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Datei: derivatives_lam.pvs Sprache: Lisp

Original von: PVS^©

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                                          ELSIF i = Imin(A!1, jj!1, N - 1) THEN
                                              jj!1
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    (Imin const-decl
     "{i: subrange(ii, jj) | (FORALL j: ii <= j AND j <= jj IMPLIES A(i) <= A(j))}"
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    (swap_perm formula-decl nil sort_array_def nil)
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    (number nonempty-type-decl nil numbers nil)
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    (number_field_pred const-decl "[number -> boolean]" number_fields
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    (bool nonempty-type-eq-decl nil booleans nil)
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    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (N formal-const-decl "nat" sort_array_def nil)
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    (permutation_of? const-decl "bool" permutations nil)
    (below_array type-eq-decl nil below_arrays nil)
    (T formal-type-decl nil sort_array_def nil)
    (below type-eq-decl nil nat_types nil)
    (pred type-eq-decl nil defined_types nil)
    (below type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil))
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(swap_P_and_Q_TCC1 0
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   ("" (skosimp*) (("" (assert) nil nil)) nil)
   ((nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
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                        (("2" (assert)
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                            (("1" (expand "Q")
                              (("1"
                                (lift-if)
                                (("1"
                                  (ground)
                                  (("1"
                                    (inst -8 "i!1" "IMIN")
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                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (lift-if)
                              (("2"
                                (ground)
                                (("1"
                                  (expand "Q")
                                  (("1"
                                    (inst -7 "i!1" "1+j!1")
                                    (("1" (assert) nil nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (expand "Q")
                                  (("2"
                                    (inst -7 "i!1" "IMIN")
                                    (("2" (assert) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
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        (("2" (skosimp*)
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              (("2" (name "IMIN" "Imin(A!1, 1 + j!1, N - 1)")
                (("2" (replace -1)
                  (("2" (hide -1)
                    (("2" (lift-if)
                      (("2" (lift-if)
                        (("2" (ground)
                          (("1" (expand "P")
                            (("1" (inst -4 "j!2")
                              (("1" (assert) nil nil)) nil))
                            nil)
                           ("2" (lift-if)
                            (("2" (ground)
                              (("1"
                                (inst -5 "1+j!1")
                                (("1" (assert) nil nil))
                                nil)
                               ("2"
                                (inst -4 "j!2")
                                (("2" (assert) nil nil))
                                nil))
                              nil))
                            nil)
                           ("3" (lift-if)
                            (("3" (ground)
                              (("1"
                                (inst -8 "i!1" "j!1+1")
                                (("1" (assert) nil nil))
                                nil)
                               ("2"
                                (inst -7 "i!1" "j!2")
                                (("2" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (below_array type-eq-decl nil below_arrays nil)
    (Imin const-decl
     "{i: subrange(ii, jj) | (FORALL j: ii <= j AND j <= jj IMPLIES A(i) <= A(j))}"
     min_array_def nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (subrange type-eq-decl nil integers nil)
    (abv type-eq-decl nil min_array_def nil)
    (<= formal-const-decl "(total_order?[T])" sort_array_def nil)
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    (pred type-eq-decl nil defined_types nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-type-decl nil sort_array_def nil)
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    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
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    (numfield nonempty-type-eq-decl nil number_fields 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
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    (real_ge_is_total_order name-judgement "(total_order?[real])"
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    (Q const-decl "bool" sort_array_def nil)
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    (swap const-decl "below_array" sort_array_def nil)
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   nil))
(asort_P_and_Q 0
  (asort_P_and_Q-1 nil 3281095548
   ("" (induct "jj")
    (("1" (flatten)
      (("1" (skosimp*)
        (("1" (assert)
          (("1" (expand "asort")
            (("1" (prop)
              (("1" (expand "P")
                (("1" (skosimp*)
                  (("1" (case-replace "i!1 = 0")
                    (("1" (case-replace "j!1 = 0")
                      (("1" (expand "swap")
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                        nil)
                       ("2" (assert) nil nil))
                      nil)
                     ("2" (assert) nil nil))
                    nil))
                  nil))
                nil)
               ("2" (expand "Q")
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                      (("1" (lift-if)
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                            (("1" (typepred "Imin(A!1, 0, N - 1)")
                              (("1"
                                (inst -3 "j!1")
                                (("1" (assert) nil nil))
                                nil))
                              nil)
                             ("2" (lift-if)
                              (("2"
                                (assert)
                                (("2"
                                  (ground)
                                  (("1"
                                    (typepred "Imin(A!1, 0, N - 1)")
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                                      (inst?)
                                      (("1" (assert) nil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (typepred "Imin(A!1, 0, N - 1)")
                                    (("2"
                                      (inst -3 "j!1")
                                      (("2" (assert) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assert) nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
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          (("2" (inst?)
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              (("2" (lemma "swap_P_and_Q")
                (("2" (assert)
                  (("2" (inst -1 "asort(A!1,jb!1)" "jb!1")
                    (("2" (ground) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (swap_P_and_Q formula-decl nil sort_array_def nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
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    (real_gt_is_strict_total_order name-judgement
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    (real_lt_is_strict_total_order name-judgement
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     real_props nil)
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    (int_minus_int_is_int application-judgement "int" integers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (total_order? const-decl "bool" orders nil)
    (<= formal-const-decl "(total_order?[T])" sort_array_def nil)
    (abv type-eq-decl nil min_array_def nil)
    (subrange type-eq-decl nil integers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (Imin const-decl
     "{i: subrange(ii, jj) | (FORALL j: ii <= j AND j <= jj IMPLIES A(i) <= A(j))}"
     min_array_def nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (below_induction formula-decl nil bounded_nat_inductions 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)
    (N formal-const-decl "nat" sort_array_def nil)
    (Q const-decl "bool" sort_array_def nil)
    (asort def-decl "below_array" sort_array_def nil)
    (P const-decl "bool" sort_array_def nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (below_array type-eq-decl nil below_arrays nil)
    (T formal-type-decl nil sort_array_def nil)
    (below type-eq-decl nil nat_types nil)
    (pred type-eq-decl nil defined_types nil)
    (below type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil))
   nil)))

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