Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/scott/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 218 kB image not shown  

Quellcode-Bibliothek scott_continuity.prf

  Sprache: Lisp
 

(scott_continuity
 (lub_set_image 0
  (lub_set_image-1 nil 3330742497
   ("" (skosimp)
    (("" (typepred "D!1")
      (("" (typepred "g!1")
        (("" (expand "lub_set?")
          (("" (typepred "le2")
            (("" (expand "directed_complete_partial_order?")
              (("" (flatten)
                (("" (expand "directed_complete?")
                  (("" (inst - "image(g!1,D!1)")
                    (("" (hide 2)
                      (("" (split)
                        (("1" (expand "nonempty?")
                          (("1" (expand "empty?")
                            (("1" (skosimp)
                              (("1"
                                (inst - "g!1(x!1)")
                                (("1"
                                  (expand "member")
                                  (("1"
                                    (expand "image")
                                    (("1" (inst + "x!1"nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "directed?")
                          (("2" (skosimp*)
                            (("2" (expand "image")
                              (("2"
                                (skosimp*)
                                (("2"
                                  (replace -1)
                                  (("2"
                                    (replace -2)
                                    (("2"
                                      (expand "directed?")
                                      (("2"
                                        (inst - "x!2" "x!3")
                                        (("2"
                                          (assert)
                                          (("2"
                                            (skosimp)
                                            (("2"
                                              (inst + "g!1(z!1)")
                                              (("1"
                                                (expand "increasing?")
                                                (("1"
                                                  (inst-cp
                                                   -
                                                   "x!2"
                                                   "z!1")
                                                  (("1"
                                                    (inst
                                                     -
                                                     "x!3"
                                                     "z!1")
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (expand "image")
                                                (("2"
                                                  (inst + "z!1")
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (lub_set? const-decl "bool" bounded_order_props "orders/")
    (directed_complete? const-decl "bool" directed_orders "orders/")
    (z!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (member const-decl "bool" sets nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (empty? const-decl "bool" sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (g!1 skolem-const-decl "(increasing?)" scott_continuity nil)
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (image const-decl "set[R]" function_image nil)
    (directed? const-decl "bool" directed_orders "orders/")
    (partial_order? const-decl "bool" orders nil)
    (increasing? const-decl "bool" fun_preds_partial nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil))
   shostak))
 (directed_image_TCC1 0
  (directed_image_TCC1-1 nil 3352749145
   ("" (skosimp)
    (("" (typepred "D!1")
      (("" (expand "nonempty?")
        (("" (expand "empty?")
          (("" (skosimp)
            (("" (expand "member")
              (("" (inst - "g!1(x!1)")
                (("" (expand "image") (("" (inst + "x!1"nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (image const-decl "set[R]" function_image nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (increasing? const-decl "bool" fun_preds_partial nil))
   nil))
 (directed_image 0
  (directed_image-1 nil 3352750482
   ("" (skosimp)
    (("" (typepred "D!1")
      (("" (typepred "g!1")
        (("" (expand "directed?")
          (("" (expand "directed?")
            (("" (skosimp*)
              (("" (expand "image")
                (("" (skosimp*)
                  (("" (replace -4)
                    (("" (replace -5)
                      (("" (hide -4 -5 -2)
                        (("" (inst - "x!2" "x!3")
                          (("" (assert)
                            (("" (skosimp)
                              ((""
                                (inst + "g!1(z!1)")
                                (("1"
                                  (expand "increasing?")
                                  (("1"
                                    (inst-cp - "x!2" "z!1")
                                    (("1"
                                      (inst - "x!3" "z!1")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (expand "image")
                                  (("2" (inst + "z!1"nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (g!1 skolem-const-decl "(increasing?)" scott_continuity nil)
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (z!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (image const-decl "set[R]" function_image nil)
    (directed? const-decl "bool" directed_orders "orders/")
    (increasing? const-decl "bool" fun_preds_partial nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil))
   shostak))
 (lub_image_TCC1 0
  (lub_image_TCC1-1 nil 3352749145
   ("" (skosimp)
    (("" (lemma "lub_set_image" ("g" "g!1" "D" "D!1"))
      (("" (propax) nil nil)) nil))
    nil)
   ((increasing? const-decl "bool" fun_preds_partial nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (lub_set_image formula-decl nil scott_continuity nil))
   nil))
 (lub_image 0
  (lub_image-1 nil 3352750819
   ("" (skosimp)
    (("" (typepred "D!1")
      (("" (lemma "directed_image" ("g" "g!1" "D" "D!1"))
        (("" (typepred "lub(image(g!1, D!1))")
          (("1" (expand "least_upper_bound?")
            (("1" (flatten)
              (("1" (inst - "g!1(lub(D!1))")
                (("1" (split -2)
                  (("1" (propax) nil nil)
                   ("2" (hide 2)
                    (("2" (expand "upper_bound?" 1)
                      (("2" (skosimp)
                        (("2" (typepred "r!1")
                          (("2" (expand "image")
                            (("2" (skosimp)
                              (("2"
                                (replace -1)
                                (("2"
                                  (hide -1 -2 -3)
                                  (("2"
                                    (typepred "g!1")
                                    (("2"
                                      (typepred "lub(D!1)")
                                      (("1"
                                        (expand "least_upper_bound?")
                                        (("1"
                                          (flatten)
                                          (("1"
                                            (hide -2)
                                            (("1"
                                              (expand "upper_bound?")
                                              (("1"
                                                (inst - "x!1")
                                                (("1"
                                                  (expand
                                                   "increasing?")
                                                  (("1"
                                                    (inst
                                                     -
                                                     "x!1"
                                                     "lub(D!1)")
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (typepred "le1")
                                                      (("2"
                                                        (expand
                                                         "directed_complete_partial_order?")
                                                        (("2"
                                                          (flatten)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (typepred "le1")
                                        (("2"
                                          (expand
                                           "directed_complete_partial_order?")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (typepred "le1")
                  (("2" (expand "directed_complete_partial_order?")
                    (("2" (flatten) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (typepred "le2")
            (("2" (expand "directed_complete_partial_order?")
              (("2" (flatten) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (image const-decl "set[R]" function_image nil)
    (lub const-decl
     "{x | bounded_orders[T].least_upper_bound?(x, SA, <=)}"
     bounded_order_props "orders/")
    (lub_set? const-decl "bool" bounded_order_props "orders/")
    (least_upper_bound? const-decl "bool" bounded_orders "orders/")
    (upper_bound? const-decl "bool" bounded_orders "orders/")
    (directed_image formula-decl nil scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (increasing? const-decl "bool" fun_preds_partial nil))
   shostak))
 (lub_image_increasing_TCC1 0
  (lub_image_increasing_TCC1-1 nil 3353382632
   ("" (skosimp)
    (("" (typepred "D!1")
      (("" (expand "nonempty?")
        (("" (expand "empty?")
          (("" (skosimp)
            (("" (inst - "f!1(x!1)")
              (("" (expand "member")
                (("" (expand "image") (("" (inst + "x!1"nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (empty? const-decl "bool" sets nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (image const-decl "set[R]" function_image nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (member const-decl "bool" sets nil))
   nil))
 (lub_image_increasing_TCC2 0
  (lub_image_increasing_TCC2-1 nil 3353382632
   ("" (skosimp)
    (("" (typepred "D!1")
      (("" (expand "lub_set?")
        (("" (typepred "le2")
          (("" (expand "directed_complete_partial_order?")
            (("" (flatten)
              (("" (expand "directed_complete?")
                (("" (inst - "image[T1,T2](f!1,D!1)")
                  (("" (expand "directed?" -4) (("" (propax) nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (partial_order? const-decl "bool" orders nil)
    (directed? const-decl "bool" directed_orders "orders/")
    (image const-decl "set[R]" function_image nil)
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (f!1 skolem-const-decl "[T1 -> T2]" scott_continuity nil)
    (directed_complete? const-decl "bool" directed_orders "orders/")
    (lub_set? const-decl "bool" bounded_order_props "orders/"))
   nil))
 (lub_image_increasing 0
  (lub_image_increasing-1 nil 3353382128
   ("" (skolem + ("f"))
    (("" (expand "increasing?")
      (("" (flatten)
        (("" (skolem + ("x" "y"))
          (("" (flatten)
            (("" (name "X" "lambda (z:T1): z=x OR z=y")
              (("" (case "lub_set?(X)")
                (("1" (case "lub[T1,le1](X)=y")
                  (("1" (inst - "X")
                    (("1" (replace -1)
                      (("1" (case-replace "f(x)=f(y)")
                        (("1" (typepred "le2")
                          (("1"
                            (expand "directed_complete_partial_order?")
                            (("1" (expand "partial_order?")
                              (("1"
                                (expand "preorder?")
                                (("1"
                                  (flatten)
                                  (("1"
                                    (expand "reflexive?")
                                    (("1" (inst - "f(y)"nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2"
                          (case-replace
                           "image(f,X) = LAMBDA (z:T2): z = f(x) OR z = f(y)")
                          (("1"
                            (name-replace "Y"
                             "LAMBDA (z: T2): z = f(x) OR z = f(y)")
                            (("1" (flatten)
                              (("1"
                                (case "lub_set?(Y)")
                                (("1"
                                  (case
                                   "lub[T2,le2](Y)=f(x) OR lub[T2,le2](Y)=f(y)")
                                  (("1"
                                    (split -1)
                                    (("1" (assertnil nil)
                                     ("2"
                                      (typepred "lub[T2, le2](Y)")
                                      (("2"
                                        (expand "least_upper_bound?")
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (expand "upper_bound?")
                                            (("2"
                                              (inst - "f(x)")
                                              (("1" (assertnil nil)
                                               ("2"
                                                (expand "Y")
                                                (("2"
                                                  (propax)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (flatten)
                                    (("2"
                                      (typepred "lub[T2, le2](Y)")
                                      (("2"
                                        (expand "least_upper_bound?")
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (expand "upper_bound?")
                                            (("2"
                                              (inst - "f(y)")
                                              (("1"
                                                (typepred "le2")
                                                (("1"
                                                  (expand
                                                   "directed_complete_partial_order?")
                                                  (("1"
                                                    (expand
                                                     "partial_order?")
                                                    (("1"
                                                      (expand
                                                       "antisymmetric?")
                                                      (("1"
                                                        (flatten)
                                                        (("1"
                                                          (inst
                                                           -
                                                           "lub[T2, le2](Y)"
                                                           "f(y)")
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (expand "Y")
                                                (("2"
                                                  (propax)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("3"
                                    (assert)
                                    (("3"
                                      (typepred "le2")
                                      (("3"
                                        (expand
                                         "directed_complete_partial_order?")
                                        (("3" (propax) nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (expand "lub_set?")
                                  (("2"
                                    (typepred "le2")
                                    (("2"
                                      (expand
                                       "directed_complete_partial_order?")
                                      (("2"
                                        (expand "directed_complete?")
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (inst - "Y")
                                            (("2"
                                              (expand "directed?")
                                              (("2"
                                                (skosimp)
                                                (("2"
                                                  (expand "directed?")
                                                  (("2"
                                                    (inst
                                                     -8
                                                     "x!1"
                                                     "y!1")
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (typepred "le2")
                                  (("3"
                                    (expand
                                     "directed_complete_partial_order?")
                                    (("3" (flatten) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (apply-extensionality 1 :hide? t)
                            (("2" (expand "image")
                              (("2"
                                (hide-all-but 1)
                                (("2"
                                  (case-replace
                                   "EXISTS (x: (X)): x!1 = f(x)")
                                  (("1"
                                    (skosimp)
                                    (("1"
                                      (typepred "x!2")
                                      (("1"
                                        (expand "X")
                                        (("1"
                                          (split -1)
                                          (("1" (assertnil nil)
                                           ("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (replace 1 2)
                                    (("2"
                                      (assert)
                                      (("2"
                                        (split -1)
                                        (("1"
                                          (inst + "x")
                                          (("1"
                                            (expand "X")
                                            (("1" (propax) nil nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (inst + "y")
                                          (("2"
                                            (expand "X")
                                            (("2" (propax) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide 2 -3)
                      (("2" (split)
                        (("1" (expand "nonempty?")
                          (("1" (expand "empty?")
                            (("1" (inst - "x")
                              (("1"
                                (expand "member")
                                (("1"
                                  (expand "X")
                                  (("1" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "directed?")
                          (("2" (expand "directed?")
                            (("2" (skosimp)
                              (("2"
                                (typepred "lub[T1, le1](X)")
                                (("2"
                                  (replace -4)
                                  (("2"
                                    (expand "least_upper_bound?")
                                    (("2"
                                      (flatten)
                                      (("2"
                                        (expand "upper_bound?")
                                        (("2"
                                          (inst + "y")
                                          (("1"
                                            (inst-cp - "x!1")
                                            (("1"
                                              (inst - "y!1")
                                              (("1" (assertnil nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (expand "X")
                                            (("2" (propax) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide -3 2)
                    (("2" (typepred "lub[T1, le1](X)")
                      (("2" (expand "least_upper_bound?")
                        (("2" (flatten)
                          (("2" (expand "upper_bound?")
                            (("2" (inst -2 "y")
                              (("2"
                                (inst -1 "y")
                                (("1"
                                  (split -2)
                                  (("1"
                                    (typepred "le1")
                                    (("1"
                                      (expand
                                       "directed_complete_partial_order?")
                                      (("1"
                                        (expand "partial_order?")
                                        (("1"
                                          (flatten)
                                          (("1"
                                            (expand "antisymmetric?")
                                            (("1"
                                              (inst
                                               -
                                               "lub[T1, le1](X)"
                                               "y")
                                              (("1" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (skosimp)
                                    (("2"
                                      (typepred "r!1")
                                      (("2"
                                        (expand "X")
                                        (("2"
                                          (split -1)
                                          (("1" (assertnil nil)
                                           ("2"
                                            (typepred "le1")
                                            (("2"
                                              (expand
                                               "directed_complete_partial_order?")
                                              (("2"
                                                (expand
                                                 "partial_order?")
                                                (("2"
                                                  (expand "preorder?")
                                                  (("2"
                                                    (flatten)
                                                    (("2"
                                                      (expand
                                                       "reflexive?")
                                                      (("2"
                                                        (inst - "y")
                                                        (("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (expand "X")
                                  (("2" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("3" (propax) nil nil))
                  nil)
                 ("2" (hide 2 -2 -1)
                  (("2" (expand "lub_set?")
                    (("2" (expand "least_bounded_above?")
                      (("2" (inst + "y")
                        (("2" (expand "least_upper_bound?")
                          (("2" (expand "upper_bound?")
                            (("2" (split)
                              (("1"
                                (skosimp)
                                (("1"
                                  (typepred "r!1")
                                  (("1"
                                    (expand "X")
                                    (("1"
                                      (split -1)
                                      (("1" (assertnil nil)
                                       ("2"
                                        (typepred "le1")
                                        (("2"
                                          (expand
                                           "directed_complete_partial_order?")
                                          (("2"
                                            (expand "partial_order?")
                                            (("2"
                                              (expand "preorder?")
                                              (("2"
                                                (flatten)
                                                (("2"
                                                  (expand "reflexive?")
                                                  (("2"
                                                    (inst - "y")
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (skosimp*)
                                (("2"
                                  (inst - "y")
                                  (("2"
                                    (expand "X")
                                    (("2" (propax) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("3" (typepred "le1")
                  (("3" (expand "directed_complete_partial_order?")
                    (("3" (flatten) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((increasing? const-decl "bool" fun_preds_partial nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (least_bounded_above? const-decl "bool" bounded_orders "orders/")
    (lub const-decl
     "{x | bounded_orders[T].least_upper_bound?(x, SA, <=)}"
     bounded_order_props "orders/")
    (least_upper_bound? const-decl "bool" bounded_orders "orders/")
    (y!1 skolem-const-decl "T1" scott_continuity nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (member const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (image const-decl "set[R]" function_image nil)
    (directed? const-decl "bool" directed_orders "orders/")
    (directed_complete? const-decl "bool" directed_orders "orders/")
    (Y skolem-const-decl "[T2 -> boolean]" scott_continuity nil)
    (f skolem-const-decl "[T1 -> T2]" scott_continuity nil)
    (x skolem-const-decl "T1" scott_continuity nil)
    (upper_bound? const-decl "bool" bounded_orders "orders/")
    (y skolem-const-decl "T1" scott_continuity nil)
    (antisymmetric? const-decl "bool" relations nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (partial_order? const-decl "bool" orders nil)
    (reflexive? const-decl "bool" relations nil)
    (preorder? const-decl "bool" orders nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (X skolem-const-decl "[T1 -> boolean]" scott_continuity nil)
    (nonempty? const-decl "bool" sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (partial_order_antisymmetric formula-decl nil partial_order_props
     "orders/")
    (partial_order_reflexive formula-decl nil partial_order_props
     "orders/")
    (set type-eq-decl nil sets nil)
    (pred type-eq-decl nil defined_types nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (lub_set? const-decl "bool" bounded_order_props "orders/"))
   shostak))
 (scott_continuous_def 0
  (scott_continuous_def-1 nil 3330155115
   ("" (skosimp)
    (("" (expand "scott_continuous?")
      (("" (split)
        (("1" (flatten)
          (("1" (case-replace "increasing?(f!1)")
            (("1" (expand "lub_preserving?")
              (("1" (skosimp)
                (("1" (typepred "D!1")
                  (("1" (lemma "directed_image" ("g" "f!1" "D" "D!1"))
                    (("1"
                      (case "nonempty?[T2](image[T1, T2](f!1,D!1))")
                      (("1"
                        (lemma "closure_is_down"
                         ("x" "lub[T2, le2](image(f!1, D!1))"))
                        (("1" (name "FD" "image(f!1, D!1)")
                          (("1" (replace -1)
                            (("1" (name "JFD" "lub[T2, le2](FD)")
                              (("1"
                                (replace -1)
                                (("1"
                                  (lemma
                                   "down_is_closed[T2,le2]"
                                   ("x" "JFD"))
                                  (("1"
                                    (expand "member")
                                    (("1"
                                      (rewrite
                                       "continuous_closed_sets")
                                      (("1"
                                        (inst - "down(JFD)")
                                        (("1"
                                          (expand "closed?")
                                          (("1"
                                            (expand "scott_open_sets")
                                            (("1"
                                              (expand "Complement")
                                              (("1"
                                                (expand "member")
                                                (("1"
                                                  (split -10)
                                                  (("1"
                                                    (skosimp)
                                                    (("1"
                                                      (typepred "b!1")
                                                      (("1"
                                                        (rewrite
                                                         "complement_equal")
                                                        (("1"
                                                          (case
                                                           "subset?(D!1,b!1)")
                                                          (("1"
                                                            (expand
                                                             "scott_closed_sets")
                                                            (("1"
                                                              (expand
                                                               "extend")
                                                              (("1"
                                                                (prop)
                                                                (("1"
                                                                  (inst
                                                                   -
                                                                   "D!1")
                                                                  (("1"
                                                                    (assert)
                                                                    (("1"
                                                                      (typepred
                                                                       "lub[T2,le2](FD)")
                                                                      (("1"
                                                                        (replace
                                                                         -7)
                                                                        (("1"
                                                                          (typepred
                                                                           "lub(D!1)")
                                                                          (("1"
                                                                            (name
                                                                             "JD"
                                                                             "lub(D!1)")
                                                                            (("1"
                                                                              (replace
                                                                               -1)
                                                                              (("1"
                                                                                (expand
                                                                                 "least_upper_bound?")
                                                                                (("1"
                                                                                  (flatten)
                                                                                  (("1"
                                                                                    (expand
                                                                                     "increasing?")
                                                                                    (("1"
                                                                                      (inst
                                                                                       -5
                                                                                       "f!1(JD)")
                                                                                      (("1"
                                                                                        (split
                                                                                         -5)
                                                                                        (("1"
                                                                                          (expand
                                                                                           "inverse_image")
                                                                                          (("1"
                                                                                            (expand
                                                                                             "member")
                                                                                            (("1"
                                                                                              (replace
                                                                                               -9
                                                                                               -7
                                                                                               rl)
                                                                                              (("1"
                                                                                                (assert)
                                                                                                (("1"
                                                                                                  (expand
                                                                                                   "down")
                                                                                                  (("1"
                                                                                                    (lemma
                                                                                                     "partial_order_antisymmetric"
                                                                                                     ("x"
                                                                                                      "f!1(JD)"
                                                                                                      "y"
                                                                                                      "JFD"))
                                                                                                    (("1"
                                                                                                      (assert)
                                                                                                      nil
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil)
                                                                                         ("2"
                                                                                          (expand
                                                                                           "upper_bound?")
                                                                                          (("2"
                                                                                            (skosimp)
                                                                                            (("2"
                                                                                              (typepred
                                                                                               "r!1")
                                                                                              (("2"
                                                                                                (replace
                                                                                                 -12
                                                                                                 -1
                                                                                                 rl)
                                                                                                (("2"
                                                                                                  (expand
                                                                                                   "image")
                                                                                                  (("2"
                                                                                                    (skosimp)
                                                                                                    (("2"
                                                                                                      (replace
                                                                                                       -1)
                                                                                                      (("2"
                                                                                                        (inst
                                                                                                         -18
                                                                                                         "x!1"
                                                                                                         "JD")
                                                                                                        (("2"
                                                                                                          (inst
                                                                                                           -3
                                                                                                           "x!1")
                                                                                                          (("2"
                                                                                                            (assert)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (typepred
                                                                             "le1")
                                                                            (("2"
                                                                              (expand
                                                                               "directed_complete_partial_order?")
                                                                              (("2"
                                                                                (flatten)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (expand
                                                             "subset?")
                                                            (("2"
                                                              (skosimp)
                                                              (("2"
                                                                (expand
                                                                 "member")
                                                                (("2"
                                                                  (replace
                                                                   -3
                                                                   1
                                                                   rl)
                                                                  (("2"
                                                                    (expand
                                                                     "inverse_image")
                                                                    (("2"
                                                                      (expand
                                                                       "member")
                                                                      (("2"
                                                                        (expand
                                                                         "down")
                                                                        (("2"
                                                                          (expand
                                                                           "JFD")
                                                                          (("2"
                                                                            (typepred
                                                                             "lub[T2, le2](FD)")
                                                                            (("2"
                                                                              (expand
                                                                               "least_upper_bound?")
                                                                              (("2"
                                                                                (flatten)
                                                                                (("2"
                                                                                  (expand
                                                                                   "upper_bound?")
                                                                                  (("2"
                                                                                    (inst
                                                                                     -
                                                                                     "f!1(x!1)")
                                                                                    (("2"
                                                                                      (expand
                                                                                       "FD")
                                                                                      (("2"
                                                                                        (expand
                                                                                         "image")
                                                                                        (("2"
                                                                                          (inst
                                                                                           +
                                                                                           "x!1")
                                                                                          nil
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (inst
                                                     +
                                                     "down(JFD)")
                                                    (("2"
                                                      (typepred "le2")
                                                      (("2"
                                                        (expand
                                                         "directed_complete_partial_order?")
                                                        (("2"
                                                          (flatten)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (typepred "le2")
                                          (("2"
                                            (expand
                                             "directed_complete_partial_order?")
                                            (("2" (flatten) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (typepred "le2")
                          (("2"
                            (expand "directed_complete_partial_order?")
                            (("2" (flatten) nil nil)) nil))
                          nil))
                        nil)
                       ("2" (expand "nonempty?")
                        (("2" (expand "empty?")
                          (("2" (skosimp)
                            (("2" (expand "member")
                              (("2"
                                (inst - "f!1(x!1)")
                                (("2"
                                  (expand "image")
                                  (("2" (inst + "x!1"nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (propax) nil nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide 2)
              (("2" (expand "increasing?")
                (("2" (skosimp)
                  (("2"
                    (case "inverse_image(f!1, Cl(singleton(f!1(y!1))))(y!1)")
                    (("1"
                      (lemma "Cl_closed" ("A" "singleton(f!1(y!1))"))
                      (("1" (rewrite "continuous_closed_sets")
                        (("1" (rewrite "order_iff_closure" 1)
                          (("1" (inst - "Cl(singleton(f!1(y!1)))")
                            (("1" (assert)
                              (("1"
                                (expand "closed?" -4)
                                (("1"
                                  (expand "scott_open_sets")
                                  (("1"
                                    (expand "scott_closed_sets")
                                    (("1"
                                      (expand "member")
                                      (("1"
                                        (expand "extend")
                                        (("1"
                                          (expand "Complement")
                                          (("1"
                                            (skosimp)
                                            (("1"
                                              (typepred "b!1")
                                              (("1"
                                                (rewrite
                                                 "complement_equal")
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (expand
                                                     "inverse_image")
                                                    (("1"
                                                      (expand "member")
                                                      (("1"
                                                        (expand
                                                         "lower_set?")
                                                        (("1"
                                                          (expand
                                                           "member")
                                                          (("1"
                                                            (inst
                                                             -2
                                                             "y!1"
                                                             "x!1")
                                                            (("1"
                                                              (replace
                                                               -6
                                                               -2
                                                               rl)
                                                              (("1"
                                                                (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" (expand "inverse_image")
                      (("2" (expand "member")
                        (("2"
                          (lemma "subset_of_Cl"
                           ("A" "singleton(f!1(y!1))"))
                          (("2" (expand "subset?")
                            (("2" (inst - "f!1(y!1)")
                              (("2"
                                (expand "member")
                                (("2"
                                  (expand "singleton")
                                  (("2" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("3" (typepred "le2")
              (("3" (expand "directed_complete_partial_order?")
                (("3" (flatten) nil nil)) nil))
              nil)
             ("4" (typepred "le1")
              (("4" (expand "directed_complete_partial_order?")
                (("4" (flatten) nil nil)) nil))
              nil))
            nil))
          nil)
         ("2" (flatten)
          (("2" (rewrite "continuous_open_sets")
            (("2" (skosimp)
              (("2" (case "upper_set?(inverse_image(f!1,Y!1))")
                (("1" (expand "open?")
                  (("1" (expand "member")
                    (("1" (expand "scott_open_sets")
                      (("1" (expand "Complement")
                        (("1" (skosimp)
                          (("1" (typepred "b!1")
                            (("1" (expand "scott_closed_sets")
                              (("1"
                                (expand "extend")
                                (("1"
                                  (prop)
                                  (("1"
                                    (name
                                     "X"
                                     "inverse_image(f!1, Y!1)")
                                    (("1"
                                      (replace -1)
                                      (("1"
                                        (inst + "complement(X)")
                                        (("1"
                                          (rewrite
                                           "complement_complement")
                                          nil
                                          nil)
                                         ("2"
                                          (expand "scott_closed_sets")
                                          (("2"
                                            (expand "extend")
                                            (("2"
                                              (lemma
                                               "complement_upper_set"
                                               ("U" "X"))
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (skosimp)
                                                  (("1"
                                                    (typepred "D!1")
                                                    (("1"
                                                      (replace -5 * rl)
                                                      (("1"
                                                        (expand
                                                         "complement")
                                                        (("1"
                                                          (expand
                                                           "member")
                                                          (("1"
                                                            (replace
                                                             -12)
                                                            (("1"
                                                              (typepred
                                                               "lub(D!1)")
                                                              (("1"
                                                                (inst
                                                                 -9
                                                                 "image(f!1,D!1)")
                                                                (("1"
                                                                  (expand
                                                                   "inverse_image")
                                                                  (("1"
                                                                    (expand
                                                                     "member")
                                                                    (("1"
                                                                      (expand
                                                                       "lub_preserving?")
                                                                      (("1"
                                                                        (inst
                                                                         -11
                                                                         "D!1")
                                                                        (("1"
                                                                          (split
                                                                           -11)
                                                                          (("1"
                                                                            (split
                                                                             -9)
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil)
                                                                             ("2"
                                                                              (expand
                                                                               "subset?")
                                                                              (("2"
                                                                                (skosimp)
                                                                                (("2"
                                                                                  (expand
                                                                                   "member")
                                                                                  (("2"
                                                                                    (expand
                                                                                     "image")
                                                                                    (("2"
                                                                                      (skosimp)
                                                                                      (("2"
                                                                                        (typepred
                                                                                         "x!2")
                                                                                        (("2"
                                                                                          (expand
                                                                                           "lower_set?")
                                                                                          (("2"
                                                                                            (inst
                                                                                             -8
                                                                                             "x!2")
                                                                                            (("2"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (expand
                                                                             "lub_set?")
                                                                            (("2"
                                                                              (lemma
                                                                               "directed_image"
                                                                               ("g"
                                                                                "f!1"
                                                                                "D"
                                                                                "D!1"))
                                                                              (("2"
                                                                                (typepred
                                                                                 "le2")
                                                                                (("2"
                                                                                  (expand
                                                                                   "directed_complete_partial_order?")
                                                                                  (("2"
                                                                                    (flatten)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "directed_complete?")
                                                                                      (("2"
                                                                                        (inst
                                                                                         -
                                                                                         "image(f!1, D!1)")
                                                                                        (("2"
                                                                                          (expand
                                                                                           "directed?"
                                                                                           -2)
                                                                                          (("2"
                                                                                            (propax)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (lemma
                                                                   "directed_image"
                                                                   ("g"
                                                                    "f!1"
                                                                    "D"
                                                                    "D!1"))
                                                                  (("2"
                                                                    (assert)
                                                                    (("2"
                                                                      (hide-all-but
                                                                       (-3
                                                                        1))
                                                                      (("2"
                                                                        (expand
                                                                         "nonempty?")
                                                                        (("2"
                                                                          (expand
                                                                           "empty?")
                                                                          (("2"
                                                                            (skosimp)
                                                                            (("2"
                                                                              (inst
                                                                               -
                                                                               "f!1(x!1)")
                                                                              (("2"
                                                                                (expand
                                                                                 "member")
                                                                                (("2"
                                                                                  (expand
                                                                                   "image")
                                                                                  (("2"
                                                                                    (inst
                                                                                     +
                                                                                     "x!1")
                                                                                    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))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide 2)
                  (("2" (expand "open?")
                    (("2" (expand "member")
                      (("2" (expand "scott_open_sets")
                        (("2" (expand "Complement")
                          (("2" (skosimp)
                            (("2" (typepred "b!1")
                              (("2"
                                (expand "scott_closed_sets")
                                (("2"
                                  (expand "extend")
                                  (("2"
                                    (prop)
                                    (("2"
                                      (expand "upper_set?")
                                      (("2"
                                        (expand "member")
                                        (("2"
                                          (expand "inverse_image")
                                          (("2"
                                            (skosimp)
                                            (("2"
                                              (expand "member")
                                              (("2"
                                                (replace -7)
                                                (("2"
                                                  (expand "complement")
                                                  (("2"
                                                    (expand "member")
                                                    (("2"
                                                      (expand
                                                       "lower_set?")
                                                      (("2"
                                                        (expand
                                                         "member")
                                                        (("2"
                                                          (inst
                                                           -
                                                           "f!1(y!1)"
                                                           "f!1(x!1)")
                                                          (("2"
                                                            (expand
                                                             "increasing?")
                                                            (("2"
                                                              (inst
                                                               -
                                                               "x!1"
                                                               "y!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)
                 ("3" (typepred "le1")
                  (("3" (expand "directed_complete_partial_order?")
                    (("3" (flatten) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((scott_continuous? const-decl "bool" scott_continuity nil)
    (open? const-decl "bool" topology "topology/")
    (complement_upper_set judgement-tcc nil partial_order_props
     "orders/")
    (upper_set nonempty-type-eq-decl nil partial_order_props "orders/")
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (directed_complete? const-decl "bool" directed_orders "orders/")
    (partial_order? const-decl "bool" orders nil)
    (directed? const-decl "bool" directed_orders "orders/")
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (complement_complement formula-decl nil sets_lemmas nil)
    (X skolem-const-decl "set[T1]" scott_continuity nil)
    (complement const-decl "set" sets nil)
    (upper_set? const-decl "bool" partial_order_props "orders/")
    (continuous_open_sets formula-decl nil continuity "topology/")
    (subset_of_Cl formula-decl nil topology "topology/")
    (Cl_closed formula-decl nil topology "topology/")
    (order_iff_closure formula-decl nil scott nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (FALSE const-decl "bool" booleans nil)
    (directed_singleton application-judgement "directed[T, <=]"
     scott_continuity nil)
    (scott_open_sets_is_topology name-judgement "topology"
     scott_continuity nil)
    (Cl const-decl "set[T]" topology "topology/")
    (singleton? const-decl "bool" sets nil)
    (singleton const-decl "(singleton?)" sets nil)
    (lub_preserving? const-decl "bool" scott_continuity nil)
    (directed type-eq-decl nil directed_order_props "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (image const-decl "set[R]" function_image nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (down_is_closed formula-decl nil scott nil)
    (scott_open_sets const-decl "setofsets[T]" scott nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (continuous_closed_sets formula-decl nil continuity "topology/")
    (closed? const-decl "bool" topology "topology/")
    (Complement const-decl "setofsets[T]" sets_lemmas nil)
    (scott_closed_sets const-decl "setofsets[T]" scott nil)
    (subset? const-decl "bool" sets nil)
    (extend const-decl "R" extend nil)
    (upper_bound? const-decl "bool" bounded_orders "orders/")
    (partial_order_antisymmetric formula-decl nil partial_order_props
     "orders/")
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (Cl_is_closed application-judgement "closed[T, scott_open_sets]"
     scott_continuity nil)
    (singleton_is_compact application-judgement
     "compact[T, scott_open_sets]" scott_continuity nil)
    (nonempty_singleton_finite application-judgement
     "non_empty_finite_set[T]" scott_continuity nil)
    (JFD skolem-const-decl
     "{x | bounded_orders[T2].least_upper_bound?(x, FD, le2)}"
     scott_continuity nil)
    (D!1 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (f!1 skolem-const-decl "[T1 -> T2]" scott_continuity nil)
    (FD skolem-const-decl "set[T2]" scott_continuity nil)
    (complement_equal formula-decl nil sets_lemmas nil)
    (inverse_image const-decl "set[D]" function_image nil)
    (lower_set? const-decl "bool" partial_order_props "orders/")
    (lower_set nonempty-type-eq-decl nil partial_order_props "orders/")
    (down const-decl "lower_set" partial_order_props "orders/")
    (member const-decl "bool" sets nil)
    (lub const-decl
     "{x | bounded_orders[T].least_upper_bound?(x, SA, <=)}"
     bounded_order_props "orders/")
    (least_upper_bound? const-decl "bool" bounded_orders "orders/")
    (lub_set? const-decl "bool" bounded_order_props "orders/")
    (closure_is_down formula-decl nil scott nil)
    (empty? const-decl "bool" sets nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (directed_image formula-decl nil scott_continuity nil)
    (increasing? const-decl "bool" fun_preds_partial nil)
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (T1 formal-type-decl nil scott_continuity nil))
   shostak))
 (scott_continuous_is_continuous 0
  (scott_continuous_is_continuous-1 nil 3330316834
   ("" (skosimp)
    (("" (typepred "x!1")
      (("" (rewrite "scott_continuous_def"nil nil)) nil))
    nil)
   ((scott_continuous? const-decl "bool" scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (scott_continuous_def formula-decl nil scott_continuity nil))
   shostak))
 (continuous_is_scott_continuous 0
  (continuous_is_scott_continuous-1 nil 3330316859
   ("" (skosimp)
    (("" (typepred "x!1")
      (("" (rewrite "scott_continuous_def"nil nil)) nil))
    nil)
   ((continuous? const-decl "bool" continuity_def "topology/")
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (scott_open_sets const-decl "setofsets[T]" scott nil)
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (scott_continuous_def formula-decl nil scott_continuity nil))
   shostak))
 (continuous_is_increasing 0
  (continuous_is_increasing-1 nil 3330547445
   ("" (skosimp)
    (("" (typepred "x!1")
      (("" (rewrite "scott_continuous_def")
        (("" (expand "scott_continuous?") (("" (flatten) nil nil))
          nil))
        nil))
      nil))
    nil)
   ((continuous? const-decl "bool" continuity_def "topology/")
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (scott_open_sets const-decl "setofsets[T]" scott nil)
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (scott_continuous? const-decl "bool" scott_continuity nil)
    (scott_continuous_def formula-decl nil scott_continuity nil))
   shostak))
 (continuous_is_lub_preserving 0
  (continuous_is_lub_preserving-1 nil 3330934809
   ("" (skosimp)
    (("" (typepred "x!1")
      (("" (rewrite "scott_continuous_def")
        (("" (expand "scott_continuous?") (("" (flatten) nil nil))
          nil))
        nil))
      nil))
    nil)
   ((continuous? const-decl "bool" continuity_def "topology/")
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (scott_open_sets const-decl "setofsets[T]" scott nil)
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pred type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (scott_continuous? const-decl "bool" scott_continuity nil)
    (scott_continuous_def formula-decl nil scott_continuity nil))
   shostak))
 (pointwise_complete 0
  (pointwise_complete-1 nil 3352784732
   ("" (expand "restrict")
    (("" (expand "directed_complete?")
      (("" (skosimp)
        (("" (typepred "D!1")
          (("" (expand "pointwise")
            (("" (expand "least_bounded_above?")
              ((""
                (case "FORALL (x: T1): lub_set?[T2, le2]({y: T2 | EXISTS (f: (D!1)): y=f(x)})")
                (("1"
                  (name "F"
                        "LAMBDA (x:T1): lub[T2,le2]({y: T2 | EXISTS (f: (D!1)): y=f(x)})")
                  (("1" (case "increasing?(F)")
                    (("1" (lemma "directed_is_lub_set[T1,le1]")
                      (("1" (lemma "lub_set_image" ("g" "F"))
                        (("1"
                          (name "FX"
                                "LAMBDA (D:directed[T1, le1]): {y: T2 | EXISTS (f:(D!1),x:(D)): y = f(x)}")
                          (("1"
                            (case "forall (D: directed[T1, le1]): nonempty?[T2](FX(D)) AND directed?[T2](le2)(FX(D))")
                            (("1"
                              (case "forall (D: directed[T1, le1]): lub_set?(FX(D))")
                              (("1"
                                (case
                                 "forall (D:directed[T1, le1]): F(lub(D)) = lub(FX(D))")
                                (("1"
                                  (case
                                   "FORALL (D: directed[T1, le1]): lub(image[T1,T2](F,D)) = lub(FX(D))")
                                  (("1"
                                    (case "lub_preserving?(F)")
                                    (("1"
                                      (case "scott_continuous?(F)")
                                      (("1"
                                        (inst + "F")
                                        (("1"
                                          (expand "least_upper_bound?")
                                          (("1"
                                            (split 1)
                                            (("1"
                                              (expand "upper_bound?")
                                              (("1"
                                                (skosimp)
                                                (("1"
                                                  (skosimp)
                                                  (("1"
                                                    (typepred "r!1")
                                                    (("1"
                                                      (inst -14 "x!1")
                                                      (("1"
                                                        (typepred
                                                         "lub({y: T2 | EXISTS (f: (D!1)): y = f(x!1)})")
                                                        (("1"
                                                          (expand
                                                           "least_upper_bound?")
                                                          (("1"
                                                            (flatten)
                                                            (("1"
                                                              (expand
                                                               "upper_bound?")
                                                              (("1"
                                                                (inst
                                                                 -
                                                                 "r!1(x!1)")
                                                                (("1"
                                                                  (expand
                                                                   "F")
                                                                  (("1"
                                                                    (propax)
                                                                    nil
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (inst
                                                                   +
                                                                   "r!1")
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (skosimp)
                                              (("2"
                                                (skosimp)
                                                (("2"
                                                  (typepred "r!1")
                                                  (("2"
                                                    (expand
                                                     "upper_bound?")
                                                    (("2"
                                                      (inst -14 "x!1")
                                                      (("2"
                                                        (typepred
                                                         "lub({y: T2 | EXISTS (f: (D!1)): y = f(x!1)})")
                                                        (("1"
                                                          (expand
                                                           "least_upper_bound?")
                                                          (("1"
                                                            (flatten)
                                                            (("1"
                                                              (inst
                                                               -
                                                               "r!1(x!1)")
                                                              (("1"
                                                                (split
                                                                 -2)
                                                                (("1"
                                                                  (expand
                                                                   "F")
                                                                  (("1"
                                                                    (propax)
                                                                    nil
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (hide
                                                                   2)
                                                                  (("2"
                                                                    (expand
                                                                     "upper_bound?")
                                                                    (("2"
                                                                      (skosimp)
                                                                      (("2"
                                                                        (typepred
                                                                         "r!2")
                                                                        (("2"
                                                                          (skosimp)
                                                                          (("2"
                                                                            (replace
                                                                             -1)
                                                                            (("2"
                                                                              (inst
                                                                               -4
                                                                               "f!1")
                                                                              (("2"
                                                                                (inst
                                                                                 -4
                                                                                 "x!1")
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (expand "scott_continuous?")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide 2)
                                      (("2"
                                        (expand "lub_preserving?")
                                        (("2"
                                          (skosimp)
                                          (("2"
                                            (inst - "D!2")
                                            (("2"
                                              (inst - "D!2")
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2 -1)
                                    (("2"
                                      (skosimp)
                                      (("2"
                                        (lemma
                                         "lub_set_image"
                                         ("g" "F" "D" "D!2"))
                                        (("2"
                                          (inst - "D!2")
                                          (("2"
                                            (typepred
                                             "lub(image[T1, T2](F, D!2))")
                                            (("2"
                                              (typepred "lub(FX(D!2))")
                                              (("2"
                                                (expand
                                                 "least_upper_bound?")
                                                (("2"
                                                  (flatten)
                                                  (("2"
                                                    (inst
                                                     -
                                                     "lub(image[T1, T2](F, D!2))")
                                                    (("1"
                                                      (split -2)
                                                      (("1"
                                                        (inst
                                                         -4
                                                         "lub(FX(D!2))")
                                                        (("1"
                                                          (split -4)
                                                          (("1"
                                                            (lemma
                                                             "partial_order_antisymmetric"
                                                             ("x"
                                                              "lub(image[T1, T2](F, D!2))"
                                                              "y"
                                                              "lub(FX(D!2))"))
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (hide 2 -1)
                                                            (("2"
                                                              (expand
                                                               "upper_bound?")
                                                              (("2"
                                                                (skosimp)
                                                                (("2"
                                                                  (typepred
                                                                   "r!1")
                                                                  (("2"
                                                                    (expand
                                                                     "image")
                                                                    (("2"
                                                                      (skosimp)
                                                                      (("2"
                                                                        (replace
                                                                         -1)
                                                                        (("2"
                                                                          (inst
                                                                           -12
                                                                           "x!1")
                                                                          (("2"
                                                                            (typepred
                                                                             "lub({y: T2 | EXISTS (f: (D!1)): y = f(x!1)})")
                                                                            (("1"
                                                                              (expand
                                                                               "least_upper_bound?")
                                                                              (("1"
                                                                                (flatten)
                                                                                (("1"
                                                                                  (inst
                                                                                   -
                                                                                   "lub(FX(D!2))")
                                                                                  (("1"
                                                                                    (split
                                                                                     -2)
                                                                                    (("1"
                                                                                      (expand
                                                                                       "F")
                                                                                      (("1"
                                                                                        (propax)
                                                                                        nil
                                                                                        nil))
                                                                                      nil)
                                                                                     ("2"
                                                                                      (hide
                                                                                       2)
                                                                                      (("2"
                                                                                        (expand
                                                                                         "upper_bound?")
                                                                                        (("2"
                                                                                          (skosimp)
                                                                                          (("2"
                                                                                            (typepred
                                                                                             "r!2")
                                                                                            (("2"
                                                                                              (skosimp)
                                                                                              (("2"
                                                                                                (replace
                                                                                                 -1)
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -4
                                                                                                   "f!1(x!1)")
                                                                                                  (("2"
                                                                                                    (expand
                                                                                                     "FX")
                                                                                                    (("2"
                                                                                                      (inst
                                                                                                       +
                                                                                                       "f!1"
                                                                                                       "x!1")
                                                                                                      nil
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil)
                                                                                   ("2"
                                                                                    (typepred
                                                                                     "le2")
                                                                                    (("2"
                                                                                      (expand
                                                                                       "directed_complete_partial_order?")
                                                                                      (("2"
                                                                                        (flatten)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil)
                                                                             ("2"
                                                                              (propax)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (typepred
                                                           "le2")
                                                          (("2"
                                                            (expand
                                                             "directed_complete_partial_order?")
                                                            (("2"
                                                              (flatten)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide 2)
                                                        (("2"
                                                          (expand
                                                           "upper_bound?")
                                                          (("2"
                                                            (skosimp)
                                                            (("2"
                                                              (typepred
                                                               "r!1")
                                                              (("2"
                                                                (expand
                                                                 "FX"
                                                                 -1)
                                                                (("2"
                                                                  (skosimp)
                                                                  (("2"
                                                                    (replace
                                                                     -1)
                                                                    (("2"
                                                                      (inst
                                                                       -3
                                                                       "F(x!1)")
                                                                      (("1"
                                                                        (hide-all-but
                                                                         (1
                                                                          -3
                                                                          -13))
                                                                        (("1"
                                                                          (inst
                                                                           -
                                                                           "x!1")
                                                                          (("1"
                                                                            (name-replace
                                                                             "DRL100"
                                                                             "lub(image[T1, T2](F, D!2))")
                                                                            (("1"
                                                                              (expand
                                                                               "F")
                                                                              (("1"
                                                                                (typepred
                                                                                 "lub({y: T2 | EXISTS (f: (D!1)): y = f(x!1)})")
                                                                                (("1"
                                                                                  (expand
                                                                                   "least_upper_bound?")
                                                                                  (("1"
                                                                                    (flatten)
                                                                                    (("1"
                                                                                      (hide
                                                                                       -2)
                                                                                      (("1"
                                                                                        (expand
                                                                                         "upper_bound?")
                                                                                        (("1"
                                                                                          (inst
                                                                                           -
                                                                                           "f!1(x!1)")
                                                                                          (("1"
                                                                                            (lemma
                                                                                             "partial_order_transitive"
                                                                                             ("x"
                                                                                              "f!1(x!1)"
                                                                                              "y"
                                                                                              "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(x!1)})"
                                                                                              "z"
                                                                                              "DRL100"))
                                                                                            (("1"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil))
                                                                                            nil)
                                                                                           ("2"
                                                                                            (inst
                                                                                             +
                                                                                             "f!1")
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (propax)
                                                                                  nil
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil)
                                                                       ("2"
                                                                        (expand
                                                                         "image")
                                                                        (("2"
                                                                          (inst
                                                                           +
                                                                           "x!1")
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (typepred "le2")
                                                      (("2"
                                                        (expand
                                                         "directed_complete_partial_order?")
                                                        (("2"
                                                          (flatten)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("3" (propax) nil nil))
                                  nil)
                                 ("2"
                                  (skosimp)
                                  (("2"
                                    (hide 2)
                                    (("2"
                                      (expand "F" 1)
                                      (("2"
                                        (expand "FX")
                                        (("2"
                                          (hide -3 -7)
                                          (("2"
                                            (inst - "D!2")
                                            (("2"
                                              (inst - "D!2")
                                              (("2"
                                                (flatten)
                                                (("2"
                                                  (typepred
                                                   "lub({y: T2 | EXISTS (f: (D!1), x: (D!2)): y = f(x)})")
                                                  (("1"
                                                    (typepred
                                                     "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(lub(D!2))})")
                                                    (("1"
                                                      (expand
                                                       "least_upper_bound?")
                                                      (("1"
                                                        (flatten)
                                                        (("1"
                                                          (expand
                                                           "upper_bound?")
                                                          (("1"
                                                            (inst
                                                             -2
                                                             "lub({y: T2 | EXISTS (f: (D!1), x: (D!2)): y = f(x)})")
                                                            (("1"
                                                              (split
                                                               -2)
                                                              (("1"
                                                                (inst
                                                                 -4
                                                                 "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(lub(D!2))})")
                                                                (("1"
                                                                  (split
                                                                   -4)
                                                                  (("1"
                                                                    (lemma
                                                                     "partial_order_antisymmetric"
                                                                     ("x"
                                                                      "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(lub(D!2))})"
                                                                      "y"
                                                                      "lub({y: T2 | EXISTS (f: (D!1), x: (D!2)): y = f(x)})"))
                                                                    (("1"
                                                                      (assert)
                                                                      nil
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (hide
                                                                     2)
                                                                    (("2"
                                                                      (skosimp)
                                                                      (("2"
                                                                        (typepred
                                                                         "r!1")
                                                                        (("2"
                                                                          (skosimp)
                                                                          (("2"
                                                                            (typepred
                                                                             "x!1")
                                                                            (("2"
                                                                              (inst
                                                                               -4
                                                                               "f!1(lub(D!2))")
                                                                              (("1"
                                                                                (name-replace
                                                                                 "DRL100"
                                                                                 "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(lub(D!2))})")
                                                                                (("1"
                                                                                  (replace
                                                                                   -2)
                                                                                  (("1"
                                                                                    (hide-all-but
                                                                                     (1
                                                                                      -1
                                                                                      -4))
                                                                                    (("1"
                                                                                      (typepred
                                                                                       "f!1")
                                                                                      (("1"
                                                                                        (expand
                                                                                         "scott_continuous?")
                                                                                        (("1"
                                                                                          (flatten)
                                                                                          (("1"
                                                                                            (typepred
                                                                                             "lub(D!2)")
                                                                                            (("1"
                                                                                              (expand
                                                                                               "least_upper_bound?")
                                                                                              (("1"
                                                                                                (flatten)
                                                                                                (("1"
                                                                                                  (expand
                                                                                                   "upper_bound?")
                                                                                                  (("1"
                                                                                                    (inst
                                                                                                     -
                                                                                                     "x!1")
                                                                                                    (("1"
                                                                                                      (expand
                                                                                                       "increasing?")
                                                                                                      (("1"
                                                                                                        (inst
                                                                                                         -
                                                                                                         "x!1"
                                                                                                         "lub(D!2)")
                                                                                                        (("1"
                                                                                                          (assert)
                                                                                                          (("1"
                                                                                                            (lemma
                                                                                                             "partial_order_transitive"
                                                                                                             ("x"
                                                                                                              "f!1(x!1)"
                                                                                                              "y"
                                                                                                              "f!1(lub(D!2))"
                                                                                                              "z"
                                                                                                              "DRL100"))
                                                                                                            (("1"
                                                                                                              (assert)
                                                                                                              nil
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (typepred
                                                                                                           "le1")
                                                                                                          (("2"
                                                                                                            (expand
                                                                                                             "directed_complete_partial_order?")
                                                                                                            (("2"
                                                                                                              (flatten)
                                                                                                              nil
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (inst
                                                                                 +
                                                                                 "f!1")
                                                                                nil
                                                                                nil)
                                                                               ("3"
                                                                                (typepred
                                                                                 "le1")
                                                                                (("3"
                                                                                  (expand
                                                                                   "directed_complete_partial_order?")
                                                                                  (("3"
                                                                                    (flatten)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil)
                                                                               ("4"
                                                                                (typepred
                                                                                 "le2")
                                                                                (("4"
                                                                                  (expand
                                                                                   "directed_complete_partial_order?")
                                                                                  (("4"
                                                                                    (flatten)
                                                                                    (("4"
                                                                                      (lemma
                                                                                       "directed_is_lub_set[T1,le1]")
                                                                                      (("4"
                                                                                        (inst?)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (inst
                                                                   -10
                                                                   "lub(D!2)")
                                                                  (("1"
                                                                    (typepred
                                                                     "le1")
                                                                    (("1"
                                                                      (expand
                                                                       "directed_complete_partial_order?")
                                                                      (("1"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (lemma
                                                                     "directed_is_lub_set[T1,le1]")
                                                                    (("2"
                                                                      (inst?)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("3"
                                                                  (skosimp*)
                                                                  (("3"
                                                                    (typepred
                                                                     "le1")
                                                                    (("3"
                                                                      (expand
                                                                       "directed_complete_partial_order?")
                                                                      (("3"
                                                                        (flatten)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil)
                                                                 ("4"
                                                                  (skosimp*)
                                                                  (("4"
                                                                    (lemma
                                                                     "directed_is_lub_set[T1,le1]")
                                                                    (("4"
                                                                      (inst?)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (skosimp)
                                                                (("2"
                                                                  (typepred
                                                                   "r!1")
                                                                  (("2"
                                                                    (skosimp)
                                                                    (("2"
                                                                      (replace
                                                                       -1)
                                                                      (("2"
                                                                        (hide
                                                                         2)
                                                                        (("2"
                                                                          (hide
                                                                           -1)
                                                                          (("2"
                                                                            (typepred
                                                                             "f!1")
                                                                            (("2"
                                                                              (expand
                                                                               "scott_continuous?")
                                                                              (("2"
                                                                                (flatten)
                                                                                (("2"
                                                                                  (expand
                                                                                   "lub_preserving?")
                                                                                  (("2"
                                                                                    (inst
                                                                                     -
                                                                                     "D!2")
                                                                                    (("2"
                                                                                      (lemma
                                                                                       "lub_set_image"
                                                                                       ("g"
                                                                                        "f!1"
                                                                                        "D"
                                                                                        "D!2"))
                                                                                      (("2"
                                                                                        (assert)
                                                                                        (("2"
                                                                                          (replace
                                                                                           -3)
                                                                                          (("2"
                                                                                            (expand
                                                                                             "image")
                                                                                            (("2"
                                                                                              (name-replace
                                                                                               "DRL101"
                                                                                               "{y: T2 | EXISTS (f: (D!1), x: (D!2)): y = f(x)}")
                                                                                              (("2"
                                                                                                (name-replace
                                                                                                 "DRL102"
                                                                                                 "{y: T2 | EXISTS (x: (D!2)): y = f!1(x)}")
                                                                                                (("2"
                                                                                                  (lemma
                                                                                                   "above_lub[T2,le2]")
                                                                                                  (("2"
                                                                                                    (inst
                                                                                                     -
                                                                                                     "DRL101"
                                                                                                     "DRL102")
                                                                                                    (("2"
                                                                                                      (split
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (propax)
                                                                                                        nil
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (hide
                                                                                                         2)
                                                                                                        (("2"
                                                                                                          (expand
                                                                                                           "above?")
                                                                                                          (("2"
                                                                                                            (skosimp)
                                                                                                            (("2"
                                                                                                              (typepred
                                                                                                               "x!1")
                                                                                                              (("2"
                                                                                                                (expand
                                                                                                                 "DRL102")
                                                                                                                (("2"
                                                                                                                  (skosimp)
                                                                                                                  (("2"
                                                                                                                    (replace
                                                                                                                     -1)
                                                                                                                    (("2"
                                                                                                                      (expand
                                                                                                                       "DRL101")
                                                                                                                      (("2"
                                                                                                                        (inst
                                                                                                                         +
                                                                                                                         "f!1(x!2)")
                                                                                                                        (("1"
                                                                                                                          (rewrite
                                                                                                                           "partial_order_reflexive")
                                                                                                                          nil
                                                                                                                          nil)
                                                                                                                         ("2"
                                                                                                                          (expand
                                                                                                                           "DRL101")
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             +
                                                                                                                             "f!1"
                                                                                                                             "x!2")
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (inst
                                                       -8
                                                       "lub(D!2)")
                                                      (("1"
                                                        (typepred
                                                         "le1")
                                                        (("1"
                                                          (expand
                                                           "directed_complete_partial_order?")
                                                          (("1"
                                                            (flatten)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (lemma
                                                         "directed_is_lub_set[T1,le1]")
                                                        (("2"
                                                          (inst?)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (skosimp*)
                                                      (("3"
                                                        (typepred
                                                         "le1")
                                                        (("3"
                                                          (expand
                                                           "directed_complete_partial_order?")
                                                          (("3"
                                                            (flatten)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("4"
                                                      (skosimp*)
                                                      (("4"
                                                        (lemma
                                                         "directed_is_lub_set[T1,le1]")
                                                        (("4"
                                                          (inst?)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("3" (propax) nil nil)
                                 ("4"
                                  (assert)
                                  (("4"
                                    (skosimp*)
                                    (("4"
                                      (typepred "le1")
                                      (("4"
                                        (expand
                                         "directed_complete_partial_order?")
                                        (("4" (flatten) nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("5" (propax) nil nil))
                                nil)
                               ("2"
                                (skosimp)
                                (("2"
                                  (inst - "D!2")
                                  (("2"
                                    (flatten)
                                    (("2"
                                      (lemma
                                       "directed_is_lub_set"
                                       ("x" "FX(D!2)"))
                                      (("1" (propax) nil nil)
                                       ("2"
                                        (expand "directed?" 1)
                                        (("2" (propax) nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("3"
                                (skosimp*)
                                (("3"
                                  (typepred "le2")
                                  (("3"
                                    (expand
                                     "directed_complete_partial_order?")
                                    (("3" (flatten) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide 2 -1 -5)
                              (("2"
                                (skosimp)
                                (("2"
                                  (typepred "D!2")
                                  (("2"
                                    (expand "nonempty?")
                                    (("2"
                                      (expand "empty?")
                                      (("2"
                                        (expand "member")
                                        (("2"
                                          (split 2)
                                          (("1"
                                            (skosimp*)
                                            (("1"
                                              (expand "FX")
                                              (("1"
                                                (inst - "x!2(x!1)")
                                                (("1"
                                                  (inst + "x!2" "x!1")
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2 3)
                                            (("2"
                                              (expand "directed?")
                                              (("2"
                                                (skosimp*)
                                                (("2"
                                                  (expand "FX")
                                                  (("2"
                                                    (skosimp*)
                                                    (("2"
                                                      (replace -1)
                                                      (("2"
                                                        (replace -2)
                                                        (("2"
                                                          (typepred
                                                           "f!1")
                                                          (("2"
                                                            (typepred
                                                             "f!2")
                                                            (("2"
                                                              (inst
                                                               -12
                                                               "f!1"
                                                               "f!2")
                                                              (("2"
                                                                (assert)
                                                                (("2"
                                                                  (expand
                                                                   "scott_continuous?")
                                                                  (("2"
                                                                    (flatten)
                                                                    (("2"
                                                                      (hide-all-but
                                                                       (1
                                                                        -1
                                                                        -3
                                                                        -4
                                                                        -6
                                                                        -9
                                                                        -14))
                                                                      (("2"
                                                                        (skosimp)
                                                                        (("2"
                                                                          (expand
                                                                           "directed?")
                                                                          (("2"
                                                                            (inst
                                                                             -
                                                                             "x!2"
                                                                             "x!3")
                                                                            (("2"
                                                                              (assert)
                                                                              (("2"
                                                                                (skosimp)
                                                                                (("2"
                                                                                  (expand
                                                                                   "increasing?")
                                                                                  (("2"
                                                                                    (inst
                                                                                     -
                                                                                     "x!3"
                                                                                     "z!2")
                                                                                    (("2"
                                                                                      (inst
                                                                                       -
                                                                                       "x!2"
                                                                                       "z!2")
                                                                                      (("2"
                                                                                        (assert)
                                                                                        (("2"
                                                                                          (inst
                                                                                           -
                                                                                           "z!2")
                                                                                          (("2"
                                                                                            (inst
                                                                                             -
                                                                                             "z!2")
                                                                                            (("2"
                                                                                              (lemma
                                                                                               "partial_order_transitive"
                                                                                               ("z"
                                                                                                "z!1(z!2)"))
                                                                                              (("2"
                                                                                                (inst-cp
                                                                                                 -
                                                                                                 "f!1(x!2)"
                                                                                                 "f!1(z!2)")
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -
                                                                                                   "f!2(x!3)"
                                                                                                   "f!2(z!2)")
                                                                                                  (("2"
                                                                                                    (assert)
                                                                                                    (("2"
                                                                                                      (inst
                                                                                                       +
                                                                                                       "z!1(z!2)")
                                                                                                      (("1"
                                                                                                        (assert)
                                                                                                        nil
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (expand
                                                                                                         "FX")
                                                                                                        (("2"
                                                                                                          (inst
                                                                                                           +
                                                                                                           "z!1"
                                                                                                           "z!2")
                                                                                                          nil
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("3" (skosimp*)
                              (("3"
                                (typepred "le2")
                                (("3"
                                  (expand
                                   "directed_complete_partial_order?")
                                  (("3" (flatten) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (propax) nil nil))
                        nil))
                      nil)
                     ("2" (hide 2)
                      (("2" (expand "increasing?")
                        (("2" (skosimp)
                          (("2" (expand "F")
                            (("2" (hide -2)
                              (("2"
                                (inst-cp - "x!1")
                                (("2"
                                  (inst - "y!1")
                                  (("2"
                                    (typepred
                                     "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(x!1)})")
                                    (("1"
                                      (typepred
                                       "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(y!1)})")
                                      (("1"
                                        (expand "least_upper_bound?")
                                        (("1"
                                          (flatten)
                                          (("1"
                                            (expand "upper_bound?")
                                            (("1"
                                              (inst
                                               -4
                                               "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(y!1)})")
                                              (("1"
                                                (split -4)
                                                (("1" (propax) nil nil)
                                                 ("2"
                                                  (hide 2)
                                                  (("2"
                                                    (skosimp)
                                                    (("2"
                                                      (typepred "r!1")
                                                      (("2"
                                                        (skosimp)
                                                        (("2"
                                                          (replace -1)
                                                          (("2"
                                                            (inst
                                                             -2
                                                             "f!1(y!1)")
                                                            (("1"
                                                              (typepred
                                                               "f!1")
                                                              (("1"
                                                                (expand
                                                                 "scott_continuous?")
                                                                (("1"
                                                                  (flatten)
                                                                  (("1"
                                                                    (expand
                                                                     "increasing?")
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "x!1"
                                                                       "y!1")
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (lemma
                                                                           "partial_order_transitive"
                                                                           ("x"
                                                                            "f!1(x!1)"
                                                                            "y"
                                                                            "f!1(y!1)"
                                                                            "z"
                                                                            "lub[T2, le2]({y: T2 | EXISTS (f: (D!1)): y = f(y!1)})"))
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (inst
                                                               +
                                                               "f!1")
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (typepred "le2")
                                                (("2"
                                                  (expand
                                                   "directed_complete_partial_order?")
                                                  (("2"
                                                    (flatten)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2" (propax) nil nil))
                                      nil)
                                     ("2" (propax) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("3" (typepred "le2")
                      (("3" (expand "directed_complete_partial_order?")
                        (("3" (flatten) nil nil)) nil))
                      nil)
                     ("4" (typepred "le1")
                      (("4" (expand "directed_complete_partial_order?")
                        (("4" (flatten) nil nil)) nil))
                      nil))
                    nil)
                   ("2" (propax) nil nil))
                  nil)
                 ("2" (hide 2)
                  (("2" (skosimp)
                    (("2" (expand "lub_set?")
                      (("2" (typepred "le2")
                        (("2"
                          (expand "directed_complete_partial_order?")
                          (("2" (flatten)
                            (("2" (expand "directed_complete?")
                              (("2"
                                (inst
                                 -
                                 "{y: T2 | EXISTS (f: (D!1)): y = f(x!1)}")
                                (("2"
                                  (hide 2 -1)
                                  (("2"
                                    (split)
                                    (("1"
                                      (expand "nonempty?")
                                      (("1"
                                        (expand "empty?")
                                        (("1"
                                          (skosimp)
                                          (("1"
                                            (expand "member")
                                            (("1"
                                              (inst - "x!2(x!1)")
                                              (("1"
                                                (inst + "x!2")
                                                nil
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (expand "directed?")
                                      (("2"
                                        (skosimp*)
                                        (("2"
                                          (replace -1)
                                          (("2"
                                            (replace -2)
                                            (("2"
                                              (typepred "f!1")
                                              (("2"
                                                (typepred "f!2")
                                                (("2"
                                                  (inst - "f!1" "f!2")
                                                  (("2"
                                                    (assert)
                                                    (("2"
                                                      (skosimp)
                                                      (("2"
                                                        (inst - "x!1")
                                                        (("2"
                                                          (inst
                                                           -
                                                           "x!1")
                                                          (("2"
                                                            (inst
                                                             +
                                                             "z!1(x!1)")
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil)
                                                             ("2"
                                                              (inst
                                                               +
                                                               "z!1")
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("3" (skosimp*)
                  (("3" (typepred "le2")
                    (("3" (expand "directed_complete_partial_order?")
                      (("3" (flatten) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((directed_complete? const-decl "bool" directed_orders "orders/")
    (le2 formal-const-decl "(directed_complete_partial_order?[T2])"
     scott_continuity nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pointwise const-decl "bool" pointwise_orders "orders/")
    (directed? const-decl "bool" directed_orders "orders/")
    (partial_order? const-decl "bool" orders nil)
    (pred type-eq-decl nil defined_types nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (scott_continuous? const-decl "bool" scott_continuity nil)
    (T2 formal-type-decl nil scott_continuity nil)
    (T1 formal-type-decl nil scott_continuity nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (least_bounded_above? const-decl "bool" bounded_orders "orders/")
    (z!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (x!2 skolem-const-decl "(scott_continuous?)" scott_continuity nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (least_upper_bound? const-decl "bool" bounded_orders "orders/")
    (lub const-decl
     "{x | bounded_orders[T].least_upper_bound?(x, SA, <=)}"
     bounded_order_props "orders/")
    (y!1 skolem-const-decl "T1" scott_continuity nil)
    (f!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (directed_is_lub_set judgement-tcc nil directed_order_props
     "orders/")
    (directed? const-decl "bool" directed_order_props "orders/")
    (directed type-eq-decl nil directed_order_props "orders/")
    (empty? const-decl "bool" sets nil)
    (D!2 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (x!2 skolem-const-decl "(scott_continuous?)" scott_continuity nil)
    (z!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (z!2 skolem-const-decl "(D!2)" scott_continuity nil)
    (member const-decl "bool" sets nil)
    (f!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (above_lub formula-decl nil bounded_order_props "orders/")
    (above? const-decl "bool" bounded_order_props "orders/")
    (DRL101 skolem-const-decl "[T2 -> boolean]" scott_continuity nil)
    (partial_order_reflexive formula-decl nil partial_order_props
     "orders/")
    (x!2 skolem-const-decl "(D!2)" scott_continuity nil)
    (f!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (DRL102 skolem-const-decl "[T2 -> boolean]" scott_continuity nil)
    (D!2 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (image const-decl "set[R]" function_image nil)
    (upper_bound? const-decl "bool" bounded_orders "orders/")
    (x!1 skolem-const-decl "T1" scott_continuity nil)
    (r!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (F skolem-const-decl "[x: T1 ->
   {x_1 |
            bounded_orders[T2].least_upper_bound?
                (x_1, {y: T2 | EXISTS (f: (D!1)): y = f(x)}, le2)}]"
     scott_continuity nil)
    (D!1 skolem-const-decl
     "(directed?(LAMBDA (s: [(scott_continuous?), (scott_continuous?)]):
             pointwise(le2)(s)))" scott_continuity nil)
    (lub_preserving? const-decl "bool" scott_continuity nil)
    (partial_order_antisymmetric formula-decl nil partial_order_props
     "orders/")
    (FX skolem-const-decl "[directed[T1, le1] -> [T2 -> boolean]]"
     scott_continuity nil)
    (D!2 skolem-const-decl "directed[T1, le1]" scott_continuity nil)
    (f!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (x!1 skolem-const-decl "(D!2)" scott_continuity nil)
    (x!1 skolem-const-decl "(D!2)" scott_continuity nil)
    (f!1 skolem-const-decl "(D!1)" scott_continuity nil)
    (partial_order_transitive formula-decl nil partial_order_props
     "orders/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (lub_set_image formula-decl nil scott_continuity nil)
    (increasing? const-decl "bool" fun_preds_partial nil)
    (le1 formal-const-decl "(directed_complete_partial_order?[T1])"
     scott_continuity nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (lub_set? const-decl "bool" bounded_order_props "orders/")
    (restrict const-decl "R" restrict nil))
   shostak))
 (scott_continuous_dcpo 0
  (scott_continuous_dcpo-1 nil 3352751261
   ("" (lemma "pointwise_complete")
    (("" (expand "directed_complete_partial_order?")
      (("" (assert)
        (("" (expand "restrict")
          (("" (typepred "le2")
            (("" (expand "directed_complete_partial_order?")
              (("" (flatten)
                (("" (assert)
                  (("" (hide -2 -3)
                    (("" (expand "partial_order?")
                      (("" (prop)
                        (("1" (expand "pointwise")
                          (("1" (expand "preorder?")
                            (("1" (prop)
                              (("1"
                                (expand "reflexive?")
                                (("1"
                                  (skosimp*)
                                  (("1" (inst?) nil nil))
                                  nil))
                                nil)
                               ("2"
                                (expand "transitive?")
                                (("2"
                                  (skosimp*)
                                  (("2"
                                    (inst -1 "x!2")
                                    (("2"
                                      (inst - "x!2")
                                      (("2"
                                        (assert)
                                        (("2"
                                          (inst
                                           -
                                           "x!1(x!2)"
                                           "y!1(x!2)"
                                           "z!1(x!2)")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (expand "antisymmetric?")
                          (("2" (skosimp*)
                            (("2" (expand "pointwise")
                              (("2"
                                (apply-extensionality 1 :hide? t)
                                (("2"
                                  (inst - "x!2")
                                  (("2"
                                    (inst - "x!2")
                                    (("2"
                                      (inst - "x!1(x!2)" "y!1(x!2)")
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((partial_order_restrict application-judgement "(partial_order?[S])"
     restrict_order_props nil)
    (preorder_restrict application-judgement "(preorder?[S])"
     restrict_order_props nil)
    (transitive_restrict application-judgement "(transitive?[S])"
     restrict_order_props nil)
    (antisymmetric_restrict application-judgement "(antisymmetric?[S])"
     restrict_order_props nil)
    (reflexive_restrict application-judgement "(reflexive?[S])"
     restrict_order_props nil)
    (directed_complete_partial_order? const-decl "bool" directed_orders
     "orders/")
    (pointwise_complete formula-decl nil scott_continuity nil))
   shostak))
 (IMP_scott_TCC1 0
  (IMP_scott_TCC1-1 nil 3352749145
   ("" (lemma "scott_continuous_dcpo") (("" (propax) nil nil)) nil)
   ((scott_continuous_dcpo formula-decl nil scott_continuity nil))
   nil)))


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.0.229Bemerkung:  (vorverarbeitet am  2026-05-05) ¤

*Bot Zugriff






Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

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

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.