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Datei: update-require   Sprache: Lisp

Original von: PVS©

(atan_approx
 (atan_pos_le1_ub_lt 0
  (atan_pos_le1_ub_lt-1 nil 3408809781
   ("" (skeep)
    (("" (expand "atan_pos_le1_ub")
      (("" (expand "atan_series_n")
        (("" (expand "sigma" 1 1)
          (("" (expand "sigma" 1 1)
            (("" (move-terms 1 r 1)
              (("1" (assert)
                (("1" (expand "atan_series_term")
                  (("1" (expand "atan_series_coef")
                    (("1" (lemma "not_even_m1_pow")
                      (("1" (inst?)
                        (("1" (split -1)
                          (("1" (replaces -1)
                            (("1" (lemma "even_m1_pow")
                              (("1"
                                (inst?)
                                (("1"
                                  (split -1)
                                  (("1"
                                    (replaces -1)
                                    (("1"
                                      (case
                                       "(x ^ (5 + 4 * n)) * (1 / (5 + 4 * n)) < (x ^ (5 + 4 * n)) / (3 + 4 * n)")
                                      (("1"
                                        (case
                                         "(x ^ (5 + 4 * n)) / (3 + 4 * n) <= (x ^ (3 + 4 * n)) / (3 + 4 * n)")
                                        (("1" (assertnil nil)
                                         ("2"
                                          (hide -1 2)
                                          (("2"
                                            (grind-reals)
                                            (("2"
                                              (rewrite
                                               "both_sides_expt_lt1_le")
                                              (("2"
                                                (case "x=1")
                                                (("1"
                                                  (replaces -1)
                                                  (("1"
                                                    (grind)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide 2)
                                        (("2"
                                          (case "x ^ (5+4 *n) > 0")
                                          (("1"
                                            (name-replace
                                             "aa"
                                             "x^(5+4*n)")
                                            (("1"
                                              (grind-reals)
                                              nil
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (rewrite "expt_pos")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but 1)
                                    (("2" (grind) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide-all-but -1)
                            (("2" (grind) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (skosimp*) (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((atan_pos_le1_ub const-decl "real" atan_approx nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (sigma def-decl "real" sigma "reals/")
    (integer nonempty-type-from-decl nil integers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (IFF const-decl "[bool, 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)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (atan_series_term const-decl "[nat -> real]" atan nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (not_even_m1_pow formula-decl nil exponentiation nil)
    (even_m1_pow formula-decl nil exponentiation nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (/= const-decl "boolean" notequal nil)
    (^ const-decl "real" exponentiation nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (both_sides_expt_lt1_le formula-decl nil exponentiation nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (odd_minus_odd_is_even application-judgement "even_int" integers
     nil)
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (le_times_le_pos formula-decl nil real_props nil)
    (expt def-decl "real" exponentiation nil)
    (nonzero_times1 formula-decl nil real_props nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (both_sides_div_pos_le1 formula-decl nil real_props nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (times_div1 formula-decl nil real_props nil)
    (lt_div_lt_pos1 formula-decl nil real_props nil)
    (expt_pos formula-decl nil exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (even? const-decl "bool" integers nil)
    (atan_series_coef const-decl "rat" atan nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (int_exp application-judgement "int" exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (atan_series_n const-decl "real" atan nil))
   nil))
 (atan_pos_le1_lb_lt 0
  (atan_pos_le1_lb_lt-1 nil 3408809815
   ("" (skeep)
    (("" (expand "atan_pos_le1_lb")
      (("" (expand "atan_series_n")
        (("" (expand "sigma" 1 2)
          (("" (expand "sigma" 1 2)
            (("" (move-terms 1 r 3)
              (("1" (assert)
                (("1" (expand "atan_series_term")
                  (("1" (expand "atan_series_coef")
                    (("1" (lemma "not_even_m1_pow")
                      (("1" (inst -1 "3+2*n")
                        (("1" (split -1)
                          (("1" (replaces -1)
                            (("1" (lemma "even_m1_pow")
                              (("1"
                                (inst?)
                                (("1"
                                  (split -1)
                                  (("1"
                                    (replaces -1)
                                    (("1"
                                      (case
                                       "(x ^ (5 + 4 * n))/(7 + 4 * n) < (x ^ (5 + 4 * n)) * (1 / (5 + 4 * n))")
                                      (("1"
                                        (case
                                         "(x ^ (7 + 4 * n))/(7+4*n) <= (x ^ (5 + 4 * n))/(7+4*n)")
                                        (("1" (assertnil nil)
                                         ("2"
                                          (hide -1 2)
                                          (("2"
                                            (grind-reals)
                                            (("2"
                                              (rewrite
                                               "both_sides_expt_lt1_le")
                                              (("2"
                                                (case-replace "x=1")
                                                (("1" (grind) nil nil)
                                                 ("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide 2)
                                        (("2"
                                          (case "x ^ (5+4*n) > 0")
                                          (("1"
                                            (name-replace
                                             "aa"
                                             "x^(5+4*n)")
                                            (("1"
                                              (grind-reals)
                                              nil
                                              nil))
                                            nil)
                                           ("2"
                                            (hide 2)
                                            (("2"
                                              (rewrite "expt_pos")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide-all-but 1)
                                    (("2" (grind) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide-all-but -1)
                            (("2" (grind) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (skosimp*) (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((atan_pos_le1_lb const-decl "real" atan_approx nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (sigma def-decl "real" sigma "reals/")
    (integer nonempty-type-from-decl nil integers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (IFF const-decl "[bool, 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)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (<= const-decl "bool" reals nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (atan_series_term const-decl "[nat -> real]" atan nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (not_even_m1_pow formula-decl nil exponentiation nil)
    (even_m1_pow formula-decl nil exponentiation nil)
    (real_div_nzreal_is_real application-judgement "real" reals 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 "real" exponentiation nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (both_sides_expt_lt1_le formula-decl nil exponentiation nil)
    (nonzero_times1 formula-decl nil real_props nil)
    (expt def-decl "real" exponentiation nil)
    (le_times_le_pos formula-decl nil real_props nil)
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (odd_minus_odd_is_even application-judgement "even_int" integers
     nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (both_sides_div_pos_le1 formula-decl nil real_props nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (times_div1 formula-decl nil real_props nil)
    (lt_div_lt_pos1 formula-decl nil real_props nil)
    (expt_pos formula-decl nil exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (even? const-decl "bool" integers nil)
    (atan_series_coef const-decl "rat" atan nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (int_exp application-judgement "int" exponentiation nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (atan_series_n const-decl "real" atan nil))
   nil))
 (atan_pos_le1_bounds 0
  (atan_pos_le1_bounds-1 nil 3295292840
   ("" (skosimp)
    (("" (expand "atan_pos_le1_lb")
      (("" (expand "atan_pos_le1_ub")
        (("" (case "FORALL (n:posnat): 0 < x!1^n/n")
          (("1"
            (case "FORALL (n,m:posnat): n < m => x!1^m/m < x!1^n/n")
            (("1"
              (case-replace
               "atan_series_n(x!1, 1 + 2 * n!1) < atan(x!1)")
              (("1" (lemma "atan_series" ("x" "x!1" "n" "1+2*n!1"))
                (("1" (rewrite "abs_expt" -1 :dir rl)
                  (("1" (expand "abs" -1)
                    (("1" (assert)
                      (("1" (expand "atan_series_n")
                        (("1" (expand "sigma" -1)
                          (("1"
                            (name-replace "UB"
                             "sigma(0, 2 * n!1, atan_series_term(x!1))")
                            (("1" (hide -2)
                              (("1"
                                (expand "atan_series_term")
                                (("1"
                                  (expand "atan_series_coef")
                                  (("1"
                                    (lemma
                                     "not_even_m1_pow"
                                     ("i" "1+2*n!1"))
                                    (("1"
                                      (split -1)
                                      (("1"
                                        (replace -1)
                                        (("1"
                                          (inst - "3+4*n!1" "5+4*n!1")
                                          (("1" (assertnil nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide-all-but -1)
                                        (("2"
                                          (expand "even?")
                                          (("2" (propax) nil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (lemma "atan_series" ("x" "x!1" "n" "2+2*n!1"))
                  (("2" (rewrite "abs_expt" -1 :dir rl)
                    (("2" (expand "abs")
                      (("2" (assert)
                        (("2" (expand "atan_series_n")
                          (("2" (expand "sigma" -1)
                            (("2"
                              (name-replace "LB"
                               "sigma(0, 1 + 2 * n!1, atan_series_term(x!1))")
                              (("2"
                                (expand "atan_series_term")
                                (("2"
                                  (expand "atan_series_coef")
                                  (("2"
                                    (lemma
                                     "even_m1_pow"
                                     ("i" "2+2*n!1"))
                                    (("2"
                                      (split -1)
                                      (("1"
                                        (replace -1)
                                        (("1"
                                          (hide -1)
                                          (("1"
                                            (inst
                                             -
                                             "5 + 4 * n!1"
                                             "7 + 4 * n!1")
                                            (("1"
                                              (assert)
                                              (("1"
                                                (inst-cp
                                                 -
                                                 "7 + 4 * n!1")
                                                (("1"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (expand "even?")
                                        (("2"
                                          (inst + "n!1+1")
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide 2)
              (("2" (skosimp)
                (("2" (case-replace "x!1=1")
                  (("1" (rewrite "expt_1i")
                    (("1" (rewrite "expt_1i")
                      (("1" (lemma "both_sides_div_pos_lt2")
                        (("1" (inst - "m!1" "n!2" "1")
                          (("1" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil)
                   ("2"
                    (lemma "both_sides_expt_lt1_lt"
                     ("lt1x" "x!1" "i" "m!1" "j" "n!2"))
                    (("1" (assert)
                      (("1"
                        (lemma "lt_div_lt_pos1"
                         ("px" "x!1 ^ m!1" "y" "x!1 ^ n!2" "pz" "n!2"
                          "w" "m!1"))
                        (("1" (assertnil nil)
                         ("2" (lemma "expt_pos" ("px" "x!1" "i" "m!1"))
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil)
                     ("2" (assertnil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide 2)
            (("2" (skosimp)
              (("2" (lemma "expt_pos" ("px" "x!1" "i" "n!2"))
                (("1" (rewrite "div_mult_pos_lt2")
                  (("1" (assertnil nil)) nil)
                 ("2" (assertnil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((atan const-decl "real_abs_lt_pi2" atan nil)
    (atan_series_n const-decl "real" atan nil)
    (sigma def-decl "real" sigma "reals/")
    (atan_series_coef const-decl "rat" atan nil)
    (atan_series_term const-decl "[nat -> real]" atan nil)
    (atan_series formula-decl nil atan nil))
   shostak))
 (atan_pos_le1_lb_inc 0
  (atan_pos_le1_lb_inc-1 nil 3408809885
   (""
    (case "FORALL (n: nat, px: posreal):
                      px <= 1 => atan_pos_le1_lb(n, px) < atan_pos_le1_lb(n+1, px)")
    (("1"
      (case "FORALL (n: nat,pn:posnat, px: posreal):
                          px <= 1 => atan_pos_le1_lb(n, px) < atan_pos_le1_lb(n+pn, px)")
      (("1" (skosimp)
        (("1" (inst - "n!1" "m!1-n!1" "px!1")
          (("1" (assertnil nil) ("2" (assertnil nil)) nil))
        nil)
       ("2" (hide 2)
        (("2" (induct "pn")
          (("1" (assertnil nil) ("2" (assertnil nil)
           ("3" (skosimp*)
            (("3" (case-replace "j!1=0")
              (("1" (assert)
                (("1" (hide -1 -2 -3)
                  (("1" (inst - "n!1" "px!1") (("1" (assertnil nil))
                    nil))
                  nil))
                nil)
               ("2" (assert)
                (("2" (inst - "n!1" "px!1")
                  (("2" (assert)
                    (("2" (inst - "j!1+n!1" "px!1")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2" (skosimp)
        (("2" (expand "atan_pos_le1_lb")
          (("2" (expand "atan_series_n")
            (("2" (expand "sigma" 1 2)
              (("2" (expand "sigma" 1 2)
                (("2"
                  (name-replace "SUM"
                   "sigma(0, 1 + 2 * n!1, atan_series_term(px!1))")
                  (("1" (expand "atan_series_term")
                    (("1" (expand "atan_series_coef")
                      (("1" (expand "^")
                        (("1" (expand "expt" 1 3)
                          (("1" (expand "expt" 1 3)
                            (("1" (expand "expt" 1 4)
                              (("1"
                                (case-replace
                                 "expt((-1), 2 + 2 * n!1)=1")
                                (("1"
                                  (typepred "expt(px!1, 5 + 4 * n!1)")
                                  (("1"
                                    (hide -1 -3)
                                    (("1"
                                      (name-replace
                                       "PX"
                                       "expt(px!1, 5 + 4 * n!1)")
                                      (("1"
                                        (lemma
                                         "posreal_times_posreal_is_posreal"
                                         ("px"
                                          "PX"
                                          "py"
                                          "1 / (5 + 4 * n!1)-sq(px!1)/(7 + 4 * n!1)"))
                                        (("1"
                                          (expand "sq")
                                          (("1" (assertnil nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (lemma
                                             "div_mult_pos_lt1"
                                             ("py"
                                              "7+4*n!1"
                                              "z"
                                              "sq(px!1)"
                                              "x"
                                              "1/(5+4*n!1)"))
                                            (("2"
                                              (lemma
                                               "sq_nz_pos"
                                               ("nz" "px!1"))
                                              (("2"
                                                (lemma
                                                 "div_mult_pos_lt2"
                                                 ("py"
                                                  "5 + 4 * n!1"
                                                  "z"
                                                  "7 + 4 * n!1"
                                                  "x"
                                                  "1"))
                                                (("2"
                                                  (lemma
                                                   "sq_le"
                                                   ("nna"
                                                    "px!1"
                                                    "nnb"
                                                    "1"))
                                                  (("2"
                                                    (assert)
                                                    (("2"
                                                      (name-replace
                                                       "LHS"
                                                       "sq(px!1) / (7 + 4 * n!1)")
                                                      (("2"
                                                        (name-replace
                                                         "RHS"
                                                         "1 / (5 + 4 * n!1)")
                                                        (("2"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide-all-but 1)
                                  (("2"
                                    (lemma
                                     "expt_times"
                                     ("n0x" "-1" "i" "2" "j" "1+n!1"))
                                    (("2"
                                      (expand "^" -1 1)
                                      (("2"
                                        (expand "^" -1 1)
                                        (("2"
                                          (replace -1)
                                          (("2"
                                            (hide -1)
                                            (("2"
                                              (expand "expt")
                                              (("2"
                                                (expand "expt")
                                                (("2"
                                                  (expand "expt")
                                                  (("2"
                                                    (rewrite "sq_rew")
                                                    (("2"
                                                      (rewrite
                                                       "sq_neg")
                                                      (("2"
                                                        (rewrite
                                                         "expt_1i")
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (skosimp*) (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (sigma def-decl "real" sigma "reals/")
    (integer nonempty-type-from-decl nil integers nil)
    (atan_series_term const-decl "[nat -> real]" atan nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (int_exp application-judgement "int" exponentiation nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (atan_series_coef const-decl "rat" atan nil)
    (nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (expt def-decl "real" exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (expt_1i formula-decl nil exponentiation nil)
    (expt_times formula-decl nil exponentiation nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (odd_times_odd_is_odd application-judgement "odd_int" integers nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (sq_nz_pos judgement-tcc nil sq "reals/")
    (nzreal nonempty-type-eq-decl nil reals nil)
    (sq_le formula-decl nil sq "reals/")
    (nonneg_rat nonempty-type-eq-decl nil rationals nil)
    (posrat nonempty-type-eq-decl nil rationals nil)
    (sq_1 formula-decl nil sq "reals/")
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (div_mult_pos_lt2 formula-decl nil real_props nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (nprat_div_posrat_is_nprat application-judgement "nprat" rationals
     nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (posreal_times_posreal_is_posreal judgement-tcc nil real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (^ const-decl "real" exponentiation nil)
    (posreal_expt application-judgement "posreal" exponentiation nil)
    (nzreal_expt application-judgement "nzreal" exponentiation nil)
    (int_expt application-judgement "int" exponentiation nil)
    (posreal_exp application-judgement "posreal" exponentiation nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (atan_series_n const-decl "real" atan nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (m!1 skolem-const-decl "nat" atan_approx nil)
    (n!1 skolem-const-decl "nat" atan_approx nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil) (< const-decl "bool" reals nil)
    (atan_pos_le1_lb const-decl "real" atan_approx nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields
       nil))
   nil))
 (atan_pos_le1_ub_dec 0
  (atan_pos_le1_ub_dec-1 nil 3408809912
   (""
    (case "FORALL (n: nat, px: posreal):
                                                  px <= 1 => atan_pos_le1_ub(n, px) > atan_pos_le1_ub(n+1, px)")
    (("1"
      (case "FORALL (n: nat,pn:posnat, px: posreal):
                                                              px <= 1 => atan_pos_le1_ub(n, px) > atan_pos_le1_ub(n+pn, px)")
      (("1" (skosimp)
        (("1" (inst - "n!1" "m!1-n!1" "px!1")
          (("1" (assertnil nil) ("2" (assertnil nil)) nil))
        nil)
       ("2" (hide 2)
        (("2" (induct "pn")
          (("1" (assertnil nil) ("2" (assertnil nil)
           ("3" (skosimp*)
            (("3" (case-replace "j!1=0")
              (("1" (assert)
                (("1" (hide -1 -2 -3)
                  (("1" (inst - "n!1" "px!1") (("1" (assertnil nil))
                    nil))
                  nil))
                nil)
               ("2" (assert)
                (("2" (inst - "n!1" "px!1")
                  (("2" (assert)
                    (("2" (inst - "j!1+n!1" "px!1")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2" (skosimp)
        (("2" (expand "atan_pos_le1_ub")
          (("2" (expand "atan_series_n")
            (("2" (expand "sigma" 1 2)
              (("2" (expand "sigma" 1 2)
                (("2"
                  (name-replace "SUM"
                   "sigma(0, 2 * n!1, atan_series_term(px!1))")
                  (("1" (move-terms 1 l)
                    (("1" (assert)
                      (("1" (expand "atan_series_term")
                        (("1" (expand "atan_series_coef")
                          (("1" (expand "^")
                            (("1" (expand "expt" 1 3)
                              (("1"
                                (expand "expt" 1 3)
                                (("1"
                                  (expand "expt" 1 4)
                                  (("1"
                                    (case-replace
                                     "expt((-1), 1 + 2 * n!1)= -1")
                                    (("1"
                                      (hide -1)
                                      (("1"
                                        (cancel-by
                                         1
                                         "expt(px!1, 3 + 4 * n!1)")
                                        (("1"
                                          (hide -1 2)
                                          (("1"
                                            (case
                                             "5 > 3 * (px!1 * px!1)")
                                            (("1"
                                              (case
                                               "4 * n!1 >= 4 * (n!1 * px!1 * px!1)")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (grind-reals)
                                                  nil
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide -1 2)
                                                (("2"
                                                  (cancel-by 1 "n!1")
                                                  (("2"
                                                    (cancel-by 2 "4")
                                                    (("2"
                                                      (rewrite
                                                       "sq"
                                                       :dir
                                                       rl)
                                                      (("2"
                                                        (lemma "sq_le")
                                                        (("2"
                                                          (inst?)
                                                          (("2"
                                                            (rewrite
                                                             "sq_1")
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (hide 2)
                                              (("2"
                                                (rewrite "sq" :dir rl)
                                                (("2"
                                                  (lemma "sq_le")
                                                  (("2"
                                                    (inst?)
                                                    (("2"
                                                      (assert)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide-all-but 1)
                                      (("2"
                                        (lemma "not_even_m1_pow")
                                        (("2"
                                          (inst -1 "1+2*n!1")
                                          (("2"
                                            (split -1)
                                            (("1"
                                              (expand "^" -1 1)
                                              (("1" (propax) nil nil))
                                              nil)
                                             ("2"
                                              (hide 1)
                                              (("2" (grind) nil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (skosimp*) (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (sigma def-decl "real" sigma "reals/")
    (integer nonempty-type-from-decl nil integers nil)
    (atan_series_term const-decl "[nat -> real]" atan nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (int_exp application-judgement "int" exponentiation nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (atan_series_coef const-decl "rat" atan nil)
    (nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (expt def-decl "real" exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (even? const-decl "bool" integers nil)
    (not_even_m1_pow formula-decl nil exponentiation nil)
    (ge_times_ge_pos formula-decl nil real_props nil)
    (minus_div1 formula-decl nil real_props nil)
    (neg_div_gt formula-decl nil real_props nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (CBD_2 skolem-const-decl "nat" atan_approx nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (pos_div_gt formula-decl nil real_props nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (zero_times1 formula-decl nil real_props nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (sq_1 formula-decl nil sq "reals/")
    (sq_le formula-decl nil sq "reals/")
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (posint nonempty-type-eq-decl nil integers nil)
    (add_neg formula-decl nil extra_tegies nil)
    (neg_div formula-decl nil extra_tegies nil)
    (zero_div formula-decl nil extra_tegies nil)
    (minus_nzrat_is_nzrat application-judgement "nzrat" rationals nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (both_sides_times_pos_le1_imp formula-decl nil extra_real_props
     nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (bijective? const-decl "bool" functions nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (TRUE const-decl "bool" booleans nil)
    (odd_times_odd_is_odd application-judgement "odd_int" integers nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (^ const-decl "real" exponentiation nil)
    (posreal_expt application-judgement "posreal" exponentiation nil)
    (nzreal_expt application-judgement "nzreal" exponentiation nil)
    (int_expt application-judgement "int" exponentiation nil)
    (posreal_exp application-judgement "posreal" exponentiation nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (atan_series_n const-decl "real" atan nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (m!1 skolem-const-decl "nat" atan_approx nil)
    (n!1 skolem-const-decl "nat" atan_approx nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (even_plus_odd_is_odd application-judgement "odd_int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (atan_pos_le1_ub const-decl "real" atan_approx nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields
       nil))
   nil))
 (pi_lbn_lt 0
  (pi_lbn_lt-1 nil 3408810052
   ("" (skeep)
    (("" (expand "pi_lbn" :assert? none)
      (("" (div-by 1 "4")
        (("" (lemma "atan_pos_le1_lb_lt")
          (("" (inst?)
            (("" (lemma "atan_pos_le1_ub_lt")
              (("" (inst?) (("" (assertnil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((pi_lbn const-decl "real" atan_approx nil)
    (atan_pos_le1_lb_lt formula-decl nil atan_approx nil)
    (atan_pos_le1_ub_lt formula-decl nil atan_approx 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (times_div_cancel1 formula-decl nil extra_real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals 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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields 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)
    (atan_pos_le1_lb const-decl "real" atan_approx nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (atan_pos_le1_ub const-decl "real" atan_approx nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (both_sides_div_pos_lt1 formula-decl nil real_props nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   nil))
 (pi_lbn_LT 0
  (pi_lbn_LT-1 nil 3408810072
   ("" (skeep)
    (("" (case "forall(i,n:nat):pi_lbn(n) < pi_lbn(n+1+i)")
      (("1" (inst -1 "k-n-1" "n") (("1" (assertnil nil)) nil)
       ("2" (hide 2)
        (("2" (induct "i")
          (("1" (skosimp)
            (("1" (assert) (("1" (rewrite "pi_lbn_lt"nil nil)) nil))
            nil)
           ("2" (skeep)
            (("2" (skosimp)
              (("2" (inst -1 "n!1+1")
                (("2" (lemma "pi_lbn_lt")
                  (("2" (inst -1 "n!1") (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (pi_lbn const-decl "real" atan_approx nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (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)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (> const-decl "bool" reals nil)
    (above nonempty-type-eq-decl nil integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (pred type-eq-decl nil defined_types nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (pi_lbn_lt formula-decl nil atan_approx nil))
   nil))
 (pi_bounds 0
  (pi_bounds-3 "closer bounds" 3307959333
   ("" (skosimp)
    (("" (expand "pi_lbn")
      (("" (expand "pi_ubn")
        (("" (lemma "atan_1")
          (("" (lemma "atan_pos_le1_bounds" ("x" "1/5" "n" "n!1"))
            (("" (lemma "atan_pos_le1_bounds" ("x" "1/239" "n" "n!1"))
              (("" (assert)
                (("" (flatten)
                  (("" (expand "pi") (("" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   nil shostak)
  (pi_bounds-2 nil 3307959315
   ("" (skosimp)
    (("" (expand "pi_lbn")
      (("" (expand "pi_ubn")
        (("" (lemma "atan_1")
          (("" (lemma "atan_pos_le1_bounds" ("x" "1/5" "n" "n!1"))
            (("" (lemma "atan_pos_le1_bounds" ("x" "1/239" "n" "n!1"))
              (("" (assert)
                (("" (flatten)
                  (("" (expand "pi")
                    (("" (assertnil))))))))))))))))))
    nil)
   ((atan_1 formula-decl nil atan nil)) nil)
  (pi_bounds-1 nil 3295296628
   ("" (skosimp)
    (("" (expand "pi_lb")
      (("" (expand "pi_ub")
        (("" (lemma "atan_1")
          (("" (lemma "atan_pos_le1_bounds" ("x" "1/5" "n" "n!1"))
            (("" (lemma "atan_pos_le1_bounds" ("x" "1/239" "n" "n!1"))
              (("" (assert)
                (("" (flatten)
                  (("" (expand "pi") (("" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((atan_1 formula-decl nil atan nil)) shostak))
 (pi_lb_pos 0
  (pi_lb_pos-2 nil 3408810176
   ("" (skeep)
    (("" (case "pi_lbn(n) > 0")
      (("1" (assertnil nil)
       ("2" (hide 2)
        (("2" (case "pi_lbn(0) > 0")
          (("1" (lemma "pi_lbn_LT")
            (("1" (inst -1 "0" "n")
              (("1" (assertnil nil) ("2" (assertnil nil)) nil))
            nil)
           ("2" (hide 2) (("2" (grind) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((pi_lbn const-decl "real" atan_approx nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (> const-decl "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_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (n skolem-const-decl "nat" atan_approx nil)
    (above nonempty-type-eq-decl nil integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (pi_lbn_LT formula-decl nil atan_approx nil)
    (expt def-decl "real" exponentiation nil)
    (^ const-decl "real" exponentiation nil)
    (atan_series_coef const-decl "rat" atan nil)
    (atan_series_term const-decl "[nat -> real]" atan nil)
    (sigma def-decl "real" sigma "reals/")
    (atan_series_n const-decl "real" atan nil)
    (atan_pos_le1_lb const-decl "real" atan_approx nil)
    (atan_pos_le1_ub const-decl "real" atan_approx nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (int_expt application-judgement "int" exponentiation nil)
    (nzreal_expt application-judgement "nzreal" exponentiation nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (int_exp application-judgement "int" exponentiation nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (posrat_expt application-judgement "posrat" exponentiation nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.115 Sekunden  (vorverarbeitet)  ¤





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