Quelle finite_bags.prf
Sprache: Lisp
(finite_bagsnil 3249305393 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil finite_bags nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil )"bool" functions nil )"bool" finite_sets nil ))nil ))nil 3289826684"" (skosimp*)"" (expand "is_finite" )"" (expand "is_finite" )"" (skolem -1 ("n!1" "f!1" ))"" (inst + "n!1" "f!1" )"1" (expand "injective?" )"1" (skosimp*)"1" (inst?)"1" (assert ) nil nil ) ("2" (grind) nil nil )"3" (expand "bag_to_set" ) (("3" (assert ) nil nil ))nil ))nil ))nil ))nil )"2" (skosimp*) (("2" (grind) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"bool" finite_sets nil )"bool" finite_bags nil )"(strict_total_order?[real])" real_props nil )nil application-judgement "finite_set" finite_bags nil )"bool" functions nil )nil nat_types nil )"bool" reals nil )nil finite_bags nil )nil booleans nil )nil booleans nil )"[bool, bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil ) (set type-eq-decl nil sets nil )"set[T]" bags_to_sets nil )nil finite_bags nil )"finite_bag" finite_bags nil )"bool" reals nil ))nil 3249305393"" (expand "emptybag" )"" (lemma "finite_emptyset" )"" (expand "emptyset" )"" (expand "is_finite" 1)"" (expand "bag_to_set" ) (("" (propax) nil nil )) nil ))nil ))nil ))nil ))nil )nil finite_bags nil )nil finite_sets nil )"bool" finite_bags nil )"set[T]" bags_to_sets nil )"set" sets nil )"bag" bags nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "singleton_bag" )"" (expand "is_finite" )"" (expand "bag_to_set" )"" (expand "is_finite" )"" "1" "(LAMBDA (t: {t: T | IF t = t!1 THEN 1 ELSE 0 ENDIF > 0}): 0)" )"" (expand "injective?" )"" (skosimp*)"" (typepred "x1!1" )"" (typepred "x2!1" )"" (ground)"" (lift-if)"" (prop)"1" (assert ) nil nil )"2" (assert ) nil nil )"3" (assert ) nil nil )"4" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bag" bags nil )"set[T]" bags_to_sets nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil finite_bags nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"[T, T -> boolean]" equalities nil )"bool" reals nil )nil nat_types nil )"(strict_total_order?[real])" real_props nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" functions nil )"bool" finite_sets nil )"bool" finite_bags nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "is_finite" )"" (rewrite "insert_bag_lem" )"" (rewrite "finite_add" ) nil nil )) nil ))nil ))nil )"bool" finite_bags nil )nil finite_sets nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )"set[T]" bags_to_sets nil )"non_empty_finite_set[T]" finite_bags nil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )nil finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bags_to_sets nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "is_finite" )"" (rewrite "purge_bag_lem" )"" (rewrite "finite_remove" ) nil nil )) nil ))nil ))nil )"bool" finite_bags nil )nil finite_sets nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )"set[T]" bags_to_sets nil )"finite_set[T]" finite_bagsnil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )nil finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bags_to_sets nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "delete" )"" (typepred "B!1" )"" (expand "is_finite" )"" (expand "bag_to_set" )"" (expand "is_finite" )"" (skosimp*)"" "N!1" "(LAMBDA (t:{t: T | IF t!1 = t THEN IF B!1(t) >= n!1 THEN B!1(t) - n!1") "1" (expand "injective?" )"1" (skosimp*)"1" (inst?)"1" (assert ) nil nil )"2" (assert )"2" (assert )"2" (typepred "x2!1" )"2" "2" "2" (ground) nil nil ))nil ))nil ))nil ))nil ))nil )"3" (typepred "x1!1" )"3" (lift-if) (("3" (ground) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (skosimp*)"2" (typepred "t!2" )"2" (lift-if) (("2" (ground) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"bag" bags nil )"bool" finite_sets nil )"int" integers nil )"[numfield, numfield -> numfield]" number_fields nil )"nat" finite_bags nil )"finite_bag" finite_bags nil )"T" finite_bags nil )"[T, T -> boolean]" equalities nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil number_fields nil )"bool" reals nil ) (< const-decl "bool" reals nil )nil nat_types nil )"bool" functions nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"set[T]" bags_to_sets nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil finite_bags nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "is_finite" )"" (rewrite "bag_union_lem" )"" (rewrite "finite_union" ) nil nil )) nil ))nil ))nil )"bool" finite_bags nil )nil finite_sets nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )"set[T]" bags_to_sets nil )"finite_set[T]" finite_bagsnil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )nil finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bags_to_sets nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "is_finite" )"" (rewrite "bag_intersection_lem" )"" (lemma "finite_intersection" ) (("" (inst?) nil nil )) nil ))nil ))nil ))nil )"bool" finite_bags nil )nil finite_sets nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )"set[T]" bags_to_sets nil )"finite_set[T]" nil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )nil finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bags_to_sets nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "is_finite" )"" (rewrite "bag_plus_lem" )"" (rewrite "finite_union" ) nil nil )) nil ))nil ))nil )"bool" finite_bags nil )nil finite_sets nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )"set[T]" bags_to_sets nil )"finite_set[T]" finite_bagsnil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )nil finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil bags_to_sets nil ))nil ))nil 3249305393"" (skosimp*)"" (typepred "B!1" )"" (expand "is_finite" )"" (case "B!1(x!1) = 0" )"1" (case "xn!1 = 0 " )"1" (replace -1)"1" (hide -1)"1" (case-replace "B!1 WITH [x!1 := 0] = B!1" )"1" (assert )"1" (apply-extensionality :hide? t)"1" (lift-if) (("1" (ground) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" "bag_to_set(B!1 WITH [x!1 := xn!1]) = add(x!1, bag_to_set(B!1))" )"1" (hide -1) (("1" (rewrite "finite_add" ) nil nil ))nil )"2" (hide -1 -2 3)"2" (apply-extensionality :hide? t)"2" (grind) nil nil )) nil ))nil ))nil ))nil )"2" (case "xn!1 = 0 " )"1" (replace -1)"1" (hide -1)"1" (lemma "finite_subset" )"1" "bag_to_set(B!1)" "bag_to_set(B!1 WITH [x!1 := 0])" )"1" (assert )"1" (hide -1 3) (("1" (grind) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"2" "bag_to_set(B!1 WITH [x!1 := xn!1]) = bag_to_set(B!1)" )"2" (apply-extensionality :hide? t)"2" (hide -1 4) (("2" (grind) nil nil )) nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )nil finite_bags nil )"bool" finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )nil finite_bags nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )"[T, T -> boolean]" equalities nil )"non_empty_finite_set[T]" finite_bags nil )set type-eq-decl nil sets nil )"set[T]" bags_to_sets nil )"bool" sets nil )"(nonempty?)" sets nil )nil finite_sets nil )"bool" finite_sets nil )nil finite_sets nil )"(strict_total_order?[real])" real_props nil )"bool" sets nil )nil application-judgement "finite_set" finite_bags nil )nil finite_sets nil )"(partial_order?[set[T]])" nil )"(partial_order?[set[T]])" nil )"bool" sets nil ))nil ))nil 3286293268 ("" (grind) nil nil )nil application-judgement "finite_set" finite_bags nil )) nil ))nil 3286293280"" (skosimp*)"" (expand "is_finite" )"" (expand "extract" )"" (lemma "finite_subset" )"" (inst?)"" (inst - "bag_to_set(B!1)" )"" (assert )"" (expand "subset?" )"" (skosimp*)"" (expand "member" )"" (expand "bag_to_set" )"" (lift-if)"" (assert ) nil ))))))))))))))))))))))))nil )"bool" finite_bags nil )nil finite_bags nil )nil finite_sets nil )nil application-judgement "finite_set" finite_bags nil )nil finite_sets nil )"bool" finite_sets nil )"bool" sets nil )"bool" sets nil )"(strict_total_order?[real])" real_props nil )"(partial_order?[set[T]])" nil )"(partial_order?[set[T]])" nil )nil finite_bags nil )"[T, T -> boolean]" equalities nil )IF const-decl "[boolean, T, T -> T]" if_def nil )"set[T]" bags_to_sets nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )"bag" bags nil ))nil ))nil 3249305418"" (skosimp*) (("" (typepred "x!1" ) (("" (grind) nil nil )) nil ))nil )nil finite_bags nil )"bool" bags nil )nil finite_bags nil )"bool" finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )nil finite_bags nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )nil application-judgement "finite_set" finite_bags nil )"bool" bags nil ))nil 3249305393"" (skosimp*) (("" (rewrite "finite_singleton_bag" ) nil nil )) nil )nil finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (skosimp*) (("" (rewrite "finite_bag_union" ) nil nil )) nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (skosimp*) (("" (rewrite "finite_bag_intersection" ) nil nil ))nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (skosimp*) (("" (rewrite "finite_bag_plus" ) nil nil )) nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (skosimp*)"" (typepred "NB!1" ) (("" (hide -1) (("" (grind) nil nil )) nil ))nil ))nil )nil finite_bags nil )"bool" bags nil )nil finite_bags nil )"bool" finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )nil finite_bags nil )NOT const-decl "[bool -> bool]" booleans nil )nil booleans nil )nil booleans nil )nil application-judgement "finite_bag" finite_bags nil )max const-decl "{p: real | p >= m AND p >= n}" real_defs nil )"bag" bags nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3249305393 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil )nil finite_bags nil )"nonneg_int" nil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )"bag" bags nil )"bool" bags nil )nil application-judgement "finite_bag" finite_bags nil ))nil ))nil 3249305393 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil )nil finite_bags nil )"{k: nat | i <= k AND j <= k}" nil )"{s: nonneg_rat | s >= q AND s >= r}" real_defs nil )"(strict_total_order?[real])" real_props nil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )"bag" bags nil )"bool" bags nil )nil application-judgement "finite_bag" finite_bags nil ))nil ))nil 3249305393 ("" (subtype-tcc) nil nil )nil booleans nil )nil booleans nil )NOT const-decl "[bool -> bool]" booleans nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil )nil finite_bags nil )"nonneg_int" nil )nil application-judgement "finite_set" finite_bags nil )nil finite_bags nil )"bag" bags nil )"bool" bags nil )nil application-judgement "finite_bag" finite_bags nil ))nil ))nil 3249305393"" (skosimp*)"" (rewrite "finite_insert" )"" (assert ) (("" (grind) nil nil )) nil )) nil ))nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil )"bag" bags nil )"bool" bags nil )"posint" nil )"even_int" integersnil )"int" integers nil )"odd_int" integersnil ))nil ))nil 3249305393"" (skosimp*) (("" (rewrite "finite_purge" ) nil nil )) nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (skosimp*) (("" (rewrite "finite_delete" ) nil nil )) nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (rewrite "finite_emptybag" ) nil nil )nil finite_bags nil )) nil ))nil 3286293715"" (skosimp*) (("" (rewrite "finite_extract" ) nil nil )) nil )nil finite_bags nil )nil finite_bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil ))nil )nil 3286293634 ("" (postpone) nil nil ) nil shostak))nil 3249305393"" (expand "emptybag" )"" (expand "card" )"" (expand "bag_to_set" )"" (lemma "sum_emptyset" )"" (inst?)"" (expand "emptyset" ) (("" (propax) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"nat" finite_bags nil )"[numfield, numfield -> numfield]" number_fields nil )"R" restrict nil )nil number_fields nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil finite_bags nil )nil finite_sets_sum "finite_sets/" )"set" sets nil )"set[T]" bags_to_sets nil )"bag" bags nil ))nil ))nil 3249305393"" (skosimp*)"" (rewrite "emptybag_is_empty?" )"" (iff)"" (split)"1" (flatten)"1" (expand "card" )"1" (expand "emptybag" )"1" (use "sum_is_null" )"1" (assert )"1" (case "EXISTS x: B!1(x) > 0" )"1" (skosimp*)"1" (inst -2 "x!1" )"1" (assert )"1" (expand "bag_to_set" )"1" (propax) nil nil )) nil ))nil ))nil ))nil )"2" (grind)"2" (use "emptybag_is_empty?" )"2" (expand "emptybag" )"2" (expand "empty?" )"2" "2" "2" "2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (use "card_emptybag" )"2" (flatten) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil ))nil )nil bags nil )nil numbers nil )nil booleans nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil )nil bags nil )"bool" finite_bags nil )nil finite_bags nil )nil finite_bags nil )nil name-judgement "finite_bag" finite_bags nil )"nat" finite_bags nil )nil finite_sets_sum_real "finite_sets/" )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )"set[T]" bags_to_sets nil )nil application-judgement "finite_set" finite_bags nil )"bool" reals nil )"(strict_total_order?[real])" real_props nil )"bool" bags nil )"bool" sets nil )"bool" sets nil )nil application-judgement "int" finite_bags nil )nil application-judgement "nat" finite_bags nil )"bag" bags nil )nil finite_bags nil ))nil ))nil 3249305393"" (expand "singleton_bag" )"" (expand "card" )"" (expand "bag_to_set" )"" (skosimp*)"" case "{t_3: T | IF t_3 = t!1 THEN 1 ELSE 0 ENDIF > 0} = ") "1" (replace -1)"1" (hide -1)"1" (lemma "sum_singleton" )"1" (inst?) (("1" (assert ) nil nil )) nil )) nil ))nil ))nil )"2" (hide 2)"2" (apply-extensionality 1 :hide? t)"2" (grind) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil )"nat" finite_bags nil )"(strict_total_order?[real])" real_props nil )nil finite_sets_sum "finite_sets/" )nil number_fields nil )"R" restrict nil )"[numfield, numfield -> numfield]" number_fields nil )"odd_int" integers nil )"posint" nil )nil application-judgement "nat" finite_bags nil )nil application-judgement "int" finite_bags nil )"odd_int" integersnil )"non_empty_finite_set[T]" finite_bags nil )nil finite_bags nil )nil booleans nil )nil booleans nil )"[T, T -> boolean]" equalities nil )nil numbers nil )"[number -> boolean]" number_fieldsnil )nil number_fields nil )"[number_field -> boolean]" reals nil )nil reals nil )"bool" reals nil )IF const-decl "[boolean, T, T -> T]" if_def nil )set type-eq-decl nil sets nil )"bool" sets nil )"(singleton?)" sets nil )"set[T]" bags_to_sets nil )"bag" bags nil ))nil ))nil 3249305393"" (skosimp*)"" (expand "subbag?" )"" (expand "card" )"" (lemma "sum_order_sub" )"" (inst?)"" (split -1)"1" (propax) nil nil )"2" (hide 2)"2" (expand "subset?" )"2" (expand "member" )"2" (expand "bag_to_set" )"2" (skosimp*)"2" (inst?) (("2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil )"3" (propax) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"bool" bags nil )nil finite_bags nil )nil finite_sets_sum_real"finite_sets/" )"bool" sets nil )"(strict_total_order?[real])" real_props nil )"(total_order?[real])" nil )"bool" sets nil )nil real_types nil )nil finite_bags nil )"bool" finite_bags nil )"set[T]" bags_to_sets nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil numbers nil )nil finite_sets nil )"bool" finite_sets nil )set type-eq-decl nil sets nil )nil booleans nil )nil booleans nil )nil application-judgement "finite_set" finite_bags nil )"nat" finite_bags nil ))nil ))nil 3249305393"" (skosimp*) (("" (use "finite_update" ) nil nil )) nil )nil finite_bags nil )"bool" finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil booleans nil )nil numbers nil )nil finite_bags nil )nil finite_bags nil ))nil ))nil 3495462629"" (skosimp*)"" (case "B!1(x!1) = 0" )"1" (expand "card" )"1" (case "xn!1 = 0" )"1" (replace * -1 -2)"1" (assert )"1" (case "B!1 WITH [x!1 := 0] = B!1" )"1" (replace -1) (("1" (propax) nil nil )) nil )"2" (apply-extensionality 1)"2" (lift-if) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"2" case "bag_to_set(B!1 WITH [x!1 := xn!1]) = add(x!1,bag_to_set(B!1))" )"1" (replace -1)"1" (rewrite "sum_add" )"1" (case "NOT member(x!1, bag_to_set(B!1))" )"1" (replace 1)"1" (assert )"1" (replace -2)"1" (apply-eta "B!1" )"1" (replace -1 :hide? t)"1" (apply-eta "B!1 WITH [x!1 := xn!1]" )"1" replace -1 :hide? t)"1" "sum_particular_gen" )"1" replace -1)"1" assert )"1" "1" expand "member" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "bag_to_set" )"2" (expand "member" ) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 3)"2" (apply-extensionality 1 :hide? t)"2" (grind) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (expand "card" )"2" (apply-eta " B!1 WITH [x!1 := xn!1]" )"2" (replace -1 :hide? t)"2" (apply-eta " B!1" )"2" (replace -1 :hide? t)"2" (case "xn!1 = 0" )"1" case "bag_to_set(B!1) = add(x!1,bag_to_set(B!1 WITH [x!1 := xn!1]))" )"1" (replace -1 :hide? t)"1" case "is_finite[T](bag_to_set[T](B!1 WITH [x!1 := xn!1]))" )"1" (rewrite "sum_add" )"1" (expand "member" )"1" (lift-if)"1" "1" "1" assert )"1" expand "bag_to_set" )"1" (assert ) nil nil ))nil ))nil ))nil )"2" "2" assert )"2" "sum_particular_gen" )"2" replace -1 :hide? t)"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 3)"2" (rewrite "bag_to_set_TCC1" )"2" (hide 2)"2" (rewrite "finite_update" ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "bag_to_set" )"2" (expand "add" )"2" (expand "member" )"2" (apply-extensionality 1 :hide? t)"2" (case "x!1 = x!2" )"1" (ground) nil nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "bag_to_set(B!1 WITH [x!1 := xn!1]) = bag_to_set(B!1)" )"1" (replace -1 :hide? t)"1" (use "sum_particular_gen" )"1" (replace -1 :hide? t)"1" (lift-if)"1" (expand "bag_to_set" )"1" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (expand "bag_to_set" )"2" (apply-extensionality 1 :hide? t)"2" (lift-if) (("2" (assert ) nil nil )) nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )nil finite_bags nil )"bool" finite_bags nil )nil bags nil )nil naturalnumbers nil )"bool" reals nil )nil booleans nil )nil integers nil )"[rational -> boolean]" integers nil )nil rationals nil )"[real -> boolean]" rationals nil )nil reals nil )"[number_field -> boolean]" reals nil )nil number_fields nil )"[number -> boolean]" number_fieldsnil )nil finite_bags nil )"[T, T -> boolean]" equalities nil )nil booleans nil )nil numbers nil )nil application-judgement "finite_set" finite_bags nil )"int" integers nil )nil application-judgement "int" finite_bags nil )nil application-judgement "nat" finite_bags nil )"int" integers nil )"odd_int" integers nil )NOT const-decl "[bool -> bool]" booleans nil )"bool" sets nil )nil finite_sets_sum_real"finite_sets/" )"real" reals nil )"(strict_total_order?[real])" real_props nil )"nonneg_int" nil )"[numfield, numfield -> numfield]" number_fields nil )"R" restrict nil )nil number_fields nil )nil finite_sets nil )"bool" finite_sets nil )nil finite_sets_sum "finite_sets/" )"non_empty_finite_set[T]" finite_bags nil )set type-eq-decl nil sets nil )"set[T]" bags_to_sets nil )"bool" sets nil )"(nonempty?)" sets nil )"nat" finite_bags nil )OR const-decl "[bool, bool -> bool]" booleans nil )"bool" reals nil )nil finite_bags nil )nil finite_bags nil )"real" reals nil ))nil )nil 3249305393"" (skosimp*)"" (case "B!1(x!1) = 0" )"1" (expand "card" )"1" (case "xn!1 = 0" )"1" (replace * -1 -2)"1" (assert )"1" (case "B!1 WITH [x!1 := 0] = B!1" )"1" (replace -1) (("1" (propax) nil nil )) nil )"2" (apply-extensionality 1)"2" (lift-if) (("2" (assert ) nil nil )) nil )) nil ))nil ))nil ))nil )"2" case "bag_to_set(B!1 WITH [x!1 := xn!1]) = add(x!1,bag_to_set(B!1))" )"1" (replace -1)"1" (rewrite "sum_add" )"1" (case "NOT member(x!1, bag_to_set(B!1))" )"1" (replace 1)"1" (assert )"1" (replace -2)"1" (apply-eta "B!1" )"1" (replace -1 :hide? t)"1" (apply-eta "B!1 WITH [x!1 := xn!1]" )"1" replace -1 :hide? t)"1" "sum_particular_gen" )"1" replace -1)"1" assert )"1" "1" expand "member" )"1" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "bag_to_set" )"2" (expand "member" ) (("2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 3)"2" (apply-extensionality 1 :hide? t)"2" (grind) nil nil )) nil ))nil ))nil ))nil ))nil )"2" (expand "card" )"2" (apply-eta " B!1 WITH [x!1 := xn!1]" )"2" (replace -1 :hide? t)"2" (apply-eta " B!1" )"2" (replace -1 :hide? t)"2" (case "xn!1 = 0" )"1" case "bag_to_set(B!1) = add(x!1,bag_to_set(B!1 WITH [x!1 := xn!1]))" )"1" (replace -1 :hide? t)"1" case "is_finite[T](bag_to_set[T](B!1 WITH [x!1 := xn!1]))" )"1" (rewrite "sum_add" )"1" (expand "member" )"1" (lift-if)"1" "1" "1" assert )"1" expand "bag_to_set" )"1" (assert ) nil nil ))nil ))nil ))nil )"2" "2" assert )"2" "sum_particular_gen" )"2" replace -1 :hide? t)"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 3)"2" (rewrite "bag_to_set_TCC1" )"2" (hide 2)"2" (rewrite "finite_update" ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (expand "bag_to_set" )"2" (expand "add" )"2" (expand "member" )"2" (apply-extensionality 1 :hide? t)"2" (case-old-lift-if 1)"1" (assert ) nil nil )"2" (assert ) nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" case "bag_to_set(B!1 WITH [x!1 := xn!1]) = bag_to_set(B!1)" )quality 100%
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