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Datei: epsilon_lemmas.pvs   Sprache: Lisp

Original von: PVS©

(integral_cont
 (IMP_integral_cont_scaf_TCC1 0
  (IMP_integral_cont_scaf_TCC1-1 nil 3292324347
   ("" (skosimp*)
    (("" (lemma "connected_domain")
      (("" (inst?) (("" (inst?) (("" (assertnil nil)) nil)) nil))
      nil))
    nil)
   ((connected_domain formula-decl nil integral_cont nil)) shostak))
 (IMP_integral_cont_scaf_TCC2 0
  (IMP_integral_cont_scaf_TCC2-1 nil 3292324347
   ("" (skosimp*) (("" (rewrite "not_one_element"nil nil)) nil)
   ((not_one_element formula-decl nil integral_cont nil)) shostak))
 (in_interval_TCC1 0
  (in_interval_TCC1-1 nil 3262523543 ("" (subtype-tcc) nil nilnil
   nil))
 (in_interval_TCC2 0
  (in_interval_TCC2-1 nil 3262523543 ("" (subtype-tcc) nil nilnil
   nil))
 (in_interval 0
  (in_interval-1 nil 3262523543
   ("" (skosimp*)
    (("" (typepred "x!1")
      (("" (typepred "y!1")
        (("" (expand "abs")
          (("" (lift-if) (("" (ground) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((partition type-eq-decl nil integral_def nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (below type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil) (>= const-decl "bool" reals 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)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (> const-decl "bool" reals nil)
    (finite_sequence type-eq-decl nil finite_sequences nil)
    (closed_interval type-eq-decl nil intervals_real "reals/")
    (below type-eq-decl nil nat_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (T formal-subtype-decl nil integral_cont nil)
    (T_pred const-decl "[real -> boolean]" integral_cont 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)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
 (continuous_integrable 0
  (continuous_integrable-3 nil 3477657179
   ("" (skosimp*)
    (("" (lemma "unif_cont_interval[closed_interval(a!1,b!1)]")
      (("1" (inst?)
        (("1" (inst -1 "f!1")
          (("1" (assert)
            (("1" (expand "restrict")
              (("1"
                (case " continuous_on?(LAMBDA (s: closed_interval[T](a!1, b!1)): f!1(s),
                                                   {xx: closed_interval[T](a!1, b!1) | TRUE})")
                (("1" (assert)
                  (("1" (hide -1)
                    (("1" (rewrite "step_to_integrable")
                      (("1" (skosimp*)
                        (("1" (expand "uniformly_continuous?")
                          (("1" (inst -1 "(eps!1/2)/(b!1-a!1)")
                            (("1" (skosimp*)
                              (("1"
                                (name
                                 "PP"
                                 "eq_partition(a!1,b!1,floor((b!1-a!1)/delta!1)+2)")
                                (("1"
                                  (case
                                   "step_function?(a!1, b!1, fmin(a!1, b!1, PP, f!1))")
                                  (("1"
                                    (case
                                     "step_function?(a!1, b!1, fmax(a!1, b!1, PP, f!1))")
                                    (("1"
                                      (inst
                                       +
                                       "fmin(a!1,b!1,PP,f!1)"
                                       "fmax(a!1,b!1,PP,f!1)")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (case
                                           "integrable?(a!1, b!1, fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1))")
                                          (("1"
                                            (assert)
                                            (("1"
                                              (case
                                               " integral(a!1, b!1, fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1)) < eps!1")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (skosimp*)
                                                  (("1"
                                                    (hide-all-but
                                                     (-7 1))
                                                    (("1"
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                                                      (("1"
                                                        (typepred
                                                         "fmin(a!1, b!1, PP, f!1)")
                                                        (("1"
                                                          (lemma
                                                           "part_induction")
                                                          (("1"
                                                            (inst
                                                             -1
                                                             "(LAMBDA x: fmin(a!1, b!1, PP, f!1)(x) <= f!1(x))"
                                                             "a!1"
                                                             "b!1"
                                                             "PP"
                                                             "xx!1")
                                                            (("1"
                                                              (assert)
                                                              (("1"
                                                                (assert)
                                                                (("1"
                                                                  (skosimp*)
                                                                  (("1"
                                                                    (assert)
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "ii!1")
                                                                      (("1"
                                                                        (inst
                                                                         -
                                                                         "xx!1")
                                                                        (("1"
                                                                          (flatten)
                                                                          (("1"
                                                                            (split
                                                                             -4)
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil)
                                                                             ("2"
                                                                              (flatten)
                                                                              (("2"
                                                                                (assert)
                                                                                (("2"
                                                                                  (replace
                                                                                   -3)
                                                                                  (("2"
                                                                                    (hide
                                                                                     -3)
                                                                                    (("2"
                                                                                      (assert)
                                                                                      (("2"
                                                                                        (typepred
                                                                                         "min_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)")
                                                                                        (("2"
                                                                                          (hide
                                                                                           -1
                                                                                           -2
                                                                                           -3)
                                                                                          (("2"
                                                                                            (inst
                                                                                             -1
                                                                                             "xx!1")
                                                                                            (("2"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (typepred
                                                         "fmax(a!1, b!1, PP, f!1)")
                                                        (("2"
                                                          (lemma
                                                           "part_induction")
                                                          (("2"
                                                            (inst
                                                             -1
                                                             "(LAMBDA x: fmax(a!1, b!1, PP, f!1)(x) >= f!1(x))"
                                                             "a!1"
                                                             "b!1"
                                                             "PP"
                                                             "xx!1")
                                                            (("2"
                                                              (assert)
                                                              (("2"
                                                                (assert)
                                                                (("2"
                                                                  (skosimp*)
                                                                  (("2"
                                                                    (inst
                                                                     -
                                                                     "ii!1")
                                                                    (("2"
                                                                      (inst
                                                                       -
                                                                       "xx!1")
                                                                      (("2"
                                                                        (flatten)
                                                                        (("2"
                                                                          (split
                                                                           -4)
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil)
                                                                           ("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (flatten)
                                                                              (("2"
                                                                                (assert)
                                                                                (("2"
                                                                                  (replace
                                                                                   -3)
                                                                                  (("2"
                                                                                    (hide
                                                                                     -3)
                                                                                    (("2"
                                                                                      (typepred
                                                                                       "max_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)")
                                                                                      (("2"
                                                                                        (hide
                                                                                         -1
                                                                                         -2
                                                                                         -3)
                                                                                        (("2"
                                                                                          (inst
                                                                                           -
                                                                                           "xx!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)
                                               ("2"
                                                (hide 2)
                                                (("2"
                                                  (lemma
                                                   "width_eq_part")
                                                  (("2"
                                                    (inst?)
                                                    (("2"
                                                      (assert)
                                                      (("2"
                                                        (replace -5)
                                                        (("2"
                                                          (case
                                                           "width(a!1,b!1,PP) < delta!1")
                                                          (("1"
                                                            (hide
                                                             -2
                                                             -6)
                                                            (("1"
                                                              (lemma
                                                               "integral_bound_abs")
                                                              (("1"
                                                                (inst
                                                                 -1
                                                                 "eps!1/(2*(b!1-a!1))"
                                                                 "a!1"
                                                                 "b!1"
                                                                 "fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1)")
                                                                (("1"
                                                                  (name-replace
                                                                   "BMA"
                                                                   "b!1-a!1")
                                                                  (("1"
                                                                    (assert)
                                                                    (("1"
                                                                      (split
                                                                       -1)
                                                                      (("1"
                                                                        (name
                                                                         "III"
                                                                         "integral(a!1, b!1, fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1))")
                                                                        (("1"
                                                                          (case
                                                                           "III >= 0")
                                                                          (("1"
                                                                            (replace
                                                                             -2)
                                                                            (("1"
                                                                              (expand
                                                                               "abs"
                                                                               -3)
                                                                              (("1"
                                                                                (assert)
                                                                                nil
                                                                                nil))
                                                                              nil))
                                                                            nil)
                                                                           ("2"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil)
                                                                       ("2"
                                                                        (hide
                                                                         2)
                                                                        (("2"
                                                                          (skosimp*)
                                                                          (("2"
                                                                            (expand
                                                                             "-")
                                                                            (("2"
                                                                              (lemma
                                                                               "part_induction")
                                                                              (("2"
                                                                                (inst
                                                                                 -1
                                                                                 "(LAMBDA x: (fmax(a!1, b!1, PP, f!1) - fmin(a!1, b!1, PP, f!1))(x) <= eps!1 / (2 * BMA))"
                                                                                 "a!1"
                                                                                 "b!1"
                                                                                 "PP"
                                                                                 "x!1")
                                                                                (("2"
                                                                                  (assert)
                                                                                  (("2"
                                                                                    (expand
                                                                                     "-")
                                                                                    (("2"
                                                                                      (case
                                                                                       "fmax(a!1, b!1, PP, f!1)(x!1) - fmin(a!1, b!1, PP, f!1)(x!1) >= 0")
                                                                                      (("1"
                                                                                        (expand
                                                                                         "abs"
                                                                                         1)
                                                                                        (("1"
                                                                                          (assert)
                                                                                          (("1"
                                                                                            (assert)
                                                                                            (("1"
                                                                                              (skosimp*)
                                                                                              (("1"
                                                                                                (typepred
                                                                                                 "fmax(a!1, b!1, PP, f!1)")
                                                                                                (("1"
                                                                                                  (typepred
                                                                                                   "fmin(a!1, b!1, PP, f!1)")
                                                                                                  (("1"
                                                                                                    (inst
                                                                                                     -1
                                                                                                     "ii!1")
                                                                                                    (("1"
                                                                                                      (inst
                                                                                                       -2
                                                                                                       "ii!1")
                                                                                                      (("1"
                                                                                                        (inst
                                                                                                         -1
                                                                                                         "x!1")
                                                                                                        (("1"
                                                                                                          (inst
                                                                                                           -2
                                                                                                           "x!1")
                                                                                                          (("1"
                                                                                                            (flatten)
                                                                                                            (("1"
                                                                                                              (case
                                                                                                               "(seq(PP)(ii!1) = x!1 OR x!1 = seq(PP)(1 + ii!1))")
                                                                                                              (("1"
                                                                                                                (assert)
                                                                                                                (("1"
                                                                                                                  (split
                                                                                                                   -1)
                                                                                                                  (("1"
                                                                                                                    (assert)
                                                                                                                    (("1"
                                                                                                                      (hide
                                                                                                                       -2
                                                                                                                       -4)
                                                                                                                      (("1"
                                                                                                                        (replace
                                                                                                                         -2)
                                                                                                                        (("1"
                                                                                                                          (replace
                                                                                                                           -3)
                                                                                                                          (("1"
                                                                                                                            (assert)
                                                                                                                            (("1"
                                                                                                                              (hide-all-but
                                                                                                                               1)
                                                                                                                              (("1"
                                                                                                                                (reveal
                                                                                                                                 -21)
                                                                                                                                (("1"
                                                                                                                                  (mult-by
                                                                                                                                   1
                                                                                                                                   "(2*BMA)")
                                                                                                                                  (("1"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (assert)
                                                                                                                    (("2"
                                                                                                                      (hide
                                                                                                                       -2
                                                                                                                       -4)
                                                                                                                      (("2"
                                                                                                                        (replace
                                                                                                                         -2)
                                                                                                                        (("2"
                                                                                                                          (replace
                                                                                                                           -3)
                                                                                                                          (("2"
                                                                                                                            (assert)
                                                                                                                            (("2"
                                                                                                                              (hide-all-but
                                                                                                                               1)
                                                                                                                              (("2"
                                                                                                                                (reveal
                                                                                                                                 -21)
                                                                                                                                (("2"
                                                                                                                                  (mult-by
                                                                                                                                   1
                                                                                                                                   "(2*BMA)")
                                                                                                                                  (("2"
                                                                                                                                    (assert)
                                                                                                                                    nil
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
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                                                                                                                nil)
                                                                                                               ("2"
                                                                                                                (flatten)
                                                                                                                (("2"
                                                                                                                  (assert)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     -2
                                                                                                                     -4)
                                                                                                                    (("2"
                                                                                                                      (replace
                                                                                                                       -1)
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                                                                                                                        (hide
                                                                                                                         -1)
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                                                                                                                          (replace
                                                                                                                           -1)
                                                                                                                          (("2"
                                                                                                                            (hide
                                                                                                                             -1)
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                                                                                                                              (name
                                                                                                                               "MIN_x"
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                                                                                                                              (("2"
                                                                                                                                (replace
                                                                                                                                 -1)
                                                                                                                                (("2"
                                                                                                                                  (name
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                                                                                                                                  (("2"
                                                                                                                                    (replace
                                                                                                                                     -1)
                                                                                                                                    (("2"
                                                                                                                                      (case
                                                                                                                                       "(FORALL (x,y: closed_interval(seq(PP)(ii!1),seq(PP)(ii!1+1))): abs(x-y) <= seq(PP)(1 + ii!1) - seq(PP)(ii!1))")
                                                                                                                                      (("1"
                                                                                                                                        (case
                                                                                                                                         "abs(MAX_x - MIN_x) <= seq(PP)(1+ii!1) - seq(PP)(ii!1)")
                                                                                                                                        (("1"
                                                                                                                                          (inst
                                                                                                                                           -12
                                                                                                                                           "MAX_x"
                                                                                                                                           "MIN_x")
                                                                                                                                          (("1"
                                                                                                                                            (assert)
                                                                                                                                            (("1"
                                                                                                                                              (lemma
                                                                                                                                               "width_lem")
                                                                                                                                              (("1"
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                                                                                                                                                 "finseq_appl")
                                                                                                                                                (("1"
                                                                                                                                                  (inst?)
                                                                                                                                                  (("1"
                                                                                                                                                    (assert)
                                                                                                                                                    nil
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil)
                                                                                                                                           ("2"
                                                                                                                                            (ground)
                                                                                                                                            nil
                                                                                                                                            nil)
                                                                                                                                           ("3"
                                                                                                                                            (ground)
                                                                                                                                            nil
                                                                                                                                            nil))
                                                                                                                                          nil)
                                                                                                                                         ("2"
                                                                                                                                          (assert)
                                                                                                                                          (("2"
                                                                                                                                            (hide-all-but
                                                                                                                                             (-2
                                                                                                                                              -3
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                                                                                                                                              1))
                                                                                                                                            (("2"
                                                                                                                                              (typepred
                                                                                                                                               "MAX_x")
                                                                                                                                              (("2"
                                                                                                                                                (typepred
                                                                                                                                                 "MIN_x")
                                                                                                                                                (("2"
                                                                                                                                                  (hide
                                                                                                                                                   -1
                                                                                                                                                   -4
                                                                                                                                                   -5
                                                                                                                                                   -8
                                                                                                                                                   -9
                                                                                                                                                   -10)
                                                                                                                                                  (("2"
                                                                                                                                                    (lemma
                                                                                                                                                     "in_interval")
                                                                                                                                                    (("2"
                                                                                                                                                      (inst?)
                                                                                                                                                      nil
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--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.30 Sekunden  (vorverarbeitet)  ¤





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