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Datei: well_nat.prf Sprache: Lisp

Original von: PVS^©

(well_nat
(well_lt_nat 0
  (well_lt_nat-1 nil 3318616417
   ("" (skolem!)
    (("" (lemma "wf_nat")
      (("" (grind :if-match nil)
        (("" (inst - "p!1")
          (("" (split)
            (("1" (skolem-typepred)
              (("1" (expand "extend")
                (("1" (prop)
                  (("1" (inst + "y!2")
                    (("1" (skolem!)
                      (("1" (inst - "x!1")
                        (("1" (expand "extend")
                          (("1" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (inst + "y!1")
              (("2" (expand "extend") (("2" (propax) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((wf_nat formula-decl nil naturalnumbers nil)
    (extend const-decl "R" extend nil)
    (FALSE const-decl "bool" booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (x!1 skolem-const-decl "(p!1)" well_nat nil)
    (y!2 skolem-const-decl "(extend[nat, (S!1), bool, FALSE](p!1))"
     well_nat nil)
    (p!1 skolem-const-decl "pred[(S!1)]" well_nat nil)
    (S!1 skolem-const-decl "set[nat]" well_nat nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (set type-eq-decl nil sets 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)
    (well_founded? const-decl "bool" orders nil)
    (restrict const-decl "R" restrict nil)
    (well_ordered? const-decl "bool" orders nil))
   shostak))
(well_gt_nat 0
  (well_gt_nat-1 nil 3318616896
   ("" (expand* "well_ordered?" "well_founded?")
    (("" (skosimp* t)
      (("" (lemma "non_empty_finite_has_greatest")
        (("" (inst - "p!1" "<=")
          (("1" (grind :if-match nil)
            (("1" (inst + "t!1")
              (("1" (skolem!)
                (("1" (inst - "x!1") (("1" (assert) nil nil)) nil))
                nil))
              nil))
            nil)
           ("2" (split)
            (("1" (expand "is_finite")
              (("1" (skolem!)
                (("1"
                  (inst + "N!1"
                   "LAMBDA (x: (extend[nat, (F!1), bool, FALSE](p!1))): f!1(x)")
                  (("1" (expand "injective?")
                    (("1" (skosimp :preds? t)
                      (("1" (expand "extend")
                        (("1" (prop)
                          (("1" (inst - "x1!1" "x2!1")
                            (("1" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (skolem-typepred)
                    (("2" (expand "extend") (("2" (prop) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (grind) nil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((finite_set type-eq-decl nil finite_sets nil)
    (is_finite const-decl "bool" finite_sets nil)
    (set type-eq-decl nil sets 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)
    (reflexive_restrict application-judgement "(reflexive?[S])"
     restrict_order_props nil)
    (preorder_restrict application-judgement "(preorder?[S])"
     restrict_order_props nil)
    (partial_order_restrict application-judgement "(partial_order?[S])"
     restrict_order_props nil)
    (dichotomous_restrict application-judgement "(dichotomous?[S])"
     restrict_order_props nil)
    (total_order_restrict application-judgement "(total_order?[S])"
     restrict_order_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (F!1 skolem-const-decl "finite_set[nat]" well_nat nil)
    (FALSE const-decl "bool" booleans nil)
    (extend const-decl "R" extend nil)
    (pred type-eq-decl nil defined_types nil)
    (p!1 skolem-const-decl "pred[(F!1)]" well_nat nil)
    (empty? const-decl "bool" sets nil)
    (<= const-decl "bool" reals nil)
    (restrict const-decl "R" restrict nil)
    (total_order? const-decl "bool" orders nil)
    (non_empty_finite_set type-eq-decl nil finite_sets nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (upper_bound? const-decl "bool" bounded_orders nil)
    (greatest? const-decl "bool" minmax_orders nil)
    (has_greatest? const-decl "bool" minmax_orders nil)
    (injective? const-decl "bool" functions nil)
    (member const-decl "bool" sets nil)
    (below type-eq-decl nil nat_types nil)
    (< const-decl "bool" reals nil)
    (non_empty_finite_has_greatest formula-decl nil minmax_orders nil)
    (well_ordered? const-decl "bool" orders 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)
    (well_founded? const-decl "bool" orders nil))
   shostak)))

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