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Datei: integral_bounded.prf   Sprache: Lisp

Original von: PVS©

(rs_integral_cont
 (IMP_rs_integral_prep_TCC1 0
  (IMP_rs_integral_prep_TCC1-1 nil 3494863474
   ("" (lemma "connected_domain") (("" (propax) nil nil)) nil)
   ((connected_domain formula-decl nil rs_integral_cont nil)) nil))
 (IMP_rs_integral_prep_TCC2 0
  (IMP_rs_integral_prep_TCC2-1 nil 3494863474
   ("" (lemma "not_one_element") (("" (propax) nil nil)) nil)
   ((not_one_element formula-decl nil rs_integral_cont nil)) nil))
 (RS_integrable_cont_inc_TCC1 0
  (RS_integrable_cont_inc_TCC1-1 nil 3494864317
   ("" (skeep)
    (("" (skeep) (("" (hide -) (("" (grind) nil nil)) nil)) nil)) nil)
   ((real_minus_real_is_real application-judgement "real" reals nil)
    (metric_space? const-decl "bool" metric_spaces_def nil)
    (space_triangle? const-decl "bool" metric_spaces_def nil)
    (space_symmetric? const-decl "bool" metric_spaces_def nil)
    (space_zero? const-decl "bool" metric_spaces_def nil)
    (real_dist const-decl "nnreal" real_metric_space nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (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)
    (T_pred const-decl "[real -> boolean]" rs_integral_cont nil)
    (T formal-nonempty-subtype-decl nil rs_integral_cont nil)
    (set type-eq-decl nil sets nil) (fullset const-decl "set" sets nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_real_is_real application-judgement "real" reals nil))
   nil))
 (RS_integrable_cont_inc_TCC2 0
  (RS_integrable_cont_inc_TCC2-1 nil 3494864317
   ("" (skeep)
    (("" (skeep)
      (("" (hide -)
        (("" (lemma "real_metric_space") (("" (inst?) nil nil)) nil))
        nil))
      nil))
    nil)
   ((real_metric_space formula-decl nil real_metric_space nil)
    (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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (fullset const-decl "set" sets nil))
   nil))
 (RS_integrable_cont_inc 0
  (RS_integrable_cont_inc-1 nil 3492526014
   ("" (skolem 1 ("a" "b" "f" "gg"))
    (("" (flatten)
      (("" (assert)
        (("" (label "altb" -1)
          (("" (flatten)
            (("" (label "fcont" -2)
              (("" (label "ginc" -3)
                (("" (name "CI" "closed_intv[T](a,b)")
                  (("" (replace -1)
                    (("" (label "CIname" -1)
                      ((""
                        (case "NOT uniformly_continuous?[T,real_dist,real,real_dist](f,CI)")
                        (("1"
                          (name "ff"
                                "(LAMBDA (x:real): IF T_pred(x) THEN f(x) ELSE 0 ENDIF)")
                          (("1"
                            (lemma
                             "Heine[real,real_dist,real,real_dist]")
                            (("1"
                              (inst - "LAMBDA (x:real): a<=x AND x<=b"
                               "ff")
                              (("1"
                                (assert)
                                (("1"
                                  (split -)
                                  (("1"
                                    (hide-all-but (-1 1))
                                    (("1"
                                      (expand "uniformly_continuous?")
                                      (("1"
                                        (skosimp*)
                                        (("1"
                                          (inst - "epsilon!1")
                                          (("1"
                                            (skosimp*)
                                            (("1"
                                              (inst + "delta!1")
                                              (("1"
                                                (skosimp*)
                                                (("1"
                                                  (inst - "x!1" "p!1")
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (expand "ff")
                                                      (("1"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (assert)
                                                    (("2"
                                                      (typepred "p!1")
                                                      (("2"
                                                        (expand "CI")
                                                        (("2"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("3"
                                                    (typepred "x!1")
                                                    (("3"
                                                      (expand "CI")
                                                      (("3"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but ("fcont" 1))
                                    (("2"
                                      (expand "continuous?")
                                      (("2"
                                        (skosimp*)
                                        (("2"
                                          (inst - "x!1")
                                          (("1"
                                            (expand "continuous_at?")
                                            (("1"
                                              (skosimp*)
                                              (("1"
                                                (inst - "epsilon!1")
                                                (("1"
                                                  (skosimp*)
                                                  (("1"
                                                    (inst + "delta!1")
                                                    (("1"
                                                      (skosimp*)
                                                      (("1"
                                                        (inst - "y!1")
                                                        (("1"
                                                          (assert)
                                                          (("1"
                                                            (expand
                                                             "ball")
                                                            (("1"
                                                              (case
                                                               "T_pred(x!1) AND T_pred(y!1)")
                                                              (("1"
                                                                (flatten)
                                                                (("1"
                                                                  (expand
                                                                   "ff")
                                                                  (("1"
                                                                    (assert)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (hide-all-but
                                                                 1)
                                                                (("2"
                                                                  (lemma
                                                                   "connected_domain")
                                                                  (("2"
                                                                    (expand
                                                                     "connected?")
                                                                    (("2"
                                                                      (inst-cp
                                                                       -
                                                                       "a"
                                                                       "b"
                                                                       "x!1")
                                                                      (("2"
                                                                        (inst
                                                                         -
                                                                         "a"
                                                                         "b"
                                                                         "y!1")
                                                                        (("2"
                                                                          (assert)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (hide-all-but
                                                           1)
                                                          (("2"
                                                            (expand
                                                             "CI")
                                                            (("2"
                                                              (lemma
                                                               "connected_domain")
                                                              (("2"
                                                                (expand
                                                                 "connected?")
                                                                (("2"
                                                                  (inst
                                                                   -
                                                                   "a"
                                                                   "b"
                                                                   "y!1")
                                                                  (("2"
                                                                    (assert)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide-all-but 1)
                                            (("2"
                                              (expand "CI")
                                              (("2"
                                                (lemma
                                                 "connected_domain")
                                                (("2"
                                                  (expand "connected?")
                                                  (("2"
                                                    (inst
                                                     -
                                                     "a"
                                                     "b"
                                                     "x!1")
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("3"
                                    (lemma "closed_intervals_compact")
                                    (("3" (inst - "a" "b"nil nil))
                                    nil)
                                   ("4"
                                    (assert)
                                    (("4"
                                      (hide-all-but (-1))
                                      (("4"
                                        (expand "empty?")
                                        (("4"
                                          (inst - "a")
                                          (("4"
                                            (expand "member")
                                            (("4" (propax) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (label "funif" -1)
                          (("2" (lemma "integrable_lem2")
                            (("2" (inst - "a" "b" "f" "gg")
                              (("2"
                                (assert)
                                (("2"
                                  (split -)
                                  (("1"
                                    (hide 2)
                                    (("1"
                                      (skolem 1 "eps")
                                      (("1"
                                        (expand
                                         "uniformly_continuous?")
                                        (("1"
                                          (name
                                           "newep"
                                           "eps/(2*(abs(gg(b)-gg(a)) + 1))")
                                          (("1"
                                            (label "newepname" -1)
                                            (("1"
                                              (inst "funif" "newep")
                                              (("1"
                                                (skolem
                                                 "funif"
                                                 "deltaf")
                                                (("1"
                                                  (inst + "deltaf/2")
                                                  (("1"
                                                    (skeep)
                                                    (("1"
                                                      (label
                                                       "Pwidth"
                                                       -2)
                                                      (("1"
                                                        (skolem
                                                         1
                                                         ("RSS1"
                                                          "RSS2"))
                                                        (("1"
                                                          (typepred
                                                           "RSS1")
                                                          (("1"
                                                            (label
                                                             "RSS1name"
                                                             -1)
                                                            (("1"
                                                              (typepred
                                                               "RSS2")
                                                              (("1"
                                                                (label
                                                                 "RSS2name"
                                                                 -1)
                                                                (("1"
                                                                  (expand
                                                                   "Riemann_sum?")
                                                                  (("1"
                                                                    (skolem
                                                                     "RSS2name"
                                                                     "xis2")
                                                                    (("1"
                                                                      (skolem
                                                                       "RSS1name"
                                                                       "xis1")
                                                                      (("1"
                                                                        (replace
                                                                         "RSS1name"
                                                                         +)
                                                                        (("1"
                                                                          (replace
                                                                           "RSS2name"
                                                                           +)
                                                                          (("1"
                                                                            (expand
                                                                             "Rie_sum"
                                                                             +)
                                                                            (("1"
                                                                              (rewrite
                                                                               "sigma_minus")
                                                                              (("1"
                                                                                (lemma
                                                                                 "sigma_abs[below(P`length-1)]")
                                                                                (("1"
                                                                                  (inst?)
                                                                                  (("1"
                                                                                    (case
                                                                                     "sigma(0, P`length - 2,
                                                                                            LAMBDA (n: below(P`length - 1)):
                                                                                              abs(f(xis1`seq(n)) * gg(P`seq(1 + n)) +
                                                                                                   f(xis2`seq(n)) * gg(P`seq(n))
                                                                                                   - f(xis1`seq(n)) * gg(P`seq(n))
                                                                                                   - f(xis2`seq(n)) * gg(P`seq(1 + n)))) < eps")
                                                                                    (("1"
                                                                                      (assert)
                                                                                      nil
                                                                                      nil)
                                                                                     ("2"
                                                                                      (hide
                                                                                       -1)
                                                                                      (("2"
                                                                                        (hide
                                                                                         2)
                                                                                        (("2"
                                                                                          (lemma
                                                                                           "sigma_le[below(P`length-1)]")
                                                                                          (("2"
                                                                                            (inst
                                                                                             -
                                                                                             "LAMBDA (n: below(P`length - 1)):
                                                                              abs(f(xis1`seq(n)) * gg(P`seq(1 + n)) +
                                                                                   f(xis2`seq(n)) * gg(P`seq(n))
                                                                                   - f(xis1`seq(n)) * gg(P`seq(n))
                                                                                   - f(xis2`seq(n)) * gg(P`seq(1 + n)))"
                                                                                             "LAMBDA (n: below(P`length - 1)):
                                                                              newep * (gg(P`seq(1 + n))- gg(P`seq(n)))"
                                                                                             "P`length-2"
                                                                                             "0")
                                                                                            (("2"
                                                                                              (assert)
                                                                                              (("2"
                                                                                                (case
                                                                                                 "sigma(0, P`length - 2,
                                                                                                       LAMBDA (n: below(P`length - 1)):
                                                                                                         gg(P`seq(1 + n)) * newep - gg(P`seq(n)) * newep) < eps")
                                                                                                (("1"
                                                                                                  (assert)
                                                                                                  (("1"
                                                                                                    (hide
                                                                                                     2)
                                                                                                    (("1"
                                                                                                      (skolem
                                                                                                       1
                                                                                                       "nn")
                                                                                                      (("1"
                                                                                                        (case
                                                                                                         "abs(f(xis1`seq(nn))-f(xis2`seq(nn)))< newep")
                                                                                                        (("1"
                                                                                                          (mult-by
                                                                                                           -1
                                                                                                           "abs(gg(P`seq(1+nn))-gg(P`seq(nn)))")
                                                                                                          (("1"
                                                                                                            (case
                                                                                                             "abs(gg(P`seq(1 + nn)) - gg(P`seq(nn))) = gg(P`seq(1 + nn)) - gg(P`seq(nn))")
                                                                                                            (("1"
                                                                                                              (rewrite
                                                                                                               "abs_mult"
                                                                                                               -2
                                                                                                               :dir
                                                                                                               rl)
                                                                                                              (("1"
                                                                                                                (assert)
                                                                                                                nil
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (hide
                                                                                                               2)
                                                                                                              (("2"
                                                                                                                (copy
                                                                                                                 "ginc")
                                                                                                                (("2"
                                                                                                                  (expand
                                                                                                                   "increasing?"
                                                                                                                   -1)
                                                                                                                  (("2"
                                                                                                                    (expand
                                                                                                                     "restrict")
                                                                                                                    (("2"
                                                                                                                      (inst
                                                                                                                       -
                                                                                                                       "P`seq(nn)"
                                                                                                                       "P`seq(1+nn)")
                                                                                                                      (("1"
                                                                                                                        (assert)
                                                                                                                        (("1"
                                                                                                                          (expand
                                                                                                                           "abs"
                                                                                                                           +)
                                                                                                                          (("1"
                                                                                                                            (lift-if)
                                                                                                                            (("1"
                                                                                                                              (ground)
                                                                                                                              (("1"
                                                                                                                                (typepred
                                                                                                                                 "P")
                                                                                                                                (("1"
                                                                                                                                  (expand
                                                                                                                                   "increasing?")
                                                                                                                                  (("1"
                                                                                                                                    (inst
                                                                                                                                     -
                                                                                                                                     "nn"
                                                                                                                                     "1+nn")
                                                                                                                                    (("1"
                                                                                                                                      (assert)
                                                                                                                                      nil
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("2"
                                                                                                                        (hide
                                                                                                                         2)
                                                                                                                        (("2"
                                                                                                                          (typepred
                                                                                                                           "P")
                                                                                                                          (("2"
                                                                                                                            (inst
                                                                                                                             -
                                                                                                                             "1+nn")
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil)
                                                                                                                       ("3"
                                                                                                                        (typepred
                                                                                                                         "P")
                                                                                                                        (("3"
                                                                                                                          (inst
                                                                                                                           -
                                                                                                                           "nn")
                                                                                                                          nil
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil)
                                                                                                         ("2"
                                                                                                          (hide
                                                                                                           2)
                                                                                                          (("2"
                                                                                                            (inst
                                                                                                             "funif"
                                                                                                             "xis1`seq(nn)"
                                                                                                             "xis2`seq(nn)")
                                                                                                            (("1"
                                                                                                              (assert)
                                                                                                              (("1"
                                                                                                                (expand
                                                                                                                 "real_dist")
                                                                                                                (("1"
                                                                                                                  (hide-all-but
                                                                                                                   ("funif"
                                                                                                                    "Pwidth"))
                                                                                                                  (("1"
                                                                                                                    (typepred
                                                                                                                     "xis1")
                                                                                                                    (("1"
                                                                                                                      (inst
                                                                                                                       -
                                                                                                                       "nn")
                                                                                                                      (("1"
                                                                                                                        (typepred
                                                                                                                         "xis2")
                                                                                                                        (("1"
                                                                                                                          (inst
                                                                                                                           -
                                                                                                                           "nn")
                                                                                                                          (("1"
                                                                                                                            (assert)
                                                                                                                            (("1"
                                                                                                                              (flatten)
                                                                                                                              (("1"
                                                                                                                                (lemma
                                                                                                                                 "width_lem")
                                                                                                                                (("1"
                                                                                                                                  (inst
                                                                                                                                   -
                                                                                                                                   "a"
                                                                                                                                   "b"
                                                                                                                                   "P"
                                                                                                                                   "nn")
                                                                                                                                  (("1"
                                                                                                                                    (grind
                                                                                                                                     :exclude
                                                                                                                                     "width")
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("2"
                                                                                                              (typepred
                                                                                                               "xis2")
                                                                                                              (("2"
                                                                                                                (inst
                                                                                                                 -
                                                                                                                 "nn")
                                                                                                                (("2"
                                                                                                                  (flatten)
                                                                                                                  (("2"
                                                                                                                    (typepred
                                                                                                                     "P")
                                                                                                                    (("2"
                                                                                                                      (inst-cp
                                                                                                                       -
                                                                                                                       "nn")
                                                                                                                      (("2"
                                                                                                                        (inst-cp
                                                                                                                         -
                                                                                                                         "nn+1")
                                                                                                                        (("2"
                                                                                                                          (ground)
                                                                                                                          (("2"
                                                                                                                            (expand
                                                                                                                             "CI")
                                                                                                                            (("2"
                                                                                                                              (propax)
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil)
                                                                                                             ("3"
                                                                                                              (typepred
                                                                                                               "xis1")
                                                                                                              (("3"
                                                                                                                (inst
                                                                                                                 -
                                                                                                                 "nn")
                                                                                                                (("3"
                                                                                                                  (flatten)
                                                                                                                  (("3"
                                                                                                                    (typepred
                                                                                                                     "P")
                                                                                                                    (("3"
                                                                                                                      (inst-cp
                                                                                                                       -
                                                                                                                       "nn")
                                                                                                                      (("3"
                                                                                                                        (inst-cp
                                                                                                                         -
                                                                                                                         "nn+1")
                                                                                                                        (("3"
                                                                                                                          (ground)
                                                                                                                          (("3"
                                                                                                                            (expand
                                                                                                                             "CI")
                                                                                                                            (("3"
                                                                                                                              (propax)
                                                                                                                              nil
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil)
                                                                                                 ("2"
                                                                                                  (hide
                                                                                                   2)
                                                                                                  (("2"
                                                                                                    (case
                                                                                                     "FORALL (nn:below(P`length-1)): sigma(0, nn,
                                                                         LAMBDA (n: below(P`length - 1)):
                                                                           gg(P`seq(1 + n)) * newep - gg(P`seq(n)) * newep) = newep*(gg(P`seq(1+nn))-gg(P`seq(0)))")
                                                                                                    (("1"
                                                                                                      (inst
                                                                                                       -
                                                                                                       "P`length-2")
                                                                                                      (("1"
                                                                                                        (assert)
                                                                                                        (("1"
                                                                                                          (replace
                                                                                                           -1)
                                                                                                          (("1"
                                                                                                            (typepred
                                                                                                             "P")
                                                                                                            (("1"
                                                                                                              (replace
                                                                                                               -2)
                                                                                                              (("1"
                                                                                                                (replace
                                                                                                                 -3)
                                                                                                                (("1"
                                                                                                                  (case
                                                                                                                   "(gg(b)-gg(a))*newep < eps")
                                                                                                                  (("1"
                                                                                                                    (assert)
                                                                                                                    nil
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (name
                                                                                                                     "CC1"
                                                                                                                     "gg(b)-gg(a)")
                                                                                                                    (("2"
                                                                                                                      (replace
                                                                                                                       -1)
                                                                                                                      (("2"
                                                                                                                        (expand
                                                                                                                         "newep"
                                                                                                                         1)
                                                                                                                        (("2"
                                                                                                                          (cross-mult
                                                                                                                           1)
                                                                                                                          (("2"
                                                                                                                            (replace
                                                                                                                             -1
                                                                                                                             :dir
                                                                                                                             rl)
                                                                                                                            (("2"
                                                                                                                              (assert)
                                                                                                                              (("2"
--> --------------------

--> maximum size reached

--> --------------------

¤ Dauer der Verarbeitung: 0.65 Sekunden  (vorverarbeitet)  ¤





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