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Datei: bernstein_minmax.prf Sprache: Lisp

Original von: PVS^©

(homeomorphism_def
(homeomorphism?_TCC1 0
  (homeomorphism?_TCC1-1 nil 3300961357
   ("" (skosimp)
    (("" (skosimp)
      (("" (expand "bijective?")
        (("" (flatten)
          (("" (expand "surjective?")
            (("" (inst - "r!1")
              (("" (skosimp) (("" (inst + "x!1") nil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T2 formal-type-decl nil homeomorphism_def nil)
    (T1 formal-type-decl nil homeomorphism_def nil)
    (surjective? const-decl "bool" functions nil)
    (bijective? const-decl "bool" functions nil))
   shostak))
(homeomorphism_def 0
  (homeomorphism_def-1 nil 3346477468
   ("" (skosimp)
    (("" (expand "homeomorphism?")
      (("" (case-replace "bijective?[T1, T2](f!1)")
        (("1" (case "nonempty?(fullset[T1])")
          (("1" (case "nonempty?(fullset[T2])")
            (("1" (case "EXISTS (x:T1): TRUE")
              (("1"
                (case "forall (y:T2): f!1(inverse_alt(f!1)(y)) = y")
                (("1"
                  (case "forall (x:T1): inverse_alt(f!1)(f!1(x)) = x")
                  (("1" (split 1)
                    (("1" (flatten)
                      (("1" (skosimp)
                        (("1" (split)
                          (("1" (flatten)
                            (("1" (rewrite "continuous_open_sets" -3)
                              (("1"
                                (inst - "X!1")
                                (("1"
                                  (assert)
                                  (("1"
                                    (lemma
                                     "extensionality"
                                     ("f"
                                      "inverse_image(inverse_alt(f!1), X!1)"
                                      "g"
                                      "image(f!1, X!1)"))
                                    (("1"
                                      (split -1)
                                      (("1" (assert) nil nil)
                                       ("2"
                                        (hide-all-but (-4 -5 -6 1 -9))
                                        (("2"
                                          (skosimp)
                                          (("2"
                                            (expand "inverse_image")
                                            (("2"
                                              (expand "member")
                                              (("2"
                                                (case-replace
                                                 "image(f!1, X!1)(x!1)")
                                                (("1"
                                                  (expand "image")
                                                  (("1"
                                                    (skosimp)
                                                    (("1"
                                                      (replace -1)
                                                      (("1"
                                                        (typepred
                                                         "x!2")
                                                        (("1"
                                                          (inst
                                                           -3
                                                           "x!2")
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (replace 1 2)
                                                  (("2"
                                                    (assert)
                                                    (("2"
                                                      (expand "image")
                                                      (("2"
                                                        (inst
                                                         +
                                                         "inverse_alt(f!1)(x!1)")
                                                        (("1"
                                                          (inst
                                                           -3
                                                           "x!1")
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (flatten)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2" (flatten) nil nil))
                                    nil))
                                  nil))
                                nil)
                               ("2" (flatten) nil nil))
                              nil))
                            nil)
                           ("2" (flatten)
                            (("2" (rewrite "continuous_open_sets" -2)
                              (("2"
                                (inst - "image(f!1,X!1)")
                                (("2"
                                  (assert)
                                  (("2"
                                    (rewrite
                                     "injective_inverse_image_image")
                                    (("2"
                                      (expand "bijective?")
                                      (("2" (propax) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (flatten)
                      (("2" (split)
                        (("1" (rewrite "continuous_open_sets" 1)
                          (("1" (skosimp)
                            (("1" (inst - "inverse_image(f!1, Y!1)")
                              (("1"
                                (replace -2)
                                (("1"
                                  (rewrite
                                   "surjective_image_inverse_image"
                                   1)
                                  (("1"
                                    (expand "bijective?")
                                    (("1" (propax) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (rewrite "continuous_open_sets" 1)
                          (("1" (skosimp)
                            (("1" (inst - "Y!1")
                              (("1"
                                (assert)
                                (("1"
                                  (lemma
                                   "extensionality"
                                   ("f"
                                    "image(f!1, Y!1)"
                                    "g"
                                    "inverse_image(inverse_alt(f!1), Y!1)"))
                                  (("1"
                                    (split -1)
                                    (("1" (assert) nil nil)
                                     ("2"
                                      (hide 2 -1 -2)
                                      (("2"
                                        (skosimp)
                                        (("2"
                                          (case-replace
                                           "image(f!1, Y!1)(x!1)")
                                          (("1"
                                            (expand "image")
                                            (("1"
                                              (skosimp)
                                              (("1"
                                                (replace -1)
                                                (("1"
                                                  (expand
                                                   "inverse_image")
                                                  (("1"
                                                    (expand "member")
                                                    (("1"
                                                      (inst -2 "x!2")
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (replace 1 2)
                                            (("2"
                                              (assert)
                                              (("2"
                                                (expand
                                                 "inverse_image")
                                                (("2"
                                                  (expand "member")
                                                  (("2"
                                                    (expand "image")
                                                    (("2"
                                                      (inst
                                                       +
                                                       "inverse_alt(f!1)(x!1)")
                                                      (("1"
                                                        (inst -3 "x!1")
                                                        (("1"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (flatten)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2" (flatten) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (flatten) nil nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (skosimp)
                    (("2" (expand "bijective?")
                      (("2" (flatten)
                        (("2" (expand "injective?")
                          (("2" (inst - "f!1(x!1)")
                            (("2"
                              (inst - "inverse_alt(f!1)(f!1(x!1))"
                               "x!1")
                              (("1" (assert) nil nil)
                               ("2" (flatten) nil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("3" (skosimp) nil nil))
                  nil)
                 ("2" (hide-all-but (-1 -4 1))
                  (("2" (skosimp)
                    (("2" (typepred "inverse_alt(f!1)")
                      (("1" (expand "inverse?")
                        (("1" (inst - "y!1")
                          (("1" (split -1)
                            (("1" (propax) nil nil)
                             ("2" (expand "bijective?")
                              (("2"
                                (flatten)
                                (("2"
                                  (expand "surjective?")
                                  (("2" (inst -3 "y!1") nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (flatten) nil nil))
                      nil))
                    nil))
                  nil)
                 ("3" (skosimp) nil nil))
                nil)
               ("2" (hide-all-but (-2 1))
                (("2" (expand "fullset")
                  (("2" (expand "nonempty?")
                    (("2" (expand "empty?")
                      (("2" (skosimp) (("2" (inst + "x!1") nil nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide-all-but (-1 1))
              (("2" (expand "fullset")
                (("2" (expand "nonempty?")
                  (("2" (expand "empty?")
                    (("2" (skosimp)
                      (("2" (inst - "f!1(x!1)")
                        (("2" (expand "member")
                          (("2" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (expand "nonempty?")
            (("2" (case "empty?(fullset[T2])")
              (("1" (rewrite "emptyset_is_empty?")
                (("1" (rewrite "emptyset_is_empty?")
                  (("1" (split)
                    (("1" (flatten)
                      (("1" (skosimp)
                        (("1" (case-replace "X!1=emptyset[T1]")
                          (("1" (rewrite "image_emptyset")
                            (("1" (expand "open?")
                              (("1"
                                (expand "member")
                                (("1"
                                  (typepred "S")
                                  (("1"
                                    (typepred "T")
                                    (("1"
                                      (expand "topology?")
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (expand "topology_empty?")
                                          (("1"
                                            (expand "member")
                                            (("1" (assert) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (apply-extensionality 1 :hide? t)
                            (("2" (expand "emptyset")
                              (("2"
                                (rewrite
                                 "extensionality_postulate"
                                 -5
                                 :dir
                                 rl)
                                (("2"
                                  (inst -5 "x!1")
                                  (("2"
                                    (expand "fullset")
                                    (("2" (propax) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (skosimp*)
                      (("2" (expand "continuous?")
                        (("2" (split)
                          (("1" (skosimp)
                            (("1"
                              (rewrite "extensionality_postulate" -3
                               :dir rl)
                              (("1"
                                (inst -3 "x!1")
                                (("1"
                                  (expand "emptyset")
                                  (("1"
                                    (expand "fullset")
                                    (("1" (propax) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (skosimp)
                            (("2"
                              (rewrite "extensionality_postulate" -2
                               :dir rl)
                              (("2"
                                (inst -2 "x!1")
                                (("2"
                                  (expand "emptyset")
                                  (("2"
                                    (expand "fullset")
                                    (("2" (propax) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (expand "fullset")
                  (("2" (expand "empty?")
                    (("2" (skosimp)
                      (("2" (expand "member")
                        (("2" (expand "bijective?")
                          (("2" (flatten)
                            (("2" (expand "surjective?")
                              (("2"
                                (inst -3 "x!1")
                                (("2"
                                  (skosimp)
                                  (("2" (inst - "x!2") nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assert) nil nil))
        nil))
      nil))
    nil)
   ((homeomorphism? const-decl "bool" homeomorphism_def nil)
    (fullset_is_clopen name-judgement "clopen" homeomorphism_def nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (fullset const-decl "set" sets nil)
    (TRUE const-decl "bool" booleans nil)
    (Y!1 skolem-const-decl "set[T1]" homeomorphism_def nil)
    (f!1 skolem-const-decl "[T1 -> T2]" homeomorphism_def nil)
    (x!1 skolem-const-decl "T2" homeomorphism_def nil)
    (surjective? const-decl "bool" functions nil)
    (surjective_image_inverse_image formula-decl nil function_image_bis
     "structures/")
    (S formal-const-decl "topology[T1]" homeomorphism_def nil)
    (T formal-const-decl "topology[T2]" homeomorphism_def nil)
    (topology nonempty-type-eq-decl nil topology_prelim nil)
    (topology? const-decl "bool" topology_prelim nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (continuous_open_sets formula-decl nil continuity nil)
    (member const-decl "bool" sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (image const-decl "set[R]" function_image nil)
    (inverse_image const-decl "set[D]" function_image nil)
    (extensionality formula-decl nil functions nil)
    (injective_inverse_image_image formula-decl nil function_image_bis
     "structures/")
    (injective? const-decl "bool" functions nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (FALSE const-decl "bool" booleans nil)
    (inverse_alt const-decl "inverses(f)" function_inverse_alt nil)
    (inverses nonempty-type-eq-decl nil function_inverse_alt nil)
    (inverse? const-decl "bool" function_inverse_def nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (empty? const-decl "bool" sets nil)
    (fullset_is_clopen name-judgement "clopen" homeomorphism_def nil)
    (emptyset_is_compact name-judgement "compact" homeomorphism_def
     nil)
    (emptyset_is_clopen name-judgement "clopen" homeomorphism_def nil)
    (continuous? const-decl "bool" continuity_def nil)
    (emptyset const-decl "set" sets nil)
    (open? const-decl "bool" topology nil)
    (topology_empty? const-decl "bool" topology_prelim nil)
    (image_emptyset formula-decl nil function_image_bis "structures/")
    (extensionality_postulate formula-decl nil functions nil)
    (finite_emptyset name-judgement "finite_set" finite_sets nil)
    (emptyset_is_clopen name-judgement "clopen" homeomorphism_def nil)
    (emptyset_is_compact name-judgement "compact" homeomorphism_def
     nil)
    (emptyset_is_empty? formula-decl nil sets_lemmas nil)
    (bijective? const-decl "bool" functions nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T2 formal-type-decl nil homeomorphism_def nil)
    (T1 formal-type-decl nil homeomorphism_def nil))
   shostak)))

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