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Datei: pointwise_orders.prf Sprache: Lisp

Original von: PVS^©

(relation_iterate
(comp_equal_r 0
  (comp_equal_r-1 nil 3314540085
   ("" (skolem!)
    (("" (decompose-equality)
      (("" (iff)
        (("" (expand "o")
          (("" (prop)
            (("1" (skosimp) (("1" (assert) nil nil)) nil)
             ("2" (inst?) nil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bool nonempty-type-eq-decl nil booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (O const-decl "bool" relation_props nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil relation_iterate nil))
   shostak))
(comp_equal_l 0
  (comp_equal_l-1 nil 3314540148
   ("" (skolem!)
    (("" (decompose-equality)
      (("" (iff)
        (("" (expand "o")
          (("" (prop)
            (("1" (skosimp) (("1" (assert) nil nil)) nil)
             ("2" (inst?) nil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bool nonempty-type-eq-decl nil booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (O const-decl "bool" relation_props nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil relation_iterate nil))
   shostak))
(iterate_TCC1 0
  (iterate_TCC1-1 nil 3314540067 ("" (subtype-tcc) nil 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)
    (>= const-decl "bool" 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
(iterate_TCC2 0
  (iterate_TCC2-1 nil 3314540067 ("" (termination-tcc) nil nil) nil
   nil))
(iterate_add 0
  (iterate_add-1 nil 3314540217
   ("" (induct "n")
    (("1" (skolem!)
      (("1" (expand "iterate" + 3)
        (("1" (rewrite "comp_equal_r") nil nil)) nil))
      nil)
     ("2" (skosimp*)
      (("2" (expand "iterate" + (1 3))
        (("2" (rewrite "assoc" :dir rl)
          (("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((assoc formula-decl nil relation_props2 nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (comp_equal_r formula-decl nil relation_iterate nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (O const-decl "bool" relation_props nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (iterate def-decl "pred[[T, T]]" relation_iterate nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (T formal-type-decl nil relation_iterate nil)
    (pred type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   shostak))
(iterate_add_applied 0
  (iterate_add_applied-1 nil 3314540329
   ("" (skosimp)
    (("" (rewrite "iterate_add")
      (("" (expand "o") (("" (inst?) (("" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   ((iterate_add formula-decl nil relation_iterate nil)
    (T formal-type-decl nil relation_iterate nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (pred type-eq-decl nil defined_types 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)
    (O const-decl "bool" relation_props nil))
   shostak))
(iterate_1 0
  (iterate_1-1 nil 3314540378
   ("" (skolem!)
    (("" (expand* "iterate" "iterate")
      (("" (rewrite "comp_equal_l") nil nil)) nil))
    nil)
   ((iterate def-decl "pred[[T, T]]" relation_iterate 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-type-decl nil relation_iterate nil)
    (comp_equal_l formula-decl nil relation_iterate nil))
   shostak))
(iterate_add_one 0
  (iterate_add_one-1 nil 3314540410
   ("" (skolem!)
    (("" (use "iterate_add" ("m" "1" "n" "n!1"))
      (("" (replace -1 :hide? t) (("" (rewrite "iterate_1") nil nil))
        nil))
      nil))
    nil)
   ((iterate_add formula-decl nil relation_iterate 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T formal-type-decl nil relation_iterate nil)
    (pred type-eq-decl nil defined_types nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (iterate_1 formula-decl nil relation_iterate nil))
   shostak))
(iterate_mult 0
  (iterate_mult-1 nil 3314540483
   ("" (induct "n")
    (("1" (skolem!)
      (("1" (expand "iterate" + (1 3)) (("1" (propax) nil nil)) nil))
      nil)
     ("2" (skosimp*)
      (("2" (expand "iterate" + 1)
        (("2" (inst?)
          (("2" (rewrite "iterate_add") (("2" (assert) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((iterate_add formula-decl nil relation_iterate nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (iterate def-decl "pred[[T, T]]" relation_iterate nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (T formal-type-decl nil relation_iterate nil)
    (pred type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil))
   shostak))
(converse_equal 0
  (converse_equal-1 nil 3314540542
   ("" (decompose-equality)
    (("" (expand "converse")
      (("" (rewrite "symmetry_of_equals") nil nil)) nil))
    nil)
   ((symmetry_of_equals formula-decl nil equality_props nil)
    (T formal-type-decl nil relation_iterate nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil))
   shostak))
(converse_comp 0
  (converse_comp-1 nil 3314540571
   ("" (skolem!)
    (("" (decompose-equality)
      (("" (iff)
        (("" (expand* "o" "converse")
          (("" (prop)
            (("1" (skosimp)
              (("1" (inst?) (("1" (assert) nil nil)) nil)) nil)
             ("2" (skosimp)
              (("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bool nonempty-type-eq-decl nil booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (O const-decl "bool" relation_props nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil relation_iterate nil))
   shostak))
(iterate_converse 0
  (iterate_converse-1 nil 3314540613
   ("" (induct "n")
    (("1" (skolem!)
      (("1" (expand "iterate")
        (("1" (rewrite "converse_equal") nil nil)) nil))
      nil)
     ("2" (skosimp*)
      (("2" (rewrite "iterate_add_one")
        (("2" (expand "iterate" + 2)
          (("2" (rewrite "converse_comp")
            (("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((converse_comp formula-decl nil relation_iterate nil)
    (iterate_add_one formula-decl nil relation_iterate nil)
    (converse_equal formula-decl nil relation_iterate nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (iterate def-decl "pred[[T, T]]" relation_iterate nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (T formal-type-decl nil relation_iterate nil)
    (pred type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
(comp_is_monotone 0
  (comp_is_monotone-1 nil 3314540754
   ("" (expand* "subset?" "member" "o")
    (("" (skosimp*)
      (("" (inst?)
        (("" (inst?) (("" (inst?) (("" (assert) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil relation_iterate nil)
    (subset? const-decl "bool" sets nil)
    (O const-decl "bool" relation_props nil)
    (member const-decl "bool" sets nil))
   shostak))
(iterate_is_monotone 0
  (iterate_is_monotone-1 nil 3314540812
   ("" (induct "n")
    (("1" (skosimp)
      (("1" (expand* "iterate" "subset?" "member")
        (("1" (skosimp) nil nil)) nil))
      nil)
     ("2" (skosimp*)
      (("2" (expand "iterate" +)
        (("2" (rewrite "comp_is_monotone")
          (("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((comp_is_monotone formula-decl nil relation_iterate nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (member const-decl "bool" sets nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (iterate def-decl "pred[[T, T]]" relation_iterate nil)
    (subset? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-type-decl nil relation_iterate nil)
    (pred type-eq-decl nil defined_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak)))

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