(measure_def
(IMP_generalized_measure_def_TCC1 0
(IMP_generalized_measure_def_TCC1-1 nil 3458457144
("" (assuming-tcc) nil nil )
((finite_emptyset name-judgement "finite_set" finite_sets nil )
(finite_emptyset name-judgement "finite_set[T]" countable_props
"sets_aux/" )
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nil ))
(increasing_indexed_measurable_TCC1 0
(increasing_indexed_measurable_TCC1-1 nil 3450166483
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(("" (expand "increasing_indexed?" )
(("" (split)
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(("1" (expand "IUnion" ) (("1" (propax) nil nil )) nil ))
nil ))
nil )
("2" (expand "increasing?" )
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nil ))
nil ))
nil ))
nil ))
nil )
((increasing_indexed? const-decl "bool" nat_indexed_sets
"sets_aux/" )
(increasing? const-decl "bool" fun_preds_partial "structures/" )
(member const-decl "bool" sets nil )
(subset? const-decl "bool" sets nil )
(boolean nonempty-type-decl nil booleans nil )
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(IUnion const-decl "set[T]" indexed_sets nil )
(set type-eq-decl nil sets nil )
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(int nonempty-type-eq-decl nil integers nil )
(integer_pred const-decl "[rational -> boolean]" integers nil )
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(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 )
(number nonempty-type-decl nil numbers nil )
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(subset_algebra_fullset name-judgement "(S)" measure_def nil ))
nil ))
(sigma_finite_measure_TCC1 0
(sigma_finite_measure_TCC1-1 nil 3396670582
("" (expand "sigma_finite_measure?" )
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("2" (expand "measure_sigma_finite?" )
(("2" (expand "zero_measure" )
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nil ))
nil ))
nil )
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
((measure_type_TCC1 subtype-tcc nil generalized_measure_def nil )
(T formal-type-decl nil measure_def nil )
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(bool nonempty-type-eq-decl nil booleans nil )
(setof type-eq-decl nil defined_types nil )
(setofsets type-eq-decl nil sets nil )
(subset_algebra? const-decl "bool" subset_algebra_def nil )
(subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil )
(S formal-const-decl "subset_algebra" measure_def nil )
(zero_measure const-decl "extended_nnreal" generalized_measure_def
nil )
(intersection const-decl "set" sets nil )
(member const-decl "bool" 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 )
(disjoint_indexed_measurable nonempty-type-eq-decl nil
generalized_measure_def 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 )
(disjoint_indexed_measurable? const-decl "bool"
generalized_measure_def nil )
(set type-eq-decl nil sets nil )
(IF const-decl "[boolean, T, T -> T]" if_def nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(fullset const-decl "set" sets nil )
(emptyset const-decl "set" sets nil )
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(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/" )
(subset_algebra_emptyset name-judgement "(S)" measure_def nil )
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nil ))
(discrete_measure_TCC1 0
(discrete_measure_TCC1-1 nil 3396670582
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(("" (split)
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(("1" (rewrite "card_emptyset[T]" ) nil nil )) nil )
("2" (expand "measure_countably_additive?" )
(("2"
(name-replace "MM" "LAMBDA a:
IF is_finite[T](a) THEN (TRUE, card[T](a)) ELSE (FALSE, 0) ENDIF")
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(case-replace "(LAMBDA (i_1: nat):
IF is_finite[T](X!1(i_1))
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(("1"
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-
"epsilon!1" )
(("1"
(skosimp)
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"n!1" )
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nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(expand
"bounded_above?" )
(("2"
(inst
+
"card[T](IUnion(X!1))" )
nil
nil ))
nil ))
nil )
("3"
(hide 2)
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(expand
"fullset" )
(("3"
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"image" )
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1)
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-1)
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nil ))
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("2"
(inst
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"i!1" )
nil
nil ))
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nil )
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(hide
-1
-2)
(("2"
(lemma
"convergence_cauchy1"
("u"
"CS" ))
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(assert )
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(expand
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(inst
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"1/2" )
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(x!1)")
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(("1"
(skosimp)
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(prop)
(("1"
(inst
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"i!1" )
nil
nil ))
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nil )
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1
2)
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1)
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(inst
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"n!1" )
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(inst
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"i!1" )
(("2"
(replace
-9)
(("2"
(replace
-8)
(("2"
(hide
-9
-8)
(("2"
(lemma
"same_card_subset[T]"
("A"
"{x: T |
EXISTS (i_1: nat):
IF i_1 <= n!1 THEN X!1(i_1)(x) ELSE FALSE ENDIF}"
"B"
"{x_1: T |
EXISTS (i_2: nat):
IF i_2 <= i!1 THEN X!1(i_2)(x_1) ELSE FALSE ENDIF}"))
(("1"
(replace
-2)
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(hide
-2)
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(split
-1)
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(lemma
"extensionality_postulate"
("f"
"{x_1: T |
EXISTS (i_2: nat):
IF i_2 <= n!1 THEN X!1(i_2)(x_1) ELSE FALSE ENDIF}"
"g"
"{x_2: T |
EXISTS (i_1: nat):
IF i_1 <= i!1 THEN X!1(i_1)(x_2) ELSE FALSE ENDIF}"))
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rl)
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-1)
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"member" )
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(prop)
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nil )
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(hide -2 2)
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(lemma
"card_subset[T]"
("A"
"IUnion(LAMBDA (n: nat):
IF n <= i!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"
"B"
"IUnion(X!1)" ))
(("2"
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(expand
"subset?" )
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(prop)
(("2"
(inst
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nil
nil ))
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nil ))
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nil ))
nil ))
nil )
("2"
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("i!1" "j!1" ))
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-4
-5)
(("2"
(lemma
"card_subset[T]"
("A"
"IUnion(LAMBDA (n_1: nat):
IF n_1 <= i!1 THEN X!1(n_1) ELSE emptyset[T] ENDIF)"
"B"
"IUnion(LAMBDA (n: nat):
IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
(("2"
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2)
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nil ))
nil )
("2"
(hide
2)
(("2"
(expand
"bounded_above?" )
(("2"
(inst
+
"card[T](IUnion(X!1))" )
nil
nil ))
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2)
(("3"
(expand
"fullset" )
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(-5
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1))
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2)
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-
"x!1" )
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nil ))
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nil ))
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("S"
"IUnion(Y!1)" ))
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(assert )
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(lemma
"choose_rest[T]"
("a"
"IUnion(Y!1)" ))
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(assert )
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(lemma
"card_rest[T]"
("S"
"IUnion(Y!1)" ))
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(lemma
"choose_member[T]"
("a"
"IUnion(Y!1)" ))
(("2"
(name-replace
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1
-1)
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(name
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"lambda i: remove[T](CC,Y!1(i))" )
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-4)
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"A"
"Y!1(i!1)" ))
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nil
nil )
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nil )
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(lemma
"card_subset[T]"
("A"
"IUnion(LAMBDA (n_1: nat):
IF n_1 <= i!1 THEN X!1(n_1) ELSE emptyset[T] ENDIF)"
"B"
"IUnion(LAMBDA (n: nat):
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2)
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(lemma
"card_subset[T]"
("A"
"IUnion(LAMBDA (n: nat):
IF n <= i!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"
"B"
"IUnion(X!1)" ))
(("2"
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nil
nil )
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2)
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nil
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(lemma
"convergence_cauchy1"
("u" "CS" ))
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(inst - "1/2" )
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nil
nil ))
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nil )
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(hide 2)
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(apply-extensionality
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t)
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(expand
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1)
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(prop)
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(inst
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nil
nil )
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2)
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1)
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"i!1" )
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(inst
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"n!1" )
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(name-replace
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("A"
"IU_n"
"B"
"IU_i" ))
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(split
-1)
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(expand
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nil )
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(inst
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(inst
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"x!1" )
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(lemma
"same_card_subset[T]"
("A"
"IU_n"
"B"
"IU_i" ))
(("2"
(split)
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(lemma
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("f"
"IU_n"
"g"
"IU_i" ))
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-2
rl)
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(hide
-1)
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(inst
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"x!1" )
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nil ))
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nil ))
nil ))
nil ))
nil ))
nil )
("2"
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1
-1)
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(assert )
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(inst
2
"i!1" )
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(assert )
nil
nil ))
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nil ))
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nil ))
nil ))
nil ))
nil )
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(-2
1))
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nil ))
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nil ))
nil ))
nil ))
nil ))
nil )
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(-1
1))
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(expand
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nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
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nil ))
nil )
("2"
(inst
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nil ))
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nil ))
nil ))
nil )
("2" (assert ) nil nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide-all-but (-2 1))
(("2"
(copy -1)
(("2"
(induct "i" )
(("1"
(case-replace
"(IUnion(LAMBDA (n: nat):
IF n <= 0 THEN X!1(n) ELSE emptyset[T] ENDIF))=X!1(0)")
(("1" (inst - "0" ) nil nil )
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(hide-all-but 1)
(("2"
(apply-extensionality
:hide?
t)
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(expand "emptyset" )
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(expand "IUnion" )
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(case-replace
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(inst + "0" )
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nil
nil ))
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nil
nil ))
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nil ))
nil )
("2"
(skosimp*)
(("2"
(case-replace
"IUnion(LAMBDA (n: nat):
IF n <= j!1 + 1 THEN X!1(n) ELSE emptyset[T] ENDIF)=union(X!1(1+j!1),IUnion(LAMBDA (n: nat):
IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF))")
(("1"
(hide -1)
(("1"
(lemma
"finite_union[T]"
("A"
"X!1(1 + j!1)"
"B"
"IUnion(LAMBDA (n_1: nat):
IF n_1 <= j!1 THEN X!1(n_1) ELSE emptyset[T] ENDIF)"))
(("1" (propax) nil nil )
("2" (propax) nil nil )
("3"
(inst - "1+j!1" )
nil
nil ))
nil ))
nil )
("2"
(hide-all-but 1)
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(apply-extensionality
:hide?
t)
(("2"
(expand "union" )
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(expand "member" )
(("2"
(expand "IUnion" )
(("2"
(expand
"emptyset" )
(("2"
(case-replace
"X!1(1 + j!1)(x!1)" )
(("1"
(inst
+
"1+j!1" )
(("1"
(assert )
nil
nil ))
nil )
("2"
(assert )
(("2"
(case-replace
"EXISTS (i: nat): IF i <= j!1 THEN X!1(i)(x!1) ELSE FALSE ENDIF" )
(("1"
(skosimp)
(("1"
(prop)
(("1"
(inst
+
"i!1" )
(("1"
(assert )
nil
nil ))
nil ))
nil ))
nil )
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(replace
1
3)
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(assert )
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(skosimp)
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(prop)
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(expand
"<="
-1)
(("2"
(split
-1)
(("1"
(inst
+
"i!1" )
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(assert )
nil
nil ))
nil )
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(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(expand "series" )
(("2"
(copy -1)
(("2"
(induct "i" )
(("1"
(case-replace
"IUnion(LAMBDA (n: nat):
IF n <= 0 THEN X!1(n) ELSE emptyset[T] ENDIF) =X!1(0)")
(("1"
(expand "sigma" )
(("1"
(expand "sigma" )
(("1" (propax) nil nil ))
nil ))
nil )
("2"
(hide 2)
(("2"
(apply-extensionality
:hide?
t)
(("2"
(expand "IUnion" )
(("2"
(expand "emptyset" )
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(case-replace
"X!1(0)(x!1)" )
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(inst + "0" )
(("1"
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nil
nil ))
nil )
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(assert )
nil
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(skosimp*)
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(expand "sigma" 1)
(("2"
(case-replace
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IF n <= 1 + j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)=union(X!1(1+j!1),IUnion(LAMBDA (n: nat):
IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF))")
(("1"
(hide -1)
(("1"
(case
"disjoint?(X!1(1+j!1),IUnion(LAMBDA (n: nat):
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(("1"
(lemma
"card_disj_union[T]"
("A"
"X!1(1 + j!1)"
"B"
"IUnion(LAMBDA (n: nat):
IF n <= j!1 THEN X!1(n) ELSE emptyset[T] ENDIF)"))
(("1"
(assert )
nil
nil ))
nil )
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(hide -1 2)
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(hide-all-but 1)
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(typepred
"X!1" )
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(expand
"disjoint_indexed_measurable?" )
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(expand
"disjoint?" )
(("2"
(expand
"disjoint?" )
(("2"
(expand
"intersection" )
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(expand
"IUnion" )
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(expand
"emptyset" )
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(expand
"empty?" )
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(expand
"member" )
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(skosimp*)
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(prop)
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(inst
-
"i!1"
"1+j!1" )
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nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil ))
nil )
("2"
(hide -1 2)
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(apply-extensionality
:hide?
t)
(("2"
(expand "union" )
(("2"
(expand "member" )
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(expand
"IUnion" )
(("2"
(case-replace
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(("1"
(inst
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"1+j!1" )
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nil ))
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(assert )
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(case-replace
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(("1"
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nil ))
nil )
("3"
(hide 2)
(("3"
(induct "i" )
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(case-replace
"IUnion[nat, T]
(LAMBDA (n: nat):
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(inst - "0" )
nil
nil )
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(hide-all-but 1)
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shostak))
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shostak))
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(expand
"<="
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(("2"
(split)
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"n!1" )
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(inst
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"n!1" )
(("1"
(replace
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(("1"
(replace
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2
:dir
rl)
(("1"
(expand
"IUnion"
2)
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(expand
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(("1"
(expand
"Union" )
(("1"
(inst
+
"B!1(n!2)" )
(("1"
(inst
+
"n!2" )
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nil ))
nil ))
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nil ))
nil ))
nil ))
nil ))
nil ))
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(("1" (inst + "n!1" ) nil nil ))
nil ))
nil ))
nil ))
nil )
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(inst
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"n!2"
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(("2"
(expand
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(assert )
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(inst
-
"x!1" )
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(assert )
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nil ))
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("2" (propax) nil nil ))
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(("2" (propax) nil nil )) nil ))
nil ))
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("2" (assert ) nil nil ))
nil ))
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nil )
((sigma_finite_def1 formula-decl nil measure_def nil )
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(setof type-eq-decl nil defined_types nil )
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(subset_algebra? const-decl "bool" subset_algebra_def nil )
(subset_algebra nonempty-type-eq-decl nil subset_algebra_def nil )
(S formal-const-decl "subset_algebra" measure_def 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 )
(nnreal type-eq-decl nil real_types nil )
(extended_nnreal nonempty-type-eq-decl nil extended_nnreal
"extended_nnreal/" )
(disjoint_indexed_measurable nonempty-type-eq-decl nil
generalized_measure_def nil )
(disjoint_indexed_measurable? const-decl "bool"
generalized_measure_def nil )
(sequence type-eq-decl nil sequences nil )
(set type-eq-decl nil sets nil )
(nat nonempty-type-eq-decl nil naturalnumbers 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 )
(increasing_IUnion formula-decl nil nat_indexed_sets "sets_aux/" )
(increasing_indexed? const-decl "bool" nat_indexed_sets
"sets_aux/" )
(subset_algebra_fullset name-judgement "(S)" measure_def nil )
(int_minus_int_is_int application-judgement "int" integers nil )
(nnint_plus_posint_is_posint application-judgement "posint"
integers nil )
(even_minus_odd_is_odd application-judgement "odd_int" integers
nil )
(measure_union formula-decl nil measure_def nil )
(x_add const-decl "extended_nnreal" extended_nnreal
"extended_nnreal/" )
(x_le const-decl "bool" extended_nnreal "extended_nnreal/" )
(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 )
(nat_induction formula-decl nil naturalnumbers nil )
(pred type-eq-decl nil defined_types nil )
(increasing_indexed_measurable nonempty-type-eq-decl nil
measure_def nil )
(increasing_indexed_measurable? const-decl "bool" measure_def nil )
(B!1 skolem-const-decl "sequence[set[T]]" measure_def nil )
(AND const-decl "[bool, bool -> bool]" booleans nil )
(subset_algebra_union judgement-tcc nil subset_algebra nil )
(disjoint_IUnion formula-decl nil nat_indexed_sets "sets_aux/" )
(B!1 skolem-const-decl "sequence[set[T]]" measure_def nil )
(NOT const-decl "[bool -> bool]" booleans nil )
(measure_monotone formula-decl nil measure_def nil )
(= const-decl "[T, T -> boolean]" equalities nil )
(IUnion const-decl "set[T]" nat_indexed_sets "sets_aux/" )
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(member const-decl "bool" sets nil )
(image const-decl "set[R]" function_image nil )
(<= const-decl "bool" reals nil )
(i!1 skolem-const-decl "nat" measure_def nil )
(real_le_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(Union const-decl "set" sets nil )
(increasing? const-decl "bool" fun_preds_partial "structures/" )
(n!1 skolem-const-decl "nat" measure_def nil )
(P!1 skolem-const-decl "increasing_indexed_measurable" measure_def
nil )
(subset_is_partial_order name-judgement "(partial_order?[set[T]])"
sets_lemmas nil )
(n!1 skolem-const-decl "nat" measure_def nil )
(subset_algebra_difference application-judgement "(S)" measure_def
nil )
(difference const-decl "set" sets nil )
(int_plus_int_is_int application-judgement "int" integers nil )
(x1!1 skolem-const-decl "nat" measure_def nil )
(- const-decl "[numfield, numfield -> numfield]" number_fields nil )
(real_ge_is_total_order name-judgement "(total_order?[real])"
real_props nil )
(n!1 skolem-const-decl "nat" measure_def nil )
(disjoint? const-decl "bool" indexed_sets_aux "sets_aux/" )
(disjoint? const-decl "bool" sets nil )
(empty? const-decl "bool" sets nil )
(intersection const-decl "set" sets nil )
(/= const-decl "boolean" notequal nil )
(real_lt_is_strict_total_order name-judgement
"(strict_total_order?[real])" real_props nil )
(n!2 skolem-const-decl "({i | i <= 1 + n!1})" measure_def nil )
(measure? const-decl "bool" generalized_measure_def nil ))
shostak)))
Messung V0.5 in Prozent C=100 H=100 G=100
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