Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/vect_analysis/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 53 kB image not shown  

Quelle  deriv_sigma.prf   Sprache: Lisp

 
(deriv_sigma
 (IMP_derivatives_TCC1 0
  (IMP_derivatives_TCC1-1 nil 3514223777
   ("" (lemma "deriv_domain") (("" (propax) nil nil)) nil)
   ((deriv_domain formula-decl nil deriv_sigma nil)) nil))
 (IMP_derivatives_TCC2 0
  (IMP_derivatives_TCC2-1 nil 3514223777
   ("" (lemma "not_one_element") (("" (propax) nil nil)) nil)
   ((not_one_element formula-decl nil deriv_sigma nil)) nil))
 (sigma_derivable_TCC1 0
  (sigma_derivable_TCC1-1 nil 3460460862 ("" (subtype-tcc) nil nil)
   ((fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
 (sigma_derivable_TCC2 0
  (sigma_derivable_TCC2-1 nil 3460460862 ("" (subtype-tcc) nil nil)
   ((fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
 (sigma_derivable_TCC3 0
  (sigma_derivable_TCC3-1 nil 3460460862 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (> const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_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)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (Index type-eq-decl nil vectors "vectors/")
    (integer nonempty-type-from-decl nil integers nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (sigma_derivable 0
  (sigma_derivable-1 nil 3460463493
   ("" (induct "m")
    (("1" (assertnil nil) ("2" (assertnil nil)
     ("3" (skosimp*)
      (("3" (case "j!1 > 0")
        (("1" (assert)
          (("1"
            (case-replace "(LAMBDA (x: T):
                   sigma(0, j!1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i))) =
              (LAMBDA (x: T):
                   sigma(0, j!1-1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i)) + h!1(x)(j!1))")
            (("1" (hide -1)
              (("1"
                (inst - "y!1"
                 "LAMBDA (x:T): LAMBDA (i: Index[j!1]): h!1(x)(i)")
                (("1" (assert)
                  (("1" (split -2)
                    (("1" (lemma "sum_derivable[T]")
                      (("1"
                        (inst - "(LAMBDA (x: T): h!1(x)(j!1))"
                         "(LAMBDA (x: T):
                     sigma(0, j!1 - 1,
                           LAMBDA (i: Index[1 + j!1]): h!1(x)(i)))"
                         "y!1")
                        (("1" (assert)
                          (("1" (split -1)
                            (("1"
                              (expand "+
")
                              (("1" (propax) nil nil)) nil)
                             ("2" (hide 2) (("2" (inst?) nil nil)) nil)
                             ("3" (hide 2)
                              (("3"
                                (case-replace
                                 "(LAMBDA (x_1: T):
                   sigma(0, j!1 - 1,
                         LAMBDA (i_1: Index[j!1]): h!1(x_1)(i_1))) =
(LAMBDA (x: T):
                    sigma(0, j!1 - 1,
                          LAMBDA (i: Index[1 + j!1]): h!1(x)(i)))
")
                                (("1"
                                  (hide 2)
                                  (("1"
                                    (apply-extensionality 1 :hide? t)
                                    (("1"
                                      (hide -1)
                                      (("1"
                                        (lemma
                                         "sigma_diff_eq[j!1,j!1+1]")
                                        (("1"
                                          (inst?)
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (skosimp*)
                                      (("2" (assertnil nil))
                                      nil)
                                     ("3"
                                      (skosimp*)
                                      (("3" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (skosimp*)
                                  (("2" (assertnil nil))
                                  nil)
                                 ("3"
                                  (skosimp*)
                                  (("3" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (skosimp*) (("2" (assertnil nil)) nil))
                        nil))
                      nil)
                     ("2" (skosimp*) (("2" (inst?) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide -2 2)
              (("2" (apply-extensionality 1 :hide? t)
                (("1" (lemma "sigma_last[Index[j!1+1]]")
                  (("1" (inst?) (("1" (assertnil nil)) nil)
                   ("2" (skosimp*) (("2" (assertnil nil)) nil))
                  nil)
                 ("2" (skosimp*) (("2" (assertnil nil)) nil))
                nil))
              nil)
             ("3" (skosimp*) (("3" (assertnil nil)) nil))
            nil))
          nil)
         ("2" (hide -1)
          (("2" (assert)
            (("2" (expand "sigma")
              (("2" (expand "sigma") (("2" (inst?) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("4" (hide 2) (("4" (skosimp*) (("4" (assertnil nil)) nil)) nil)
     ("5" (hide 2) (("5" (skosimp*) (("5" (assertnil nil)) nil)) nil)
     ("6" (skosimp*) (("6" (assertnil nil)) nil))
    nil)
   ((posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (sigma_last formula-decl nil sigma "reals/")
    (sum_derivable formula-decl nil derivatives_def "analysis/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (below type-eq-decl nil naturalnumbers nil)
    (sigma_diff_eq formula-decl nil sigma_below_sub "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (+ const-decl "[T -> real]" real_fun_ops "reals/")
    (j!1 skolem-const-decl "nat" deriv_sigma nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (pred type-eq-decl nil defined_types nil)
    (integer nonempty-type-from-decl nil integers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (> const-decl "bool" reals nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (< const-decl "bool" reals nil)
    (Index type-eq-decl nil vectors "vectors/")
    (below type-eq-decl nil nat_types nil)
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   shostak))
 (deriv_sigma_TCC1 0
  (deriv_sigma_TCC1-1 nil 3460386482 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (deriv_sigma_TCC2 0
  (deriv_sigma_TCC2-1 nil 3460461487 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (deriv_sigma_TCC3 0
  (deriv_sigma_TCC3-1 nil 3460461487 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (integer nonempty-type-from-decl nil integers nil)
    (Index type-eq-decl nil vectors "vectors/")
    (< const-decl "bool" reals nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (deriv_sigma_TCC4 0
  (deriv_sigma_TCC4-1 nil 3460461487
   ("" (skosimp*)
    (("" (lemma "sigma_derivable")
      (("" (inst - "n!1" "y!1" "h!1") (("" (assertnil nil)) nil))
      nil))
    nil)
   ((sigma_derivable formula-decl nil deriv_sigma nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (Index type-eq-decl nil vectors "vectors/")
    (< const-decl "bool" reals nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (> const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (n!1 skolem-const-decl "nat" deriv_sigma nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
   nil))
 (deriv_sigma_TCC5 0
  (deriv_sigma_TCC5-1 nil 3460462472 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (deriv_sigma_TCC6 0
  (deriv_sigma_TCC6-1 nil 3460463371 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (deriv_sigma_TCC7 0
  (deriv_sigma_TCC7-1 nil 3460463371 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (fullset const-decl "set" sets nil)
    (adh const-decl "setof[real]" convergence_functions "analysis/")
    (NQ const-decl "real" derivatives_def "analysis/")
    (convergence const-decl "bool" convergence_functions "analysis/")
    (convergence const-decl "bool" lim_of_functions "analysis/")
    (convergent? const-decl "bool" lim_of_functions "analysis/")
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (derivable? const-decl "bool" deriv_real_vect_def nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   nil))
 (deriv_sigma_TCC8 0
  (deriv_sigma_TCC8-1 nil 3460463371
   ("" (skosimp*) (("" (assertnil nil)) nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
   nil))
 (deriv_sigma 0
  (deriv_sigma-1 nil 3460387081
   ("" (induct "n")
    (("1" (skosimp*)
      (("1" (expand "sigma")
        (("1" (lemma "deriv_const[T]")
          (("1" (inst?)
            (("1" (expand "const_fun") (("1" (inst?) nil nil)) nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (skosimp*)
      (("2" (case "j!1 = 0")
        (("1" (replace -1)
          (("1" (hide -2)
            (("1" (expand "sigma")
              (("1" (expand "sigma")
                (("1" (expand "deriv" 1 2) (("1" (propax) nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (inst - "y!1")
          (("2" (assert)
            (("1"
              (inst -
               "LAMBDA (x: T): LAMBDA (i: Index[j!1]): h!1(x)(i)")
              (("1" (split -1)
                (("1" (lemma "sigma_last[below(j!1+1)]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replace -1)
                        (("1" (hide -1)
                          (("1"
                            (case-replace
                             "(LAMBDA (x: T): sigma(0, j!1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i))) =
(LAMBDA (x: T): sigma(0, j!1-1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i)) + h!1(x)(j!1))")
                            (("1" (hide -1)
                              (("1"
                                (lemma "deriv_sum[T]")
                                (("1"
                                  (inst
                                   -
                                   "(LAMBDA (x: T): sigma(0, j!1 - 1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i)))"
                                   "(LAMBDA (x: T): h!1(x)(j!1))"
                                   "y!1")
                                  (("1"
                                    (assert)
                                    (("1"
                                      (split -1)
                                      (("1"
                                        (expand "+ " -1)
                                        (("1"
                                          (replace -1)
                                          (("1"
                                            (hide -1)
                                            (("1"
                                              (case-replace
                                               "(LAMBDA (x: T): sigma(0, j!1 - 1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i))) =
             (LAMBDA (x: T): sigma(0, j!1 - 1, LAMBDA (i: Index[j!1]): h!1(x)(i)))")
                                              (("1"
                                                (hide -1)
                                                (("1"
                                                  (replace -1)
                                                  (("1"
                                                    (hide -1)
                                                    (("1"
                                                      (case
                                                       "sigma(0, j!1 - 1, LAMBDA (i: Index[1 + j!1]): deriv(h!1, y!1)(i)) =
                            sigma(0, j!1 - 1, LAMBDA (i: Index[j!1]): deriv(h!1, y!1)(i))")
                                                      (("1"
                                                        (case-replace
                                                         "sigma(0, j!1 - 1, LAMBDA (i: Index[1 + j!1]): deriv(h!1, y!1)(i)) =
                         sigma(0, j!1 - 1, LAMBDA (i: Index[j!1]): deriv(h!1, y!1)(i))")
                                                        (("1"
                                                          (hide -1)
                                                          (("1"
                                                            (expand
                                                             "deriv"
                                                             2
                                                             3)
                                                            (("1"
                                                              (expand
                                                               "deriv"
                                                               2
                                                               2)
                                                              (("1"
                                                                (assert)
                                                                (("1"
                                                                  (expand
                                                                   "deriv"
                                                                   2
                                                                   1)
                                                                  (("1"
                                                                    (propax)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide 3)
                                                        (("2"
                                                          (lemma
                                                           "sigma_diff_eq[1+j!1,j!1]")
                                                          (("2"
                                                            (inst?)
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil)
                                                             ("2"
                                                              (skosimp*)
                                                              (("2"
                                                                (lemma
                                                                 "not_one_element")
                                                                (("2"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("3"
                                                              (skosimp*)
                                                              (("3"
                                                                (lemma
                                                                 "deriv_domain")
                                                                (("3"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("4"
                                                              (skosimp*)
                                                              (("4"
                                                                (expand
                                                                 "derivable?"
                                                                 1)
                                                                (("4"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("5"
                                                              (skosimp*)
                                                              (("5"
                                                                (lemma
                                                                 "not_one_element")
                                                                (("5"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("6"
                                                              (skosimp*)
                                                              (("6"
                                                                (lemma
                                                                 "deriv_domain")
                                                                (("6"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("7"
                                                              (skosimp*)
                                                              (("7"
                                                                (expand
                                                                 "derivable?"
                                                                 1)
                                                                (("7"
                                                                  (propax)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("3"
                                                        (skosimp*)
                                                        (("3"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil)
                                                       ("4"
                                                        (skosimp*)
                                                        (("4"
                                                          (lemma
                                                           "not_one_element")
                                                          (("4"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("5"
                                                        (skosimp*)
                                                        (("5"
                                                          (assert)
                                                          (("5"
                                                            (lemma
                                                             "deriv_domain")
                                                            (("5"
                                                              (propax)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("6"
                                                        (skosimp*)
                                                        (("6"
                                                          (lemma
                                                           "not_one_element")
                                                          (("6"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("7"
                                                        (skosimp*)
                                                        (("7"
                                                          (lemma
                                                           "deriv_domain")
                                                          (("7"
                                                            (propax)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide 3)
                                                (("2"
                                                  (apply-extensionality
                                                   1
                                                   :hide?
                                                   t)
                                                  (("1"
                                                    (lemma
                                                     "sigma_diff_eq[1+j!1,j!1]")
                                                    (("1"
                                                      (inst?)
                                                      (("1"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (skosimp*)
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil)
                                                   ("3"
                                                    (skosimp*)
                                                    (("3"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (skosimp*)
                                                (("3"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil)
                                               ("4"
                                                (skosimp*)
                                                (("4"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide 3)
                                        (("2"
                                          (hide -1)
                                          (("2"
                                            (case-replace
                                             "(LAMBDA (x: T): sigma(0, j!1 - 1, LAMBDA (i: Index[1 + j!1]): h!1(x)(i))) =
(LAMBDA (x: T): sigma(0, j!1 - 1, LAMBDA (i: Index[ j!1]): h!1(x)(i)))")
                                            (("1"
                                              (hide -1)
                                              (("1"
                                                (lemma
                                                 "sigma_derivable")
                                                (("1"
                                                  (hide -1)
                                                  (("1"
                                                    (lemma
                                                     "sigma_derivable")
                                                    (("1"
                                                      (inst
                                                       -
                                                       "j!1"
                                                       "y!1"
                                                       "(LAMBDA (x: T): LAMBDA (i: Index[j!1]): h!1(x)(i))")
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (skosimp*)
                                                          (("1"
                                                            (inst?)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide 2)
                                              (("2"
                                                (apply-extensionality
                                                 1
                                                 :hide?
                                                 t)
                                                (("1"
                                                  (lemma
                                                   "sigma_diff_eq[1+j!1,j!1]")
                                                  (("1"
                                                    (inst?)
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (skosimp*)
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("3"
                                                  (skosimp*)
                                                  (("3"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("3"
                                              (skosimp*)
                                              (("3" (assertnil nil))
                                              nil)
                                             ("4"
                                              (skosimp*)
                                              (("4" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("3"
                                        (hide 3)
                                        (("3"
                                          (hide -1)
                                          (("3" (inst?) nil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (skosimp*)
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide 3)
                              (("2"
                                (apply-extensionality 1 :hide? t)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (lemma "sigma_last[below(j!1+1)]")
                                    (("1"
                                      (inst?)
                                      (("1" (assertnil nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (skosimp*)
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil)
                             ("3" (skosimp*) (("3" (assertnil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (skosimp*)
                      (("2" (hide 3)
                        (("2" (hide -1)
                          (("2" (lemma "not_one_element")
                            (("2" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil)
                     ("3" (skosimp*)
                      (("3" (lemma "deriv_domain")
                        (("3" (propax) nil nil)) nil))
                      nil)
                     ("4" (skosimp*)
                      (("4" (expand "derivable?" 1)
                        (("4" (propax) nil nil)) nil))
                      nil))
                    nil)
                   ("2" (skosimp*) (("2" (assertnil nil)) nil))
                  nil)
                 ("2" (hide 3)
                  (("2" (expand "derivable?")
                    (("2" (skosimp*) (("2" (inst?) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (assertnil nil))
            nil))
          nil))
        nil))
      nil)
     ("3" (skosimp*) (("3" (assertnil nil)) nil)
     ("4" (skosimp*)
      (("4" (lemma "not_one_element") (("4" (propax) nil nil)) nil))
      nil)
     ("5" (skosimp*)
      (("5" (assert)
        (("5" (lemma "deriv_domain") (("5" (propax) nil nil)) nil))
        nil))
      nil)
     ("6" (skosimp*)
      (("6" (assert)
        (("6" (hide 2)
          (("6" (expand "derivable?" 1) (("6" (propax) nil nil)) nil))
          nil))
        nil))
      nil)
     ("7" (skosimp*) (("7" (assertnil nil)) nil)
     ("8" (skosimp*) (("8" (assertnil nil)) nil)
     ("9" (skosimp*)
      (("9" (assert)
        (("9" (hide 2)
          (("9" (lemma "sigma_derivable")
            (("9" (inst - "n!2" "y!1" "h!1") (("9" (assertnil nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("10" (hide 2) (("10" (skosimp*) (("10" (assertnil nil)) nil))
      nil)
     ("11" (skosimp*) (("11" (assertnil nil)) nil)
     ("12" (skosimp*) (("12" (assertnil nil)) nil))
    nil)
   ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sigma_last formula-decl nil sigma "reals/")
    (below type-eq-decl nil naturalnumbers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sigma_diff_eq formula-decl nil sigma_below_sub "reals/")
    (deriv_domain formula-decl nil deriv_sigma nil)
    (not_one_element formula-decl nil deriv_sigma nil)
    (+ const-decl "[T -> real]" real_fun_ops "reals/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (sigma_derivable formula-decl nil deriv_sigma nil)
    (deriv_sum formula-decl nil derivatives_def "analysis/")
    (real_plus_real_is_real application-judgement "real" reals nil)
    (y!1 skolem-const-decl "T" deriv_sigma nil)
    (h!1 skolem-const-decl "[T -> [Index[1 + j!1] -> real]]"
     deriv_sigma nil)
    (j!1 skolem-const-decl "nat" deriv_sigma nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (deriv_const formula-decl nil derivatives_def "analysis/")
    (const_fun const-decl "[T -> real]" real_fun_ops "reals/")
    (nat_induction formula-decl nil naturalnumbers nil)
    (deriv const-decl "Vector[n]" deriv_real_vect_def nil)
    (deriv const-decl "real" derivatives_def "analysis/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (pred type-eq-decl nil defined_types nil)
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/")
    (deriv_domain? const-decl "bool" deriv_domain_def "analysis/")
    (Vector type-eq-decl nil vectors "vectors/")
    (derivable? const-decl "bool" deriv_real_vect_def nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (sigma def-decl "real" sigma "reals/")
    (integer nonempty-type-from-decl nil integers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (T_pred const-decl "[real -> boolean]" deriv_sigma nil)
    (T formal-subtype-decl nil deriv_sigma nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (> const-decl "bool" reals nil) (< const-decl "bool" reals nil)
    (Index type-eq-decl nil vectors "vectors/")
    (below type-eq-decl nil nat_types nil)
    (derivable? const-decl "bool" derivatives_def "analysis/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   shostak)))

93%


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Versionsinformation zu Columbo

Bemerkung:

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Anfrage:

Dauer der Verarbeitung:

Sekunden

sprechenden Kalenders