Quelle majority_array.prf Sprache: Lisp

(majority_array
(is_majority_TCC1 0
  (is_majority_TCC1-1 nil 3506271671
   ("" (skosimp*) (("" (rewrite "finite_below") nil)) nil)
   ((finite_below formula-decl nil finite_sets_below "finite_sets/")
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (below type-eq-decl nil naturalnumbers nil)
    (set type-eq-decl nil sets nil)
    (below type-eq-decl nil nat_types nil)
    (T formal-nonempty-type-decl nil majority_array nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (N formal-const-decl "posnat" majority_array nil))
   nil))
(maj_TCC1 0
  (maj_TCC1-1 nil 3506271671
   (""
    (inst 1 "(LAMBDA A:
             IF maj_exists(A) THEN
                choose({mv: T | is_majority(mv, A)})
             ELSE
                epsilon({mv: T | TRUE})
             ENDIF)")
    (("1" (skosimp*) nil) ("2" (skosimp*) (("2" (assert) nil)))
     ("3" (grind) nil))
    nil)
   ((member const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (TRUE const-decl "bool" booleans nil)
    (epsilon const-decl "T" epsilons nil)
    (pred type-eq-decl nil defined_types nil)
    (choose const-decl "(p)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (set type-eq-decl nil sets nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (is_majority const-decl "bool" majority_array nil)
    (maj_exists const-decl "bool" majority_array nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-nonempty-type-decl nil majority_array nil)
    (below type-eq-decl nil nat_types nil)
    (N formal-const-decl "posnat" majority_array nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_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)
    (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))
(is_majority_unique 0
  (is_majority_unique-1 nil 3506271671
   ("" (skosimp*)
    (("" (expand "is_majority")
      ((""
        (case "disjoint?({ii | A!1(ii) = mv1!1},{ii | A!1(ii) = mv2!1})")
        (("1" (lemma "card_disj_union[below(N)]")
          (("1" (inst?)
            (("1" (assert)
              (("1" (hide -2)
                (("1"
                  (case "card(union({ii | A!1(ii) = mv1!1}, {ii | A!1(ii) = mv2!1})) <= N")
                  (("1" (replace -2)
                    (("1" (hide -2 -3 -4 1) (("1" (assert) nil)))))
                   ("2" (hide -1 -2 -3)
                    (("2" (rewrite "card_below") nil)))
                   ("3" (rewrite "finite_below") nil)))))))
             ("2" (rewrite "finite_below") nil)
             ("3" (rewrite "finite_below") nil)))))
         ("2" (expand "disjoint?")
          (("2" (expand "intersection")
            (("2" (expand "empty?")
              (("2" (expand "member")
                (("2" (skosimp*) (("2" (assert) nil))))))))))))))))
    nil)
   ((is_majority const-decl "bool" majority_array nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (card_disj_union formula-decl nil finite_sets nil)
    (below type-eq-decl nil naturalnumbers nil)
    (finite_below formula-decl nil finite_sets_below "finite_sets/")
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (<= const-decl "bool" reals nil)
    (Card const-decl "nat" finite_sets nil)
    (card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
    (union const-decl "set" sets nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (card_below formula-decl nil finite_sets_below "finite_sets/")
    (finite_set type-eq-decl nil finite_sets nil)
    (mv2!1 skolem-const-decl "T" majority_array nil)
    (is_finite const-decl "bool" finite_sets nil)
    (A!1 skolem-const-decl "[below[N] -> T]" majority_array nil)
    (mv1!1 skolem-const-decl "T" majority_array 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)
    (< const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (N formal-const-decl "posnat" majority_array nil)
    (below type-eq-decl nil nat_types nil)
    (set type-eq-decl nil sets nil)
    (disjoint? const-decl "bool" sets nil)
    (T formal-nonempty-type-decl nil majority_array nil)
    (= const-decl "[T, T -> boolean]" equalities nil))
   nil))
(maj_lem 0
  (maj_lem-1 nil 3506271671
   ("" (skosimp*)
    (("" (prop)
      (("1" (expand "maj_exists") (("1" (inst?) nil)))
       ("2" (typepred "maj(A!1)")
        (("2" (split -1)
          (("1" (lemma "is_majority_unique")
            (("1" (inst -1 "A!1" "mv!1" "maj(A!1)")
              (("1" (assert) nil)))))
           ("2" (expand "maj_exists") (("2" (inst?) nil)))))))
       ("3" (typepred "maj(A!1)") (("3" (assert) nil))))))
    nil)
   ((T formal-nonempty-type-decl nil majority_array nil)
    (maj_exists const-decl "bool" majority_array nil)
    (is_majority_unique formula-decl nil majority_array nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (IMPLIES const-decl "[bool, 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)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (N formal-const-decl "posnat" majority_array nil)
    (below type-eq-decl nil nat_types nil)
    (is_majority const-decl "bool" majority_array nil)
    (maj const-decl "{mv | maj_exists(A) => is_majority(mv, A)}"
     majority_array nil))
   nil))
(maj_subset 0
  (maj_subset-1 nil 3506271671
   ("" (skosimp*)
    (("" (expand "is_majority")
      (("" (case "subset?(IDS!1,{ii | A!1(ii) = mv!1})")
        (("1" (lemma "card_subset[below(N)]")
          (("1" (inst?)
            (("1" (assert) nil) ("2" (rewrite "finite_below") nil)))))
         ("2" (hide -1 2)
          (("2" (expand "subset?")
            (("2" (skosimp*)
              (("2" (expand "member")
                (("2" (inst -2 "x!1")
                  (("2" (assert) nil))))))))))))))))
    nil)
   ((is_majority const-decl "bool" majority_array nil)
    (member const-decl "bool" sets nil)
    (card_subset formula-decl nil finite_sets nil)
    (finite_below formula-decl nil finite_sets_below "finite_sets/")
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (mv!1 skolem-const-decl "T" majority_array nil)
    (A!1 skolem-const-decl "[below[N] -> T]" majority_array 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)
    (< const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (N formal-const-decl "posnat" majority_array nil)
    (below type-eq-decl nil naturalnumbers nil)
    (set type-eq-decl nil sets nil)
    (subset? const-decl "bool" sets nil)
    (is_finite const-decl "bool" finite_sets nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (below type-eq-decl nil nat_types nil)
    (T formal-nonempty-type-decl nil majority_array nil)
    (= const-decl "[T, T -> boolean]" equalities nil))
   nil))
(maj_in_array 0
  (maj_in_array-1 nil 3506271671
   ("" (skosimp*)
    (("" (expand "is_majority")
      (("" (lemma "card_empty?[below(N)]")
        (("" (inst?)
          (("1" (expand "empty?")
            (("1" (expand "member")
              (("1" (iff -1)
                (("1" (flatten)
                  (("1" (split -1)
                    (("1" (assert) nil)
                     ("2" (split -1)
                      (("1" (propax) nil)
                       ("2" (skosimp*) (("2" (inst?) nil)))))))))))))))
           ("2" (rewrite "finite_below") nil))))))))
    nil)
   ((is_majority const-decl "bool" majority_array nil)
    (mv!1 skolem-const-decl "T" majority_array nil)
    (A!1 skolem-const-decl "[below[N] -> T]" majority_array nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (T formal-nonempty-type-decl nil majority_array nil)
    (below type-eq-decl nil nat_types nil)
    (is_finite const-decl "bool" finite_sets nil)
    (set type-eq-decl nil sets nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (member const-decl "bool" sets nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (empty? const-decl "bool" sets nil)
    (finite_below formula-decl nil finite_sets_below "finite_sets/")
    (card_empty? formula-decl nil finite_sets 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)
    (< const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (N formal-const-decl "posnat" majority_array nil)
    (below type-eq-decl nil naturalnumbers nil))
   nil))
(is_majority_const 0
  (is_majority_const-1 nil 3506271671
   ("" (skosimp*)
    (("" (lemma "card_below_fullset")
      (("" (expand "constant_array")
        (("" (expand "is_majority")
          (("" (expand "fullset") (("" (assert) nil))))))))))
    nil)
   ((N formal-const-decl "posnat" majority_array nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (card_below_fullset formula-decl nil finite_sets_below
     "finite_sets/")
    (is_majority const-decl "bool" majority_array nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (fullset const-decl "set" sets nil)
    (constant_array const-decl "below_array" majority_array nil))
   nil))
(maj_const 0
  (maj_const-1 nil 3506271671
   ("" (skosimp*)
    (("" (lemma "is_majority_const")
      (("" (inst?)
        (("" (lemma "maj_lem") (("" (inst?) (("" (ground) nil))))))))))
    nil)
   ((is_majority_const formula-decl nil majority_array nil)
    (maj_lem formula-decl nil majority_array nil)
    (constant_array const-decl "below_array" majority_array nil)
    (below_array type-eq-decl nil below_arrays nil)
    (below type-eq-decl nil naturalnumbers nil)
    (below type-eq-decl nil nat_types nil)
    (N formal-const-decl "posnat" majority_array nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_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)
    (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)
    (T formal-nonempty-type-decl nil majority_array nil))
   nil)))

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