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Datei: countability_aux.pvs Sprache: Lisp

Original von: PVS^©

(monotone_sequences
(reflexive_closure_ascending 0
  (reflexive_closure_ascending-1 nil 3314543892
   ("" (skolem-typepred)
    (("" (expand* "ascending?" "reflexive_closure" "union" "member")
      (("" (skosimp) (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((reflexive_closure const-decl "(reflexive?)" closure_ops nil)
    (member const-decl "bool" sets nil)
    (union const-decl "set" sets nil)
    (ascending? const-decl "bool" monotone_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (pred type-eq-decl nil defined_types nil)
    (T formal-type-decl nil monotone_sequences nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(reflexive_closure_descending 0
  (reflexive_closure_descending-1 nil 3314543932
   ("" (skolem-typepred)
    (("" (expand* "descending?" "reflexive_closure" "union" "member")
      (("" (skosimp) (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((reflexive_closure const-decl "(reflexive?)" closure_ops nil)
    (member const-decl "bool" sets nil)
    (union const-decl "set" sets nil)
    (descending? const-decl "bool" monotone_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (pred type-eq-decl nil defined_types nil)
    (T formal-type-decl nil monotone_sequences nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(ascending_lem 0
  (ascending_lem-1 nil 3314543958
   ("" (skolem-typepred)
    (("" (expand* "preserves" "restrict")
      (("" (induct "x2")
        (("1" (skosimp) (("1" (assert) nil nil)) nil)
         ("2" (skosimp*)
          (("2" (case-replace "x1!1 = j!1")
            (("1" (hide -2)
              (("1" (expand "ascending?")
                (("1" (inst?) (("1" (assert) nil nil)) nil)) nil))
              nil)
             ("2" (inst?)
              (("2" (assert)
                (("2" (expand "transitive?")
                  (("2"
                    (inst - "seq!1(x1!1)" "seq!1(j!1)"
                     "seq!1(1 + j!1)")
                    (("2" (assert)
                      (("2" (hide 2 -1)
                        (("2" (expand "ascending?")
                          (("2" (inst?) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((restrict const-decl "R" restrict nil)
    (preserves const-decl "bool" functions nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
    (ascending? const-decl "bool" monotone_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (pred type-eq-decl nil defined_types nil)
    (transitive? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (T formal-type-decl nil monotone_sequences nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(descending_lem 0
  (descending_lem-1 nil 3314544287
   ("" (skolem-typepred)
    (("" (rewrite "ascending_lem")
      (("" (expand* "ascending?" "converse" "descending?") nil nil))
      nil))
    nil)
   ((transitive_converse application-judgement "(transitive?[T])"
     relation_converse_props nil)
    (ascending_lem formula-decl nil monotone_sequences nil)
    (converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (ascending? const-decl "bool" monotone_sequences nil)
    (strict_total_order_restrict application-judgement
     "(strict_total_order?[S])" restrict_order_props nil)
    (trichotomous_restrict application-judgement "(trichotomous?[S])"
     restrict_order_props nil)
    (strict_order_restrict application-judgement "(strict_order?[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)
    (irreflexive_restrict application-judgement "(irreflexive?[S])"
     restrict_order_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (descending? const-decl "bool" monotone_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (pred type-eq-decl nil defined_types nil)
    (transitive? const-decl "bool" relations nil)
    (PRED type-eq-decl nil defined_types nil)
    (T formal-type-decl nil monotone_sequences nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(ascending_subsequence 0
  (ascending_subsequence-1 nil 3314544355
   ("" (skolem-typepred)
    (("" (lemma "preserves_composition[nat, nat, T]")
      (("" (inst?)
        (("" (invoke (inst - "%1" "%1" "lt!1") (! -2 0 1))
          (("" (expand "preserves" -2)
            (("" (assert)
              (("" (hide -2)
                (("" (split)
                  (("1" (hide -3)
                    (("1" (expand* "ascending?" "preserves" "restrict")
                      (("1" (skolem!)
                        (("1" (inst?) (("1" (assert) nil nil)) nil))
                        nil))
                      nil))
                    nil)
                   ("2" (rewrite "ascending_lem") nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((preserves_composition formula-decl nil function_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (irreflexive_restrict application-judgement "(irreflexive?[S])"
     restrict_order_props nil)
    (antisymmetric_restrict application-judgement "(antisymmetric?[S])"
     restrict_order_props nil)
    (transitive_restrict application-judgement "(transitive?[S])"
     restrict_order_props nil)
    (strict_order_restrict application-judgement "(strict_order?[S])"
     restrict_order_props nil)
    (trichotomous_restrict application-judgement "(trichotomous?[S])"
     restrict_order_props nil)
    (strict_total_order_restrict application-judgement
     "(strict_total_order?[S])" restrict_order_props nil)
    (preserves const-decl "bool" functions nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (ascending_lem formula-decl nil monotone_sequences nil)
    (ascending? const-decl "bool" monotone_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (pred type-eq-decl nil defined_types nil)
    (transitive? const-decl "bool" relations nil)
    (T formal-type-decl nil monotone_sequences nil)
    (< const-decl "bool" reals nil)
    (restrict const-decl "R" restrict nil)
    (preserves const-decl "bool" functions nil)
    (PRED type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
(descending_subsequence 0
  (descending_subsequence-1 nil 3314544571
   ("" (skolem-typepred)
    (("" (use "ascending_subsequence" ("lt" "converse(lt!1)"))
      (("1" (expand* "ascending?" "converse" "descending?") nil nil)
       ("2" (expand* "ascending?" "converse" "descending?") nil nil))
      nil))
    nil)
   ((converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (ascending_subsequence formula-decl nil monotone_sequences nil)
    (transitive_converse application-judgement "(transitive?[T])"
     relation_converse_props nil)
    (seq!1 skolem-const-decl "(descending?(lt!1))" monotone_sequences
     nil)
    (lt!1 skolem-const-decl "(transitive?[T])" monotone_sequences nil)
    (ascending? const-decl "bool" monotone_sequences nil)
    (descending? const-decl "bool" monotone_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (pred type-eq-decl nil defined_types nil)
    (transitive? const-decl "bool" relations nil)
    (T formal-type-decl nil monotone_sequences nil)
    (< const-decl "bool" reals nil)
    (restrict const-decl "R" restrict nil)
    (preserves const-decl "bool" functions nil)
    (PRED type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak)))

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