SSL complementary_orders.prf Sprache: Lisp

(complementary_orders
(complement_complement 0
  (complement_complement-1 nil 3318701342
   ("" (skolem-typepred)
    (("" (expand* "complement?" "o" "id")
      (("" (flatten)
        (("" (decompose-equality) (("" (inst?) nil nil)) nil)) nil))
      nil))
    nil)
   ((O const-decl "T3" function_props nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (complement? const-decl "bool" complementary_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(image_complement 0
  (image_complement-1 nil 3318701387
   ("" (skosimp* t)
    (("" (decompose-equality)
      (("" (expand "image")
        (("" (iff)
          (("" (prop)
            (("1" (skosimp* t)
              (("1" (use "complement_complement")
                (("1" (assert) nil nil)) nil))
              nil)
             ("2" (inst + "C!1(x!1)")
              (("1" (rewrite "complement_complement") nil nil)
               ("2" (inst?) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((set type-eq-decl nil sets nil)
    (image const-decl "set[R]" function_image nil)
    (x!1 skolem-const-decl "T" complementary_orders nil)
    (C!1 skolem-const-decl "(complement?(rel!1))" complementary_orders
     nil)
    (rel!1 skolem-const-decl "pred[[T, T]]" complementary_orders nil)
    (S!1 skolem-const-decl "set[T]" complementary_orders nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (complement? const-decl "bool" complementary_orders nil))
   shostak))
(inverse_complement 0
  (inverse_complement-1 nil 3318701457
   ("" (skolem-typepred)
    (("" (decompose-equality)
      (("" (rewrite "bijective_inverse[T, T]")
        (("1" (rewrite "complement_complement") nil nil)
         ("2" (lemma "left_inj_surj[T, T]")
          (("2" (inst - "C!1" "C!1")
            (("1" (expand "bijective?") (("1" (propax) nil nil)) nil)
             ("2" (expand* "left_inverse?" "complement?" "o" "id")
              (("2" (flatten) (("2" (decompose-equality) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((inverse const-decl "D" function_inverse nil)
    (left_inj_surj formula-decl nil function_inverse_def nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (O const-decl "T3" function_props nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (C!1 skolem-const-decl "(complement?(rel!1))" complementary_orders
     nil)
    (rel!1 skolem-const-decl "pred[[T, T]]" complementary_orders nil)
    (left_inverse? const-decl "bool" function_inverse_def nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (bijective? const-decl "bool" functions nil)
    (bijective_inverse formula-decl nil function_inverse nil)
    (complement? const-decl "bool" complementary_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(le_complement 0
  (le_complement-1 nil 3318701551
   ("" (skolem-typepred)
    (("" (prop)
      (("1" (expand* "complement?" "preserves" "converse")
        (("1" (flatten)
          (("1" (inst - "C!1(t1!1)" "C!1(t2!1)")
            (("1" (assert)
              (("1" (rewrite "complement_complement")
                (("1" (rewrite "complement_complement") nil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (expand* "complement?" "preserves" "converse")
        (("2" (flatten) (("2" (inst?) (("2" (assert) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((complement_complement formula-decl nil complementary_orders nil)
    (converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (preserves const-decl "bool" functions nil)
    (complement? const-decl "bool" complementary_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(upper_bound_complement 0
  (upper_bound_complement-1 nil 3318701679
   ("" (skolem!)
    (("" (prop)
      (("1" (expand* "lower_bound?" "upper_bound?")
        (("1" (skolem-typepred)
          (("1" (inst - "C!1(r!1)")
            (("1" (use "le_complement" ("t2" "C!1(t!1)"))
              (("1" (assert)
                (("1" (rewrite "complement_complement") nil nil)) nil))
              nil)
             ("2" (expand "image")
              (("2" (skolem-typepred)
                (("2" (use "complement_complement" ("t" "x!1"))
                  (("2" (assert) nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (expand* "lower_bound?" "upper_bound?")
        (("2" (skolem-typepred)
          (("2" (inst - "C!1(r!1)")
            (("1" (rewrite "le_complement") nil nil)
             ("2" (expand "image") (("2" (inst?) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (set type-eq-decl nil sets nil)
    (image const-decl "set[R]" function_image nil)
    (pred type-eq-decl nil defined_types nil)
    (complement? const-decl "bool" complementary_orders nil)
    (le_complement formula-decl nil complementary_orders nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (S!1 skolem-const-decl "set[T]" complementary_orders nil)
    (rel!1 skolem-const-decl "pred[[T, T]]" complementary_orders nil)
    (C!1 skolem-const-decl "(complement?(rel!1))" complementary_orders
     nil)
    (r!1 skolem-const-decl "(image(C!1, S!1))" complementary_orders
     nil)
    (lower_bound? const-decl "bool" bounded_orders nil)
    (upper_bound? const-decl "bool" bounded_orders nil)
    (r!1 skolem-const-decl "(S!1)" complementary_orders nil))
   shostak))
(bounded_above_complement 0
  (bounded_above_complement-1 nil 3318701876
   ("" (skolem!)
    (("" (expand* "bounded_above?" "bounded_below?")
      (("" (prop)
        (("1" (skolem!)
          (("1" (inst + "C!1(t!1)")
            (("1" (use "upper_bound_complement")
              (("1" (assert) nil nil)) nil))
            nil))
          nil)
         ("2" (skolem!)
          (("2" (inst + "C!1(t!1)")
            (("2" (use "upper_bound_complement")
              (("2" (rewrite "complement_complement")
                (("2" (assert) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bounded_below? const-decl "bool" bounded_orders nil)
    (bounded_above? const-decl "bool" bounded_orders nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (upper_bound_complement formula-decl nil complementary_orders nil)
    (set type-eq-decl nil sets nil)
    (complement? const-decl "bool" complementary_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil))
   shostak))
(least_upper_bound_complement 0
  (least_upper_bound_complement-1 nil 3318701959
   ("" (skolem!)
    (("" (expand* "least_upper_bound?" "greatest_lower_bound?")
      (("" (prop)
        (("1" (use "upper_bound_complement") (("1" (assert) nil nil))
          nil)
         ("2" (skosimp)
          (("2" (hide -2)
            (("2" (inst - "C!1(r!1)")
              (("2" (use "upper_bound_complement")
                (("2" (rewrite "complement_complement")
                  (("2" (assert)
                    (("2"
                      (use "le_complement"
                           ("t1" "C!1(r!1)" "t2" "t!1"))
                      (("2" (assert)
                        (("2" (rewrite "complement_complement") nil
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("3" (use "upper_bound_complement") (("3" (assert) nil nil))
          nil)
         ("4" (skosimp)
          (("4" (hide -2)
            (("4" (inst - "C!1(r!1)")
              (("4" (use "upper_bound_complement")
                (("4" (assert)
                  (("4" (rewrite "le_complement") nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((greatest_lower_bound? const-decl "bool" bounded_orders nil)
    (least_upper_bound? const-decl "bool" bounded_orders nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (le_complement formula-decl nil complementary_orders nil)
    (upper_bound_complement formula-decl nil complementary_orders nil)
    (complement? const-decl "bool" complementary_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil))
   shostak))
(least_bounded_above_complement 0
  (least_bounded_above_complement-1 nil 3318702440
   ("" (skolem!)
    (("" (expand* "least_bounded_above?" "greatest_bounded_below?")
      (("" (prop)
        (("1" (skolem!)
          (("1" (use "least_upper_bound_complement")
            (("1" (inst?) (("1" (assert) nil nil)) nil)) nil))
          nil)
         ("2" (skolem!)
          (("2" (inst + "C!1(t!1)")
            (("2" (use "least_upper_bound_complement")
              (("2" (rewrite "complement_complement")
                (("2" (assert) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((greatest_bounded_below? const-decl "bool" bounded_orders nil)
    (least_bounded_above? const-decl "bool" bounded_orders nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (complement? const-decl "bool" complementary_orders nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (pred type-eq-decl nil defined_types nil)
    (least_upper_bound_complement formula-decl nil complementary_orders
     nil))
   shostak))
(lub_complement_TCC1 0
  (lub_complement_TCC1-1 nil 3318701312
   ("" (skolem!)
    (("" (use "least_bounded_above_complement") (("" (assert) nil nil))
      nil))
    nil)
   ((least_bounded_above_complement formula-decl nil
     complementary_orders nil)
    (complement? const-decl "bool" complementary_orders nil)
    (antisymmetric? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (least_bounded_above? const-decl "bool" bounded_orders nil)
    (pred type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil))
   nil))
(lub_complement 0
  (lub_complement-1 nil 3318702643
   ("" (skolem!)
    (("" (invoke (name-replace "T" "%1" :hide? nil) (! 1 r 1))
      (("" (use "unique_least_upper_bound")
        (("" (rewrite "glb_def")
          (("" (use "least_upper_bound_complement")
            (("" (rewrite "complement_complement")
              (("" (assert) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((least_bounded_above? const-decl "bool" bounded_orders nil)
    (complement? const-decl "bool" complementary_orders nil)
    (image const-decl "set[R]" function_image nil)
    (antisymmetric? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (glb const-decl "(LAMBDA (t): greatest_lower_bound?(t, S, <=))"
     bounded_orders nil)
    (greatest_lower_bound? const-decl "bool" bounded_orders nil)
    (greatest_bounded_below? const-decl "bool" bounded_orders nil)
    (set type-eq-decl nil sets nil)
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (glb_def formula-decl nil bounded_orders nil)
    (complement_complement formula-decl nil complementary_orders nil)
    (least_upper_bound_complement formula-decl nil complementary_orders
     nil)
    (lub const-decl "(LAMBDA (t): least_upper_bound?(t, S, <=))"
     bounded_orders nil)
    (least_upper_bound? const-decl "bool" bounded_orders nil)
    (T skolem-const-decl
     "(LAMBDA (t: T): greatest_lower_bound?(t, image(C!1, S!1), rel!1))"
     complementary_orders nil)
    (S!1 skolem-const-decl
     "(LAMBDA (S: set[T]): least_bounded_above?(S, rel!1))"
     complementary_orders nil)
    (C!1 skolem-const-decl "(complement?(rel!1))" complementary_orders
     nil)
    (rel!1 skolem-const-decl "(antisymmetric?[T])" complementary_orders
     nil)
    (unique_least_upper_bound formula-decl nil bounded_orders nil))
   shostak))
(glb_complement_TCC1 0
  (glb_complement_TCC1-1 nil 3318701312
   ("" (skolem!)
    ((""
      (use "least_bounded_above_complement"
           ("S" "image[T, T](C!1, S!1)"))
      (("" (rewrite "image_complement") (("" (assert) nil nil)) nil))
      nil))
    nil)
   ((least_bounded_above_complement formula-decl nil
     complementary_orders nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (image const-decl "set[R]" function_image nil)
    (pred type-eq-decl nil defined_types nil)
    (complement? const-decl "bool" complementary_orders nil)
    (PRED type-eq-decl nil defined_types nil)
    (antisymmetric? const-decl "bool" relations nil)
    (greatest_bounded_below? const-decl "bool" bounded_orders nil)
    (image_complement formula-decl nil complementary_orders nil))
   nil))
(glb_complement 0
  (glb_complement-1 nil 3318702699
   ("" (skolem!)
    (("" (invoke (name-replace "T" "%1" :hide? nil) (! 1 r 1))
      (("" (use "unique_greatest_lower_bound")
        (("" (rewrite "lub_def")
          (("" (use "least_upper_bound_complement")
            (("" (rewrite "image_complement") (("" (assert) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((greatest_bounded_below? const-decl "bool" bounded_orders nil)
    (complement? const-decl "bool" complementary_orders nil)
    (image const-decl "set[R]" function_image nil)
    (antisymmetric? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (lub const-decl "(LAMBDA (t): least_upper_bound?(t, S, <=))"
     bounded_orders nil)
    (least_upper_bound? const-decl "bool" bounded_orders nil)
    (least_bounded_above? const-decl "bool" bounded_orders nil)
    (set type-eq-decl nil sets nil)
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-nonempty-type-decl nil complementary_orders nil)
    (lub_def formula-decl nil bounded_orders nil)
    (image_complement formula-decl nil complementary_orders nil)
    (least_upper_bound_complement formula-decl nil complementary_orders
     nil)
    (glb const-decl "(LAMBDA (t): greatest_lower_bound?(t, S, <=))"
     bounded_orders nil)
    (greatest_lower_bound? const-decl "bool" bounded_orders nil)
    (T skolem-const-decl
     "(LAMBDA (t: T): least_upper_bound?(t, image(C!1, S!1), rel!1))"
     complementary_orders nil)
    (S!1 skolem-const-decl
     "(LAMBDA (S: set[T]): greatest_bounded_below?(S, rel!1))"
     complementary_orders nil)
    (C!1 skolem-const-decl "(complement?(rel!1))" complementary_orders
     nil)
    (rel!1 skolem-const-decl "(antisymmetric?[T])" complementary_orders
     nil)
    (unique_greatest_lower_bound formula-decl nil bounded_orders nil))
   shostak)))

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