SSL countable_set.prf
Interaktion und
(countable_setnil 3317054588 ("" (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 )"int" integers nil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil ))nil ))nil 3317054588 ("" (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 )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"(total_order?[real])" nil )"(strict_total_order?[real])" real_props nil ))nil ))nil 3317054588 ("" (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 )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil naturalnumbers nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"bool" integers nil )"nonneg_rat" nil ))nil ))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 )nil naturalnumbers 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 )"int" integers nil )"[numfield, numfield -> numfield]" number_fields nil )"even_int" integersnil )"int" integers nil )"odd_int" integers nil )"posint" nil )"nzrat" rationalsnil )"even_int" integers nil )"nzint" integers nil )"posint" nil )"rat" rationals nil )"[numfield -> numfield]" number_fields nil )nil integers nil )"boolean" notequal nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )nil integer_props nil )"bool" integers nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil ))nil ))nil 3317054588"" (grind :if-match nil )"" (inst + "nat_to_int(y!1)" ) (("" (grind) nil nil )) nil )) nil )"int" countable_set nil )"rat" rationals nil )"rat" rationals nil )"nzrat" rationalsnil )"nzrat" rationalsnil )"nonneg_rat" nil )"bool" integers nil )"posint" nil )"nzint" integers nil )"even_int" integers nil )"(total_order?[real])" nil )"bool" reals nil )nil naturalnumbers 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 )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"(strict_total_order?[real])" real_props nil )"bool" functions nil )"bool" functions nil )"bool" functions nil )"nat" countable_set nil ))nil ))nil 3317054588"" (grind :if-match nil )"" (inst + "int_to_nat(y!1)" ) (("" (grind) nil nil )) nil )) nil )"nat" countable_set nil )"(strict_total_order?[real])" real_props nil )"odd_int" integers 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 )"bool" reals nil )"[real -> boolean]" rationals nil )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil naturalnumbers nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"posint" nil )"nzint" integers nil )"even_int" integers nil )"(total_order?[real])" nil )"bool" functions nil )"bool" functions nil )"bool" functions nil )"int" countable_set nil )"bool" integers nil ))nil ))nil 3317054743"" (inst + "bit_encoding" ) (("" (assert ) nil nil )) nil )"(bijective?[finite_set[nat], nat])" bits nil )"nat" bits nil )nil finite_sets nil )"bool" finite_sets nil )set type-eq-decl nil sets 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 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 )"bool" functions nil )"bool" functions nil )"(bijective?[int, nat])" nil )"bool" functions nil )"nonneg_rat" nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"int" integers nil )"(strict_total_order?[real])" real_props nil )"even_int" integers nil )"nzint" integers nil )"posint" nil )"nzrat" rationalsnil )"[numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )"boolean" notequal nil )"bool" integers nil )IF const-decl "[boolean, T, T -> T]" if_def nil )nil number_fields nil )NOT const-decl "[bool -> bool]" booleans nil )"[bool, bool -> bool]" booleans nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"nzrat" rationalsnil )"nprat" rationalsnil )"nonneg_rat" nil )"rat" rationals nil )"rat" rationals nil )"finite_set[nat]" countable_set nil )"nat" countable_set nil )"int" countable_set nil )"(bijective?[nat, int])" nil )"finite_set[int]" countable_setnil )"set[R]" function_image nil )"nat" countable_set nil )"finite_set[nat]" countable_set nil )nil finite_sets nil )"bool" finite_sets nil )set type-eq-decl nil sets nil )"[T, T -> boolean]" equalities 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 countable_set nil ))nil ))nil 3317054966"" (inst + "bit_encoding o intset_to_natset" )"" (assert ) nil nil )) nil )"(bijective?[finite_set[int], finite_set[nat]])" countable_setnil )"(injective?[T1, T3])" nil )"(surjective?[T1, T3])" function_props nil )"(bijective?[T1, T3])" nil )"finite_set[nat]" countable_set nil )"nat" bits nil )"T3" function_props nil )nil naturalnumbers nil )"bool" reals nil )nil finite_sets nil )"bool" finite_sets nil )set type-eq-decl nil sets 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 )"(bijective?[finite_set[nat], nat])" bits nil ))nil 3317054588"" (skolem!)"" (expand "is_finite" )"" "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 )"nonneg_int" nil )"even_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"bool" finite_sets nil )"bool" functions nil )nil modulo_arithmetic nil )"int" integers nil )"posint" nil )"odd_int" integers nil )"bool" divides nil )NOT const-decl "[bool -> bool]" booleans nil )"{r: mod(b) | EXISTS q: x = b * q + r}" nil )nil euclidean_division nil )nil integers nil )"bool" reals nil )nil integers nil )nil nat_types nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"[T, T -> boolean]" equalities nil )OR const-decl "[bool, bool -> bool]" booleans 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 )"even_int" integersnil )"posint" nil )"odd_int" integersnil ))nil ))nil 3317055072"" (lemma "inj_inj_bij[[nat, nat], nat]" )"" (prop)"1" (lemma "countable_family_nat" )"1" (expand "bijective?" )"1" (skosimp)"1" (assert )"1" "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 )"bool" functions nil )"[T, T -> boolean]" equalities nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"posint" nil )"odd_int" integers nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"nonneg_int" nil )"T3" function_props nil )nil function_props nil )"bool" functions nil )"finite_set[nat]" countable_set nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )nil countable_set nil )nil set_antisymmetric "orders/" )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 3317054588"" (skolem!)"" (expand "is_finite" )"" "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_int" integersnil )"(divides(n))" divides nil )"(divides(m))" divides nil )"bool" finite_sets nil )"bool" functions nil )nil modulo_arithmetic nil )"int" integers nil )"odd_int" integers nil )"bool" divides nil )NOT const-decl "[bool -> bool]" booleans nil )"{r: mod(b) | EXISTS q: x = b * q + r}" nil )nil euclidean_division nil )nil integers nil )"bool" reals nil )nil integers nil )nil nat_types nil )"bool" reals nil )"[numfield, numfield -> numfield]" number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"[T, T -> boolean]" equalities nil )OR const-decl "[bool, bool -> bool]" booleans 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 )"even_int" integersnil )"odd_int" integersnil ))nil ))nil 3317055262"" (lemma "inj_inj_bij[[int, int], nat]" )"" (prop)"1" (lemma "countable_family_int" )"1" (expand "bijective?" )"1" (skosimp)"1" (assert )"1" "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 )"bool" functions nil )"[T, T -> boolean]" equalities nil )"[numfield, numfield -> numfield]" number_fields nil )nil number_fields nil )"[numfield, numfield -> numfield]" number_fields nil )"odd_int" integers nil )"(divides(m))" divides nil )"(divides(n))" divides nil )"even_int" integersnil )"T3" function_props nil )nil function_props nil )"bool" functions nil )"finite_set[int]" countable_set nil )set type-eq-decl nil sets nil )"bool" finite_sets nil )nil finite_sets nil )nil countable_set nil )nil set_antisymmetric "orders/" )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 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" "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 )"bool" functions nil )"[numfield, nznum -> numfield]" number_fields nil )nil number_fields nil )nil number_fields nil )"[T, T -> boolean]" equalities nil )"boolean" notequal nil )AND const-decl "[bool, bool -> bool]" booleans nil )"bool" sets nil )set type-eq-decl nil sets nil )"rat" rationals nil )nil function_props nil )"bool" functions nil )"(p)" sets nil )"T3" function_props nil )"bool" sets nil )"bool" sets nil )nil integers nil )nil rational_props nil )nil countable_set nil )nil set_antisymmetric "orders/" )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 )nil rationals nil )"[rational -> boolean]" integers nil )nil integers nil )nil booleans nil )"bool" reals nil )nil naturalnumbers nil ))quality 96%
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2026-03-28
