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products/sources/formale sprachen/PVS/exact_real_arith/

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Datei: rat.prf Sprache: Lisp

Original von: PVS^©

(rat
(denominators_TCC1 0
  (denominators_TCC1-1 nil 3251057939
   ("" (skosimp*)
    (("" (typepred "x!1")
      (("" (lemma "rational_pred_ax2" ("r" "x!1"))
        (("" (expand "nonempty?")
          (("" (expand "empty?")
            (("" (skosimp*)
              (("" (inst - "p!1")
                (("" (expand "member") (("" (inst + "i!1") nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((rat nonempty-type-eq-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)
    (nonempty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (empty? const-decl "bool" sets nil)
    (rational_pred_ax2 formula-decl nil rational_props nil))
   shostak))
(numerator_TCC1 0
  (numerator_TCC1-1 nil 3251057939
   ("" (skosimp)
    (("" (typepred "denominator(x!1)")
      (("" (expand "denominator")
        (("" (lemma "denominators_TCC1" ("x" "x!1"))
          (("" (expand "denominators")
            ((""
              (typepred "min[posnat]({pn | EXISTS i: x!1 = i / pn})")
              (("1" (skosimp)
                (("1" (rewrite "div_cancel4" -2)
                  (("1" (assert) nil nil)) nil))
                nil)
               ("2" (propax) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((denominator const-decl "posnat" rat nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (rat nonempty-type-eq-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (> const-decl "bool" reals 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)
    (denominators_TCC1 subtype-tcc nil rat nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (min const-decl "{a | S(a) AND (FORALL x: S(x) IMPLIES a <= x)}"
         min_nat nil)
    (<= const-decl "bool" reals nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (div_cancel4 formula-decl nil real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (denominators const-decl "(nonempty?[posnat])" rat nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil))
   shostak))
(cauchy_rat_TCC1 0
  (cauchy_rat_TCC1-1 nil 3475497570
   ("" (skosimp)
    (("" (expand "cauchy_nzreal?")
      (("" (inst + "denominator(x!1)")
        (("" (rewrite "int_lemma") nil nil)) nil))
      nil))
    nil)
   ((cauchy_nzreal? const-decl "bool" cauchy nil)
    (int_lemma formula-decl nil int nil)
    (denominator const-decl "posnat" rat nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (rat nonempty-type-eq-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal 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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   nil))
(rat_lemma 0
  (rat_lemma-1 nil 3475497761
   ("" (skosimp)
    (("" (expand "cauchy_rat")
      (("" (lemma "int_lemma" ("n" "numerator(x!1)"))
        (("" (lemma "int_lemma" ("n" "denominator(x!1)"))
          (("" (typepred "denominator(x!1)")
            ((""
              (lemma "div_lemma"
               ("x" "numerator(x!1)" "nzy" "denominator(x!1)" "cx"
                "cauchy_int(numerator(x!1))" "nzcy"
                "cauchy_int(denominator(x!1))"))
              (("" (assert)
                ((""
                  (case-replace
                   "numerator(x!1) / denominator(x!1)=x!1")
                  (("" (hide-all-but (1 -2))
                    (("" (rewrite "div_cancel3")
                      (("" (expand "numerator") (("" (propax) nil nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cauchy_rat const-decl "cauchy_real" rat nil)
    (denominator const-decl "posnat" rat nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal nil)
    (cauchy_nzreal nonempty-type-eq-decl nil cauchy nil)
    (cauchy_nzreal? const-decl "bool" cauchy nil)
    (cauchy_int const-decl "cauchy_real" int nil)
    (cauchy_real nonempty-type-eq-decl nil cauchy nil)
    (cauchy_real? const-decl "bool" cauchy nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (div_lemma formula-decl nil div nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (div_cancel3 formula-decl nil real_props nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (int_lemma formula-decl nil int nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans 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)
    (rat nonempty-type-eq-decl nil rationals nil)
    (numerator const-decl "int" rat nil))
   shostak)))

¤ Dauer der Verarbeitung: 0.20 Sekunden (vorverarbeitet) ¤

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