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Datei: sort_array.prf Sprache: Lisp

Original von: PVS^©

(sort_array
(sort_TCC1 0
  (sort_TCC1-1 nil 3282575942
   ("" (case "N = 0")
    (("1" (inst + "(LAMBDA (A: [below(N) -> T]): A)")
      (("1" (skosimp*)
        (("1" (rewrite "perm_reflexive")
          (("1" (expand "sorted?")
            (("1" (skosimp*) (("1" (assert) nil nil)) nil)) nil))
          nil))
        nil))
      nil)
     ("2" (inst + "(LAMBDA A: asort(A,N-1))")
      (("1" (skosimp*)
        (("1" (rewrite "asort_perm")
          (("1" (assert)
            (("1" (lemma "asort_P_and_Q")
              (("1" (inst?)
                (("1" (expand "sorted?")
                  (("1" (skosimp*)
                    (("1" (expand "P")
                      (("1" (inst?) (("1" (assert) nil nil)) nil))
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                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (assert) (("2" (skosimp*) (("2" (assert) nil nil)) nil))
        nil))
      nil))
    nil)
   ((int_minus_int_is_int application-judgement "int" integers nil)
    (below_array type-eq-decl nil below_arrays nil)
    (below type-eq-decl nil nat_types nil)
    (pred type-eq-decl nil defined_types nil)
    (total_order? const-decl "bool" orders nil)
    (<= formal-const-decl "(total_order?[T])" sort_array nil)
    (asort def-decl "below_array" sort_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)
    (asort_perm formula-decl nil sort_array_def nil)
    (asort_P_and_Q formula-decl nil sort_array_def nil)
    (P const-decl "bool" sort_array_def nil)
    (< const-decl "bool" reals nil)
    (below type-eq-decl nil naturalnumbers nil)
    (T formal-type-decl nil sort_array nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (permutation_of? const-decl "bool" permutations nil)
    (sorted? const-decl "bool" sort_array nil)
    (perm_reflexive formula-decl nil permutations nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities 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 nil))
   nil))
(sort_lem 0
  (sort_lem-1 nil 3282575942
   ("" (skosimp*)
    (("" (typepred "sort(A!1)") (("" (assert) nil nil)) nil)) nil)
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      sort_array nil)
    (sorted? const-decl "bool" sort_array nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (permutation_of? const-decl "bool" permutations nil)
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    (below type-eq-decl nil naturalnumbers nil)
    (N formal-const-decl "nat" sort_array nil)
    (< const-decl "bool" reals 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)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   nil))
(sort_in? 0
  (sort_in?-1 nil 3282575942
   ("" (skosimp*)
    (("" (lemma "sort_lem")
      (("" (inst?)
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              (("" (inst -1 "sort(A!1)")
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                    (("" (hide -1) (("" (ground) nil nil)) nil)) nil))
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      nil))
    nil)
   ((sort_lem formula-decl nil sort_array nil)
    (sort const-decl "{a | permutation_of?(A, a) AND sorted?(a)}"
      sort_array nil)
    (sorted? const-decl "bool" sort_array nil)
    (permutation_of? const-decl "bool" permutations nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (perm_in? formula-decl nil permutations nil)
    (T formal-type-decl nil sort_array nil)
    (below type-eq-decl nil naturalnumbers nil)
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    (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))
   nil))
(sort_map_TCC1 0
  (sort_map_TCC1-1 nil 3282575942
   (""
    (inst + "(LAMBDA (A: [below(N) -> T]):
        choose({map: [below(N) -> below(N)] |
                             bijective?[below(N), below(N)](map) AND
                                  FORALL i: A(i) = sort(A)(map(i))}))")
    (("" (skosimp*)
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                    (("" (skosimp*)
                      (("" (inst -4 "f!1") (("" (assert) nil nil))
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        nil))
      nil))
    nil)
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    (choose const-decl "(p)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (sort const-decl "{a | permutation_of?(A, a) AND sorted?(a)}"
      sort_array nil)
    (sorted? const-decl "bool" sort_array nil)
    (permutation_of? const-decl "bool" permutations nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (below type-eq-decl nil nat_types nil)
    (bijective? const-decl "bool" functions nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-type-decl nil sort_array nil)
    (below type-eq-decl nil naturalnumbers nil)
    (N formal-const-decl "nat" sort_array nil)
    (< const-decl "bool" reals 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))
   nil))
(sort_map_def 0
  (sort_map_def-1 nil 3282575942
   ("" (skosimp*)
    (("" (assert)
      (("" (typepred "sort_map(A!1)") (("" (inst?) nil nil)) nil))
      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)
    (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)
    (< const-decl "bool" reals nil)
    (N formal-const-decl "nat" sort_array nil)
    (below type-eq-decl nil naturalnumbers nil)
    (bijective? const-decl "bool" functions nil)
    (T formal-type-decl nil sort_array nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (below type-eq-decl nil nat_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (permutation_of? const-decl "bool" permutations nil)
    (sorted? const-decl "bool" sort_array nil)
    (sort const-decl "{a | permutation_of?(A, a) AND sorted?(a)}"
      sort_array nil)
    (sort_map const-decl "{map: [below(N) -> below(N)] |
         bijective?(map) AND (FORALL i: A(i) = sort(A)(map(i)))}"
     sort_array nil))
   nil))
(sort_map_bij 0
  (sort_map_bij-1 nil 3282575942
   ("" (skosimp*) (("" (assert) nil nil)) nil) nil nil))
(isort_map_TCC1 0
  (isort_map_TCC1-1 nil 3282575942
   (""
    (inst + "(LAMBDA (A: [below(N) -> T]):
             choose({map: [below(N) -> below(N)] |
                              bijective?[below(N), below(N)](map) AND
                                 FORALL i: A(map(i)) = sort(A)(i)}))")
    (("" (skosimp*)
      (("" (expand "nonempty?")
        (("" (expand "empty?")
          (("" (expand "member")
            (("" (lemma "sort_lem")
              (("" (inst?)
                (("" (flatten)
                  (("" (expand "permutation_of?")
                    (("" (skosimp*)
                      (("" (case "N = 0")
                        (("1" (inst -5 "f!1")
                          (("1" (assert) (("1" (skosimp*) nil nil))
                            nil))
                          nil)
                         ("2" (inst -4 "inverse(f!1)")
                          (("1" (prop)
                            (("1"
                              (lemma
                               "bij_inv_is_bij[below(N), below(N)]")
                              (("1"
                                (inst?)
                                (("1" (assert) nil nil))
                                nil)
                               ("2"
                                (inst + "0")
                                (("2" (assert) nil nil))
                                nil))
                              nil)
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                              (("2"
                                (inst -2 "inverse(f!1)(i!1)")
                                (("1"
                                  (assert)
                                  (("1"
                                    (expand "bijective?")
                                    (("1"
                                      (flatten)
                                      (("1"
                                        (lemma
                                         "comp_inverse_right_surj[below(N),below(N)]")
                                        (("1"
                                          (inst -1 "i!1" "f!1")
                                          (("1" (assert) nil nil))
                                          nil)
                                         ("2" (inst + "0") nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2" (inst + "0") nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (inst + "0") (("2" (assert) nil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((empty? const-decl "bool" sets nil)
    (sort_lem formula-decl nil sort_array nil)
    (inverse const-decl "D" function_inverse nil)
    (surjective? const-decl "bool" functions nil)
    (f!1 skolem-const-decl "[below(N) -> below(N)]" sort_array nil)
    (comp_inverse_right_surj formula-decl nil function_inverse nil)
    (bij_inv_is_bij formula-decl nil function_inverse nil)
    (member const-decl "bool" sets nil)
    (choose const-decl "(p)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (sort const-decl "{a | permutation_of?(A, a) AND sorted?(a)}"
      sort_array nil)
    (sorted? const-decl "bool" sort_array nil)
    (permutation_of? const-decl "bool" permutations nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (below type-eq-decl nil nat_types nil)
    (bijective? const-decl "bool" functions nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-type-decl nil sort_array nil)
    (below type-eq-decl nil naturalnumbers nil)
    (N formal-const-decl "nat" sort_array nil)
    (< const-decl "bool" reals 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))
   nil))
(isort_map_def 0
  (isort_map_def-1 nil 3282575942
   ("" (skosimp*)
    (("" (assert)
      (("" (typepred "isort_map(A!1)") (("" (inst?) nil nil)) nil))
      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)
    (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)
    (< const-decl "bool" reals nil)
    (N formal-const-decl "nat" sort_array nil)
    (below type-eq-decl nil naturalnumbers nil)
    (bijective? const-decl "bool" functions nil)
    (T formal-type-decl nil sort_array nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (below type-eq-decl nil nat_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (permutation_of? const-decl "bool" permutations nil)
    (sorted? const-decl "bool" sort_array nil)
    (sort const-decl "{a | permutation_of?(A, a) AND sorted?(a)}"
      sort_array nil)
    (isort_map const-decl "{map: [below(N) -> below(N)] |
         bijective?(map) AND (FORALL i: A(map(i)) = sort(A)(i))}"
     sort_array nil))
   nil))
(isort_map_bij 0
  (isort_map_bij-1 nil 3282575942
   ("" (skosimp*) (("" (assert) nil nil)) nil) nil nil)))

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