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Datei: Configuration.vdmpp Sprache: Lisp

Original von: PVS^©

(csequence_induction
(cseq_induction 0
  (cseq_induction-1 nil 3513553237
   ("" (skosimp)
    (("" (lemma "every_weak_coinduction")
      ((""
        (inst - "p!1"
         "LAMBDA cseq: (index?(cseq)(0) IMPLIES p!1(nth(cseq, 0))) AND (FORALL n: (index?(cseq)(n) IMPLIES p!1(nth(cseq, n))) IMPLIES (index?(cseq)(n + 1) IMPLIES p!1(nth(cseq, n + 1))))")
        (("" (split)
          (("1" (inst - "cseq!1") (("1" (prop) nil nil)) nil)
           ("2" (delete -1 -2 2)
            (("2" (skosimp)
              (("2" (expand "nth" -)
                (("2" (expand "index?" -)
                  (("2" (expand "nth" 1 1)
                    (("2" (expand "index?" 1 1)
                      (("2" (lift-if)
                        (("2" (ground)
                          (("1" (inst - 0)
                            (("1" (expand* "nth" "index?") nil nil))
                            nil)
                           ("2" (skosimp)
                            (("2" (inst - "1 + n!1")
                              (("2" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil csequence_induction nil)
    (every_weak_coinduction formula-decl nil csequence_codt nil)
    (int_minus_int_is_int application-judgement "int" integers 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)
    (nth def-decl "T" csequence_nth nil)
    (indexes type-eq-decl nil csequence_nth nil)
    (index? def-decl "bool" csequence_nth 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)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (pred type-eq-decl nil defined_types nil)
    (PRED type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil))
   shostak))
(cseq_infinite_induction_TCC1 0
  (cseq_infinite_induction_TCC1-1 nil 3513553168
   ("" (skolem!) (("" (use "index?_infinite") nil nil)) 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)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (index?_infinite formula-decl nil csequence_nth nil)
    (T formal-type-decl nil csequence_induction nil))
   nil))
(cseq_infinite_induction_TCC2 0
  (cseq_infinite_induction_TCC2-1 nil 3513553168
   ("" (skosimp*) (("" (use "index?_infinite") nil nil)) 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)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (index?_infinite formula-decl nil csequence_nth nil)
    (T formal-type-decl nil csequence_induction nil))
   nil))
(cseq_infinite_induction_TCC3 0
  (cseq_infinite_induction_TCC3-1 nil 3513553168
   ("" (skosimp*) (("" (use "index?_infinite") nil nil)) nil)
   ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields 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)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (index?_infinite formula-decl nil csequence_nth nil)
    (T formal-type-decl nil csequence_induction nil))
   nil))
(cseq_infinite_induction 0
  (cseq_infinite_induction-1 nil 3513553548
   ("" (skosimp)
    (("" (use "cseq_induction")
      (("" (assert)
        (("" (skosimp)
          (("" (use "index?_infinite")
            (("" (assert)
              (("" (inst - "n!1") (("" (assert) nil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((cseq_induction formula-decl nil csequence_induction nil)
    (pred type-eq-decl nil defined_types nil)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (T formal-type-decl nil csequence_induction nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers 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)
    (index?_infinite formula-decl nil csequence_nth nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil))
   shostak))
(CSEQ_induction 0
  (CSEQ_induction-1 nil 3513553706
   ("" (skosimp)
    (("" (lemma "every_coinduction")
      ((""
        (inst - "p!1"
         "LAMBDA cseq: (index?(cseq)(0) IMPLIES p!1(nth(cseq, 0))) AND (FORALL n: (FORALL m: m < n IMPLIES (index?(cseq)(m) IMPLIES p!1(nth(cseq, m)))) IMPLIES (index?(cseq)(n) IMPLIES p!1(nth(cseq, n))))")
        (("" (split)
          (("1" (inst - "cseq!1")
            (("1" (prop)
              (("1" (inst - 0)
                (("1" (prop)
                  (("1" (skosimp) (("1" (assert) nil nil)) nil)) nil))
                nil))
              nil))
            nil)
           ("2" (delete -1 2)
            (("2" (skosimp)
              (("2" (lift-if)
                (("2" (expand "nth" -1)
                  (("2" (expand "index?" -1)
                    (("2" (expand "index?" 1 1)
                      (("2" (ground)
                        (("1" (inst - 1)
                          (("1" (expand "nth" -3 2)
                            (("1" (expand "index?" -3 2)
                              (("1"
                                (expand "index?" -3 2)
                                (("1"
                                  (skosimp)
                                  (("1"
                                    (expand "nth")
                                    (("1" (assert) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (skosimp)
                          (("2" (inst -4 "1 + n!1")
                            (("2" (expand "index?" -4 2)
                              (("2"
                                (expand "nth" -4 2)
                                (("2"
                                  (prop)
                                  (("2"
                                    (skosimp)
                                    (("2"
                                      (expand "nth" 1)
                                      (("2"
                                        (ground)
                                        (("2"
                                          (inst - "m!1 - 1")
                                          (("2"
                                            (expand "index?" -2)
                                            (("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)
   ((T formal-type-decl nil csequence_induction nil)
    (every_coinduction formula-decl nil csequence_codt nil)
    (real_lt_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)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (< const-decl "bool" reals nil)
    (nth def-decl "T" csequence_nth nil)
    (indexes type-eq-decl nil csequence_nth nil)
    (index? def-decl "bool" csequence_nth 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)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (pred type-eq-decl nil defined_types 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))
(CSEQ_infinite_induction_TCC1 0
  (CSEQ_infinite_induction_TCC1-1 nil 3513553168
   ("" (skosimp) (("" (use "index?_infinite") nil nil)) 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)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (index?_infinite formula-decl nil csequence_nth nil)
    (T formal-type-decl nil csequence_induction nil))
   nil))
(CSEQ_infinite_induction_TCC2 0
  (CSEQ_infinite_induction_TCC2-1 nil 3513553168
   ("" (skosimp) (("" (use "index?_infinite") nil nil)) 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)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (index?_infinite formula-decl nil csequence_nth nil)
    (T formal-type-decl nil csequence_induction nil))
   nil))
(CSEQ_infinite_induction 0
  (CSEQ_infinite_induction-1 nil 3513554388
   ("" (skosimp)
    (("" (use "CSEQ_induction")
      (("" (assert)
        (("" (skosimp)
          (("" (inst -3 "n!1")
            (("" (assert)
              (("" (skosimp)
                (("" (inst - "m!1")
                  (("" (assert) (("" (use "index?_infinite") nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((CSEQ_induction formula-decl nil csequence_induction nil)
    (pred type-eq-decl nil defined_types nil)
    (infinite_csequence type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (T formal-type-decl nil csequence_induction nil)
    (index?_infinite formula-decl nil csequence_nth nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props 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))
   shostak))
(finite_sequence_induction 0
  (finite_sequence_induction-1 nil 3513554466
   ("" (skosimp*)
    (("" (lemma "nat_induction")
      ((""
        (inst -
         "LAMBDA (n: nat): FORALL fseq: length(fseq) = n IMPLIES sp!1(fseq)")
        (("" (prop)
          (("1" (inst - "length(fseq!1)")
            (("1" (inst - "fseq!1") nil nil)) nil)
           ("2" (skosimp)
            (("2" (lemma "length_empty?_rew" ("cseq" "fseq!2"))
              (("2" (assert) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nat_induction formula-decl nil naturalnumbers nil)
    (length_empty?_rew formula-decl nil csequence_length nil)
    (length def-decl "{n | has_length(fseq, n)}" csequence_length nil)
    (has_length def-decl "bool" csequence_props nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (finite_csequence nonempty-type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (csequence type-decl nil csequence_codt nil)
    (T formal-type-decl nil csequence_induction 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))
(FINITE_SEQUENCE_induction 0
  (FINITE_SEQUENCE_induction-1 nil 3513554747
   ("" (skosimp*)
    (("" (lemma "NAT_induction")
      ((""
        (inst -
         "LAMBDA (n: nat): FORALL fseq: length(fseq) = n IMPLIES sp!1(fseq)")
        (("" (prop)
          (("1" (inst - "length(fseq!1)")
            (("1" (inst - "fseq!1") nil nil)) nil)
           ("2" (skosimp*)
            (("2" (inst -3 "j!1")
              (("2" (split)
                (("1" (inst - "fseq!2") (("1" (assert) nil nil)) nil)
                 ("2" (skosimp)
                  (("2" (inst - "length(fseq!3)")
                    (("2" (assert) (("2" (inst - "fseq!3") nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((NAT_induction formula-decl nil naturalnumbers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (length def-decl "{n | has_length(fseq, n)}" csequence_length nil)
    (has_length def-decl "bool" csequence_props nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (finite_csequence nonempty-type-eq-decl nil csequence_props nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (csequence type-decl nil csequence_codt nil)
    (T formal-type-decl nil csequence_induction 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))
(sequence_induction 0
  (sequence_induction-1 nil 3513554918
   ("" (skosimp*)
    (("" (use "finite_sequence_induction")
      (("" (prop)
        (("" (inst - "cseq!1") (("" (inst - "cseq!1") nil nil)) nil))
        nil))
      nil))
    nil)
   ((finite_sequence_induction formula-decl nil csequence_induction
     nil)
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (T formal-type-decl nil csequence_induction nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (finite_csequence nonempty-type-eq-decl nil csequence_props nil)
    (cseq!1 skolem-const-decl "csequence[T]" csequence_induction nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (infinite_csequence type-eq-decl nil csequence_props nil))
   shostak))
(SEQUENCE_induction 0
  (SEQUENCE_induction-1 nil 3513554944
   ("" (skosimp*)
    (("" (use "FINITE_SEQUENCE_induction")
      (("" (prop)
        (("" (inst - "cseq!1") (("" (inst - "cseq!1") nil nil)) nil))
        nil))
      nil))
    nil)
   ((FINITE_SEQUENCE_induction formula-decl nil csequence_induction
     nil)
    (pred type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (csequence type-decl nil csequence_codt nil)
    (T formal-type-decl nil csequence_induction nil)
    (is_finite inductive-decl "bool" csequence_props nil)
    (finite_csequence nonempty-type-eq-decl nil csequence_props nil)
    (cseq!1 skolem-const-decl "csequence[T]" csequence_induction nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (infinite_csequence type-eq-decl nil csequence_props nil))
   shostak)))

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