| products/sources/formale sprachen/PVS/lebesgue/ |
|
Quellcode-Bibliothek© Kompilation durch diese Firma[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]Datei: fundamental_algebra.prf Sprache: LispOriginal von: PVS© |
|
||||||
|
(real_lebesgue_scaf
(limit_TCC1 0 (limit_TCC1-1 nil 3396062613 ("" (skosimp*) (("" (expand "convergent") (("" (expand "convergent?") (("" (expand "convergence") (("" (split) (("1" (skosimp*) (("1" (inst + "l!1") (("1" (rewrite "metric_convergence_def" *) (("1" (expand "metric_converges_to") (("1" (skosimp) (("1" (inst - "r!1") (("1" (skosimp) (("1" (inst + "n!1") (("1" (skosimp) (("1" (inst - "i!1") (("1" (expand "ball") (("1" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (skosimp*) (("2" (inst + "l!1") (("2" (skosimp) (("2" (rewrite "metric_convergence_def" *) (("2" (expand "metric_converges_to") (("2" (inst - "epsilon!1") (("2" (skosimp) (("2" (inst + "n!1") (("2" (skosimp) (("2" (inst - "i!1") (("2" (expand "ball") (("2" (grind) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ((convergent? const-decl "bool" topological_convergence "topology/") (convergent? const-decl "bool" convergence_sequences "analysis/") (number nonempty-type-decl nil numbers nil) (boolean nonempty-type-decl nil booleans 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) (metric_converges_to const-decl "bool" metric_space_def "metric_space/") (posreal nonempty-type-eq-decl nil real_types nil) (> const-decl "bool" reals nil) (minus_real_is_real application-judgement "real" reals nil) (member const-decl "bool" 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) (ball const-decl "set[T]" metric_space_def "metric_space/") (- const-decl "[numfield, numfield -> numfield]" number_fields nil) (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs nil) (- const-decl "[numfield -> numfield]" number_fields nil) (numfield nonempty-type-eq-decl nil number_fields nil) (AND const-decl "[bool, bool -> bool]" booleans nil) (nonneg_real nonempty-type-eq-decl nil real_types nil) (>= const-decl "bool" reals nil) (bool nonempty-type-eq-decl nil booleans nil) (sequence type-eq-decl nil sequences 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) (metric_convergence_def formula-decl nil metric_space "metric_space/") (convergence const-decl "bool" convergence_sequences "analysis/") (real_minus_real_is_real application-judgement "real" reals nil)) nil)) (lebesgue_outer_measure_TCC1 0 (lebesgue_outer_measure_TCC1-1 nil 3397030001 ("" (skosimp) (("" (skolem + ("n!1")) (("" (typepred "I!1(n!1)") (("" (expand "bounded_open_interval?") (("" (expand "bounded_interval?") (("" (flatten) nil nil)) nil)) nil)) nil)) nil)) nil) ((bounded_interval? const-decl "bool" real_topology "metric_space/") (boolean nonempty-type-decl nil booleans nil) (bool nonempty-type-eq-decl nil booleans nil) (NOT const-decl "[bool -> bool]" booleans 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) (set type-eq-decl nil sets nil) (bounded_open_interval? const-decl "bool" real_topology "metric_space/") (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) (bounded_open_interval nonempty-type-eq-decl nil real_topology "metric_space/") (sequence type-eq-decl nil sequences nil)) nil)) (lebesgue_outer_measure_TCC2 0 (lebesgue_outer_measure_TCC2-1 nil 3407841244 ("" (name-replace "L" "(LAMBDA A: x_inf({z | EXISTS I: x_le(x_sum(o[nat, interval, extended_nnreal] (x_length, I)), z) AND subset?[real](A, IUnion[nat, real](I))}))") (("1" (expand "outer_measure?") (("1" (case-replace "om_empty?(L)") (("1" (case-replace "om_increasing?(L)") (("1" (expand "om_countably_subadditive?") (("1" (skosimp) (("1" (expand "x_le") (("1" (flatten) (("1" (assert) (("1" (case "forall (n:nat): L(X!1(n))`1") (("1" (case "forall (n:nat,r:posreal): exists I: subset?[real](X!1(n),IUnion[nat,real]( (("1" (case "forall (r:posreal): EXISTS I: subset?[real](IUnion(X!1), IUnion[nat, real](I)) & x_lt(L(IUnion(X!1)), x_add((TRUE, r),x_sum(L o X!1)))") (("1" (hide -2) (("1" (inst-cp - "1") (("1" (skosimp) (("1" (expand "x_add" -3) (("1" (expand "x_lt" -3) (("1" (flatten) (("1" (assert) (("1" (case "x_sum(L o X!1)`2 (("1" (hide -3 -5 1) (("1" (inst - "L(IUnion(X!1))`2-x_sum(L o X!1)`2") (("1" (skosimp) (("1" (expand "x_add") (("1" (expand "x_lt") (("1" (propax) nil nil)) nil)) nil)) nil)) nil)) nil) ("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (skosimp) (("2" (case "EXISTS I: subset?[real](IUnion(X!1), IUnion[nat, real](I)) & x_lt(L(IUnion(X!1)),x_sum(LAMBDA n: x_add((TRUE, (r!1/2) / 2 ^ n), L(X!1(n)))))") (("1" (skosimp) (("1" (inst + "I!1") (("1" (hide -3) (("1" (assert) (("1" (case "x_eq(x_sum(LAMBDA n: x_add((TRUE, (r!1 / 2) / 2 ^ n), L(X!1(n)))),x_add((TRUE, (("1" (name-replace "LHS" "L(IUnion(X!1))") (("1" (name-replace "MID" "x_sum(LAMBDA n: x_add((TRUE, (r!1 / 2) / 2 ^ n), L(X!1(n))))") (("1" (name-replace "RHS" "x_add((TRUE, r!1), x_sum(L o X!1))") (("1" (hide-all-but (-1 -3 1)) (("1" (expand "x_lt") (("1" (expand "x_eq") (("1" (flatten) (("1" (assert) (("1" (flatten) (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide -1 -2 2) (("2" (expand "x_add") (("2" (expand "x_eq") (("2" (expand "o ") (("2" (case-replace "x_sum(LAMBDA n: IF L(X!1(n))`1 THEN (TRUE, (r!1 / 2) / 2 ^ n + L(X!1(n))`2) ELSE (FALSE, 0) ENDIF)`1") (("1" (expand "x_sum") (("1" (prop) (("1" (replace -3) (("1" (replace -5) (("1" (replace -3) (("1" (replace -1) (("1" (lemma "const_geometric_series" ("x" "1/2" "c" "r!1/2")) (("1" (case-replace "r!1 / 2 / (1 - 1 / 2)=r!1") (("1" (expand "abs" -2) (("1" (hide -1) (("1" (expand "convergent?") (("1" (skosimp*) (("1" (lemma "convergence_sequences.limit_def" ("v" "series(LAMBDA (i:nat): IF L(X!1(i))`1 THEN (r!1 / 2) / 2 ^ i + L(X!1(i))`2 ELSE 0 ENDIF)" "l" "l!2")) (("1" (assert) (("1" (replace -1) (("1" (hide -1 -4 -7 -8) (("1" (lemma "convergence_sequences.limit_def" ("v" "series(LAMBDA (i:nat): L(X!1(i))`2)" "l" "l!1")) (("1" (assert) (("1" (replace -1) (("1" (hide -1) (("1" (lemma "cnv_seq_sum" ("s1" "series(LAMBDA (i:nat): L(X!1(i))`2)" "l1" "l!1" "s2" "series(r!1 / 2 * geometric(1 / 2))" "l2" "r!1")) (("1" (assert) (("1" (rewrite "series_sum") (("1" (expand "+" -1) (("1" (case-replace "(LAMBDA (i:nat): IF L(X!1(i))`1 THEN (r!1 / 2) / 2 ^ i + L(X!1(i))`2 ELSE 0 ENDIF)=(LAMBDA (x: nat): (r!1 / 2 * geometric(1 / 2))(x) + L(X!1(x))`2)") (("1" (hide -1 -6) (("1" (lemma "convergence_sequences.unique_limit" ("u" "series(LAMBDA (x: nat): (r!1 / 2 * geometric(1 / 2))(x) + L(X!1(x))`2)" "l1" "l!2" "l2" "l!1+r!1")) (("1" (assert) nil nil)) nil)) nil) ("2" (apply-extensionality :hide? t) (("2" (inst -5 "x!1") (("2" (assert) (("2" (expand "geometric") (("2" (expand "*") (("2" (rewrite "inv_expt") (("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" (hide-all-but 1) (("2" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (expand "x_sum") (("2" (prop) (("1" (skosimp) (("1" (inst - "i!1") (("1" (assert) nil nil)) nil)) nil) ("2" (lemma "geometric_conv" ("x" "1/2")) (("2" (expand "abs") (("2" (lemma "series_scal" ("c" "r!1/2" "a" "geometric(1 / 2)")) (("2" (lemma "convergent_scal" ("s1" "series(geometric(1 / 2))" "a" "r!1/2")) (("2" (assert) (("2" (replace -2) (("2" (hide -2 -3) (("2" (lemma "convergent_sum" ("s1" "series(LAMBDA n: r!1 / 2 * geometric(1 / 2)(n))" "s2" "series(LAMBDA n: L(X!1(n))`2)")) (("2" (assert) (("2" (hide -2 -3) (("2" (rewrite "series_sum") (("2" (expand "+" -1) (("2" (case-replace "(LAMBDA (x: nat): r!1 / 2 * geometric(1 / 2)(x) + L(X!1(x))`2)=(LAMBDA n: IF L(X!1(n))`1 THEN (r!1 / 2) / 2 ^ n + L(X!1(n))`2 ELSE 0 ENDIF)") (("2" (apply-extensionality :hide? t) (("2" (inst - "x!1") (("2" (assert) (("2" (expand "geometric") (("2" (rewrite "inv_expt" 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)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide 2) (("2" (lemma "posreal_div_posreal_is_posreal" ("px" "r!1" "py" "2")) (("2" (inst - "_" "r!1/2") (("2" (name-replace "R" "r!1/2") (("2" (name "I_n" "lambda n: choose({I | subset?[real](X!1(n), IUnion[nat, real](I)) & x_lt(x_sum(LAMBDA (n: nat): x_length(I(n))), x_add((TRUE, R / 2 ^ n), L(X!1(n))))})") (("1" (case "forall n: subset?[real](X!1(n), IUnion[nat, real](I_n(n))) & x_lt(x_sum(LAMBDA (m: nat): x_length(I_n(n)(m))), x_add((TRUE, R / 2 ^ n), L(X!1(n))))") (("1" (lemma "double_index_n_ij") (("1" (lemma "double_index_ij_n") (("1" (name-replace "NN" "double_index_n") (("1" (name-replace "II" "double_index_i") (("1" (name-replace "JJ" "double_index_j") (("1" (inst + "lambda (n:nat): I_n(II(n))(JJ(n))") (("1" (case-replace "subset?[real] (IUnion(X!1), IUnion[nat, real](LAMBDA (n: nat): I_n(II(n))(JJ(n))))") (("1" (hide -5) (("1" (case "x_lt(x_sum(LAMBDA n:x_sum(LAMBDA (m: nat): x_length(I_n(n)(m)))),x_sum(LAMBDA n (("1" (case "x_le(L(IUnion(X!1)),x_sum(LAMBDA n: x_sum(LAMBDA (m: nat): x_length(I_n(n)(m))) (("1" (name-replace "LHS" "L(IUnion(X!1))") (("1" (name-replace "RHS" "x_sum(LAMBDA n: x_add((TRUE, R / 2 ^ n), L(X!1(n))))") (("1" (name-replace "MID" "x_sum(LAMBDA n: x_sum(LAMBDA (m: nat): x_length(I_n(n)(m))))") (("1" (hide-all-but (-1 -2 1)) (("1" (expand "x_le") (("1" (expand "x_lt") (("1" (flatten) (("1" (assert) (("1" (flatten) (("1" (assert) (("1" (flatten) (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide -1 2) (("2" (lemma "double_x_sum" ("U" "lambda (i,j:nat): x_length(I_n(i)(j))")) (("2" (case "x_le(L(IUnion(X!1)),x_sum(single_index(LAMBDA (i, j: nat): x_length(I_n(i)(j))) (("1" (assert) (("1" (name-replace "LHS" "L(IUnion(X!1))") (("1" (name-replace "RHS" "x_sum(LAMBDA n: x_sum(LAMBDA (m: nat): x_length(I_n(n)(m))))") (("1" (name-replace "MID" "x_sum(single_index(LAMBDA (i, j: nat): x_length(I_n(i)(j))))") (("1" (hide-all-but (-1 -2 1)) (("1" (expand "x_eq") (("1" (expand "x_le") (("1" (flatten) (("1" (assert) (("1" (flatten) (("1" (assert) nil nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide -1 2) (("2" (case "forall A,I: subset?(A,IUnion(I)) => x_le(L(A),x_sum(x_length o I))") (("1" (expand "single_index") (("1" (inst - "IUnion(X!1)" "lambda n: I_n(II(n))(JJ(n))") (("1" (replace -2) (("1" (expand "o" -1) (("1" (expand "JJ") (("1" (expand "II") (("1" (propax) nil nil)) nil)) nil)) nil)) nil)) nil)) nil) ("2" (hide-all-but 1) (("2" (skosimp) (("2" (expand "x_le") (("2" (expand "o ") (("2" (expand "L") (("2" (expand "o ") (("2" (case-replace "x_sum(LAMBDA (x: nat): x_length(I!1(x)))`1") (("1" (case-replace "x_inf({z | EXISTS I: x_le(x_sum(LAMBDA (x: nat): x_length(I(x))), z) AND subset?[real](A!1, IUnion[nat, real](I))})`1") (("1" (expand "x_inf") (("1" (case-replace "FORALL (x_1: ({z | EXISTS I: x_le(x_sum(LAMBDA (x: nat): x_length(I(x))), z) AND subset?[real](A!1, IUnion[nat, real](I))})): NOT x_1`1") (("1" (replace 1 2) (("1" (skosimp) (("1" (hide -2) (("1" (typepred "x!1") (("1" (skosimp) (("1" (typepred "glb({z_1: real | EXISTS (x_1: extended_nnreal): (EXISTS I: x_le(x_sum(LAMBDA (x: nat): x_length(I(x))), x_1) AND subset?[real](A!1, IUnion[nat, real](I))) AND x_1`1 AND x_1`2 = z_1})") (("1" (name-replace "GLB" "glb({z_1: real | EXISTS (x_1: extended_nnreal): (EXISTS I: x_le(x_sum(LAMBDA (x: nat): x_length(I(x))), x_1) AND subset?[real](A!1, IUnion[nat, real](I))) AND x_1`1 AND x_1`2 = z_1})") (("1" (expand "greatest_lower_bound?") (("1" (flatten) (("1" (expand "lower_bound?") (("1" (hide -2) (("1" (inst - "x_sum(LAMBDA (x: nat): x_length(I!1(x)))`2") (("1" (hide 2) (("1" (inst + "x_sum(LAMBDA (x: nat): x_length(I!1(x)))") (("1" --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.72 Sekunden (vorverarbeitet) ¤ |
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:Die farbliche Syntaxdarstellung ist noch experimentell.Bot Zugriff |
