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Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: int63.v   Sprache: Lisp

Original von: PVS©

(closest_approach
 (sq_dist_lem 0
  (sq_dist_lem-1 nil 3256049050
   ("" (skosimp*)
    (("" (assert)
      (("" (rewrite "sq_dist_is_dist_sq")
        (("" (rewrite "dist_norm")
          (("" (expand "norm")
            (("" (rewrite "sq_sqrt")
              (("" (expand "sqv")
                (("" (expand "*")
                  (("" (expand "+")
                    (("" (expand "-") (("" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((dist_norm formula-decl nil distance nil)
    (sq_sqrt formula-decl nil sqrt "reals/")
    (nnreal type-eq-decl nil real_types nil)
    (sqv const-decl "nnreal" vectors nil)
    (- const-decl "real" vectors nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (* const-decl "real" vectors nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (int_minus_int_is_int application-judgement "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_times_real_is_real application-judgement "real" reals nil)
    (norm const-decl "nnreal" vectors nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (* const-decl "Vector" vectors nil)
    (+ const-decl "real" vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (sq_dist_is_dist_sq formula-decl nil distance nil))
   nil))
 (sq_dist_quad 0
  (sq_dist_quad-1 nil 3257266340
   ("" (skosimp*)
    (("" (replace -1)
      (("" (hide -1)
        (("" (replace -1)
          (("" (hide -1)
            (("" (replace -1)
              (("" (hide -1)
                (("" (replace -1)
                  (("" (hide -1)
                    (("" (expand "sq_dist")
                      (("" (expand "*")
                        ((""
                          (expand "+
")
                          ((""
                            (case-replace "sigma(0, n - 1,
             LAMBDA (i: Index[n]): (p0!1 - q0!1)(i) * (p0!1 - q0!1)(i))
        +
        sigma(0, n - 1,
              LAMBDA (i_1: Index[n]):
                2 * ((p0!1 - q0!1)(i_1) * (u!1 - v!1)(i_1)))
         * t!1
        +
        sigma(0, n - 1,
              LAMBDA (i: Index[n]): (u!1 - v!1)(i) * (u!1 - v!1)(i))
         * sq(t!1) =

         sigma(0, n - 1,
             (LAMBDA (i: Index[n]): (p0!1 - q0!1)(i) * (p0!1 - q0!1)(i)
               +  2 * t!1 *((p0!1 - q0!1)(i) * (u!1 - v!1)(i))
               + sq(t!1)*(u!1 - v!1)(i) * (u!1 - v!1)(i)))")
                            (("1" (hide -1)
                              (("1"
                                (case-replace
                                 "(LAMBDA (i: Index[n]):
                (p0!1 - q0!1)(i) * (p0!1 - q0!1)(i) +
                 2 * t!1 * ((p0!1 - q0!1)(i) * (u!1 - v!1)(i))
                 + sq(t!1) * (u!1 - v!1)(i) * (u!1 - v!1)(i))
 =             (LAMBDA (i: Index[n]):
              sq(p0!1(i) - q0!1(i) - t!1 * v!1(i) + t!1 * u!1(i)))")
                                (("1"
                                  (hide 2)
                                  (("1"
                                    (apply-extensionality 1 :hide? t)
                                    (("1" (grind) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide 2)
                              (("2"
                                (lemma "sigma_scal[Index[n]]")
                                (("1"
                                  (inst -1 "_" "t!1" "n-1" "0")
                                  (("1"
                                    (inst
                                     -1
                                     "LAMBDA (i_1: Index[n]):
               2 * ((p0!1 - q0!1)(i_1) * (u!1 - v!1)(i_1))")
                                    (("1"
                                      (replace -1 * rl)
                                      (("1"
                                        (hide -1)
                                        (("1"
                                          (lemma
                                           "sigma_scal[Index[n]]")
                                          (("1"
                                            (inst
                                             -1
                                             "_"
                                             "sq(t!1)"
                                             "n-1"
                                             "0")
                                            (("1"
                                              (inst
                                               -1
                                               "(LAMBDA (i: Index[n]): (u!1 - v!1)(i) * (u!1 - v!1)(i))")
                                              (("1"
                                                (replace -1 * rl)
                                                (("1"
                                                  (hide -1)
                                                  (("1"
                                                    (rewrite
                                                     "sigma_sum[Index[n]]")
                                                    (("1"
                                                      (rewrite
                                                       "sigma_sum[Index[n]]")
                                                      (("1"
                                                        (rewrite
                                                         "sigma_eq[Index[n]]")
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2" (assertnil nil)
                                             ("3" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2" (assertnil nil)
                                   ("3" (assertnil nil))
                                  nil)
                                 ("2"
                                  (skosimp*)
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil)
                             ("3" (hide 2)
                              (("3"
                                (skosimp*)
                                (("3" (assertnil nil))
                                nil))
                              nil)
                             ("4" (hide 2) (("4" (assertnil nil))
                              nil)
                             ("5" (hide 2) (("5" (assertnil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((real_times_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (sq_dist const-decl "nnreal" distance nil)
    (+ const-decl "real" vectors nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sigma_eq formula-decl nil sigma "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)
    (sigma_sum formula-decl nil sigma "reals/")
    (sigma_scal formula-decl nil sigma "reals/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (- const-decl "real" vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sigma def-decl "real" sigma "reals/")
    (T_high type-eq-decl nil sigma "reals/")
    (T_low type-eq-decl nil sigma "reals/")
    (<= const-decl "bool" reals nil)
    (Index type-eq-decl nil vectors nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (< const-decl "bool" reals nil) (>= const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (OR const-decl "[bool, bool -> bool]" booleans 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)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-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)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (integer nonempty-type-from-decl nil integers nil)
    (* const-decl "Vector" vectors nil)
    (* const-decl "real" vectors nil))
   shostak))
 (dist_eq_vel 0
  (dist_eq_vel-1 nil 3255968102 ("" (grind) nil nil)
   ((* const-decl "Vector" vectors nil)
    (+ const-decl "real" vectors nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers 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)
    (sq_dist const-decl "nnreal" distance nil)
    (dist const-decl "nnreal" distance nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   shostak))
 (norm_diff_eq_0 0
  (norm_diff_eq_0-1 nil 3255967833
   ("" (skosimp*)
    (("" (rewrite "norm_eq_0")
      (("" (rewrite "sub_eq_zero")
        (("" (replaces -1)
          (("" (rewrite "dist_eq_vel")
            (("" (rewrite "dist_eq_vel"nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((norm_eq_0 formula-decl nil vectors nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (Index type-eq-decl nil vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (- const-decl "real" vectors 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)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (dist_eq_vel formula-decl nil closest_approach nil)
    (sub_eq_zero formula-decl nil vectors nil))
   shostak))
 (time_closest_TCC1 0
  (time_closest_TCC1-1 nil 3255962785
   ("" (skosimp*)
    (("" (rewrite "sq_eq_0")
      (("" (rewrite "norm_eq_0")
        (("" (assert) (("" (postpone) nil nil)) nil)) nil))
      nil))
    nil)
   ((sq_eq_0 formula-decl nil sq "reals/")
    (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)
    (< const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (Index type-eq-decl nil vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (nnreal type-eq-decl nil real_types nil)
    (norm const-decl "nnreal" vectors nil)
    (- const-decl "real" vectors nil))
   shostak))
 (time_closest_lem_TCC1 0
  (time_closest_lem_TCC1-1 nil 3256051573
   ("" (skosimp*)
    (("" (rewrite "dot_sq_norm")
      (("" (assert)
        (("" (replace -2)
          (("" (hide -2)
            (("" (hide -1)
              (("" (lemma "sq_eq_0")
                (("" (inst?) (("" (assertnil nil)) nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((dot_sq_norm formula-decl nil vectors nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (Index type-eq-decl nil vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (- const-decl "real" vectors 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)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (norm const-decl "nnreal" vectors nil)
    (nnreal type-eq-decl nil real_types nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sq_eq_0 formula-decl nil sq "reals/")
    (dot_scal_left formula-decl nil vectors nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   shostak))
 (time_closest_lem 0
  (time_closest_lem-1 nil 3256049208
   ("" (skosimp*)
    (("" (expand "time_closest")
      (("" (assert)
        (("" (rewrite "dot_sq_norm") (("" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((time_closest const-decl "real" closest_approach nil)
    (dot_sq_norm formula-decl nil vectors nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (Index type-eq-decl nil vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (- const-decl "real" vectors 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)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (dot_scal_left formula-decl nil vectors nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (minus_real_is_real application-judgement "real" reals nil))
   shostak))
 (cpa_prep_mono 0
  (cpa_prep_mono-1 nil 3401715712
   ("" (skosimp*)
    (("" (expand "time_closest")
      (("" (assert)
        (("" (case "a!1 > 0")
          (("1" (lemma "quad_min_mono_inc")
            (("1" (inst - "a!1" "b!1" "c!1" "t2!1" "t1!1")
              (("1" (assert)
                (("1" (prop)
                  (("1" (lemma "sq_dist_quad")
                    (("1"
                      (inst -1 "a!1" "b!1" "c!1" "p0!1" "q0!1" "t1!1"
                       "u!1" "v!1" "w0!1")
                      (("1" (assert)
                        (("1" (replace -1)
                          (("1" (lemma "sq_dist_quad")
                            (("1"
                              (inst -1 "a!1" "b!1" "c!1" "p0!1" "q0!1"
                               "t2!1" "u!1" "v!1" "w0!1")
                              (("1"
                                (assert)
                                (("1"
                                  (replace -1)
                                  (("1"
                                    (expand "quadratic")
                                    (("1" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (case-replace "sq(norm(u!1 - v!1)) = a!1")
                    (("1" (assertnil nil)
                     ("2" (hide 2 3 4)
                      (("2" (lemma "dot_sq_norm[n]")
                        (("2" (inst?) (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (assertnil nil))
              nil))
            nil)
           ("2" (assert)
            (("2" (rewrite "dot_sq_norm")
              (("2" (lemma "sq_eq_0")
                (("2" (inst?) (("2" (assertnil nil)) nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((time_closest const-decl "real" closest_approach 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)
    (/= const-decl "boolean" notequal nil)
    (a!1 skolem-const-decl "real" closest_approach nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (< const-decl "bool" reals 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)
    (quadratic const-decl "real" quadratic "reals/")
    (real_plus_real_is_real application-judgement "real" reals nil)
    (sq_dist_quad formula-decl nil closest_approach nil)
    (dot_sq_norm formula-decl nil vectors nil)
    (- const-decl "real" vectors nil)
    (norm const-decl "nnreal" vectors nil)
    (nnreal type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (quad_min_mono_inc formula-decl nil quad_minmax "reals/")
    (sq_eq_0 formula-decl nil sq "reals/")
    (dot_scal_left formula-decl nil vectors nil)
    (real_times_real_is_real application-judgement "real" reals 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_div_nzreal_is_real application-judgement "real" reals nil)
    (minus_real_is_real application-judgement "real" reals nil))
   nil))
 (time_cpa 0
  (time_cpa-2 nil 3421667533
   ("" (skosimp*)
    (("" (case "norm(u!1 - v!1) = 0")
      (("1" (expand "is_minimum?")
        (("1" (skosimp*)
          (("1" (lemma "norm_diff_eq_0")
            (("1" (inst -1 "p0!1" "q0!1" "t_cpa!1" "y!1" "u!1" "v!1")
              (("1" (assert)
                (("1" (assert)
                  (("1" (expand "dist")
                    (("1" (rewrite "sqrt_eq"nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (expand "time_closest")
        (("2" (name "A" "(u!1-v!1)*(u!1-v!1)")
          (("2" (name "B" "2*(p0!1-q0!1)*(u!1-v!1)")
            (("2" (name "C" "(p0!1-q0!1)*(p0!1-q0!1)")
              (("2" (case "A > 0")
                (("1" (lemma "quad_min")
                  (("1" (inst -1 "A" "B" "C")
                    (("1" (assert)
                      (("1" (assert)
                        (("1" (lemma "dot_sq_norm[n]")
                          (("1" (inst - "u!1 - v!1")
                            (("1" (assert)
                              (("1"
                                (replace -1 * lr)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (replace -5 * lr)
                                    (("1"
                                      (hide -5)
                                      (("1"
                                        (lemma "sq_dist_quad")
                                        (("1"
                                          (case-replace
                                           "(LAMBDA t: sq_dist(p0!1 + t * u!1, q0!1 + t * v!1)) = quadratic(A, B, C)")
                                          (("1" (assertnil nil)
                                           ("2"
                                            (apply-extensionality
                                             1
                                             :hide?
                                             t)
                                            (("1"
                                              (inst
                                               -
                                               "A"
                                               "B"
                                               "C"
                                               "p0!1"
                                               "q0!1"
                                               "x!1"
                                               "u!1"
                                               "v!1"
                                               "p0!1-q0!1")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (expand "quadratic")
                                                  (("1"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (skosimp*)
                                              (("2"
                                                (assert)
                                                (("2"
                                                  (expand "quadratic")
                                                  (("2"
                                                    (assert)
                                                    (("2"
                                                      (hide -1)
                                                      (("2"
                                                        (lemma
                                                         "sq_dist_quad")
                                                        (("2"
                                                          (inst
                                                           -1
                                                           "A"
                                                           "B"
                                                           "C"
                                                           "p0!1"
                                                           "q0!1"
                                                           "x1!1"
                                                           "u!1"
                                                           "v!1"
                                                           "p0!1-q0!1")
                                                          (("2"
                                                            (assert)
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (assertnil nil))
                    nil))
                  nil)
                 ("2" (lemma "dot_sq_norm[n]")
                  (("2" (inst - "u!1-v!1")
                    (("2" (lemma "sq_eq_0")
                      (("2" (inst?) (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((- const-decl "real" vectors nil)
    (norm const-decl "nnreal" vectors nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (* const-decl "Vector" vectors nil)
    (+ const-decl "real" vectors nil)
    (sq_dist const-decl "nnreal" distance nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sqrt_eq formula-decl nil sqrt "reals/")
    (dist const-decl "nnreal" distance nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (norm_diff_eq_0 formula-decl nil closest_approach nil)
    (is_minimum? const-decl "bool" quad_minmax "reals/")
    (* const-decl "real" vectors nil)
    (sq_eq_0 formula-decl nil sq "reals/")
    (quad_min formula-decl nil quad_minmax "reals/")
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (dot_scal_left formula-decl nil vectors nil)
    (dot_sq_norm formula-decl nil vectors nil)
    (quadratic const-decl "real" quadratic "reals/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (B skolem-const-decl "real" closest_approach nil)
    (C skolem-const-decl "real" closest_approach nil)
    (sq_dist_quad formula-decl nil closest_approach nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (A skolem-const-decl "real" closest_approach nil)
    (/= const-decl "boolean" notequal nil)
    (time_closest const-decl "real" closest_approach nil))
   nil)
  (time_cpa-1 nil 3255788628
   ("" (skosimp*)
    (("" (lemma "cpa_prep")
      (("" (inst?)
        (("" (name "W0" "p0!1-q0!1")
          ((""
            (inst - "(u!1 - v!1) * (u!1 - v!1)" "2 * W0 * (u!1 - v!1)"
             " W0 * W0" "W0")
            (("" (assertnil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((- const-decl "real" vectors nil)
    (* const-decl "Vector" vectors nil)
    (* const-decl "real" vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil))
   shostak))
 (dist_cpa 0
  (dist_cpa-1 nil 3255788937
   ("" (skosimp*)
    (("" (lemma "time_cpa")
      (("" (inst?)
        (("" (inst?)
          (("" (expand "is_minimum?")
            (("" (inst -1 "t!1")
              (("" (expand "dist") (("" (rewrite "sqrt_le"nil nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((time_cpa formula-decl nil closest_approach nil)
    (time_closest const-decl "real" closest_approach nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sqrt_le formula-decl nil sqrt "reals/")
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nnreal type-eq-decl nil real_types nil)
    (sq_dist const-decl "nnreal" distance nil)
    (+ const-decl "real" vectors nil)
    (* const-decl "Vector" vectors nil)
    (dist const-decl "nnreal" distance nil)
    (is_minimum? const-decl "bool" quad_minmax "reals/")
    (Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
 (dist_cpa_lt 0
  (dist_cpa_lt-2 nil 3401719067
   ("" (skosimp*)
    (("" (assert)
      (("" (lemma "cpa_prep_mono")
        (("" (inst?)
          (("" (name "W0" "p0!1-q0!1")
            ((""
              (inst - "(u!1 - v!1) * (u!1 - v!1)"
               "2 * W0 * (u!1 - v!1)" " W0 * W0" "t1!1" "t2!1" "W0")
              (("" (assertnil nil)) nil))
            nil))
          nil))
        nil))
      nil))
    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)
    (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)
    (< const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (Index type-eq-decl nil vectors nil)
    (Vector type-eq-decl nil vectors nil)
    (* const-decl "Vector" vectors nil)
    (* const-decl "real" vectors nil) (- const-decl "real" vectors nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (cpa_prep_mono formula-decl nil closest_approach nil))
   nil)
  (dist_cpa_lt-1 nil 3401703645
   ("" (skosimp*)
    (("" (lemma "time_cpa_lt")
      (("" (inst?)
        (("" (inst - "t_cpa!1") (("" (assertnil nil)) nil)) nil))
      nil))
    nil)
   ((Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil))
   nil))
 (dist_min 0
  (dist_min-1 nil 3255958397
   ("" (skosimp*)
    (("" (assert)
      (("" (flatten)
        (("" (lemma "dist_cpa")
          (("" (inst -1 "p0!1" "q0!1" "t1!1" "u!1" "v!1")
            (("" (assertnil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (dist_cpa formula-decl nil closest_approach nil)
    (Vector type-eq-decl nil vectors nil)
    (Index type-eq-decl nil vectors nil)
    (n formal-const-decl "posnat" closest_approach nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (> const-decl "bool" reals nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
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