Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/analysis_ax/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 2 kB image not shown  

Quellcode-Bibliothek integral_cont.prf   Sprache: unbekannt

 
(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"
                                                      (split 1)
                                                      (("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))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil)
                                                                                                               ("2"
                                                                                                                (flatten)
                                                                                                                (("2"
                                                                                                                  (assert)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     -2
                                                                                                                     -4)
                                                                                                                    (("2"
                                                                                                                      (replace
                                                                                                                       -1)
                                                                                                                      (("2"
                                                                                                                        (hide
                                                                                                                         -1)
                                                                                                                        (("2"
                                                                                                                          (replace
                                                                                                                           -1)
                                                                                                                          (("2"
                                                                                                                            (hide
                                                                                                                             -1)
                                                                                                                            (("2"
                                                                                                                              (name
                                                                                                                               "MIN_x"
                                                                                                                               "min_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)")
                                                                                                                              (("2"
                                                                                                                                (replace
                                                                                                                                 -1)
                                                                                                                                (("2"
                                                                                                                                  (name
                                                                                                                                   "MAX_x"
                                                                                                                                   "max_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)")
                                                                                                                                  (("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"
                                                                                                                                                (expand
                                                                                                                                                 "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
                                                                                                                                              -16
                                                                                                                                              1))
                                                                                                                                            (("2"
                                                                                                                                              (typepred
                                                                                                                                               "MAX_x")
                                                                                                                                              (("2"
                                                                                                                                                (typepred
                                                                                                                                                 "MIN_x")
                                                                                                                                                (("2"
                                                                                                                                                  (hide
                                                                                                                                                   -1
                                                                                                                                                   -4
                                                                                                                                                   -5
                                                                                                                                                   -8
                                                                                                                                                   -9
                                                                                                                                                   -10)
                                                                                                                                                  (("2"
                                                                                                                                                    (lemma
                                                                                                                                                     "in_interval")
                                                                                                                                                    (("2"
                                                                                                                                                      (inst?)
                                                                                                                                                      nil
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil)
                                                                                                                                       ("2"
                                                                                                                                        (assert)
                                                                                                                                        (("2"
                                                                                                                                          (hide-all-but
                                                                                                                                           1)
                                                                                                                                          (("2"
                                                                                                                                            (lemma
                                                                                                                                             "in_interval")
                                                                                                                                            (("2"
                                                                                                                                              (skosimp*)
                                                                                                                                              (("2"
                                                                                                                                                (inst?)
                                                                                                                                                nil
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide
                                                                                         2)
                                                                                        (("2"
                                                                                          (rewrite
                                                                                           "fmax_ge")
                                                                                          nil
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (hide
                                                             -3
                                                             -4
                                                             -6
                                                             -8
                                                             2)
                                                            (("2"
                                                              (replace
                                                               -1)
                                                              (("2"
                                                                (hide
                                                                 -1
                                                                 -2
                                                                 -3)
                                                                (("2"
                                                                  (name-replace
                                                                   "BMA"
                                                                   "b!1-a!1")
                                                                  (("2"
                                                                    (mult-by
                                                                     1
                                                                     "(1 + floor(BMA / delta!1))")
                                                                    (("1"
                                                                      (div-by
                                                                       1
                                                                       "delta!1")
                                                                      (("1"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (hide
                                                                       2)
                                                                      (("2"
                                                                        (reveal
                                                                         -3)
                                                                        (("2"
                                                                          (assert)
                                                                          (("2"
                                                                            (case-replace
                                                                             "BMA / delta!1 >= 0")
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil)
                                                                             ("2"
                                                                              (hide
                                                                               2)
                                                                              (("2"
                                                                                (mult-by
                                                                                 1
                                                                                 "delta!1")
                                                                                (("2"
                                                                                  (assert)
                                                                                  (("2"
                                                                                    (reveal
                                                                                     -4)
                                                                                    (("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)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (lemma "integral_diff")
                                              (("2"
                                                (inst
                                                 -1
                                                 "a!1"
                                                 "b!1"
                                                 "fmax(a!1, b!1, PP, f!1)"
                                                 "fmin(a!1, b!1, PP, f!1)")
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (split -1)
                                                    (("1"
                                                      (flatten)
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (expand "-")
                                                          (("1"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide 2)
                                                      (("2"
                                                        (rewrite
                                                         "step_function_integrable?")
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide 2)
                                                      (("3"
                                                        (rewrite
                                                         "step_function_integrable?")
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide 2)
                                      (("2"
                                        (rewrite "fmax_step")
                                        nil
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (rewrite "fmin_step")
                                      nil
                                      nil))
                                    nil)
                                   ("3"
                                    (hide 2)
                                    (("3"
                                      (skosimp*)
                                      (("3"
                                        (hide -1 -2 -3)
                                        (("3"
                                          (inst?)
                                          (("3"
                                            (lemma "cont_restrict")
                                            (("3"
                                              (inst?)
                                              (("3" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (case
                                       "(b!1 - a!1) / delta!1 > 0")
                                      (("1" (assertnil nil)
                                       ("2"
                                        (mult-by 1 "delta!1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide 2)
                              (("2"
                                (split +)
                                (("1"
                                  (mult-by 1 "b!1-a!1")
                                  (("1" (assertnil nil))
                                  nil)
                                 ("2"
                                  (mult-by 1 "b!1-a!1")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide -1 2)
                  (("2" (rewrite "continuous_on_def")
                    (("2" (skosimp*)
                      (("2" (inst -2 "y!1")
                        (("2" (rewrite "continuity_def")
                          (("2" (expand "convergence")
                            (("2" (expand "convergence")
                              (("2"
                                (flatten)
                                (("2"
                                  (expand "extend")
                                  (("2"
                                    (expand "adh")
                                    (("2"
                                      (split +)
                                      (("1"
                                        (skosimp*)
                                        (("1"
                                          (inst + "y!1")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (skosimp*)
                                        (("2"
                                          (inst -3 "epsilon!1")
                                          (("2"
                                            (skosimp*)
                                            (("2"
                                              (inst + "delta!1")
                                              (("2"
                                                (skosimp*)
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (inst -4 "x!1")
                                                    (("2"
                                                      (expand
                                                       "fullset")
                                                      (("2"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assertnil nil) ("3" (assertnil nil))
        nil)
       ("2" (expand "not_one_element?")
        (("2" (skosimp*)
          (("2" (typepred "x!1")
            (("2" (inst-cp 1 "a!1")
              (("2" (inst-cp 1 "b!1") (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("3" (lemma "connected_domain")
        (("3" (expand "connected?")
          (("3" (assert)
            (("3" (skosimp*)
              (("3" (lemma "connected_domain")
                (("3" (expand "connected?")
                  (("3" (inst -1 "a!1" "b!1" "z!1")
                    (("3" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((closed_interval type-eq-decl nil intervals_real "reals/")
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil 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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (unif_cont_interval formula-decl nil unif_cont_fun nil)
    (connected? const-decl "bool" deriv_domain_def nil)
    (not_one_element? const-decl "bool" deriv_domain_def nil)
    (restrict const-decl "R" restrict nil)
    (continuity_def formula-decl nil continuous_functions nil)
    (extend const-decl "R" extend nil)
    (abs_0 formula-decl nil abs_lems "reals/")
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions nil)
    (convergence const-decl "bool" convergence_functions nil)
    (convergence const-decl "bool" lim_of_functions nil)
    (continuous_on_def formula-decl nil continuous_functions nil)
    (step_to_integrable formula-decl nil integral_step nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (uniformly_continuous? const-decl "bool" unif_cont_fun nil)
    (both_sides_times_pos_le1_imp formula-decl nil extra_real_props
     nil)
    (step_function? const-decl "bool" step_fun_def nil)
    (continuous? const-decl "bool" continuous_functions nil)
    (fun_cont_on type-eq-decl nil integral_cont_scaf nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (fun_cont_on type-eq-decl nil interval_minmax nil)
    (min_x const-decl "{mx: T |
         a <= mx AND
          mx <= b AND
           (FORALL (x: T): a <= x AND x <= b IMPLIES f(mx) <= f(x))}"
     interval_minmax nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (fmin const-decl "{ff: [T -> real] |
         LET xx = seq(P) IN
           FORALL (ii: below(length(P) - 1)):
             FORALL (x: T):
               (xx(ii) < x AND x < xx(ii + 1) IMPLIES
                 ff(x) = f(min_x(xx(ii), xx(ii + 1), f)))
                AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
     integral_cont_scaf nil)
    (fmax_step formula-decl nil integral_cont_scaf nil)
    (integrable? const-decl "bool" integral_def nil)
    (- const-decl "[T -> real]" real_fun_ops "reals/")
    (integral? const-decl "bool" integral_def nil)
    (integral const-decl "{S: real | integral?(a, b, ff, S)}"
     integral_def nil)
    (part_induction formula-decl nil integral_def nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (width_eq_part formula-decl nil integral_def nil)
    (width const-decl "posreal" integral_def nil)
    (integral_bound_abs formula-decl nil integral_prep nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (BMA skolem-const-decl "real" integral_cont nil)
    (div_cancel2 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (finite_sequence type-eq-decl nil finite_sequences nil)
    (partition type-eq-decl nil integral_def nil)
    (in_interval formula-decl nil integral_cont nil)
    (PP skolem-const-decl
     "{fs: finite_sequence[{x | a!1 <= x AND x <= b!1}] |
         length(fs) > 1 AND
          seq(fs)(0) = a!1 AND
           seq(fs)(length(fs) - 1) = b!1 AND
            (FORALL (ii: below(length(fs) - 1)):
               seq(fs)(ii) < seq(fs)(1 + ii))}" integral_cont nil)
    (ii!1 skolem-const-decl "below(length(PP) - 1)" integral_cont nil)
    (f!1 skolem-const-decl "[T -> real]" integral_cont nil)
    (MAX_x skolem-const-decl "{mx: T |
         seq(PP)(ii!1) <= mx AND
          mx <= seq(PP)(1 + ii!1) AND
           (FORALL (x: T):
              seq(PP)(ii!1) <= x AND x <= seq(PP)(1 + ii!1) IMPLIES
               f!1(mx) >= f!1(x))}" integral_cont nil)
    (MIN_x skolem-const-decl "{mx: T |
         seq(PP)(ii!1) <= mx AND
          mx <= seq(PP)(1 + ii!1) AND
           (FORALL (x: T):
              seq(PP)(ii!1) <= x AND x <= seq(PP)(1 + ii!1) IMPLIES
               f!1(mx) <= f!1(x))}" integral_cont nil)
    (width_lem formula-decl nil integral_def nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (fmax_ge formula-decl nil integral_cont_scaf nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (both_sides_div_pos_lt1 formula-decl nil real_props nil)
    (times_div_cancel1 formula-decl nil extra_real_props nil)
    (BMA skolem-const-decl "real" integral_cont nil)
    (delta!1 skolem-const-decl "posreal" integral_cont nil)
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (integral_diff formula-decl nil integral_prep nil)
    (step_function_integrable? formula-decl nil integral_step nil)
    (fmax const-decl "{ff: [T -> real] |
         LET xx = seq(P) IN
           FORALL (ii: below(length(P) - 1)):
             FORALL (x: T):
               (xx(ii) < x AND x < xx(ii + 1) IMPLIES
                 ff(x) = f(max_x(xx(ii), xx(ii + 1), f)))
                AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
     integral_cont_scaf nil)
    (max_x const-decl "{mx: T |
         a <= mx AND
          mx <= b AND
           (FORALL (x: T): a <= x AND x <= b IMPLIES f(mx) >= f(x))}"
     interval_minmax nil)
    (fmin_step formula-decl nil integral_cont_scaf nil)
    (cont_restrict formula-decl nil integral_cont_scaf nil)
    (floor const-decl "{i | i <= x & x < i + 1}" floor_ceil nil)
    (integer nonempty-type-from-decl nil integers nil)
    (eq_partition const-decl "partition(a, b)" integral_def nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (below type-eq-decl nil naturalnumbers nil)
    (above nonempty-type-eq-decl nil integers 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 "bool" reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (below type-eq-decl nil nat_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (eps!1 skolem-const-decl "posreal" integral_cont nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (>= const-decl "bool" reals nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (TRUE const-decl "bool" booleans nil)
    (continuous_on? const-decl "bool" continuous_functions nil)
    (setof type-eq-decl nil defined_types nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (b!1 skolem-const-decl "T" integral_cont nil)
    (a!1 skolem-const-decl "T" integral_cont nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (connected_domain formula-decl nil integral_cont nil))
   nil)
  (continuous_integrable-2 nil 3292320666
   ("" (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"
                                                      (split 1)
                                                      (("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))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil)
                                                                                                               ("2"
                                                                                                                (flatten)
                                                                                                                (("2"
                                                                                                                  (assert)
                                                                                                                  (("2"
                                                                                                                    (hide
                                                                                                                     -2
                                                                                                                     -4)
                                                                                                                    (("2"
                                                                                                                      (replace
                                                                                                                       -1)
                                                                                                                      (("2"
                                                                                                                        (hide
                                                                                                                         -1)
                                                                                                                        (("2"
                                                                                                                          (replace
                                                                                                                           -1)
                                                                                                                          (("2"
                                                                                                                            (hide
                                                                                                                             -1)
                                                                                                                            (("2"
                                                                                                                              (name
                                                                                                                               "MIN_x"
                                                                                                                               "min_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)")
                                                                                                                              (("2"
                                                                                                                                (replace
                                                                                                                                 -1)
                                                                                                                                (("2"
                                                                                                                                  (name
                                                                                                                                   "MAX_x"
                                                                                                                                   "max_x(seq(PP)(ii!1), seq(PP)(1 + ii!1), f!1)")
                                                                                                                                  (("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"
                                                                                                                                                (expand
                                                                                                                                                 "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
                                                                                                                                              -16
                                                                                                                                              1))
                                                                                                                                            (("2"
                                                                                                                                              (typepred
                                                                                                                                               "MAX_x")
                                                                                                                                              (("2"
                                                                                                                                                (typepred
                                                                                                                                                 "MIN_x")
                                                                                                                                                (("2"
                                                                                                                                                  (hide
                                                                                                                                                   -1
                                                                                                                                                   -4
                                                                                                                                                   -5
                                                                                                                                                   -8
                                                                                                                                                   -9
                                                                                                                                                   -10)
                                                                                                                                                  (("2"
                                                                                                                                                    (lemma
                                                                                                                                                     "in_interval")
                                                                                                                                                    (("2"
                                                                                                                                                      (inst?)
                                                                                                                                                      nil
                                                                                                                                                      nil))
                                                                                                                                                    nil))
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil)
                                                                                                                                       ("2"
                                                                                                                                        (assert)
                                                                                                                                        (("2"
                                                                                                                                          (hide-all-but
                                                                                                                                           1)
                                                                                                                                          (("2"
                                                                                                                                            (lemma
                                                                                                                                             "in_interval")
                                                                                                                                            (("2"
                                                                                                                                              (skosimp*)
                                                                                                                                              (("2"
                                                                                                                                                (inst?)
                                                                                                                                                nil
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide
                                                                                         2)
                                                                                        (("2"
                                                                                          (rewrite
                                                                                           "fmax_ge")
                                                                                          nil
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (hide
                                                             -3
                                                             -4
                                                             -6
                                                             -8
                                                             2)
                                                            (("2"
                                                              (replace
                                                               -1)
                                                              (("2"
                                                                (hide
                                                                 -1
                                                                 -2
                                                                 -3)
                                                                (("2"
                                                                  (name-replace
                                                                   "BMA"
                                                                   "b!1-a!1")
                                                                  (("2"
                                                                    (mult-by
                                                                     1
                                                                     "(1 + floor(BMA / delta!1))")
                                                                    (("1"
                                                                      (div-by
                                                                       1
                                                                       "delta!1")
                                                                      (("1"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (hide
                                                                       2)
                                                                      (("2"
                                                                        (reveal
                                                                         -3)
                                                                        (("2"
                                                                          (assert)
                                                                          (("2"
                                                                            (case-replace
                                                                             "BMA / delta!1 >= 0")
                                                                            (("1"
                                                                              (assert)
                                                                              nil
                                                                              nil)
                                                                             ("2"
                                                                              (hide
                                                                               2)
                                                                              (("2"
                                                                                (mult-by
                                                                                 1
                                                                                 "delta!1")
                                                                                (("2"
                                                                                  (assert)
                                                                                  (("2"
                                                                                    (reveal
                                                                                     -4)
                                                                                    (("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)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (lemma "integral_diff")
                                              (("2"
                                                (inst
                                                 -1
                                                 "a!1"
                                                 "b!1"
                                                 "fmax(a!1, b!1, PP, f!1)"
                                                 "fmin(a!1, b!1, PP, f!1)")
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (split -1)
                                                    (("1"
                                                      (flatten)
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (expand "-")
                                                          (("1"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide 2)
                                                      (("2"
                                                        (rewrite
                                                         "step_function_integrable?")
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide 2)
                                                      (("3"
                                                        (rewrite
                                                         "step_function_integrable?")
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide 2)
                                      (("2"
                                        (rewrite "fmax_step")
                                        nil
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (rewrite "fmin_step")
                                      nil
                                      nil))
                                    nil)
                                   ("3"
                                    (hide 2)
                                    (("3"
                                      (skosimp*)
                                      (("3"
                                        (hide -1 -2 -3)
                                        (("3"
                                          (inst?)
                                          (("3"
                                            (lemma "cont_restrict")
                                            (("3"
                                              (inst?)
                                              (("3" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (case
                                       "(b!1 - a!1) / delta!1 > 0")
                                      (("1" (assertnil nil)
                                       ("2"
                                        (mult-by 1 "delta!1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide 2)
                              (("2"
                                (split +)
                                (("1"
                                  (mult-by 1 "b!1-a!1")
                                  (("1" (assertnil nil))
                                  nil)
                                 ("2"
                                  (mult-by 1 "b!1-a!1")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide -1 2)
                  (("2" (rewrite "continuous_on_def")
                    (("2" (skosimp*)
                      (("2" (inst -2 "y!1")
                        (("2" (rewrite "continuity_def")
                          (("2" (expand "convergence")
                            (("2" (expand "convergence")
                              (("2"
                                (flatten)
                                (("2"
                                  (expand "extend")
                                  (("2"
                                    (expand "adh")
                                    (("2"
                                      (split +)
                                      (("1"
                                        (skosimp*)
                                        (("1"
                                          (inst + "y!1")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (skosimp*)
                                        (("2"
                                          (inst -3 "epsilon!1")
                                          (("2"
                                            (skosimp*)
                                            (("2"
                                              (inst + "delta!1")
                                              (("2"
                                                (skosimp*)
                                                (("2"
                                                  (assert)
                                                  (("2"
                                                    (inst -4 "x!1")
                                                    (("2"
                                                      (expand
                                                       "fullset")
                                                      (("2"
                                                        (propax)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assertnil nil) ("3" (assertnil nil))
        nil)
       ("2" (skosimp*)
        (("2" (typepred "x!1")
          (("2" (inst-cp 1 "a!1")
            (("2" (inst-cp 1 "b!1") (("2" (assertnil nil)) nil))
            nil))
          nil))
        nil)
       ("3" (skosimp*)
        (("3" (lemma "connected_domain")
          (("3" (inst -1 "a!1" "b!1" "z!1") (("3" (assertnil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((closed_interval type-eq-decl nil intervals_real "reals/")
    (unif_cont_interval formula-decl nil unif_cont_fun nil)
    (continuity_def formula-decl nil continuous_functions nil)
    (abs_0 formula-decl nil abs_lems "reals/")
    (adh const-decl "setof[real]" convergence_functions nil)
    (convergence const-decl "bool" convergence_functions nil)
    (convergence const-decl "bool" lim_of_functions nil)
    (continuous_on_def formula-decl nil continuous_functions nil)
    (step_to_integrable formula-decl nil integral_step nil)
    (uniformly_continuous? const-decl "bool" unif_cont_fun nil)
    (step_function? const-decl "bool" step_fun_def nil)
    (fun_cont_on type-eq-decl nil integral_cont_scaf nil)
    (fun_cont_on type-eq-decl nil interval_minmax nil)
    (min_x const-decl "{mx: T |
         a <= mx AND
          mx <= b AND
           (FORALL (x: T): a <= x AND x <= b IMPLIES f(mx) <= f(x))}"
     interval_minmax nil)
    (fmin const-decl "{ff: [T -> real] |
         LET xx = seq(P) IN
           FORALL (ii: below(length(P) - 1)):
             FORALL (x: T):
               (xx(ii) < x AND x < xx(ii + 1) IMPLIES
                 ff(x) = f(min_x(xx(ii), xx(ii + 1), f)))
                AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
     integral_cont_scaf nil)
    (fmax_step formula-decl nil integral_cont_scaf nil)
    (integrable? const-decl "bool" integral_def nil)
    (integral? const-decl "bool" integral_def nil)
    (integral const-decl "{S: real | integral?(a, b, ff, S)}"
     integral_def nil)
    (part_induction formula-decl nil integral_def nil)
    (width_eq_part formula-decl nil integral_def nil)
    (width const-decl "posreal" integral_def nil)
    (integral_bound_abs formula-decl nil integral_prep nil)
    (partition type-eq-decl nil integral_def nil)
    (width_lem formula-decl nil integral_def nil)
    (fmax_ge formula-decl nil integral_cont_scaf nil)
    (integral_diff formula-decl nil integral_prep nil)
    (step_function_integrable? formula-decl nil integral_step nil)
    (fmax const-decl "{ff: [T -> real] |
         LET xx = seq(P) IN
           FORALL (ii: below(length(P) - 1)):
             FORALL (x: T):
               (xx(ii) < x AND x < xx(ii + 1) IMPLIES
                 ff(x) = f(max_x(xx(ii), xx(ii + 1), f)))
                AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
     integral_cont_scaf nil)
    (max_x const-decl "{mx: T |
         a <= mx AND
          mx <= b AND
           (FORALL (x: T): a <= x AND x <= b IMPLIES f(mx) >= f(x))}"
     interval_minmax nil)
    (fmin_step formula-decl nil integral_cont_scaf nil)
    (cont_restrict formula-decl nil integral_cont_scaf nil)
    (eq_partition const-decl "partition(a, b)" integral_def nil)
    (continuous_on? const-decl "bool" continuous_functions nil))
   nil)
  (continuous_integrable-1 nil 3262523543
   ("" (skosimp*)
    (("" (lemma "unif_cont_interval[closed_interval(a!1,b!1)]")
      (("1" (inst?)
        (("1" (inst -1 "f!1")
          (("1" (assert)
            (("1"
              (case "continuous[closed_interval(a!1, b!1)](restrict(f!1),                 extend[real, closed_interval(a!1, b!1), bool, FALSE]                     ({xx: closed_interval(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" (expand "restrict")
                              (("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"
                                                      (split 1)
                                                      (("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)")
                                                                                        (("1"
                                                                                          (hide
                                                                                           -1
                                                                                           -2
                                                                                           -3)
                                                                                          (("1"
                                                                                            (inst
                                                                                             -1
                                                                                             "xx!1")
                                                                                            (("1"
                                                                                              (assert)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil)
                                                                                         ("2"
                                                                                          (hide-all-but
                                                                                           1)
                                                                                          (("2"
                                                                                            (skosimp*)
                                                                                            (("2"
                                                                                              (reveal
                                                                                               -14)
                                                                                              (("2"
                                                                                                (lemma
                                                                                                 "cont_interv")
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -1
                                                                                                   "a!1"
                                                                                                   "b!1"
                                                                                                   "f!1"
                                                                                                   "PP"
                                                                                                   "seq(PP)")
                                                                                                  (("2"
                                                                                                    (assert)
                                                                                                    (("2"
                                                                                                      (split
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (inst?)
                                                                                                        (("1"
                                                                                                          (expand
                                                                                                           "continuous?"
                                                                                                           -1)
                                                                                                          (("1"
                                                                                                            (inst?)
                                                                                                            nil
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil)
                                                                                                       ("2"
                                                                                                        (hide
                                                                                                         2)
                                                                                                        (("2"
                                                                                                          (expand
                                                                                                           "continuous?"
                                                                                                           1)
                                                                                                          (("2"
                                                                                                            (skosimp*)
                                                                                                            (("2"
                                                                                                              (inst?)
                                                                                                              (("2"
                                                                                                                (lemma
                                                                                                                 "cont_restrict")
                                                                                                                (("2"
                                                                                                                  (inst?)
                                                                                                                  (("2"
                                                                                                                    (assert)
                                                                                                                    nil
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (hide-all-but
                                                               1)
                                                              (("2"
                                                                (skosimp*)
                                                                (("2"
                                                                  (reveal
                                                                   -10)
                                                                  (("2"
                                                                    (inst?)
                                                                    (("2"
                                                                      (lemma
                                                                       "cont_restrict")
                                                                      (("2"
                                                                        (inst?)
                                                                        (("2"
                                                                          (assert)
                                                                          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")
                                                            (("1"
                                                              (assert)
                                                              (("1"
                                                                (assert)
                                                                (("1"
                                                                  (skosimp*)
                                                                  (("1"
                                                                    (inst
                                                                     -
                                                                     "ii!1")
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "xx!1")
                                                                      (("1"
                                                                        (flatten)
                                                                        (("1"
                                                                          (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)")
                                                                                      (("1"
                                                                                        (hide
                                                                                         -1
                                                                                         -2
                                                                                         -3)
                                                                                        (("1"
                                                                                          (inst
                                                                                           -
                                                                                           "xx!1")
                                                                                          (("1"
                                                                                            (assert)
                                                                                            nil
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide-all-but
                                                                                         1)
                                                                                        (("2"
                                                                                          (skosimp*)
                                                                                          (("2"
                                                                                            (typepred
                                                                                             "x!1")
                                                                                            (("2"
                                                                                              (reveal
                                                                                               -14)
                                                                                              (("2"
                                                                                                (lemma
                                                                                                 "cont_interv")
                                                                                                (("2"
                                                                                                  (inst
                                                                                                   -1
                                                                                                   "a!1"
                                                                                                   "b!1"
                                                                                                   "f!1"
                                                                                                   "PP"
                                                                                                   "seq(PP)")
                                                                                                  (("2"
                                                                                                    (assert)
                                                                                                    (("2"
                                                                                                      (split
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (inst?)
                                                                                                        (("1"
                                                                                                          (assert)
                                                                                                          (("1"
                                                                                                            (expand
                                                                                                             "continuous?"
                                                                                                             -1)
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                                                                                 -1
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                                                                                                       -2
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                                                                                                                                 -1)
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                                                                                                                                              -3
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                                                             -8
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                                                              (replace
                                                               -1)
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                                                                (hide
                                                                 -1
                                                                 -2
                                                                 -3)
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                                                                     1
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                                                                       1
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                                                                       2)
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                                                                         -3)
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                                                                                 1
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                                                                       2)
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                                                                          nil
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                                                                           2)
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                                                                             1
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                                                                                 -4)
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                                                                        (assert)
                                                                        nil
                                                                        nil)
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                                                                             1
                                                                             "delta!1")
                                                                            (("2"
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                                                        nil)
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                                                         1
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                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (lemma "integral_diff")
                                              (("2"
                                                (inst
                                                 -1
                                                 "a!1"
                                                 "b!1"
                                                 "fmax(a!1, b!1, PP, f!1)"
                                                 "fmin(a!1, b!1, PP, f!1)")
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (split -1)
                                                    (("1"
                                                      (flatten)
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (expand "-")
                                                          (("1"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide 2)
                                                      (("2"
                                                        (rewrite
                                                         "step_function_integrable?")
                                                        nil
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide 2)
                                                      (("3"
                                                        (rewrite
                                                         "step_function_integrable?")
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (skosimp*)
                                                  (("2"
                                                    (assert)
                                                    (("2"
                                                      (hide 2)
                                                      (("2"
                                                        (hide-all-but
                                                         (-6 1))
                                                        (("2"
                                                          (inst?)
                                                          (("2"
                                                            (lemma
                                                             "cont_restrict")
                                                            (("2"
                                                              (inst?)
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide-all-but (-6 1))
                                        (("2"
                                          (skosimp*)
                                          (("2"
                                            (inst?)
                                            (("2"
                                              (lemma "cont_restrict")
                                              (("2"
                                                (inst?)
                                                (("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide 2)
                                      (("2"
                                        (rewrite "fmax_step")
                                        nil
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (rewrite "fmin_step")
                                      nil
                                      nil))
                                    nil)
                                   ("3"
                                    (hide 2)
                                    (("3"
                                      (skosimp*)
                                      (("3"
                                        (hide -1 -2 -3)
                                        (("3"
                                          (inst?)
                                          (("3"
                                            (lemma "cont_restrict")
                                            (("3"
                                              (inst?)
                                              (("3" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (case
                                       "(b!1 - a!1) / delta!1 > 0")
                                      (("1" (assertnil nil)
                                       ("2"
                                        (mult-by 1 "delta!1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide 2)
                            (("2" (split +)
                              (("1"
                                (mult-by 1 "b!1-a!1")
                                (("1" (assertnil nil))
                                nil)
                               ("2"
                                (mult-by 1 "b!1-a!1")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide -1 2)
                (("2" (expand "continuous?")
                  (("2" (skosimp*)
                    (("2" (inst -2 "y!1")
                      (("1" (assert)
                        (("1" (expand "extend")
                          (("1" (expand "convergence")
                            (("1" (expand "convergence")
                              (("1"
                                (flatten)
                                (("1"
                                  (prop)
                                  (("1"
                                    (hide -3)
                                    (("1"
                                      (expand "adh")
                                      (("1"
                                        (skosimp*)
                                        (("1"
                                          (inst - "e!1")
                                          (("1"
                                            (skosimp*)
                                            (("1"
                                              (typepred "y!1")
                                              (("1"
                                                (expand "extend")
                                                (("1"
                                                  (ground)
                                                  (("1"
                                                    (case
                                                     "y!1 + e!1/2 <= b!1")
                                                    (("1"
                                                      (inst
                                                       +
                                                       "y!1+e!1/2")
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (expand
                                                           "abs"
                                                           1)
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (assert)
                                                        (("2"
                                                          (lemma
                                                           "connected_domain")
                                                          (("2"
                                                            (inst
                                                             -1
                                                             "y!1"
                                                             "b!1"
                                                             "y!1+e!1/2")
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (inst + "b!1")
                                                      (("2"
                                                        (expand "abs")
                                                        (("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (skosimp*)
                                    (("2"
                                      (inst -3 "epsilon!1")
                                      (("2"
                                        (skosimp*)
                                        (("2"
                                          (inst + "delta!1")
                                          (("2"
                                            (skosimp*)
                                            (("2"
                                              (inst -4 "x!1")
                                              (("2"
                                                (assert)
                                                (("2"
                                                  (expand "restrict")
                                                  (("2"
                                                    (expand "fullset")
                                                    (("2"
                                                      (propax)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (typepred "y!1")
                        (("2" (expand "extend")
                          (("2" (ground) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (hide -1 -3 2)
                (("3" (expand "subset?")
                  (("3" (skosimp*)
                    (("3" (expand "member")
                      (("3" (expand "extend") (("3" (ground) nil nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assertnil nil) ("3" (assertnil nil))
        nil)
       ("2" (skosimp*)
        (("2" (typepred "x!1")
          (("2" (inst-cp 1 "a!1")
            (("2" (inst-cp 1 "b!1") (("2" (assertnil nil)) nil))
            nil))
          nil))
        nil)
       ("3" (skosimp*)
        (("3" (lemma "connected_domain")
          (("3" (inst -1 "a!1" "b!1" "z!1") (("3" (assertnil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((step_function? const-decl "bool" step_fun_def nil)
    (fmin const-decl "{ff: [T -> real] |
         LET xx = seq(P) IN
           FORALL (ii: below(length(P) - 1)):
             FORALL (x: T):
               (xx(ii) < x AND x < xx(ii + 1) IMPLIES
                 ff(x) = f(min_x(xx(ii), xx(ii + 1), f)))
                AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
     integral_cont_scaf nil)
    (integrable? const-decl "bool" integral_def nil)
    (integral? const-decl "bool" integral_def nil)
    (integral const-decl "{S: real | integral?(a, b, ff, S)}"
     integral_def nil)
    (part_induction formula-decl nil integral_def nil)
    (cont_restrict formula-decl nil integral_cont_scaf nil)
    (cont_interv formula-decl nil integral_cont_scaf nil)
    (width_eq_part formula-decl nil integral_def nil)
    (width const-decl "posreal" integral_def nil)
    (integral_bound_abs formula-decl nil integral_prep nil)
    (width_lem formula-decl nil integral_def nil)
    (integral_diff formula-decl nil integral_prep nil)
    (step_function_integrable? formula-decl nil integral_step nil)
    (fmax const-decl "{ff: [T -> real] |
         LET xx = seq(P) IN
           FORALL (ii: below(length(P) - 1)):
             FORALL (x: T):
               (xx(ii) < x AND x < xx(ii + 1) IMPLIES
                 ff(x) = f(max_x(xx(ii), xx(ii + 1), f)))
                AND ((xx(ii) = x OR x = xx(ii + 1)) IMPLIES ff(x) = f(x))}"
     integral_cont_scaf nil)
    (partition type-eq-decl nil integral_def nil)
    (eq_partition const-decl "partition(a, b)" integral_def nil)
    (step_to_integrable formula-decl nil integral_step nil)
    (adh const-decl "setof[real]" convergence_functions nil)
    (convergence const-decl "bool" convergence_functions nil))
   nil))
 (cont_fun_integrable 0
  (cont_fun_integrable-1 nil 3313938280
   ("" (skosimp*)
    (("" (lemma "continuous_integrable")
      (("" (inst?)
        (("" (assert)
          (("" (skosimp*)
            (("" (expand "continuous?" -2) (("" (inst?) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((continuous_integrable formula-decl nil integral_cont nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (continuous? const-decl "bool" continuous_functions nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (closed_interval type-eq-decl nil intervals_real "reals/")
    (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)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak)))


Messung V0.5 in Prozent
C=100 H=100 G=100

[0.310QuellennavigatorsProjekt 2026-04-25]