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Datei: power_series_conv.prf   Sprache: Lisp

Original von: PVS©

(power_series_conv
 (Inf_sum_TCC1 0
  (Inf_sum_TCC1-1 nil 3298729967 ("" (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)
    (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)
    (sequence type-eq-decl nil sequences nil)
    (conv_powerseries? const-decl "bool" power_series_conv nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (series const-decl "sequence[real]" series nil)
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil))
   shostak))
 (Inf_sum_TCC2 0
  (Inf_sum_TCC2-1 nil 3298729967 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (>= const-decl "bool" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (sequence type-eq-decl nil sequences nil)
    (conv_powerseries? const-decl "bool" power_series_conv nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (series const-decl "sequence[real]" series nil)
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (real_minus_real_is_real application-judgement "real" reals nil))
   shostak))
 (end_powerseries_conv 0
  (end_powerseries_conv-1 nil 3298368454
   ("" (skosimp*)
    (("" (lemma "end_series_conv")
      (("" (expand "conv_powerseries?")
        (("" (expand "powerseries")
          (("" (split +)
            (("1" (ground)
              (("1" (skosimp*)
                (("1" (inst?)
                  (("1" (inst?) (("1" (assertnil nil)) nil)) nil))
                nil))
              nil)
             ("2" (flatten)
              (("2" (skosimp*)
                (("2" (inst?)
                  (("2" (inst?) (("2" (assertnil nil)) nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((end_series_conv formula-decl nil series nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (powerseq const-decl "sequence[real]" power_series nil)
    (sequence type-eq-decl nil sequences 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)
    (conv_powerseries? const-decl "bool" power_series_conv nil)
    (conv_powerseries? const-decl "bool" power_series_conv nil))
   shostak))
 (Inf_inf_TCC1 0
  (Inf_inf_TCC1-1 nil 3298731432
   ("" (skosimp*)
    (("" (expand "conv_series?")
      (("" (expand "conv_powerseries?")
        (("" (inst?)
          (("" (expand "powerseries") (("" (propax) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((conv_series? const-decl "bool" series nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (conv_powerseries? const-decl "bool" power_series_conv nil))
   shostak))
 (Inf_inf 0
  (Inf_inf-1 nil 3298730864
   ("" (skosimp*)
    (("" (apply-extensionality 1 :hide? t)
      (("1" (expand "Inf_sum")
        (("1" (expand "powerseries")
          (("1" (expand "inf_sum") (("1" (propax) nil nil)) nil)) nil))
        nil)
       ("2" (skosimp*)
        (("2" (expand "conv_powerseries?")
          (("2" (expand "conv_series?")
            (("2" (expand "powerseries") (("2" (inst?) nil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (inf_sum const-decl "real" series nil)
    (Inf_sum const-decl "real" power_series_conv nil)
    (conv_powerseries? const-decl "bool" power_series_conv 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)
    (sequence type-eq-decl nil sequences nil)
    (conv_series? const-decl "bool" series nil)
    (powerseq const-decl "sequence[real]" power_series nil)
    (a!1 skolem-const-decl "sequence[real]" power_series_conv nil)
    (powerseries const-decl "sequence[real]" power_series nil))
   shostak))
 (Inf_inf_m_TCC1 0
  (Inf_inf_m_TCC1-1 nil 3298731432
   ("" (skosimp*)
    (("" (expand "conv_series?")
      (("" (expand "conv_powerseries?")
        (("" (expand "powerseries") (("" (inst?) nil nil)) nil)) nil))
      nil))
    nil)
   ((conv_series? const-decl "bool" series nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (conv_powerseries? const-decl "bool" power_series_conv nil))
   shostak))
 (Inf_inf_m 0
  (Inf_inf_m-1 nil 3298731378
   ("" (skosimp*)
    (("" (expand "Inf_sum")
      (("" (apply-extensionality 1 :hide? t)
        (("1" (expand "inf_sum")
          (("1" (expand "powerseries") (("1" (propax) nil nil)) nil))
          nil)
         ("2" (skosimp*)
          (("2" (expand "conv_powerseries?")
            (("2" (expand "powerseries")
              (("2" (expand "conv_series?") (("2" (inst?) nil nil))
                nil))
              nil))
            nil))
          nil)
         ("3" (skosimp*)
          (("3" (expand "conv_powerseries?") (("3" (inst?) nil nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((Inf_sum const-decl "real" power_series_conv nil)
    (conv_powerseries? const-decl "bool" power_series_conv nil)
    (m!1 skolem-const-decl "nat" power_series_conv nil)
    (a!1 skolem-const-decl "sequence[real]" power_series_conv nil)
    (powerseries const-decl "sequence[real]" power_series nil)
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (sequence type-eq-decl nil sequences 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)
    (powerseq const-decl "sequence[real]" power_series nil)
    (conv_series? const-decl "bool" series nil)
    (limit const-decl "real" convergence_sequences "analysis/")
    (inf_sum const-decl "real" series nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
 (powerseries_abs_conv 0
  (powerseries_abs_conv-1 nil 3297171631
   ("" (skosimp*)
    (("" (case-replace "EXISTS (y:T): abs(y) > abs(x!1)")
      (("1" (skosimp*)
        (("1" (lemma "powerseries_conv_point")
          (("1" (expand "absolutely_convergent_series?")
            (("1" (inst?)
              (("1" (assert)
                (("1" (inst -1 "y!1")
                  (("1" (assert) (("1" (inst -2 "y!1"nil nil)) nil)
                   ("2" (assert)
                    (("2" (hide -2 2) (("2" (grind) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide -1 2)
        (("2" (case "x!1 >= 0")
          (("1" (lemma "open")
            (("1" (inst -1 "x!1")
              (("1" (skosimp*)
                (("1" (inst + "x!1 + delta!1/2")
                  (("1" (hide -1) (("1" (grind) nil nil)) nil)
                   ("2" (inst?)
                    (("2" (assert)
                      (("2" (hide 1) (("2" (grind) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (lemma "open")
            (("2" (inst -1 "x!1")
              (("2" (skosimp*)
                (("2" (inst + "x!1 - delta!1/2")
                  (("1" (hide -1) (("1" (grind) nil nil)) nil)
                   ("2" (inst?)
                    (("2" (assert) (("2" (grind) nil nil)) nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (> const-decl "bool" reals nil) (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (powerseries_conv_point formula-decl nil power_series nil)
    (sequence type-eq-decl nil sequences nil)
    (nat nonempty-type-eq-decl nil naturalnumbers 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 "boolean" notequal nil)
    (y!1 skolem-const-decl "T" power_series_conv nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (absolutely_convergent_series? const-decl "boolean" power_series
     nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (x!1 skolem-const-decl "T" power_series_conv nil)
    (delta!1 skolem-const-decl "posreal" power_series_conv nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (minus_nzreal_is_nzreal application-judgement "nzreal" real_types
     nil)
    (nonzero_abs_is_pos application-judgement "{y: posreal | y >= x}"
     real_defs nil)
    (nzreal_times_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (open formula-decl nil power_series_conv nil)
    (delta!1 skolem-const-decl "posreal" power_series_conv nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (minus_real_is_real application-judgement "real" reals nil))
   nil))
 (simple_TCC1 0
  (simple_TCC1-1 nil 3297159301 ("" (subtype-tcc) nil nil)
   ((/= const-decl "boolean" notequal nil)) nil))
 (simple_0_conv 0
  (simple_0_conv-1 nil 3297159301
   ("" (expand "simple")
    (("" (flatten)
      (("" (lemma "tail_series_conv2")
        (("" (inst -1 "1" "_")
          (("" (inst?)
            (("" (assert)
              (("" (hide 2)
                (("" (expand "series")
                  (("" (expand "^")
                    (("" (expand "expt")
                      (("" (assert)
                        (("" (expand "convergent?")
                          (("" (inst 1 "0")
                            (("" (expand "convergence")
                              ((""
                                (skosimp*)
                                ((""
                                  (inst 1 "100")
                                  ((""
                                    (skosimp*)
                                    ((""
                                      (assert)
                                      ((""
                                        (case-replace
                                         "(LAMBDA (n_1962: nat):
                    0 * expt(0, n_1962) / factorial(1 + n_1962)) = (LAMBDA n: 0)")
                                        (("1"
                                          (rewrite "sigma_const")
                                          (("1"
                                            (expand "abs")
                                            (("1" (assertnil nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (apply-extensionality
                                             1
                                             :hide?
                                             t)
                                            nil
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (zero_hat formula-decl nil exponent_props "reals/")
    (sigma_nnreal application-judgement "nnreal" sigma_nat "reals/")
    (series const-decl "sequence[real]" series nil)
    (convergent? const-decl "bool" convergence_sequences "analysis/")
    (convergence const-decl "bool" convergence_sequences "analysis/")
    (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)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (T_low type-eq-decl nil sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (<= const-decl "bool" reals nil)
    (sigma_const formula-decl nil sigma "reals/")
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (nat_expt application-judgement "nat" exponentiation nil)
    (sigma_nat application-judgement "nat" sigma_nat "reals/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (expt def-decl "real" exponentiation nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (factorial def-decl "posnat" factorial "ints/")
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (^ const-decl "real" exponentiation nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (sequence type-eq-decl nil sequences nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (tail_series_conv2 formula-decl nil series nil)
    (simple const-decl "sequence[real]" power_series_conv nil)
    (nat_exp application-judgement "nat" exponentiation nil))
   nil))
 (simple_conv 0
  (simple_conv-1 nil 3297159301
   ("" (skosimp*)
    (("" (case-replace "x!1 = 0")
      (("1" (rewrite "simple_0_conv"nil nil)
       ("2" (lemma "ratio_test_gen")
        (("2" (inst?)
          (("2" (assert)
            (("2" (hide 3)
              (("2" (split +)
                (("1" (skosimp*)
                  (("1" (expand "simple")
                    (("1" (rewrite "div_eq_zero")
                      (("1" (assert)
                        (("1" (expand "^")
                          (("1" (lemma "expt_nonzero_aux")
                            (("1" (inst?) (("1" (assertnil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (inst + "1/2")
                  (("2" (assert)
                    (("2" (inst + "floor(abs(2*x!1))")
                      (("2" (skosimp*)
                        (("2"
                          (case-replace
                           "simple(x!1)(1 + n!1) / simple(x!1)(n!1) = x!1/(1+n!1)")
                          (("1" (hide -1)
                            (("1" (typepred "floor(abs(2 * x!1))")
                              (("1"
                                (hide -1)
                                (("1"
                                  (case "n!1 > abs(2 * x!1) - 1")
                                  (("1"
                                    (hide -2 -3 -4)
                                    (("1"
                                      (rewrite "abs_div")
                                      (("1"
                                        (expand "abs" 1 2)
                                        (("1"
                                          (lemma
                                           "both_sides_div_pos_le1")
                                          (("1"
                                            (inst
                                             -1
                                             "1+n!1"
                                             "abs(x!1)"
                                             "1/2*(1+n!1)")
                                            (("1"
                                              (flatten)
                                              (("1"
                                                (hide -1)
                                                (("1"
                                                  (case-replace
                                                   "1 / 2 * (1 + n!1) / (1 + n!1) = 1/2")
                                                  (("1"
                                                    (split -2)
                                                    (("1"
                                                      (assert)
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (hide -1 2)
                                                      (("2"
                                                        (lemma
                                                         "both_sides_times_pos_le1")
                                                        (("2"
                                                          (inst
                                                           -1
                                                           "2"
                                                           "abs(x!1)"
                                                           "1 / 2 * (1 + n!1)")
                                                          (("2"
                                                            (flatten)
                                                            (("2"
                                                              (hide -2)
                                                              (("2"
                                                                (split
                                                                 -1)
                                                                (("1"
                                                                  (propax)
                                                                  nil
                                                                  nil)
                                                                 ("2"
                                                                  (hide
                                                                   2)
                                                                  (("2"
                                                                    (case-replace
                                                                     "1 / 2 * (1 + n!1) * 2 = 1+n!1")
                                                                    (("1"
                                                                      (hide
                                                                       -1)
                                                                      (("1"
                                                                        (rewrite
                                                                         "abs_mult")
                                                                        (("1"
                                                                          (expand
                                                                           "abs"
                                                                           -1
                                                                           1)
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil)
                                                                     ("2"
                                                                      (hide
                                                                       2)
                                                                      (("2"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide -1 -2 2)
                                                    (("2"
                                                      (name-replace
                                                       "NN"
                                                       "(1 + n!1)")
                                                      (("2"
                                                        (assert)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (hide -1 2)
                            (("2" (expand "simple")
                              (("2"
                                (case
                                 "(forall (a: real),(b,c,d: nzreal): (a/b)/(c/d) = (a/c)/(b/d))")
                                (("1"
                                  (inst?)
                                  (("1"
                                    (replace -1)
                                    (("1"
                                      (hide -1)
                                      (("1"
                                        (case-replace
                                         "(x!1 ^ (1 + n!1)) / x!1 ^ n!1 = x!1")
                                        (("1"
                                          (case-replace
                                           "(factorial(1 + n!1) / factorial(n!1)) = 1 + n!1")
                                          (("1"
                                            (hide 2)
                                            (("1"
                                              (expand "factorial" 1 1)
                                              (("1" (assertnil nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (hide 2)
                                          (("2"
                                            (expand "^")
                                            (("2"
                                              (expand "expt" 1 1)
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide 2)
                                    (("2"
                                      (expand "^")
                                      (("2"
                                        (lemma "expt_nonzero_aux")
                                        (("2"
                                          (inst?)
                                          (("2" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2 3)
                                  (("2"
                                    (skosimp*)
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (simple_0_conv formula-decl nil power_series_conv nil)
    (real_div_nzreal_is_real application-judgement "real" 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)
    (sequence type-eq-decl nil sequences nil)
    (simple const-decl "sequence[real]" power_series_conv nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (floor const-decl "{i | i <= x & x < i + 1}" floor_ceil nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (< const-decl "bool" reals nil) (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (integer nonempty-type-from-decl nil integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (abs_div formula-decl nil real_props nil)
    (nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
     real_defs nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (both_sides_div_pos_le1 formula-decl nil real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (abs_mult formula-decl nil real_props nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (posint nonempty-type-eq-decl nil integers nil)
    (rat_minus_rat_is_rat application-judgement "rat" rationals nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (x!1 skolem-const-decl "T" power_series_conv nil)
    (n!1 skolem-const-decl "nat" power_series_conv nil)
    (expt def-decl "real" exponentiation nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (div_eq_zero formula-decl nil real_props nil)
    (/= const-decl "boolean" notequal nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (factorial def-decl "posnat" factorial "ints/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (^ const-decl "real" exponentiation nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt_nonzero_aux formula-decl nil exponentiation nil)
    (ratio_test_gen formula-decl nil series nil))
   nil))
 (m1_conv 0
  (m1_conv-1 nil 3321109289
   ("" (skosimp*)
    (("" (expand "powerseries")
      (("" (expand "powerseq")
        (("" (lemma "geometric_conv")
          (("" (expand "geometric")
            (("" (inst - "-x!1")
              (("" (rewrite "abs_neg")
                (("" (assert)
                  ((""
                    (case-replace
                     "(LAMBDA (k: nat): (-1) ^ k * x!1 ^ k) = (LAMBDA n: (-x!1) ^ n)")
                    (("" (hide -1 2)
                      (("" (apply-extensionality 1 :hide? t)
                        (("" (lemma "mult_expt")
                          (("" (inst?)
                            (("1" (assertnil nil)
                             ("2" (flatten)
                              (("2"
                                (replace -1)
                                (("2"
                                  (typepred "x!2")
                                  (("2"
                                    (case-replace "x!2 = 0")
                                    (("1" (assertnil nil)
                                     ("2"
                                      (assert)
                                      (("2" (grind) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((powerseries const-decl "sequence[real]" power_series nil)
    (geometric_conv formula-decl nil series nil)
    (minus_real_is_real application-judgement "real" reals 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)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (T_pred const-decl "[real -> boolean]" power_series_conv nil)
    (T formal-subtype-decl nil power_series_conv nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (mult_expt formula-decl nil exponentiation nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (zero_hat formula-decl nil exponent_props "reals/")
    (expt_x0 formula-decl nil exponentiation nil)
    (minus_even_is_even application-judgement "even_int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nat_exp application-judgement "nat" exponentiation nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (x!1 skolem-const-decl "T" power_series_conv nil)
    (^ const-decl "real" exponentiation nil)
    (/= const-decl "boolean" notequal nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (nzreal_exp application-judgement "nzreal" exponentiation nil)
    (int_exp application-judgement "int" exponentiation nil)
    (abs_neg formula-decl nil abs_lems "reals/")
    (geometric const-decl "sequence[real]" series nil)
    (powerseq const-decl "sequence[real]" power_series nil))
   nil)))


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