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Datei: lagrange_index.prf   Sprache: PVS

Original von: PVS©

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    (union const-decl "set" sets 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 ordered_subset nil)
    (PRED type-eq-decl nil defined_types nil)
    (transitive? const-decl "bool" relations nil)
    (member const-decl "bool" sets nil)
    (reflexive_closure const-decl "(reflexive?)" closure_ops nil))
   nil))
 (up_subset 0
  (up_subset-1 nil 3406647310
   ("" (expand "reflexive_closure")
    (("" (expand "union")
      (("" (expand "member")
        (("" (expand "subset?")
          (("" (expand "member")
            (("" (skosimp*)
              (("" (typepred "tr!1")
                (("" (expand "transitive?")
                  (("" (inst - "x!1" "y!1" "x!2")
                    (("" (split -3)
                      (("1" (assertnil nil) ("2" (assertnil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((union const-decl "set" sets nil)
    (subset? const-decl "bool" sets 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 ordered_subset nil)
    (PRED type-eq-decl nil defined_types nil)
    (transitive? const-decl "bool" relations nil)
    (member const-decl "bool" sets nil)
    (reflexive_closure const-decl "(reflexive?)" closure_ops nil))
   nil))
 (complement_lower_set 0
  (complement_lower_set-1 nil 3406647692
   ("" (skosimp)
    (("" (typepred "X!1")
      (("" (expand "suffix?")
        (("" (skosimp*)
          (("" (expand "member")
            (("" (expand "complement")
              (("" (expand "member")
                (("" (expand "prefix?")
                  (("" (typepred "r!1")
                    (("" (expand "complement")
                      (("" (expand "member")
                        (("" (inst - "r!1" "t!1")
                          (("" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((prefix? const-decl "bool" ordered_subset nil)
    (pred type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (complement const-decl "set" sets nil)
    (lessp!1 skolem-const-decl "pred[[T, T]]" ordered_subset nil)
    (X!1 skolem-const-decl "(LAMBDA (S): prefix?(S, lessp!1))"
     ordered_subset nil)
    (t!1 skolem-const-decl "T" ordered_subset nil)
    (member const-decl "bool" sets nil)
    (suffix? const-decl "bool" ordered_subset nil))
   nil))
 (complement_upper_set 0
  (complement_upper_set-1 nil 3406647708
   ("" (skosimp)
    (("" (typepred "X!1")
      (("" (expand "complement")
        (("" (expand "member")
          (("" (expand "prefix?")
            (("" (expand "suffix?")
              (("" (expand "member")
                (("" (skosimp)
                  (("" (typepred "r!1")
                    (("" (inst - "r!1" "t!1") (("" (assertnil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((suffix? const-decl "bool" ordered_subset nil)
    (pred type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (member const-decl "bool" sets nil)
    (lessp!1 skolem-const-decl "pred[[T, T]]" ordered_subset nil)
    (X!1 skolem-const-decl "(LAMBDA (S): suffix?(S, lessp!1))"
     ordered_subset nil)
    (t!1 skolem-const-decl "T" ordered_subset nil)
    (prefix? const-decl "bool" ordered_subset nil)
    (complement const-decl "set" sets nil))
   nil))
 (lower_set_def 0
  (lower_set_def-1 nil 3406647530
   ("" (skosimp)
    (("" (typepred "tr!1")
      (("" (split)
        (("1" (flatten)
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            (("1" (expand "union")
              (("1" (expand "member")
                (("1" (expand "image")
                  (("1" (expand "extend")
                    (("1" (apply-extensionality 1 :hide? t)
                      (("1" (case-replace "X!1(x!1)")
                        (("1" (expand "Union")
                          (("1" (inst + "down(tr!1)(x!1)")
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                              (("1"
                                (expand "union")
                                (("1"
                                  (expand "member")
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                                  nil))
                                nil))
                              nil)
                             ("2" (split)
                              (("1"
                                (flatten)
                                (("1"
                                  (inst + "x!1")
                                  (("1"
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                                    (("1"
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                                      (("1"
                                        (expand "member")
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                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
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                                (("2"
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                                      (("2"
                                        (expand "member")
                                        (("2"
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                                            (("2"
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                                                 "reflexive_closure")
                                                (("2"
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                                                         -5
                                                         "t!1"
                                                         "r!1"
                                                         "x!1")
                                                        (("1"
                                                          (assert)
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                                                        nil)
                                                       ("2"
                                                        (replace -1)
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                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (replace 1 2)
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                              (("2"
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                                (("2"
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                                  (("2"
                                    (assert)
                                    (("2"
                                      (skosimp)
                                      (("2"
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                                        (("2"
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                                          (("2"
                                            (assert)
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                                              (expand "prefix?")
                                              (("2"
                                                (expand "member")
                                                (("2"
                                                  (inst -5 "x!1" "x!2")
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (skosimp*)
                        (("2" (expand "prefix?")
                          (("2" (skosimp)
                            (("2" (expand "member")
                              (("2"
                                (flatten)
                                (("2"
                                  (typepred "x!1")
                                  (("2"
                                    (typepred "r!1")
                                    (("2"
                                      (split -1)
                                      (("1"
                                        (expand "transitive?")
                                        (("1"
                                          (inst -6 "t!2" "r!1" "x!1")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (replace -1)
                                        (("2" (propax) nil nil))
                                        nil))
                                      nil))
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                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (flatten)
          (("2" (replace -1)
            (("2" (hide -1)
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                    (("2" (expand "image")
                      (("2" (expand "extend")
                        (("2" (expand "Union")
                          (("2" (expand "prefix?")
                            (("2" (expand "member")
                              (("2"
                                (skosimp)
                                (("2"
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                                  (("2"
                                    (skosimp)
                                    (("2"
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                                        (assert)
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                                          (skosimp)
                                          (("2"
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                                            (("2"
                                              (skosimp)
                                              (("2"
                                                (typepred "a!2")
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (typepred "x!2")
                                                      (("2"
                                                        (inst + "a!2")
                                                        (("2"
                                                          (expand
                                                           "prefix?"
                                                           -3)
                                                          (("2"
                                                            (inst
                                                             -
                                                             "t!1"
                                                             "r!1")
                                                            (("2"
                                                              (expand
                                                               "member")
                                                              (("2"
                                                                (propax)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((transitive? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (T formal-type-decl nil ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (union const-decl "set" sets nil)
    (image const-decl "set[R]" function_image nil)
    (FALSE const-decl "bool" booleans nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (Union const-decl "set" sets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (pred type-eq-decl nil defined_types nil)
    (prefix? const-decl "bool" ordered_subset nil)
    (tr!1 skolem-const-decl "(transitive?[T])" ordered_subset nil)
    (X!1 skolem-const-decl "set[T]" ordered_subset nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (x!1 skolem-const-decl "T" ordered_subset nil)
    (reflexive? const-decl "bool" relations nil)
    (extend const-decl "R" extend nil)
    (member const-decl "bool" sets nil)
    (reflexive_closure const-decl "(reflexive?)" closure_ops nil))
   nil))
 (upper_set_def 0
  (upper_set_def-1 nil 3406647552
   ("" (skosimp)
    (("" (expand "reflexive_closure")
      (("" (expand "union")
        (("" (expand "member")
          (("" (expand "image")
            (("" (expand "extend")
              (("" (split)
                (("1" (flatten)
                  (("1" (apply-extensionality 1 :hide? t)
                    (("1" (case-replace "X!1(x!1)")
                      (("1" (expand "Union")
                        (("1" (inst + "up(tr!1)(x!1)")
                          (("1" (expand "reflexive_closure")
                            (("1" (expand "union")
                              (("1"
                                (expand "member")
                                (("1" (propax) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (assert)
                            (("2" (prop)
                              (("1"
                                (inst + "x!1")
                                (("1"
                                  (expand "reflexive_closure")
                                  (("1"
                                    (expand "union")
                                    (("1"
                                      (expand "member")
                                      (("1" (propax) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (expand "reflexive_closure")
                                (("2"
                                  (expand "union")
                                  (("2"
                                    (expand "member")
                                    (("2"
                                      (expand "suffix?")
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                                        (skosimp)
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                                          (expand "member")
                                          (("2"
                                            (flatten)
                                            (("2"
                                              (typepred "r!1")
                                              (("2"
                                                (split)
                                                (("1"
                                                  (typepred "tr!1")
                                                  (("1"
                                                    (expand
                                                     "transitive?")
                                                    (("1"
                                                      (inst
                                                       -
                                                       "x!1"
                                                       "r!1"
                                                       "t!1")
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (replace 1 2)
                        (("2" (assert)
                          (("2" (expand "Union")
                            (("2" (skosimp)
                              (("2"
                                (typepred "a!1")
                                (("2"
                                  (assert)
                                  (("2"
                                    (skosimp)
                                    (("2"
                                      (typepred "x!2")
                                      (("2"
                                        (expand "suffix?")
                                        (("2"
                                          (replace -2)
                                          (("2"
                                            (assert)
                                            (("2"
                                              (inst -5 "x!1" "x!2")
                                              (("2"
                                                (expand "member")
                                                (("2"
                                                  (propax)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (skosimp*)
                      (("2" (expand "suffix?")
                        (("2" (skosimp)
                          (("2" (expand "member")
                            (("2" (flatten)
                              (("2"
                                (typepred "x!1")
                                (("2"
                                  (typepred "r!1")
                                  (("2"
                                    (split -1)
                                    (("1"
                                      (typepred "tr!1")
                                      (("1"
                                        (expand "transitive?")
                                        (("1"
                                          (inst - "x!1" "r!1" "t!2")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (flatten)
                  (("2" (replace -1)
                    (("2" (hide -1)
                      (("2" (expand "Union")
                        (("2" (expand "suffix?")
                          (("2" (expand "member")
                            (("2" (skosimp*)
                              (("2"
                                (typepred "r!1")
                                (("2"
                                  (skosimp)
                                  (("2"
                                    (typepred "a!1")
                                    (("2"
                                      (assert)
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                                        (skosimp)
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                                          (typepred "x!1")
                                          (("2"
                                            (inst + "a!1")
                                            (("2"
                                              (expand "suffix?")
                                              (("2"
                                                (inst - "t!1" "r!1")
                                                (("2"
                                                  (expand "member")
                                                  (("2"
                                                    (propax)
                                                    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)
   ((reflexive_closure const-decl "(reflexive?)" closure_ops nil)
    (member const-decl "bool" sets nil)
    (extend const-decl "R" extend nil)
    (reflexive? const-decl "bool" relations nil)
    (x!1 skolem-const-decl "T" ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (X!1 skolem-const-decl "set[T]" ordered_subset nil)
    (tr!1 skolem-const-decl "(transitive?[T])" ordered_subset nil)
    (transitive? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (suffix? const-decl "bool" ordered_subset nil)
    (pred type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setofsets type-eq-decl nil sets nil)
    (Union const-decl "set" sets nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (FALSE const-decl "bool" booleans nil)
    (T formal-type-decl nil ordered_subset nil)
    (boolean nonempty-type-decl nil booleans nil)
    (image const-decl "set[R]" function_image nil)
    (union const-decl "set" sets nil))
   nil))
 (Union_lower_set 0
  (Union_lower_set-1 nil 3406647579
   ("" (skosimp)
    (("" (expand "every")
      (("" (expand "prefix?")
        (("" (skosimp)
          (("" (expand "member")
            (("" (expand "Union")
              (("" (typepred "r!1")
                (("" (expand "Union")
                  (("" (skosimp)
                    (("" (inst + "a!1")
                      (("" (inst - "a!1")
                        (("" (inst - "t!1" "r!1")
                          (("" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((every const-decl "bool" sets nil)
    (Union const-decl "set" sets nil)
    (U!1 skolem-const-decl "setofsets[T]" ordered_subset nil)
    (a!1 skolem-const-decl "(U!1)" ordered_subset nil)
    (r!1 skolem-const-decl "(Union(U!1))" ordered_subset 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 ordered_subset nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (set type-eq-decl nil sets nil) (member const-decl "bool" sets nil)
    (prefix? const-decl "bool" ordered_subset nil))
   nil))
 (Union_upper_set 0
  (Union_upper_set-1 nil 3406647751
   ("" (expand "every")
    (("" (expand "Union")
      (("" (expand "suffix?")
        (("" (expand "member")
          (("" (skosimp*)
            (("" (typepred "r!1")
              (("" (skosimp)
                (("" (typepred "a!1")
                  (("" (inst - "a!1")
                    (("" (inst - "t!1" "r!1")
                      (("" (inst + "a!1") (("" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((Union const-decl "set" sets nil)
    (member const-decl "bool" sets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (U!1 skolem-const-decl "setofsets[T]" ordered_subset nil)
    (a!1 skolem-const-decl "(U!1)" ordered_subset nil)
    (r!1 skolem-const-decl "({x | EXISTS (a: (U!1)): a(x)})"
     ordered_subset nil)
    (suffix? const-decl "bool" ordered_subset nil)
    (every const-decl "bool" sets nil))
   nil))
 (union_lower_set 0
  (union_lower_set-1 nil 3406647775
   ("" (skosimp)
    ((""
      (lemma "Union_lower_set"
       ("U" "add(X!1,singleton(Y!1))" "tr" "tr!1"))
      (("" (split -1)
        (("1"
          (lemma "extensionality"
           ("f" "Union(extend
                             [setof[T], (LAMBDA (S): prefix?(S, tr!1)), bool,
                              FALSE]
                             (add(X!1, singleton(Y!1))))" "g"
            "union(X!1,Y!1)"))
          (("1" (split -1)
            (("1" (assertnil nil)
             ("2" (hide-all-but 1)
              (("2" (skosimp)
                (("2" (expand "union")
                  (("2" (expand "singleton")
                    (("2" (expand "add")
                      (("2" (expand "extend")
                        (("2" (expand "Union")
                          (("2" (expand "member")
                            (("2" (case-replace "X!1(x!1)")
                              (("1" (inst + "X!1"nil nil)
                               ("2"
                                (assert)
                                (("2"
                                  (case-replace "Y!1(x!1)")
                                  (("1"
                                    (inst + "Y!1")
                                    (("1"
                                      (assert)
                                      (("1"
                                        (expand "member")
                                        (("1" (propax) nil nil))
                                        nil))
                                      nil))
                                    nil)
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                                    (assert)
                                    (("2"
                                      (skosimp)
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                                        (("2"
                                          (assert)
                                          (("2"
                                            (expand "member")
                                            (("2" (assertnil nil))
                                            nil))
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                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
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                        nil))
                      nil))
                    nil))
                  nil))
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              nil))
            nil))
          nil)
         ("2" (hide 2)
          (("2" (expand "singleton")
            (("2" (expand "add")
              (("2" (expand "member")
                (("2" (expand "extend")
                  (("2" (expand "every") (("2" (propax) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((singleton const-decl "(singleton?)" sets nil)
    (singleton? const-decl "bool" sets nil)
    (add const-decl "(nonempty?)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (transitive? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (prefix? const-decl "bool" ordered_subset nil)
    (pred type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets 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 ordered_subset nil)
    (Union_lower_set formula-decl nil ordered_subset nil)
    (nonempty_add_finite application-judgement "non_empty_finite_set"
     finite_sets nil)
    (nonempty_singleton_finite application-judgement
     "non_empty_finite_set" finite_sets nil)
    (every const-decl "bool" sets nil)
    (nonempty_extend application-judgement "(nonempty?[T])"
     extend_set_props nil)
    (finite_extend application-judgement "finite_set[T]"
     extend_set_props nil)
    (extensionality formula-decl nil functions nil)
    (Union const-decl "set" sets nil) (union const-decl "set" sets nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (Y!1 skolem-const-decl "(LAMBDA (S): prefix?(S, tr!1))"
     ordered_subset nil)
    (X!1 skolem-const-decl "(LAMBDA (S): prefix?(S, tr!1))"
     ordered_subset nil)
    (tr!1 skolem-const-decl "(transitive?[T])" ordered_subset nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (member const-decl "bool" sets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil))
   nil))
 (union_upper_set 0
  (union_upper_set-1 nil 3406647799
   ("" (skosimp)
    ((""
      (lemma "Union_upper_set"
       ("U" "add(X!1,singleton(Y!1))" "<" "lessp!1"))
      (("" (split -1)
        (("1"
          (lemma "extensionality"
           ("f" "Union(extend
                             [setof[T], (LAMBDA (S): suffix?(S, lessp!1)), bool,
                              FALSE]
                             (add(X!1, singleton(Y!1))))" "g"
            "union(X!1, Y!1)"))
          (("1" (split -1)
            (("1" (assertnil nil)
             ("2" (hide-all-but 1)
              (("2" (skosimp)
                (("2" (expand "union")
                  (("2" (expand "member")
                    (("2" (expand "singleton")
                      (("2" (expand "add")
                        (("2" (expand "member")
                          (("2" (expand "extend")
                            (("2" (expand "Union")
                              (("2"
                                (case-replace "X!1(x!1)")
                                (("1" (inst + "X!1"nil nil)
                                 ("2"
                                  (case-replace "Y!1(x!1)")
                                  (("1" (inst + "Y!1"nil nil)
                                   ("2"
                                    (assert)
                                    (("2"
                                      (skosimp)
                                      (("2"
                                        (typepred "a!1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (hide 2)
          (("2" (expand "singleton")
            (("2" (expand "add")
              (("2" (expand "member")
                (("2" (expand "extend")
                  (("2" (expand "every") (("2" (propax) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((singleton const-decl "(singleton?)" sets nil)
    (singleton? const-decl "bool" sets nil)
    (add const-decl "(nonempty?)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (suffix? const-decl "bool" ordered_subset nil)
    (set type-eq-decl nil sets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (pred 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 ordered_subset nil)
    (Union_upper_set formula-decl nil ordered_subset nil)
    (nonempty_add_finite application-judgement "non_empty_finite_set"
     finite_sets nil)
    (nonempty_singleton_finite application-judgement
     "non_empty_finite_set" finite_sets nil)
    (every const-decl "bool" sets nil)
    (nonempty_extend application-judgement "(nonempty?[T])"
     extend_set_props nil)
    (finite_extend application-judgement "finite_set[T]"
     extend_set_props nil)
    (extensionality formula-decl nil functions nil)
    (Union const-decl "set" sets nil) (union const-decl "set" sets nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (member const-decl "bool" sets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil))
   nil))
 (Intersection_lower_set 0
  (Intersection_lower_set-1 nil 3406647819
   ("" (expand "every")
    (("" (expand "Intersection")
      (("" (expand "prefix?")
        (("" (expand "member")
          (("" (skosimp*)
            (("" (typepred "a!1")
              (("" (typepred "r!1")
                (("" (inst - "a!1")
                  (("" (inst - "a!1")
                    (("" (inst - "t!1" "r!1") (("" (assertnil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((Intersection const-decl "set" sets nil)
    (member const-decl "bool" sets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T formal-type-decl nil ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (U!1 skolem-const-decl "setofsets[T]" ordered_subset nil)
    (a!1 skolem-const-decl "(U!1)" ordered_subset nil)
    (r!1 skolem-const-decl "({x | FORALL (a: (U!1)): a(x)})"
     ordered_subset nil)
    (prefix? const-decl "bool" ordered_subset nil)
    (every const-decl "bool" sets nil))
   nil))
 (Intersection_upper_set 0
  (Intersection_upper_set-1 nil 3406647838
   ("" (skosimp)
    (("" (lemma "Union_lower_set" ("tr" "tr!1" "U" "Complement(U!1)"))
      (("" (split -1)
        (("1" (rewrite "Demorgan2" -1 :dir rl)
          (("1" (lemma "complement_lower_set" ("<" "tr!1"))
            (("1" (inst - "complement(Intersection(U!1))")
              (("1" (rewrite "complement_complement"nil nil)) nil))
            nil))
          nil)
         ("2" (hide 2)
          (("2" (expand "every")
            (("2" (skosimp)
              (("2" (typepred "x!1")
                (("2" (expand "Complement")
                  (("2" (skosimp)
                    (("2" (replace -1)
                      (("2" (typepred "b!1")
                        (("2" (hide -2)
                          (("2" (inst - "b!1")
                            (("2"
                              (lemma "complement_upper_set"
                               ("<" "tr!1" "X" "b!1"))
                              (("1" (propax) nil nil)
                               ("2" (propax) nil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((transitive? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (Complement const-decl "setofsets[T]" sets_lemmas 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 ordered_subset nil)
    (Union_lower_set formula-decl nil ordered_subset nil)
    (suffix? const-decl "bool" ordered_subset nil)
    (complement_upper_set formula-decl nil ordered_subset nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (every const-decl "bool" sets nil)
    (Demorgan2 formula-decl nil sets_lemmas nil)
    (Intersection_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (tr!1 skolem-const-decl "(transitive?[T])" ordered_subset nil)
    (U!1 skolem-const-decl "setofsets[T]" ordered_subset nil)
    (Intersection const-decl "set" sets nil)
    (complement const-decl "set" sets nil)
    (prefix? const-decl "bool" ordered_subset nil)
    (set type-eq-decl nil sets nil)
    (complement_complement formula-decl nil sets_lemmas nil)
    (pred type-eq-decl nil defined_types nil)
    (complement_lower_set formula-decl nil ordered_subset nil))
   nil))
 (intersection_lower_set 0
  (intersection_lower_set-1 nil 3406647858
   ("" (skosimp)
    ((""
      (lemma "Intersection_lower_set"
       ("<" "lessp!1" "U" "add(X!1,singleton(Y!1))"))
      (("" (split -1)
        (("1"
          (lemma "extensionality"
           ("f" "Intersection(extend
                                    [setof[T],
                                     (LAMBDA (S): prefix?(S, lessp!1)), bool,
                                     FALSE]
                                    (add(X!1, singleton(Y!1))))" "g"
            "intersection(X!1, Y!1)"))
          (("1" (split -1)
            (("1" (assertnil nil)
             ("2" (hide-all-but 1)
              (("2" (skosimp)
                (("2" (expand "intersection")
                  (("2" (expand "member")
                    (("2" (expand "singleton")
                      (("2" (expand "add")
                        (("2" (expand "member")
                          (("2" (expand "extend")
                            (("2" (expand "Intersection")
                              (("2"
                                (case-replace "X!1(x!1)")
                                (("1"
                                  (case-replace "Y!1(x!1)")
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (typepred "a!1")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (assert)
                                    (("2" (inst - "Y!1"nil nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (assert)
                                  (("2" (inst - "X!1"nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
--> --------------------

--> maximum size reached

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

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