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((IUnion_n_def formula-decl nil nat_indexed_sets "sets_aux/" )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(difference_difference1 formula-decl nil sets_lemmas nil )
(union const-decl "set" sets nil )
(j!2 skolem-const-decl "nat" ast_def nil )
(n!2 skolem-const-decl "nat" ast_def nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(AA skolem-const-decl "set[T]" ast_def nil )
(BB skolem-const-decl "set[T]" ast_def nil )
(E!2 skolem-const-decl "sequence[set[T]]" ast_def nil )
(n!1 skolem-const-decl "nat" ast_def nil )
(i!1 skolem-const-decl "nat" ast_def nil )
(/= const-decl "boolean" notequal nil )
(finite_intersection2 application-judgement "finite_set[T]"
countable_setofsets "sets_aux/" )
(difference_emptyset2 formula-decl nil sets_lemmas nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(union_commutative formula-decl nil sets_lemmas nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(A_difference formula-decl nil ast_def nil )
(IUnion_0_def formula-decl nil nat_indexed_sets "sets_aux/" )
(nat_induction formula-decl nil naturalnumbers nil )
(pred type-eq-decl nil defined_types nil )
(> const-decl "bool" reals nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(A_empty formula-decl nil ast_def nil )
(emptyset_is_empty? formula-decl nil sets_lemmas nil )
(< const-decl "bool" reals nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(n!1 skolem-const-decl "nat" ast_def nil )
(E!1 skolem-const-decl "sequence[set[T]]" ast_def nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(a!1 skolem-const-decl "set[T]" ast_def nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(image const-decl "set[R]" function_image nil )
(<= const-decl "bool" reals nil )
(i!1 skolem-const-decl "nat" ast_def nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(Union const-decl "set" sets nil )
(member const-decl "bool" sets nil )
(difference_emptyset1 formula-decl nil sets_lemmas nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(intersection const-decl "set" sets nil )
(empty? const-decl "bool" sets nil )
(disjoint? const-decl "bool" sets nil )
(disjoint? const-decl "bool" indexed_sets_aux "sets_aux/" )
(IUnion const-decl "set[T]" indexed_sets nil )
(real_lt_is_strict_total_order name-judgement
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(finite_emptyset name-judgement "finite_set" finite_sets nil )
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("2" (hide -6 -7 -8 2)
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((measure_type nonempty-type-eq-decl nil generalized_measure_def
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(extended_nnreal nonempty-type-eq-decl nil extended_nnreal
"extended_nnreal/" )
(nnreal type-eq-decl nil real_types nil )
(>= const-decl "bool" reals nil )
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(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 )
(A formal-const-decl "(nonempty?[set[T]])" ast_def nil )
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(set type-eq-decl nil sets nil )
(setof type-eq-decl nil defined_types nil )
(T formal-type-decl nil ast_def nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(bool nonempty-type-eq-decl nil booleans nil )
(boolean nonempty-type-decl nil booleans nil )
(j!1 skolem-const-decl "nat" ast_def nil )
(nat_induction formula-decl nil naturalnumbers nil )
(IUnion_0_def formula-decl nil nat_indexed_sets "sets_aux/" )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(BB skolem-const-decl "set[T]" ast_def nil )
(Union const-decl "set" sets nil )
(image const-decl "set[R]" function_image nil )
(AA skolem-const-decl "set[T]" ast_def nil )
(n!1 skolem-const-decl "nat" ast_def nil )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(even_minus_odd_is_odd application-judgement "odd_int" integers
nil )
(odd_plus_odd_is_even application-judgement "even_int" integers
nil )
(odd_minus_odd_is_even application-judgement "even_int" integers
nil )
(posint_plus_nnint_is_posint application-judgement "posint"
integers nil )
(IUnion_n_def formula-decl nil nat_indexed_sets "sets_aux/" )
(pred type-eq-decl nil defined_types nil )
(NAT_induction formula-decl nil naturalnumbers nil )
(disjoint_indexed_measurable? const-decl "bool"
generalized_measure_def nil )
(disjoint_indexed_measurable nonempty-type-eq-decl nil
generalized_measure_def nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(IUnion const-decl "set[T]" indexed_sets nil )
(subset? const-decl "bool" sets nil )
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil )
(difference const-decl "set" sets nil )
(XX skolem-const-decl "[nat -> set[T]]" ast_def nil )
(union const-decl "set" sets nil )
(x_sum_eq formula-decl nil extended_nnreal "extended_nnreal/" )
(x_add const-decl "extended_nnreal" extended_nnreal
"extended_nnreal/" )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(limit_series_shift formula-decl nil series "series/" )
(nonneg_int nonempty-type-eq-decl nil integers nil )
(> const-decl "bool" reals nil )
(posnat nonempty-type-eq-decl nil integers nil )
(real_gt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(sigma_eq formula-decl nil sigma "reals/" )
(OR const-decl "[bool, bool -> bool]" booleans nil )
(<= const-decl "bool" reals nil )
(T_high type-eq-decl nil sigma "reals/" )
(T_low type-eq-decl nil sigma "reals/" )
(sigma_zero formula-decl nil sigma "reals/" )
(real_plus_real_is_real application-judgement "real" reals nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(conv_series_shift formula-decl nil series "series/" )
(convergent? const-decl "bool" convergence_sequences "analysis/" )
(series const-decl "sequence[real]" series "series/" )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(i!1 skolem-const-decl "nat" ast_def nil )
(n!1 skolem-const-decl "nat" ast_def nil )
(nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(TRUE const-decl "bool" booleans nil )
(x_add_sum formula-decl nil extended_nnreal "extended_nnreal/" )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(numfield nonempty-type-eq-decl nil number_fields nil )
(< const-decl "bool" reals nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(difference_disjoint formula-decl nil sets_lemmas nil )
(union_diff_subset formula-decl nil sets_lemmas nil )
(A_difference_union formula-decl nil ast_def nil )
(j!1 skolem-const-decl "nat" ast_def nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(x_sum_le formula-decl nil extended_nnreal "extended_nnreal/" )
(UU skolem-const-decl "[[nat, nat] -> (A)]" ast_def nil )
(single_index const-decl "[nat -> T]" double_index
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(measure_countably_additive? const-decl "bool"
generalized_measure_def nil )
(disjoint? const-decl "bool" indexed_sets_aux "sets_aux/" )
(nnint_times_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(mult_divides1 application-judgement "(divides(n))" divides nil )
(mult_divides2 application-judgement "(divides(m))" divides nil )
(nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(nnrat_plus_nnrat_is_nnrat application-judgement "nonneg_rat"
rationals nil )
(nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
rationals nil )
(/= const-decl "boolean" notequal nil )
(intersection const-decl "set" sets nil )
(member const-decl "bool" sets nil )
(empty? const-decl "bool" sets nil )
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(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(x_eq const-decl "bool" extended_nnreal "extended_nnreal/" )
(double_index_n_ij formula-decl nil code_product
"extended_nnreal/" )
(double_index_j const-decl "nat" code_product "extended_nnreal/" )
(double_index_i const-decl "nat" code_product "extended_nnreal/" )
(IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/" )
(double_index_ij_n formula-decl nil code_product
"extended_nnreal/" )
(double_index_n const-decl "nat" code_product "extended_nnreal/" )
(double_x_sum formula-decl nil extended_nnreal "extended_nnreal/" )
(O const-decl "T3" function_props nil )
(x_sum const-decl "extended_nnreal" extended_nnreal
"extended_nnreal/" )
(x_le const-decl "bool" extended_nnreal "extended_nnreal/" )
(choose_member formula-decl nil sets_lemmas nil )
(choose const-decl "(p)" sets nil )
(emptyset const-decl "set" sets nil )
(finite_emptyset name-judgement "finite_set[T]" countable_setofsets
"sets_aux/" )
(finite_emptyset name-judgement "finite_set[T]" countable_props
"sets_aux/" )
(finite_emptyset name-judgement "finite_set" finite_sets nil )
(A_empty formula-decl nil ast_def nil )
(E!1 skolem-const-decl "sequence[set[T]]" ast_def nil )
(finite_disjoint_union? const-decl "bool" subset_algebra_def nil )
(setofsets type-eq-decl nil sets nil )
(disjoint_IUnion formula-decl nil nat_indexed_sets "sets_aux/" )
(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 )
(sequence type-eq-decl nil sequences nil ))
shostak))
(generalized_monotonicity 0
(generalized_monotonicity-1 nil 3458063268
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((""
(lemma "A_difference_union" ("a" "Y!1" "b" "IUnion(n!1, I!1)" ))
(("" (assert )
(("" (split)
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(("1"
(lemma "union_diff_subset"
("a" "IUnion(n!1, I!1)" "b" "Y!1" ))
(("1" (assert )
(("1" (replace -3)
(("1"
(name "XX"
"lambda (i:nat): if i<n!1 then I!1(i) else E!1(i-n!1) endif" )
(("1" (case "disjoint_indexed_measurable?(XX)" )
(("1" (typepred "mu!1" )
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(expand "measure_countably_additive?" )
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(flatten)
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(inst - "XX" )
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(case-replace "IUnion(XX)=Y!1" )
(("1"
(assert )
(("1"
(lemma
"x_add_sum"
("S"
"lambda (i:nat): if i<n!1 then mu!1(I!1(i)) else (TRUE,0) endif"
"T"
"lambda (i:nat): if i<n!1 then (TRUE,0) else mu!1(E!1(i-n!1)) endif" ))
(("1"
(lemma
"x_sum_eq"
("S"
"LAMBDA (i_1: nat):
x_add((LAMBDA (i: nat):
IF i < n!1 THEN mu!1(I!1(i))
ELSE (TRUE, 0)
ENDIF)
(i_1),
(LAMBDA (i: nat):
IF i < n!1 THEN (TRUE, 0)
ELSE mu!1(E!1(i - n!1))
ENDIF)
(i_1))"
"T"
"mu!1 o XX" ))
(("1"
(split -1)
(("1"
(name-replace
"DRL1"
"x_sum(LAMBDA (i_1: nat):
x_add((LAMBDA (i: nat):
IF i < n!1 THEN mu!1(I!1(i))
ELSE (TRUE, 0)
ENDIF)
(i_1),
(LAMBDA (i: nat):
IF i < n!1 THEN (TRUE, 0)
ELSE mu!1(E!1(i - n!1))
ENDIF)
(i_1)))")
(("1"
(case
"x_eq(x_sum(mu!1 o I!1),x_sum(LAMBDA (i: nat):
IF i < n!1 THEN mu!1(I!1(i))
ELSE (TRUE, 0)
ENDIF))")
(("1"
(name-replace
"LHS"
"x_sum(mu!1 o I!1)" )
(("1"
(name-replace
"DRL2"
"x_sum(LAMBDA (i: nat):
IF i < n!1 THEN mu!1(I!1(i)) ELSE (TRUE, 0) ENDIF)")
(("1"
(hide-all-but
(1 -1 -2 -3 -6))
(("1"
(expand "x_eq" )
(("1"
(expand
"x_le" )
(("1"
(expand
"x_add" )
(("1"
(flatten)
(("1"
(assert )
(("1"
(assert )
(("1"
(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 )
("2"
(hide 2 -2)
(("2"
(lemma
"x_sum_eq"
("S"
"mu!1 o I!1"
"T"
"LAMBDA (i: nat):
IF i < n!1 THEN mu!1(I!1(i)) ELSE (TRUE, 0) ENDIF"))
(("2"
(assert )
(("2"
(hide 2)
(("2"
(skosimp)
(("2"
(expand
"o " )
(("2"
(case-replace
"i!1 < n!1" )
(("1"
(expand
"x_eq" )
(("1"
(propax)
nil
nil ))
nil )
("2"
(assert )
(("2"
(inst
-13
"i!1" )
(("2"
(assert )
(("2"
(expand
"measure_empty?" )
(("2"
(case-replace
"I!1(i!1)=emptyset" )
(("1"
(replace
-4)
(("1"
(expand
"x_eq" )
(("1"
(propax)
nil
nil ))
nil ))
nil )
("2"
(apply-extensionality
:hide?
t)
(("2"
(expand
"emptyset" )
(("2"
(expand
"empty?" )
(("2"
(inst
-14
"x!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp)
(("2"
(case-replace
"i!1 < n!1" )
(("1"
(assert )
nil
nil )
("2"
(assert )
nil
nil ))
nil ))
nil )
("3"
(skosimp)
(("3"
(case-replace
"i!1 < n!1" )
(("1"
(assert )
nil
nil )
("2"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("2"
(hide 2 -1)
(("2"
(skosimp)
(("2"
(expand "o" 1)
(("2"
(expand "XX" )
(("2"
(case-replace
"i!1 < n!1" )
(("1"
(hide-all-but
1)
(("1"
(expand
"x_add" )
(("1"
(expand
"x_eq" )
(("1"
(case-replace
"mu!1(I!1(i!1))`1" )
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
(("2"
(hide-all-but
(1 2 -9))
(("2"
(expand
"x_add" )
(("2"
(expand
"x_eq" )
(("2"
(inst
-
"i!1 - n!1" )
(("2"
(flatten)
(("2"
(lift-if
2)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp)
(("2"
(lift-if 1)
(("2"
(assert )
(("2"
(prop)
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil )
("3"
(skosimp)
(("3"
(replace -1)
(("3"
(inst -11 "i!1-n!1" )
(("1"
(flatten)
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(replace 2)
(("1"
(assert )
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(case-replace
"E!1(i!1 - n!1)=emptyset" )
(("1"
(rewrite
"A_empty" )
nil
nil )
("2"
(apply-extensionality
:hide?
t)
(("2"
(expand
"emptyset" )
(("2"
(expand
"empty?" )
(("2"
(inst
-12
"x!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(assert )
nil
nil ))
nil ))
nil ))
nil )
("4"
(skosimp)
(("4" (assert ) nil nil ))
nil )
("5"
(skosimp)
(("5"
(lift-if 1)
(("5"
(assert )
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(prop)
(("5"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp)
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(inst -9 "i!1-n!1" )
(("1"
(flatten)
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(assert )
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(case-replace
"E!1(i!1 - n!1)=emptyset" )
(("1"
(rewrite "A_empty" )
nil
nil )
("2"
(apply-extensionality
:hide?
t)
(("2"
(expand
"emptyset" )
(("2"
(expand
"empty?" )
(("2"
(inst
-10
"x!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(apply-extensionality :hide? t)
(("2"
(replace -5 1 rl)
(("2"
(expand "XX" )
(("2"
(hide-all-but (1 -11 -8))
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(expand "union" )
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(expand "member" )
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(expand "IUnion" )
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(expand "image" )
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(expand
"Union" )
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(case-replace
"EXISTS (i_1: nat):
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1
"I!1(i!1)" )
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(inst
1
"i!1" )
nil
nil ))
nil ))
nil )
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(assert )
(("2"
(inst
3
"i!1-n!1" )
nil
nil ))
nil ))
nil ))
nil )
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(replace
1
2)
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(split)
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(skosimp)
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(typepred
"a!1" )
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(typepred
"j!1" )
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(expand
"<="
-1)
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(split)
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(inst
+
"j!1" )
(("1"
(assert )
nil
nil ))
nil )
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(inst
-5
"j!1" )
(("2"
(assert )
(("2"
(expand
"empty?" )
(("2"
(inst
-5
"x!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(skosimp)
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(inst
+
"i!1+n!1" )
(("2"
(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
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(("2" (expand "XX" )
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(expand "disjoint?" )
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(lemma
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(replace -4)
(("2"
(skosimp)
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(case-replace "i!1 < n!1" )
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(case-replace "j!1 < n!1" )
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(expand "empty?" )
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(expand "member" )
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(skosimp)
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(inst -2 "x!1" )
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(expand
"IUnion" )
(("2"
(expand
"image" )
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(expand
"Union" )
(("2"
(split
2)
(("1"
(inst
+
"I!1(i!1)" )
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(inst
+
"i!1" )
nil
nil ))
nil )
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(inst
+
"j!1 - n!1" )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil ))
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nil )
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(case-replace "j!1 < n!1" )
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"IUnion"
2)
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+
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+
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nil
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+
"i!1-n!1" )
nil
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nil ))
nil )
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(assert )
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(inst
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"i!1-n!1"
"j!1-n!1" )
(("2" (assert ) nil nil ))
nil ))
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nil
nil )
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(apply-extensionality
:hide?
t)
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(expand "emptyset" )
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(expand "empty?" )
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(inst -6 "x!1" )
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(expand "member" )
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nil ))
nil ))
nil ))
nil ))
nil ))
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nil ))
nil )
("2" (skosimp) (("2" (assert ) nil nil )) nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2" (hide 2)
(("2" (expand "finite_disjoint_union?" )
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nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/" )
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(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 )
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(set type-eq-decl nil sets nil )
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(numfield nonempty-type-eq-decl nil number_fields nil )
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(NOT const-decl "[bool -> bool]" booleans nil )
(IMPLIES const-decl "[bool, bool -> bool]" booleans nil )
(difference_disjoint formula-decl nil sets_lemmas nil )
(j!1 skolem-const-decl "nat" ast_def nil )
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(i!1 skolem-const-decl "nat" ast_def nil )
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"extended_nnreal/" )
(measure? const-decl "bool" generalized_measure_def nil )
(measure_type nonempty-type-eq-decl nil generalized_measure_def
nil )
(measure_countably_additive? const-decl "bool"
generalized_measure_def nil )
(XX skolem-const-decl "[nat -> set[T]]" ast_def nil )
(disjoint_indexed_measurable nonempty-type-eq-decl nil
generalized_measure_def nil )
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(<= const-decl "bool" reals nil )
(I!1 skolem-const-decl "sequence[(A)]" ast_def nil )
(i!1 skolem-const-decl "nat" ast_def nil )
(+ const-decl "[numfield, numfield -> numfield]" number_fields nil )
(nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
integers nil )
(Union const-decl "set" sets nil ) (union const-decl "set" sets nil )
(i!1 skolem-const-decl "nat" ast_def nil )
(x_sum_eq formula-decl nil extended_nnreal "extended_nnreal/" )
(x_add const-decl "extended_nnreal" extended_nnreal
"extended_nnreal/" )
(O const-decl "T3" function_props nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(x_sum const-decl "extended_nnreal" extended_nnreal
"extended_nnreal/" )
(measure_empty? const-decl "bool" generalized_measure_def nil )
(empty? const-decl "bool" sets nil )
(finite_emptyset name-judgement "finite_set" finite_sets nil )
(finite_emptyset name-judgement "finite_set[T]" countable_props
"sets_aux/" )
(finite_emptyset name-judgement "finite_set[T]" countable_setofsets
"sets_aux/" )
(emptyset const-decl "set" sets 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 )
(x_le const-decl "bool" extended_nnreal "extended_nnreal/" )
(nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
real_types nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(x_eq const-decl "bool" extended_nnreal "extended_nnreal/" )
(minus_odd_is_odd application-judgement "odd_int" integers nil )
(A_empty formula-decl nil ast_def nil )
(n!1 skolem-const-decl "nat" ast_def nil )
(i!1 skolem-const-decl "nat" ast_def nil )
(TRUE const-decl "bool" booleans nil )
(x_add_sum formula-decl nil extended_nnreal "extended_nnreal/" )
(IUnion const-decl "set[T]" indexed_sets nil )
(disjoint_indexed_measurable? const-decl "bool"
generalized_measure_def nil )
(setof type-eq-decl nil defined_types nil )
(union_diff_subset formula-decl nil sets_lemmas nil )
(finite_disjoint_union? const-decl "bool" subset_algebra_def nil )
(Union_surjective name-judgement
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sets_lemmas nil ))
shostak))
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(generalized_measure_monotone-1 nil 3458455064
("" (skosimp)
((""
(lemma "generalized_monotonicity"
("mu" "mu!1" "Y" "b!1" "n" "1" "I"
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(("1" (split)
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mu!1(a!1))")
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nil ))
nil ))
nil )
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nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(lemma
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("n"
"0"
"a"
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nil ))
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nil ))
nil )
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(("3" (expand "emptyset" )
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("4" (hide-all-but 1)
(("4" (skosimp) (("4" (assert ) (("4" (grind) nil nil )) nil ))
nil ))
nil ))
nil )
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nil ))
nil )
((measure_type nonempty-type-eq-decl nil generalized_measure_def
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(T formal-type-decl nil ast_def nil )
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(/= const-decl "boolean" notequal nil )
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shostak))
(ast_TCC1 0
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(("" (typepred "mu!1" )
(("" (expand "outer_measure?" )
((""
(name "F" "LAMBDA X:
x_inf({z
|
EXISTS
I:
x_eq
(x_sum
(o[nat, (A), extended_nnreal]
(mu!1, I)),
z)
AND
subset?[T](X, IUnion[nat, T](I))})")
(("" (replace -1)
(("" (case-replace "om_empty?(F)" )
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nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
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nil )
("2" (hide 2)
(("2" (skosimp)
(("2"
(case "x_lt(F(IUnion(X!1)),
x_sum(LAMBDA (n: nat):
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(("1"
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(("1"
(case-replace
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(("1"
(lemma
"convergence_sequences.limit_lemma"
("v"
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(("1"
(hide -2)
(("1"
(lemma
"cnv_seq_sum"
("s1"
"series(LAMBDA (i: nat): F(X!1(i))`2)"
"l1"
"convergence_sequences.limit(series(LAMBDA (i:nat): F(X!1(i))`2))"
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"l2"
"epsilon!1" ))
(("1"
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IF F(X!1(i))`1
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ELSE 0
ENDIF)=series(LAMBDA (i: nat): F(X!1(i))`2) +
series(LAMBDA (i: nat): epsilon!1 / expt(2, 1 + i))")
(("1"
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(name-replace
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(("1"
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rl)
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1
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nil
nil ))
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2)
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nil
nil ))
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nil ))
nil ))
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nil ))
nil ))
nil ))
nil )
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(hide-all-but
(-4 1))
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t)
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(("2"
(hide
2)
(("2"
(apply-extensionality
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t)
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nil ))
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+
F(X!1(i))`2
ELSE 0
ENDIF))")
(("1"
(lemma
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("s1"
"series(LAMBDA (i: nat):
IF F(X!1(i))`1
THEN epsilon!1 / expt(2, 1 + i) + F(X!1(i))`2
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sigma(0, x,
LAMBDA (i: nat):
IF F(X!1(i))`1
THEN epsilon!1 / expt(2, 1 + i) + F(X!1(i))`2
ELSE 0
ENDIF)
-
sigma(0, x,
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(rewrite
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ELSE 0
ENDIF
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-1)
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F(X!1(n))))})")
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F(X!1(n))))})")
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-2
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--> --------------------
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--> --------------------
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