Quelle bags_to_sets.prf Sprache: Lisp

(bags_to_sets
(insert_bag_lem 0
  (insert_bag_lem-1 nil 3286292907
   ("" (skosimp*) (("" (apply-extensionality 1) (("" (grind) nil))))
    nil)
   ((T formal-type-decl nil bags_to_sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (add const-decl "(nonempty?)" sets nil)
    (nonempty? const-decl "bool" sets nil)
    (insert const-decl "bag" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (set type-eq-decl nil sets nil) (bag type-eq-decl nil bags 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)
    (number nonempty-type-decl nil numbers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (member const-decl "bool" sets nil))
   nil))
(purge_bag_lem 0
  (purge_bag_lem-1 nil 3286292907
   ("" (skosimp*)
    (("" (apply-extensionality 1 :hide? t) (("" (grind) nil)))) nil)
   ((T formal-type-decl nil bags_to_sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (remove const-decl "set" sets nil)
    (purge const-decl "bag" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (set type-eq-decl nil sets nil) (bag type-eq-decl nil bags 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)
    (number nonempty-type-decl nil numbers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (/= const-decl "boolean" notequal nil)
    (member const-decl "bool" sets nil))
   nil))
(bag_union_lem 0
  (bag_union_lem-1 nil 3286292907
   ("" (skosimp*)
    (("" (expand "bag_to_set")
      (("" (expand "union")
        (("" (expand "member")
          (("" (apply-extensionality 1 :hide? t)
            (("" (expand "max")
              (("" (lift-if)
                (("" (case "A!1(x!1) > 0")
                  (("1" (assert) nil) ("2" (assert) nil))))))))))))))))
    nil)
   ((bag_to_set const-decl "set[T]" bags_to_sets nil)
    (member const-decl "bool" sets nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nonneg_rat_max application-judgement
     "{s: nonneg_rat | s >= q AND s >= r}" real_defs nil)
    (nat_max application-judgement "{k: nat | i <= k AND j <= k}"
     real_defs nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (> const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (>= const-decl "bool" reals nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (bag type-eq-decl nil bags nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-type-decl nil bags_to_sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (union const-decl "bag" bags nil)
    (union const-decl "set" sets nil))
   nil))
(bag_intersection_lem 0
  (bag_intersection_lem-1 nil 3286292907
   ("" (skosimp*)
    (("" (expand "intersection")
      (("" (apply-extensionality 1 :hide? t)
        (("" (expand "member")
          (("" (expand "bag_to_set")
            (("" (expand "min")
              (("" (lift-if)
                (("" (case "A!1(x!1) > 0")
                  (("1" (ground) nil) ("2" (ground) nil))))))))))))))))
    nil)
   ((intersection const-decl "set" sets nil)
    (intersection const-decl "bag" bags nil)
    (> const-decl "bool" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nat_min application-judgement "{k: nat | k <= i AND k <= j}"
     real_defs 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (bag type-eq-decl nil bags nil) (set type-eq-decl nil sets nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (member const-decl "bool" sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil bags_to_sets nil))
   nil))
(subbag_lem 0
  (subbag_lem-1 nil 3286292907
   ("" (skosimp*)
    (("" (expand "subbag?")
      (("" (expand "bag_to_set")
        (("" (expand "subset?")
          (("" (skosimp*)
            (("" (expand "member")
              (("" (inst?) (("" (assert) nil))))))))))))))
    nil)
   ((subbag? const-decl "bool" bags nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (T formal-type-decl nil bags_to_sets nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil))
   nil))
(bag_to_set_emptybag 0
  (bag_to_set_emptybag-1 nil 3286292907
   ("" (expand "bag_to_set")
    (("" (expand "emptybag")
      (("" (expand "emptyset") (("" (propax) nil))))))
    nil)
   ((emptybag const-decl "bag" bags nil)
    (emptyset const-decl "set" sets nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil))
   nil))
(empty_bts_bag 0
  (empty_bts_bag-1 nil 3286292907
   ("" (skosimp*)
    (("" (expand "bag_to_set")
      (("" (expand "empty?")
        (("" (expand "member")
          (("" (skosimp*)
            (("" (inst?)
              (("" (typepred "b!1(x!1)")
                (("" (ground) nil))))))))))))))
    nil)
   ((bag_to_set const-decl "set[T]" bags_to_sets nil)
    (member const-decl "bool" sets nil)
    (T formal-type-decl nil bags_to_sets nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props 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)
    (>= const-decl "bool" 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)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (bag type-eq-decl nil bags nil) (empty? const-decl "bool" sets nil)
    (empty? const-decl "bool" bags nil))
   nil))
(bag_intersection_commutative 0
  (bag_intersection_commutative-1 nil 3286292907
   ("" (skosimp*)
    (("" (expand "intersection")
      (("" (apply-extensionality 1 :hide? t)
        (("1" (grind) nil) ("2" (skosimp*) (("2" (grind) nil))))))))
    nil)
   ((intersection const-decl "bag" bags nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nat_min application-judgement "{k: nat | k <= i AND k <= j}"
     real_defs nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (bag type-eq-decl nil bags 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)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans 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-type-decl nil bags_to_sets nil))
   nil))
(bag_union_commutative 0
  (bag_union_commutative-1 nil 3286292907
   ("" (skosimp*)
    (("" (expand "union")
      (("" (apply-extensionality 1 :hide? t)
        (("1" (grind) nil) ("2" (skosimp*) (("2" (grind) nil))))))))
    nil)
   ((union const-decl "bag" bags nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nonneg_rat_max application-judgement
     "{s: nonneg_rat | s >= q AND s >= r}" real_defs nil)
    (nat_max application-judgement "{k: nat | i <= k AND j <= k}"
     real_defs nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (bag type-eq-decl nil bags nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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)
    (>= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans 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-type-decl nil bags_to_sets nil))
   nil))
(bag_plus_lem 0
  (bag_plus_lem-1 nil 3286292907
   ("" (skosimp*)
    (("" (apply-extensionality 1 :hide? t)
      (("" (expand "bag_to_set")
        (("" (expand "union")
          (("" (expand "member")
            (("" (typepred "A!1(x!1)")
              (("" (typepred "B!1(x!1)") (("" (grind) nil))))))))))))))
    nil)
   ((T formal-type-decl nil bags_to_sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (union const-decl "set" sets nil) (plus const-decl "bag" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (set type-eq-decl nil sets nil) (bag type-eq-decl nil bags 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)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (member const-decl "bool" sets nil))
   nil))
(extract_empty_or_singlton_set 0
  (extract_empty_or_singlton_set-1 nil 3286292927
   ("" (skosimp*) (("" (grind) nil)) nil)
   ((T formal-type-decl nil bags_to_sets nil)
    (extract const-decl "bag" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (singleton? const-decl "bool" sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (> const-decl "bool" 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)
    (bag type-eq-decl nil bags nil)
    (A!1 skolem-const-decl "bag[T]" bags_to_sets nil)
    (t!1 skolem-const-decl "T" bags_to_sets nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
   nil))
(extract_singleton 0
  (extract_singleton-1 nil 3286292940
   ("" (skosimp*)
    (("" (grind) (("" (decompose-equality) (("" (grind) nil)))))) nil)
   ((T formal-type-decl nil bags_to_sets nil)
    (extract const-decl "bag" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (singleton? const-decl "bool" sets nil)
    (nonempty_singleton_finite application-judgement
     "non_empty_finite_set" finite_sets nil)
    (set type-eq-decl nil sets nil) (bag type-eq-decl nil bags 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)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (t!1 skolem-const-decl "T" bags_to_sets nil)
    (A!1 skolem-const-decl "bag[T]" bags_to_sets nil)
    (> const-decl "bool" reals nil)
    (singleton const-decl "(singleton?)" sets nil))
   nil))
(bag_disj_set 0
  (bag_disj_set-1 nil 3286292964 ("" (grind) nil nil)
   ((real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (T formal-type-decl nil bags_to_sets nil)
    (intersection const-decl "bag" bags nil)
    (empty? const-decl "bool" bags nil)
    (disjoint? const-decl "bool" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (member const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil)
    (disjoint? const-decl "bool" sets nil))
   nil))
(bag_set_dist_union 0
  (bag_set_dist_union-1 nil 3286293066
   ("" (skosimp*)
    (("" (decompose-equality +)
      (("" (expand "bag_to_set") (("" (grind) nil))))))
    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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (bag type-eq-decl nil bags nil) (set type-eq-decl nil sets nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (union const-decl "bag" bags nil) (union const-decl "set" sets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil bags_to_sets nil)
    (member const-decl "bool" sets nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nat_max application-judgement "{k: nat | i <= k AND j <= k}"
     real_defs nil)
    (nonneg_rat_max application-judgement
     "{s: nonneg_rat | s >= q AND s >= r}" real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
   nil))
(bag_non_empty 0
  (bag_non_empty-1 nil 3286292999 ("" (grind) nil nil)
   ((real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (T formal-type-decl nil bags_to_sets nil)
    (empty? const-decl "bool" bags nil)
    (bag_to_set const-decl "set[T]" bags_to_sets nil)
    (member const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (nonempty? const-decl "bool" sets nil))
   nil)))

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