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Datei: convergence_sequences.prf Sprache: Lisp

Original von: PVS^©

(convergence_sequences
(unique_limit 0
  (unique_limit-1 nil 3253549181
   ("" (skosimp)
    (("" (expand "convergence")
      (("" (rewrite "abs_diff_0")
        (("" (name "eps" "abs(l1!1 - l2!1)/3")
          (("" (assert)
            (("" (inst -2 "eps")
              (("" (inst -3 "eps")
                (("" (skolem!)
                  (("" (skolem!)
                    (("" (inst -2 "n!1+n!2")
                      (("" (inst -3 "n!1+n!2")
                        (("" (assert)
                          (("" (lemma "triangle2")
                            ((""
                              (inst -1 "eps" "eps" "l1!1"
                               "u!1(n!1+n!2)" "l2!1")
                              (("" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((real_minus_real_is_real application-judgement "real" reals nil)
    (convergence const-decl "bool" convergence_sequences nil)
    (nnreal type-eq-decl nil real_types 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)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil 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)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields 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)
    (sequence type-eq-decl nil sequences nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (triangle2 formula-decl nil abs_lems "reals/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (abs_diff_0 formula-decl nil abs_lems "reals/"))
   nil))
(limit_lemma 0
  (limit_lemma-1 nil 3253549181
   ("" (skolem-typepred)
    (("" (name-replace "ll" "limit(v!1)" :hide? nil)
      (("" (expand "limit")
        (("" (expand "convergent?")
          ((""
            (lemma "epsilon_ax"
             ("p" "LAMBDA (l: real): convergence(v!1, l)"))
            (("" (assert) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((= const-decl "[T, T -> boolean]" equalities nil)
    (limit const-decl "real" convergence_sequences nil)
    (epsilon_ax formula-decl nil epsilons nil)
    (pred type-eq-decl nil defined_types nil)
    (convergence const-decl "bool" convergence_sequences nil)
    (convergent? const-decl "bool" convergence_sequences nil)
    (sequence type-eq-decl nil sequences 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)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   nil))
(limit_def 0
  (limit_def-1 nil 3253549181
   ("" (skolem!)
    (("" (use "limit_lemma")
      (("" (ground)
        (("" (use "unique_limit") (("" (assert) nil nil)) nil)) nil))
      nil))
    nil)
   ((limit_lemma formula-decl nil convergence_sequences nil)
    (convergent? const-decl "bool" convergence_sequences 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)
    (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)
    (unique_limit formula-decl nil convergence_sequences nil)
    (limit const-decl "real" convergence_sequences nil))
   nil))
(convergence_subsequence 0
  (convergence_subsequence-1 nil 3253549181
   (""
    (grind :defs nil :rewrites ("subseq" "convergence") :if-match nil)
    (("" (delete -1 -2 -3 -4)
      (("" (inst -1 "epsilon!1")
        (("" (skolem!)
          (("" (inst 1 "n!1")
            (("" (skosimp)
              (("" (inst? -2)
                (("" (replace -2)
                  (("" (inst?)
                    (("" (assert)
                      (("" (use "extract_incr3")
                        (("" (assert) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((minus_odd_is_odd application-judgement "odd_int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (extract_incr3 formula-decl nil sequence_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types 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)
    (strict_increasing? const-decl "bool" real_fun_preds "reals/")
    (extraction type-eq-decl nil sequence_props 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)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (subseq const-decl "bool" sequence_props nil)
    (convergence const-decl "bool" convergence_sequences nil))
   nil))
(limit_accumulation 0
  (limit_accumulation-1 nil 3253549181
   (""
    (grind :defs nil :rewrites ("convergence" "accumulation") :if-match
     nil)
    (("" (inst?)
      (("" (skolem!)
        (("" (inst -5 "n!1+n!2")
          (("" (inst 1 "n!1+n!2") (("" (assert) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil 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)
    (>= 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)
    (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)
    (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)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (accumulation const-decl "bool" convergence_sequences nil)
    (convergence const-decl "bool" convergence_sequences nil))
   nil))
(g_TCC1 0
  (g_TCC1-1 nil 3253549181 ("" (skosimp) (("" (assert) nil nil)) nil)
   ((int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
(g_TCC2 0
  (g_TCC2-1 nil 3253549181 ("" (skosimp) (("" (assert) nil nil)) nil)
   ((int_minus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil))
   nil))
(g_TCC3 0 (g_TCC3-1 nil 3253549181 ("" (skosimp*) nil nil) nil nil))
(g_prop 0
  (g_prop-1 nil 3253549181
   ("" (skosimp*)
    (("" (expand "accumulation")
      (("" (name-replace "y" "g(u!1, a!1)(n!1 + 1)" :hide? nil)
        (("" (expand "g" -1)
          (("" (assert)
            ((""
              (lemma "epsilon_ax"
               ("p"
                "LAMBDA (i: nat): g(u!1, a!1)(n!1) < i AND abs(u!1(i) - a!1) < 1 / (1 + n!1)"))
              (("" (replace -2)
                (("" (split -1)
                  (("1" (assert) nil nil)
                   ("2" (inst -2 "1/(1+n!1)" "g(u!1, a!1)(n!1) + 1")
                    (("2" (skosimp)
                      (("2" (inst 1 "i!1") (("2" (assert) nil nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((accumulation const-decl "bool" convergence_sequences nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal 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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (pred type-eq-decl nil defined_types nil)
    (epsilon_ax formula-decl nil epsilons nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types 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)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (g def-decl "nat" convergence_sequences 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)
    (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)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
(g_increasing 0
  (g_increasing-1 nil 3253549181
   ("" (skosimp)
    (("" (rewrite "strict_incr_condition")
      (("" (skolem!)
        (("" (forward-chain "g_prop")
          (("" (inst?) (("" (ground) nil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((strict_incr_condition formula-decl nil sequence_props 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)
    (sequence type-eq-decl nil sequences nil)
    (g def-decl "nat" convergence_sequences nil)
    (g_prop formula-decl nil convergence_sequences nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers 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))
   nil))
(g_convergence 0
  (g_convergence-1 nil 3253549181
   ("" (skosimp*)
    (("" (forward-chain "g_prop")
      (("" (inst -1 "n!1") (("" (flatten) nil nil)) nil)) nil))
    nil)
   ((g_prop formula-decl nil convergence_sequences 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)
    (sequence type-eq-decl nil sequences nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
(accumulation_subsequence 0
  (accumulation_subsequence-1 nil 3253549181
   ("" (skolem!)
    (("" (prop)
      (("1" (inst 1 "LAMBDA (i : nat) : u!1(g(u!1, a!1)(i))")
        (("1" (split)
          (("1" (forward-chain "g_increasing")
            (("1" (expand "subseq")
              (("1" (inst 1 "g(u!1, a!1)") (("1" (skolem!) nil nil))
                nil))
              nil))
            nil)
           ("2" (forward-chain "g_convergence")
            (("2" (expand "convergence")
              (("2" (skolem!)
                (("2" (lemma "archimedean2" ("x" "epsilon!1"))
                  (("2" (skolem!)
                    (("2" (inst 1 "a!2")
                      (("2" (skosimp)
                        (("2" (assert)
                          (("2" (inst -2 "i!1 - 1")
                            (("2" (assert)
                              (("2"
                                (case "1/i!1 <= 1/a!2")
                                (("1" (assert) nil nil)
                                 ("2"
                                  (rewrite "both_sides_div_pos_le2")
                                  nil
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2"
        (grind :defs nil :rewrites
         ("subseq" "convergence" "accumulation") :if-match nil)
        (("2" (inst? -6)
          (("2" (skolem!)
            (("2" (inst -5 "n!1 + n!2")
              (("2" (inst -6 "n!1+n!2")
                (("2" (inst?)
                  (("2" (assert)
                    (("2" (assert)
                      (("2" (use "extract_incr3")
                        (("2" (assert) nil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subseq const-decl "bool" sequence_props nil)
    (extraction type-eq-decl nil sequence_props nil)
    (strict_increasing? const-decl "bool" real_fun_preds "reals/")
    (u!1 skolem-const-decl "sequence[real]" convergence_sequences nil)
    (a!1 skolem-const-decl "real" convergence_sequences nil)
    (g_increasing formula-decl nil convergence_sequences nil)
    (convergence const-decl "bool" convergence_sequences nil)
    (archimedean2 formula-decl nil real_facts "reals/")
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil 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)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nzrat_div_nzrat_is_nzrat application-judgement "nzrat" rationals
     nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (both_sides_div_pos_le2 formula-decl nil real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (<= const-decl "bool" reals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (g_convergence formula-decl nil convergence_sequences nil)
    (g def-decl "nat" convergence_sequences 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)
    (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)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (extract_incr3 formula-decl nil sequence_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (accumulation const-decl "bool" convergence_sequences nil))
   nil))
(cauchy_accumulation 0
  (cauchy_accumulation-1 nil 3253549181
   (""
    (grind :defs nil :rewrites ("cauchy" "accumulation" "convergence")
     :if-match nil)
    (("" (inst -4 "epsilon!1/2")
      (("" (skolem!)
        (("" (inst -5 "epsilon!1/2" "n!1")
          (("" (skosimp)
            (("" (inst 1 "i!1")
              (("" (skosimp)
                (("" (inst -4 "i!1" "i!2")
                  (("" (assert)
                    (("" (lemma "triangle2")
                      (("" (assert)
                        ((""
                          (inst -1 "epsilon!1/2" "epsilon!1/2"
                           "u!1(i!2)" "u!1(i!1)" "a!1")
                          (("" (assert) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types 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)
    (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)
    (triangle2 formula-decl nil abs_lems "reals/")
    (sequence type-eq-decl nil sequences nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (>= 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)
    (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)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (convergence const-decl "bool" convergence_sequences nil)
    (accumulation const-decl "bool" convergence_sequences nil)
    (cauchy const-decl "bool" convergence_sequences nil))
   nil))
(cauchy_subsequence 0
  (cauchy_subsequence-1 nil 3253549181
   ("" (skosimp)
    (("" (rewrite "cauchy_accumulation" 1)
      (("" (rewrite "accumulation_subsequence")
        (("" (inst?) (("" (assert) nil nil)) nil)) nil))
      nil))
    nil)
   ((cauchy_accumulation formula-decl nil convergence_sequences 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)
    (sequence type-eq-decl nil sequences nil)
    (accumulation_subsequence formula-decl nil convergence_sequences
     nil))
   nil))
(increasing_bounded_convergence 0
  (increasing_bounded_convergence-1 nil 3253549181
   ("" (skosimp)
    (("" (assert)
      ((""
        (grind :defs nil :if-match nil :rewrites
         ("increasing?" "convergence"))
        (("" (use "supfun_is_sup[nat]")
          (("" (skolem!)
            (("" (inst 1 "x!1")
              (("" (skosimp)
                (("" (inst -4 "x!1" "i!1")
                  (("" (assert)
                    (("" (use "supfun_is_bound" ("x" "i!1"))
                      (("" (assert)
                        (("" (expand "abs")
                          (("" (lift-if) (("" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (supfun_is_sup formula-decl nil real_fun_supinf nil)
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (sequence type-eq-decl nil sequences nil)
    (supfun_is_bound formula-decl nil real_fun_supinf nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_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)
    (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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (convergence const-decl "bool" convergence_sequences nil)
    (increasing? const-decl "bool" real_fun_preds "reals/"))
   nil))
(decreasing_bounded_convergence 0
  (decreasing_bounded_convergence-1 nil 3253549181
   ("" (skosimp)
    (("" (assert)
      ((""
        (grind :defs nil :if-match nil :rewrites
         ("decreasing?" "convergence"))
        (("" (use "inffun_is_inf[nat]")
          (("" (skolem!)
            (("" (inst 1 "x!1")
              (("" (skosimp)
                (("" (inst -4 "x!1" "i!1")
                  (("" (assert)
                    (("" (use "inffun_is_bound" ("x" "i!1"))
                      (("" (assert)
                        (("" (expand "abs") (("" (assert) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (inffun_is_inf formula-decl nil real_fun_supinf nil)
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (sequence type-eq-decl nil sequences nil)
    (inffun_is_bound formula-decl nil real_fun_supinf nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (convergence const-decl "bool" convergence_sequences nil)
    (decreasing? const-decl "bool" real_fun_preds "reals/"))
   nil))
(bolzano_weierstrass1 0
  (bolzano_weierstrass1-1 nil 3253549181
   ("" (skolem!)
    (("" (use "monotone_subsequence")
      (("" (skosimp)
        (("" (use* "bounded_above_subseq" "bounded_below_subseq")
          (("" (assert)
            (("" (expand "subseq")
              (("" (skolem!)
                ((""
                  (auto-rewrite "supfun_is_sup2[nat]"
                                "inffun_is_inf2[nat]"
                                "supfun_is_bound[nat]"
                                "inffun_is_bound[nat]"
                                "accumulation_subsequence")
                  (("" (ground)
                    (("1"
                      (forward-chain "increasing_bounded_convergence")
                      (("1" (inst? +)
                        (("1" (ground)
                          (("1" (inst - "0")
                            (("1" (use "supfun_is_bound" ("g" "s!1"))
                              (("1"
                                (use "inffun_is_bound" ("h" "w!1"))
                                (("1" (assert) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (skolem!)
                            (("2" (inst?)
                              (("2"
                                (replace*)
                                (("2" (assert) nil nil))
                                nil))
                              nil))
                            nil)
                           ("3" (inst?) (("3" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil)
                     ("2"
                      (forward-chain "decreasing_bounded_convergence")
                      (("2" (inst? +)
                        (("2" (ground)
                          (("1" (skolem!)
                            (("1" (inst?)
                              (("1"
                                (replace -5)
                                (("1" (assert) nil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (inst - "0")
                            (("2" (use "inffun_is_bound" ("h" "s!1"))
                              (("2"
                                (use "supfun_is_bound" ("g" "w!1"))
                                (("2" (assert) nil nil))
                                nil))
                              nil))
                            nil)
                           ("3" (inst?) (("3" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((monotone_subsequence formula-decl nil monotone_subsequence nil)
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (AND const-decl "[bool, bool -> bool]" booleans 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)
    (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)
    (bounded_above_subseq formula-decl nil sequence_props nil)
    (bounded_below_subseq formula-decl nil sequence_props nil)
    (subseq const-decl "bool" sequence_props nil)
    (decreasing_bounded_convergence formula-decl nil
     convergence_sequences nil)
    (inffun_is_inf2 formula-decl nil real_fun_supinf nil)
    (inf const-decl "real" real_fun_supinf nil)
    (increasing_bounded_convergence formula-decl nil
     convergence_sequences nil)
    (supfun_is_sup2 formula-decl nil real_fun_supinf nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (supfun_is_bound formula-decl nil real_fun_supinf nil)
    (extraction type-eq-decl nil sequence_props nil)
    (strict_increasing? const-decl "bool" real_fun_preds "reals/")
    (inffun_is_bound formula-decl nil real_fun_supinf nil)
    (sup const-decl "real" real_fun_supinf nil)
    (accumulation_subsequence formula-decl nil convergence_sequences
     nil))
   nil))
(bolzano_weierstrass2 0
  (bolzano_weierstrass2-1 nil 3253549181
   ("" (skolem!)
    (("" (use "bolzano_weierstrass1")
      (("" (skosimp) (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((bolzano_weierstrass1 formula-decl nil convergence_sequences nil)
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (AND const-decl "[bool, bool -> bool]" booleans 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)
    (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))
   nil))
(bolzano_weierstrass3 0
  (bolzano_weierstrass3-1 nil 3253549181
   ("" (skosimp)
    (("" (use "bolzano_weierstrass1")
      (("" (skosimp)
        (("" (rewrite "accumulation_subsequence")
          (("" (skosimp)
            (("" (inst?)
              (("" (assert)
                (("" (expand "convergent?") (("" (inst?) nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bolzano_weierstrass1 formula-decl nil convergence_sequences nil)
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (AND const-decl "[bool, bool -> bool]" booleans 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)
    (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)
    (accumulation_subsequence formula-decl nil convergence_sequences
     nil)
    (convergent? const-decl "bool" convergence_sequences nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil))
   nil))
(bolzano_weierstrass4 0
  (bolzano_weierstrass4-1 nil 3253549181
   ("" (skosimp)
    (("" (case "bounded_above?(u!1) AND bounded_below?(u!1)")
      (("1" (ground)
        (("1" (case "a!1 <= inf(u!1) AND sup(u!1) <= b!1")
          (("1" (ground)
            (("1" (use "bolzano_weierstrass1")
              (("1" (skosimp)
                (("1" (inst + "a!2") (("1" (assert) nil nil)) nil))
                nil))
              nil))
            nil)
           ("2" (ground)
            (("1" (rewrite "inffun_is_inf2[nat]")
              (("1" (skolem!)
                (("1" (inst?) (("1" (assert) nil nil)) nil)) nil))
              nil)
             ("2" (rewrite "supfun_is_sup2[nat]")
              (("2" (skolem!)
                (("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (delete 2)
        (("2" (grind :if-match nil)
          (("1" (inst + "a!1")
            (("1" (skolem!)
              (("1" (inst?) (("1" (assert) nil nil)) nil)) nil))
            nil)
           ("2" (inst + "b!1")
            (("2" (skolem!)
              (("2" (inst?) (("2" (assert) nil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bounded_below? const-decl "bool" real_fun_preds "reals/")
    (sequence type-eq-decl nil sequences nil)
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (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)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (sup const-decl "real" real_fun_supinf nil)
    (inf const-decl "real" real_fun_supinf nil)
    (<= const-decl "bool" reals nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (bolzano_weierstrass1 formula-decl nil convergence_sequences nil)
    (supfun_is_sup2 formula-decl nil real_fun_supinf nil)
    (inffun_is_inf2 formula-decl nil real_fun_supinf nil))
   nil))
(prefix_bounded1 0
  (prefix_bounded1-1 nil 3253549181
   ("" (skolem 1 (_ "u!1"))
    (("" (induct "n")
      (("1" (inst 1 "u!1(0)")
        (("1" (skosimp) (("1" (assert) nil nil)) nil)) nil)
       ("2" (skosimp*)
        (("2" (inst 1 "max(a!1, u!1(j!1+1))")
          (("2" (skosimp) (("2" (inst?) (("2" (assert) 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)
    (pred type-eq-decl nil defined_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (sequence type-eq-decl nil sequences nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (max const-decl "{p: real | p >= m AND p >= n}" real_defs nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil))
   nil))
(prefix_bounded2 0
  (prefix_bounded2-1 nil 3253549181
   ("" (skolem 1 (_ "u!1"))
    (("" (induct "n")
      (("1" (inst 1 "u!1(0)")
        (("1" (skosimp) (("1" (assert) nil nil)) nil)) nil)
       ("2" (skosimp*)
        (("2" (inst 1 "min(a!1, u!1(j!1+1))")
          (("2" (skosimp)
            (("2" (inst -1 "i!1") (("2" (assert) 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)
    (pred type-eq-decl nil defined_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (sequence type-eq-decl nil sequences nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields
       nil))
   nil))
(cauchy_bounded 0
  (cauchy_bounded-1 nil 3253549181
   ("" (skosimp)
    (("" (expand "cauchy")
      (("" (inst - "1")
        (("" (skolem!)
          (("" (inst - "n!1" _)
            (("" (auto-rewrite "bounded_above?" "bounded_below?" "abs")
              (("" (ground)
                (("1" (use "prefix_bounded1" ("n" "n!1"))
                  (("1" (skolem!)
                    (("1" (inst + "a!1 + 1")
                      (("1" (skolem!)
                        (("1" (inst-cp - "n!1")
                          (("1" (inst - "x!1")
                            (("1" (inst - "x!1")
                              (("1" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (use "prefix_bounded2" ("n" "n!1"))
                  (("2" (skolem!)
                    (("2" (inst + "a!1 - 1")
                      (("2" (skolem!)
                        (("2" (inst-cp - "n!1")
                          (("2" (inst - "x!1")
                            (("2" (inst - "x!1")
                              (("2" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((real_minus_real_is_real application-judgement "real" reals nil)
    (cauchy const-decl "bool" convergence_sequences nil)
    (prefix_bounded2 formula-decl nil convergence_sequences nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (prefix_bounded1 formula-decl nil convergence_sequences nil)
    (sequence type-eq-decl nil sequences nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (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)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   nil))
(convergence_cauchy1 0
  (convergence_cauchy1-1 nil 3253549181
   (""
    (grind :defs nil :rewrites ("convergent?" "convergence" "cauchy")
     :if-match nil)
    (("" (delete -1 -2 -3)
      (("" (inst -1 "epsilon!1/2")
        (("" (skolem!)
          (("" (inst? 1)
            (("" (skosimp)
              (("" (inst-cp -1 "i!1")
                (("" (inst -1 "j!1")
                  (("" (assert)
                    (("" (rewrite "abs_diff_commute" -1)
                      (("" (rewrite "abs_diff_commute" -2)
                        (("" (lemma "triangle2")
                          ((""
                            (inst -1 "epsilon!1/2" "epsilon!1/2"
                             "u!1(j!1)" "l!1" "u!1(i!1)")
                            (("" (assert)
                              ((""
                                (rewrite "abs_diff_commute" +)
                                (("" (assert) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((sequence type-eq-decl nil sequences nil)
    (abs_diff_commute formula-decl nil abs_lems "reals/")
    (triangle2 formula-decl nil abs_lems "reals/")
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props 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)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (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)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (cauchy const-decl "bool" convergence_sequences nil)
    (convergent? const-decl "bool" convergence_sequences nil)
    (convergence const-decl "bool" convergence_sequences nil))
   nil))
(convergence_cauchy2 0
  (convergence_cauchy2-1 nil 3253549181
   ("" (skosimp)
    (("" (use "bolzano_weierstrass2")
      (("1" (skolem!)
        (("1" (expand "convergent?")
          (("1" (inst?) (("1" (rewrite "cauchy_accumulation") nil nil))
            nil))
          nil))
        nil)
       ("2" (rewrite "cauchy_bounded") nil nil))
      nil))
    nil)
   ((bolzano_weierstrass2 formula-decl nil convergence_sequences nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (AND 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)
    (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)
    (bounded_above? const-decl "bool" real_fun_preds "reals/")
    (sequence type-eq-decl nil sequences nil)
    (u!1 skolem-const-decl "sequence[real]" convergence_sequences nil)
    (bounded_below? const-decl "bool" real_fun_preds "reals/")
    (convergent? const-decl "bool" convergence_sequences nil)
    (cauchy_accumulation formula-decl nil convergence_sequences nil)
    (cauchy_bounded formula-decl nil convergence_sequences nil))
   nil))
(convergence_cauchy 0
  (convergence_cauchy-1 nil 3253549181
   ("" (skolem!)
    (("" (prop)
      (("1" (rewrite "convergence_cauchy1") nil nil)
       ("2" (rewrite "convergence_cauchy2") nil nil))
      nil))
    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)
    (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)
    (convergence_cauchy1 formula-decl nil convergence_sequences nil)
    (convergence_cauchy2 formula-decl nil convergence_sequences nil))
   nil)))

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