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Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei:   Sprache: SML

Original von: PVS©

(sqrt_approx
 (sq_sq_aux 0
  (sq_sq_aux-1 nil 3320768920 ("" (grind) nil nil)
   ((nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posreal_plus_nnreal_is_posreal application-judgement "posreal"
     real_types nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (sq const-decl "nonneg_real" sq nil))
   shostak))
 (sqrt_newton_aux_TCC1 0
  (sqrt_newton_aux_TCC1-1 nil 3320768919 ("" (grind-reals) 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)
    (nnreal type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (posreal_times_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_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)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
 (sqrt_newton_aux_TCC2 0
  (sqrt_newton_aux_TCC2-1 nil 3320768919 ("" (grind-reals) nil nil)
   ((posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_ge_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)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil))
 (sqrt_newton_aux_TCC3 0
  (sqrt_newton_aux_TCC3-1 nil 3320768919 ("" (grind) 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)
    (nnreal 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)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_ge_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_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (sq const-decl "nonneg_real" sq nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil))
   nil))
 (sqrt_newton_aux_TCC4 0
  (sqrt_newton_aux_TCC4-1 nil 3320768919 ("" (grind) 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)
    (nnreal 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)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_ge_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)
    (sq const-decl "nonneg_real" sq nil)
    (/= const-decl "boolean" notequal nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil))
   nil))
 (sqrt_newton_aux_TCC5 0
  (sqrt_newton_aux_TCC5-2 nil 3320769110
   ("" (skosimp* :preds? t)
    (("" (case "y!1 > 0")
      (("1" (split)
        (("1" (cross-mult 1) (("1" (field) nil nil)) nil)
         ("2" (move-terms 1 l)
          (("2" (rewrite "sq_sq_aux")
            (("2" (rewrite "sq_div")
              (("2" (grind-reals)
                (("1" (rewrite "sq_times") (("1" (assertnil nil))
                  nil)
                 ("2" (name-replace "AA" "sq(y!1) - x!1")
                  (("2" (expand "sq") (("2" (grind-reals) nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (case "sq(y!1) > 0")
        (("1" (expand "sq" -1) (("1" (grind-reals) nil nil)) nil)
         ("2" (assertnil nil))
        nil))
      nil))
    nil)
   ((real_minus_real_is_real application-judgement "real" reals nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sq_div formula-decl nil sq nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (sq_times formula-decl nil sq nil)
    (sq_nz_pos application-judgement "posreal" sq nil)
    (pos_times_lt formula-decl nil real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (pos_div_lt formula-decl nil real_props nil)
    (sq_sq_aux formula-decl nil sqrt_approx nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (div_mult_pos_ge1 formula-decl nil 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 "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (FDX_87 skolem-const-decl "{y | x!1 < sq(y)}" sqrt_approx nil)
    (x!1 skolem-const-decl "nnreal" sqrt_approx nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (bijective? const-decl "bool" functions nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (TRUE const-decl "bool" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities 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)
    (nnreal type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil))
   nil))
 (sqrt_newton_aux_TCC6 0
  (sqrt_newton_aux_TCC6-2 nil 3320772107
   ("" (skosimp :preds? t)
    (("" (assert)
      (("" (lift-if)
        (("" (split 2)
          (("1" (flatten) (("1" (assertnil nil)) nil)
           ("2" (flatten)
            (("2" (case "y!1 > 0")
              (("1" (lemma "sq_sq_aux")
                (("1" (inst -1 "x!1" "y!1")
                  (("1" (replaces -1)
                    (("1" (rewrite "sq_div")
                      (("1" (rewrite "sq_times")
                        (("1" (case-replace "sq(2) = 4")
                          (("1" (lemma "log_nat_incr")
                            (("1" (inst?)
                              (("1"
                                (inst -1 "(sq(y!1)-x!1)/eps!1")
                                (("1"
                                  (assert)
                                  (("1"
                                    (hide 3)
                                    (("1"
                                      (field 1)
                                      (("1"
                                        (factor 1 r)
                                        (("1"
                                          (name-replace
                                           "AA"
                                           "sq(y!1)-x!1"
                                           :hide?
                                           nil)
                                          (("1"
                                            (expand "sq" 1 1)
                                            (("1"
                                              (cancel-by 1 "AA")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (expand "sq" 1)
                            (("2" (propax) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (wrap-formula 1 "sq")
                (("1" (rewrite "sq_0") (("1" (grind) nil nil)) nil)
                 ("2" (grind) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_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)
    (x!1 skolem-const-decl "nnreal" sqrt_approx nil)
    (y!1 skolem-const-decl "{y | x!1 < sq(y)}" sqrt_approx nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (sq_div formula-decl nil sq nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/= const-decl "boolean" notequal nil)
    (above nonempty-type-eq-decl nil integers 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_times_real_is_real application-judgement "real" reals nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nonzero_times3 formula-decl nil real_props nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (pos_times_gt formula-decl nil real_props nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (FDX_89 skolem-const-decl "posreal" sqrt_approx nil)
    (FDX_88 skolem-const-decl "nonneg_real" sqrt_approx nil)
    (div_cancel2 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (bijective? const-decl "bool" functions nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (TRUE const-decl "bool" booleans nil)
    (zero_times1 formula-decl nil real_props nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (pos_div_gt formula-decl nil real_props nil)
    (CBD_90 skolem-const-decl "real" sqrt_approx nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (log_nat_incr formula-decl nil log_nat nil)
    (sq_nz_pos application-judgement "posreal" sq nil)
    (sq_times formula-decl nil sq nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sq_sq_aux formula-decl nil sqrt_approx nil)
    (sq_0 formula-decl nil sq nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals 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)
    (nnreal type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil))
   nil))
 (sqrt_newton_aux_NOT_increasing 0
  (sqrt_newton_aux_NOT_increasing-1 nil 3557592665
   ("" (eval-expr "sq(2.9)")
    (("" (eval-expr "sq(2.91)")
      (("" (inst + "8.32" "8.31681" "2.9" "2.91" "0.1")
        (("1" (assertnil nil) ("2" (grind) nil nil)
         ("3" (grind) nil nil))
        nil))
      nil))
    nil)
   ((nnreal type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sq_nz_pos application-judgement "posreal" sq nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (number nonempty-type-decl nil numbers 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)
    (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq 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)
    (TRUE const-decl "bool" booleans nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (bijective? const-decl "bool" functions nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   nil))
 (sqrt_newton_TCC1 0
  (sqrt_newton_TCC1-1 nil 3320768919 ("" (subtype-tcc) nil nil)
   ((nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posreal_plus_nnreal_is_posreal application-judgement "posreal"
     real_types nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sq const-decl "nonneg_real" sq nil))
   nil))
 (sqrt_newton_is_NOT_increasing 0
  (sqrt_newton_is_NOT_increasing-1 nil 3557676691
   ("" (eval-formula) nil nil)
   ((TRUE const-decl "bool" booleans nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (bijective? const-decl "bool" functions nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   shostak))
 (sq_sqrt_newton 0
  (sq_sqrt_newton-1 nil 3321016010
   ("" (skeep)
    (("" (typepred "sqrt_newton(x, eps)") (("" (assertnil nil)) nil))
    nil)
   ((sqrt_newton const-decl "{z | x < sq(z) AND sq(z) - x < eps}"
                 sqrt_approx nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (sq const-decl "nonneg_real" sq nil)
    (< const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans 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)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (real_ge_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_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (sq_nz_pos application-judgement "posreal" sq nil))
   shostak))
 (sqrt_newton_gt 0
  (sqrt_newton_gt-1 nil 3321016041
   ("" (skeep)
    (("" (lemma "sq_sqrt_newton")
      (("" (inst -1 "eps" "x")
        (("" (wrap-formula -1 "sqrt")
          (("1" (rewrite "sqrt_sq"nil nil)
           ("2" (hide 2) (("2" (rewrite "sqrt_gt"nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((sq_sqrt_newton formula-decl nil sqrt_approx nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sqrt const-decl "{nnz: nnreal | nnz * nnz = nnx}" sqrt nil)
    (sq const-decl "nonneg_real" sq nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sqrt_newton const-decl "{z | x < sq(z) AND sq(z) - x < eps}"
                 sqrt_approx nil)
    (bijective? const-decl "bool" functions nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (TRUE const-decl "bool" booleans nil)
    (sqrt_sq formula-decl nil sqrt nil)
    (real_ge_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)
    (sqrt_gt formula-decl nil sqrt nil)
    (nnreal type-eq-decl nil real_types 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)
    (sq_nz_pos application-judgement "posreal" sq nil))
   shostak))
 (sqrt_newton_lt 0
  (sqrt_newton_lt-1 nil 3321016482
   ("" (skeep)
    (("" (cross-mult 2)
      (("" (lemma "sqrt_newton_gt")
        (("" (inst -1 "eps" "x")
          (("" (mult-by -1 "sqrt(x)") (("" (assertnil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (sqrt const-decl "{nnz: nnreal | nnz * nnz = nnx}" sqrt nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (sqrt_newton const-decl "{z | x < sq(z) AND sq(z) - x < eps}"
                 sqrt_approx nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (sq const-decl "nonneg_real" sq nil)
    (< const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (nnreal type-eq-decl nil real_types 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)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (x skolem-const-decl "nnreal" sqrt_approx nil)
    (both_sides_times_pos_gt1 formula-decl nil real_props nil)
    (sqrt_newton_gt formula-decl nil sqrt_approx nil)
    (sq_nz_pos application-judgement "posreal" sq nil))
   shostak))
 (sqrt_newton_le 0
  (sqrt_newton_le-1 nil 3321016611
   ("" (skeep)
    (("" (lemma "sqrt_newton_lt")
      (("" (inst -1 "eps" "x")
        (("" (split -1) (("1" (assertnil nil) ("2" (assertnil nil))
          nil))
        nil))
      nil))
    nil)
   ((sqrt_newton_lt formula-decl nil sqrt_approx nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnreal type-eq-decl nil real_types 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)
    (sq_nz_pos application-judgement "posreal" sq nil))
   shostak))
 (sqrt_approx_TCC1 0
  (sqrt_approx_TCC1-2 nil 3557832683
   ("" (skosimp*)
    (("" (assert)
      (("" (replaces -4)
        (("" (cross-mult 1)
          (("" (ground)
            (("1" (case "x!1*N!1>=0")
              (("1" (assertnil)
               ("2" (assert)
                (("2" (lemma "nnreal_times_nnreal_is_nnreal")
                  (("2" (inst?) nil)))))))
             ("2" (case "2*10^n!1>=1")
              (("1" (assertnil)
               ("2" (case "FORALL (nn:nat): 2*10^nn >=1")
                (("1" (inst?) nil)
                 ("2" (hide-all-but 1)
                  (("2" (induct "nn")
                    (("1" (grind) nil)
                     ("2" (skeep)
                      (("2" (expand "^")
                        (("2" (expand "expt" +)
                          (("2" (assertnil))))))))))))))))))))))))
    nil)
   ((real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (posrat_exp application-judgement "posrat" exponentiation nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (< const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (<= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (integer nonempty-type-from-decl nil integers nil)
    (posint nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
  (sqrt_approx_TCC1-1 nil 3557745411
   ("" (skeep)
    (("" (skeep)
      (("" (skosimp*)
        (("" (assert)
          (("" (typepred "floor(2*10^n)")
            (("" (assert)
              (("" (typepred "floor(1)")
                (("" (assert)
                  (("" (case "2*10^n >= 1")
                    (("1" (assertnil nil)
                     ("2" (hide-all-but 1)
                      (("2" (case "FORALL (nn:nat): 2*10^nn >=1")
                        (("1" (inst?) nil nil)
                         ("2" (induct "nn")
                          (("1" (hide 2) (("1" (grind) nil nil)) nil)
                           ("2" (hide 2)
                            (("2" (skeep)
                              (("2"
                                (expand "^")
                                (("2"
                                  (expand "expt" +)
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   nil nil))
 (sqrt_approx_TCC2 0
  (sqrt_approx_TCC2-2 nil 3557832760
   ("" (skeep)
    (("" (skeep)
      (("" (skeep)
        (("" (skeep)
          (("" (name "qq" "sqrt_newton(xx, 10 ^ (-n)/2)")
            (("1" (replaces -1)
              (("1" (typepred "qq")
                (("1" (hide (-1 -2))
                  (("1" (case "x >= xx AND abs(x-xx)<10^(-n)/2")
                    (("1" (flatten)
                      (("1" (grind :exclude ("sq" "floor" "^")) nil
                        nil))
                      nil)
                     ("2" (hide 2)
                      (("2" (split)
                        (("1" (mult-by 1 "N") (("1" (assertnil nil))
                          nil)
                         ("2" (case "1/N = 10^(-n)/2")
                          (("1" (case "abs(x-xx)<1/N")
                            (("1" (assert)
                              (("1" (replaces -2) nil nil)) nil)
                             ("2" (hide 2)
                              (("2"
                                (cross-mult 1)
                                (("2"
                                  (ground)
                                  (("2"
                                    (lemma "abs_mult")
                                    (("2"
                                      (inst - "x-xx" "N")
                                      (("2"
                                        (assert)
                                        (("2"
                                          (expand "abs" - 3)
                                          (("2"
                                            (replaces -1 :dir rl)
                                            (("2"
                                              (assert)
                                              (("2"
                                                (case "N*xx = k")
                                                (("1"
                                                  (replace -1)
                                                  (("1"
                                                    (assert)
                                                    (("1"
                                                      (replace -7)
                                                      (("1"
                                                        (hide-all-but
                                                         1)
                                                        (("1"
                                                          (grind
                                                           :exclude
                                                           ("sq"
                                                            "floor"
                                                            "^"))
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (cross-mult -7)
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (cross-mult 1)
                            (("2" (assert)
                              (("2"
                                (replace -4)
                                (("2"
                                  (assert)
                                  (("2"
                                    (lemma "expt_plus")
                                    (("2"
                                      (inst - "n" "-n" "10")
                                      (("2"
                                        (assert)
                                        (("2"
                                          (replace -1 :dir rl)
                                          (("2"
                                            (hide-all-but 1)
                                            (("2" (grind) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (replace -4)
              (("2" (cross-mult 1) (("2" (assertnil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (div_cancel4 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (<= const-decl "bool" reals nil)
    (integer nonempty-type-from-decl nil integers nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (nzint_abs_is_pos application-judgement "{j: posint | j >= i}"
     real_defs nil)
    (abs_mult formula-decl nil real_props nil)
    (div_mult_pos_lt2 formula-decl nil real_props nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt def-decl "real" exponentiation nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (expt_plus formula-decl nil exponentiation nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (times_div2 formula-decl nil real_props nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (div_cancel3 formula-decl nil real_props nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posint nonempty-type-eq-decl nil integers nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (= 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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types 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)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
    (sq const-decl "nonneg_real" sq nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sqrt_newton const-decl "{z | x < sq(z) AND sq(z) - x < eps}"
                 sqrt_approx nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rat nonempty-type-eq-decl nil rationals nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (^ const-decl "real" exponentiation nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (sq_nz_pos application-judgement "posreal" sq nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (posrat_exp application-judgement "posrat" exponentiation nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
   nil)
  (sqrt_approx_TCC2-1 nil 3557745411
   ("" (skosimp*)
    (("" (assert)
      (("" (replaces -4)
        (("" (cross-mult 1)
          (("" (ground)
            (("1" (case "x!1*N!1>=0")
              (("1" (assertnil nil)
               ("2" (assert)
                (("2" (lemma "nnreal_times_nnreal_is_nnreal")
                  (("2" (inst?) nil nil)) nil))
                nil))
              nil)
             ("2" (case "2*10^n!1>=1")
              (("1" (assertnil nil)
               ("2" (case "FORALL (nn:nat): 2*10^nn >=1")
                (("1" (inst?) nil nil)
                 ("2" (hide-all-but 1)
                  (("2" (induct "nn")
                    (("1" (grind) nil nil)
                     ("2" (skeep)
                      (("2" (expand "^")
                        (("2" (expand "expt" +)
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   nil nil))
 (sqrt_approx_increasing 0
  (sqrt_approx_increasing-1 nil 3557827887
   ("" (skeep)
    (("" (name "qx" "sqrt_approx(x, n)")
      (("" (name "qy" "sqrt_approx(y, n)")
        (("" (replace -1)
          (("" (replace -2)
            (("" (expand "sqrt_approx")
              (("" (invoke (name "yy" "%1") (! -1 1 1))
                (("" (replace -1)
                  (("" (invoke (name "xx" "%1") (! -3 1 1))
                    (("" (replace -1)
                      (("" (case "xx<=yy")
                        (("1"
                          (name "qxx" "sqrt_newton(xx, 10 ^ (-n) / 2)")
                          (("1"
                            (name "qyy"
                                  "sqrt_newton(yy, 10 ^ (-n) / 2)")
                            (("1" (assert)
                              (("1"
                                (case "xx = yy")
                                (("1" (assertnil nil)
                                 ("2"
                                  (case "NOT xx)
                                  (("1" (assertnil nil)
                                   ("2"
                                    (hide 1)
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                                      (name "N" "2*10^n")
                                      (("2"
                                        (replace -1)
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                                          (case "yy-xx>=1/N")
                                          (("1"
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                                              (("1"
                                                (typepred "qyy")
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (case
                                                     "1/N = 10^(-n)/2")
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (replace -1)
                                                        (("1"
                                                          (name
                                                           "epsi"
                                                           "10^(-n)/2")
                                                          (("1"
                                                            (replace
                                                             -1)
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                                                                (assert)
                                                                nil
                                                                nil)
                                                               ("2"
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                                                                 (-5
                                                                  -6
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                                                                  (("1"
                                                                    (assert)
                                                                    nil
                                                                    nil)
                                                                   ("2"
                                                                    (lemma
                                                                     "sq_le")
                                                                    (("2"
                                                                      (inst?)
                                                                      (("2"
                                                                        (assert)
                                                                        nil
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
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                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (assert)
                                                      (("2"
                                                        (expand "N" 1)
                                                        (("2"
                                                          (cross-mult
                                                           1)
                                                          (("2"
                                                            (lemma
                                                             "expt_plus")
                                                            (("2"
                                                              (inst
                                                               -
                                                               "n"
                                                               "-n"
                                                               "10")
                                                              (("2"
                                                                (assert)
                                                                (("2"
                                                                  (replace
                                                                   -1
                                                                   :dir
                                                                   rl)
                                                                  (("2"
                                                                    (hide-all-but
                                                                     1)
                                                                    (("2"
                                                                      (grind)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
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                                                    nil))
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                                            nil)
                                           ("2"
                                            (assert)
                                            (("2"
                                              (hide-all-but (-2 1))
                                              (("2"
                                                (expand "xx")
                                                (("2"
                                                  (expand "yy")
                                                  (("2"
                                                    (expand "N")
                                                    (("2"
                                                      (case
                                                       "floor(2 * (10 ^ n * x)) <=
       floor(2 * (10 ^ n * y))-1")
                                                      (("1"
                                                        (mult-by
                                                         -1
                                                         "1/(2*10^n)")
                                                        (("1"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (hide 2)
                                                        (("2"
                                                          (case
                                                           "floor(2 * (10 ^ n * x)) < floor(2 * (10 ^ n * y))")
                                                          (("1"
                                                            (assert)
                                                            nil
                                                            nil)
                                                           ("2"
                                                            (mult-by
                                                             1
                                                             "1/(2*10^n)")
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (assert)
                          (("2"
                            (case "floor(2 * (10 ^ n * x)) <= floor(2 * (10 ^ n * y))")
                            (("1" (mult-by -1 "1/(2*10^n)")
                              (("1" (assertnil nil)) nil)
                             ("2" (mult-by -5 "2*10^n")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((sqrt_approx const-decl "{z | abs(sq(z) - x) < 10 ^ (-n) / 2}"
     sqrt_approx 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)
    (sq const-decl "nonneg_real" sq 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)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals 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)
    (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)
    (nnreal 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)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (rat_div_nzrat_is_rat application-judgement "rat" rationals nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (posint_exp application-judgement "posint" exponentiation nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (minus_int_is_int application-judgement "int" integers nil)
    (posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (sqrt_newton const-decl "{z | x < sq(z) AND sq(z) - x < eps}"
                 sqrt_approx nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posint nonempty-type-eq-decl nil integers nil)
    (rat_minus_rat_is_rat application-judgement "rat" rationals nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (div_cancel3 formula-decl nil real_props nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (posrat_times_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (times_div2 formula-decl nil real_props nil)
    (div_cancel4 formula-decl nil real_props nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (expt def-decl "real" exponentiation nil)
    (posnat_expt application-judgement "posnat" exponentiation nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (expt_plus formula-decl nil exponentiation nil)
    (N skolem-const-decl "posint" sqrt_approx nil)
    (posrat nonempty-type-eq-decl nil rationals nil)
    (sq_le formula-decl nil sq nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (yy skolem-const-decl "nonneg_rat" sqrt_approx nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (rat_times_rat_is_rat application-judgement "rat" rationals nil)
    (nnrat_times_nnrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (both_sides_times_pos_le1_imp formula-decl nil extra_real_props
     nil)
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil)
    (xx skolem-const-decl "nonneg_rat" sqrt_approx nil)
    (integer nonempty-type-from-decl nil integers nil)
    (<= const-decl "bool" reals nil)
    (+ const-decl "[numfield, 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)
    (nonneg_rat nonempty-type-eq-decl nil rationals nil)
    (sq_nz_pos application-judgement "posreal" sq nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (nnrat_exp application-judgement "nnrat" exponentiation nil)
    (posrat_exp application-judgement "posrat" exponentiation nil))
   shostak))
 (sqrt_old_bounds 0
  (sqrt_old_bounds-1 nil 3576862732
   ("" (skeep)
    (("" (split)
      (("1" (expand "sqrt_lb_old")
        (("1" (name "q" "sqrt_approx(x,2*n)")
          (("1" (replaces -1)
            (("1" (typepred "q")
              (("1" (hide -1)
                (("1" (case "NOT x > sq(q)-10 ^ (-(2 * n)) / 2")
                  (("1" (hide 2) (("1" (grind :exclude "^"nil nil))
                    nil)
                   ("2" (lemma "sqrt_gt")
                    (("2" (expand "max")
                      (("2" (lift-if)
                        (("2" (ground)
                          (("2"
                            (inst - "x" "sq(q)-10 ^ (-(2 * n)) / 2")
                            (("1" (assert)
                              (("1"
                                (case
                                 "sqrt(sq(q)-10 ^ (-(2 * n)) / 2)>=q-10 ^ (-n)")
                                (("1" (assertnil nil)
                                 ("2"
                                  (lemma "sq_ge")
                                  (("2"
                                    (inst?)
                                    (("2"
                                      (assert)
                                      (("2"
                                        (rewrite "sq_minus")
                                        (("2"
                                          (assert)
                                          (("2"
                                            (case
                                             "2 * (10 ^ (-n) * q) >= 10 ^ (-(2 * n)) / 2 + sq(10 ^ (-n))")
                                            (("1" (assertnil nil)
                                             ("2"
                                              (hide 2)
                                              (("2"
                                                (case
                                                 "2 * 10 ^ (-n) >= 10 ^ (-n) / 2 + 10 ^ (-n)")
                                                (("1"
                                                  (case "q>=10^(-n)")
                                                  (("1"
                                                    (mult-ineq -1 -2)
                                                    (("1"
                                                      (assert)
                                                      (("1"
                                                        (expand "sq")
                                                        (("1"
                                                          (lemma
                                                           "expt_plus")
                                                          (("1"
                                                            (inst
                                                             -
                                                             "-n"
                                                             "-n"
                                                             "10")
                                                            (("1"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (lemma "sq_ge")
                              (("2"
                                (inst - "q" "10^(-n)")
                                (("2"
                                  (assert)
                                  (("2"
                                    (expand "sq")
                                    (("2"
                                      (rewrite "expt_plus" :dir rl)
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (expand "sqrt_ub_old")
        (("2" (name "q" "sqrt_approx(x,2*n)")
          (("2" (replaces -1)
            (("2" (typepred "q")
              (("2" (hide -1)
                (("2" (case "NOT x < sq(q)+10 ^ (-(2 * n)) / 2")
                  (("1" (hide 2) (("1" (grind :exclude "^"nil nil))
                    nil)
                   ("2" (lemma "sqrt_lt")
                    (("2" (inst - "x" "sq(q)+10 ^ (-(2 * n)) / 2")
                      (("2" (assert)
                        (("2"
                          (case "sqrt(sq(q)+10 ^ (-(2 * n)) / 2)<=q+10 ^ (-n)")
                          (("1" (assertnil nil)
                           ("2" (hide 2)
                            (("2" (lemma "sq_le")
                              (("2"
                                (inst?)
                                (("2"
                                  (assert)
                                  (("2"
                                    (rewrite "sq_plus")
                                    (("2"
--> --------------------

--> maximum size reached

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

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