|
(set_of_functions
(empty_finite_domain1 0
(empty_finite_domain1-1 nil 3306495597
("" (skosimp)
(("" (rewrite "empty_card" :dir rl)
(("" (inst + "LAMBDA (x: (A!1)): epsilon! (y: (B!1)): TRUE")
(("" (grind) nil nil)) nil))
nil))
nil)
((empty_card formula-decl nil finite_sets nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(T1 formal-type-decl nil set_of_functions nil)
(member const-decl "bool" sets nil)
(empty? const-decl "bool" sets nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(A!1 skolem-const-decl "finite_set[T1]" set_of_functions nil)
(B!1 skolem-const-decl "finite_set[T2]" set_of_functions nil)
(TRUE const-decl "bool" booleans nil)
(epsilon const-decl "T" epsilons nil)
(pred type-eq-decl nil defined_types nil)
(T2 formal-type-decl nil set_of_functions nil))
shostak))
(empty_finite_domain2 0
(empty_finite_domain2-1 nil 3306495668
("" (skosimp*)
(("" (rewrite "empty_card" :dir rl) (("" (grind-with-ext) nil nil))
nil))
nil)
((empty_card formula-decl nil finite_sets nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(T1 formal-type-decl nil set_of_functions nil)
(T2 formal-type-decl nil set_of_functions nil)
(empty? const-decl "bool" sets nil)
(member const-decl "bool" sets nil))
shostak))
(empty_finite_range 0
(empty_finite_range-1 nil 3306495707
("" (skosimp*)
(("" (rewrite "empty_card" :dir rl)
(("" (rewrite "empty_card" :dir rl)
(("" (grind :if-match nil)
(("" (inst - "f!1(x!1)") (("" (assert) nil nil)) nil)) nil))
nil))
nil))
nil)
((empty_card formula-decl nil finite_sets nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(T2 formal-type-decl nil set_of_functions nil)
(empty? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(T1 formal-type-decl nil set_of_functions nil))
shostak))
(finite_funset_bijection1 0
(finite_funset_bijection1-1 nil 3306495777
("" (skosimp)
(("" (case-replace "n!1 = 0")
(("1" (lemma "empty_domain1" ("m" "m!1"))
(("1" (skolem!)
(("1" (inst + "LAMBDA (h: [(A!1) -> (B!1)]): f!1")
(("1" (expand* "bijective?" "injective?" "surjective?")
(("1" (ground)
(("1" (rewrite "empty_finite_domain2") nil nil)
("2" (use "empty_finite_domain1" ("B" "B!1"))
(("2" (assert)
(("2" (skosimp*)
(("2" (inst + "f!2")
(("2" (apply-extensionality) nil nil)) nil))
nil))
nil))
nil))
nil))
nil)
("2" (assert) nil nil))
nil))
nil))
nil)
("2" (case "EXISTS (x: (A!1)): TRUE")
(("1" (assert)
(("1" (hide -1)
(("1" (case-replace "m!1 = 0")
(("1" (use "empty_finite_range")
(("1" (assert)
(("1" (lemma "empty_range" ("m" "m!1" "p" "n!1"))
(("1" (split)
(("1"
(inst +
"LAMBDA (h: [(A!1) -> (B!1)]): (LAMBDA (x: below[n!1]): 0)")
(("1"
(expand* "bijective?" "injective?"
"surjective?")
(("1" (grind :if-match nil :defs nil)
(("1" (inst - "y!1") nil nil)
("2" (inst -2 "x1!1") nil nil))
nil))
nil)
("2" (skolem!) (("2" (inst -2 "h!1") nil nil))
nil))
nil)
("2" (propax) nil nil))
nil))
nil))
nil))
nil)
("2" (use "empty_card[T1]")
(("2" (use "empty_card[T2]")
(("2" (assert)
(("2" (hide 1 2)
(("2" (rewrite "card_bij")
(("2" (rewrite "card_bij")
(("2" (skosimp*)
(("2" (assert)
(("2"
(inst
+
"LAMBDA (h: [(A!1) -> (B!1)]): f!2 o h o inverse(f!1)")
(("1"
(expand "bijective?" +)
(("1"
(expand*
"injective?"
"surjective?")
(("1"
(grind :if-match nil :defs nil)
(("1"
(inst
+
"inverse(f!2) o y!1 o f!1")
(("1"
(apply-extensionality
3
:hide?
t)
(("1"
(expand "o")
(("1"
(use
"comp_inverse_right"
("f" "f!1"))
(("1"
(use
"comp_inverse_right"
("f" "f!2"))
(("1" (assert) nil nil)
("2"
(reveal 3)
(("2"
(expand "empty?")
(("2"
(skosimp*)
(("2"
(expand
"member")
(("2"
(inst
+
"x!2")
nil
nil))
nil))
nil))
nil))
nil))
nil)
("2"
(reveal -4)
(("2"
(propax)
nil
nil))
nil))
nil))
nil)
("2"
(reveal 3)
(("2"
(expand*
"empty?"
"member")
(("2"
(skolem!)
(("2" (inst?) nil nil))
nil))
nil))
nil))
nil)
("2"
(reveal 3)
(("2"
(expand* "empty?" "member")
(("2"
(skolem!)
(("2" (inst?) nil nil))
nil))
nil))
nil))
nil)
("2"
(rewrite
"composition_right_cancel2")
(("1"
(rewrite
"composition_left_cancel2")
nil
nil)
("2"
(rewrite "bij_inv_is_bij")
nil
nil))
nil))
nil))
nil))
nil)
("2"
(skolem!)
(("2"
(reveal 3)
(("2"
(expand* "empty?" "member")
(("2"
(skolem!)
(("2" (inst?) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (delete -2 3)
(("2" (rewrite "nonempty_exists" :dir rl)
(("2" (rewrite "nonempty_card") (("2" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
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)
(injective? const-decl "bool" functions nil)
(surjective? const-decl "bool" functions nil)
(bijective? const-decl "bool" functions nil)
(empty_finite_domain1 formula-decl nil set_of_functions nil)
(empty_finite_domain2 formula-decl nil set_of_functions nil)
(FALSE const-decl "bool" booleans nil)
(below type-eq-decl nil nat_types nil)
(T1 formal-type-decl nil set_of_functions nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(A!1 skolem-const-decl "finite_set[T1]" set_of_functions nil)
(T2 formal-type-decl nil set_of_functions nil)
(B!1 skolem-const-decl "finite_set[T2]" set_of_functions nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(< const-decl "bool" reals nil)
(n!1 skolem-const-decl "nat" set_of_functions nil)
(empty_domain1 formula-decl nil fun_below_props nil)
(nonempty_card formula-decl nil finite_sets nil)
(nonempty_exists formula-decl nil sets_lemmas nil)
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(empty_range formula-decl nil fun_below_props nil)
(empty_finite_range formula-decl nil set_of_functions nil)
(x!1 skolem-const-decl "T1" set_of_functions nil)
(x!1 skolem-const-decl "T2" set_of_functions nil)
(comp_inverse_right formula-decl nil function_inverse nil)
(x!2 skolem-const-decl "T2" set_of_functions nil)
(member const-decl "bool" sets nil)
(empty? const-decl "bool" sets nil)
(x!1 skolem-const-decl "T2" set_of_functions nil)
(bij_inv_is_bij formula-decl nil function_inverse nil)
(composition_left_cancel2 formula-decl nil func_composition
"finite_sets/")
(composition_right_cancel2 formula-decl nil func_composition
"finite_sets/")
(inverse const-decl "D" function_inverse nil)
(O const-decl "T3" function_props nil)
(card_bij formula-decl nil finite_sets nil)
(empty_card formula-decl nil finite_sets nil)
(TRUE const-decl "bool" booleans nil))
shostak))
(finite_funset_bijection2_TCC1 0
(finite_funset_bijection2_TCC1-1 nil 3306495580
("" (subtype-tcc) nil nil) ((/= const-decl "boolean" notequal nil))
nil))
(finite_funset_bijection2 0
(finite_funset_bijection2-1 nil 3306496435
("" (skosimp)
(("" (forward-chain "finite_funset_bijection1")
(("" (lemma "funset_bijection" ("n" "n!1" "m" "m!1"))
(("" (skosimp*)
(("" (inst + "f!1 o f!2")
(("" (rewrite "composition_bijective") nil nil)) nil))
nil))
nil))
nil))
nil)
((finite_funset_bijection1 formula-decl nil set_of_functions nil)
(T1 formal-type-decl nil set_of_functions nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(T2 formal-type-decl nil set_of_functions 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)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(composition_bijective formula-decl nil func_composition
"finite_sets/")
(nat_exp application-judgement "nat" exponentiation nil)
(O const-decl "T3" function_props nil)
(below type-eq-decl nil nat_types nil)
(^ const-decl "real" exponentiation nil)
(/= const-decl "boolean" notequal nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(< const-decl "bool" reals nil)
(funset_bijection formula-decl nil fun_below_props nil))
shostak))
(finite_funset 0
(finite_funset-1 nil 3306496495
("" (skolem!)
(("" (expand "is_finite_type")
(("" (name "n1" "card(A!1)")
(("" (name "n2" "card(B!1)")
(("" (forward-chain "finite_funset_bijection2")
(("" (expand "bijective?")
(("" (skosimp)
(("" (inst + "n2 ^ n1" "f!1") (("" (assert) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((is_finite_type const-decl "bool" finite_sets nil)
(T2 formal-type-decl nil set_of_functions nil)
(bijective? const-decl "bool" functions nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(nat_exp application-judgement "nat" exponentiation nil)
(below type-eq-decl nil nat_types nil)
(< const-decl "bool" reals nil)
(^ const-decl "real" exponentiation nil)
(/= const-decl "boolean" notequal nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(n1 skolem-const-decl "{n: nat | n = Card[T1](A!1)}"
set_of_functions nil)
(A!1 skolem-const-decl "finite_set[T1]" set_of_functions nil)
(n2 skolem-const-decl "{n: nat | n = Card[T2](B!1)}"
set_of_functions nil)
(B!1 skolem-const-decl "finite_set[T2]" set_of_functions nil)
(finite_funset_bijection2 formula-decl nil set_of_functions nil)
(number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(T1 formal-type-decl nil set_of_functions nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets 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)
(Card const-decl "nat" finite_sets nil)
(card const-decl "{n: nat | n = Card(S)}" finite_sets nil))
shostak))
(funset_TCC1 0
(funset_TCC1-1 nil 3306495580
("" (skolem!)
(("" (rewrite "finite_full" :dir rl)
(("" (rewrite "finite_funset") nil nil)) nil))
nil)
((finite_full formula-decl nil finite_sets nil)
(T1 formal-type-decl nil set_of_functions nil)
(boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(T2 formal-type-decl nil set_of_functions nil)
(finite_funset formula-decl nil set_of_functions nil))
nil))
(card_funset_TCC1 0
(card_funset_TCC1-1 nil 3306495580 ("" (subtype-tcc) nil nil)
((boolean nonempty-type-decl nil booleans nil)
(bool nonempty-type-eq-decl nil booleans nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(T1 formal-type-decl nil set_of_functions nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets nil)
(T2 formal-type-decl nil set_of_functions nil)
(injective? const-decl "bool" functions nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(/= const-decl "boolean" notequal nil))
nil))
(card_funset 0
(card_funset-1 nil 3306496635
("" (skosimp)
(("" (name-replace "n1" "card(A!1)" :hide? nil)
(("" (name-replace "n2" "card(B!1)" :hide? nil)
(("" (rewrite "card_bij" +)
(("" (forward-chain "finite_funset_bijection2")
(("" (skolem!)
(("" (inst + "LAMBDA (x : (funset(A!1, B!1))) : f!1(x)")
(("" (delete -2 -3)
(("" (expand "bijective?")
(("" (flatten)
(("" (prop)
(("1" (grind :exclude "^") nil nil)
("2" (expand "surjective?")
(("2" (skosimp*)
(("2" (inst?)
(("2"
(skosimp*)
(("2"
(inst?)
(("2"
(expand "funset")
(("2"
(expand "fullset")
(("2" (propax) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((number nonempty-type-decl nil numbers nil)
(boolean nonempty-type-decl nil booleans nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(T1 formal-type-decl nil set_of_functions nil)
(bool nonempty-type-eq-decl nil booleans nil)
(set type-eq-decl nil sets nil)
(is_finite const-decl "bool" finite_sets nil)
(finite_set type-eq-decl nil finite_sets 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)
(Card const-decl "nat" finite_sets nil)
(card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
(nat_exp application-judgement "nat" exponentiation nil)
(card_bij formula-decl nil finite_sets nil)
(OR const-decl "[bool, bool -> bool]" booleans nil)
(/= const-decl "boolean" notequal nil)
(^ const-decl "real" exponentiation nil)
(funset const-decl "finite_set[[(A) -> (B)]]" set_of_functions nil)
(x!1 skolem-const-decl "[(A!1) -> (B!1)]" set_of_functions nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(A!1 skolem-const-decl "finite_set[T1]" set_of_functions nil)
(n1 skolem-const-decl "{n: nat | n = Card[T1](A!1)}"
set_of_functions nil)
(B!1 skolem-const-decl "finite_set[T2]" set_of_functions nil)
(n2 skolem-const-decl "{n: nat | n = Card[T2](B!1)}"
set_of_functions nil)
(fullset const-decl "set" sets nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(surjective? const-decl "bool" functions nil)
(injective? const-decl "bool" functions nil)
(bijective? const-decl "bool" functions nil)
(< const-decl "bool" reals nil)
(below type-eq-decl nil nat_types nil)
(finite_funset_bijection2 formula-decl nil set_of_functions nil)
(T2 formal-type-decl nil set_of_functions nil))
shostak)))
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