(integral_convergencenil 3409506492"" (skosimp*)"" "(EXISTS f: ae_convergence?(F!1, f)) IFF bounded?(integral o F!1)" )"1" (assert )"1" (skolem!)"1" (flatten)"1" (hide -2)"1" (split -1)"1" (expand "ae_increasing?" )"1" (expand "ae_convergence?" )"1" (expand "fullset" )"1" (expand "ae_convergence_in?" )"1" (expand "ae_in?" )"1" (skolem - "E1" )"1" (skolem - "E2" )"1" "union(E1,E2)" )"1" expand "negligible_set?" )"1" "1" "null_set_is_measurable" )"1" "X!1" )"1" "monotone_convergence_scaf" "f" "phi(complement(X!1))*f!1" "F" "lambda n: phi(complement(X!1))*F!1(n)" ))"1" "integral o (LAMBDA n: phi(complement(X!1)) * F!1(n))=integral o F!1" )"1" "1" replace -5)"1" "integral(phi(complement(X!1)) * f!1)=integral(f!1)" )"1" "1" nil nil )"2" "2" expand "pointwise_converges_upto?" )"2" "pointwise_increasing?(LAMBDA n: phi(complement(X!1)) * F!1(n))" )"1" expand "pointwise_convergence?" )"1" "1" "x!1" )"1" expand "complement" )"1" expand "member" )"1" expand "phi" )"1" expand "member" )"1" "1" expand "subset?" )"1" expand "union" )"1" expand "member" )"1" "x!1" )"1" expand "*" "1" "E2(x!1)" )"1" assert )"1" "1" expand "convergence?" )"1" "1" "0" )"1" "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" assert )"2" "X!1(x!1)" )"1" "1" expand "convergence?" )"1" "1" "0" )"1" "1" assert )nil nil ))nil ))nil ))nil ))nil ))nil )"2" assert )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" "2" expand "pointwise_increasing?" )"2" "2" expand "*" )"2" expand "complement" )"2" expand "phi" )"2" expand "member" )"2" "x!1" )"2" expand "subset?" )"2" expand "union" )"2" expand "member" )"2" "x!1" )"2" "X!1(x!1)" )"1" expand "increasing?" )"1" nil nil ))nil )"2" assert )"2" expand "increasing?" )"2" "2" assert )"2" "x!2" "y!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 ))nil )"2" "2" "integral_ae_eq" "f" "f!1" "h" "phi(complement(X!1)) * f!1" ))"1" "1" "1" assert )nil nil ))nil )"2" "2" expand "ae_eq?" )"2" expand "restrict" )"2" expand "pointwise_ae?" )"2" expand "*" )"2" expand "complement" )"2" expand "phi" )"2" expand "member" )"2" expand "ae?" )"2" expand "fullset" )"2" expand "ae_in?" )"2" "X!1" )"1" "1" assert )nil nil ))nil )"2" expand "negligible_set?" )"2" "X!1" )"2" assert )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "integrable_is_measurable" )"2" "2" rewrite "indefinite_integrable" )"2" "2" rewrite "measurable_complement" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"3" "3" rewrite "indefinite_integrable" )"3" rewrite "measurable_complement" )nil nil ))nil ))nil ))nil ))nil ))nil )"2" "2" expand "o " )"2" "1" "integral_ae_eq" "f" "F!1(x!1)" "h" "phi(complement(X!1)) * F!1(x!1)" ))"1" "1" "1" assert )nil nil ))nil )"2" "2" expand "ae_eq?" )"2" expand "restrict" )"2" expand "pointwise_ae?" )"2" expand "*" "2" expand "complement" )"2" expand "phi" )"2" expand "member" )"2" expand "ae?" )"2" expand "fullset" )"2" expand "ae_in?" )"2" "X!1" )"1" "1" assert )nil nil ))nil )"2" expand "negligible_set?" )"2" "X!1" )"2" assert )"2" "2" nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "integrable_is_measurable" )"2" rewrite "indefinite_integrable" )"2" rewrite "measurable_complement" )nil nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "indefinite_integrable" )"2" rewrite "measurable_complement" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" "2" rewrite "indefinite_integrable" )"2" rewrite "measurable_complement" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" (inst + "f!1" ) nil nil ))nil ))nil ))nil ))nil ))nil )"2" (hide 2)"2" (split)"1" (skosimp*)"1" (expand "bounded?" )"1" (split)"1" (expand "bounded_above?" )"1" (inst + "integral(f!1)" )"1" (skolem + ("n!1" ))"1" (expand "o " )"1" (expand "ae_convergence?" )"1" (expand "ae_increasing?" )"1" (expand "ae_convergence_in?" )"1" expand "fullset" )"1" expand "ae_in?" )"1" "1" "union(E!1,E!2)" )"1" expand "negligible_set?" )"1" "1" "integral_ae_le" "f1" "phi(complement(X!1))*F!1(n!1)" "f2" "phi(complement(X!1))*f!1" ))"1" "1" "integral_ae_eq" "f" "f!1" "h" "phi(complement(X!1)) * f!1" ))"1" "1" "1" replace -2 * rl)"1" "integral_ae_eq" "f" "F!1(n!1)" "h" "phi(complement(X!1)) * F!1(n!1)" ))"1" "1" "1" assert )nil nil ))nil )"2" "2" expand "ae_eq?" )"2" expand "restrict" )"2" expand "pointwise_ae?" )"2" expand "ae?" )"2" expand "*" )"2" expand "complement" )"2" expand "phi" )"2" expand "member" )"2" expand "fullset" )"2" expand "ae_in?" )"2" "X!1" )"1" "1" assert )nil nil ))nil )"2" rewrite "null_is_negligible" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "integrable_is_measurable" )nil nil ))nil ))nil ))nil )"2" "2" expand "ae_eq?" )"2" expand "restrict" )"2" expand "pointwise_ae?" )"2" expand "complement" )"2" expand "phi" )"2" expand "member" )"2" expand "ae?" )"2" expand "fullset" )"2" expand "ae_in?" )"2" "X!1" )"1" "1" assert )"1" expand "*" )"1" assert )nil nil ))nil ))nil ))nil )"2" rewrite "null_is_negligible" )nil nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil ))nil )"2" rewrite "integrable_is_measurable" )nil nil ))nil )"2" "2" expand "ae_le?" )"2" expand "*" )"2" expand "pointwise_ae?" )"2" expand "complement" )"2" expand "phi" )"2" expand "member" )"2" expand "ae?" )"2" expand "fullset" )"2" expand "ae_in?" )"2" "X!1" )"1" "1" assert )"1" expand "union" )"1" expand "subset?" )"1" expand "member" )"1" "x!1" )"1" "x!1" )"1" "x!1" )"1" "E!1(x!1)" )"1" "E!2(x!1)" )"1" assert )"1" "1" rewrite "metric_convergence_def" )"1" case "F!1(n!1)(x!1)-f!1(x!1)>0" )"1" expand "metric_converges_to" )"1" "F!1(n!1)(x!1) - f!1(x!1)" )"1" "1" "1" "n!1+n!2" )"1" assert )"1" quality 100%
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