|
(countable_set
(int_to_nat_TCC1 0
(int_to_nat_TCC1-1 nil 3317054588 ("" (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)
(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)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props 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))
nil))
(int_to_nat_TCC2 0
(int_to_nat_TCC2-1 nil 3317054588 ("" (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)
(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)
(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)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil))
nil))
(nat_to_int_TCC1 0
(nat_to_int_TCC1-1 nil 3317054588 ("" (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)
(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)
(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)
(even? const-decl "bool" integers nil)
(nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
rationals nil))
nil))
(nat_to_int_TCC2 0
(nat_to_int_TCC2-1 nil 3317054588
("" (skosimp)
(("" (rewrite "even_or_odd")
(("" (expand "odd?")
(("" (skolem!)
(("" (use "div_simple")
(("" (iff)
(("" (ground)
(("" (inst + "-j!1 - 1") (("" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((even_or_odd formula-decl nil naturalnumbers 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)
(int_minus_int_is_int application-judgement "int" integers nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(int_times_even_is_even application-judgement "even_int" integers
nil)
(minus_int_is_int application-judgement "int" integers nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil)
(minus_even_is_even application-judgement "even_int" integers nil)
(minus_nzint_is_nzint application-judgement "nzint" integers nil)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil)
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(nzint nonempty-type-eq-decl nil integers nil)
(/= const-decl "boolean" notequal nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(div_simple formula-decl nil integer_props nil)
(odd? const-decl "bool" integers 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))
nil))
(int_to_nat_bijective 0
(int_to_nat_bijective-1 nil 3317054588
("" (grind :if-match nil)
(("" (inst + "nat_to_int(y!1)") (("" (grind) nil nil)) nil)) nil)
((nat_to_int const-decl "int" countable_set nil)
(rat_times_rat_is_rat application-judgement "rat" rationals nil)
(rat_minus_rat_is_rat application-judgement "rat" rationals nil)
(nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals
nil)
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil)
(nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
rationals nil)
(even? const-decl "bool" integers nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(minus_nzint_is_nzint application-judgement "nzint" integers nil)
(minus_even_is_even application-judgement "even_int" integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(>= const-decl "bool" reals nil)
(nat nonempty-type-eq-decl nil naturalnumbers 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)
(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)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(bijective? const-decl "bool" functions nil)
(surjective? const-decl "bool" functions nil)
(injective? const-decl "bool" functions nil)
(int_to_nat const-decl "nat" countable_set nil))
nil))
(nat_to_int_bijective 0
(nat_to_int_bijective-1 nil 3317054588
("" (grind :if-match nil)
(("" (inst + "int_to_nat(y!1)") (("" (grind) nil nil)) nil)) nil)
((int_to_nat const-decl "nat" countable_set nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers 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)
(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)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(minus_nzint_is_nzint application-judgement "nzint" integers nil)
(minus_even_is_even application-judgement "even_int" integers nil)
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil)
(bijective? const-decl "bool" functions nil)
(surjective? const-decl "bool" functions nil)
(injective? const-decl "bool" functions nil)
(nat_to_int const-decl "int" countable_set nil)
(even? const-decl "bool" integers nil))
nil))
(countable_family_nat 0
(countable_family_nat-1 nil 3317054743
("" (inst + "bit_encoding") (("" (assert) nil nil)) nil)
((encoding_bijection name-judgement
"(bijective?[finite_set[nat], nat])" bits nil)
(bit_encoding def-decl "nat" bits nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets 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))
shostak))
(intset_to_natset_bijective 0
(intset_to_natset_bijective-1 nil 3317054588
("" (lemma "int_to_nat_bijective")
(("" (expand* "bijective?" "injective?" "surjective?")
(("" (prop)
(("1" (skosimp)
(("1" (decompose-equality)
(("1" (expand* "intset_to_natset" "image")
(("1" (apply-extensionality :hide? t)
(("1" (inst - "int_to_nat(x!1)")
(("1" (iff -)
(("1" (prop)
(("1" (skosimp*)
(("1" (inst-cp - "x!1" "x!3")
(("1" (inst-cp - "x!1" "x!2")
(("1" (assert) nil nil)) nil))
nil))
nil)
("2" (smash)
(("1" (inst 2 "x!1") nil nil)
("2" (inst + "x!1") nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (skolem!)
(("2" (inst + "image(nat_to_int, y!1)")
(("2" (apply-extensionality :hide? t)
(("2"
(expand* "nat_to_int" "intset_to_natset" "image"
"int_to_nat")
(("2" (smash)
(("1" (skosimp* t)
(("1" (lift-if) (("1" (ground) nil nil)) nil)) nil)
("2" (inst + "nat_to_int(x!1)")
(("1" (grind) nil nil)
("2" (inst + "x!1") (("2" (grind) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
((injective? const-decl "bool" functions nil)
(surjective? const-decl "bool" functions nil)
(int_to_nat_bijective name-judgement "(bijective?[int, nat])"
countable_set nil)
(bijective? const-decl "bool" functions nil)
(nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
rationals 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)
(int_minus_int_is_int application-judgement "int" integers nil)
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil)
(minus_even_is_even application-judgement "even_int" integers nil)
(minus_nzint_is_nzint application-judgement "nzint" integers nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
nil)
(- const-decl "[numfield -> numfield]" number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(/= const-decl "boolean" notequal nil)
(even? const-decl "bool" integers nil)
(IF const-decl "[boolean, T, T -> T]" if_def nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
(< const-decl "bool" reals nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(- const-decl "[numfield, numfield -> numfield]" number_fields nil)
(nzrat_times_nzrat_is_nzrat application-judgement "nzrat" rationals
nil)
(nnrrat_div_negrat_is_nprat application-judgement "nprat" rationals
nil)
(nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat"
rationals nil)
(rat_minus_rat_is_rat application-judgement "rat" rationals nil)
(rat_times_rat_is_rat application-judgement "rat" rationals nil)
(y!1 skolem-const-decl "finite_set[nat]" countable_set nil)
(x!1 skolem-const-decl "nat" countable_set nil)
(nat_to_int const-decl "int" countable_set nil)
(nat_to_int_bijective name-judgement "(bijective?[nat, int])"
countable_set nil)
(finite_image application-judgement "finite_set[int]" countable_set
nil)
(image const-decl "set[R]" function_image nil)
(int_to_nat const-decl "nat" countable_set nil)
(intset_to_natset const-decl "finite_set[nat]" countable_set nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets nil)
(= const-decl "[T, T -> boolean]" equalities 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)
(int_to_nat_bijective judgement-tcc nil countable_set nil))
nil))
(countable_family_int 0
(countable_family_int-1 nil 3317054966
("" (inst + "bit_encoding o intset_to_natset")
(("" (assert) nil nil)) nil)
((intset_to_natset_bijective name-judgement
"(bijective?[finite_set[int], finite_set[nat]])" countable_set
nil)
(composition_injective application-judgement "(injective?[T1, T3])"
function_props nil)
(composition_surjective application-judgement
"(surjective?[T1, T3])" function_props nil)
(composition_bijective application-judgement "(bijective?[T1, T3])"
function_props nil)
(intset_to_natset const-decl "finite_set[nat]" countable_set nil)
(bit_encoding def-decl "nat" bits nil)
(O const-decl "T3" function_props nil)
(nat nonempty-type-eq-decl nil naturalnumbers nil)
(>= const-decl "bool" reals nil)
(finite_set type-eq-decl nil finite_sets nil)
(is_finite const-decl "bool" finite_sets nil)
(set type-eq-decl nil sets 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)
(encoding_bijection name-judgement
"(bijective?[finite_set[nat], nat])" bits nil))
shostak))
(tuple_to_natset_TCC1 0
(tuple_to_natset_TCC1-1 nil 3317054588
("" (skolem!)
(("" (expand "is_finite")
((""
(inst + "2"
"LAMBDA (x: ({i: nat | i = n!1`1 * 2 OR i = n!1`2 * 2 + 1})): rem(2)(x)")
(("" (expand "injective?")
(("" (skosimp :preds? t)
(("" (rewrite "same_remainder")
(("" (expand "divides") (("" (ground) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
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)
(is_finite const-decl "bool" finite_sets nil)
(injective? const-decl "bool" functions nil)
(same_remainder formula-decl nil modulo_arithmetic nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers nil)
(divides const-decl "bool" divides nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(rem const-decl "{r: mod(b) | EXISTS q: x = b * q + r}"
modulo_arithmetic nil)
(mod nonempty-type-eq-decl nil euclidean_division nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(below type-eq-decl nil nat_types nil)
(< const-decl "bool" reals nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(OR const-decl "[bool, bool -> bool]" booleans 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)
(int_times_even_is_even application-judgement "even_int" integers
nil)
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil)
(even_plus_odd_is_odd application-judgement "odd_int" integers
nil))
nil))
(countable_nat_tuple 0
(countable_nat_tuple-1 nil 3317055072
("" (lemma "inj_inj_bij[[nat, nat], nat]")
(("" (prop)
(("1" (lemma "countable_family_nat")
(("1" (expand "bijective?")
(("1" (skosimp)
(("1" (assert)
(("1"
(lemma
"composition_injective[[nat, nat], finite_set[nat], nat]"
("f1" "tuple_to_natset" "f2" "f!1"))
(("1" (inst?) nil nil)
("2" (expand "injective?" 1)
(("2" (skosimp)
(("2" (decompose-equality)
(("2" (expand "tuple_to_natset")
(("2" (inst-cp - "2 * x1!1`1")
(("2" (inst - "1 + 2 * x1!1`2")
(("2" (assert)
(("2" (apply-extensionality) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (inst + "LAMBDA (n: nat): (n, n)")
(("2" (expand "injective?" 1) (("2" (skosimp) nil nil)) nil))
nil))
nil))
nil)
((bijective? const-decl "bool" functions nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers 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)
(O const-decl "T3" function_props nil)
(composition_injective judgement-tcc nil function_props nil)
(injective? const-decl "bool" functions nil)
(tuple_to_natset const-decl "finite_set[nat]" countable_set 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)
(countable_family_nat formula-decl nil countable_set nil)
(inj_inj_bij formula-decl nil set_antisymmetric "orders/")
(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))
shostak))
(tuple_to_intset_TCC1 0
(tuple_to_intset_TCC1-1 nil 3317054588
("" (skolem!)
(("" (expand "is_finite")
((""
(inst + "2"
"LAMBDA (x: ({i: int | i = n!1`1 * 2 OR i = n!1`2 * 2 + 1})): rem(2)(x)")
(("" (expand "injective?")
(("" (skosimp :preds? t)
(("" (rewrite "same_remainder")
(("" (expand "divides") (("" (ground) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
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)
(is_finite const-decl "bool" finite_sets nil)
(injective? const-decl "bool" functions nil)
(same_remainder formula-decl nil modulo_arithmetic nil)
(int_minus_int_is_int application-judgement "int" integers nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers nil)
(divides const-decl "bool" divides nil)
(NOT const-decl "[bool -> bool]" booleans nil)
(rem const-decl "{r: mod(b) | EXISTS q: x = b * q + r}"
modulo_arithmetic nil)
(mod nonempty-type-eq-decl nil euclidean_division nil)
(posnat nonempty-type-eq-decl nil integers nil)
(> const-decl "bool" reals nil)
(nonneg_int nonempty-type-eq-decl nil integers nil)
(below type-eq-decl nil nat_types nil)
(< const-decl "bool" reals nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(OR const-decl "[bool, bool -> bool]" booleans 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)
(int_times_even_is_even application-judgement "even_int" integers
nil)
(even_plus_odd_is_odd application-judgement "odd_int" integers
nil))
nil))
(countable_int_tuple 0
(countable_int_tuple-1 nil 3317055262
("" (lemma "inj_inj_bij[[int, int], nat]")
(("" (prop)
(("1" (lemma "countable_family_int")
(("1" (expand "bijective?")
(("1" (skosimp)
(("1" (assert)
(("1"
(lemma
"composition_injective[[int, int], finite_set[int], nat]"
("f1" "tuple_to_intset" "f2" "f!1"))
(("1" (inst?) nil nil)
("2" (expand "injective?" 1)
(("2" (skosimp)
(("2" (decompose-equality)
(("2" (expand "tuple_to_intset")
(("2" (inst-cp - "2 * x1!1`1")
(("2" (inst - "1 + 2 * x1!1`2")
(("2" (assert)
(("2" (apply-extensionality) nil nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (inst + "LAMBDA (n: nat): (n, n)")
(("2" (expand "injective?") (("2" (skosimp) nil nil)) nil))
nil))
nil))
nil)
((bijective? const-decl "bool" functions nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(* const-decl "[numfield, numfield -> numfield]" number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
(odd_plus_even_is_odd application-judgement "odd_int" integers 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)
(O const-decl "T3" function_props nil)
(composition_injective judgement-tcc nil function_props nil)
(injective? const-decl "bool" functions nil)
(tuple_to_intset const-decl "finite_set[int]" countable_set 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)
(countable_family_int formula-decl nil countable_set nil)
(inj_inj_bij formula-decl nil set_antisymmetric "orders/")
(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))
shostak))
(countable_rat 0
(countable_rat-1 nil 3317055513
("" (lemma "inj_inj_bij[rat, nat]")
(("" (prop)
(("1" (lemma "countable_int_tuple")
(("1" (expand "bijective?")
(("1" (skosimp)
(("1"
(case "FORALL (r: rat): nonempty?[[int, int]]({t: [int, int] | t`2 /= 0 AND r = t`1 / t`2})")
(("1" (assert)
(("1"
(lemma "composition_injective[rat, [int, int], nat]"
("f1"
"LAMBDA (r: rat): choose({t: [int, int] | t`2 /= 0 AND r = t`1 / t`2})"
"f2" "f!1"))
(("1" (inst? +) nil nil)
("2" (expand "injective?" 1)
(("2" (skosimp) (("2" (assert) nil nil)) nil)) nil)
("3" (propax) nil nil))
nil))
nil)
("2" (skolem!)
(("2" (expand* "nonempty?" "empty?" "member")
(("2" (lemma "rational_pred_ax" ("r" "r!1"))
(("2" (skolem!)
(("2" (inst - "(i!1, n0j!1)")
(("2" (assert) nil nil)) nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil))
nil)
("2" (inst + "LAMBDA (n: nat): n")
(("2" (expand "injective?") (("2" (skosimp) nil nil)) nil))
nil))
nil))
nil)
((bijective? const-decl "bool" functions nil)
(/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
(nznum nonempty-type-eq-decl nil number_fields nil)
(numfield nonempty-type-eq-decl nil number_fields nil)
(= const-decl "[T, T -> boolean]" equalities nil)
(/= const-decl "boolean" notequal nil)
(AND const-decl "[bool, bool -> bool]" booleans nil)
(nonempty? const-decl "bool" sets nil)
(set type-eq-decl nil sets nil)
(rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
(composition_injective judgement-tcc nil function_props nil)
(injective? const-decl "bool" functions nil)
(choose const-decl "(p)" sets nil)
(O const-decl "T3" function_props nil)
(empty? const-decl "bool" sets nil)
(member const-decl "bool" sets nil)
(nzint nonempty-type-eq-decl nil integers nil)
(rational_pred_ax formula-decl nil rational_props nil)
(countable_int_tuple formula-decl nil countable_set nil)
(inj_inj_bij formula-decl nil set_antisymmetric "orders/")
(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)
(rat nonempty-type-eq-decl nil 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))
shostak)))
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