| products/sources/formale Sprachen/PVS/measure_integration/ |
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Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: bug_2945.v Sprache: CoqOriginal von: PVS© |
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(outer_measure_props
(m_outer_empty 0 (m_outer_empty-1 nil 3396688507 ("" (typepred "m") (("" (expand "outer_measure?") (("" (flatten) (("" (expand "om_empty?") (("" (propax) nil nil)) nil)) nil)) nil)) nil) ((om_empty? const-decl "bool" outer_measure_def nil) (boolean nonempty-type-decl nil booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (T formal-type-decl nil outer_measure_props nil) (set type-eq-decl nil sets 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/") (outer_measure? const-decl "bool" outer_measure_def nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (m formal-const-decl "outer_measure" outer_measure_props nil)) shostak)) (m_outer_increasing 0 (m_outer_increasing-1 nil 3396688522 ("" (typepred "m") (("" (expand "outer_measure?") (("" (flatten) (("" (expand "om_increasing?") (("" (propax) nil nil)) nil)) nil)) nil)) nil) ((om_increasing? const-decl "bool" outer_measure_def nil) (boolean nonempty-type-decl nil booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (T formal-type-decl nil outer_measure_props nil) (set type-eq-decl nil sets 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/") (outer_measure? const-decl "bool" outer_measure_def nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (m formal-const-decl "outer_measure" outer_measure_props nil)) shostak)) (m_outer_subadditive 0 (m_outer_subadditive-1 nil 3396688484 ("" (typepred "m") (("" (expand "outer_measure?") (("" (flatten) (("" (expand "om_countably_subadditive?") (("" (propax) nil nil)) nil)) nil)) nil)) nil) ((om_countably_subadditive? const-decl "bool" outer_measure_def nil) (boolean nonempty-type-decl nil booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans nil) (T formal-type-decl nil outer_measure_props nil) (set type-eq-decl nil sets 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/") (outer_measure? const-decl "bool" outer_measure_def nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (m formal-const-decl "outer_measure" outer_measure_props nil)) shostak)) (outer_negligible_TCC1 0 (outer_negligible_TCC1-1 nil 3396763601 ("" (expand "outer_negligible?") (("" (typepred "m") (("" (expand "outer_measure?") (("" (flatten) (("" (expand "om_empty?") (("" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ((m formal-const-decl "outer_measure" outer_measure_props nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (outer_measure? const-decl "bool" outer_measure_def nil) (extended_nnreal nonempty-type-eq-decl nil extended_nnreal "extended_nnreal/") (nnreal type-eq-decl nil real_types nil) (>= const-decl "bool" reals 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) (set type-eq-decl nil sets nil) (T formal-type-decl nil outer_measure_props nil) (NOT const-decl "[bool -> bool]" booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (om_empty? const-decl "bool" outer_measure_def nil) (outer_negligible? const-decl "bool" outer_measure_props nil)) nil)) (outer_measurable_TCC1 0 (outer_measurable_TCC1-1 nil 3396585602 ("" (expand "outer_measurable?") (("" (skosimp*) (("" (case-replace "intersection(y!1, emptyset[T])=emptyset[T]") (("1" (case-replace "intersection(y!1, complement(emptyset[T]))=y!1") (("1" (lemma "m_outer_empty") (("1" (replace -1) (("1" (expand "x_add") (("1" (expand "x_eq") (("1" (hide-all-but 1) (("1" (grind) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (apply-extensionality :hide? t) (("2" (grind) nil nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (apply-extensionality :hide? t) (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil) ((complement const-decl "set" sets nil) (x_eq const-decl "bool" extended_nnreal "extended_nnreal/") (x_add const-decl "extended_nnreal" extended_nnreal "extended_nnreal/") (m_outer_empty formula-decl nil outer_measure_props nil) (member const-decl "bool" sets nil) (emptyset const-decl "set" sets nil) (intersection const-decl "set" sets nil) (= const-decl "[T, T -> boolean]" equalities nil) (set type-eq-decl nil sets nil) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (T formal-type-decl nil outer_measure_props nil) (finite_intersection1 application-judgement "finite_set[T]" bounded_nats "orders/") (finite_emptyset name-judgement "finite_set[T]" bounded_nats "orders/") (finite_emptyset name-judgement "finite_set[T]" countable_props "sets_aux/") (finite_emptyset name-judgement "finite_set" finite_sets nil) (outer_measurable? const-decl "bool" outer_measure_props nil)) nil)) (pairwise_subadditive 0 (pairwise_subadditive-1 nil 3396787341 ("" (skosimp*) (("" (lemma "m_outer_subadditive" ("A" "lambda i: if i=0 then intersection(y!1, x!1) elsif i=1 then intersection(y!1, c (("" (case-replace "IUnion(LAMBDA i: IF i = 0 THEN intersection(y!1, x!1) ELSIF i = 1 THEN intersection(y!1, complement(x!1)) ELSE emptyset[T] ENDIF) = y!1") (("1" (hide -1) (("1" (expand "o") (("1" (case-replace "x_sum(LAMBDA (i: nat): IF i = 0 THEN m(intersection(y!1, x!1)) ELSE IF i = 1 THEN m(intersection(y!1, complement(x!1))) ELSE m(emptyset[T]) ENDIF ENDIF)=x_add(m(intersection(y!1, x!1)), m(intersection(y!1, complement(x!1))))") (("1" (hide -1 2) (("1" (expand "x_sum") (("1" (name-replace "AA" "intersection(y!1, x!1)") (("1" (name-replace "BB" "intersection(y!1, complement(x!1))") (("1" (lemma "m_outer_empty") (("1" (replace -1) (("1" (assert) (("1" (expand "x_add") (("1" (case-replace "m(AA)`1") (("1" (case-replace "m(BB)`1") (("1" (lift-if 1) (("1" (assert) (("1" (lemma "zero_tail_series_conv" ("a" "(LAMBDA (i_1: nat): IF i_1 = 0 THEN m(AA)`2 ELSE IF i_1 = 1 THEN m(BB)`2 ELSE 0 ENDIF ENDIF)" "n" "1")) (("1" (case-replace "(FORALL (m_1: nat): 1 < m_1 => (LAMBDA (i_1: nat): IF i_1 = 0 THEN m(AA)`2 ELSE IF i_1 = 1 THEN m(BB)`2 ELSE 0 ENDIF ENDIF) (m_1) = 0)") (("1" (replace -2) (("1" (hide -2) (("1" (lemma "zero_tail_series_limit" ("a" "(LAMBDA (i_1: nat): IF i_1 = 0 THEN m(AA)`2 ELSE IF i_1 = 1 THEN m(BB)`2 ELSE 0 ENDIF ENDIF)" "n" "1")) (("1" (replace -2 -1) (("1" (replace -1) (("1" (hide -1 -2) (("1" (expand "series") (("1" (expand "sigma") (("1" (expand "sigma") (("1" (expand "sigma") (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2 -1) (("2" (skosimp*) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (lift-if) (("2" (assert) (("2" (prop) (("2" (inst - "1") (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) (("2" (lift-if) (("2" (assert) (("2" (prop) (("2" (assert) (("2" (inst - "0") nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (apply-extensionality :hide? t) (("2" (expand "IUnion") (("2" (expand "intersection") (("2" (expand "complement") (("2" (expand "emptyset") (("2" (expand "member") (("2" (case-replace "y!1(x!2)") (("1" (case-replace "x!1(x!2)") (("1" (inst + "0") nil nil) ("2" (assert) (("2" (inst + "1") (("2" (assert) nil nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((emptyset const-decl "set" sets nil) (complement const-decl "set" sets nil) (intersection const-decl "set" sets nil) (= const-decl "[T, T -> boolean]" equalities nil) (IF const-decl "[boolean, T, T -> T]" if_def nil) (sequence type-eq-decl nil sequences nil) (set type-eq-decl nil sets nil) (T formal-type-decl nil outer_measure_props 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) (m_outer_subadditive formula-decl nil outer_measure_props nil) (member const-decl "bool" sets nil) (nnreal type-eq-decl nil real_types nil) (extended_nnreal nonempty-type-eq-decl nil extended_nnreal "extended_nnreal/") (x_sum const-decl "extended_nnreal" extended_nnreal "extended_nnreal/") (outer_measure? const-decl "bool" outer_measure_def nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (m formal-const-decl "outer_measure" outer_measure_props nil) (x_add const-decl "extended_nnreal" extended_nnreal "extended_nnreal/") (< const-decl "bool" reals nil) (IMPLIES const-decl "[bool, bool -> bool]" booleans nil) (sigma def-decl "real" sigma "reals/") (real_gt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (sigma_0_neg formula-decl nil sigma_nat "reals/") (series const-decl "sequence[real]" series "series/") (zero_tail_series_limit formula-decl nil series_lems "series/") (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (zero_tail_series_conv formula-decl nil series_lems "series/") (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal" real_types 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]" bounded_nats "orders/") (m_outer_empty formula-decl nil outer_measure_props nil) (O const-decl "T3" function_props nil) (IUnion const-decl "set[T]" indexed_sets nil)) shostak)) (outer_measurable_def 0 (outer_measurable_def-1 nil 3396788683 ("" (skosimp*) (("" (expand "outer_measurable?") (("" (split) (("1" (skosimp*) (("1" (inst - "y!1") (("1" (grind) nil nil)) nil)) nil) ("2" (skosimp*) (("2" (inst - "y!1") (("2" (lemma "pairwise_subadditive" ("x" "x!1" "y" "y!1")) (("2" (lemma "x_le_antisymmetric" ("x" "m(y!1)" "y" "x_add(m(intersection(y!1, x!1)), m(intersection(y!1, complement(x!1))))")) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((outer_measurable? const-decl "bool" outer_measure_props nil) (pairwise_subadditive formula-decl nil outer_measure_props nil) (x_le_antisymmetric formula-decl nil extended_nnreal "extended_nnreal/") (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/") (outer_measure? const-decl "bool" outer_measure_def nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (m formal-const-decl "outer_measure" outer_measure_props nil) (intersection const-decl "set" sets nil) (complement const-decl "set" sets nil) (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (x_add const-decl "extended_nnreal" extended_nnreal "extended_nnreal/") (x_eq const-decl "bool" extended_nnreal "extended_nnreal/") (x_le const-decl "bool" extended_nnreal "extended_nnreal/") (set type-eq-decl nil sets nil) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (T formal-type-decl nil outer_measure_props nil)) shostak)) (outer_negligible_is_outer_measurable 0 (outer_negligible_is_outer_measurable-1 nil 3396763601 ("" (skosimp) (("" (typepred "x!1") (("" (rewrite "outer_measurable_def") (("" (skosimp) (("" (expand "outer_negligible?") (("" (lemma "m_outer_increasing" ("x" "intersection(y!1, x!1)" "y" "x!1")) (("" (split -1) (("1" (expand "x_le" -1) (("1" (replace -2) (("1" (assert) (("1" (flatten) (("1" (expand "x_add") (("1" (replace -1) (("1" (assert) (("1" (case-replace "m(intersection(y!1, x!1))`2 = 0") (("1" (hide -1 -3 -2) (("1" (assert) (("1" (lemma "m_outer_increasing" ("x" "intersection(y!1, complement(x!1))" "y" "y!1")) (("1" (split -1) (("1" (expand "x_le") (("1" (case-replace "m(y!1)`1") (("1" (flatten) (("1" (assert) (("1" (replace -2) (("1" (propax) nil nil)) nil)) nil)) nil) ("2" (replace 1) (("2" (propax) nil nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2 -3 -1) (("2" (typepred "m(intersection(y!1, x!1))`2") (("2" (name-replace "DRL" "m(intersection(y!1, x!1))`2") (("2" (lemma "reals_totally_ordered") (("2" (expand "strict_total_order?") (("2" (expand "trichotomous?") (("2" (expand "strict_order?") (("2" (flatten) (("2" (expand ">=") (("2" (expand "<=") (("2" (replace 1) (("2" (split -4) (("1" (expand "transitive?") (("1" (inst - "DRL" "0" "DRL") (("1" (assert) (("1" (replace -4) (("1" (propax) 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)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((outer_negligible nonempty-type-eq-decl nil outer_measure_props nil) (outer_negligible? const-decl "bool" outer_measure_props nil) (set type-eq-decl nil sets nil) (T formal-type-decl nil outer_measure_props nil) (NOT const-decl "[bool -> bool]" booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (boolean nonempty-type-decl nil booleans nil) (intersection const-decl "set" sets nil) (m_outer_increasing formula-decl nil outer_measure_props nil) (x_le const-decl "bool" extended_nnreal "extended_nnreal/") (real_le_is_total_order name-judgement "(total_order?[real])" real_props nil) (x_add const-decl "extended_nnreal" extended_nnreal "extended_nnreal/") (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal" real_types nil) (real_lt_is_strict_total_order name-judgement "(strict_total_order?[real])" real_props nil) (strict_total_order? const-decl "bool" orders nil) (strict_order? const-decl "bool" orders nil) (transitive? const-decl "bool" relations nil) (<= const-decl "bool" reals nil) (trichotomous? const-decl "bool" orders nil) (reals_totally_ordered formula-decl nil reals nil) (complement const-decl "set" sets nil) (subset? const-decl "bool" sets nil) (member const-decl "bool" sets nil) (subset_is_partial_order name-judgement "(partial_order?[set[T]])" sets_lemmas nil) (m formal-const-decl "outer_measure" outer_measure_props nil) (outer_measure nonempty-type-eq-decl nil outer_measure_def nil) (outer_measure? const-decl "bool" outer_measure_def nil) (extended_nnreal nonempty-type-eq-decl nil extended_nnreal "extended_nnreal/") (nnreal type-eq-decl nil real_types nil) (>= const-decl "bool" reals 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) (= const-decl "[T, T -> boolean]" equalities nil) (number nonempty-type-decl nil numbers nil) (outer_measurable_def formula-decl nil outer_measure_props nil)) nil)) (outer_measurable_complement 0 (outer_measurable_complement-1 nil 3396585602 ("" (skosimp) (("" (typepred "a!1") (("" (expand "outer_measurable?") 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