Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/measure_integration/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 454 kB image not shown  

Impressum measure_def.prf

  Sprache: Lisp
 

(measure_def
 (IMP_generalized_measure_def_TCC1 0
  (IMP_generalized_measure_def_TCC1-1 nil 3458457144
   ("" (assuming-tcc) nil nil)
   ((finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (subset_algebra_emptyset name-judgement "(S)" measure_def nil))
   nil))
 (increasing_indexed_measurable_TCC1 0
  (increasing_indexed_measurable_TCC1-1 nil 3450166483
   ("" (expand "increasing_indexed_measurable?")
    (("" (expand "increasing_indexed?")
      (("" (split)
        (("1" (apply-extensionality :hide? t)
          (("1" (expand "fullset")
            (("1" (expand "IUnion") (("1" (propax) nil nil)) nil))
            nil))
          nil)
         ("2" (expand "increasing?")
          (("2" (expand "subset?")
            (("2" (expand "fullset")
              (("2" (expand "member") (("2" (propax) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((increasing_indexed? const-decl "bool" nat_indexed_sets
     "sets_aux/")
    (increasing? const-decl "bool" fun_preds_partial "structures/")
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (fullset const-decl "set" sets nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (set type-eq-decl nil sets nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (increasing_indexed_measurable? const-decl "bool" measure_def nil)
    (subset_algebra_fullset name-judgement "(S)" measure_def nil))
   nil))
 (sigma_finite_measure_TCC1 0
  (sigma_finite_measure_TCC1-1 nil 3396670582
   ("" (expand "sigma_finite_measure?")
    (("" (split)
      (("1" (lemma "measure_type_TCC1") (("1" (propax) nil nil)) nil)
       ("2" (expand "measure_sigma_finite?")
        (("2" (expand "zero_measure")
          (("2"
            (inst +
             "lambda i: if i = 0 then fullset[T] else emptyset[T] endif")
            (("1" (apply-extensionality :hide? t)
              (("1" (expand "fullset")
                (("1" (expand "IUnion") (("1" (inst + "0"nil nil))
                  nil))
                nil))
              nil)
             ("2" (expand "disjoint_indexed_measurable?")
              (("2" (expand "fullset")
                (("2" (expand "emptyset")
                  (("2" (expand "disjoint?")
                    (("2" (skosimp)
                      (("2" (expand "disjoint?")
                        (("2" (expand "intersection")
                          (("2" (expand "empty?")
                            (("2" (expand "member")
                              (("2"
                                (skosimp*)
                                (("2"
                                  (case-replace "i!1=0")
                                  (("1" (assertnil nil)
                                   ("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((measure_type_TCC1 subtype-tcc nil generalized_measure_def nil)
    (T formal-type-decl nil measure_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (zero_measure const-decl "extended_nnreal" generalized_measure_def
     nil)
    (intersection const-decl "set" sets nil)
    (member const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (disjoint? const-decl "bool" sets nil)
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (IUnion const-decl "set[T]" indexed_sets nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (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)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (set type-eq-decl nil sets nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (fullset const-decl "set" sets nil)
    (emptyset const-decl "set" sets nil)
    (subset_algebra_fullset name-judgement "(S)" measure_def nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (subset_algebra_emptyset name-judgement "(S)" measure_def nil)
    (measure_sigma_finite? const-decl "bool" measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil))
   nil))
 (discrete_measure_TCC1 0
  (discrete_measure_TCC1-1 nil 3396670582
   ("" (expand "measure?")
    (("" (split)
      (("1" (expand "measure_empty?")
        (("1" (rewrite "card_emptyset[T]"nil nil)) nil)
       ("2" (expand "measure_countably_additive?")
        (("2"
          (name-replace "MM" "LAMBDA a:
             IF is_finite[T](a) THEN (TRUE, card[T](a)) ELSE (FALSE, 0) ENDIF")
          (("2" (skosimp)
            (("2" (expand "x_eq")
              (("2" (expand "o ")
                (("2" (expand "MM")
                  (("2" (expand "x_sum")
                    (("2"
                      (case-replace
                       "FORALL i: IF is_finite[T](X!1(i)) THEN TRUE ELSE FALSE ENDIF")
                      (("1" (case "FORALL i: is_finite[T](X!1(i))")
                        (("1" (hide -2)
                          (("1"
                            (case-replace "(LAMBDA (i_1: nat):
                              IF is_finite[T](X!1(i_1))
                                THEN card[T](X!1(i_1))
                              ELSE 0
                              ENDIF)=LAMBDA (i_1: nat):card[T](X!1(i_1))")
                            (("1" (hide -1)
                              (("1"
                                (case
                                 "FORALL i: series(LAMBDA (i_1: nat): card[T](X!1(i_1)))(i) = card[T](IUnion(lambda (n:nat): if n <= i then X!1(n) else emptyset[T] endif))")
                                (("1"
                                  (case
                                   "FORALL i: is_finite[T](IUnion(LAMBDA (n: nat):
                       IF n <= i THEN X!1(n) ELSE emptyset[T] ENDIF))")
                                  (("1"
                                    (name-replace
                                     "CS"
                                     "series(LAMBDA (i_1: nat): card[T](X!1(i_1)))")
                                    (("1"
                                      (case-replace
                                       "is_finite[T](IUnion(X!1))")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (case-replace
                                           "convergent(CS)")
                                          (("1"
                                            (case "increasing?(CS)")
                                            (("1"
                                              (case
                                               "FORALL i: CS(i) <= card[T](IUnion(X!1))")
                                              (("1"
                                                (lemma
                                                 "converges_upto_is_lub"
                                                 ("u"
                                                  "CS"
                                                  "x"
                                                  "card[T](IUnion(X!1))"))
                                                (("1"
                                                  (expand
                                                   "converges_upto?"
                                                   -1)
                                                  (("1"
                                                    (replace -3)
                                                    (("1"
                                                      (flatten -1)
                                                      (("1"
                                                        (hide -1)
                                                        (("1"
                                                          (split -1)
                                                          (("1"
                                                            (rewrite
                                                             "convergence_sequences.limit_def")
                                                            (("1"
                                                              (hide-all-but
                                                               (-1 1))
                                                              (("1"
                                                                (rewrite
                                                                 "metric_convergence_def")
                                                                (("1"
                                                                  (expand
                                                                   "metric_converges_to")
                                                                  (("1"
                                                                    (expand
                                                                     "convergence")
                                                                    (("1"
                                                                      (expand
                                                                       "ball")
                                                                      (("1"
                                                                        (expand
                                                                         "member")
                                                                        (("1"
                                                                          (skosimp)
                                                                          (("1"
                                                                            (inst
                                                                             -
                                                                             "epsilon!1")
                                                                            (("1"
                                                                              (skosimp)
                                                                              (("1"
                                                                                (inst
                                                                                 +
                                                                                 "n!1")
                                                                                (("1"
                                                                                  (skosimp)
                                                                                  (("1"
                                                                                    (inst
                                                                                     -
                                                                                     "i!1")
                                                                                    (("1"
                                                                                      (grind)
                                                                                      nil
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (hide 2)
                                                            (("2"
                                                              (expand
                                                               "bounded_above?")
                                                              (("2"
                                                                (inst
                                                                 +
                                                                 "card[T](IUnion(X!1))")
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("3"
                                                            (hide 2)
                                                            (("3"
                                                              (expand
                                                               "fullset")
                                                              (("3"
                                                                (expand
                                                                 "image")
                                                                (("3"
                                                                  (expand
                                                                   "least_upper_bound?")
                                                                  (("3"
                                                                    (split)
                                                                    (("1"
                                                                      (expand
                                                                       "upper_bound?")
                                                                      (("1"
                                                                        (skosimp)
                                                                        (("1"
                                                                          (typepred
                                                                           "s!1")
                                                                          (("1"
                                                                            (skosimp)
                                                                            (("1"
                                                                              (inst
                                                                               -
                                                                               "x!1")
                                                                              (("1"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (skosimp*)
                                                                      (("2"
                                                                        (expand
                                                                         "upper_bound?")
                                                                        (("2"
                                                                          (case
                                                                           "y!1<card[T](IUnion(X!1))")
                                                                          (("1"
                                                                            (hide
                                                                             1)
                                                                            (("1"
                                                                              (case
                                                                               "exists i: IUnion(LAMBDA (n: nat):
                      IF n <= i THEN X!1(n) ELSE emptyset[T] ENDIF) = IUnion(X!1)")
                                                                              (("1"
                                                                                (skosimp)
                                                                                (("1"
                                                                                  (inst
                                                                                   -3
                                                                                   "CS(i!1)")
                                                                                  (("1"
                                                                                    (inst
                                                                                     -9
                                                                                     "i!1")
                                                                                    (("1"
                                                                                      (replace
                                                                                       -1)
                                                                                      (("1"
                                                                                        (assert)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (inst
                                                                                     +
                                                                                     "i!1")
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (hide
                                                                                 -1
                                                                                 -2)
                                                                                (("2"
                                                                                  (lemma
                                                                                   "convergence_cauchy1"
                                                                                   ("u"
                                                                                    "CS"))
                                                                                  (("2"
                                                                                    (assert)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "cauchy")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -
                                                                                         "1/2")
                                                                                        (("2"
                                                                                          (skosimp)
                                                                                          (("2"
                                                                                            (inst
                                                                                             +
                                                                                             "n!1")
                                                                                            (("2"
                                                                                              (apply-extensionality
                                                                                               :hide?
                                                                                               t)
                                                                                              (("2"
                                                                                                (inst
                                                                                                 -
                                                                                                 "_"
                                                                                                 "n!1")
                                                                                                (("2"
                                                                                                  (assert)
                                                                                                  (("2"
                                                                                                    (case-replace
                                                                                                     "IUnion(LAMBDA (n: nat):
               IF n <= n!1 THEN X!1(n) ELSE emptyset[T] ENDIF)
            (x!1)")
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "IUnion")
                                                                                                      (("1"
                                                                                                        (skosimp)
                                                                                                        (("1"
                                                                                                          (expand
                                                                                                           "emptyset")
                                                                                                          (("1"
                                                                                                            (prop)
                                                                                                            (("1"
                                                                                                              (inst
                                                                                                               +
                                                                                                               "i!1")
                                                                                                              nil
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (replace
                                                                                                       1
                                                                                                       2)
                                                                                                      (("2"
                                                                                                        (assert)
                                                                                                        (("2"
                                                                                                          (expand
                                                                                                           "IUnion")
                                                                                                          (("2"
                                                                                                            (skosimp)
                                                                                                            (("2"
                                                                                                              (expand
                                                                                                               "emptyset")
                                                                                                              (("2"
                                                                                                                (case
                                                                                                                 "i!1>n!1")
                                                                                                                (("1"
                                                                                                                  (inst
                                                                                                                   -
                                                                                                                   "i!1")
                                                                                                                  (("1"
                                                                                                                    (assert)
                                                                                                                    (("1"
                                                                                                                      (hide
                                                                                                                       1)
                                                                                                                      (("1"
                                                                                                                        (expand
                                                                                                                         "increasing?")
                                                                                                                        (("1"
                                                                                                                          (inst
                                                                                                                           -5
                                                                                                                           "n!1"
                                                                                                                           "i!1")
                                                                                                                          (("1"
                                                                                                                            (assert)
                                                                                                                            (("1"
                                                                                                                              (expand
                                                                                                                               "abs"
                                                                                                                               -3)
                                                                                                                              (("1"
                                                                                                                                (assert)
                                                                                                                                (("1"
                                                                                                                                  (expand
                                                                                                                                   "<="
                                                                                                                                   -5)
                                                                                                                                  (("1"
                                                                                                                                    (split
                                                                                                                                     -5)
                                                                                                                                    (("1"
                                                                                                                                      (assert)
                                                                                                                                      (("1"
                                                                                                                                        (inst-cp
                                                                                                                                         -9
                                                                                                                                         "n!1")
                                                                                                                                        (("1"
                                                                                                                                          (inst
                                                                                                                                           -9
                                                                                                                                           "i!1")
                                                                                                                                          (("1"
                                                                                                                                            (replace
                                                                                                                                             -10)
                                                                                                                                            (("1"
                                                                                                                                              (replace
                                                                                                                                               -9)
                                                                                                                                              (("1"
                                                                                                                                                (assert)
                                                                                                                                                nil
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil)
                                                                                                                                     ("2"
                                                                                                                                      (hide
                                                                                                                                       -4)
                                                                                                                                      (("2"
                                                                                                                                        (inst-cp
                                                                                                                                         -8
                                                                                                                                         "n!1")
                                                                                                                                        (("2"
                                                                                                                                          (inst
                                                                                                                                           -8
                                                                                                                                           "i!1")
                                                                                                                                          (("2"
                                                                                                                                            (replace
                                                                                                                                             -9)
                                                                                                                                            (("2"
                                                                                                                                              (replace
                                                                                                                                               -8)
                                                                                                                                              (("2"
                                                                                                                                                (hide
                                                                                                                                                 -9
                                                                                                                                                 -8)
                                                                                                                                                (("2"
                                                                                                                                                  (lemma
                                                                                                                                                   "same_card_subset[T]"
                                                                                                                                                   ("A"
                                                                                                                                                    "{x: T |
              EXISTS (i_1: nat):
                IF i_1 <= n!1 THEN X!1(i_1)(x) ELSE FALSE ENDIF}"
                                                                                                                                                    "B"
                                                                                                                                                    "{x_1: T |
               EXISTS (i_2: nat):
                 IF i_2 <= i!1 THEN X!1(i_2)(x_1) ELSE FALSE ENDIF}"))
                                                                                                                                                  (("1"
                                                                                                                                                    (replace
                                                                                                                                                     -2)
                                                                                                                                                    (("1"
                                                                                                                                                      (hide
                                                                                                                                                       -2)
                                                                                                                                                      (("1"
                                                                                                                                                        (split
                                                                                                                                                         -1)
                                                                                                                                                        (("1"
                                                                                                                                                          (lemma
                                                                                                                                                           "extensionality_postulate"
                                                                                                                                                           ("f"
                                                                                                                                                            "{x_1: T |
          EXISTS (i_2: nat):
            IF i_2 <= n!1 THEN X!1(i_2)(x_1) ELSE FALSE ENDIF}"
                                                                                                                                                            "g"
                                                                                                                                                            "{x_2: T |
           EXISTS (i_1: nat):
             IF i_1 <= i!1 THEN X!1(i_1)(x_2) ELSE FALSE ENDIF}"))
                                                                                                                                                          (("1"
                                                                                                                                                            (replace
                                                                                                                                                             -1
                                                                                                                                                             -2
                                                                                                                                                             rl)
                                                                                                                                                            (("1"
                                                                                                                                                              (hide
                                                                                                                                                               -1)
                                                                                                                                                              (("1"
                                                                                                                                                                (inst
                                                                                                                                                                 -
                                                                                                                                                                 "x!1")
                                                                                                                                                                (("1"
                                                                                                                                                                  (case-replace
                                                                                                                                                                   "(EXISTS (i_2: nat):
         IF i_2 <= n!1 THEN X!1(i_2)(x!1) ELSE FALSE ENDIF)")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (skosimp)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (prop)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (typepred
                                                                                                                                                                         "X!1")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (expand
                                                                                                                                                                           "disjoint_indexed_measurable?")
                                                                                                                                                                          (("1"
                                                                                                                                                                            (expand
                                                                                                                                                                             "disjoint?")
                                                                                                                                                                            (("1"
                                                                                                                                                                              (expand
                                                                                                                                                                               "disjoint?")
                                                                                                                                                                              (("1"
                                                                                                                                                                                (expand
                                                                                                                                                                                 "intersection")
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (expand
                                                                                                                                                                                   "empty?")
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (expand
                                                                                                                                                                                     "member")
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (inst
                                                                                                                                                                                       -
                                                                                                                                                                                       "i!2"
                                                                                                                                                                                       "i!1")
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (assert)
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (inst
                                                                                                                                                                                           -
                                                                                                                                                                                           "x!1")
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (assert)
                                                                                                                                                                                            nil
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil)
                                                                                                                                                                   ("2"
                                                                                                                                                                    (replace
                                                                                                                                                                     1
                                                                                                                                                                     -1)
                                                                                                                                                                    (("2"
                                                                                                                                                                      (assert)
                                                                                                                                                                      (("2"
                                                                                                                                                                        (inst
                                                                                                                                                                         2
                                                                                                                                                                         "i!1")
                                                                                                                                                                        (("2"
                                                                                                                                                                          (assert)
                                                                                                                                                                          nil
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil)
                                                                                                                                                         ("2"
                                                                                                                                                          (expand
                                                                                                                                                           "subset?")
                                                                                                                                                          (("2"
                                                                                                                                                            (expand
                                                                                                                                                             "member")
                                                                                                                                                            (("2"
                                                                                                                                                              (skosimp*)
                                                                                                                                                              (("2"
                                                                                                                                                                (prop)
                                                                                                                                                                (("2"
                                                                                                                                                                  (inst
                                                                                                                                                                   +
                                                                                                                                                                   "i!2")
                                                                                                                                                                  (("2"
                                                                                                                                                                    (assert)
                                                                                                                                                                    nil
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil)
                                                                                                                                                   ("2"
                                                                                                                                                    (inst
                                                                                                                                                     -7
                                                                                                                                                     "i!1")
                                                                                                                                                    nil
                                                                                                                                                    nil)
                                                                                                                                                   ("3"
                                                                                                                                                    (inst
                                                                                                                                                     -7
                                                                                                                                                     "n!1")
                                                                                                                                                    nil
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("2"
                                                                                                                  (inst
                                                                                                                   +
                                                                                                                   "i!1")
                                                                                                                  (("2"
                                                                                                                    (assert)
                                                                                                                    nil
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                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))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide -2 2)
                                                (("2"
                                                  (skosimp)
                                                  (("2"
                                                    (inst -4 "i!1")
                                                    (("2"
                                                      (replace -4)
                                                      (("2"
                                                        (lemma
                                                         "card_subset[T]"
                                                         ("A"
                                                          "IUnion(LAMBDA (n: nat):
                    IF n <= i!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"
                                                          "B"
                                                          "IUnion(X!1)"))
                                                        (("2"
                                                          (assert)
                                                          (("2"
                                                            (hide 2)
                                                            (("2"
                                                              (expand
                                                               "subset?")
                                                              (("2"
                                                                (expand
                                                                 "IUnion")
                                                                (("2"
                                                                  (expand
                                                                   "emptyset")
                                                                  (("2"
                                                                    (expand
                                                                     "member")
                                                                    (("2"
                                                                      (skosimp*)
                                                                      (("2"
                                                                        (prop)
                                                                        (("2"
                                                                          (inst
                                                                           +
                                                                           "i!2")
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide -1 2)
                                              (("2"
                                                (expand "increasing?")
                                                (("2"
                                                  (skolem
                                                   +
                                                   ("i!1" "j!1"))
                                                  (("2"
                                                    (flatten)
                                                    (("2"
                                                      (inst-cp
                                                       -4
                                                       "i!1")
                                                      (("2"
                                                        (inst -4 "j!1")
                                                        (("2"
                                                          (replace -4)
                                                          (("2"
                                                            (replace
                                                             -5)
                                                            (("2"
                                                              (hide
                                                               -4
                                                               -5)
                                                              (("2"
                                                                (lemma
                                                                 "card_subset[T]"
                                                                 ("A"
                                                                  "IUnion(LAMBDA (n_1: nat):
                    IF n_1 <= i!1 THEN X!1(n_1) ELSE emptyset[T] ENDIF)"
                                                                  "B"
                                                                  "IUnion(LAMBDA (n: nat):
                     IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
                                                                (("2"
                                                                  (assert)
                                                                  (("2"
                                                                    (hide
                                                                     2)
                                                                    (("2"
                                                                      (expand
                                                                       "IUnion")
                                                                      (("2"
                                                                        (expand
                                                                         "subset?")
                                                                        (("2"
                                                                          (expand
                                                                           "emptyset")
                                                                          (("2"
                                                                            (expand
                                                                             "member")
                                                                            (("2"
                                                                              (skosimp*)
                                                                              (("2"
                                                                                (prop)
                                                                                (("2"
                                                                                  (inst
                                                                                   +
                                                                                   "i!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)
                                           ("2"
                                            (assert)
                                            (("2"
                                              (hide 2)
                                              (("2"
                                                (case
                                                 "forall i: CS(i) <= card[T](IUnion(X!1))")
                                                (("1"
                                                  (case
                                                   "increasing?(CS)")
                                                  (("1"
                                                    (lemma
                                                     "converges_upto_is_lub"
                                                     ("u"
                                                      "CS"
                                                      "x"
                                                      "card[T](IUnion(X!1))"))
                                                    (("1"
                                                      (expand
                                                       "converges_upto?")
                                                      (("1"
                                                        (replace -2)
                                                        (("1"
                                                          (flatten -1)
                                                          (("1"
                                                            (hide -1)
                                                            (("1"
                                                              (split
                                                               -1)
                                                              (("1"
                                                                (rewrite
                                                                 "metric_convergence_def")
                                                                (("1"
                                                                  (hide-all-but
                                                                   (-1
                                                                    1))
                                                                  (("1"
                                                                    (expand
                                                                     "metric_converges_to")
                                                                    (("1"
                                                                      (expand
                                                                       "ball")
                                                                      (("1"
                                                                        (expand
                                                                         "member")
                                                                        (("1"
                                                                          (assert)
                                                                          (("1"
                                                                            (expand
                                                                             "convergent?")
                                                                            (("1"
                                                                              (expand
                                                                               "convergence")
                                                                              (("1"
                                                                                (inst
                                                                                 +
                                                                                 "card[T](IUnion(X!1))")
                                                                                (("1"
                                                                                  (skosimp)
                                                                                  (("1"
                                                                                    (inst
                                                                                     -
                                                                                     "epsilon!1")
                                                                                    (("1"
                                                                                      (skosimp)
                                                                                      (("1"
                                                                                        (inst
                                                                                         +
                                                                                         "n!1")
                                                                                        (("1"
                                                                                          (skosimp)
                                                                                          (("1"
                                                                                            (inst
                                                                                             -
                                                                                             "i!1")
                                                                                            (("1"
                                                                                              (assert)
                                                                                              (("1"
                                                                                                (grind)
                                                                                                nil
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (hide
                                                                 2)
                                                                (("2"
                                                                  (expand
                                                                   "bounded_above?")
                                                                  (("2"
                                                                    (inst
                                                                     +
                                                                     "card[T](IUnion(X!1))")
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("3"
                                                                (hide
                                                                 2)
                                                                (("3"
                                                                  (expand
                                                                   "fullset")
                                                                  (("3"
                                                                    (expand
                                                                     "image")
                                                                    (("3"
                                                                      (expand
                                                                       "least_upper_bound?")
                                                                      (("3"
                                                                        (split)
                                                                        (("1"
                                                                          (expand
                                                                           "upper_bound?")
                                                                          (("1"
                                                                            (skosimp)
                                                                            (("1"
                                                                              (typepred
                                                                               "s!1")
                                                                              (("1"
                                                                                (skosimp)
                                                                                (("1"
                                                                                  (inst
                                                                                   -
                                                                                   "x!1")
                                                                                  (("1"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (skosimp)
                                                                          (("2"
                                                                            (case
                                                                             "y!1<card[T](IUnion(X!1))")
                                                                            (("1"
                                                                              (hide
                                                                               1)
                                                                              (("1"
                                                                                (expand
                                                                                 "upper_bound?")
                                                                                (("1"
                                                                                  (case
                                                                                   "exists i: IUnion(LAMBDA (n: nat):
                      IF n <= i THEN X!1(n) ELSE emptyset[T] ENDIF) = IUnion(X!1)")
                                                                                  (("1"
                                                                                    (skosimp)
                                                                                    (("1"
                                                                                      (inst
                                                                                       -8
                                                                                       "i!1")
                                                                                      (("1"
                                                                                        (replace
                                                                                         -1)
                                                                                        (("1"
                                                                                          (inst
                                                                                           -3
                                                                                           "CS(i!1)")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil)
                                                                                           ("2"
                                                                                            (inst
                                                                                             +
                                                                                             "i!1")
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (hide-all-but
                                                                                     (-5
                                                                                      -8
                                                                                      1))
                                                                                    (("2"
                                                                                      (case
                                                                                       "forall (Y:sequence[set[T]],n:nat): (FORALL i: is_finite[T](Y(i)))& is_finite(IUnion(Y)) & card(IUnion(Y))=n => EXISTS i:
        IUnion(LAMBDA (m: nat):
                 IF m <= i THEN Y(m) ELSE emptyset[T] ENDIF)
         = IUnion(Y)")
                                                                                      (("1"
                                                                                        (inst
                                                                                         -
                                                                                         "X!1"
                                                                                         "card(IUnion(X!1))")
                                                                                        (("1"
                                                                                          (replace
                                                                                           -2)
                                                                                          (("1"
                                                                                            (replace
                                                                                             -3)
                                                                                            (("1"
                                                                                              (propax)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide
                                                                                         -1
                                                                                         -2
                                                                                         2)
                                                                                        (("2"
                                                                                          (induct
                                                                                           "n")
                                                                                          (("1"
                                                                                            (skosimp)
                                                                                            (("1"
                                                                                              (inst
                                                                                               +
                                                                                               "0")
                                                                                              (("1"
                                                                                                (apply-extensionality
                                                                                                 :hide?
                                                                                                 t)
                                                                                                (("1"
                                                                                                  (lemma
                                                                                                   "card_is_0[T]"
                                                                                                   ("S"
                                                                                                    "IUnion(Y!1)"))
                                                                                                  (("1"
                                                                                                    (assert)
                                                                                                    (("1"
                                                                                                      (replace
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (expand
                                                                                                         "emptyset"
                                                                                                         1)
                                                                                                        (("1"
                                                                                                          (expand
                                                                                                           "IUnion"
                                                                                                           -2)
                                                                                                          (("1"
                                                                                                            (skosimp)
                                                                                                            (("1"
                                                                                                              (prop)
                                                                                                              (("1"
                                                                                                                (rewrite
                                                                                                                 "emptyset_is_empty?"
                                                                                                                 -3
                                                                                                                 :dir
                                                                                                                 rl)
                                                                                                                (("1"
                                                                                                                  (expand
                                                                                                                   "empty?")
                                                                                                                  (("1"
                                                                                                                    (inst
                                                                                                                     -
                                                                                                                     "x!1")
                                                                                                                    (("1"
                                                                                                                      (expand
                                                                                                                       "member")
                                                                                                                      (("1"
                                                                                                                        (expand
                                                                                                                         "IUnion")
                                                                                                                        (("1"
                                                                                                                          (inst
                                                                                                                           +
                                                                                                                           "i!1")
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil)
                                                                                           ("2"
                                                                                            (skosimp*)
                                                                                            (("2"
                                                                                              (lemma
                                                                                               "nonempty_card[T]"
                                                                                               ("S"
                                                                                                "IUnion(Y!1)"))
                                                                                              (("2"
                                                                                                (assert)
                                                                                                (("2"
                                                                                                  (lemma
                                                                                                   "choose_rest[T]"
                                                                                                   ("a"
                                                                                                    "IUnion(Y!1)"))
                                                                                                  (("2"
                                                                                                    (expand
                                                                                                     "nonempty?")
                                                                                                    (("2"
                                                                                                      (assert)
                                                                                                      (("2"
                                                                                                        (lemma
                                                                                                         "card_rest[T]"
                                                                                                         ("S"
                                                                                                          "IUnion(Y!1)"))
                                                                                                        (("2"
                                                                                                          (replace
                                                                                                           -6)
                                                                                                          (("2"
                                                                                                            (assert)
                                                                                                            (("2"
                                                                                                              (lemma
                                                                                                               "choose_member[T]"
                                                                                                               ("a"
                                                                                                                "IUnion(Y!1)"))
                                                                                                              (("2"
                                                                                                                (name-replace
                                                                                                                 "CC"
                                                                                                                 "choose(IUnion(Y!1))")
                                                                                                                (("2"
                                                                                                                  (replace
                                                                                                                   1
                                                                                                                   -1)
                                                                                                                  (("2"
                                                                                                                    (name
                                                                                                                     "YY"
                                                                                                                     "lambda i: remove[T](CC,Y!1(i))")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       -5
                                                                                                                       "YY")
                                                                                                                      (("2"
                                                                                                                        (case
                                                                                                                         "IUnion(YY)=rest(IUnion(Y!1))")
                                                                                                                        (("1"
                                                                                                                          (replace
                                                                                                                           -1)
                                                                                                                          (("1"
                                                                                                                            (replace
                                                                                                                             -4)
                                                                                                                            (("1"
                                                                                                                              (lemma
                                                                                                                               "finite_rest[T]"
                                                                                                                               ("A"
                                                                                                                                "IUnion(Y!1)"))
                                                                                                                              (("1"
                                                                                                                                (replace
                                                                                                                                 -1)
                                                                                                                                (("1"
                                                                                                                                  (split
                                                                                                                                   -7)
                                                                                                                                  (("1"
                                                                                                                                    (skosimp)
                                                                                                                                    (("1"
                                                                                                                                      (expand
                                                                                                                                       "member"
                                                                                                                                       -5)
                                                                                                                                      (("1"
                                                                                                                                        (expand
                                                                                                                                         "IUnion"
                                                                                                                                         -5)
                                                                                                                                        (("1"
                                                                                                                                          (skosimp)
                                                                                                                                          (("1"
                                                                                                                                            (inst
                                                                                                                                             +
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                                                                                                                                                   -8
                                                                                                                                                   -1
                                                                                                                                                   rl)
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                                                                                                                                                           +
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                                                                                                                                    nil)
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                                                                                                                                      1
                                                                                                                                      2))
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                                                                                                                                      (skosimp)
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                                                                                                                                        (("2"
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                                                                                                                                          (("2"
                                                                                                                                            (lemma
                                                                                                                                             "finite_remove[T]"
                                                                                                                                             ("x"
                                                                                                                                              "CC"
                                                                                                                                              "A"
                                                                                                                                              "Y!1(i!1)"))
                                                                                                                                            (("1"
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                                                                                                                                              nil
                                                                                                                                              nil)
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                                                                                                                         ("2"
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                                                                                                                           3
                                                                                                                           -5)
                                                                                                                          (("2"
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                                                                                                                             :hide?
                                                                                                                             t)
                                                                                                                            (("2"
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                                                                                                                               "YY")
                                                                                                                              (("2"
                                                                                                                                (expand
                                                                                                                                 "rest"
                                                                                                                                 1)
                                                                                                                                (("2"
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                                                                                                                                   1)
                                                                                                                                  (("2"
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                                                                                                                                     "IUnion"
                                                                                                                                     1)
                                                                                                                                    (("2"
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                                                                                                                                      (("2"
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                                                                                                                                         "member")
                                                                                                                                        (("2"
                                                                                                                                          (case-replace
                                                                                                                                           "choose({x: T | EXISTS (i: nat): Y!1(i)(x)}) /= x!1")
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                                                                                                                                            nil
                                                                                                                                            nil)
                                                                                                                                           ("2"
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                                                                                                                                               "IUnion")
                                                                                                                                              (("2"
                                                                                                                                                (propax)
                                                                                                                                                nil
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                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
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                                                                                                nil))
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                                                                                        nil))
                                                                                      nil))
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                                                                              nil)
                                                                             ("2"
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                                                        nil))
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                                                    nil)
                                                   ("2"
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                                                       "increasing?")
                                                      (("2"
                                                        (skolem
                                                         +
                                                         ("i!1" "j!1"))
                                                        (("2"
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                                                            (inst-cp
                                                             -5
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                                                            (("2"
                                                              (inst
                                                               -5
                                                               "j!1")
                                                              (("2"
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                                                                 -5)
                                                                (("2"
                                                                  (replace
                                                                   -6)
                                                                  (("2"
                                                                    (hide
                                                                     -5
                                                                     -6)
                                                                    (("2"
                                                                      (lemma
                                                                       "card_subset[T]"
                                                                       ("A"
                                                                        "IUnion(LAMBDA (n_1: nat):
                    IF n_1 <= i!1 THEN X!1(n_1) ELSE emptyset[T] ENDIF)"
                                                                        "B"
                                                                        "IUnion(LAMBDA (n: nat):
                     IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
                                                                      (("2"
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                                                                          (hide
                                                                           2)
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                                                                               "IUnion")
                                                                              (("2"
                                                                                (expand
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                                                                                (("2"
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                                                                                  (("2"
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                                                                                        (inst
                                                                                         +
                                                                                         "i!2")
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                                                                          nil))
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                                                                    nil))
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                                                              nil))
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                                                      nil))
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                                                  nil)
                                                 ("2"
                                                  (hide 2)
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                                                        (inst - "i!1")
                                                        (("2"
                                                          (replace -3)
                                                          (("2"
                                                            (lemma
                                                             "card_subset[T]"
                                                             ("A"
                                                              "IUnion(LAMBDA (n: nat):
                    IF n <= i!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"
                                                              "B"
                                                              "IUnion(X!1)"))
                                                            (("2"
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                                                               -1)
                                                              (("1"
                                                                (propax)
                                                                nil
                                                                nil)
                                                               ("2"
                                                                (hide
                                                                 -3
                                                                 2)
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                                                                   "subset?")
                                                                  (("2"
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                                                                     "IUnion")
                                                                    (("2"
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                                                                       "member")
                                                                      (("2"
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                                                                         "emptyset")
                                                                        (("2"
                                                                          (skosimp*)
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                                                                            (prop)
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                                                                              (inst
                                                                               +
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                                                                              nil))
                                                                            nil))
                                                                          nil))
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                                        nil)
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                                             "convergence_cauchy1"
                                             ("u" "CS"))
                                            (("1"
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                                                (expand "cauchy")
                                                (("1"
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                                                  (("1"
                                                    (skosimp)
                                                    (("1"
                                                      (case
                                                       "IUnion(LAMBDA (n: nat):
                      IF n <= n!1 THEN X!1(n) ELSE emptyset[T] ENDIF) = IUnion(X!1)")
                                                      (("1"
                                                        (inst -4 "n!1")
                                                        (("1"
                                                          (replace -1)
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                                                            (propax)
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                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide 2)
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                                                           :hide?
                                                           t)
                                                          (("2"
                                                            (expand
                                                             "IUnion"
                                                             1)
                                                            (("2"
                                                              (case-replace
                                                               "(EXISTS (i: nat):
         IF i <= n!1 THEN X!1(i)(x!1) ELSE emptyset[T](x!1) ENDIF)")
                                                              (("1"
                                                                (skosimp)
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                                                                     +
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                                                                    nil)
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                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
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                                                                 1
                                                                 2)
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                                                                  (assert)
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                                                                    (("2"
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                                                                       "i!1>n!1")
                                                                      (("1"
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                                                                         -
                                                                         "i!1"
                                                                         "n!1")
                                                                        (("1"
                                                                          (assert)
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                                                                            (inst-cp
                                                                             -6
                                                                             "i!1")
                                                                            (("1"
                                                                              (inst
                                                                               -6
                                                                               "n!1")
                                                                              (("1"
                                                                                (replace
                                                                                 -6)
                                                                                (("1"
                                                                                  (replace
                                                                                   -7)
                                                                                  (("1"
                                                                                    (hide
                                                                                     -6
                                                                                     -7)
                                                                                    (("1"
                                                                                      (hide
                                                                                       1)
                                                                                      (("1"
                                                                                        (inst-cp
                                                                                         -
                                                                                         "i!1")
                                                                                        (("1"
                                                                                          (inst
                                                                                           -
                                                                                           "n!1")
                                                                                          (("1"
                                                                                            (name-replace
                                                                                             "IU_n"
                                                                                             "IUnion(LAMBDA (n: nat):
                         IF n <= n!1 THEN X!1(n) ELSE emptyset[T] ENDIF)")
                                                                                            (("1"
                                                                                              (name-replace
                                                                                               "IU_i"
                                                                                               "IUnion(LAMBDA (n: nat):
                         IF n <= i!1 THEN X!1(n) ELSE emptyset[T] ENDIF)")
                                                                                              (("1"
                                                                                                (lemma
                                                                                                 "card_subset[T]"
                                                                                                 ("A"
                                                                                                  "IU_n"
                                                                                                  "B"
                                                                                                  "IU_i"))
                                                                                                (("1"
                                                                                                  (split
                                                                                                   -1)
                                                                                                  (("1"
                                                                                                    (expand
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                                                                                                      (expand
                                                                                                       "<="
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                                                                                                         -1)
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                                                                                                          nil)
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                                                                                                            (("2"
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                                                                                                               "disjoint_indexed_measurable?")
                                                                                                              (("2"
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                                                                                                                 "disjoint?")
                                                                                                                (("2"
                                                                                                                  (inst
                                                                                                                   -
                                                                                                                   "i!1"
                                                                                                                   "n!1")
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                                                                                                                       "disjoint?")
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                                                                                                                        (expand
                                                                                                                         "intersection")
                                                                                                                        (("2"
                                                                                                                          (expand
                                                                                                                           "empty?")
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             -
                                                                                                                             "x!1")
                                                                                                                            (("2"
                                                                                                                              (expand
                                                                                                                               "member")
                                                                                                                              (("2"
                                                                                                                                (lemma
                                                                                                                                 "same_card_subset[T]"
                                                                                                                                 ("A"
                                                                                                                                  "IU_n"
                                                                                                                                  "B"
                                                                                                                                  "IU_i"))
                                                                                                                                (("2"
                                                                                                                                  (split)
                                                                                                                                  (("1"
                                                                                                                                    (lemma
                                                                                                                                     "extensionality_postulate"
                                                                                                                                     ("f"
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                                                                                                                                      "g"
                                                                                                                                      "IU_i"))
                                                                                                                                    (("1"
                                                                                                                                      (replace
                                                                                                                                       -1
                                                                                                                                       -2
                                                                                                                                       rl)
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                                                                                                                                        (hide
                                                                                                                                         -1)
                                                                                                                                        (("1"
                                                                                                                                          (inst
                                                                                                                                           -
                                                                                                                                           "x!1")
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                                                                                                                                             "IU_n"
                                                                                                                                             -1)
                                                                                                                                            (("1"
                                                                                                                                              (expand
                                                                                                                                               "IU_i"
                                                                                                                                               -1)
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                                                                                                                                                (expand
                                                                                                                                                 "IUnion"
                                                                                                                                                 -1)
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                                                                                                                                                  (expand
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                                                                                                                                                  (("1"
                                                                                                                                                    (case-replace
                                                                                                                                                     "EXISTS (i: nat): IF i <= n!1 THEN X!1(i)(x!1) ELSE FALSE ENDIF")
                                                                                                                                                    (("1"
                                                                                                                                                      (skosimp)
                                                                                                                                                      (("1"
                                                                                                                                                        (prop)
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                                                                                                                                                          (typepred
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                                                                                                                                                          (("1"
                                                                                                                                                            (expand
                                                                                                                                                             "disjoint_indexed_measurable?")
                                                                                                                                                            (("1"
                                                                                                                                                              (expand
                                                                                                                                                               "disjoint?")
                                                                                                                                                              (("1"
                                                                                                                                                                (expand
                                                                                                                                                                 "disjoint?")
                                                                                                                                                                (("1"
                                                                                                                                                                  (expand
                                                                                                                                                                   "intersection")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (expand
                                                                                                                                                                     "empty?")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (expand
                                                                                                                                                                       "member")
                                                                                                                                                                      (("1"
                                                                                                                                                                        (inst
                                                                                                                                                                         -
                                                                                                                                                                         "i!2"
                                                                                                                                                                         "i!1")
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                                                                                                                                                                          (assert)
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                                                                                                                                                                            (inst
                                                                                                                                                                             -
                                                                                                                                                                             "x!1")
                                                                                                                                                                            (("1"
                                                                                                                                                                              (assert)
                                                                                                                                                                              nil
                                                                                                                                                                              nil))
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                                                                                                                                                                        nil))
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                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
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                                                                                                                                                        nil))
                                                                                                                                                      nil)
                                                                                                                                                     ("2"
                                                                                                                                                      (replace
                                                                                                                                                       1
                                                                                                                                                       -1)
                                                                                                                                                      (("2"
                                                                                                                                                        (assert)
                                                                                                                                                        (("2"
                                                                                                                                                          (inst
                                                                                                                                                           2
                                                                                                                                                           "i!1")
                                                                                                                                                          (("2"
                                                                                                                                                            (assert)
                                                                                                                                                            nil
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil)
                                                                                                                                   ("2"
                                                                                                                                    (hide-all-but
                                                                                                                                     (-2
                                                                                                                                      1))
                                                                                                                                    (("2"
                                                                                                                                      (expand
                                                                                                                                       "subset?")
                                                                                                                                      (("2"
                                                                                                                                        (expand
                                                                                                                                         "IU_n")
                                                                                                                                        (("2"
                                                                                                                                          (expand
                                                                                                                                           "IU_i")
                                                                                                                                          (("2"
                                                                                                                                            (expand
                                                                                                                                             "member")
                                                                                                                                            (("2"
                                                                                                                                              (skosimp)
                                                                                                                                              (("2"
                                                                                                                                                (expand
                                                                                                                                                 "IUnion")
                                                                                                                                                (("2"
                                                                                                                                                  (expand
                                                                                                                                                   "emptyset")
                                                                                                                                                  (("2"
                                                                                                                                                    (skosimp)
                                                                                                                                                    (("2"
                                                                                                                                                      (prop)
                                                                                                                                                      (("2"
                                                                                                                                                        (inst
                                                                                                                                                         +
                                                                                                                                                         "i!2")
                                                                                                                                                        (("2"
                                                                                                                                                          (assert)
                                                                                                                                                          nil
                                                                                                                                                          nil))
                                                                                                                                                        nil))
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil)
                                                                                                                                   ("3"
                                                                                                                                    (propax)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil)
                                                                                                   ("2"
                                                                                                    (hide-all-but
                                                                                                     (-1
                                                                                                      1))
                                                                                                    (("2"
                                                                                                      (expand
                                                                                                       "subset?")
                                                                                                      (("2"
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                                                                                                        (("2"
                                                                                                          (expand
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                                                                                                          (("2"
                                                                                                            (expand
                                                                                                             "member")
                                                                                                            (("2"
                                                                                                              (expand
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                                                                                                              (("2"
                                                                                                                (skosimp*)
                                                                                                                (("2"
                                                                                                                  (expand
                                                                                                                   "emptyset")
                                                                                                                  (("2"
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                                                                                                                      (inst
                                                                                                                       +
                                                                                                                       "i!2")
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                                                                                                                        (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
                                                                         +
                                                                         "i!1")
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                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but (-2 1))
                                    (("2"
                                      (copy -1)
                                      (("2"
                                        (induct "i")
                                        (("1"
                                          (case-replace
                                           "(IUnion(LAMBDA (n: nat):
                    IF n <= 0 THEN X!1(n) ELSE emptyset[T] ENDIF))=X!1(0)")
                                          (("1" (inst - "0"nil nil)
                                           ("2"
                                            (hide-all-but 1)
                                            (("2"
                                              (apply-extensionality
                                               :hide?
                                               t)
                                              (("2"
                                                (expand "emptyset")
                                                (("2"
                                                  (expand "IUnion")
                                                  (("2"
                                                    (case-replace
                                                     "X!1(0)(x!1)")
                                                    (("1"
                                                      (inst + "0")
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (skosimp*)
                                          (("2"
                                            (case-replace
                                             "IUnion(LAMBDA (n: nat):
                    IF n <= j!1 + 1 THEN X!1(n) ELSE emptyset[T] ENDIF)=union(X!1(1+j!1),IUnion(LAMBDA (n: nat):
                    IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF))")
                                            (("1"
                                              (hide -1)
                                              (("1"
                                                (lemma
                                                 "finite_union[T]"
                                                 ("A"
                                                  "X!1(1 + j!1)"
                                                  "B"
                                                  "IUnion(LAMBDA (n_1: nat):
                    IF n_1 <= j!1 THEN X!1(n_1) ELSE emptyset[T] ENDIF)"))
                                                (("1" (propax) nil nil)
                                                 ("2" (propax) nil nil)
                                                 ("3"
                                                  (inst - "1+j!1")
                                                  nil
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide-all-but 1)
                                              (("2"
                                                (apply-extensionality
                                                 :hide?
                                                 t)
                                                (("2"
                                                  (expand "union")
                                                  (("2"
                                                    (expand "member")
                                                    (("2"
                                                      (expand "IUnion")
                                                      (("2"
                                                        (expand
                                                         "emptyset")
                                                        (("2"
                                                          (case-replace
                                                           "X!1(1 + j!1)(x!1)")
                                                          (("1"
                                                            (inst
                                                             +
                                                             "1+j!1")
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (assert)
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                                                              (case-replace
                                                               "EXISTS (i: nat): IF i <= j!1 THEN X!1(i)(x!1) ELSE FALSE ENDIF")
                                                              (("1"
                                                                (skosimp)
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                                                                  (prop)
                                                                  (("1"
                                                                    (inst
                                                                     +
                                                                     "i!1")
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                                                                      (assert)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (replace
                                                                 1
                                                                 3)
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                                                                  (assert)
                                                                  (("2"
                                                                    (skosimp)
                                                                    (("2"
                                                                      (prop)
                                                                      (("2"
                                                                        (expand
                                                                         "<="
                                                                         -1)
                                                                        (("2"
                                                                          (split
                                                                           -1)
                                                                          (("1"
                                                                            (inst
                                                                             +
                                                                             "i!1")
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                                                                              (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))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (expand "series")
                                    (("2"
                                      (copy -1)
                                      (("2"
                                        (induct "i")
                                        (("1"
                                          (case-replace
                                           "IUnion(LAMBDA (n: nat):
                     IF n <= 0 THEN X!1(n) ELSE emptyset[T] ENDIF) =X!1(0)")
                                          (("1"
                                            (expand "sigma")
                                            (("1"
                                              (expand "sigma")
                                              (("1" (propax) nil nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (apply-extensionality
                                               :hide?
                                               t)
                                              (("2"
                                                (expand "IUnion")
                                                (("2"
                                                  (expand "emptyset")
                                                  (("2"
                                                    (case-replace
                                                     "X!1(0)(x!1)")
                                                    (("1"
                                                      (inst + "0")
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (skosimp*)
                                          (("2"
                                            (expand "sigma" 1)
                                            (("2"
                                              (case-replace
                                               "IUnion(LAMBDA (n: nat):
                     IF n <= 1 + j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)=union(X!1(1+j!1),IUnion(LAMBDA (n: nat):
                     IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF))")
                                              (("1"
                                                (hide -1)
                                                (("1"
                                                  (case
                                                   "disjoint?(X!1(1+j!1),IUnion(LAMBDA (n: nat):
                     IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF))")
                                                  (("1"
                                                    (lemma
                                                     "card_disj_union[T]"
                                                     ("A"
                                                      "X!1(1 + j!1)"
                                                      "B"
                                                      "IUnion(LAMBDA (n: nat):
                         IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide -1 2)
                                                    (("2"
                                                      (hide-all-but 1)
                                                      (("2"
                                                        (typepred
                                                         "X!1")
                                                        (("2"
                                                          (expand
                                                           "disjoint_indexed_measurable?")
                                                          (("2"
                                                            (expand
                                                             "disjoint?")
                                                            (("2"
                                                              (expand
                                                               "disjoint?")
                                                              (("2"
                                                                (expand
                                                                 "intersection")
                                                                (("2"
                                                                  (expand
                                                                   "IUnion")
                                                                  (("2"
                                                                    (expand
                                                                     "emptyset")
                                                                    (("2"
                                                                      (expand
                                                                       "empty?")
                                                                      (("2"
                                                                        (expand
                                                                         "member")
                                                                        (("2"
                                                                          (skosimp*)
                                                                          (("2"
                                                                            (prop)
                                                                            (("2"
                                                                              (inst
                                                                               -
                                                                               "i!1"
                                                                               "1+j!1")
                                                                              (("2"
                                                                                (assert)
                                                                                (("2"
                                                                                  (inst
                                                                                   -
                                                                                   "x!1")
                                                                                  (("2"
                                                                                    (assert)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide -1 2)
                                                (("2"
                                                  (apply-extensionality
                                                   :hide?
                                                   t)
                                                  (("2"
                                                    (expand "union")
                                                    (("2"
                                                      (expand "member")
                                                      (("2"
                                                        (expand
                                                         "IUnion")
                                                        (("2"
                                                          (case-replace
                                                           "X!1(1 + j!1)(x!1)")
                                                          (("1"
                                                            (inst
                                                             +
                                                             "1+j!1")
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (assert)
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                                                              (expand
                                                               "emptyset")
                                                              (("2"
                                                                (case-replace
                                                                 "(EXISTS (i: nat): IF i <= j!1 THEN X!1(i)(x!1) ELSE FALSE ENDIF)")
                                                                (("1"
                                                                  (skosimp)
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                                                                    (prop)
                                                                    (("1"
                                                                      (inst
                                                                       +
                                                                       "i!1")
                                                                      (("1"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (replace
                                                                   1
                                                                   3)
                                                                  (("2"
                                                                    (assert)
                                                                    (("2"
                                                                      (skosimp)
                                                                      (("2"
                                                                        (prop)
                                                                        (("2"
                                                                          (inst
                                                                           +
                                                                           "i!1")
                                                                          (("2"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("3"
                                          (hide 2)
                                          (("3"
                                            (induct "i")
                                            (("1"
                                              (case-replace
                                               "IUnion[nat, T]
               (LAMBDA (n: nat):
                  IF n <= 0 THEN X!1(n) ELSE emptyset[T] ENDIF)=X!1(0)")
                                              (("1"
                                                (inst - "0")
                                                nil
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (expand "emptyset")
                                                  (("2"
                                                    (expand "IUnion")
                                                    (("2"
                                                      (apply-extensionality
                                                       :hide?
                                                       t)
                                                      (("2"
                                                        (case-replace
                                                         "X!1(0)(x!1)")
                                                        (("1"
                                                          (inst + "0")
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (skosimp*)
                                              (("2"
                                                (case-replace
                                                 "IUnion[nat, T]
               (LAMBDA (n: nat):
                  IF n <= j!1 + 1 THEN X!1(n) ELSE emptyset[T] ENDIF)=union(X!1(1+j!1),IUnion[nat, T]
               (LAMBDA (n: nat):
                  IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF))")
                                                (("1"
                                                  (hide -1)
                                                  (("1"
                                                    (lemma
                                                     "finite_union[T]"
                                                     ("A"
                                                      "X!1(1 + j!1)"
                                                      "B"
                                                      "IUnion[nat, T]
                     (LAMBDA (n: nat):
                        IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
                                                    (("1"
                                                      (propax)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (propax)
                                                      nil
                                                      nil)
                                                     ("3"
                                                      (inst - "1+j!1")
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide-all-but 1)
                                                  (("2"
                                                    (expand "union")
                                                    (("2"
                                                      (expand
                                                       "emptyset")
                                                      (("2"
                                                        (expand
                                                         "IUnion")
                                                        (("2"
                                                          (expand
                                                           "member")
                                                          (("2"
                                                            (apply-extensionality
                                                             :hide?
                                                             t)
                                                            (("2"
                                                              (case-replace
                                                               "X!1(1 + j!1)(x!1)")
                                                              (("1"
                                                                (inst
                                                                 +
                                                                 "1+j!1")
                                                                (("1"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (assert)
                                                                (("2"
                                                                  (case-replace
                                                                   "(EXISTS (i: nat): IF i <= j!1 THEN X!1(i)(x!1) ELSE FALSE ENDIF)")
                                                                  (("1"
                                                                    (skosimp)
                                                                    (("1"
                                                                      (prop)
                                                                      (("1"
                                                                        (inst
                                                                         +
                                                                         "i!2")
                                                                        (("1"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (replace
                                                                     1
                                                                     3)
                                                                    (("2"
                                                                      (assert)
                                                                      (("2"
                                                                        (skosimp)
                                                                        (("2"
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                                                                          (("2"
                                                                            (expand
                                                                             "<="
                                                                             -1)
                                                                            (("2"
                                                                              (split
                                                                               -1)
                                                                              (("1"
                                                                                (inst
                                                                                 +
                                                                                 "i!2")
                                                                                (("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))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide 2)
                                  (("3"
                                    (induct "i")
                                    (("1"
                                      (expand "emptyset")
                                      (("1"
                                        (expand "IUnion")
                                        (("1"
                                          (case-replace
                                           "{x_1: T |
              EXISTS (i: nat): IF i <= 0 THEN X!1(i)(x_1) ELSE FALSE ENDIF}=X!1(0)")
                                          (("1" (inst - "0"nil nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (apply-extensionality
                                               :hide?
                                               t)
                                              (("2"
                                                (case-replace
                                                 "X!1(0)(x!1)")
                                                (("1"
                                                  (inst + "0")
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (skosimp*)
                                      (("2"
                                        (case-replace
                                         "(IUnion[nat, T]
               (LAMBDA (n: nat):
                  IF n <= j!1 + 1 THEN X!1(n) ELSE emptyset[T] ENDIF))=union(X!1(j!1+1),(IUnion[nat, T]
               (LAMBDA (n: nat):
                  IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)))")
                                        (("1"
                                          (hide -1)
                                          (("1"
                                            (lemma
                                             "finite_union[T]"
                                             ("A"
                                              "X!1(j!1 + 1)"
                                              "B"
                                              "IUnion[nat, T]
                      (LAMBDA (n: nat):
                         IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
                                            (("1" (propax) nil nil)
                                             ("2" (propax) nil nil)
                                             ("3"
                                              (inst - "1+j!1")
                                              nil
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide -1 2)
                                          (("2"
                                            (apply-extensionality
                                             :hide?
                                             t)
                                            (("2"
                                              (expand "union")
                                              (("2"
                                                (expand "IUnion")
                                                (("2"
                                                  (expand "emptyset")
                                                  (("2"
                                                    (expand "member")
                                                    (("2"
                                                      (case-replace
                                                       "X!1(1 + j!1)(x!1)")
                                                      (("1"
                                                        (inst
                                                         +
                                                         "1+j!1")
                                                        (("1"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (assert)
                                                        (("2"
                                                          (case-replace
                                                           "(EXISTS (i: nat): IF i <= j!1 THEN X!1(i)(x!1) ELSE FALSE ENDIF)")
                                                          (("1"
                                                            (skosimp)
                                                            (("1"
                                                              (prop)
                                                              (("1"
                                                                (inst
                                                                 +
                                                                 "i!1")
                                                                (("1"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (replace
                                                             1
                                                             3)
                                                            (("2"
                                                              (assert)
                                                              (("2"
                                                                (skosimp)
                                                                (("2"
                                                                  (prop)
                                                                  (("2"
                                                                    (expand
                                                                     "<="
                                                                     -1)
                                                                    (("2"
                                                                      (split
                                                                       -1)
                                                                      (("1"
                                                                        (inst
                                                                         +
                                                                         "i!1")
                                                                        (("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))
                                  nil)
                                 ("4" (propax) nil nil))
                                nil))
                              nil)
                             ("2" (hide 2)
                              (("2"
                                (apply-extensionality :hide? t)
                                (("2"
                                  (inst - "x!1")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide 2)
                          (("2" (skosimp)
                            (("2" (inst - "i!1") (("2" (prop) nil nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (replace 1 2)
                        (("2" (assert)
                          (("2" (skosimp)
                            (("2" (prop)
                              (("2"
                                (lemma
                                 "finite_subset[T]"
                                 ("A" "IUnion(X!1)" "s" "X!1(i!1)"))
                                (("1"
                                  (assert)
                                  (("1"
                                    (expand "IUnion")
                                    (("1"
                                      (expand "subset?")
                                      (("1"
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                                          (expand "member")
                                          (("1"
                                            (inst + "i!1")
                                            nil
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2" (propax) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil measure_def nil)
    (card_emptyset formula-decl nil finite_sets nil)
    (measure_empty? const-decl "bool" generalized_measure_def nil)
    (subset_algebra_emptyset name-judgement "(S)" measure_def nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (set type-eq-decl nil sets nil)
    (is_finite const-decl "bool" finite_sets nil)
    (TRUE const-decl "bool" booleans nil)
    (finite_set type-eq-decl nil finite_sets 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)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (Card const-decl "nat" finite_sets nil)
    (card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
    (FALSE const-decl "bool" booleans nil)
    (naturalnumber type-eq-decl nil naturalnumbers nil)
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (MM skolem-const-decl "[(S) -> [bool, naturalnumber]]" measure_def
     nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (sigma_nat application-judgement "nat" sigma_nat "reals/")
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (card_disj_union formula-decl nil finite_sets nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (sigma def-decl "real" sigma "reals/")
    (X!1 skolem-const-decl "disjoint_indexed_measurable[T, S]"
     measure_def nil)
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (converges_upto? const-decl "bool" convergence_aux "metric_space/")
    (metric_converges_to const-decl "bool" metric_space_def
     "metric_space/")
    (ball const-decl "set[T]" metric_space_def "metric_space/")
    (minus_real_is_real application-judgement "real" reals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (member const-decl "bool" sets nil)
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (metric_convergence_def formula-decl nil metric_space
     "metric_space/")
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (limit_def formula-decl nil convergence_sequences "analysis/")
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (fullset const-decl "set" sets nil)
    (least_upper_bound? const-decl "bool" bounded_real_defs nil)
    (< const-decl "bool" reals nil)
    (CS skolem-const-decl "sequence[real]" measure_def nil)
    (i!1 skolem-const-decl "nat" measure_def nil)
    (convergence_cauchy1 formula-decl nil convergence_sequences
     "analysis/")
    (cauchy const-decl "bool" convergence_sequences "analysis/")
    (same_card_subset formula-decl nil finite_sets nil)
    (subset? const-decl "bool" sets nil)
    (extensionality_postulate formula-decl nil functions nil)
    (subset_algebra_intersection application-judgement "(S)"
     measure_def nil)
    (disjoint? const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (upper_bound? const-decl "bool" bounded_real_defs nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (image const-decl "set[R]" function_image nil)
    (converges_upto_is_lub formula-decl nil convergence_aux
     "metric_space/")
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (card_subset formula-decl nil finite_sets nil)
    (increasing? const-decl "bool" real_fun_preds "reals/")
    (i!1 skolem-const-decl "nat" measure_def nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (card_is_0 formula-decl nil finite_sets nil)
    (emptyset_is_empty? formula-decl nil sets_lemmas nil)
    (nonempty_card formula-decl nil finite_sets nil)
    (choose_rest formula-decl nil sets_lemmas nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (choose_member formula-decl nil sets_lemmas nil)
    (CC skolem-const-decl "(IUnion(Y!1))" measure_def nil)
    (finite_rest judgement-tcc nil finite_sets nil)
    (YY skolem-const-decl "[nat -> set[T]]" measure_def nil)
    (add const-decl "(nonempty?)" sets nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (nat_max application-judgement "{k: nat | i <= k AND j <= k}"
     real_defs nil)
    (nonneg_rat_max application-judgement
     "{s: nonneg_rat | s >= q AND s >= r}" real_defs nil)
    (finite_remove judgement-tcc nil finite_sets nil)
    (rest const-decl "set" sets nil) (remove const-decl "set" sets nil)
    (choose const-decl "(p)" sets nil)
    (card_rest formula-decl nil finite_sets nil)
    (nonempty? const-decl "bool" sets nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (IU_i skolem-const-decl "set[T]" measure_def nil)
    (IU_n skolem-const-decl "set[T]" measure_def nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (finite_union judgement-tcc nil finite_sets nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (union const-decl "set" sets nil)
    (emptyset const-decl "set" sets nil)
    (<= const-decl "bool" reals nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (series const-decl "sequence[real]" series "series/")
    (sequence type-eq-decl nil sequences nil)
    (finite_subset formula-decl nil finite_sets nil)
    (x_sum const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (O const-decl "T3" function_props nil)
    (measure_countably_additive? const-decl "bool"
     generalized_measure_def nil)
    (measure? const-decl "bool" generalized_measure_def nil))
   nil))
 (sigma_finite_measure_is_measure 0
  (sigma_finite_measure_is_measure-1 nil 3396670582
   ("" (judgement-tcc) nil nil)
   ((NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (measure_sigma_finite? const-decl "bool" measure_def nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (subset_algebra_emptyset name-judgement "(S)" measure_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (measure_empty? const-decl "bool" generalized_measure_def nil)
    (O const-decl "T3" function_props nil)
    (series const-decl "sequence[real]" series "series/")
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (measure_countably_additive? const-decl "bool"
     generalized_measure_def nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/"))
   nil))
 (complete_sigma_finite_is_complete_measure 0
  (complete_sigma_finite_is_complete_measure-1 nil 3421359236
   ("" (skolem + ("mu!1"))
    (("" (typepred "mu!1")
      (("" (expand "complete_sigma_finite?")
        (("" (flatten)
          (("" (expand "complete_measure?")
            (("" (split)
              (("1" (propax) nil nil) ("2" (propax) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((complete_sigma_finite type-eq-decl nil measure_def nil)
    (complete_sigma_finite? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil measure_def nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (complete_measure? const-decl "bool" generalized_measure_def nil))
   nil))
 (complete_sigma_finite_is_sigma_finite_measure 0
  (complete_sigma_finite_is_sigma_finite_measure-1 nil 3421359236
   ("" (skolem + ("mu!1"))
    (("" (typepred "mu!1")
      (("" (expand "complete_sigma_finite?")
        (("" (expand "sigma_finite_measure?")
          (("" (flatten) (("" (assertnil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((complete_sigma_finite type-eq-decl nil measure_def nil)
    (complete_sigma_finite? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil measure_def nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil))
   nil))
 (measure_monotone 0
  (measure_monotone-1 nil 3396804295
   ("" (skosimp)
    ((""
      (lemma "measure_disjoint_union"
       ("f" "f!1" "a" "a!1" "b" "difference(b!1,a!1)"))
      (("" (assert)
        (("" (split -1)
          (("1" (case-replace "union(a!1, difference(b!1, a!1)) = b!1")
            (("1" (hide -1)
              (("1" (expand "x_add")
                (("1" (expand "x_eq")
                  (("1" (expand "x_le")
                    (("1" (flatten)
                      (("1" (case-replace "f!1(a!1)`1")
                        (("1" (assert)
                          (("1" (prop) (("1" (assertnil nil)) nil))
                          nil)
                         ("2" (assertnil nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but (-3 1))
              (("2" (apply-extensionality :hide? t)
                (("2" (expand "difference")
                  (("2" (expand "union")
                    (("2" (expand "subset?")
                      (("2" (expand "member")
                        (("2" (assert)
                          (("2" (inst - "x!1") (("2" (grind) nil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide-all-but (-2 1))
            (("2" (expand "difference")
              (("2" (expand "disjoint?")
                (("2" (expand "intersection")
                  (("2" (expand "empty?")
                    (("2" (expand "subset?")
                      (("2" (expand "member")
                        (("2" (skosimp*) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (difference const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (measure_disjoint_union formula-decl nil generalized_measure_def
     nil)
    (subset_algebra_difference application-judgement "(S)" measure_def
     nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (x_add const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (union const-decl "set" sets nil)
    (intersection const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil)
    (disjoint? const-decl "bool" sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (subset_algebra_union application-judgement "(S)" measure_def nil))
   shostak))
 (measure_union 0
  (measure_union-1 nil 3453271429
   ("" (skosimp)
    ((""
      (lemma "measure_disjoint_union"
       ("f" "f!1" "a" "a!1" "b" "difference(b!1,a!1)"))
      (("" (rewrite "union_difference" -1 :dir rl)
        (("" (replace -2)
          (("" (rewrite "difference_disjoint")
            (("" (name-replace "LHS" "f!1(union(a!1, b!1))")
              ((""
                (lemma "measure_monotone"
                 ("f" "f!1" "a" "difference(b!1, a!1)" "b" "b!1"))
                (("" (replace -3)
                  (("" (rewrite "difference_subset")
                    ((""
                      (name-replace "FMID" "f!1(difference(b!1, a!1))")
                      (("" (expand "x_eq")
                        (("" (expand "x_le")
                          (("" (assert)
                            (("" (flatten)
                              ((""
                                (assert)
                                ((""
                                  (expand "x_add")
                                  ((""
                                    (split)
                                    (("1"
                                      (flatten)
                                      (("1"
                                        (assert)
                                        (("1"
                                          (prop)
                                          (("1" (assertnil nil)
                                           ("2" (assertnil nil)
                                           ("3" (assertnil nil)
                                           ("4" (assertnil nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (difference const-decl "set" sets nil)
    (set type-eq-decl nil sets nil)
    (measure_disjoint_union formula-decl nil generalized_measure_def
     nil)
    (subset_algebra_difference application-judgement "(S)" measure_def
     nil)
    (union const-decl "set" sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (x_add const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (difference_subset formula-decl nil sets_lemmas nil)
    (measure_monotone formula-decl nil measure_def nil)
    (difference_disjoint formula-decl nil sets_lemmas nil)
    (subset_algebra_union application-judgement "(S)" measure_def nil)
    (union_difference formula-decl nil sets_lemmas nil))
   shostak))
 (measure_def 0
  (measure_def-1 nil 3453287567
   ("" (skosimp)
    (("" (split)
      (("1" (flatten)
        (("1" (split)
          (("1" (expand "measure?") (("1" (flatten) nil nil)) nil)
           ("2" (skosimp)
            (("2"
              (lemma "measure_disjoint_union"
               ("f" "f!1" "a" "a!1" "b" "b!1"))
              (("2" (assertnil nil)) nil))
            nil)
           ("3" (skosimp)
            (("3" (expand "measure?")
              (("3" (flatten)
                (("3" (expand "measure_countably_additive?")
                  (("3" (inst - "X!1")
                    (("3" (expand "x_le")
                      (("3" (expand "x_eq")
                        (("3" (assert)
                          (("3" (flatten) (("3" (assertnil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (flatten)
        (("2" (expand "measure?")
          (("2" (assert)
            (("2" (expand "measure_countably_additive?")
              (("2" (skosimp)
                (("2" (inst - "X!1")
                  (("2" (assert)
                    (("2"
                      (case "x_le(x_sum(f!1 o X!1), f!1(IUnion(X!1)))")
                      (("1" (name-replace "LHS" "x_sum(f!1 o X!1)")
                        (("1" (name-replace "RHS" "f!1(IUnion(X!1))")
                          (("1" (hide-all-but (-1 -4 1))
                            (("1" (grind) nil nil)) nil))
                          nil))
                        nil)
                       ("2" (hide -3 2)
                        (("2"
                          (case "forall (a,b:(S)): subset?(a,b) => x_le(f!1(a),f!1(b))")
                          (("1" (expand "x_le" 1)
                            (("1" (flatten)
                              (("1"
                                (assert)
                                (("1"
                                  (expand "x_sum")
                                  (("1"
                                    (expand "o ")
                                    (("1"
                                      (case-replace
                                       "FORALL i: f!1(X!1(i))`1")
                                      (("1"
                                        (case
                                         "FORALL (n: nat): S(IUnion[T](n, X!1))")
                                        (("1"
                                          (case
                                           "forall (n:nat): x_eq(f!1(IUnion(n,X!1)),x_sigma(0,n,f!1 o X!1))")
                                          (("1"
                                            (case
                                             "forall (nna:sequence[nnreal],nnb:nnreal): (forall (n:nat): series(nna)(n) <= nnb)=>convergent(series(nna)) & convergence_sequences.limit(series(nna))<=nnb")
                                            (("1"
                                              (inst
                                               -
                                               "LAMBDA i: f!1(X!1(i))`2"
                                               "f!1(IUnion(X!1))`2")
                                              (("1"
                                                (split -1)
                                                (("1"
                                                  (flatten)
                                                  (("1"
                                                    (replace -1)
                                                    (("1"
                                                      (propax)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide 2)
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (inst - "n!1")
                                                      (("2"
                                                        (expand
                                                         "series")
                                                        (("2"
                                                          (expand "o ")
                                                          (("2"
                                                            (inst
                                                             -2
                                                             "n!1")
                                                            (("2"
                                                              (inst
                                                               -4
                                                               "IUnion(n!1, X!1)"
                                                               "IUnion(X!1)")
                                                              (("2"
                                                                (split
                                                                 -4)
                                                                (("1"
                                                                  (name-replace
                                                                   "MID1"
                                                                   "f!1(IUnion(n!1, X!1))")
                                                                  (("1"
                                                                    (expand
                                                                     "x_sigma")
                                                                    (("1"
                                                                      (name-replace
                                                                       "LHS"
                                                                       "sigma(0, n!1, LAMBDA i: f!1(X!1(i))`2)")
                                                                      (("1"
                                                                        (case-replace
                                                                         "FORALL i: i <= n!1 => f!1(X!1(i))`1")
                                                                        (("1"
                                                                          (hide-all-but
                                                                           (-2
                                                                            -3
                                                                            1
                                                                            -6))
                                                                          (("1"
                                                                            (name-replace
                                                                             "RHS"
                                                                             "f!1(IUnion(X!1))")
                                                                            (("1"
                                                                              (grind)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (skosimp)
                                                                          (("2"
                                                                            (inst
                                                                             -5
                                                                             "i!1")
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (hide-all-but
                                                                   1)
                                                                  (("2"
                                                                    (expand
                                                                     "subset?")
                                                                    (("2"
                                                                      (expand
                                                                       "member")
                                                                      (("2"
                                                                        (skosimp)
                                                                        (("2"
                                                                          (expand
                                                                           "IUnion")
                                                                          (("2"
                                                                            (expand
                                                                             "image")
                                                                            (("2"
                                                                              (expand
                                                                               "Union")
                                                                              (("2"
                                                                                (skosimp)
                                                                                (("2"
                                                                                  (typepred
                                                                                   "a!1")
                                                                                  (("2"
                                                                                    (skosimp)
                                                                                    (("2"
                                                                                      (inst
                                                                                       +
                                                                                       "x!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)
                                             ("2"
                                              (hide-all-but 1)
                                              (("2"
                                                (skosimp)
                                                (("2"
                                                  (case
                                                   "nonempty?[real](image[nat, real](series(nna!1), fullset[nat])) AND
       above_bounded[real](image[nat, real](series(nna!1), fullset[nat]))")
                                                  (("1"
                                                    (typepred
                                                     "sup(image(series(nna!1),fullset[nat]))")
                                                    (("1"
                                                      (name-replace
                                                       "LIMIT"
                                                       "sup(image(series(nna!1), fullset[nat]))")
                                                      (("1"
                                                        (expand
                                                         "least_upper_bound")
                                                        (("1"
                                                          (expand
                                                           "fullset")
                                                          (("1"
                                                            (expand
                                                             "image")
                                                            (("1"
                                                              (expand
                                                               "upper_bound")
                                                              (("1"
                                                                (flatten)
                                                                (("1"
                                                                  (case
                                                                   "LIMIT <= nnb!1")
                                                                  (("1"
                                                                    (case
                                                                     "convergence(series(nna!1),LIMIT)")
                                                                    (("1"
                                                                      (split
                                                                       1)
                                                                      (("1"
                                                                        (expand
                                                                         "convergent?")
                                                                        (("1"
                                                                          (inst
                                                                           +
                                                                           "LIMIT")
                                                                          nil
                                                                          nil))
                                                                        nil)
                                                                       ("2"
                                                                        (rewrite
                                                                         "limit_def"
                                                                         -1
                                                                         :dir
                                                                         rl)
                                                                        (("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (hide
                                                                       2)
                                                                      (("2"
                                                                        (expand
                                                                         "convergence")
                                                                        (("2"
                                                                          (skosimp)
                                                                          (("2"
                                                                            (inst
                                                                             -3
                                                                             "LIMIT-epsilon!1/2")
                                                                            (("2"
                                                                              (assert)
                                                                              (("2"
                                                                                (skosimp)
                                                                                (("2"
                                                                                  (typepred
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                                                                                  (("2"
                                                                                    (skolem
                                                                                     -
                                                                                     "n!1")
                                                                                    (("2"
                                                                                      (replace
                                                                                       -1)
                                                                                      (("2"
                                                                                        (hide
                                                                                         -1)
                                                                                        (("2"
                                                                                          (inst
                                                                                           +
                                                                                           "n!1")
                                                                                          (("2"
                                                                                            (skosimp)
                                                                                            (("2"
                                                                                              (case
                                                                                               "LIMIT-series(nna!1)(n!1)<epsilon!1/2")
                                                                                              (("1"
                                                                                                (hide
                                                                                                 2)
                                                                                                (("1"
                                                                                                  (case
                                                                                                   "series(nna!1)(i!1) <= LIMIT")
                                                                                                  (("1"
                                                                                                    (case
                                                                                                     "LIMIT-series(nna!1)(i!1)<epsilon!1")
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "abs")
                                                                                                      (("1"
                                                                                                        (expand
                                                                                                         "<="
                                                                                                         -2)
                                                                                                        (("1"
                                                                                                          (split
                                                                                                           -2)
                                                                                                          (("1"
                                                                                                            (assert)
                                                                                                            nil
                                                                                                            nil)
                                                                                                           ("2"
                                                                                                            (assert)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (case
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                                                                                                      (("1"
                                                                                                        (assert)
                                                                                                        nil
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (hide-all-but
                                                                                                         (-3
                                                                                                          1))
                                                                                                        (("2"
                                                                                                          (lemma
                                                                                                           "sigma_split"
                                                                                                           ("low"
                                                                                                            "0"
                                                                                                            "high"
                                                                                                            "i!1"
                                                                                                            "z"
                                                                                                            "n!1"
                                                                                                            "F"
                                                                                                            "nna!1"))
                                                                                                          (("2"
                                                                                                            (expand
                                                                                                             "series")
                                                                                                            (("2"
                                                                                                              (assert)
                                                                                                              (("2"
                                                                                                                (replace
                                                                                                                 -1
                                                                                                                 1)
                                                                                                                (("2"
                                                                                                                  (assert)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     -1)
                                                                                                                    (("2"
                                                                                                                      (lemma
                                                                                                                       "sigma_ge_0"
                                                                                                                       ("low"
                                                                                                                        "1+n!1"
                                                                                                                        "high"
                                                                                                                        "i!1"
                                                                                                                        "F"
                                                                                                                        "nna!1"))
                                                                                                                      (("2"
                                                                                                                        (split)
                                                                                                                        (("1"
                                                                                                                          (assert)
                                                                                                                          nil
                                                                                                                          nil)
                                                                                                                         ("2"
                                                                                                                          (hide
                                                                                                                           2)
                                                                                                                          (("2"
                                                                                                                            (skosimp)
                                                                                                                            (("2"
                                                                                                                              (assert)
                                                                                                                              nil
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                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil)
                                                                                                   ("2"
                                                                                                    (inst
                                                                                                     -4
                                                                                                     "series(nna!1)(i!1)")
                                                                                                    (("2"
                                                                                                      (inst
                                                                                                       +
                                                                                                       "i!1")
                                                                                                      nil
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil)
                                                                                               ("2"
                                                                                                (assert)
                                                                                                nil
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (inst
                                                                     -2
                                                                     "nnb!1")
                                                                    (("2"
                                                                      (assert)
                                                                      (("2"
                                                                        (skosimp)
                                                                        (("2"
                                                                          (typepred
                                                                           "z!1")
                                                                          (("2"
                                                                            (skosimp)
                                                                            (("2"
                                                                              (inst
                                                                               -5
                                                                               "x!1")
                                                                              (("2"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (propax)
                                                      nil
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide 2)
                                                    (("2"
                                                      (split)
                                                      (("1"
                                                        (expand
                                                         "fullset")
                                                        (("1"
                                                          (expand
                                                           "image")
                                                          (("1"
                                                            (expand
                                                             "nonempty?")
                                                            (("1"
                                                              (expand
                                                               "empty?")
                                                              (("1"
                                                                (expand
                                                                 "member")
                                                                (("1"
                                                                  (inst
                                                                   -
                                                                   "series(nna!1)(0)")
                                                                  (("1"
                                                                    (inst
                                                                     +
                                                                     "0")
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (expand
                                                         "fullset")
                                                        (("2"
                                                          (expand
                                                           "above_bounded")
                                                          (("2"
                                                            (inst
                                                             +
                                                             "nnb!1")
                                                            (("2"
                                                              (expand
                                                               "image")
                                                              (("2"
                                                                (expand
                                                                 "upper_bound")
                                                                (("2"
                                                                  (skosimp)
                                                                  (("2"
                                                                    (typepred
                                                                     "z!1")
                                                                    (("2"
                                                                      (skosimp)
                                                                      (("2"
                                                                        (inst
                                                                         -
                                                                         "x!1")
                                                                        (("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (induct "n")
                                              (("1"
                                                (rewrite
                                                 "IUnion_0_def")
                                                (("1"
                                                  (expand "x_sigma")
                                                  (("1"
                                                    (expand "x_eq")
                                                    (("1"
                                                      (expand "o ")
                                                      (("1"
                                                        (case-replace
                                                         "FORALL i: i <= 0 => f!1(X!1(i))`1")
                                                        (("1"
                                                          (expand
                                                           "sigma")
                                                          (("1"
                                                            (assert)
                                                            (("1"
                                                              (inst
                                                               -
                                                               "0")
                                                              (("1"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (replace 1 2)
                                                          (("2"
                                                            (assert)
                                                            (("2"
                                                              (skosimp)
                                                              (("2"
                                                                (expand
                                                                 "<="
                                                                 -1)
                                                                (("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (skosimp)
                                                (("2"
                                                  (rewrite
                                                   "IUnion_n_def")
                                                  (("2"
                                                    (typepred "X!1")
                                                    (("2"
                                                      (inst - "j!1")
                                                      (("2"
                                                        (inst
                                                         -8
                                                         "IUnion(j!1, X!1)"
                                                         "X!1(1 + j!1)")
                                                        (("2"
                                                          (split -8)
                                                          (("1"
                                                            (name-replace
                                                             "LHS"
                                                             "f!1(union(IUnion(j!1, X!1), X!1(1 + j!1)))")
                                                            (("1"
                                                              (hide-all-but
                                                               (-1
                                                                -3
                                                                1))
                                                              (("1"
                                                                (expand
                                                                 "x_eq")
                                                                (("1"
                                                                  (expand
                                                                   "x_sigma")
                                                                  (("1"
                                                                    (expand
                                                                     "o ")
                                                                    (("1"
                                                                      (flatten)
                                                                      (("1"
                                                                        (expand
                                                                         "x_add")
                                                                        (("1"
                                                                          (case-replace
                                                                           "FORALL i: i <= 1 + j!1 => f!1(X!1(i))`1")
                                                                          (("1"
                                                                            (case-replace
                                                                             "FORALL i: i <= j!1 => f!1(X!1(i))`1")
                                                                            (("1"
                                                                              (assert)
                                                                              (("1"
                                                                                (expand
                                                                                 "sigma"
                                                                                 1)
                                                                                (("1"
                                                                                  (assert)
                                                                                  (("1"
                                                                                    (case-replace
                                                                                     "f!1(X!1(1 + j!1))`1")
                                                                                    (("1"
                                                                                      (assert)
                                                                                      nil
                                                                                      nil)
                                                                                     ("2"
                                                                                      (inst
                                                                                       -2
                                                                                       "1+j!1")
                                                                                      (("2"
                                                                                        (assert)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("2"
                                                                              (skosimp)
                                                                              (("2"
                                                                                (inst
                                                                                 -
                                                                                 "i!1")
                                                                                (("2"
                                                                                  (assert)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (replace
                                                                             1)
                                                                            (("2"
                                                                              (skosimp)
                                                                              (("2"
                                                                                (expand
                                                                                 "<="
                                                                                 -1)
                                                                                (("2"
                                                                                  (split)
                                                                                  (("1"
                                                                                    (case-replace
                                                                                     "FORALL i: i <= j!1 => f!1(X!1(i))`1")
                                                                                    (("1"
                                                                                      (assert)
                                                                                      (("1"
                                                                                        (assert)
                                                                                        (("1"
                                                                                          (inst
                                                                                           -
                                                                                           "i!1")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil)
                                                                                     ("2"
                                                                                      (replace
                                                                                       1)
                                                                                      (("2"
                                                                                        (assert)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (replace
                                                                                     -1)
                                                                                    (("2"
                                                                                      (assert)
                                                                                      nil
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (expand
                                                             "disjoint_indexed_measurable?")
                                                            (("2"
                                                              (hide-all-but
                                                               (-1 1))
                                                              (("2"
                                                                (expand
                                                                 "disjoint?")
                                                                (("2"
                                                                  (expand
                                                                   "disjoint?")
                                                                  (("2"
                                                                    (expand
                                                                     "intersection")
                                                                    (("2"
                                                                      (expand
                                                                       "empty?")
                                                                      (("2"
                                                                        (expand
                                                                         "member")
                                                                        (("2"
                                                                          (skosimp)
                                                                          (("2"
                                                                            (expand
                                                                             "IUnion")
                                                                            (("2"
                                                                              (expand
                                                                               "image")
                                                                              (("2"
                                                                                (expand
                                                                                 "Union")
                                                                                (("2"
                                                                                  (skosimp)
                                                                                  (("2"
                                                                                    (typepred
                                                                                     "a!1")
                                                                                    (("2"
                                                                                      (skolem
                                                                                       -
                                                                                       "j!2")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -
                                                                                         "1+j!1"
                                                                                         "j!2")
                                                                                        (("2"
                                                                                          (assert)
                                                                                          (("2"
                                                                                            (inst
                                                                                             -
                                                                                             "x!1")
                                                                                            (("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)
                                           ("3" (propax) nil nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (hide-all-but 1)
                                            (("2"
                                              (induct "n")
                                              (("1"
                                                (rewrite
                                                 "IUnion_0_def")
                                                (("1"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil)
                                               ("2"
                                                (skosimp)
                                                (("2"
                                                  (rewrite
                                                   "IUnion_n_def")
                                                  (("2"
                                                    (lemma
                                                     "subset_algebra_union"
                                                     ("x"
                                                      "IUnion[T](j!1, X!1)"
                                                      "y"
                                                      "X!1(1 + j!1)"))
                                                    (("1"
                                                      (propax)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (propax)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (replace 1)
                                        (("2"
                                          (skosimp)
                                          (("2"
                                            (inst
                                             -
                                             "X!1(i!1)"
                                             "IUnion(X!1)")
                                            (("2"
                                              (split)
                                              (("1"
                                                (expand "x_le")
                                                (("1"
                                                  (flatten)
                                                  nil
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (expand "subset?")
                                                  (("2"
                                                    (expand "IUnion")
                                                    (("2"
                                                      (expand "member")
                                                      (("2"
                                                        (skosimp)
                                                        (("2"
                                                          (inst
                                                           +
                                                           "i!1")
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide 2)
                            (("2" (skosimp)
                              (("2"
                                (inst -3 "a!1" "difference(b!1,a!1)")
                                (("2"
                                  (lemma
                                   "union_diff_subset"
                                   ("a" "a!1" "b" "b!1"))
                                  (("2"
                                    (assert)
                                    (("2"
                                      (replace -1)
                                      (("2"
                                        (rewrite "difference_disjoint")
                                        (("2"
                                          (hide-all-but (-4 1))
                                          (("2" (grind) nil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((measure? const-decl "bool" generalized_measure_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (measure_disjoint_union formula-decl nil generalized_measure_def
     nil)
    (subset_algebra_union application-judgement "(S)" measure_def nil)
    (measure_countably_additive? const-decl "bool"
     generalized_measure_def nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (set type-eq-decl nil sets nil)
    (O const-decl "T3" function_props nil)
    (x_sum const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (subset? const-decl "bool" sets nil)
    (subset_algebra_union judgement-tcc nil subset_algebra nil)
    (x_sigma const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (nonempty? const-decl "bool" sets nil)
    (fullset const-decl "set" sets nil)
    (above_bounded const-decl "bool" bounded_reals "reals/")
    (upper_bound const-decl "bool" bound_defs "reals/")
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (TRUE const-decl "bool" booleans nil)
    (< const-decl "bool" reals nil)
    (sigma_split formula-decl nil sigma "reals/")
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (i!1 skolem-const-decl "nat" measure_def nil)
    (nna!1 skolem-const-decl "sequence[nnreal]" measure_def nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (limit_def formula-decl nil convergence_sequences "analysis/")
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (sup const-decl "{x | least_upper_bound(<=)(x, Su)}" bounded_reals
     "reals/")
    (sup_set type-eq-decl nil bounded_reals "reals/")
    (least_upper_bound const-decl "bool" bound_defs "reals/")
    (pred type-eq-decl nil defined_types nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (Union const-decl "set" sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (image const-decl "set[R]" function_image nil)
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (limit const-decl "real" convergence_sequences "analysis/")
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (series const-decl "sequence[real]" series "series/")
    (<= const-decl "bool" reals nil)
    (X!1 skolem-const-decl "disjoint_indexed_measurable[T, S]"
     measure_def nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (sigma_0_neg formula-decl nil sigma_nat "reals/")
    (IUnion_0_def formula-decl nil nat_indexed_sets "sets_aux/")
    (IUnion_n_def formula-decl nil nat_indexed_sets "sets_aux/")
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (x_add const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (union const-decl "set" sets nil)
    (subset_algebra_intersection application-judgement "(S)"
     measure_def nil)
    (intersection const-decl "set" sets nil)
    (disjoint? const-decl "bool" sets nil)
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/")
    (sequence type-eq-decl nil sequences nil)
    (union_diff_subset formula-decl nil sets_lemmas nil)
    (difference_disjoint formula-decl nil sets_lemmas nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (difference const-decl "set" sets nil)
    (subset_algebra_difference application-judgement "(S)" measure_def
     nil))
   shostak))
 (finite_measure_def 0
  (finite_measure_def-1 nil 3454804407
   ("" (skosimp)
    (("" (split)
      (("1" (flatten)
        (("1" (lemma "measure_def" ("f" "to_measure(g!1)"))
          (("1" (assert)
            (("1" (flatten)
              (("1" (split)
                (("1" (expand "measure_empty?")
                  (("1" (expand "to_measure") (("1" (propax) nil nil))
                    nil))
                  nil)
                 ("2" (skosimp)
                  (("2" (inst - "a!1" "b!1")
                    (("2" (assert)
                      (("2" (expand "to_measure")
                        (("2" (expand "x_add")
                          (("2" (expand "x_eq")
                            (("2" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("3" (skosimp)
                  (("3" (inst -5 "X!1")
                    (("3" (assert)
                      (("3" (expand "to_measure")
                        (("3" (expand "x_le")
                          (("3" (assert)
                            (("3" (expand "o ")
                              (("3"
                                (expand "x_sum")
                                (("3"
                                  (replace -2)
                                  (("3" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (propax) nil nil))
          nil))
        nil)
       ("2" (flatten)
        (("2" (lemma "measure_def" ("f" "lambda a: (TRUE,g!1(a))"))
          (("2" (flatten -1)
            (("2" (hide -1)
              (("2" (split -1)
                (("1" (hide-all-but (-1 1))
                  (("1" (expand "measure?")
                    (("1" (expand "finite_measure?")
                      (("1" (flatten)
                        (("1" (expand "measure_empty?")
                          (("1" (assert)
                            (("1" (skosimp)
                              (("1"
                                (expand "measure_countably_additive?")
                                (("1"
                                  (inst - "X!1")
                                  (("1"
                                    (assert)
                                    (("1"
                                      (expand "x_eq")
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (assert)
                                          (("1"
                                            (expand "x_sum")
                                            (("1"
                                              (expand "o")
                                              (("1"
                                                (prop)
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (rewrite
                                                     "limit_def"
                                                     1
                                                     :dir
                                                     rl)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (expand "measure_empty?")
                  (("2" (propax) nil nil)) nil)
                 ("3" (hide 2)
                  (("3" (skosimp)
                    (("3" (inst -3 "a!1" "b!1")
                      (("3" (assert)
                        (("3" (expand "x_add")
                          (("3" (expand "x_eq")
                            (("3" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("4" (skosimp)
                  (("4" (inst -4 "X!1")
                    (("4" (assert)
                      (("4" (hide 2 -3)
                        (("4" (expand "x_sum")
                          (("4" (expand "x_le")
                            (("4" (expand "o ")
                              (("4"
                                (flatten)
                                (("4"
                                  (prop)
                                  (("4" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((to_measure const-decl "measure_type" generalized_measure_def nil)
    (measure_type nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (finite_measure nonempty-type-eq-decl nil generalized_measure_def
     nil)
    (finite_measure? const-decl "bool" generalized_measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (measure_def formula-decl nil measure_def nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (O const-decl "T3" function_props nil)
    (x_sum const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (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)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (subset_algebra_union application-judgement "(S)" measure_def nil)
    (x_add const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (x_eq const-decl "bool" extended_nnreal "extended_nnreal/")
    (measure_empty? const-decl "bool" generalized_measure_def nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (finite_emptyset name-judgement "finite_set[T]" countable_props
     "sets_aux/")
    (finite_emptyset name-judgement "finite_set[T]" countable_setofsets
     "sets_aux/")
    (subset_algebra_emptyset name-judgement "(S)" measure_def nil)
    (TRUE const-decl "bool" booleans nil)
    (limit_def formula-decl nil convergence_sequences "analysis/")
    (set type-eq-decl nil sets nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (sequence type-eq-decl nil sequences nil)
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (series const-decl "sequence[real]" series "series/")
    (measure_countably_additive? const-decl "bool"
     generalized_measure_def nil))
   shostak))
 (A_of_TCC1 0
  (A_of_TCC1-1 nil 3431320334
   ("" (skosimp)
    (("" (expand "nonempty?")
      (("" (expand "empty?")
        (("" (expand "member")
          (("" (typepred "f!1")
            (("" (expand "sigma_finite_measure?")
              (("" (flatten)
                (("" (expand "measure_sigma_finite?")
                  (("" (skosimp)
                    (("" (inst -4 "X!1") (("" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nonempty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (measure_sigma_finite? const-decl "bool" measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (subset_algebra_fullset name-judgement "(S)" measure_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-type-decl nil measure_def nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (empty? const-decl "bool" sets nil))
   nil))
 (P_of_TCC1 0
  (P_of_TCC1-1 nil 3453281846
   (""
    (case "forall (f: sigma_finite_measure, n: nat): S(IUnion(n,A_of(f)))")
    (("1"
      (case "forall (f: sigma_finite_measure, n: nat):
        IUnion[nat, T]
              (LAMBDA i: IF i <= n THEN A_of(f)(i) ELSE emptyset[T] ENDIF)= IUnion(n, A_of(f))")
      (("1" (skosimp)
        (("1" (inst - "f!1" "n!1")
          (("1" (inst - "f!1" "n!1")
            (("1" (replace -1) (("1" (propax) nil nil)) nil)) nil))
          nil))
        nil)
       ("2" (hide-all-but 1)
        (("2" (induct "n")
          (("1" (skosimp)
            (("1" (rewrite "IUnion_0_def")
              (("1" (apply-extensionality :hide? t)
                (("1" (expand "IUnion")
                  (("1" (case-replace "A_of(f!1)(0)(x!1)")
                    (("1" (inst + "0") (("1" (assertnil nil)) nil)
                     ("2" (assert)
                      (("2" (skosimp)
                        (("2" (expand "emptyset")
                          (("2" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (skosimp*)
            (("2" (rewrite "IUnion_n_def" 1)
              (("2" (apply-extensionality :hide? t)
                (("2" (expand "union")
                  (("2" (expand "member")
                    (("2" (case-replace "A_of(f!1)(1 + j!1)(x!1)")
                      (("1" (hide -2)
                        (("1" (expand "IUnion")
                          (("1" (inst + "1+j!1")
                            (("1" (assertnil nil)) nil))
                          nil))
                        nil)
                       ("2" (assert)
                        (("2" (inst - "f!1")
                          (("2"
                            (case-replace
                             "IUnion(j!1, A_of(f!1))(x!1)")
                            (("1" (replace -2 -1 rl)
                              (("1"
                                (hide -2)
                                (("1"
                                  (expand "IUnion")
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (expand "emptyset")
                                      (("1"
                                        (prop)
                                        (("1"
                                          (inst + "i!1")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (assert)
                              (("2"
                                (replace -2 1 rl)
                                (("2"
                                  (hide -2)
                                  (("2"
                                    (expand "IUnion")
                                    (("2"
                                      (skosimp)
                                      (("2"
                                        (expand "emptyset")
                                        (("2"
                                          (prop)
                                          (("2"
                                            (expand "<=" -1)
                                            (("2"
                                              (split)
                                              (("1"
                                                (inst + "i!1")
                                                (("1"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil)
                                               ("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2" (induct "n")
        (("1" (skosimp)
          (("1" (rewrite "IUnion_0_def") (("1" (assertnil nil)) nil))
          nil)
         ("2" (skosimp)
          (("2" (skosimp)
            (("2" (inst - "f!1")
              (("2" (rewrite "IUnion_n_def")
                (("2" (assert)
                  (("2" (typepred "S")
                    (("2" (expand "subset_algebra?")
                      (("2" (flatten)
                        (("2" (expand "subset_algebra_union?")
                          (("2"
                            (inst - "IUnion(j!1, A_of(f!1))"
                             "A_of(f!1)(1 + j!1)")
                            (("2" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subset_algebra_union? const-decl "bool" subset_algebra_def nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (emptyset const-decl "set" sets nil)
    (<= const-decl "bool" reals nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (IUnion_0_def formula-decl nil nat_indexed_sets "sets_aux/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (IUnion_n_def formula-decl nil nat_indexed_sets "sets_aux/")
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (member const-decl "bool" sets nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (union const-decl "set" sets nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (T formal-type-decl nil measure_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (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)
    (set type-eq-decl nil sets nil)
    (sequence type-eq-decl nil sequences nil)
    (IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/")
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (A_of const-decl "disjoint_indexed_measurable" measure_def nil))
   nil))
 (A_of_def1 0
  (A_of_def1-1 nil 3459221383
   ("" (skosimp)
    (("" (name "AA" "A_of(mu!1)")
      (("" (replace -1)
        (("" (expand "A_of")
          ((""
            (case "nonempty?({X | IUnion(X) = fullset[T] AND (FORALL i: mu!1(X(i))`1)})")
            (("1"
              (lemma "choose_member"
               ("a"
                "{X | IUnion(X) = fullset[T] AND (FORALL i: mu!1(X(i))`1)}"))
              (("1" (replace -3)
                (("1" (expand "nonempty?") (("1" (assertnil nil))
                  nil))
                nil))
              nil)
             ("2" (typepred "mu!1")
              (("2" (expand "sigma_finite_measure?")
                (("2" (flatten)
                  (("2" (expand "measure_sigma_finite?")
                    (("2" (skosimp)
                      (("2" (expand "nonempty?")
                        (("2" (expand "empty?")
                          (("2" (inst -4 "X!1")
                            (("2" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((A_of const-decl "disjoint_indexed_measurable" measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil measure_def nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (measure_sigma_finite? const-decl "bool" measure_def nil)
    (choose_member formula-decl nil sets_lemmas nil)
    (subset_algebra_fullset name-judgement "(S)" measure_def nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (fullset const-decl "set" sets nil))
   shostak))
 (A_of_def2 0
  (A_of_def2-1 nil 3459221692
   ("" (skosimp)
    (("" (name "AA" "A_of(mu!1)")
      (("" (replace -1)
        (("" (expand "A_of")
          ((""
            (case "nonempty?({X | IUnion(X) = fullset[T] AND (FORALL i: mu!1(X(i))`1)})")
            (("1"
              (lemma "choose_member"
               ("a"
                "{X | IUnion(X) = fullset[T] AND (FORALL i: mu!1(X(i))`1)}"))
              (("1" (replace -3)
                (("1" (expand "nonempty?")
                  (("1" (assert) (("1" (inst - "n!1"nil nil)) nil))
                  nil))
                nil))
              nil)
             ("2" (typepred "mu!1")
              (("2" (expand "sigma_finite_measure?")
                (("2" (flatten)
                  (("2" (hide-all-but (-2 1))
                    (("2" (expand "measure_sigma_finite?")
                      (("2" (skosimp)
                        (("2" (expand "nonempty?")
                          (("2" (expand "empty?")
                            (("2" (inst -3 "X!1")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((A_of const-decl "disjoint_indexed_measurable" measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil measure_def nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (measure_sigma_finite? const-decl "bool" measure_def nil)
    (empty? const-decl "bool" sets nil)
    (choose_member formula-decl nil sets_lemmas nil)
    (member const-decl "bool" sets nil)
    (subset_algebra_fullset name-judgement "(S)" measure_def nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (fullset const-decl "set" sets nil))
   shostak))
 (P_of_def1 0
  (P_of_def1-1 nil 3459221497
   ("" (skosimp)
    (("" (apply-extensionality :hide? t)
      (("" (expand "fullset")
        (("" (expand "P_of")
          (("" (expand "IUnion")
            (("" (lemma "A_of_def1" ("mu" "mu!1"))
              ((""
                (lemma "extensionality_postulate"
                 ("f" "IUnion(A_of(mu!1))" "g" "fullset[T]"))
                (("" (flatten)
                  (("" (hide -1)
                    (("" (split)
                      (("1" (inst - "x!1")
                        (("1" (hide -2)
                          (("1" (expand "emptyset")
                            (("1" (expand "fullset")
                              (("1"
                                (expand "IUnion")
                                (("1"
                                  (skosimp)
                                  (("1"
                                    (inst + "i!1")
                                    (("1"
                                      (inst + "i!1")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (propax) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil measure_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (fullset const-decl "set" sets nil)
    (P_of const-decl "(S)" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (set type-eq-decl nil sets nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (subset_algebra_fullset name-judgement "(S)" measure_def nil)
    (A_of_def1 formula-decl nil measure_def nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (emptyset const-decl "set" sets nil)
    (extensionality_postulate formula-decl nil functions nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (A_of const-decl "disjoint_indexed_measurable" measure_def nil))
   shostak))
 (P_of_def2 0
  (P_of_def2-1 nil 3459221799
   ("" (induct "n")
    (("1" (skosimp)
      (("1" (case-replace "P_of(mu!1)(0)=A_of(mu!1)(0)")
        (("1" (rewrite "A_of_def2"nil nil)
         ("2" (hide 2)
          (("2" (expand "P_of")
            (("2" (apply-extensionality :hide? t)
              (("2" (expand "IUnion")
                (("2" (case-replace "A_of(mu!1)(0)(x!1)")
                  (("1" (inst + "0") (("1" (assertnil nil)) nil)
                   ("2" (assert)
                    (("2" (skosimp)
                      (("2" (expand "emptyset")
                        (("2" (propax) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (skosimp)
      (("2" (skosimp)
        (("2" (inst - "mu!1")
          (("2"
            (case "P_of(mu!1)(j!1 + 1)=union(P_of(mu!1)(j!1),A_of(mu!1)(j!1+1))")
            (("1" (replace -1)
              (("1" (hide -1)
                (("1"
                  (lemma "measure_union"
                   ("a" "P_of(mu!1)(j!1)" "b" "A_of(mu!1)(j!1 + 1)" "f"
                    "mu!1"))
                  (("1" (assert)
                    (("1" (typepred "mu!1")
                      (("1" (expand "sigma_finite_measure?")
                        (("1" (flatten)
                          (("1" (assert)
                            (("1"
                              (lemma "A_of_def2"
                               ("mu" "mu!1" "n" "1+j!1"))
                              (("1"
                                (expand "x_le")
                                (("1"
                                  (expand "x_add")
                                  (("1" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but 1)
              (("2" (apply-extensionality :hide? t)
                (("2" (expand "P_of")
                  (("2" (expand "union")
                    (("2" (expand "member")
                      (("2"
                        (case-replace "IUnion(LAMBDA i:
                 IF i <= 1+j!1 THEN A_of(mu!1)(i) ELSE emptyset[T] ENDIF)
              (x!1)")
                        (("1" (flatten)
                          (("1" (expand "IUnion")
                            (("1" (skosimp)
                              (("1"
                                (expand "emptyset")
                                (("1"
                                  (prop)
                                  (("1"
                                    (expand "<=" -1)
                                    (("1"
                                      (split)
                                      (("1"
                                        (inst + "i!1")
                                        (("1" (assertnil nil))
                                        nil)
                                       ("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (replace 1 2)
                          (("2" (assert)
                            (("2" (expand "IUnion")
                              (("2"
                                (expand "emptyset")
                                (("2"
                                  (split)
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (inst + "i!1")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (prop)
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (inst + "1+j!1")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((member const-decl "bool" sets nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (measure_union formula-decl nil measure_def nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (x_add const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (subset_algebra_union application-judgement "(S)" measure_def nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (union const-decl "set" sets nil)
    (emptyset const-decl "set" sets nil)
    (<= const-decl "bool" reals nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (set type-eq-decl nil sets nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (A_of_def2 formula-decl nil measure_def nil)
    (A_of const-decl "disjoint_indexed_measurable" measure_def nil)
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (P_of const-decl "(S)" measure_def nil)
    (sigma_finite_measure nonempty-type-eq-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil measure_def nil)
    (pred type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
 (P_of_def3 0
  (P_of_def3-1 nil 3459222257
   ("" (skosimp)
    (("" (expand "subset?")
      (("" (expand "member")
        (("" (skosimp)
          (("" (expand "P_of")
            (("" (expand "IUnion")
              (("" (expand "emptyset")
                (("" (skosimp)
                  (("" (prop)
                    (("" (inst + "i!2") (("" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subset? const-decl "bool" sets nil)
    (IUnion const-decl "set[T]" indexed_sets nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (emptyset const-decl "set" sets nil)
    (P_of const-decl "(S)" measure_def nil)
    (member const-decl "bool" sets nil))
   shostak))
 (sigma_finite_def1 0
  (sigma_finite_def1-1 nil 3453268789
   ("" (expand "sigma_finite_measure?")
    (("" (skosimp)
      (("" (case-replace "measure?(f!1)")
        (("1" (expand "measure_sigma_finite?") (("1" (propax) nil nil))
          nil)
         ("2" (assertnil nil))
        nil))
      nil))
    nil)
   ((measure_sigma_finite? const-decl "bool" measure_def nil)
    (measure? const-decl "bool" generalized_measure_def nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil measure_def nil)
    (sigma_finite_measure? const-decl "bool" measure_def nil))
   shostak))
 (sigma_finite_def2 0
  (sigma_finite_def2-1 nil 3453268923
   ("" (skosimp)
    (("" (rewrite "sigma_finite_def1")
      (("" (case-replace "measure?(f!1)")
        (("1" (split)
          (("1" (skosimp*)
            (("1" (lemma "increasing_IUnion" ("A" "X!1"))
              (("1" (skosimp)
                (("1" (case "forall n: S(B!1(n))")
                  (("1" (inst + "B!1")
                    (("1" (assert)
                      (("1" (induct "i")
                        (("1" (inst -7 "0") (("1" (assertnil nil))
                          nil)
                         ("2" (skosimp)
                          (("2" (inst -5 "j!1")
                            (("2" (replace -5)
                              (("2"
                                (inst -8 "1+j!1")
                                (("2"
                                  (lemma
                                   "measure_union"
                                   ("f"
                                    "f!1"
                                    "a"
                                    "B!1(j!1)"
                                    "b"
                                    "X!1(1 + j!1)"))
                                  (("2"
                                    (assert)
                                    (("2"
                                      (expand "x_add")
                                      (("2"
                                        (expand "x_le")
                                        (("2" (propax) nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assert)
                      (("2" (replace -1)
                        (("2" (expand "increasing_indexed_measurable?")
                          (("2" (expand "increasing_indexed?")
                            (("2" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide 2)
                    (("2" (induct "n")
                      (("1" (assertnil nil)
                       ("2" (skosimp)
                        (("2" (inst -4 "j!1")
                          (("2" (replace -4)
                            (("2"
                              (lemma "subset_algebra_union"
                               ("x" "B!1(j!1)" "y" "X!1(j!1+1)"))
                              (("1" (propax) nil nil)
                               ("2" (propax) nil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (skosimp*)
            (("2" (lemma "disjoint_IUnion" ("A" "P!1"))
              (("2" (skosimp)
                (("2" (inst + "B!1")
                  (("1" (assert)
                    (("1" (typepred "P!1")
                      (("1" (skosimp)
                        (("1"
                          (lemma "measure_monotone"
                           ("f" "f!1" "a" "B!1(i!1)" "b" "P!1(i!1)"))
                          (("1" (assert)
                            (("1" (split)
                              (("1"
                                (expand "x_le")
                                (("1" (inst -8 "i!1"nil nil))
                                nil)
                               ("2"
                                (hide 2)
                                (("2"
                                  (expand
                                   "increasing_indexed_measurable?")
                                  (("2"
                                    (expand "increasing_indexed?")
                                    (("2"
                                      (case
                                       "forall n: IUnion(n, P!1)=P!1(n)")
                                      (("1"
                                        (inst - "i!1")
                                        (("1"
                                          (case
                                           "subset?(B!1(i!1), IUnion(i!1, B!1))")
                                          (("1"
                                            (inst - "i!1")
                                            (("1"
                                              (replace -5)
                                              (("1"
                                                (replace -2)
                                                (("1"
                                                  (propax)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide-all-but 1)
                                            (("2"
                                              (expand "IUnion")
                                              (("2"
                                                (expand "subset?")
                                                (("2"
                                                  (expand "member")
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (expand "image")
                                                      (("2"
                                                        (expand
                                                         "Union")
                                                        (("2"
                                                          (inst
                                                           +
                                                           "B!1(i!1)")
                                                          (("2"
                                                            (inst
                                                             +
                                                             "i!1")
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide-all-but (-1 1))
                                        (("2"
                                          (skosimp)
                                          (("2"
                                            (apply-extensionality
                                             :hide?
                                             t)
                                            (("2"
                                              (expand "IUnion")
                                              (("2"
                                                (expand "image")
                                                (("2"
                                                  (expand "Union")
                                                  (("2"
                                                    (case-replace
                                                     "P!1(n!1)(x!1)")
                                                    (("1"
                                                      (inst
                                                       +
                                                       "P!1(n!1)")
                                                      (("1"
                                                        (inst + "n!1")
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (assert)
                                                      (("2"
                                                        (skosimp)
                                                        (("2"
                                                          (typepred
                                                           "a!1")
                                                          (("2"
                                                            (skolem
                                                             -
                                                             "n!2")
                                                            (("2"
                                                              (replace
                                                               -1)
                                                              (("2"
                                                                (hide
                                                                 -1)
                                                                (("2"
                                                                  (typepred
                                                                   "n!2")
                                                                  (("2"
                                                                    (expand
                                                                     "increasing?")
                                                                    (("2"
                                                                      (inst
                                                                       -
                                                                       "n!2"
                                                                       "n!1")
                                                                      (("2"
                                                                        (assert)
                                                                        (("2"
                                                                          (expand
                                                                           "subset?")
                                                                          (("2"
                                                                            (inst
                                                                             -
                                                                             "x!1")
                                                                            (("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)
                   ("2" (split)
                    (("1" (case "B!1(0)=P!1(0)")
                      (("1" (case "forall n: P!1(n) = IUnion(n, P!1)")
                        (("1"
                          (case "forall n: B!1(n+1)=difference(P!1(n+1),P!1(n))")
                          (("1" (skosimp)
                            (("1" (case-replace "x1!1=0")
                              (("1" (assertnil nil)
                               ("2"
                                (inst - "x1!1-1")
                                (("1" (assertnil nil)
                                 ("2" (assertnil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide 2)
                            (("2" (skosimp)
                              (("2"
                                (apply-extensionality :hide? t)
                                (("2"
                                  (expand "difference")
                                  (("2"
                                    (expand "member")
                                    (("2"
                                      (case-replace
                                       "B!1(1 + n!1)(x!1)")
                                      (("1"
                                        (case-replace
                                         "P!1(1 + n!1)(x!1)")
                                        (("1"
                                          (inst-cp -4 "1+n!1")
                                          (("1"
                                            (inst -4 "n!1")
                                            (("1"
                                              (replace -5 -1)
                                              (("1"
                                                (hide -5)
                                                (("1"
                                                  (expand "IUnion" -1)
                                                  (("1"
                                                    (expand "image")
                                                    (("1"
                                                      (expand "Union")
                                                      (("1"
                                                        (skosimp)
                                                        (("1"
                                                          (replace
                                                           -4
                                                           -3)
                                                          (("1"
                                                            (hide -4)
                                                            (("1"
                                                              (inst
                                                               -6
                                                               "n!1")
                                                              (("1"
                                                                (replace
                                                                 -6
                                                                 -3
                                                                 :dir
                                                                 rl)
                                                                (("1"
                                                                  (hide-all-but
                                                                   (-2
                                                                    -3
                                                                    -5))
                                                                  (("1"
                                                                    (expand
                                                                     "IUnion")
                                                                    (("1"
                                                                      (expand
                                                                       "image")
                                                                      (("1"
                                                                        (expand
                                                                         "Union")
                                                                        (("1"
                                                                          (skosimp)
                                                                          (("1"
                                                                            (typepred
                                                                             "a!2")
                                                                            (("1"
                                                                              (skolem
                                                                               -
                                                                               "n!2")
                                                                              (("1"
                                                                                (typepred
                                                                                 "n!2")
                                                                                (("1"
                                                                                  (replace
                                                                                   -2)
                                                                                  (("1"
                                                                                    (hide
                                                                                     -2)
                                                                                    (("1"
                                                                                      (expand
                                                                                       "disjoint?")
                                                                                      (("1"
                                                                                        (inst
                                                                                         -
                                                                                         "n!2"
                                                                                         "1+n!1")
                                                                                        (("1"
                                                                                          (grind)
                                                                                          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"
                                          (assert)
                                          (("2"
                                            (hide 2)
                                            (("2"
                                              (inst -2 "1+n!1")
                                              (("2"
                                                (replace -2)
                                                (("2"
                                                  (hide -2)
                                                  (("2"
                                                    (inst -4 "1+n!1")
                                                    (("2"
                                                      (replace -4 1 rl)
                                                      (("2"
                                                        (hide-all-but
                                                         (-1 1))
                                                        (("2"
                                                          (expand
                                                           "IUnion")
                                                          (("2"
                                                            (expand
                                                             "image")
                                                            (("2"
                                                              (expand
                                                               "Union")
                                                              (("2"
                                                                (inst
                                                                 +
                                                                 "B!1(1+n!1)")
                                                                (("2"
                                                                  (inst
                                                                   +
                                                                   "1+n!1")
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (assert)
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (inst-cp -2 "1+n!1")
                                            (("2"
                                              (replace -3)
                                              (("2"
                                                (inst-cp -6 "1+n!1")
                                                (("2"
                                                  (replace -7 -1 rl)
                                                  (("2"
                                                    (expand
                                                     "IUnion"
                                                     -1)
                                                    (("2"
                                                      (expand "image")
                                                      (("2"
                                                        (expand
                                                         "Union")
                                                        (("2"
                                                          (skosimp)
                                                          (("2"
                                                            (typepred
                                                             "a!1")
                                                            (("2"
                                                              (skolem
                                                               -
                                                               "n!2")
                                                              (("2"
                                                                (replace
                                                                 -1)
                                                                (("2"
                                                                  (hide
                                                                   -1)
                                                                  (("2"
                                                                    (typepred
                                                                     "n!2")
                                                                    (("2"
                                                                      (expand
                                                                       "<="
                                                                       -1)
                                                                      (("2"
                                                                        (split)
                                                                        (("1"
                                                                          (inst
                                                                           -7
                                                                           "n!1")
                                                                          (("1"
                                                                            (inst
                                                                             -3
                                                                             "n!1")
                                                                            (("1"
                                                                              (replace
                                                                               -3)
                                                                              (("1"
                                                                                (replace
                                                                                 -7
                                                                                 2
                                                                                 :dir
                                                                                 rl)
                                                                                (("1"
                                                                                  (expand
                                                                                   "IUnion"
                                                                                   2)
                                                                                  (("1"
                                                                                    (expand
                                                                                     "image")
                                                                                    (("1"
                                                                                      (expand
                                                                                       "Union")
                                                                                      (("1"
                                                                                        (inst
                                                                                         +
                                                                                         "B!1(n!2)")
                                                                                        (("1"
                                                                                          (inst
                                                                                           +
                                                                                           "n!2")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil)
                                                                         ("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but 1)
                          (("2" (skosimp)
                            (("2" (apply-extensionality :hide? t)
                              (("2"
                                (case-replace "P!1(n!1)(x!1)")
                                (("1"
                                  (expand "IUnion")
                                  (("1"
                                    (expand "image")
                                    (("1"
                                      (expand "Union")
                                      (("1"
                                        (inst + "P!1(n!1)")
                                        (("1" (inst + "n!1"nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (assert)
                                  (("2"
                                    (expand "IUnion")
                                    (("2"
                                      (expand "image")
                                      (("2"
                                        (expand "Union")
                                        (("2"
                                          (skosimp)
                                          (("2"
                                            (typepred "a!1")
                                            (("2"
                                              (skolem - "n!2")
                                              (("2"
                                                (typepred "n!2")
                                                (("2"
                                                  (typepred "P!1")
                                                  (("2"
                                                    (expand
                                                     "increasing_indexed_measurable?")
                                                    (("2"
                                                      (expand
                                                       "increasing_indexed?")
                                                      (("2"
                                                        (flatten)
                                                        (("2"
                                                          (expand
                                                           "increasing?")
                                                          (("2"
                                                            (inst
                                                             -
                                                             "n!2"
                                                             "n!1")
                                                            (("2"
                                                              (expand
                                                               "subset?")
                                                              (("2"
                                                                (assert)
                                                                (("2"
                                                                  (inst
                                                                   -
                                                                   "x!1")
                                                                  (("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)
                       ("2" (propax) nil nil))
                      nil)
                     ("2" (expand "disjoint_indexed_measurable?")
                      (("2" (propax) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assertnil nil))
        nil))
      nil))
    nil)
   ((sigma_finite_def1 formula-decl nil measure_def nil)
    (T formal-type-decl nil measure_def nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (subset_algebra? const-decl "bool" subset_algebra_def nil)
    (subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil)
    (S formal-const-decl "subset_algebra" measure_def nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (extended_nnreal nonempty-type-eq-decl nil extended_nnreal
     "extended_nnreal/")
    (disjoint_indexed_measurable nonempty-type-eq-decl nil
     generalized_measure_def nil)
    (disjoint_indexed_measurable? const-decl "bool"
     generalized_measure_def nil)
    (sequence type-eq-decl nil sequences nil)
    (set type-eq-decl nil sets nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (increasing_IUnion formula-decl nil nat_indexed_sets "sets_aux/")
    (increasing_indexed? const-decl "bool" nat_indexed_sets
     "sets_aux/")
    (subset_algebra_fullset name-judgement "(S)" measure_def nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (measure_union formula-decl nil measure_def nil)
    (x_add const-decl "extended_nnreal" extended_nnreal
     "extended_nnreal/")
    (x_le const-decl "bool" extended_nnreal "extended_nnreal/")
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types nil)
    (increasing_indexed_measurable nonempty-type-eq-decl nil
     measure_def nil)
    (increasing_indexed_measurable? const-decl "bool" measure_def nil)
    (B!1 skolem-const-decl "sequence[set[T]]" measure_def nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (subset_algebra_union judgement-tcc nil subset_algebra nil)
    (disjoint_IUnion formula-decl nil nat_indexed_sets "sets_aux/")
    (B!1 skolem-const-decl "sequence[set[T]]" measure_def nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (measure_monotone formula-decl nil measure_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/")
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (image const-decl "set[R]" function_image nil)
    (<= const-decl "bool" reals nil)
    (i!1 skolem-const-decl "nat" measure_def nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (Union const-decl "set" sets nil)
    (increasing? const-decl "bool" fun_preds_partial "structures/")
    (n!1 skolem-const-decl "nat" measure_def nil)
    (P!1 skolem-const-decl "increasing_indexed_measurable" measure_def
     nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (n!1 skolem-const-decl "nat" measure_def nil)
    (subset_algebra_difference application-judgement "(S)" measure_def
     nil)
    (difference const-decl "set" sets nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (x1!1 skolem-const-decl "nat" measure_def nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (n!1 skolem-const-decl "nat" measure_def nil)
    (disjoint? const-decl "bool" indexed_sets_aux "sets_aux/")
    (disjoint? const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (/= const-decl "boolean" notequal nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (n!2 skolem-const-decl "({i | i <= 1 + n!1})" measure_def nil)
    (measure? const-decl "bool" generalized_measure_def nil))
   shostak)))


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

¤ 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.0.769Bemerkung:  (vorverarbeitet am  2026-05-02) ¤

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Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

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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:

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