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Datei: fun_preds_partial.pvs   Sprache: Lisp

Untersuchung PVS©

(results_normal_form
 (NF_doesnot_rewrite 0
  (NF_doesnot_rewrite-1 nil 3374065344
   ("" (skeep)
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      (("" (expand"RTC" "IUnion")
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    (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)
    (= const-decl "[T, T -> boolean]" equalities nil)
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    (number nonempty-type-decl nil numbers nil)
    (iterate def-decl "pred[[T, T]]" relation_iterate "orders/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
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    (IUnion const-decl "set[T]" indexed_sets nil))
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    (NF_doesnot_rewrite formula-decl nil results_normal_form nil)
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    (PRED type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
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 (NFs_implies_Equal 0
  (NFs_implies_Equal-1 nil 3374065575
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    (joinable? const-decl "bool" ars_terminology nil)
    (NF_doesnot_rewrite formula-decl nil results_normal_form nil)
    (PRED type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
 (Norm_and_Confl_implies_UNF 0
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    (normal_form? const-decl "bool" ars_terminology nil)
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    (joinable? const-decl "bool" ars_terminology nil)
    (NF_doesnot_rewrite formula-decl nil results_normal_form nil)
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    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
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  (Normalizing_and_Confl-1 nil 3374065828
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                                                             -1
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    (T formal-type-decl nil results_normal_form nil)
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    (church_rosser? const-decl "bool" ars_terminology nil)
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                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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 results_normal_form nil)
    (Norm_and_Confl_implies_UNF formula-decl nil results_normal_form
     nil)
    (has_unique_nf? const-decl "bool" ars_terminology nil)
    (normalizing? const-decl "bool" ars_terminology nil)
    (normal_form? const-decl "bool" ars_terminology nil)
    (RTC const-decl "reflexive_transitive" relations_closure nil)
    (reflexive_transitive type-eq-decl nil relations_closure nil)
    (reflexive_transitive? const-decl "bool" relations_closure nil)
    (pred type-eq-decl nil defined_types nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (transitive? const-decl "bool" relations nil)
    (joinable? const-decl "bool" ars_terminology nil)
    (confluent? const-decl "bool" ars_terminology nil))
   shostak))
 (Noetherian_implies_normalizing 0
  (Noetherian_implies_normalizing-1 nil 3374068005
   ("" (skeep)
    (("" (expand "normalizing?")
      (("" (skeep)
        (("" (expand "noetherian?")
          (("" (expand "well_founded?")
            (("" (inst -1 "reduct?(R)(x)")
              (("" (split)
                (("1" (skolem * "a")
                  (("1" (inst 1 "a")
                    (("1" (typepred "a")
                      (("1" (expand "reduct?")
                        (("1" (expand "normal_form?")
                          (("1" (split)
                            (("1" (propax) nil nil)
                             ("2" (expand "is_normal_form?")
                              (("2"
                                (expand "reducible?")
                                (("2"
                                  (skolem * "b")
                                  (("2"
                                    (inst -3 "b")
                                    (("1"
                                      (expand "converse")
                                      (("1" (propax) nil nil))
                                      nil)
                                     ("2"
                                      (lemma "R_subset_RTC")
                                      (("2"
                                        (inst?)
                                        (("2"
                                          (expand"subset?" "member")
                                          (("2"
                                            (inst -1 "(a, b)")
                                            (("2"
                                              (assert)
                                              (("2"
                                                (typepred "RTC(R)")
                                                (("2"
                                                  (expand
                                                   "reflexive_transitive?")
                                                  (("2"
                                                    (flatten)
                                                    (("2"
                                                      (hide -1)
                                                      (("2"
                                                        (expand
                                                         "transitive?")
                                                        (("2"
                                                          (inst
                                                           -1
                                                           "x"
                                                           "a"
                                                           "b")
                                                          (("2"
                                                            (assert)
                                                            (("2"
                                                              (expand
                                                               "reduct?")
                                                              (("2"
                                                                (propax)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide 2)
                  (("2" (inst 1 "x")
                    (("2" (expand "reduct?")
                      (("2" (expand"RTC" "IUnion")
                        (("2" (inst 1 "0")
                          (("2" (expand "iterate")
                            (("2" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((normalizing? const-decl "bool" ars_terminology nil)
    (noetherian? const-decl "bool" noetherian nil)
    (T formal-type-decl nil results_normal_form nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (PRED type-eq-decl nil defined_types nil)
    (reduct? const-decl "PRED[T]" ars_terminology 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)
    (iterate def-decl "pred[[T, T]]" relation_iterate "orders/")
    (IUnion const-decl "set[T]" indexed_sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_form? const-decl "bool" ars_terminology nil)
    (is_normal_form? const-decl "bool" ars_terminology nil)
    (R_subset_RTC formula-decl nil relations_closure nil)
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (transitive? const-decl "bool" relations nil)
    (RTC const-decl "reflexive_transitive" relations_closure nil)
    (reflexive_transitive type-eq-decl nil relations_closure nil)
    (reflexive_transitive? const-decl "bool" relations_closure nil)
    (converse const-decl "pred[[T2, T1]]" relation_defs nil)
    (b skolem-const-decl "T" results_normal_form nil)
    (x skolem-const-decl "T" results_normal_form nil)
    (R skolem-const-decl "PRED[[T, T]]" results_normal_form nil)
    (reducible? const-decl "bool" ars_terminology nil)
    (well_founded? const-decl "bool" orders nil))
   shostak))
 (Convergent_UNF 0
  (Convergent_UNF-1 nil 3374068143
   ("" (skeep)
    (("" (expand "convergent?")
      (("" (flatten)
        (("" (lemma "Noetherian_implies_normalizing")
          (("" (inst?)
            (("" (assert)
              (("" (lemma "Norm_and_Confl_implies_UNF")
                (("" (inst?) (("" (assertnil nil)) nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((convergent? const-decl "bool" noetherian nil)
    (Noetherian_implies_normalizing formula-decl nil
     results_normal_form nil)
    (Norm_and_Confl_implies_UNF formula-decl nil results_normal_form
     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 results_normal_form nil))
   shostak))
 (Noet_and_Confl_iff_UNF 0
  (Noet_and_Confl_iff_UNF-1 nil 3374068168
   ("" (skeep)
    (("" (lemma "Noetherian_implies_normalizing")
      (("" (inst?)
        (("" (assert)
          (("" (hide -2)
            (("" (lemma "Normal_Confl_iff_UNF")
              (("" (inst?) (("" (assertnil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((Noetherian_implies_normalizing formula-decl nil
     results_normal_form nil)
    (Normal_Confl_iff_UNF formula-decl nil results_normal_form 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 results_normal_form nil))
   shostak))
 (Convergent_iff_eqNF 0
  (Convergent_iff_eqNF-1 nil 3374068205
   ("" (skeep)
    (("" (split)
      (("1" (flatten)
        (("1" (skeep)
          (("1" (expand "normal_form?")
            (("1" (flatten)
              (("1" (lemma "RTC_subset_EC")
                (("1" (inst?)
                  (("1" (copy -1)
                    (("1" (expand"subset?" "member")
                      (("1" (inst -1 "(x, u)")
                        (("1" (inst -2 "(y, v)")
                          (("1" (assert)
                            (("1" (hide (-4 -6))
                              (("1"
                                (typepred "EC(R)")
                                (("1"
                                  (expand "equivalence?")
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (hide -1)
                                      (("1"
                                        (expand*
                                         "symmetric?"
                                         "transitive?")
                                        (("1"
                                          (inst -1 "x" "u")
                                          (("1"
                                            (assert)
                                            (("1"
                                              (copy -2)
                                              (("1"
                                                (inst -1 "u" "x" "y")
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (inst
                                                     -3
                                                     "u"
                                                     "y"
                                                     "v")
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (expand
                                                         "convergent?")
                                                        (("1"
                                                          (flatten)
                                                          (("1"
                                                            (hide-all-but
                                                             (-3
                                                              -7
                                                              -8
                                                              -9
                                                              1))
                                                            (("1"
                                                              (lemma
                                                               "NFs_implies_Equal")
                                                              (("1"
                                                                (inst?)
                                                                (("1"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (flatten)
        (("2" (expand "convergent?")
          (("2" (flatten)
            (("2" (lemma "Noetherian_implies_normalizing")
              (("2" (inst?)
                (("2" (assert)
                  (("2" (expand "normalizing?")
                    (("2" (copy -1)
                      (("2" (inst -1 "x")
                        (("2" (inst -2 "y")
                          (("2" (skolem * "u")
                            (("2" (skolem * "v")
                              (("2"
                                (inst -3 "u" "v")
                                (("2"
                                  (assert)
                                  (("2"
                                    (expand "normal_form?")
                                    (("2"
                                      (flatten)
                                      (("2"
                                        (replaces -5)
                                        (("2"
                                          (hide-all-but (-1 -3 1))
                                          (("2"
                                            (case "joinable?(R)(x, y)")
                                            (("1"
                                              (rewrite
                                               "Joinable_implies_Equiv")
                                              nil
                                              nil)
                                             ("2"
                                              (hide 2)
                                              (("2"
                                                (expand "joinable?")
                                                (("2"
                                                  (inst 1 "v")
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (PRED type-eq-decl nil defined_types nil)
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (NFs_implies_Equal formula-decl nil results_normal_form nil)
    (convergent? const-decl "bool" noetherian nil)
    (symmetric? const-decl "bool" relations nil)
    (transitive? const-decl "bool" relations nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (equivalence? const-decl "bool" relations nil)
    (equivalence type-eq-decl nil relations_closure nil)
    (EC const-decl "equivalence" relations_closure nil)
    (RTC_subset_EC formula-decl nil relations_closure nil)
    (T formal-type-decl nil results_normal_form nil)
    (normal_form? const-decl "bool" ars_terminology nil)
    (Noetherian_implies_normalizing formula-decl nil
     results_normal_form nil)
    (Joinable_implies_Equiv formula-decl nil results_confluence nil)
    (joinable? const-decl "bool" ars_terminology nil)
    (normalizing? const-decl "bool" ars_terminology nil))
   shostak)))


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