products/sources/formale sprachen/PVS/lnexp_fnd image not shown  

Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: ln_series.prf   Sprache: Lisp

Original von: PVS©

(ln_series
 (noa_posreal 0
  (noa_posreal-1 nil 3477843250
   ("" (expand "not_one_element?")
    (("" (skosimp*)
      (("" (inst + "x!1 + 1") (("" (assertnil nil)) nil)) nil))
    nil)
   ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields 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 "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (posreal_plus_nnreal_is_posreal application-judgement "posreal"
     real_types nil)
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/"))
   shostak))
 (noa_gt_m1 0
  (noa_gt_m1-1 nil 3477843265
   ("" (expand "not_one_element?")
    (("" (skosimp*)
      (("" (inst + "x!1 + 1") (("" (assertnil nil)) nil)) nil))
    nil)
   ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (> const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans 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)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/"))
   shostak))
 (conn_gt_m1 0
  (conn_gt_m1-1 nil 3477843280
   ("" (expand "connected?")
    (("" (skosimp*) (("" (assertnil nil)) nil)) 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)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (connected? const-decl "bool" deriv_domain_def "analysis/"))
   shostak))
 (conn_posreal 0
  (conn_posreal-1 nil 3477843288
   ("" (expand "connected?")
    (("" (skosimp*) (("" (assertnil nil)) nil)) 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)
    (connected? const-decl "bool" deriv_domain_def "analysis/"))
   shostak))
 (conn_abslt1 0
  (conn_abslt1-1 nil 3477843296
   ("" (expand "connected?")
    (("" (skosimp*)
      (("" (assert)
        (("" (typepred "y!1")
          (("" (typepred "x!1") (("" (grind) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((abslt1 type-eq-decl nil ln_series nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil) (< const-decl "bool" reals 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)
    (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)
    (connected? const-decl "bool" deriv_domain_def "analysis/"))
   shostak))
 (noa_abslt1 0
  (noa_abslt1-1 nil 3477843230
   ("" (expand "not_one_element?")
    (("" (skosimp*)
      (("" (inst-cp + "1/2")
        (("1" (inst + "1/4")
          (("1" (assertnil nil) ("2" (grind) nil nil)) nil)
         ("2" (grind) nil nil))
        nil))
      nil))
    nil)
   ((/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (abslt1 type-eq-decl nil ln_series 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) (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nzrat_abs_is_pos application-judgement "{r: posrat | r >= q}"
     real_defs nil)
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/"))
   nil))
 (deriv_domain_gtm1 0
  (deriv_domain_gtm1-1 nil 3477843360
   ("" (expand "deriv_domain?")
    (("" (skosimp*) (("" (inst + "e!1/2") (("" (grind) nil nil)) nil))
      nil))
    nil)
   ((abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (nonzero_abs_is_pos application-judgement "{y: posreal | y >= x}"
     real_defs nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (> const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal 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)
    (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)
    (deriv_domain? const-decl "bool" deriv_domain_def "analysis/")
    (real_plus_real_is_real application-judgement "real" reals nil))
   shostak))
 (nderiv_ln_TCC1 0
  (nderiv_ln_TCC1-1 nil 3563622659 ("" (assertnil nil)
   ((deriv_domain_posreal formula-decl nil deriv_domain "analysis/"))
   nil))
 (nderiv_ln_TCC2 0
  (nderiv_ln_TCC2-1 nil 3563622659 ("" (assertnil nil)
   ((noa_posreal formula-decl nil ln_series nil)) nil))
 (nderiv_ln 0
  (nderiv_ln-2 nil 3445354085
   (""
    (case "FORALL (n: nat): derivable_n_times?[posreal](ln, n) & nderiv[posreal](n,ln) = IF n = 0 THEN ln ELSE (LAMBDA (x:posreal): -factorial(n-1)/(-x)^n) ENDIF")
    (("1" (skosimp*)
      (("1" (inst - "n!1") (("1" (flatten) nil nil)) nil)) nil)
     ("2" (hide 2)
      (("2" (induct "n")
        (("1" (expand "derivable_n_times?") (("1" (propax) nil nil))
          nil)
         ("2" (expand "nderiv") (("2" (propax) nil nil)) nil)
         ("3" (skosimp*)
          (("3" (case-replace "j!1=0")
            (("1" (lemma "ln_derivable")
              (("1" (flatten -1)
                (("1" (expand "derivable_n_times?")
                  (("1" (expand "derivable_n_times?")
                    (("1" (expand "nderiv" 1)
                      (("1" (expand "nderiv" 1)
                        (("1"
                          (lemma "extensionality"
                           ("f" "deriv(ln)" "g"
                            "(LAMBDA (x: posreal): -factorial(1) / (-x) ^ 1)"))
                          (("1" (split -1)
                            (("1" (assertnil nil)
                             ("2" (hide 2)
                              (("2"
                                (replace -2)
                                (("2"
                                  (skosimp*)
                                  (("2"
                                    (expand "factorial")
                                    (("2"
                                      (expand "factorial")
                                      (("2"
                                        (rewrite "expt_x1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide 2)
                            (("2" (skosimp*)
                              (("2"
                                (inst + "x!1+1")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil)
                           ("3" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (assert)
              (("2" (lemma "ln_derivable")
                (("2" (flatten -1)
                  (("2" (lemma "identity_derivable_fun[posreal]")
                    (("1"
                      (lemma "const_derivable_fun[posreal]"
                       ("b" "-factorial(j!1)"))
                      (("1" (lemma "deriv_id_fun[posreal]")
                        (("1"
                          (lemma "deriv_const_fun[posreal]"
                           ("b" "-factorial(j!1)"))
                          (("1" (expand "I")
                            (("1" (expand "const_fun")
                              (("1"
                                (lemma
                                 "neg_derivable_fun[posreal]"
                                 ("f" "LAMBDA (x: posreal): x"))
                                (("1"
                                  (assert)
                                  (("1"
                                    (expand "-")
                                    (("1"
                                      (lemma
                                       "deriv_neg_fun[posreal]"
                                       ("ff" "LAMBDA (x: posreal): x"))
                                      (("1"
                                        (replace -4)
                                        (("1"
                                          (expand "-" -1)
                                          (("1"
                                            (lemma
                                             "inv_derivable_fun[posreal]"
                                             ("g"
                                              "LAMBDA (x_1: posreal): -x_1"))
                                            (("1"
                                              (assert)
                                              (("1"
                                                (expand "/")
                                                (("1"
                                                  (lemma
                                                   "deriv_inv_fun[posreal]"
                                                   ("gg"
                                                    "LAMBDA (x_1: posreal): -x_1"))
                                                  (("1"
                                                    (replace -3)
                                                    (("1"
                                                      (expand "/")
                                                      (("1"
                                                        (expand "*")
                                                        (("1"
                                                          (expand
                                                           "-"
                                                           -1)
                                                          (("1"
                                                            (lemma
                                                             "extensionality"
                                                             ("f"
                                                              "LAMBDA (x: posreal): --1 / (-x * -x)"
                                                              "g"
                                                              "LAMBDA (x: posreal): 1 / (x * x)"))
                                                            (("1"
                                                              (split
                                                               -1)
                                                              (("1"
                                                                (replace
                                                                 -1)
                                                                (("1"
                                                                  (hide
                                                                   -1)
                                                                  (("1"
                                                                    (lemma
                                                                     "deriv_exp_fun[posreal]"
                                                                     ("f"
                                                                      "LAMBDA (x: posreal): 1 / -x"
                                                                      "n"
                                                                      "j!1"))
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (flatten
                                                                         -1)
                                                                        (("1"
                                                                          (lemma
                                                                           "scal_derivable_fun[posreal]"
                                                                           ("f"
                                                                            "(LAMBDA (x: posreal): 1 / -x) ^ j!1"
                                                                            "b"
                                                                            "-factorial(j!1 - 1)"))
                                                                          (("1"
                                                                            (replace
                                                                             -2)
                                                                            (("1"
                                                                              (lemma
                                                                               "deriv_scal_fun[posreal]"
                                                                               ("ff"
                                                                                "(LAMBDA (x: posreal): 1 / -x) ^ j!1"
                                                                                "b"
                                                                                "-factorial(j!1 - 1)"))
                                                                              (("1"
                                                                                (replace
                                                                                 -4
                                                                                 -1)
                                                                                (("1"
                                                                                  (replace
                                                                                   -5)
                                                                                  (("1"
                                                                                    (lemma
                                                                                     "extensionality"
                                                                                     ("f"
                                                                                      "-factorial(j!1 - 1) * (LAMBDA (x: posreal): 1 / -x) ^ j!1"
                                                                                      "g"
                                                                                      "LAMBDA (x: posreal): -factorial(j!1 - 1) / (-x) ^ j!1"))
                                                                                    (("1"
                                                                                      (split
                                                                                       -1)
                                                                                      (("1"
                                                                                        (replace
                                                                                         -1
                                                                                         *
                                                                                         rl)
                                                                                        (("1"
                                                                                          (replace
                                                                                           -17
                                                                                           *
                                                                                           rl)
                                                                                          (("1"
                                                                                            (lemma
                                                                                             "extensionality"
                                                                                             ("f"
                                                                                              "-factorial(j!1 - 1) *
                                                         (j!1 * (LAMBDA (x: posreal): 1 / -x) ^ (j!1 - 1) *
                                                           (LAMBDA (x: posreal): 1 / (x * x)))"
                                                                                              "g"
                                                                                              "LAMBDA (x: posreal): -factorial(j!1) / (-x) ^ (1 + j!1)"))
                                                                                            (("1"
                                                                                              (split
                                                                                               -1)
                                                                                              (("1"
                                                                                                (replace
                                                                                                 -1)
                                                                                                (("1"
                                                                                                  (lemma
                                                                                                   "nderiv_derivable_eqv[posreal]"
                                                                                                   ("f"
                                                                                                    "ln"
                                                                                                    "n"
                                                                                                    "j!1"))
                                                                                                  (("1"
                                                                                                    (replace
                                                                                                     -18
                                                                                                     -1)
                                                                                                    (("1"
                                                                                                      (replace
                                                                                                       -5
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (flatten
                                                                                                         -1)
                                                                                                        (("1"
                                                                                                          (lemma
                                                                                                           "nderiv_derivable_aux[posreal]"
                                                                                                           ("f"
                                                                                                            "ln"
                                                                                                            "n"
                                                                                                            "j!1"))
                                                                                                          (("1"
                                                                                                            (replace
                                                                                                             -2)
                                                                                                            (("1"
                                                                                                              (replace
                                                                                                               -5
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (replace
                                                                                                                 -1
                                                                                                                 2)
                                                                                                                (("1"
                                                                                                                  (propax)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil)
                                                                                               ("2"
                                                                                                (hide
                                                                                                 3)
                                                                                                (("2"
                                                                                                  (skosimp*)
                                                                                                  (("2"
                                                                                                    (expand
                                                                                                     "factorial"
                                                                                                     1
                                                                                                     2)
                                                                                                    (("2"
                                                                                                      (expand
                                                                                                       "*"
                                                                                                       1)
                                                                                                      (("2"
                                                                                                        (expand
                                                                                                         "^"
                                                                                                         1
                                                                                                         1)
                                                                                                        (("2"
                                                                                                          (rewrite
                                                                                                           "div_expt"
                                                                                                           1)
                                                                                                          (("2"
                                                                                                            (hide-all-but
                                                                                                             (1
                                                                                                              2))
                                                                                                            (("2"
                                                                                                              (lemma
                                                                                                               "expt_plus"
                                                                                                               ("n0x"
                                                                                                                "-x!1"
                                                                                                                "i"
                                                                                                                "j!1-1"
                                                                                                                "j"
                                                                                                                "2"))
                                                                                                              (("2"
                                                                                                                (replace
                                                                                                                 -1)
                                                                                                                (("2"
                                                                                                                  (case
                                                                                                                   "(-x!1) ^ 2=x!1*x!1")
                                                                                                                  (("1"
                                                                                                                    (replace
                                                                                                                     -1)
                                                                                                                    (("1"
                                                                                                                      (assert)
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil)
                                                                                                                   ("2"
                                                                                                                    (hide-all-but
                                                                                                                     1)
                                                                                                                    (("2"
                                                                                                                      (grind)
                                                                                                                      nil
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil))
                                                                                                    nil))
                                                                                                  nil))
                                                                                                nil))
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil)
                                                                                       ("2"
                                                                                        (hide
                                                                                         3)
                                                                                        (("2"
                                                                                          (skosimp*)
                                                                                          (("2"
                                                                                            (expand
                                                                                             "*"
                                                                                             1)
                                                                                            (("2"
                                                                                              (expand
                                                                                               "^"
                                                                                               1
                                                                                               1)
                                                                                              (("2"
                                                                                                (rewrite
                                                                                                 "div_expt"
                                                                                                 1)
                                                                                                (("2"
                                                                                                  (assert)
                                                                                                  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"
                                                                  (grind)
                                                                  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" (skosimp*)
                      (("2" (assert)
                        (("2" (hide 3)
                          (("2" (inst + "x!1+1")
                            (("2" (assertnil nil)) nil))
                          nil))
                        nil))
                      nil)
                     ("3" (assert)
                      (("3" (lemma "deriv_domain_posreal")
                        (("3" (propax) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("4" (skosimp*) (("4" (assertnil nil)) nil)
         ("5" (hide 1) (("5" (skosimp*) (("5" (assertnil nil)) nil))
          nil))
        nil))
      nil)
     ("3" (hide 2) (("3" (skosimp*) (("3" (assertnil nil)) nil))
      nil))
    nil)
   ((const_derivable_fun formula-decl nil derivatives "analysis/")
    (deriv_const_fun formula-decl nil derivatives "analysis/")
    (neg_derivable_fun formula-decl nil derivatives "analysis/")
    (- const-decl "[T -> real]" real_fun_ops "reals/")
    (inv_derivable_fun formula-decl nil derivatives "analysis/")
    (nzreal nonempty-type-eq-decl nil reals nil)
    (/ const-decl "[T -> real]" real_fun_ops "reals/")
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (deriv_exp_fun formula-decl nil derivatives "analysis/")
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (posint_exp application-judgement "posint" exponentiation nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (odd_minus_odd_is_even application-judgement "even_int" integers
     nil)
    (nzreal_expt application-judgement "nzreal" exponentiation nil)
    (expt def-decl "real" exponentiation nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (expt_plus formula-decl nil exponentiation nil)
    (expt_1i formula-decl nil exponentiation nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (div_expt formula-decl nil exponentiation nil)
    (nderiv_derivable_aux formula-decl nil nth_derivatives "analysis/")
    (nderiv_derivable_eqv formula-decl nil nth_derivatives "analysis/")
    (deriv_scal_fun formula-decl nil derivatives "analysis/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (^ const-decl "[T -> real]" real_fun_ops "reals/")
    (scal_derivable_fun formula-decl nil derivatives "analysis/")
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (/ const-decl "[T -> real]" real_fun_ops "reals/")
    (nz_deriv_fun type-eq-decl nil derivatives "analysis/")
    (deriv_inv_fun formula-decl nil derivatives "analysis/")
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (deriv_neg_fun formula-decl nil derivatives "analysis/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (I const-decl "(bijective?[T, T])" identity nil)
    (deriv_id_fun formula-decl nil derivatives "analysis/")
    (identity_derivable_fun formula-decl nil derivatives "analysis/")
    (ln_derivable formula-decl nil ln_exp nil)
    (extensionality formula-decl nil functions nil)
    (derivable? const-decl "bool" derivatives "analysis/")
    (deriv_fun type-eq-decl nil derivatives "analysis/")
    (deriv const-decl "[T -> real]" derivatives "analysis/")
    (deriv_domain? const-decl "bool" deriv_domain_def "analysis/")
    (not_one_element? const-decl "bool" deriv_domain_def "analysis/")
    (factorial_1 formula-decl nil factorial "ints/")
    (expt_x1 formula-decl nil exponentiation nil)
    (factorial_0 formula-decl nil factorial "ints/")
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (minus_nzint_is_nzint application-judgement "nzint" integers nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (minus_nzreal_is_nzreal application-judgement "nzreal" real_types
     nil)
    (nzreal_exp application-judgement "nzreal" exponentiation 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)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (derivable_n_times? def-decl "bool" nth_derivatives "analysis/")
    (ln const-decl "real" ln_exp nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (nderiv_fun type-eq-decl nil nth_derivatives "analysis/")
    (nderiv def-decl "[T -> real]" nth_derivatives "analysis/")
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (IF const-decl "[boolean, T, T -> T]" if_def nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (factorial def-decl "posnat" factorial "ints/")
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (^ const-decl "real" exponentiation nil))
   nil)
  (nderiv_ln-1 nil 3302280454
   (""
    (case "FORALL (n: nat): derivable_n_times[posreal](ln, n) & nderiv[posreal](n,ln) = IF n = 0 THEN ln ELSE (LAMBDA (x:posreal): -factorial(n-1)/(-x)^n) ENDIF")
    (("1" (skosimp*)
      (("1" (inst - "n!1") (("1" (flatten) nil nil)) nil)) nil)
     ("2" (hide 2)
      (("2" (induct "n")
        (("1" (expand "derivable_n_times?") (("1" (propax) nil nil))
          nil)
         ("2" (expand "nderiv") (("2" (propax) nil nil)) nil)
         ("3" (skosimp*)
          (("3" (case-replace "j!1=0")
            (("1" (lemma "ln_derivable")
              (("1" (flatten -1)
                (("1" (expand "derivable_n_times?")
                  (("1" (expand "derivable_n_times?")
                    (("1" (expand "nderiv" 1)
                      (("1" (expand "nderiv" 1)
                        (("1"
                          (lemma "extensionality"
                           ("f" "deriv(ln)" "g"
                            "(LAMBDA (x: posreal): -factorial(1) / (-x) ^ 1)"))
                          (("1" (split -1)
                            (("1" (assertnil nil)
                             ("2" (hide 2)
                              (("2"
                                (replace -2)
                                (("2"
                                  (skosimp*)
                                  (("2"
                                    (expand "factorial")
                                    (("2"
                                      (expand "factorial")
                                      (("2"
                                        (rewrite "expt_x1")
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide 2)
                            (("2" (skosimp*)
                              (("2"
                                (inst + "x!1+1")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil)
                           ("3" (skosimp*) (("3" (assertnil nil))
                            nil)
                           ("4" (propax) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (assert)
              (("2" (lemma "ln_derivable")
                (("2" (flatten -1)
                  (("2" (lemma "identity_derivable_fun[posreal]")
                    (("1"
                      (lemma "const_derivable_fun[posreal]"
                       ("b" "-factorial(j!1)"))
                      (("1" (lemma "deriv_id_fun[posreal]")
                        (("1"
                          (lemma "deriv_const_fun[posreal]"
                           ("b" "-factorial(j!1)"))
                          (("1" (expand "I")
                            (("1" (expand "const_fun")
                              (("1"
                                (lemma
                                 "neg_derivable_fun[posreal]"
                                 ("f" "LAMBDA (x: posreal): x"))
                                (("1"
                                  (assert)
                                  (("1"
                                    (expand "-")
                                    (("1"
                                      (lemma
                                       "deriv_neg_fun[posreal]"
                                       ("ff" "LAMBDA (x: posreal): x"))
                                      (("1"
                                        (replace -4)
                                        (("1"
                                          (expand "-" -1)
                                          (("1"
                                            (lemma
                                             "inv_derivable_fun[posreal]"
                                             ("g"
                                              "LAMBDA (x_1: posreal): -x_1"))
                                            (("1"
                                              (assert)
                                              (("1"
                                                (expand "/")
                                                (("1"
                                                  (lemma
                                                   "deriv_inv_fun[posreal]"
                                                   ("gg"
                                                    "LAMBDA (x_1: posreal): -x_1"))
                                                  (("1"
                                                    (replace -3)
                                                    (("1"
                                                      (expand "/")
                                                      (("1"
                                                        (expand "*")
                                                        (("1"
                                                          (expand
                                                           "-"
                                                           -1)
                                                          (("1"
                                                            (lemma
                                                             "extensionality"
                                                             ("f"
                                                              "LAMBDA (x: posreal): --1 / (-x * -x)"
                                                              "g"
                                                              "LAMBDA (x: posreal): 1 / (x * x)"))
                                                            (("1"
                                                              (split
                                                               -1)
                                                              (("1"
                                                                (replace
                                                                 -1)
                                                                (("1"
                                                                  (hide
                                                                   -1)
                                                                  (("1"
                                                                    (lemma
                                                                     "deriv_exp_fun[posreal]"
                                                                     ("f"
                                                                      "LAMBDA (x: posreal): 1 / -x"
                                                                      "n"
                                                                      "j!1"))
                                                                    (("1"
                                                                      (assert)
                                                                      (("1"
                                                                        (flatten
                                                                         -1)
                                                                        (("1"
                                                                          (lemma
                                                                           "scal_derivable_fun[posreal]"
                                                                           ("f"
                                                                            "(LAMBDA (x: posreal): 1 / -x) ^ j!1"
                                                                            "b"
                                                                            "-factorial(j!1 - 1)"))
                                                                          (("1"
                                                                            (replace
                                                                             -2)
                                                                            (("1"
                                                                              (lemma
                                                                               "deriv_scal_fun[posreal]"
                                                                               ("ff"
                                                                                "(LAMBDA (x: posreal): 1 / -x) ^ j!1"
                                                                                "b"
                                                                                "-factorial(j!1 - 1)"))
                                                                              (("1"
                                                                                (replace
                                                                                 -4
                                                                                 -1)
                                                                                (("1"
                                                                                  (replace
                                                                                   -5)
                                                                                  (("1"
                                                                                    (lemma
                                                                                     "extensionality"
                                                                                     ("f"
                                                                                      "-factorial(j!1 - 1) * (LAMBDA (x: posreal): 1 / -x) ^ j!1"
                                                                                      "g"
                                                                                      "LAMBDA (x: posreal): -factorial(j!1 - 1) / (-x) ^ j!1"))
                                                                                    (("1"
                                                                                      (split
                                                                                       -1)
                                                                                      (("1"
                                                                                        (replace
                                                                                         -1
                                                                                         *
                                                                                         rl)
                                                                                        (("1"
                                                                                          (replace
                                                                                           -17
                                                                                           *
                                                                                           rl)
                                                                                          (("1"
                                                                                            (lemma
                                                                                             "extensionality"
                                                                                             ("f"
                                                                                              "-factorial(j!1 - 1) *
                                        (j!1 * (LAMBDA (x: posreal): 1 / -x) ^ (j!1 - 1) *
                                          (LAMBDA (x: posreal): 1 / (x * x)))"
                                                                                              "g"
                                                                                              "LAMBDA (x: posreal): -factorial(j!1) / (-x) ^ (1 + j!1)"))
                                                                                            (("1"
                                                                                              (split
                                                                                               -1)
                                                                                              (("1"
                                                                                                (replace
                                                                                                 -1)
                                                                                                (("1"
                                                                                                  (lemma
                                                                                                   "nderiv_derivable_eqv[posreal]"
                                                                                                   ("f"
                                                                                                    "ln"
                                                                                                    "n"
                                                                                                    "j!1"))
                                                                                                  (("1"
                                                                                                    (replace
                                                                                                     -18
                                                                                                     -1)
                                                                                                    (("1"
                                                                                                      (replace
                                                                                                       -5
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (flatten
                                                                                                         -1)
                                                                                                        (("1"
                                                                                                          (lemma
                                                                                                           "nderiv_derivable_aux[posreal]"
                                                                                                           ("f"
                                                                                                            "ln"
                                                                                                            "n"
                                                                                                            "j!1"))
                                                                                                          (("1"
                                                                                                            (replace
                                                                                                             -2)
                                                                                                            (("1"
                                                                                                              (replace
                                                                                                               -5
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (replace
                                                                                                                 -1
                                                                                                                 2)
                                                                                                                (("1"
                                                                                                                  (propax)
                                                                                                                  nil
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.59 Sekunden  (vorverarbeitet)  ¤





Download des
Quellennavigators
Download des
sprechenden Kalenders

in der Quellcodebibliothek suchen




Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.


Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.


Bot Zugriff