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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
 (ln_estimate_TCC1 0
  (ln_estimate_TCC1-1 nil 3302450959
   ("" (skosimp*) (("" (assertnil nil)) nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (ln_estimate_TCC2 0
  (ln_estimate_TCC2-1 nil 3320163563 ("" (subtype-tcc) nil nil)
   ((/= const-decl "boolean" notequal nil)) nil))
 (ln_estimate_TCC3 0
  (ln_estimate_TCC3-1 nil 3322477326 ("" (assuming-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)
    (/= const-decl "boolean" notequal nil))
   nil))
 (ln_taylors_TCC1 0
  (ln_taylors_TCC1-1 nil 3302450959
   ("" (skosimp*) (("" (assertnil nil)) nil)
   ((real_plus_real_is_real application-judgement "real" reals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   shostak))
 (ln_taylors_TCC2 0
  (ln_taylors_TCC2-1 nil 3302450959
   ("" (skosimp*) (("" (typepred "c!1") (("" (grind) nil nil)) nil))
    nil)
   ((between type-eq-decl nil taylor_help nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (Gt_m1 type-eq-decl nil ln_series nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (< const-decl "bool" reals nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields 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)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
   shostak))
 (ln_taylors_TCC3 0
  (ln_taylors_TCC3-1 nil 3302450959
   ("" (skosimp*) (("" (assertnil nil)) nil)
   ((nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil))
   shostak))
 (ln_taylors 0
  (ln_taylors-2 nil 3302450340
   ("" (skosimp*)
    ((""
      (lemma "Taylors[posreal]"
       ("f" "ln" "n" "n!1" "bb" "xgm1!1+1" "aa" "1"))
      (("1" (lemma "nderiv_ln" ("n" "n!1+1"))
        (("1" (replace -1)
          (("1" (skosimp*)
            (("1" (inst + "c!1")
              (("1" (lemma "ln_nderiv" ("n" "n!1+1"))
                (("1" (simplify -1)
                  (("1" (replace -1 -3)
                    (("1" (simplify -3)
                      (("1" (rewrite "div_expt" 1)
                        (("1" (expand "ln_estimate" 1)
                          (("1"
                            (lemma "sigma_eq[nat]"
                             ("low" "0" "high" "n!1" "F"
                              "LAMBDA (nn:nat):
                                 IF nn > n!1 THEN 0
                                 ELSIF nn = 0 THEN ln(1)
                                 ELSE nderiv(nn, ln)(1) * xgm1!1 ^ nn / factorial(nn)
                                 ENDIF" "G" "LAMBDA (nn: nat):
                                 IF nn = 0 THEN 0 ELSE -(-xgm1!1) ^ nn / nn ENDIF"))
                            (("1" (split -1)
                              (("1"
                                (replace -1 -4)
                                (("1"
                                  (name-replace
                                   "K1"
                                   "sigma(0, n!1,
                                     LAMBDA (nn: nat):
                                       IF nn = 0 THEN 0 ELSE -(-xgm1!1) ^ nn / nn ENDIF)")
                                  (("1"
                                    (replace -4 1)
                                    (("1"
                                      (hide-all-but 1)
                                      (("1"
                                        (typepred "c!1")
                                        (("1"
                                          (hide -1 -3 -4 -5)
                                          (("1"
                                            (expand "factorial" 1 2)
                                            (("1"
                                              (name-replace
                                               "K2"
                                               "factorial(n!1)")
                                              (("1"
                                                (lemma
                                                 "div_cancel1"
                                                 ("x"
                                                  "((-1/ (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (1+n!1)"
                                                  "n0z"
                                                  "K2"))
                                                (("1"
                                                  (rewrite
                                                   "div_div2"
                                                   -1)
                                                  (("1"
                                                    (case
                                                     "K2 * (((-1 / (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (K2 + K2 * n!1)) = ((-K2 / (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (K2 + K2 * n!1)")
                                                    (("1"
                                                      (replace -1)
                                                      (("1"
                                                        (replace -2)
                                                        (("1"
                                                          (hide -1 -2)
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (hide 2)
                                                      (("2"
                                                        (name-replace
                                                         "K3"
                                                         "K2 + K2 * n!1")
                                                        (("2"
                                                          (name-replace
                                                           "K4"
                                                           "xgm1!1 ^ (1 + n!1)")
                                                          (("2"
                                                            (name-replace
                                                             "K5"
                                                             "(-c!1) ^ (1 + n!1)")
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (skosimp*)
                                  (("2"
                                    (typepred "n!2")
                                    (("2"
                                      (assert)
                                      (("2"
                                        (lift-if)
                                        (("2"
                                          (ground)
                                          (("2"
                                            (lemma "ln_nderiv")
                                            (("2"
                                              (inst?)
                                              (("2"
                                                (replace -1)
                                                (("2"
                                                  (hide -1)
                                                  (("2"
                                                    (assert)
                                                    (("2"
                                                      (simplify 2)
                                                      (("2"
                                                        (lemma
                                                         "cross_mult"
                                                         ("x"
                                                          "-factorial(n!2 - 1) / (-1) ^ n!2 * xgm1!1 ^ n!2"
                                                          "n0x"
                                                          "factorial(n!2)"
                                                          "y"
                                                          "-(-xgm1!1) ^ n!2"
                                                          "n0y"
                                                          "n!2"))
                                                        (("2"
                                                          (replace
                                                           -1
                                                           2)
                                                          (("2"
                                                            (expand
                                                             "factorial"
                                                             2
                                                             2)
                                                            (("2"
                                                              (case-replace
                                                               "xgm1!1=0")
                                                              (("1"
                                                                (expand
                                                                 "^"
                                                                 2
                                                                 3)
                                                                (("1"
                                                                  (expand
                                                                   "^"
                                                                   2
                                                                   1)
                                                                  (("1"
                                                                    (expand
                                                                     "expt")
                                                                    (("1"
                                                                      (assert)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil)
                                                               ("2"
                                                                (hide
                                                                 -1)
                                                                (("2"
                                                                  (field
                                                                   3)
                                                                  (("2"
                                                                    (flatten)
                                                                    (("2"
                                                                      (expand
                                                                       "^")
                                                                      (("2"
                                                                        (lemma
                                                                         "expt_of_mult")
                                                                        (("2"
                                                                          (inst
                                                                           -
                                                                           "n!2"
                                                                           "-xgm1!1"
                                                                           "-1")
                                                                          (("2"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide-all-but 1)
                              (("2" (skosimp*) nil nil)) nil)
                             ("3" (skosimp*)
                              (("3"
                                (lemma "nderiv_ln")
                                (("3" (inst?) nil nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (assert)
                          (("2" (flatten)
                            (("2" (replace -1)
                              (("2"
                                (rewrite "ln_1")
                                (("2"
                                  (expand "^" 1)
                                  (("2"
                                    (expand "expt" 1)
                                    (("2"
                                      (assert)
                                      (("2"
                                        (rewrite "zero_times1")
                                        (("2"
                                          (simplify 1)
                                          (("2"
                                            (expand "ln_estimate")
                                            (("2"
                                              (hide-all-but 1)
                                              (("2"
                                                (lemma
                                                 "sigma_zero[nat]"
                                                 ("low"
                                                  "0"
                                                  "high"
                                                  "n!1"))
                                                (("2"
                                                  (lemma
                                                   "sigma_eq[nat]"
                                                   ("low"
                                                    "0"
                                                    "high"
                                                    "n!1"
                                                    "F"
                                                    "LAMBDA (i: nat): 0"
                                                    "G"
                                                    "LAMBDA (nn: nat): IF nn = 0 THEN 0 ELSE -(-0) ^ nn / nn ENDIF"))
                                                  (("1"
                                                    (split -1)
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (hide -1 2)
                                                      (("2"
                                                        (skosimp*)
                                                        (("2"
                                                          (case
                                                           "n!2=0")
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil)
                                                           ("2"
                                                            (assert)
                                                            (("2"
                                                              (expand
                                                               "^")
                                                              (("2"
                                                                (expand
                                                                 "expt")
                                                                (("2"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (skosimp*)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide -1 -2)
                (("2" (typepred "xgm1!1")
                  (("2" (typepred "c!1")
                    (("2" (replace -3)
                      (("2" (replace -4) (("2" (propax) nil nil)) nil))
                      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" (hide 2) (("3" (skosimp*) (("3" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((ln const-decl "real" ln_exp nil)
    (sigma_zero formula-decl nil sigma "reals/")
    (sigma def-decl "real" sigma "reals/")
    (sigma_eq formula-decl nil sigma "reals/")
    (ln_1 formula-decl nil ln_exp nil))
   nil)
  (ln_taylors-1 nil 3302449853
   ("" (skosimp*)
    ((""
      (lemma "Taylors[posreal]"
       ("f" "ln" "n" "n!1" "bb" "xgm1!1+1" "aa" "1"))
      (("1" (lemma "ln_derivable_n_times" ("n" "n!1+1"))
        (("1" (replace -1)
          (("1" (skosimp*)
            (("1" (inst + "c!1")
              (("1" (lemma "ln_nderiv" ("n" "n!1+1"))
                (("1" (simplify -1)
                  (("1" (replace -1 -3)
                    (("1" (simplify -3)
                      (("1" (rewrite "div_expt" 1)
                        (("1" (expand "ln_series_n" 1)
                          (("1"
                            (lemma "sigma_eq[nat]"
                             ("low" "0" "high" "n!1" "F"
                              "LAMBDA (nn:nat):
                        IF nn > n!1 THEN 0
                        ELSIF nn = 0 THEN ln(1)
                        ELSE nderiv(nn, ln)(1) * xgm1!1 ^ nn / factorial(nn)
                        ENDIF" "G" "LAMBDA (nn: nat):
                        IF nn = 0 THEN 0 ELSE -(-xgm1!1) ^ nn / nn ENDIF"))
                            (("1" (split -1)
                              (("1"
                                (replace -1 -4)
                                (("1"
                                  (name-replace
                                   "K1"
                                   "sigma(0, n!1,
                         LAMBDA (nn: nat):
                           IF nn = 0 THEN 0 ELSE -(-xgm1!1) ^ nn / nn ENDIF)")
                                  (("1"
                                    (replace -4 1)
                                    (("1"
                                      (hide-all-but 1)
                                      (("1"
                                        (typepred "c!1")
                                        (("1"
                                          (hide -1 -3 -4 -5)
                                          (("1"
                                            (expand "factorial" 1 2)
                                            (("1"
                                              (name-replace
                                               "K2"
                                               "factorial(n!1)")
                                              (("1"
                                                (lemma
                                                 "div_cancel1"
                                                 ("x"
                                                  "((-1/ (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (1+n!1)"
                                                  "n0z"
                                                  "K2"))
                                                (("1"
                                                  (rewrite
                                                   "div_div2"
                                                   -1)
                                                  (("1"
                                                    (case
                                                     "K2 * (((-1 / (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (K2 + K2 * n!1)) = ((-K2 / (-c!1) ^ (1 + n!1)) * xgm1!1 ^ (1 + n!1)) / (K2 + K2 * n!1)")
                                                    (("1"
                                                      (replace -1)
                                                      (("1"
                                                        (replace -2)
                                                        (("1"
                                                          (hide -1 -2)
                                                          (("1"
                                                            (assert)
                                                            nil)))))))
                                                     ("2"
                                                      (hide 2)
                                                      (("2"
                                                        (name-replace
                                                         "K3"
                                                         "K2 + K2 * n!1")
                                                        (("2"
                                                          (name-replace
                                                           "K4"
                                                           "xgm1!1 ^ (1 + n!1)")
                                                          (("2"
                                                            (name-replace
                                                             "K5"
                                                             "(-c!1) ^ (1 + n!1)")
                                                            (("2"
                                                              (assert)
                                                              nil)))))))))))))))))))))))))))))))
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (skosimp*)
                                  (("2"
                                    (typepred "n!2")
                                    (("2"
                                      (assert)
                                      (("2"
                                        (rewrite "ln_1")
                                        (("2"
                                          (case "n!2=0")
                                          (("1" (assertnil)
                                           ("2"
                                            (assert)
                                            (("2"
                                              (lemma
                                               "ln_nderiv"
                                               ("n" "n!2"))
                                              (("2"
                                                (assert)
                                                (("2"
                                                  (replace -1 2)
                                                  (("2"
                                                    (simplify 2)
                                                    (("2"
                                                      (hide -1)
                                                      (("2"
                                                        (lemma
                                                         "cross_mult"
                                                         ("x"
                                                          "-factorial(n!2 - 1) / (-1) ^ n!2 * xgm1!1 ^ n!2"
                                                          "n0x"
                                                          "factorial(n!2)"
                                                          "y"
                                                          "-(-xgm1!1) ^ n!2"
                                                          "n0y"
                                                          "n!2"))
                                                        (("2"
                                                          (replace
                                                           -1
                                                           2)
                                                          (("2"
                                                            (expand
                                                             "factorial"
                                                             2
                                                             2)
                                                            (("2"
                                                              (case-replace
                                                               "xgm1!1=0")
                                                              (("1"
                                                                (expand
                                                                 "^"
                                                                 2
                                                                 3)
                                                                (("1"
                                                                  (expand
                                                                   "^"
                                                                   2
                                                                   1)
                                                                  (("1"
                                                                    (expand
                                                                     "expt")
                                                                    (("1"
                                                                      (assert)
                                                                      nil)))))))
                                                               ("2"
                                                                (lemma
                                                                 "div_expt"
                                                                 ("n0x"
                                                                  "xgm1!1"
                                                                  "n0y"
                                                                  "-1"
                                                                  "i"
                                                                  "n!2"))
                                                                (("1"
                                                                  (assert)
                                                                  nil)
                                                                 ("2"
                                                                  (assert)
                                                                  nil)))))))))))))))))))))))))))))))))))))
                             ("2" (hide-all-but 1)
                              (("2"
                                (skosimp*)
                                (("2"
                                  (assert)
                                  (("2"
                                    (lemma
                                     "ln_derivable_n_times"
                                     ("n" "nn!1"))
                                    (("2" (propax) nil)))))))))))))
                         ("2" (expand "/=" 1)
                          (("2" (replace -1)
                            (("2" (rewrite "ln_1")
                              (("2"
                                (expand "^" 1)
                                (("2"
                                  (expand "expt" 1)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (rewrite "zero_times1")
                                      (("2"
                                        (simplify 1)
                                        (("2"
                                          (expand "ln_series_n")
                                          (("2"
                                            (hide-all-but 1)
                                            (("2"
                                              (lemma
                                               "sigma_zero[nat]"
                                               ("low"
                                                "0"
                                                "high"
                                                "n!1"))
                                              (("2"
                                                (lemma
                                                 "sigma_eq[nat]"
                                                 ("low"
                                                  "0"
                                                  "high"
                                                  "n!1"
                                                  "F"
                                                  "LAMBDA (i: nat): 0"
                                                  "G"
                                                  "LAMBDA (nn: nat): IF nn = 0 THEN 0 ELSE -(-0) ^ nn / nn ENDIF"))
                                                (("2"
                                                  (split -1)
                                                  (("1" (assertnil)
                                                   ("2"
                                                    (hide -1 2)
                                                    (("2"
                                                      (skosimp*)
                                                      (("2"
                                                        (case "n!2=0")
                                                        (("1"
                                                          (assert)
                                                          nil)
                                                         ("2"
                                                          (assert)
                                                          (("2"
                                                            (expand
                                                             "^")
                                                            (("2"
                                                              (expand
                                                               "expt")
                                                              (("2"
                                                                (assert)
                                                                nil)))))))))))))))))))))))))))))))))))))))))))))))))
               ("2" (hide -1 -2)
                (("2" (typepred "xgm1!1")
                  (("2" (typepred "c!1")
                    (("2" (replace -3)
                      (("2" (replace -4)
                        (("2" (propax) nil)))))))))))))))))))
       ("2" (hide 2)
        (("2" (skosimp*)
          (("2" (inst + "x!1+1") (("2" (assertnil)))))))
       ("3" (hide 2) (("3" (skosimp*) (("3" (assertnil))))))))
    nil)
   nil nil))
 (lnp1_TCC1 0
  (lnp1_TCC1-1 nil 3321095836 ("" (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) (>= 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)
    (abslt1 type-eq-decl nil ln_series nil)
    (real_lt_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_plus_real_is_real application-judgement "real" reals nil))
   nil)))


¤ Dauer der Verarbeitung: 0.17 Sekunden  (vorverarbeitet)  ¤





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