "_")
                                                                                                            (("1"
                                                                                                              (replace
                                                                                                               -6)
                                                                                                              (("1"
                                                                                                                (assert)
                                                                                                                (("1"
                                                                                                                  (hide
                                                                                                                   -1)
                                                                                                                  (("1"
                                                                                                                    (rewrite
                                                                                                                     "isf_fubini_tonelli_4")
                                                                                                                    (("1"
                                                                                                                      (lemma
                                                                                                                       "integral_ae_le"
                                                                                                                       ("f1"
                                                                                                                        "GG(n!1)"
                                                                                                                        "f2"
                                                                                                                        "G"))
                                                                                                                      (("1"
                                                                                                                        (split)
                                                                                                                        (("1"
                                                                                                                          (expand
                                                                                                                           "GG"
                                                                                                                           -1)
                                                                                                                          (("1"
                                                                                                                            (propax)
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil)
                                                                                                                         ("2"
                                                                                                                          (hide
                                                                                                                           2)
                                                                                                                          (("2"
                                                                                                                            (hide-all-but
                                                                                                                             (-1
                                                                                                                              -2
                                                                                                                              -12
                                                                                                                              1))
                                                                                                                            (("2"
                                                                                                                              (expand
                                                                                                                               "ae_le?")
                                                                                                                              (("2"
                                                                                                                                (expand
                                                                                                                                 "pointwise_ae?")
                                                                                                                                (("2"
                                                                                                                                  (expand
                                                                                                                                   "ae?")
                                                                                                                                  (("2"
                                                                                                                                    (expand
                                                                                                                                     "fullset")
                                                                                                                                    (("2"
                                                                                                                                      (expand
                                                                                                                                       "ae_in?")
                                                                                                                                      (("2"
                                                                                                                                        (inst
                                                                                                                                         +
                                                                                                                                         "emptyset")
                                                                                                                                        (("2"
                                                                                                                                          (skolem
                                                                                                                                           +
                                                                                                                                           "y!1")
                                                                                                                                          (("2"
                                                                                                                                            (expand
                                                                                                                                             "member")
                                                                                                                                            (("2"
                                                                                                                                              (flatten)
                                                                                                                                              (("2"
                                                                                                                                                (expand
                                                                                                                                                 "pointwise_converges_upto?")
                                                                                                                                                (("2"
                                                                                                                                                  (expand
                                                                                                                                                   "pointwise_convergence?")
                                                                                                                                                  (("2"
                                                                                                                                                    (expand
                                                                                                                                                     "pointwise_increasing?")
                                                                                                                                                    (("2"
                                                                                                                                                      (inst
                                                                                                                                                       -
                                                                                                                                                       "y!1")
                                                                                                                                                      (("2"
                                                                                                                                                        (inst
                                                                                                                                                         -
                                                                                                                                                         "y!1")
                                                                                                                                                        (("2"
                                                                                                                                                          (inst
                                                                                                                                                           -
                                                                                                                                                           "y!1")
                                                                                                                                                          (("2"
                                                                                                                                                            (hide
                                                                                                                                                             1)
                                                                                                                                                            (("2"
                                                                                                                                                              (rewrite
                                                                                                                                                               "metric_convergence_def")
                                                                                                                                                              (("2"
                                                                                                                                                                (expand
                                                                                                                                                                 "metric_converges_to")
                                                                                                                                                                (("2"
                                                                                                                                                                  (expand
                                                                                                                                                                   "increasing?")
                                                                                                                                                                  (("2"
                                                                                                                                                                    (inst
                                                                                                                                                                     -
                                                                                                                                                                     "GG(n!1)(y!1)-G(y!1)")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (skosimp)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (inst
                                                                                                                                                                         -
                                                                                                                                                                         "n!1+n!2")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (expand
                                                                                                                                                                           "ball")
                                                                                                                                                                          (("1"
                                                                                                                                                                            (assert)
                                                                                                                                                                            (("1"
                                                                                                                                                                              (inst
                                                                                                                                                                               -
                                                                                                                                                                               "n!1"
                                                                                                                                                                               "n!1+n!2")
                                                                                                                                                                              (("1"
                                                                                                                                                                                (assert)
                                                                                                                                                                                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))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("2"
                                                                                                                        (inst
                                                                                                                         -
                                                                                                                         "n!1")
                                                                                                                        nil
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (propax)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil)
                                                                                                 ("2"
                                                                                                  (lemma
                                                                                                   "integral_nonneg"
                                                                                                   ("f"
                                                                                                    "G"))
                                                                                                  (("2"
                                                                                                    (assert)
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil)
                                                                                           ("2"
                                                                                            (hide
                                                                                             2)
                                                                                            (("2"
                                                                                              (apply-extensionality
                                                                                               :hide?
                                                                                               t)
                                                                                              (("2"
                                                                                                (expand
                                                                                                 "o ")
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -6
                                                                                                   "x!1")
                                                                                                  (("2"
                                                                                                    (rewrite
                                                                                                     "isf_integral")
                                                                                                    nil
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("3"
                                                                                        (skolem
                                                                                         +
                                                                                         "n!1")
                                                                                        (("3"
                                                                                          (inst
                                                                                           -6
                                                                                           "n!1")
                                                                                          (("3"
                                                                                            (lemma
                                                                                             "isf_is_integrable[[T1, T2], sigma_times(S1, S2), to_measure(fm_times(mu, nu))]")
                                                                                            (("3"
                                                                                              (inst
                                                                                               -
                                                                                               "HH(n!1)")
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil)
                                                                                     ("2"
                                                                                      (expand
                                                                                       "pointwise_converges_upto?")
                                                                                      (("2"
                                                                                        (assert)
                                                                                        (("2"
                                                                                          (flatten)
                                                                                          (("2"
                                                                                            (hide
                                                                                             2)
                                                                                            (("2"
                                                                                              (expand
                                                                                               "pointwise_convergence?")
                                                                                              (("2"
                                                                                                (skolem
                                                                                                 +
                                                                                                 "y!1")
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -13
                                                                                                   "y!1")
                                                                                                  (("2"
                                                                                                    (inst
                                                                                                     -14
                                                                                                     "y!1")
                                                                                                    (("2"
                                                                                                      (replace
                                                                                                       -13
                                                                                                       1
                                                                                                       rl)
                                                                                                      (("2"
                                                                                                        (expand
                                                                                                         "GG"
                                                                                                         1)
                                                                                                        (("2"
                                                                                                          (lemma
                                                                                                           "monotone_convergence_scaf[T1,S1,to_measure(mu)]"
                                                                                                           ("f"
                                                                                                            "LAMBDA (x: T1): H(x, y!1)"
                                                                                                            "F"
                                                                                                            "lambda (n:nat): lambda (x:T1): HH(n)(x, y!1)"))
                                                                                                          (("1"
                                                                                                            (split
                                                                                                             -1)
                                                                                                            (("1"
                                                                                                              (flatten)
                                                                                                              (("1"
                                                                                                                (expand
                                                                                                                 "converges_upto?")
                                                                                                                (("1"
                                                                                                                  (flatten)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "o"
                                                                                                                     -2)
                                                                                                                    (("1"
                                                                                                                      (case-replace
                                                                                                                       "(LAMBDA (x_1: nat):
                     integral(LAMBDA (x: T1): HH(x_1)(x, y!1)))=LAMBDA (n:nat): isf_integral(LAMBDA (x: T1): HH(n)(x, y!1))")
                                                                                                                      (("1"
                                                                                                                        (apply-extensionality
                                                                                                                         :hide?
                                                                                                                         t)
                                                                                                                        (("1"
                                                                                                                          (inst
                                                                                                                           -8
                                                                                                                           "x!1")
                                                                                                                          (("1"
                                                                                                                            (lemma
                                                                                                                             "isf_y_section"
                                                                                                                             ("y"
                                                                                                                              "y!1"
                                                                                                                              "i"
                                                                                                                              "HH(x!1)"))
                                                                                                                            (("1"
                                                                                                                              (rewrite
                                                                                                                               "isf_integral"
                                                                                                                               1)
                                                                                                                              nil
                                                                                                                              nil)
                                                                                                                             ("2"
                                                                                                                              (propax)
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil)
                                                                                                                         ("2"
                                                                                                                          (skosimp)
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             -8
                                                                                                                             "n!1")
                                                                                                                            (("2"
                                                                                                                              (rewrite
                                                                                                                               "isf_y_section")
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil)
                                                                                                                         ("3"
                                                                                                                          (skolem
                                                                                                                           +
                                                                                                                           "n!1")
                                                                                                                          (("3"
                                                                                                                            (inst
                                                                                                                             -8
                                                                                                                             "n!1")
                                                                                                                            (("3"
                                                                                                                              (lemma
                                                                                                                               "isf_y_section"
                                                                                                                               ("y"
                                                                                                                                "y!1"
                                                                                                                                "i"
                                                                                                                                "HH(n!1)"))
                                                                                                                              (("1"
                                                                                                                                (lemma
                                                                                                                                 "isf_is_integrable"
                                                                                                                                 ("x"
                                                                                                                                  "LAMBDA (x: T1): HH(n!1)(x, y!1)"))
                                                                                                                                (("1"
                                                                                                                                  (propax)
                                                                                                                                  nil
                                                                                                                                  nil)
                                                                                                                                 ("2"
                                                                                                                                  (propax)
                                                                                                                                  nil
                                                                                                                                  nil))
                                                                                                                                nil)
                                                                                                                               ("2"
                                                                                                                                (propax)
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("2"
                                                                                                                        (skosimp)
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -8
                                                                                                                           "n!1")
                                                                                                                          (("2"
                                                                                                                            (rewrite
                                                                                                                             "isf_y_section")
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (hide
                                                                                                               2)
                                                                                                              (("2"
                                                                                                                (expand
                                                                                                                 "pointwise_converges_upto?")
                                                                                                                (("2"
                                                                                                                  (split)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "pointwise_convergence?")
                                                                                                                    (("1"
                                                                                                                      (skosimp)
                                                                                                                      (("1"
                                                                                                                        (inst
                                                                                                                         -7
                                                                                                                         "(x!1,y!1)")
                                                                                                                        nil
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (hide-all-but
                                                                                                                     (-8
                                                                                                                      1))
                                                                                                                    (("2"
                                                                                                                      (expand
                                                                                                                       "pointwise_increasing?")
                                                                                                                      (("2"
                                                                                                                        (skosimp)
                                                                                                                        (("2"
                                                                                                                          (inst
                                                                                                                           -
                                                                                                                           "(x!1,y!1)")
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("3"
                                                                                                              (hide
                                                                                                               2)
                                                                                                              (("3"
                                                                                                                (expand
                                                                                                                 "bounded?")
                                                                                                                (("3"
                                                                                                                  (split)
                                                                                                                  (("1"
                                                                                                                    (expand
                                                                                                                     "o")
                                                                                                                    (("1"
                                                                                                                      (expand
                                                                                                                       "bounded_above?")
                                                                                                                      (("1"
                                                                                                                        (inst
                                                                                                                         +
                                                                                                                         "G(y!1)")
                                                                                                                        (("1"
                                                                                                                          (skolem
                                                                                                                           +
                                                                                                                           "n!1")
                                                                                                                          (("1"
                                                                                                                            (lemma
                                                                                                                             "integral_ae_le[T1,S1,to_measure(mu)]"
                                                                                                                             ("f1"
                                                                                                                              "LAMBDA (x: T1): HH(n!1)(x, y!1)"
                                                                                                                              "f2"
                                                                                                                              "LAMBDA (x: T1): H(x, y!1)"))
                                                                                                                            (("1"
                                                                                                                              (assert)
                                                                                                                              (("1"
                                                                                                                                (hide
                                                                                                                                 2)
                                                                                                                                (("1"
                                                                                                                                  (expand
                                                                                                                                   "ae_le?")
                                                                                                                                  (("1"
                                                                                                                                    (expand
                                                                                                                                     "pointwise_ae?")
                                                                                                                                    (("1"
                                                                                                                                      (expand
                                                                                                                                       "ae?")
                                                                                                                                      (("1"
                                                                                                                                        (expand
                                                                                                                                         "fullset")
                                                                                                                                        (("1"
                                                                                                                                          (expand
                                                                                                                                           "ae_in?")
                                                                                                                                          (("1"
                                                                                                                                            (inst
                                                                                                                                             +
                                                                                                                                             "emptyset")
                                                                                                                                            (("1"
                                                                                                                                              (skosimp)
                                                                                                                                              (("1"
                                                                                                                                                (expand
                                                                                                                                                 "member")
                                                                                                                                                (("1"
                                                                                                                                                  (hide-all-but
                                                                                                                                                   (-7
                                                                                                                                                    -8
                                                                                                                                                    2))
                                                                                                                                                  (("1"
                                                                                                                                                    (expand
                                                                                                                                                     "pointwise_increasing?")
                                                                                                                                                    (("1"
                                                                                                                                                      (inst
                                                                                                                                                       -
                                                                                                                                                       "(x!1,y!1)")
                                                                                                                                                      (("1"
                                                                                                                                                        (inst
                                                                                                                                                         -
                                                                                                                                                         "(x!1,y!1)")
                                                                                                                                                        (("1"
                                                                                                                                                          (expand
                                                                                                                                                           "increasing?")
                                                                                                                                                          (("1"
                                                                                                                                                            (rewrite
                                                                                                                                                             "metric_convergence_def")
                                                                                                                                                            (("1"
                                                                                                                                                              (expand
                                                                                                                                                               "metric_converges_to")
                                                                                                                                                              (("1"
                                                                                                                                                                (inst
                                                                                                                                                                 -
                                                                                                                                                                 "HH(n!1)(x!1, y!1)-H(x!1, y!1)")
                                                                                                                                                                (("1"
                                                                                                                                                                  (skosimp)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (inst
                                                                                                                                                                     -
                                                                                                                                                                     "n!1+n!2")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (assert)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (expand
                                                                                                                                                                         "ball")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (inst
                                                                                                                                                                           -
                                                                                                                                                                           "n!1"
                                                                                                                                                                           "n!1+n!2")
                                                                                                                                                                          (("1"
                                                                                                                                                                            (assert)
                                                                                                                                                                            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)
                                                                                                                             ("2"
                                                                                                                              (propax)
                                                                                                                              nil
                                                                                                                              nil)
                                                                                                                             ("3"
                                                                                                                              (inst
                                                                                                                               -5
                                                                                                                               "n!1")
                                                                                                                              (("3"
                                                                                                                                (lemma
                                                                                                                                 "isf_y_section"
                                                                                                                                 ("y"
                                                                                                                                  "y!1"
                                                                                                                                  "i"
                                                                                                                                  "HH(n!1)"))
                                                                                                                                (("1"
                                                                                                                                  (lemma
                                                                                                                                   "isf_is_integrable[T1, S1, to_measure(mu)]")
                                                                                                                                  (("1"
                                                                                                                                    (inst
                                                                                                                                     -
                                                                                                                                     "LAMBDA (x: T1): HH(n!1)(x, y!1)")
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil)
                                                                                                                                 ("2"
                                                                                                                                  (propax)
                                                                                                                                  nil
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (expand
                                                                                                                     "bounded_below?")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       +
                                                                                                                       "0")
                                                                                                                      (("2"
                                                                                                                        (skolem
                                                                                                                         +
                                                                                                                         "n!1")
                                                                                                                        (("2"
                                                                                                                          (expand
                                                                                                                           "o ")
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             -5
                                                                                                                             "n!1")
                                                                                                                            (("2"
                                                                                                                              (lemma
                                                                                                                               "isf_y_section"
                                                                                                                               ("y"
                                                                                                                                "y!1"
                                                                                                                                "i"
                                                                                                                                "HH(n!1)"))
                                                                                                                              (("1"
                                                                                                                                (rewrite
                                                                                                                                 "isf_integral"
                                                                                                                                 1)
                                                                                                                                (("1"
                                                                                                                                  (lemma
                                                                                                                                   "isf_integral_pos[T1,S1,to_measure(mu)]"
                                                                                                                                   ("i"
                                                                                                                                    "LAMBDA (x: T1): HH(n!1)(x, y!1)"))
                                                                                                                                  (("1"
                                                                                                                                    (assert)
                                                                                                                                    (("1"
                                                                                                                                      (skosimp)
                                                                                                                                      (("1"
                                                                                                                                        (inst
                                                                                                                                         -5
                                                                                                                                         "n!1"
                                                                                                                                         "(x!1,y!1)")
                                                                                                                                        nil
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil)
                                                                                                                                   ("2"
                                                                                                                                    (propax)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil)
                                                                                                                               ("2"
                                                                                                                                (propax)
                                                                                                                                nil
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (skosimp)
                                                                                                            (("2"
                                                                                                              (inst
                                                                                                               -5
                                                                                                               "n!1")
                                                                                                              (("2"
                                                                                                                (lemma
                                                                                                                 "isf_y_section"
                                                                                                                 ("i"
                                                                                                                  "HH(n!1)"
                                                                                                                  "y"
                                                                                                                  "y!1"))
                                                                                                                (("1"
                                                                                                                  (lemma
                                                                                                                   "isf_is_integrable"
                                                                                                                   ("x"
                                                                                                                    "LAMBDA (x: T1): HH(n!1)(x, y!1)"))
                                                                                                                  (("1"
                                                                                                                    (propax)
                                                                                                                    nil
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (propax)
                                                                                                                    nil
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("2"
                                                                                                                  (propax)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (hide
                                                                                     2
                                                                                     -2
                                                                                     -1)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "increasing_nn_simple?")
                                                                                      (("2"
                                                                                        (flatten)
                                                                                        (("2"
                                                                                          (expand
                                                                                           "pointwise_increasing?")
                                                                                          (("2"
                                                                                            (skolem
                                                                                             +
                                                                                             "y!1")
                                                                                            (("2"
                                                                                              (expand
                                                                                               "increasing?")
                                                                                              (("2"
                                                                                                (skolem
                                                                                                 +
                                                                                                 ("i!1"
                                                                                                  "j!1"))
                                                                                                (("2"
                                                                                                  (flatten)
                                                                                                  (("2"
                                                                                                    (expand
                                                                                                     "GG")
                                                                                                    (("2"
                                                                                                      (lemma
                                                                                                       "isf_integral_le"
                                                                                                       ("i1"
                                                                                                        "LAMBDA (x: T1): HH(i!1)(x, y!1)"
                                                                                                        "i2"
                                                                                                        "LAMBDA (x: T1): HH(j!1)(x, y!1)"))
                                                                                                      (("1"
                                                                                                        (assert)
                                                                                                        (("1"
                                                                                                          (hide
                                                                                                           2)
                                                                                                          (("1"
                                                                                                            (skosimp)
                                                                                                            (("1"
                                                                                                              (inst
                                                                                                               -5
                                                                                                               "(x!1, y!1)")
                                                                                                              (("1"
                                                                                                                (inst
                                                                                                                 -5
                                                                                                                 "i!1"
                                                                                                                 "j!1")
                                                                                                                (("1"
                                                                                                                  (assert)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (inst
                                                                                                         -3
                                                                                                         "j!1")
                                                                                                        (("2"
                                                                                                          (rewrite
                                                                                                           "isf_y_section")
                                                                                                          nil
                                                                                                          nil))
                                                                                                        nil)
                                                                                                       ("3"
                                                                                                        (inst
                                                                                                         -3
                                                                                                         "i!1")
                                                                                                        (("3"
                                                                                                          (rewrite
                                                                                                           "isf_y_section")
                                                                                                          nil
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (hide
                                                                                   -1
                                                                                   2)
                                                                                  (("2"
                                                                                    (skosimp)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "GG")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -2
                                                                                         "n!1")
                                                                                        (("2"
                                                                                          (rewrite
                                                                                           "isf_integral_y")
                                                                                          (("2"
                                                                                            (case-replace
                                                                                             "(LAMBDA (x: T1, y: T2): HH(n!1)(x, y))=HH(n!1)")
--> --------------------

--> maximum size reached

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

Messung V0.5 in Prozent
C=100 H=99 G=99

¤ Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.0.650Bemerkung:  (vorverarbeitet am  2026-05-01) ¤

*Bot Zugriff






Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.