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Datei: cache.scala   Sprache: VDM

Original von: PVS©

(bisection_nat_sqrt
 (fs_lt_def 0
  (fs_lt_def-1 nil 3394267943 ("" (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)
    (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)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     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)
    (sq const-decl "nonneg_real" sq "reals/")
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   shostak))
 (fs_eq_def 0
  (fs_eq_def-1 nil 3394267949 ("" (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)
    (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)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (even_times_int_is_even application-judgement "even_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)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   shostak))
 (fs_gt_def 0
  (fs_gt_def-1 nil 3394267959 ("" (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)
    (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)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   shostak))
 (fs_trichotomy 0
  (fs_trichotomy-1 nil 3394267964 ("" (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)
    (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)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     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)
    (sq const-decl "nonneg_real" sq "reals/")
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   shostak))
 (fs_le_ge 0
  (fs_le_ge-1 nil 3394269618
   ("" (skosimp)
    (("" (expand "fs_le?")
      (("" (expand "fs_ge?")
        (("" (expand "fs_lt?")
          (("" (expand "fs_eq?")
            (("" (expand "fs_gt?")
              (("" (split -1)
                (("1" (split)
                  (("1" (flatten)
                    (("1" (lemma "sq_lt" ("nna" "1+m!1" "nnb" "1+n!1"))
                      (("1" (assertnil nil)) nil))
                    nil)
                   ("2" (lemma "sq_lt" ("nna" "1+m!1" "nnb" "n!1"))
                    (("2" (assertnil nil)) nil))
                  nil)
                 ("2" (split)
                  (("1" (flatten)
                    (("1" (lemma "sq_lt" ("nna" "n!1" "nnb" "1+m!1"))
                      (("1" (lemma "sq_lt" ("nna" "m!1" "nnb" "1+n!1"))
                        (("1" (assertnil nil)) nil))
                      nil))
                    nil)
                   ("2" (flatten)
                    (("2" (lemma "sq_lt" ("nna" "m!1" "nnb" "n!1"))
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((fs_le? const-decl "bool" bisection_nat_sqrt nil)
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields 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_lt formula-decl nil sq "reals/")
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (fs_ge? const-decl "bool" bisection_nat_sqrt nil))
   shostak))
 (bisection_sqrt_aux_TCC1 0
  (bisection_sqrt_aux_TCC1-1 nil 3394269004
   ("" (skosimp)
    (("" (typepred "m!1")
      (("" (typepred "n!1")
        (("" (lemma "fs_le_ge" ("x" "x!1" "m" "m!1" "n" "n!1"))
          (("" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((fs_le? const-decl "bool" bisection_nat_sqrt 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)
    (fs_le_ge formula-decl nil bisection_nat_sqrt nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (fs_ge? const-decl "bool" bisection_nat_sqrt nil))
   nil))
 (bisection_sqrt_aux_TCC2 0
  (bisection_sqrt_aux_TCC2-1 nil 3394269004
   ("" (skosimp*)
    (("" (typepred "t!1")
      (("" (assert)
        (("" (expand "fs_le?")
          (("" (flatten)
            (("" (expand "fs_lt?")
              (("" (expand "fs_eq?") (("" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((< const-decl "bool" reals nil)
    (fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_le? const-decl "bool" bisection_nat_sqrt nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields 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)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (integer nonempty-type-from-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (<= const-decl "bool" reals nil)
    (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_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   nil))
 (bisection_sqrt_aux_TCC3 0
  (bisection_sqrt_aux_TCC3-1 nil 3394269004
   ("" (skosimp*)
    (("" (typepred "m!1")
      (("" (typepred "n!1")
        (("" (lemma "nonneg_floor_is_nat" ("x" "(n!1+m!1)/2"))
          (("" (assert)
            (("" (hide-all-but (-4 2))
              (("" (expand "fs_le?")
                (("" (expand "fs_lt?")
                  (("" (flatten)
                    (("" (expand "fs_eq?")
                      (("" (replace -1) (("" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((fs_le? const-decl "bool" bisection_nat_sqrt 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)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields 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)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nonneg_floor_is_nat judgement-tcc nil floor_ceil nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt 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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   nil))
 (bisection_sqrt_aux_TCC4 0
  (bisection_sqrt_aux_TCC4-1 nil 3394269004
   ("" (skosimp*)
    (("" (lemma "fs_le_ge" ("x" "x!1" "m" "m!1" "n" "n!1"))
      (("" (assertnil nil)) nil))
    nil)
   ((fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_le? const-decl "bool" bisection_nat_sqrt 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)
    (fs_le_ge formula-decl nil bisection_nat_sqrt nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   nil))
 (bisection_sqrt_aux_TCC5 0
  (bisection_sqrt_aux_TCC5-1 nil 3394269004
   ("" (skosimp*)
    (("" (lemma "fs_le_ge" ("x" "x!1" "m" "m!1" "n" "n!1"))
      (("" (lemma "nonneg_floor_is_nat" ("x" "(n!1+m!1)/2"))
        (("" (assert)
          (("" (expand "fs_eq?")
            (("" (expand "fs_lt?")
              (("" (expand "fs_ge?")
                (("" (expand "fs_gt?") (("" (assertnil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_le? const-decl "bool" bisection_nat_sqrt 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)
    (fs_le_ge formula-decl nil bisection_nat_sqrt nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt 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)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (nonneg_floor_is_nat judgement-tcc nil floor_ceil nil)
    (nonneg_real nonempty-type-eq-decl nil real_types 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)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   nil))
 (bisection_sqrt_aux_TCC6 0
  (bisection_sqrt_aux_TCC6-1 nil 3413715945
   ("" (skosimp*)
    (("" (lemma "fs_le_ge" ("x" "x!1" "m" "m!1" "n" "n!1"))
      (("" (assertnil nil)) nil))
    nil)
   ((fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_le? const-decl "bool" bisection_nat_sqrt 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)
    (fs_le_ge formula-decl nil bisection_nat_sqrt nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil))
   nil))
 (bisection_sqrt_aux_prop 0
  (bisection_sqrt_aux_prop-1 nil 3412657227
   (""
    (case "FORALL (k,m, n, x: nat):
        n-m<=k & fs_le?(x)(m) AND fs_ge?(x)(n) =>
         fs_eq?(x, bisection_sqrt_aux(x, m, n))")
    (("1" (skosimp)
      (("1" (inst - "n!1-m!1" "m!1" "n!1" "x!1")
        (("1" (assertnil nil)
         ("2" (assert)
          (("2" (hide 2)
            (("2" (expand "fs_le?")
              (("2" (expand "fs_ge?")
                (("2" (expand "fs_eq?")
                  (("2" (expand "fs_lt?")
                    (("2" (expand "fs_gt?")
                      (("2" (split)
                        (("1" (split)
                          (("1" (flatten)
                            (("1"
                              (lemma "sq_lt"
                               ("nna" "1+m!1" "nnb" "1+n!1"))
                              (("1" (assertnil nil)) nil))
                            nil)
                           ("2"
                            (lemma "sq_lt" ("nna" "1+m!1" "nnb" "n!1"))
                            (("2" (assertnil nil)) nil))
                          nil)
                         ("2" (split)
                          (("1" (flatten)
                            (("1"
                              (lemma "sq_lt"
                               ("nna" "m!1" "nnb" "1+n!1"))
                              (("1"
                                (lemma
                                 "sq_lt"
                                 ("nna" "n!1" "nnb" "1+m!1"))
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil)
                           ("2" (flatten)
                            (("2"
                              (lemma "sq_lt" ("nna" "m!1" "nnb" "n!1"))
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2" (induct "k")
        (("1" (skosimp)
          (("1" (expand "<=")
            (("1" (assert)
              (("1" (split)
                (("1" (assert)
                  (("1" (expand "fs_le?")
                    (("1" (hide 1)
                      (("1" (expand "fs_ge?")
                        (("1" (expand "fs_eq?")
                          (("1" (expand "fs_lt?")
                            (("1" (expand "fs_gt?")
                              (("1"
                                (split)
                                (("1"
                                  (split)
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (lemma
                                       "sq_lt"
                                       ("nna" "1+m!1" "nnb" "1+n!1"))
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (lemma
                                     "sq_lt"
                                     ("nna" "1+m!1" "nnb" "n!1"))
                                    (("2" (assertnil nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (split)
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (lemma
                                       "sq_lt"
                                       ("nna" "m!1" "nnb" "1+n!1"))
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (flatten)
                                    (("2"
                                      (lemma
                                       "sq_lt"
                                       ("nna" "m!1" "nnb" "n!1"))
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (expand "bisection_sqrt_aux")
                  (("2" (assert)
                    (("2" (expand "fs_le?")
                      (("2" (expand "fs_ge?")
                        (("2" (expand "fs_lt?")
                          (("2" (expand "fs_gt?")
                            (("2" (expand "fs_eq?")
                              (("2"
                                (lemma
                                 "sq_lt"
                                 ("nna" "m!1" "nnb" "1+m!1"))
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (skosimp*)
          (("2" (expand "bisection_sqrt_aux" 1)
            (("2" (case-replace "n!1=m!1")
              (("1" (assert)
                (("1" (expand "fs_le?")
                  (("1" (expand "fs_ge?")
                    (("1" (hide -1 -2 -3)
                      (("1" (expand "fs_gt?")
                        (("1" (expand "fs_lt?")
                          (("1" (expand "fs_eq?")
                            (("1"
                              (lemma "sq_lt"
                               ("nna" "1+m!1" "nnb" "m!1"))
                              (("1" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (assert)
                (("2" (case "n!1>m!1")
                  (("1" (hide 1)
                    (("1" (typepred "floor((m!1 + n!1) / 2)")
                      (("1" (name-replace "N" "floor((m!1 + n!1) / 2)")
                        (("1" (rewrite "div_mult_pos_le2")
                          (("1" (rewrite "div_mult_pos_lt1")
                            (("1" (case-replace "fs_lt?(x!1, N)")
                              (("1"
                                (inst - "1+N" "n!1" "x!1")
                                (("1"
                                  (assert)
                                  (("1"
                                    (hide 2)
                                    (("1"
                                      (expand "fs_le?")
                                      (("1"
                                        (flatten)
                                        (("1"
                                          (expand "fs_eq?")
                                          (("1"
                                            (expand "fs_ge?")
                                            (("1"
                                              (expand "fs_lt?")
                                              (("1"
                                                (expand "fs_eq?")
                                                (("1"
                                                  (expand "fs_gt?")
                                                  (("1"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (assert)
                                (("2"
                                  (case-replace "fs_eq?(x!1, N)")
                                  (("2"
                                    (assert)
                                    (("2"
                                      (lemma
                                       "fs_trichotomy"
                                       ("n" "x!1" "m" "N"))
                                      (("2"
                                        (assert)
                                        (("2"
                                          (hide 1 2)
                                          (("2"
                                            (inst - "m!1" "N" "x!1")
                                            (("2"
                                              (assert)
                                              (("2"
                                                (hide 2)
                                                (("2"
                                                  (expand "fs_ge?")
                                                  (("2"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (assert)
                    (("2" (hide-all-but (-3 -4 1 2))
                      (("2" (expand "fs_le?")
                        (("2" (expand "fs_ge?")
                          (("2" (expand "fs_eq?")
                            (("2" (expand "fs_lt?")
                              (("2"
                                (expand "fs_gt?")
                                (("2"
                                  (split)
                                  (("1"
                                    (split)
                                    (("1"
                                      (flatten)
                                      (("1"
                                        (lemma
                                         "sq_lt"
                                         ("nna" "1+m!1" "nnb" "1+n!1"))
                                        (("1" (assertnil nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (lemma
                                       "sq_lt"
                                       ("nna" "1+m!1" "nnb" "n!1"))
                                      (("2" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (split)
                                    (("1"
                                      (flatten)
                                      (("1"
                                        (lemma
                                         "sq_lt"
                                         ("nna" "m!1" "nnb" "1+n!1"))
                                        (("1" (assertnil nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (flatten)
                                      (("2"
                                        (lemma
                                         "sq_lt"
                                         ("nna" "m!1" "nnb" "n!1"))
                                        (("2" (assertnil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((= const-decl "[T, T -> boolean]" equalities nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (> const-decl "bool" reals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (integer nonempty-type-from-decl nil integers nil)
    (< const-decl "bool" reals nil)
    (floor const-decl "{i | i <= x & x < i + 1}" floor_ceil nil)
    (/= const-decl "boolean" notequal nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (div_mult_pos_le2 formula-decl nil real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (int_times_even_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (even_plus_even_is_even application-judgement "even_int" integers
     nil)
    (fs_trichotomy formula-decl nil bisection_nat_sqrt nil)
    (posint_times_posint_is_posint application-judgement "posint"
     integers nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (div_mult_pos_lt1 formula-decl nil real_props nil)
    (nnrat_div_posrat_is_nnrat application-judgement "nonneg_rat"
     rationals nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (nat_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (sq_lt formula-decl nil sq "reals/")
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (m!1 skolem-const-decl "nat" bisection_nat_sqrt nil)
    (n!1 skolem-const-decl "nat" bisection_nat_sqrt nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (fs_le? const-decl "bool" bisection_nat_sqrt nil)
    (fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (bisection_sqrt_aux def-decl "nat" bisection_nat_sqrt nil))
   shostak))
 (bisection_sqrt_TCC1 0
  (bisection_sqrt_TCC1-1 nil 3394269004
   ("" (skosimp)
    (("" (expand "fs_le?")
      (("" (expand "fs_lt?")
        (("" (expand "fs_eq?")
          (("" (flatten) (("" (assertnil nil)) nil)) nil))
        nil))
      nil))
    nil)
   ((fs_le? const-decl "bool" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (sq_1 formula-decl nil sq "reals/")
    (sq_0 formula-decl nil sq "reals/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil))
   nil))
 (bisection_sqrt_TCC2 0
  (bisection_sqrt_TCC2-1 nil 3394269004
   ("" (skosimp)
    (("" (expand "fs_ge?")
      (("" (expand "fs_eq?")
        (("" (expand "fs_gt?")
          (("" (flatten)
            (("" (assert)
              (("" (expand "sq") (("" (assertnil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil))
   nil))
 (bisection_sqrt_def 0
  (bisection_sqrt_def-1 nil 3394270385
   ("" (skosimp)
    (("" (expand "bisection_sqrt")
      ((""
        (lemma "bisection_sqrt_aux_prop" ("x" "n!1" "m" "0" "n" "n!1"))
        (("" (split -1)
          (("1" (name-replace "M" "bisection_sqrt_aux(n!1, 0, n!1)")
            (("1" (expand "fs_eq?")
              (("1" (typepred "floor(sqrt(n!1))")
                (("1" (name-replace "N" "floor(sqrt(n!1))")
                  (("1" (lemma "sq_le" ("nna" "N" "nnb" "sqrt(n!1)"))
                    (("1"
                      (lemma "sq_lt" ("nnb" "N+1" "nna" "sqrt(n!1)"))
                      (("1" (assert)
                        (("1" (flatten)
                          (("1" (lemma "sq_lt" ("nna" "M" "nnb" "1+N"))
                            (("1"
                              (lemma "sq_lt" ("nna" "N" "nnb" "1+M"))
                              (("1" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide 2)
            (("2" (expand "fs_le?")
              (("2" (expand "fs_lt?")
                (("2" (expand "fs_eq?")
                  (("2" (flatten)
                    (("2" (expand "sq") (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("3" (hide 2)
            (("3" (expand "fs_ge?")
              (("3" (expand "fs_eq?")
                (("3" (expand "fs_gt?")
                  (("3" (flatten)
                    (("3" (case-replace "n!1=0")
                      (("1" (expand "sq") (("1" (propax) nil nil)) nil)
                       ("2" (case-replace "n!1=1")
                        (("1" (expand "sq") (("1" (propax) nil nil))
                          nil)
                         ("2"
                          (lemma "both_sides_times_pos_le1"
                           ("x" "2" "y" "n!1" "pz" "n!1"))
                          (("1" (rewrite "sq_rew")
                            (("1" (assertnil nil)) nil)
                           ("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((bisection_sqrt const-decl "nat" bisection_nat_sqrt nil)
    (fs_eq? const-decl "bool" bisection_nat_sqrt nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (sq_lt formula-decl nil sq "reals/")
    (sq_sqrt formula-decl nil sqrt "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (sq_le formula-decl nil sq "reals/")
    (nonneg_floor_is_nat application-judgement "nat" floor_ceil nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (integer nonempty-type-from-decl nil integers nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (< const-decl "bool" reals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (floor const-decl "{i | i <= x & x < i + 1}" floor_ceil nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nnreal type-eq-decl nil real_types nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sqrt const-decl "{nnz: nnreal | nnz * nnz = nnx}" sqrt "reals/")
    (bisection_sqrt_aux def-decl "nat" bisection_nat_sqrt nil)
    (fs_ge? const-decl "bool" bisection_nat_sqrt nil)
    (fs_le? const-decl "bool" bisection_nat_sqrt nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (fs_lt? const-decl "bool" bisection_nat_sqrt nil)
    (fs_gt? const-decl "bool" bisection_nat_sqrt nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (even_times_int_is_even application-judgement "even_int" integers
     nil)
    (nnint_times_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (sq_rew formula-decl nil sq "reals/")
    (bisection_sqrt_aux_prop formula-decl nil bisection_nat_sqrt 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))
   shostak)))


¤ Dauer der Verarbeitung: 0.85 Sekunden  (vorverarbeitet)  ¤





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