Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/PVS/ACCoRD/   (Beweissystem der NASA Version 6.0.9©)  Datei vom 28.9.2014 mit Größe 3 MB image not shown  

Quelle  circle_optimum_2D.prf

  Sprache: Lisp
 

(circle_optimum_2D
 (intersects_circle_fun_scal 0
  (intersects_circle_fun_scal-1 nil 3527949260
   (""
    (case "FORALL (R: posreal, c: Vect2, k: posreal, v: Vect2):
                                    intersects_circle_fun?(v, c, R) IMPLIES
                                     intersects_circle_fun?(k * v, c, R)")
    (("1" (skeep)
      (("1" (ground)
        (("1" (inst?) (("1" (assertnil nil)) nil)
         ("2" (inst - "R" "c" "1/k" "k*v") (("2" (assertnil nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
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              (("1" (assert)
                (("1" (replace -1) (("1" (assertnil nil)) nil)) nil)
               ("2" (case "k*v /= zero")
                (("1" (assert)
                  (("1" (flatten)
                    (("1" (lemma "Delta_scal[R]")
                      (("1" (inst - "-c" "k" "v")
                        (("1" (replace -1)
                          (("1" (hide -1)
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                                (assert)
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                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
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                        nil))
                      nil))
                    nil))
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                    (("2" (expand "zero")
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                        (("2" (decompose-equality +)
                          (("1" (mult-by 1 "k")
                            (("1" (rewrite "vx_scal")
                              (("1" (assertnil nil)) nil))
                            nil)
                           ("2" (mult-by 1 "k")
                            (("2" (rewrite "vy_scal")
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((= const-decl "[T, T -> boolean]" equalities nil)
    (zero const-decl "Vector" vectors_2D "vectors/")
    (scal_zero formula-decl nil vectors_2D "vectors/")
    (dot_scal_right formula-decl nil vectors_2D "vectors/")
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (vx_scal formula-decl nil vectors_2D "vectors/")
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (both_sides_times1 formula-decl nil real_props nil)
    (vy_scal formula-decl nil vectors_2D "vectors/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (Delta_scal formula-decl nil horizontal nil)
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (Delta const-decl "real" horizontal nil)
    (* const-decl "real" vectors_2D "vectors/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (- const-decl "Vector" vectors_2D "vectors/")
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     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)
    (scal_assoc formula-decl nil vectors_2D "vectors/")
    (scal_1 formula-decl nil vectors_2D "vectors/")
    (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)
    (bool nonempty-type-eq-decl nil booleans 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)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (* const-decl "Vector" vectors_2D "vectors/"))
   shostak))
 (intersects_circle_fun_def 0
  (intersects_circle_fun_def-1 nil 3527524122
   ("" (skeep)
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              (("1" (ground)
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                  (("1" (assert)
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                    nil))
                  nil)
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                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
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          nil))
        nil)
       ("2" (ground)
        (("1" (expand "intersects_circle_fun?")
          (("1" (flatten)
            (("1" (inst + "Theta_D[R](-c,v,1)*v")
              (("1" (split +)
                (("1" (expand "circle?")
                  (("1" (lemma "Theta_D_unique[R]")
                    (("1" (inst - "v" "-c" "Theta_D[R](-c, v, 1)")
                      (("1" (assert)
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                          (("1" (rewrite "sqrt_sqv_norm")
                            (("1" (hide-all-but (-1 1))
                              (("1"
                                (grind :exclude ("norm" "Theta_D"))
                                nil
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (expand "segment?")
                  (("2" (inst?)
                    (("2" (lemma "Theta_D_ge_0[R]")
                      (("2" (case "NOT sqv(-c) <= sq(R)")
                        (("1" (assert)
                          (("1" (inst?)
                            (("1" (assert)
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                                  (("1"
                                    (inst?)
                                    (("1" (assertnil nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide-all-but (-2 1))
                                  (("2" (grind) nil nil))
                                  nil))
                                nil))
                              nil))
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                          nil)
                         ("2" (assert)
                          (("2" (hide -4)
                            (("2" (hide -2)
                              (("2"
                                (lemma
                                 "horizontal_los_inside_Theta[R]")
                                (("2"
                                  (inst?)
                                  (("2"
                                    (inst - "0")
                                    (("2"
                                      (assert)
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                                        (lemma "Theta_D_unique[R]")
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                                          (inst?)
                                          (("2"
                                            (inst - "0")
                                            (("2"
                                              (assert)
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                                                (lemma "Theta_D_le[R]")
                                                (("2"
                                                  (inst?)
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
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                    nil))
                  nil))
                nil))
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            nil))
          nil)
         ("2" (skeep -)
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              (("2" (replaces -2)
                (("2" (expand "intersects_circle_fun?")
                  (("2" (case "NOT Delta[R](-c, v) >= 0")
                    (("1" (hide 2)
                      (("1" (rewrite "Delta_ge_0")
                        (("1" (inst + "t")
                          (("1" (expand "circle?")
                            (("1" (rewrite "sqrt_eq" 1 :dir rl)
                              (("1"
                                (rewrite "sqrt_sqv_norm")
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                                nil))
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                      nil)
                     ("2" (assert)
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                        (("2" (case "t = 0")
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                                (assert)
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                                  (rewrite "sqrt_le" 2 :dir rl)
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                                    (rewrite "sqrt_sqv_norm")
                                    (("1" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (expand "circle?")
                            (("2"
                              (lemma "horizontal_conflict[norm(c)]")
                              (("1"
                                (inst - "v" "-c")
                                (("1"
                                  (flatten)
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                                      (split -)
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                                               "sqrt_lt"
                                               :dir
                                               rl)
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                                                (assert)
                                                (("1"
                                                  (rewrite
                                                   "sqrt_sqv_norm")
                                                  (("1"
                                                    (assert)
                                                    nil
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                                            nil)
                                           ("2"
                                            (hide-all-but (-1 2))
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                                       ("2"
                                        (expand "horizontal_conflict?")
                                        (("2"
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                                             "sqrt_lt"
                                             1
                                             :dir
                                             rl)
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                                              (assert)
                                              (("2"
                                                (rewrite
                                                 "sqrt_sqv_norm")
                                                (("2"
                                                  (rewrite
                                                   "sqrt_le"
                                                   4
                                                   :dir
                                                   rl)
                                                  (("2"
                                                    (rewrite
                                                     "sqrt_sqv_norm")
                                                    (("2"
                                                      (assert)
                                                      (("2"
                                                        (hide -1)
                                                        (("2"
                                                          (grind
                                                           :exclude
                                                           "norm")
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (lemma "norm_eq_0")
                                (("2"
                                  (inst?)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (replace -1)
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((zero const-decl "Vector" vectors_2D "vectors/")
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (real nonempty-type-from-decl nil reals nil)
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (segment? const-decl "bool" circle_optimum_2D nil)
    (sqrt_eq formula-decl nil sqrt "reals/")
    (sqv const-decl "nnreal" vectors_2D "vectors/")
    (sq const-decl "nonneg_real" sq "reals/")
    (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)
    (sqrt_pos application-judgement "posreal" sqrt "reals/")
    (sqrt_sq formula-decl nil sqrt "reals/")
    (sqrt_sqv_norm formula-decl nil vectors_2D "vectors/")
    (sq_eq formula-decl nil sq "reals/")
    (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)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (nnreal type-eq-decl nil real_types nil)
    (norm const-decl "nnreal" vectors_2D "vectors/")
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (sq_norm formula-decl nil vectors_2D "vectors/")
    (norm_neg formula-decl nil vectors_2D "vectors/")
    (sub_zero_left formula-decl nil vectors_2D "vectors/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (scal_zero formula-decl nil vectors_2D "vectors/")
    (circle? const-decl "bool" circle_optimum_2D nil)
    (sqrt_le formula-decl nil sqrt "reals/")
    (horizontal_conflict formula-decl nil horizontal nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sqrt_lt formula-decl nil sqrt "reals/")
    (horizontal_conflict? const-decl "bool" horizontal nil)
    (dot_zero_left formula-decl nil vectors_2D "vectors/")
    (sub_zero_right formula-decl nil vectors_2D "vectors/")
    (neg_zero formula-decl nil vectors_2D "vectors/")
    (norm_zero formula-decl nil vectors_2D "vectors/")
    (norm_eq_0 formula-decl nil vectors_2D "vectors/")
    (Delta_ge_0 formula-decl nil horizontal nil)
    (v skolem-const-decl "Vect2" circle_optimum_2D nil)
    (c skolem-const-decl "Vect2" circle_optimum_2D nil)
    (- const-decl "Vector" vectors_2D "vectors/")
    (Delta const-decl "real" horizontal nil)
    (R skolem-const-decl "posreal" circle_optimum_2D nil)
    (* const-decl "Vector" vectors_2D "vectors/")
    (/= const-decl "boolean" notequal nil)
    (Nz_vect2 type-eq-decl nil vectors_2D "vectors/")
    (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)
    (nzint nonempty-type-eq-decl nil integers nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (Sign type-eq-decl nil sign "reals/")
    (Theta_D const-decl "real" horizontal nil)
    (Theta_D_ge_0 formula-decl nil horizontal nil)
    (scal_0 formula-decl nil vectors_2D "vectors/")
    (add_zero_right formula-decl nil vectors_2D "vectors/")
    (horizontal_los_inside_Theta formula-decl nil horizontal nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (* const-decl "real" vectors_2D "vectors/")
    (Theta_D_le formula-decl nil horizontal nil)
    (Sp_vect2 type-eq-decl nil horizontal nil)
    (<= const-decl "bool" reals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (+ const-decl "Vector" vectors_2D "vectors/")
    (- const-decl "Vector" vectors_2D "vectors/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (Theta_D_unique formula-decl nil horizontal nil))
   nil))
 (intersects_circle_fun_inc 0
  (intersects_circle_fun_inc-1 nil 3527947389
   (""
    (case "FORALL (R: posreal, c: Vect2, u, v: Vect2): sqv(u) = 1 AND sqv(v) = 1 AND
                             intersects_circle_fun?(v, c, R) AND ^(v) * c <= ^(u) * c IMPLIES
                              intersects_circle_fun?(u, c, R)")
    (("1" (skeep)
      (("1" (inst - "R" "c" "^(u)" "^(v)")
        (("1" (assert)
          (("1" (split -)
            (("1" (lemma "intersects_circle_fun_scal")
              (("1" (inst - "R" "c" "1/norm(u)" "u")
                (("1" (assert)
                  (("1" (expand "^") (("1" (propax) nil nil)) nil))
                  nil)
                 ("2" (split +)
                  (("1" (cross-mult 1)
                    (("1" (ground)
                      (("1" (lemma "norm_eq_0")
                        (("1" (inst?) (("1" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil)
                   ("2" (cross-mult 1)
                    (("2" (ground)
                      (("2" (lemma "norm_eq_0")
                        (("2" (inst?) (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("3" (ground)
                  (("3" (lemma "norm_eq_0")
                    (("3" (inst?) (("3" (assertnil nil)) nil)) nil))
                  nil))
                nil))
              nil)
             ("2" (expand "^")
              (("2" (rewrite "sqv_scal")
                (("2" (rewrite "sq_div")
                  (("1" (rewrite "sq_norm") (("1" (assertnil nil))
                    nil)
                   ("2" (lemma "norm_eq_0")
                    (("2" (inst?) (("2" (assertnil nil)) nil)) nil))
                  nil))
                nil))
              nil)
             ("3" (expand "^")
              (("3" (rewrite "sqv_scal")
                (("3" (rewrite "sq_div")
                  (("1" (rewrite "sq_norm") (("1" (assertnil nil))
                    nil)
                   ("2" (lemma "norm_eq_0")
                    (("2" (inst?) (("2" (assertnil nil)) nil)) nil))
                  nil))
                nil))
              nil)
             ("4" (lemma "intersects_circle_fun_scal")
              (("4" (inst - "R" "c" "1/norm(v)" "v")
                (("1" (assert)
                  (("1" (expand "^") (("1" (propax) nil nil)) nil))
                  nil)
                 ("2" (split +)
                  (("1" (cross-mult 1)
                    (("1" (ground)
                      (("1" (lemma "norm_eq_0")
                        (("1" (inst?) (("1" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil)
                   ("2" (cross-mult 1)
                    (("2" (ground)
                      (("2" (lemma "norm_eq_0")
                        (("2" (inst?) (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("3" (lemma "norm_eq_0")
                  (("3" (inst?) (("3" (assertnil nil)) nil)) nil))
                nil))
              nil)
             ("5" (case "FORALL (zpt:Nz_vect2): ^(^(zpt)) = ^(zpt)")
              (("1" (inst-cp - "u")
                (("1" (inst - "v") (("1" (assertnil nil)) nil)) nil)
               ("2" (hide-all-but 1)
                (("2" (skeep)
                  (("2" (expand "^" + 1)
                    (("2" (typepred "^(zpt)")
                      (("2" (assert)
                        (("2" (replace -1) (("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
      (("2" (skeep)
        (("2" (label "sqvuv" (-1 -2))
          (("2" (case "^(v) = v AND ^(u) = u")
            (("1" (flatten)
              (("1" (replaces -1)
                (("1" (replaces -1)
                  (("1" (hide "sqvuv")
                    (("1" (expand "intersects_circle_fun?")
                      (("1" (case "NOT (v = zero OR u = zero)")
                        (("1" (flatten)
                          (("1" (assert)
                            (("1" (flatten)
                              (("1"
                                (label "uvnz" (1 2))
                                (("1"
                                  (hide "uvnz")
                                  (("1"
                                    (case
                                     "FORALL (bl1,bl2:bool): (bl1 OR bl2) IFF (bl1 OR (bl2 AND not bl1))")
                                    (("1"
                                      (label "final" 1)
                                      (("1"
                                        (hide "final")
                                        (("1"
                                          (rewrite -1)
                                          (("1"
                                            (hide -1)
                                            (("1"
                                              (reveal "final")
                                              (("1"
                                                (split -)
                                                (("1"
                                                  (split +)
                                                  (("1"
                                                    (expand "Delta")
                                                    (("1"
                                                      (reveal "sqvuv")
                                                      (("1"
                                                        (replace -1)
                                                        (("1"
                                                          (hide -1)
                                                          (("1"
                                                            (replace
                                                             -1)
                                                            (("1"
                                                              (hide -1)
                                                              (("1"
                                                                (case
                                                                 "sq(u*perpR(c)) <= sq(v*perpR(c))")
                                                                (("1"
                                                                  (hide
                                                                   -4)
                                                                  (("1"
                                                                    (grind)
                                                                    nil
                                                                    nil))
                                                                  nil)
                                                                 ("2"
                                                                  (hide
                                                                   (-2
                                                                    2))
                                                                  (("2"
                                                                    (lemma
                                                                     "orthogonal_basis")
                                                                    (("2"
                                                                      (inst
                                                                       -
                                                                       "c"
                                                                       "perpR(c)"
                                                                       _)
                                                                      (("2"
                                                                        (inst-cp
                                                                         -
                                                                         "u")
                                                                        (("2"
                                                                          (inst
                                                                           -
                                                                           "v")
                                                                          (("2"
                                                                            (assert)
                                                                            (("2"
                                                                              (case
                                                                               "c /= zero AND perpR(c) /= zero AND orthogonal?(c, perpR(c))")
                                                                              (("1"
                                                                                (flatten)
                                                                                (("1"
                                                                                  (assert)
                                                                                  (("1"
                                                                                    (lemma
                                                                                     "orthogonal_basis_sqv")
                                                                                    (("1"
                                                                                      (inst?)
                                                                                      (("1"
                                                                                        (assert)
                                                                                        (("1"
                                                                                          (label
                                                                                           "ddvc1"
                                                                                           -3)
                                                                                          (("1"
                                                                                            (hide
                                                                                             "ddvc1")
                                                                                            (("1"
                                                                                              (label
                                                                                               "ddvc2"
                                                                                               -1)
                                                                                              (("1"
                                                                                                (hide
                                                                                                 "ddvc2")
                                                                                                (("1"
                                                                                                  (lemma
                                                                                                   "orthogonal_basis_sqv")
                                                                                                  (("1"
                                                                                                    (inst?)
                                                                                                    (("1"
                                                                                                      (assert)
                                                                                                      (("1"
                                                                                                        (reveal
                                                                                                         "ddvc1")
                                                                                                        (("1"
                                                                                                          (reveal
                                                                                                           "ddvc2")
                                                                                                          (("1"
                                                                                                            (delabel
                                                                                                             "ddvc1")
                                                                                                            (("1"
                                                                                                              (delabel
                                                                                                               "ddvc2")
                                                                                                              (("1"
                                                                                                                (case
                                                                                                                 "sqv(perpR(c)) = sqv(c)")
                                                                                                                (("1"
                                                                                                                  (replace
                                                                                                                   -1)
                                                                                                                  (("1"
                                                                                                                    (hide
                                                                                                                     -1)
                                                                                                                    (("1"
                                                                                                                      (reveal
                                                                                                                       "sqvuv")
                                                                                                                      (("1"
                                                                                                                        (replace
                                                                                                                         -1)
                                                                                                                        (("1"
                                                                                                                          (replace
                                                                                                                           -2)
                                                                                                                          (("1"
                                                                                                                            (case
                                                                                                                             "sq(u*c) >= sq(v*c)")
                                                                                                                            (("1"
                                                                                                                              (rewrite
                                                                                                                               "sq_div")
                                                                                                                              (("1"
                                                                                                                                (rewrite
                                                                                                                                 "sq_div")
                                                                                                                                (("1"
                                                                                                                                  (rewrite
                                                                                                                                   "sq_div")
                                                                                                                                  (("1"
                                                                                                                                    (rewrite
                                                                                                                                     "sq_div")
                                                                                                                                    (("1"
                                                                                                                                      (mult-by
                                                                                                                                       3
                                                                                                                                       "sqv(c)/sq(sqv(c))")
                                                                                                                                      (("1"
                                                                                                                                        (mult-by
                                                                                                                                         -1
                                                                                                                                         "sqv(c)/sq(sqv(c))")
                                                                                                                                        (("1"
                                                                                                                                          (ground)
                                                                                                                                          nil
                                                                                                                                          nil)
                                                                                                                                         ("2"
                                                                                                                                          (real-props)
                                                                                                                                          nil
                                                                                                                                          nil))
                                                                                                                                        nil)
                                                                                                                                       ("2"
                                                                                                                                        (split
                                                                                                                                         +)
                                                                                                                                        (("1"
                                                                                                                                          (cross-mult
                                                                                                                                           1)
                                                                                                                                          nil
                                                                                                                                          nil)
                                                                                                                                         ("2"
                                                                                                                                          (cross-mult
                                                                                                                                           1)
                                                                                                                                          (("2"
                                                                                                                                            (rewrite
                                                                                                                                             "sq_ge")
                                                                                                                                            (("2"
                                                                                                                                              (lemma
                                                                                                                                               "sqv_eq_0")
                                                                                                                                              (("2"
                                                                                                                                                (inst?)
                                                                                                                                                (("2"
                                                                                                                                                  (assert)
                                                                                                                                                  nil
                                                                                                                                                  nil))
                                                                                                                                                nil))
                                                                                                                                              nil))
                                                                                                                                            nil))
                                                                                                                                          nil))
                                                                                                                                        nil))
                                                                                                                                      nil))
                                                                                                                                    nil))
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil)
                                                                                                                             ("2"
                                                                                                                              (rewrite
                                                                                                                               "sq_ge")
                                                                                                                              (("1"
                                                                                                                                (hide-all-but
                                                                                                                                 (-8
                                                                                                                                  1))
                                                                                                                                (("1"
                                                                                                                                  (grind)
                                                                                                                                  nil
                                                                                                                                  nil))
                                                                                                                                nil)
                                                                                                                               ("2"
                                                                                                                                (hide-all-but
                                                                                                                                 (-8
                                                                                                                                  -9
                                                                                                                                  1))
                                                                                                                                (("2"
                                                                                                                                  (grind)
                                                                                                                                  nil
                                                                                                                                  nil))
                                                                                                                                nil))
                                                                                                                              nil))
                                                                                                                            nil))
                                                                                                                          nil))
                                                                                                                        nil))
                                                                                                                      nil))
                                                                                                                    nil))
                                                                                                                  nil)
                                                                                                                 ("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))
                                                                                  nil))
                                                                                nil)
                                                                               ("2"
                                                                                (split
                                                                                 +)
                                                                                (("1"
                                                                                  (flatten)
                                                                                  (("1"
                                                                                    (replace
                                                                                     -1)
                                                                                    (("1"
                                                                                      (assert)
                                                                                      nil
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("2"
                                                                                  (flatten)
                                                                                  (("2"
                                                                                    (case
                                                                                     "c = zero")
                                                                                    (("1"
                                                                                      (replace
                                                                                       -1)
                                                                                      (("1"
                                                                                        (assert)
                                                                                        nil
                                                                                        nil))
                                                                                      nil)
                                                                                     ("2"
                                                                                      (hide-all-but
                                                                                       (-1
                                                                                        1))
                                                                                      (("2"
                                                                                        (expand
                                                                                         "zero")
                                                                                        (("2"
                                                                                          (expand
                                                                                           "perpR")
                                                                                          (("2"
                                                                                            (flatten)
                                                                                            (("2"
                                                                                              (decompose-equality)
                                                                                              nil
                                                                                              nil))
                                                                                            nil))
                                                                                          nil))
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil)
                                                                                 ("3"
                                                                                  (hide-all-but
                                                                                   1)
                                                                                  (("3"
                                                                                    (grind)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (flatten)
                                                    (("2"
                                                      (hide-all-but
                                                       (-1 -3 1))
                                                      (("2"
                                                        (grind)
                                                        nil
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (assert)
                                                  (("2"
                                                    (flatten)
                                                    (("2"
                                                      (assert)
                                                      (("2"
                                                        (lemma
                                                         "Delta_ge_0_2[R]")
                                                        (("2"
                                                          (inst?)
                                                          (("2"
                                                            (assert)
                                                            (("2"
                                                              (inst
                                                               +
                                                               "0")
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil)
                                     ("2"
                                      (hide-all-but 1)
                                      (("2"
                                        (skeep)
                                        (("2" (ground) nil nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but -1)
                          (("2" (reveal "sqvuv")
                            (("2" (split -)
                              (("1"
                                (replace -1)
                                (("1" (assertnil nil))
                                nil)
                               ("2"
                                (replace -1)
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (case "norm(u) = 1 and norm(v) = 1")
              (("1" (expand "^" 1)
                (("1" (flatten)
                  (("1" (replaces -1)
                    (("1" (replaces -1) (("1" (assertnil nil)) nil))
                    nil))
                  nil))
                nil)
               ("2" (hide-all-but (-1 -2 1))
                (("2" (rewrite "sqrt_eq" -1 :dir rl)
                  (("2" (rewrite "sqrt_eq" -2 :dir rl)
                    (("2" (rewrite "sqrt_sqv_norm")
                      (("2" (rewrite "sqrt_sqv_norm")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("3" (hide 2)
      (("3" (skeep) (("3" (replace -4) (("3" (assertnil nil)) nil))
        nil))
      nil)
     ("4" (hide 2)
      (("4" (skeep) (("4" (replace -4) (("4" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((posrat_div_posrat_is_posrat application-judgement "posrat"
     rationals nil)
    (sqrt_eq formula-decl nil sqrt "reals/")
    (sqrt_pos application-judgement "posreal" sqrt "reals/")
    (sqrt_1 formula-decl nil sqrt "reals/")
    (sqrt_sqv_norm formula-decl nil vectors_2D "vectors/")
    (sqv_zero formula-decl nil vectors_2D "vectors/")
    (sq const-decl "nonneg_real" sq "reals/")
    (add_zero_right formula-decl nil vectors_2D "vectors/")
    (scal_0 formula-decl nil vectors_2D "vectors/")
    (Delta_ge_0_2 formula-decl nil horizontal nil)
    (orthogonal? const-decl "bool" vectors_2D "vectors/")
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (both_sides_times_pos_le1 formula-decl nil real_props nil)
    (c skolem-const-decl "Vect2" circle_optimum_2D nil)
    (pos_div_ge formula-decl nil real_props nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil)
    (div_mult_pos_gt1 formula-decl nil extra_real_props nil)
    (sqv_eq_0 formula-decl nil vectors_2D "vectors/")
    (sq_ge formula-decl nil sq "reals/")
    (div_mult_pos_ge1 formula-decl nil real_props nil)
    (orthogonal_basis_sqv formula-decl nil basis_2D "vectors/")
    (perpR_eq_zero formula-decl nil perpendicular_2D "vectors/")
    (dot_zero_right formula-decl nil vectors_2D "vectors/")
    (sq_0 formula-decl nil sq "reals/")
    (orthogonal_basis formula-decl nil basis_2D "vectors/")
    (minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (det const-decl "real" det_2D "vectors/")
    (- const-decl "Vector" vectors_2D "vectors/")
    (perpR const-decl "Vect2" perpendicular_2D "vectors/")
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (Delta const-decl "real" horizontal nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (nz_sqv_gt_0 application-judgement "posreal" vectors_2D "vectors/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (Nz_vect2 type-eq-decl nil vectors_2D "vectors/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nz_norm_gt_0 application-judgement "posreal" vectors_2D
     "vectors/")
    (nz_nzv application-judgement "Nz_vector" vectors_2D "vectors/")
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (scal_1 formula-decl nil vectors_2D "vectors/")
    (norm_normalize formula-decl nil vectors_2D "vectors/")
    (v skolem-const-decl "Vect2" circle_optimum_2D nil)
    (sq_div formula-decl nil sq "reals/")
    (sq_1 formula-decl nil sq "reals/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (dot_scal_left formula-decl nil vectors_2D "vectors/")
    (sq_norm formula-decl nil vectors_2D "vectors/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (sqv_scal formula-decl nil vectors_2D "vectors/")
    (intersects_circle_fun_scal formula-decl nil circle_optimum_2D nil)
    (norm_eq_0 formula-decl nil vectors_2D "vectors/")
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (div_mult_pos_neg_ge1 formula-decl nil extra_real_props nil)
    (div_mult_pos_neg_gt1 formula-decl nil extra_real_props nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (u skolem-const-decl "Vect2" circle_optimum_2D nil)
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types 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)
    (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)
    (bool nonempty-type-eq-decl nil booleans 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)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (nnreal type-eq-decl nil real_types nil)
    (sqv const-decl "nnreal" vectors_2D "vectors/")
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (<= const-decl "bool" reals nil)
    (* const-decl "real" vectors_2D "vectors/")
    (/= const-decl "boolean" notequal nil)
    (zero const-decl "Vector" vectors_2D "vectors/")
    (Nz_vector type-eq-decl nil vectors_2D "vectors/")
    (norm const-decl "nnreal" vectors_2D "vectors/")
    (Normalized type-eq-decl nil vectors_2D "vectors/")
    (^ const-decl "Normalized" vectors_2D "vectors/"))
   nil))
 (inter_circle_max_time_TCC1 0
  (inter_circle_max_time_TCC1-1 nil 3527524768
   ("" (subtype-tcc) nil 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)
    (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 "boolean" notequal nil)
    (- const-decl "Vector" vectors_2D "vectors/")
    (sq const-decl "nonneg_real" sq "reals/")
    (* const-decl "real" vectors_2D "vectors/")
    (sqv const-decl "nnreal" vectors_2D "vectors/")
    (det const-decl "real" det_2D "vectors/")
    (Delta const-decl "real" horizontal nil)
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil))
   nil))
 (inter_circle_max_time_TCC2 0
  (inter_circle_max_time_TCC2-1 nil 3527524768
   ("" (skeep)
    (("" (lemma "intersects_circle_fun_def")
      (("" (inst?)
        (("" (assert)
          (("" (skeep -1)
            (("" (expand "segment?")
              (("" (skeep -2)
                (("" (replace -2)
                  (("" (hide -2)
                    (("" (expand "circle?")
                      (("" (lemma "Theta_D_unique[R]")
                        (("" (inst?)
                          (("" (inst - "-c")
                            (("" (assert)
                              ((""
                                (expand "intersects_circle_fun?")
                                ((""
                                  (flatten)
                                  ((""
                                    (assert)
                                    ((""
                                      (flatten)
                                      ((""
                                        (hide -2)
                                        ((""
                                          (split -)
                                          (("1"
                                            (lemma "Theta_D_le[R]")
                                            (("1"
                                              (inst?)
                                              (("1" (assertnil nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (rewrite
                                             "sqrt_eq"
                                             1
                                             :dir
                                             rl)
                                            (("2"
                                              (rewrite "sqrt_sqv_norm")
                                              (("2"
                                                (assert)
                                                (("2"
                                                  (hide-all-but (-1 1))
                                                  (("2"
                                                    (grind
                                                     :exclude
                                                     "norm")
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((intersects_circle_fun_def formula-decl nil circle_optimum_2D nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (segment? const-decl "bool" circle_optimum_2D nil)
    (circle? const-decl "bool" circle_optimum_2D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Nz_vect2 type-eq-decl nil vectors_2D "vectors/")
    (zero const-decl "Vector" vectors_2D "vectors/")
    (/= const-decl "boolean" notequal nil)
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (Theta_D_le formula-decl nil horizontal nil)
    (sqrt_sqv_norm formula-decl nil vectors_2D "vectors/")
    (- const-decl "Vector" vectors_2D "vectors/")
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (sqrt_sq formula-decl nil sqrt "reals/")
    (sqrt_pos application-judgement "posreal" sqrt "reals/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sqrt_eq formula-decl nil sqrt "reals/")
    (sqv const-decl "nnreal" vectors_2D "vectors/")
    (+ const-decl "Vector" vectors_2D "vectors/")
    (* const-decl "Vector" vectors_2D "vectors/")
    (sq const-decl "nonneg_real" sq "reals/")
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (- const-decl "Vector" vectors_2D "vectors/")
    (Theta_D_unique formula-decl nil horizontal nil)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (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))
 (inter_circle_max_time_scal 0
  (inter_circle_max_time_scal-1 nil 3529411095
   ("" (skeep)
    (("" (cross-mult 1)
      (("" (expand "inter_circle_max_time")
        (("" (lift-if)
          (("" (lift-if)
            (("" (lift-if)
              (("" (assert)
                (("" (ground)
                  (("1" (lemma "Theta_D_scal[R]")
                    (("1" (inst?)
                      (("1" (assert)
                        (("1" (expand "intersects_circle_fun?")
                          (("1" (flatten)
                            (("1" (assert)
                              (("1"
                                (expand "sign" -1)
                                (("1" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (replace -1) (("2" (assertnil nil)) nil)
                   ("3" (expand "intersects_circle_fun?")
                    (("3" (assert)
                      (("3" (flatten)
                        (("3" (assert)
                          (("3" (split +)
                            (("1" (flatten)
                              (("1"
                                (replace -1)
                                (("1" (assertnil nil))
                                nil))
                              nil)
                             ("2" (lemma "Delta_scal[R]")
                              (("2"
                                (inst?)
                                (("2"
                                  (assert)
                                  (("2"
                                    (mult-by 1 "sq(pt)")
                                    (("2" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("3" (flatten)
                              (("3"
                                (mult-by 1 "pt")
                                (("3" (ground) nil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("4" (case "(1/pt)*(pt*v) = zero")
                    (("1" (assertnil nil)
                     ("2" (replace -2) (("2" (assertnil nil)) nil))
                    nil)
                   ("5" (expand "intersects_circle_fun?")
                    (("5" (flatten)
                      (("5" (split 3)
                        (("1" (flatten)
                          (("1" (case "(1/pt)*(pt*v) = zero")
                            (("1" (assertnil nil)
                             ("2" (replace -1) (("2" (assertnil nil))
                              nil))
                            nil))
                          nil)
                         ("2" (lemma "Delta_scal[R]")
                          (("2" (inst?)
                            (("2" (assert)
                              (("2"
                                (mult-by -2 "sq(pt)")
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("3" (flatten)
                          (("3" (assert)
                            (("3" (mult-by -2 "pt")
                              (("3" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nnreal_div_posreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (inter_circle_max_time const-decl "nnreal" circle_optimum_2D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (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)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal 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)
    (times_div2 formula-decl nil real_props nil)
    (div_cancel4 formula-decl nil real_props nil)
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (* const-decl "Vector" vectors_2D "vectors/")
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nzint nonempty-type-eq-decl nil integers nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (Sign type-eq-decl nil sign "reals/")
    (nzreal nonempty-type-eq-decl nil reals nil)
    (zero const-decl "Vector" vectors_2D "vectors/")
    (Nz_vect2 type-eq-decl nil vectors_2D "vectors/")
    (- const-decl "Vector" vectors_2D "vectors/")
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (dot_scal_right formula-decl nil vectors_2D "vectors/")
    (sign const-decl "Sign" sign "reals/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (Theta_D_scal formula-decl nil horizontal nil)
    (scal_zero formula-decl nil vectors_2D "vectors/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (* const-decl "real" vectors_2D "vectors/")
    (Delta_scal formula-decl nil horizontal nil)
    (Delta const-decl "real" horizontal nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (scal_1 formula-decl nil vectors_2D "vectors/")
    (scal_assoc formula-decl nil vectors_2D "vectors/")
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil))
   shostak))
 (inter_circle_max_time_def 0
  (inter_circle_max_time_def-1 nil 3527525027
   ("" (skeep)
    (("" (name "tcm" "inter_circle_max_time(v, c, R)")
      (("" (replace -1)
        (("" (assert)
          (("" (split +)
            (("1" (flatten)
              (("1" (lemma "intersects_circle_fun_def")
                (("1" (inst?)
                  (("1" (assert)
                    (("1" (case "NOT circle?(c, R)(tcm * v)")
                      (("1" (hide 2)
                        (("1" (expand "circle?")
                          (("1" (expand "inter_circle_max_time")
                            (("1" (lift-if)
                              (("1"
                                (ground)
                                (("1"
                                  (lemma "Theta_D_unique[R]")
                                  (("1"
                                    (inst?)
                                    (("1"
                                      (inst - "-c")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (copy -4)
                                          (("1"
                                            (expand
                                             "intersects_circle_fun?"
                                             -1)
                                            (("1"
                                              (assert)
                                              (("1"
                                                (flatten)
                                                (("1"
                                                  (assert)
                                                  (("1"
                                                    (rewrite
                                                     "sqrt_eq"
                                                     -3
                                                     :dir
                                                     rl)
                                                    (("1"
                                                      (rewrite
                                                       "sqrt_sqv_norm")
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (hide-all-but
                                                           (-3 2))
                                                          (("1"
                                                            (grind)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (replace -1)
                                  (("2"
                                    (assert)
                                    (("2"
                                      (skeep -3)
                                      (("2"
                                        (expand "segment?")
                                        (("2"
                                          (skeep -4)
                                          (("2"
                                            (assert)
                                            (("2"
                                              (replace -4)
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (propax) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (assert)
              (("2" (skeep)
                (("2" (expand "inter_circle_max_time")
                  (("2" (lift-if)
                    (("2" (split -)
                      (("1" (flatten)
                        (("1" (lemma "Theta_D_unique[R]")
                          (("1" (inst - _ _ "nnt")
                            (("1" (inst?)
                              (("1"
                                (assert)
                                (("1"
                                  (expand "intersects_circle_fun?")
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (assert)
                                      (("1"
                                        (copy -5)
                                        (("1"
                                          (expand "circle?" -1)
                                          (("1"
                                            (case
                                             "sqv(-c+nnt*v) = sq(R)")
                                            (("1"
                                              (assert)
                                              (("1"
                                                (case "nnt = tcm")
                                                (("1" (assertnil nil)
                                                 ("2"
                                                  (assert)
                                                  (("2"
                                                    (case "nnt = 0")
                                                    (("1"
                                                      (replace -1)
                                                      (("1"
                                                        (assert)
                                                        (("1"
                                                          (rewrite
                                                           "norm_scal")
                                                          (("1"
                                                            (expand
                                                             "abs"
                                                             +)
                                                            (("1"
                                                              (lemma
                                                               "posreal_times_posreal_is_posreal")
                                                              (("1"
                                                                (inst?)
                                                                (("1"
                                                                  (assert)
                                                                  nil
                                                                  nil)
                                                                 ("2"
                                                                  (lemma
                                                                   "norm_eq_0")
                                                                  (("2"
                                                                    (inst?)
                                                                    (("2"
                                                                      (assert)
                                                                      nil
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("2"
                                                      (assert)
                                                      (("2"
                                                        (case
                                                         "nnt < tcm")
                                                        (("1"
                                                          (mult-by
                                                           -1
                                                           "norm(v)")
                                                          (("1"
                                                            (case
                                                             "forall (aannrr:nnreal): aannrr = abs(aannrr)")
                                                            (("1"
                                                              (inst-cp
                                                               -
                                                               "nnt")
                                                              (("1"
                                                                (inst
                                                                 -
                                                                 "tcm")
                                                                (("1"
                                                                  (replace
                                                                   -1
                                                                   -3)
                                                                  (("1"
                                                                    (replace
                                                                     -2
                                                                     -3)
                                                                    (("1"
                                                                      (hide
                                                                       (-1
                                                                        -2))
                                                                      (("1"
                                                                        (rewrite
                                                                         "norm_scal"
                                                                         :dir
                                                                         rl)
                                                                        (("1"
                                                                          (rewrite
                                                                           "norm_scal"
                                                                           :dir
                                                                           rl)
                                                                          (("1"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil)
                                                             ("2"
                                                              (hide-all-but
                                                               1)
                                                              (("2"
                                                                (skeep)
                                                                (("2"
                                                                  (expand
                                                                   "abs")
                                                                  (("2"
                                                                    (propax)
                                                                    nil
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil)
                                                           ("2"
                                                            (lemma
                                                             "norm_eq_0")
                                                            (("2"
                                                              (inst?)
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil)
                                                         ("2"
                                                          (lemma
                                                           "Theta_D_le[R]")
                                                          (("2"
                                                            (inst?)
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (rewrite
                                               "norm_eq_sqv_eq"
                                               1)
                                              (("2"
                                                (hide-all-but (-1 1))
                                                (("2"
                                                  (grind
                                                   :exclude
                                                   "norm")
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (assert)
                        (("2" (flatten)
                          (("2" (replace -1 :dir rl)
                            (("2" (assert)
                              (("2"
                                (case
                                 "NOT intersects_circle_fun?(v, c, R)")
                                (("1"
                                  (lemma "intersects_circle_fun_def")
                                  (("1"
                                    (inst?)
                                    (("1"
                                      (assert)
                                      (("1"
                                        (inst + "nnt*v")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (expand "segment?")
                                            (("1" (inst?) nil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (assert)
                                  (("2"
                                    (flatten)
                                    (("2"
                                      (replace -2)
                                      (("2" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((inter_circle_max_time const-decl "nnreal" circle_optimum_2D 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_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)
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (real nonempty-type-from-decl nil reals nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (norm_eq_sqv_eq formula-decl nil vectors_2D "vectors/")
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (Theta_D_le formula-decl nil horizontal nil)
    (both_sides_times_pos_lt1 formula-decl nil real_props 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)
    (norm_scal formula-decl nil vectors_2D "vectors/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (posreal_times_posreal_is_posreal judgement-tcc nil real_types nil)
    (norm_eq_0 formula-decl nil vectors_2D "vectors/")
    (v skolem-const-decl "Vect2" circle_optimum_2D nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (norm_zero formula-decl nil vectors_2D "vectors/")
    (add_zero_right formula-decl nil vectors_2D "vectors/")
    (scal_0 formula-decl nil vectors_2D "vectors/")
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (* const-decl "Vector" vectors_2D "vectors/")
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (circle? const-decl "bool" circle_optimum_2D nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (segment? const-decl "bool" circle_optimum_2D nil)
    (norm_neg formula-decl nil vectors_2D "vectors/")
    (sub_zero_left formula-decl nil vectors_2D "vectors/")
    (scal_zero formula-decl nil vectors_2D "vectors/")
    (Theta_D_unique formula-decl nil horizontal nil)
    (- const-decl "Vector" vectors_2D "vectors/")
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (sqrt_sqv_norm formula-decl nil vectors_2D "vectors/")
    (minus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (- const-decl "Vector" vectors_2D "vectors/")
    (norm const-decl "nnreal" vectors_2D "vectors/")
    (* const-decl "real" vectors_2D "vectors/")
    (sqrt_sq formula-decl nil sqrt "reals/")
    (sqrt_pos application-judgement "posreal" sqrt "reals/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (sqrt_eq formula-decl nil sqrt "reals/")
    (sqv const-decl "nnreal" vectors_2D "vectors/")
    (+ const-decl "Vector" vectors_2D "vectors/")
    (sq const-decl "nonneg_real" sq "reals/")
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (/= const-decl "boolean" notequal nil)
    (zero const-decl "Vector" vectors_2D "vectors/")
    (Nz_vect2 type-eq-decl nil vectors_2D "vectors/")
    (intersects_circle_fun_def formula-decl nil circle_optimum_2D nil))
   shostak))
 (inter_circle_max_norm_scal 0
  (inter_circle_max_norm_scal-1 nil 3527585618
   ("" (skeep)
    (("" (expand "inter_circle_max_norm")
      (("" (lift-if)
        (("" (lift-if)
          (("" (lift-if)
            (("" (ground)
              (("1" (expand "inter_circle_max_time")
                (("1" (lift-if)
                  (("1" (lift-if)
                    (("1" (lift-if)
                      (("1" (assert)
                        (("1" (ground)
                          (("1" (lemma "Theta_D_scal[R]")
                            (("1" (inst?)
                              (("1"
                                (split -)
                                (("1"
                                  (assert)
                                  (("1"
                                    (rewrite "norm_scal")
                                    (("1"
                                      (expand "abs")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (expand "sign" -1)
                                          (("1" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (expand "intersects_circle_fun?")
                                    (("2" (flatten) nil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (replace -1) (("2" (assertnil nil))
                            nil)
                           ("3" (case "v = zero")
                            (("1" (replace -1) (("1" (assertnil nil))
                              nil)
                             ("2" (hide-all-but (-1 1))
                              (("2"
                                (expand "zero")
                                (("2"
                                  (decompose-equality -)
                                  (("2"
                                    (rewrite "vx_scal")
                                    (("2"
                                      (rewrite "vy_scal")
                                      (("2"
                                        (decompose-equality +)
                                        nil
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (expand "intersects_circle_fun?")
                  (("2" (flatten)
                    (("2" (lemma "Delta_scal[R]")
                      (("2" (inst?)
                        (("2" (assert)
                          (("2" (replace -1)
                            (("2" (split -)
                              (("1"
                                (flatten)
                                (("1"
                                  (case "NOT Delta[R](-c,v) >=0")
                                  (("1"
                                    (hide 2)
                                    (("1"
                                      (mult-by 1 "sq(pt)")
                                      (("1" (assertnil nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (assert)
                                    (("2"
                                      (case "v/=zero")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (hide -1)
                                          (("1"
                                            (flatten)
                                            (("1"
                                              (assert)
                                              (("1"
                                                (mult-by 2 "pt")
                                                (("1"
                                                  (assert)
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide-all-but (1 2))
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (replace -1)
                                            (("2" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (flatten)
                                (("2"
                                  (case "v = zero")
                                  (("1" (assertnil nil)
                                   ("2"
                                    (hide-all-but (-1 1))
                                    (("2"
                                      (expand "zero")
                                      (("2"
                                        (decompose-equality -)
                                        (("2"
                                          (rewrite "vy_scal")
                                          (("2"
                                            (rewrite "vx_scal")
                                            (("2"
                                              (decompose-equality +)
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (hide 1)
                (("3" (expand "intersects_circle_fun?")
                  (("3" (flatten)
                    (("3" (split -)
                      (("1" (flatten)
                        (("1" (assert)
                          (("1" (lemma "Delta_scal[R]")
                            (("1" (inst?)
                              (("1"
                                (assert)
                                (("1"
                                  (case "NOT Delta[R](-c,pt*v) >=0")
                                  (("1"
                                    (mult-by -2 "sq(pt)")
                                    (("1" (assertnil nil))
                                    nil)
                                   ("2"
                                    (assert)
                                    (("2"
                                      (case "pt*v /= zero")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (flatten)
                                          (("1"
                                            (assert)
                                            (("1"
                                              (mult-by -4 "pt")
                                              (("1" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("2"
                                        (hide-all-but (1 2))
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (expand "zero")
                                            (("2"
                                              (decompose-equality -)
                                              (("2"
                                                (rewrite "vx_scal")
                                                (("2"
                                                  (rewrite "vy_scal")
                                                  (("2"
                                                    (decompose-equality
                                                     +)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (flatten)
                        (("2" (replace -1) (("2" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((inter_circle_max_norm const-decl "nnreal" circle_optimum_2D nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (/= const-decl "boolean" notequal nil)
    (nzint nonempty-type-eq-decl nil integers nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (Sign type-eq-decl nil sign "reals/")
    (nzreal nonempty-type-eq-decl nil reals nil)
    (Vector type-eq-decl nil vectors_2D "vectors/")
    (zero const-decl "Vector" vectors_2D "vectors/")
    (Nz_vect2 type-eq-decl nil vectors_2D "vectors/")
    (Vect2 type-eq-decl nil vectors_2D_def "vectors/")
    (- const-decl "Vector" vectors_2D "vectors/")
    (intersects_circle_fun? const-decl "bool" circle_optimum_2D nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (mult_divides2 application-judgement "(divides(m))" divides nil)
    (mult_divides1 application-judgement "(divides(n))" divides nil)
    (nzint_times_nzint_is_nzint application-judgement "nzint" integers
     nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (sign const-decl "Sign" sign "reals/")
    (nonzero_abs_is_pos application-judgement "{y: posreal | y >= x}"
     real_defs nil)
    (norm_scal formula-decl nil vectors_2D "vectors/")
    (Theta_D_scal formula-decl nil horizontal 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)
    (bool nonempty-type-eq-decl nil booleans 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)
    (scal_zero formula-decl nil vectors_2D "vectors/")
    (norm_zero formula-decl nil vectors_2D "vectors/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (* const-decl "Vector" vectors_2D "vectors/")
    (vy_scal formula-decl nil vectors_2D "vectors/")
    (vx_scal formula-decl nil vectors_2D "vectors/")
    (inter_circle_max_time const-decl "nnreal" circle_optimum_2D nil)
    (Delta_scal formula-decl nil horizontal nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (sqv_neg formula-decl nil vectors_2D "vectors/")
    (dot_scal_right formula-decl nil vectors_2D "vectors/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (Delta const-decl "real" horizontal nil)
    (both_sides_times_pos_ge1 formula-decl nil real_props nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (* const-decl "real" vectors_2D "vectors/")
    (both_sides_times_pos_ge1_imp formula-decl nil extra_real_props
     nil)
    (dot_zero_right formula-decl nil vectors_2D "vectors/"))
   shostak))
 (max_norm_increasing 0
  (max_norm_increasing-1 nil 3527584737
   (""
    (case "FORALL (R: posreal, c: Vect2, u, v: Vect2): norm(u) = 1 AND norm(v) = 1 IMPLIES
                                                                                                                                             ^(v) * c <= ^(u) * c IMPLIES
                                                                                                                                              inter_circle_max_norm(v, c, R) <= inter_circle_max_norm(u, c, R)")
    (("1" (skeep)
      (("1" (split -)
        (("1" (inst - "R" "c" "^(u)" "^(v)")
          (("1" (assert)
            (("1" (flatten)
              (("1" (split -)
                (("1" (expand "^")
                  (("1" (lemma "inter_circle_max_norm_scal")
                    (("1" (inst - "R" "c" _ _)
                      (("1" (inst-cp - "v" "1/norm(v)")
                        (("1" (inst - "u" "1/norm(u)")
                          (("1" (assertnil nil)
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                            (("1" (cross-mult 1)
                              (("1"
                                (ground)
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                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (cross-mult 1)
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                           ("2" (cross-mult 1)
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                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2"
                  (case "FORALL (vvz:Vect2): norm(vvz) = 1 IMPLIES ^(vvz) = vvz")
                  (("1" (inst-cp - "^(v)")
                    (("1" (inst - "^(u)") (("1" (assertnil nil))
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                    nil)
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                        (("2" (assert)
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                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("3" (hide-all-but 1)
                    (("3" (skeep)
                      (("3" (replace -2) (("3" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assert)
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            (("2" (flatten)
              (("2" (case "v = zero")
                (("1" (hide -2)
                  (("1" (replace -1)
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                      (("1" (lift-if)
                        (("1" (lift-if)
                          (("1" (lift-if)
                            (("1" (assert) (("1" (ground) nil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (assert)
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                    (("2" (assert)
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                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil)
     ("2" (hide 2)
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          (case "FORALL (vvz:Vect2): norm(vvz) = 1 IMPLIES ^(vvz) = vvz")
          (("1" (inst-cp - "v")
            (("1" (inst - "u")
              (("1" (assert)
                (("1" (replaces -1)
                  (("1" (replaces -1)
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                      (("1" (label "normv" -2)
                        (("1" (case "u/=zero AND v/=zero")
                          (("1" (flatten)
                            (("1" (label "uvnz" (1 2))
                              (("1"
                                (hide "uvnz")
                                (("1"
                                  (case "sqv(u) = 1 AND sqv(v) = 1")
                                  (("1"
                                    (flatten)
                                    (("1"
                                      (label "sqvu" -1)
                                      (("1"
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                                        (("1"
                                          (hide
                                           ("sqvu"
                                            "sqvv"
                                            "normu"
                                            "normv"))
                                          (("1"
                                            (expand
                                             "inter_circle_max_norm")
                                            (("1"
                                              (lift-if)
                                              (("1"
                                                (lift-if)
                                                (("1"
                                                  (lift-if)
                                                  (("1"
                                                    (ground)
                                                    (("1"
                                                      (name
                                                       "pu"
                                                       "inter_circle_max_time(u, c, R) * u")
                                                      (("1"
                                                        (name
                                                         "pv"
                                                         "inter_circle_max_time(v, c, R) * v")
                                                        (("1"
                                                          (lemma
                                                           "law_cosines_alt")
                                                          (("1"
                                                            (inst
                                                             -
                                                             "zero"
                                                             "c"
                                                             "pu")
                                                            (("1"
                                                              (lemma
                                                               "law_cosines_alt")
                                                              (("1"
                                                                (inst
                                                                 -
                                                                 "zero"
                                                                 "c"
                                                                 "pv")
                                                                (("1"
                                                                  (assert)
                                                                  (("1"
                                                                    (lemma
                                                                     "law_cosines_alt")
                                                                    (("1"
                                                                      (inst
                                                                       -
                                                                       "c"
                                                                       "pu"
                                                                       "pv")
                                                                      (("1"
                                                                        (assert)
                                                                        (("1"
                                                                          (label
                                                                           "dlem"
                                                                           -1)
                                                                          (("1"
                                                                            (hide
                                                                             "dlem")
                                                                            (("1"
                                                                              (expand
                                                                               "dist")
                                                                              (("1"
                                                                                (rewrite
                                                                                 "sq_sqrt")
                                                                                (("1"
                                                                                  (rewrite
                                                                                   "sq_sqrt")
                                                                                  (("1"
                                                                                    (rewrite
                                                                                     "sq_sqrt")
                                                                                    (("1"
                                                                                      (rewrite
                                                                                       "sq_sqrt")
                                                                                      (("1"
                                                                                        (rewrite
                                                                                         "sq_sqrt")
                                                                                        (("1"
                                                                                          (rewrite
                                                                                           "sq_dist_norm")
                                                                                          (("1"
                                                                                            (rewrite
                                                                                             "sq_dist_norm")
                                                                                            (("1"
                                                                                              (rewrite
                                                                                               "sq_dist_norm")
                                                                                              (("1"
                                                                                                (rewrite
                                                                                                 "sq_dist_norm")
                                                                                                (("1"
                                                                                                  (rewrite
                                                                                                   "sq_dist_norm")
                                                                                                  (("1"
                                                                                                    (case
                                                                                                     "sq(norm(pv)) <= sq(norm(pu))")
                                                                                                    (("1"
                                                                                                      (rewrite
                                                                                                       "sq_le"
                                                                                                       -1)
                                                                                                      (("1"
                                                                                                        (hide-all-but
                                                                                                         (-1
                                                                                                          1))
                                                                                                        (("1"
                                                                                                          (expand
                                                                                                           "pv")
                                                                                                          (("1"
                                                                                                            (expand
                                                                                                             "pu")
                                                                                                            (("1"
                                                                                                              (rewrite
                                                                                                               "norm_scal")
                                                                                                              (("1"
                                                                                                                (rewrite
                                                                                                                 "norm_scal")
                                                                                                                (("1"
                                                                                                                  (expand
                                                                                                                   "abs")
                                                                                                                  (("1"
                                                                                                                    (propax)
                                                                                                                    nil
                                                                                                                    nil))
                                                                                                                  nil))
                                                                                                                nil))
                                                                                                              nil))
                                                                                                            nil))
                                                                                                          nil))
                                                                                                        nil))
                                                                                                      nil)
                                                                                                     ("2"
                                                                                                      (hide
                                                                                                       2)
                                                                                                      (("2"
                                                                                                        (case
                                                                                                         "sq(norm(pv-c)) = sq(norm(pu-c))")
                                                                                                        (("1"
                                                                                                          (replace
                                                                                                           -1)
                                                                                                          (("1"
                                                                                                            (hide
                                                                                                             -1)
                                                                                                            (("1"
                                                                                                              (replaces
                                                                                                               -1)
                                                                                                              (("1"
                                                                                                                (replaces
                                                                                                                 -1)
                                                                                                                (("1"
                                                                                                                  (assert)
                                                                                                                  (("1"
                                                                                                                    (case
                                                                                                                     "c*(c-pu) <= c*(c-pv)")
                                                                                                                    (("1"
                                                                                                                      (assert)
                                                                                                                      nil
                                                                                                                      nil)
                                                                                                                     ("2"
                                                                                                                      (hide
                                                                                                                       2)
                                                                                                                      (("2"
                                                                                                                        (case
                                                                                                                         "c*pv <= c*pu")
                                                                                                                        (("1"
                                                                                                                          (hide-all-but
                                                                                                                           (-1
                                                                                                                            1))
                                                                                                                          (("1"
                                                                                                                            (grind)
                                                                                                                            nil
                                                                                                                            nil))
                                                                                                                          nil)
                                                                                                                         ("2"
                                                                                                                          (hide
                                                                                                                           2)
                                                                                                                          (("2"
                                                                                                                            (expand
                                                                                                                             "intersects_circle_fun?")
                                                                                                                            (("2"
                                                                                                                              (case
                                                                                                                               "FORALL (b1,b2:bool): (b1 OR b2) IFF (b1 OR (b2 AND NOT b1))")
                                                                                                                              (("1"
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                                                                                                                                 "uvnz")
                                                                                                                                (("1"
                                                                                                                                  (assert)
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                                                                                                                                    (hide
                                                                                                                                     "uvnz")
                                                                                                                                    (("1"
                                                                                                                                      (rewrite
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                                                                                                                                         "hyp1"
                                                                                                                                         -4)
                                                                                                                                        (("1"
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                                                                                                                                           "hyp1")
                                                                                                                                          (("1"
                                                                                                                                            (rewrite
                                                                                                                                             -1)
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                                                                                                                                              (reveal
                                                                                                                                               "hyp1")
                                                                                                                                              (("1"
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                                                                                                                                                 "hyp1")
                                                                                                                                                (("1"
                                                                                                                                                  (hide
                                                                                                                                                   -2)
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                                                                                                                                                    (flatten)
                                                                                                                                                    (("1"
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                                                                                                                                                      (("1"
                                                                                                                                                        (lemma
                                                                                                                                                         "Delta_discr2b[R]")
                                                                                                                                                        (("1"
                                                                                                                                                          (hide-all-but
                                                                                                                                                           (-1
                                                                                                                                                            -2
                                                                                                                                                            -6
                                                                                                                                                            1))
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                                                                                                                                                            (expand
                                                                                                                                                             "discr2b")
                                                                                                                                                            (("1"
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                                                                                                                                                               +)
                                                                                                                                                              (("1"
                                                                                                                                                                (inst?)
                                                                                                                                                                (("1"
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                                                                                                                                                                  (("1"
                                                                                                                                                                    (reveal
                                                                                                                                                                     "sqvu")
                                                                                                                                                                    (("1"
                                                                                                                                                                      (assert)
                                                                                                                                                                      nil
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil)
                                                                                                                                                               ("2"
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                                                                                                                                                                 -2)
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                                                                                                                                                                  (inst?)
                                                                                                                                                                  (("2"
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                                                                                                                                                                    (("2"
                                                                                                                                                                      (reveal
                                                                                                                                                                       "sqvv")
                                                                                                                                                                      (("2"
                                                                                                                                                                        (assert)
                                                                                                                                                                        nil
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil))
                                                                                                                                                          nil))
                                                                                                                                                        nil)
                                                                                                                                                       ("2"
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                                                                                                                                                        (("2"
                                                                                                                                                          (ground)
                                                                                                                                                          (("1"
                                                                                                                                                            (hide
                                                                                                                                                             (-6
                                                                                                                                                              -7))
                                                                                                                                                            (("1"
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                                                                                                                                                               "pv")
                                                                                                                                                              (("1"
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                                                                                                                                                                 "pu")
                                                                                                                                                                (("1"
                                                                                                                                                                  (case
                                                                                                                                                                   "inter_circle_max_time(v, c, R) <=inter_circle_max_time(u, c, R)")
                                                                                                                                                                  (("1"
                                                                                                                                                                    (mult-ineq
                                                                                                                                                                     -1
                                                                                                                                                                     -8)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (assert)
                                                                                                                                                                      nil
                                                                                                                                                                      nil))
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                                                                                                                                                                   ("2"
                                                                                                                                                                    (hide
                                                                                                                                                                     2)
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                                                                                                                                                                       "inter_circle_max_time")
                                                                                                                                                                      (("2"
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                                                                                                                                                                         "Theta_D")
                                                                                                                                                                        (("2"
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                                                                                                                                                                           "root2b")
                                                                                                                                                                          (("2"
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                                                                                                                                                                             "discr2b")
                                                                                                                                                                            (("2"
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                                                                                                                                                                               ("sqvu"
                                                                                                                                                                                "sqvv"))
                                                                                                                                                                              (("2"
                                                                                                                                                                                (replace
                                                                                                                                                                                 -1)
                                                                                                                                                                                (("2"
                                                                                                                                                                                  (replace
                                                                                                                                                                                   -2)
                                                                                                                                                                                  (("2"
                                                                                                                                                                                    (assert)
                                                                                                                                                                                    (("2"
                                                                                                                                                                                      (case
                                                                                                                                                                                       "FORALL (zzpr:real): zzpr/1 = zzpr")
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                                                                                                                                                                                         -1)
                                                                                                                                                                                        (("1"
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                                                                                                                                                                                           -1)
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                                                                                                                                                                                             -1)
                                                                                                                                                                                            (("1"
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                                                                                                                                                                                               "-(-c * v) = v*c")
                                                                                                                                                                                              (("1"
                                                                                                                                                                                                (case-replace
                                                                                                                                                                                                 "-(-c * u) = u*c")
                                                                                                                                                                                                (("1"
                                                                                                                                                                                                  (hide
                                                                                                                                                                                                   -1
                                                                                                                                                                                                   -2)
                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                    (case
                                                                                                                                                                                                     "sqrt(sq(-c * v) + sq(R) - sqv(c)) <= sqrt(sq(-c * u) + sq(R) - sqv(c))")
                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                      (assert)
                                                                                                                                                                                                      nil
                                                                                                                                                                                                      nil)
                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                      (hide
                                                                                                                                                                                                       2)
                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                         "sqrt_le")
                                                                                                                                                                                                        (("2"
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                                                                                                                                                                                                           "sq_le")
                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                            (inst
                                                                                                                                                                                                             -
                                                                                                                                                                                                             "v*c"
                                                                                                                                                                                                             "u*c")
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (grind)
                                                                                                                                                                                                                nil
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil)
                                                                                                                                                                                                             ("2"
                                                                                                                                                                                                              (hide-all-but
                                                                                                                                                                                                               (-4
                                                                                                                                                                                                                1))
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (grind)
                                                                                                                                                                                                                nil
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil)
                                                                                                                                                                                                             ("3"
                                                                                                                                                                                                              (hide-all-but
                                                                                                                                                                                                               (-3
                                                                                                                                                                                                                1))
                                                                                                                                                                                                              (("3"
                                                                                                                                                                                                                (grind)
                                                                                                                                                                                                                nil
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil)
                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                   1)
                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                    nil
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil)
                                                                                                                                                                                               ("2"
                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                 1)
                                                                                                                                                                                                (("2"
                                                                                                                                                                                                  (grind)
                                                                                                                                                                                                  nil
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil)
                                                                                                                                                                                       ("2"
                                                                                                                                                                                        (hide-all-but
                                                                                                                                                                                         1)
                                                                                                                                                                                        (("2"
                                                                                                                                                                                          (skeep)
                                                                                                                                                                                          (("2"
                                                                                                                                                                                            (assert)
                                                                                                                                                                                            nil
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil))
                                                                                                                                                                                      nil))
                                                                                                                                                                                    nil))
                                                                                                                                                                                  nil))
                                                                                                                                                                                nil))
                                                                                                                                                                              nil))
                                                                                                                                                                            nil))
                                                                                                                                                                          nil))
                                                                                                                                                                        nil))
                                                                                                                                                                      nil))
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil)
                                                                                                                                                           ("2"
                                                                                                                                                            (mult-by
                                                                                                                                                             -2
                                                                                                                                                             "inter_circle_max_time(u, c, R)")
                                                                                                                                                            (("2"
                                                                                                                                                              (mult-by
                                                                                                                                                               1
                                                                                                                                                               "inter_circle_max_time(v, c, R)")
                                                                                                                                                              (("2"
                                                                                                                                                                (expand
                                                                                                                                                                 "pv"
                                                                                                                                                                 +)
                                                                                                                                                                (("2"
                                                                                                                                                                  (expand
                                                                                                                                                                   "pu"
                                                                                                                                                                   +)
                                                                                                                                                                  (("2"
                                                                                                                                                                    (assert)
                                                                                                                                                                    nil
                                                                                                                                                                    nil))
                                                                                                                                                                  nil))
                                                                                                                                                                nil))
                                                                                                                                                              nil))
                                                                                                                                                            nil)
                                                                                                                                                           ("3"
                                                                                                                                                            (hide-all-but
                                                                                                                                                             (-1
                                                                                                                                                              -9
                                                                                                                                                              1))
                                                                                                                                                            (("3"
                                                                                                                                                              (grind)
                                                                                                                                                              nil
                                                                                                                                                              nil))
                                                                                                                                                            nil)
                                                                                                                                                           ("4"
                                                                                                                                                            (reveal
                                                                                                                                                             "dlem")
                                                                                                                                                            (("4"
                                                                                                                                                              (case
                                                                                                                                                               "sq(dist(pv, c)) = sq(R) AND sq(dist(pu, c)) = sq(R)")
                                                                                                                                                              (("1"
                                                                                                                                                                (flatten)
                                                                                                                                                                (("1"
                                                                                                                                                                  (replaces
                                                                                                                                                                   -1)
                                                                                                                                                                  (("1"
                                                                                                                                                                    (replaces
                                                                                                                                                                     -1)
                                                                                                                                                                    (("1"
                                                                                                                                                                      (assert)
                                                                                                                                                                      (("1"
                                                                                                                                                                        (case
                                                                                                                                                                         "sq(dist(pv,pu)) = sqv(pu-pv)")
                                                                                                                                                                        (("1"
                                                                                                                                                                          (replace
                                                                                                                                                                           -1)
                                                                                                                                                                          (("1"
                                                                                                                                                                            (hide
                                                                                                                                                                             -1)
                                                                                                                                                                            (("1"
                                                                                                                                                                              (both-sides
                                                                                                                                                                               "-"
                                                                                                                                                                               "sqv(pu-pv)"
                                                                                                                                                                               -1)
                                                                                                                                                                              (("1"
                                                                                                                                                                                (assert)
                                                                                                                                                                                (("1"
                                                                                                                                                                                  (mult-by
                                                                                                                                                                                   -1
                                                                                                                                                                                   "-1")
                                                                                                                                                                                  (("1"
                                                                                                                                                                                    (assert)
                                                                                                                                                                                    (("1"
                                                                                                                                                                                      (case
                                                                                                                                                                                       "(pu*c-sqv(c))^2 + (pu*perpR(c))^2 = (pv*c-sqv(c))^2 + (pv*perpR(c))^2")
                                                                                                                                                                                      (("1"
                                                                                                                                                                                        (case
                                                                                                                                                                                         "(pu * perpR(c)) ^ 2 >= (pv * perpR(c)) ^ 2 AND (pu * c - sqv(c)) ^ 2 >
                                                                                                                                                                                                                                                      (pv * c - sqv(c)) ^ 2")
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (flatten)
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (assert)
                                                                                                                                                                                            nil
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil)
                                                                                                                                                                                         ("2"
                                                                                                                                                                                          (hide
                                                                                                                                                                                           -1)
                                                                                                                                                                                          (("2"
                                                                                                                                                                                            (lemma
                                                                                                                                                                                             "orthogonal_basis")
                                                                                                                                                                                            (("2"
                                                                                                                                                                                              (case
                                                                                                                                                                                               "FORALL (e1, e2, w1,w2: Vect2):
                                                                                                                                                                                                                                            e1 /= zero AND e2 /= zero AND orthogonal?(e1, e2) AND norm(e1) = norm(e2) AND sqv(w1) = sqv(w2) IMPLIES
                                                                                                                                                                                                                                             ((sq(w1*e1) >= sq(w2*e1) AND sq(w1*e2) <= sq(w2*e2)) OR (sq(w1*e1) < sq(w2*e1) AND sq(w1*e2) > sq(w2*e2)))")
                                                                                                                                                                                              (("1"
                                                                                                                                                                                                (hide
                                                                                                                                                                                                 -2)
                                                                                                                                                                                                (("1"
                                                                                                                                                                                                  (inst
                                                                                                                                                                                                   -
                                                                                                                                                                                                   "c"
                                                                                                                                                                                                   "perpR(c)"
                                                                                                                                                                                                   _
                                                                                                                                                                                                   _)
                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                    (inst-cp
                                                                                                                                                                                                     -
                                                                                                                                                                                                     "v"
                                                                                                                                                                                                     "u")
                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                      (inst
                                                                                                                                                                                                       -
                                                                                                                                                                                                       "pv-c"
                                                                                                                                                                                                       "pu-c")
                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                          (case
                                                                                                                                                                                                           "perpR(c) /= zero AND
                                                                                                                                                                                                                                                         orthogonal?(c, perpR(c))")
                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                            (flatten)
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (case
                                                                                                                                                                                                                 "norm(c) = norm(perpR(c)) AND sqv(v) = sqv(u) AND sqv(pv - c) = sqv(pu - c)")
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (flatten)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (assert)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (split
                                                                                                                                                                                                                       -)
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (flatten)
                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                          (case
                                                                                                                                                                                                                           "c*pv <= 0")
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (lemma
                                                                                                                                                                                                                             "sq_gt")
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (inst
                                                                                                                                                                                                                               -
                                                                                                                                                                                                                               "-((pu-c)*c)"
                                                                                                                                                                                                                               "-((pv-c)*c)")
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (rewrite
                                                                                                                                                                                                                                 "sq_neg")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (rewrite
                                                                                                                                                                                                                                   "sq_neg")
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (assert)
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (hide-all-but
                                                                                                                                                                                                                                       (1
                                                                                                                                                                                                                                        6))
                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                        (grind)
                                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil)
                                                                                                                                                                                                                               ("2"
                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                 (-1
                                                                                                                                                                                                                                  1))
                                                                                                                                                                                                                                (("2"
                                                                                                                                                                                                                                  (typepred
                                                                                                                                                                                                                                   "sqv(c)")
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil)
                                                                                                                                                                                                                               ("3"
                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                 (-1
                                                                                                                                                                                                                                  1
                                                                                                                                                                                                                                  6))
                                                                                                                                                                                                                                (("3"
                                                                                                                                                                                                                                  (typepred
                                                                                                                                                                                                                                   "sqv(c)")
                                                                                                                                                                                                                                  (("3"
                                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                            (mult-by
                                                                                                                                                                                                                             4
                                                                                                                                                                                                                             "inter_circle_max_time(v, c, R)")
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                 (1
                                                                                                                                                                                                                                  4))
                                                                                                                                                                                                                                (("2"
                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                   "pv")
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (grind
                                                                                                                                                                                                                                     :exclude
                                                                                                                                                                                                                                     "inter_circle_max_time")
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                        (flatten)
                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                          (split
                                                                                                                                                                                                                           -)
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (flatten)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (name
                                                                                                                                                                                                                               "puc"
                                                                                                                                                                                                                               "inter_circle_max_time(u, c, R)")
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (name
                                                                                                                                                                                                                                 "pvc"
                                                                                                                                                                                                                                 "inter_circle_max_time(v, c, R)")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (replaces
                                                                                                                                                                                                                                   -1)
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (replaces
                                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (case
                                                                                                                                                                                                                                       "pvc <= puc")
                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                        (case
                                                                                                                                                                                                                                         "sq(pvc) <= sq(puc)")
                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                          (mult-ineq
                                                                                                                                                                                                                                           -1
                                                                                                                                                                                                                                           -4)
                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                            (rewrite
                                                                                                                                                                                                                                             "sq_times"
                                                                                                                                                                                                                                             :dir
                                                                                                                                                                                                                                             rl)
                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                                                               "sq_times"
                                                                                                                                                                                                                                               :dir
                                                                                                                                                                                                                                               rl)
                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                (split
                                                                                                                                                                                                                                                 +)
                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                                                                   (-1
                                                                                                                                                                                                                                                    1))
                                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                                    (expand
                                                                                                                                                                                                                                                     "pvc")
                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                      (expand
                                                                                                                                                                                                                                                       "pu")
                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                        (expand
                                                                                                                                                                                                                                                         "puc")
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (expand
                                                                                                                                                                                                                                                           "pv")
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (grind
                                                                                                                                                                                                                                                             :exclude
                                                                                                                                                                                                                                                             ("inter_circle_max_time"
                                                                                                                                                                                                                                                              "perpR"))
                                                                                                                                                                                                                                                            nil
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                                  (lemma
                                                                                                                                                                                                                                                   "sq_gt")
                                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                                    (inst
                                                                                                                                                                                                                                                     -
                                                                                                                                                                                                                                                     "- (pu * c - sqv(c))"
                                                                                                                                                                                                                                                     "-(pv * c - sqv(c))")
                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                                       "sq_neg")
                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                                                                         "sq_neg")
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (flatten)
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (hide
                                                                                                                                                                                                                                                             -1)
                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                              (split
                                                                                                                                                                                                                                                               -)
                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                                                 (-1
                                                                                                                                                                                                                                                                  1))
                                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                                  (grind)
                                                                                                                                                                                                                                                                  nil
                                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                                nil)
                                                                                                                                                                                                                                                               ("2"
                                                                                                                                                                                                                                                                (case
                                                                                                                                                                                                                                                                 "c*pu <=0")
                                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                                                                                   (-1
                                                                                                                                                                                                                                                                    1
                                                                                                                                                                                                                                                                    6))
                                                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                                                  (mult-by
                                                                                                                                                                                                                                                                   6
                                                                                                                                                                                                                                                                   "puc")
                                                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                                                    (hide-all-but
                                                                                                                                                                                                                                                                     (1
                                                                                                                                                                                                                                                                      6))
                                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                                      (expand
                                                                                                                                                                                                                                                                       "pu")
                                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                                        (expand
                                                                                                                                                                                                                                                                         "puc")
                                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                                          (grind
                                                                                                                                                                                                                                                                           :exclude
                                                                                                                                                                                                                                                                           ("inter_circle_max_time"))
                                                                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                                    nil)
                                                                                                                                                                                                                                                                   ("2"
                                                                                                                                                                                                                                                                    (case
                                                                                                                                                                                                                                                                     "puc = 0")
                                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                                      (expand
                                                                                                                                                                                                                                                                       "puc"
                                                                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                                        (expand
                                                                                                                                                                                                                                                                         "pu"
                                                                                                                                                                                                                                                                         2)
                                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                                          (replace
                                                                                                                                                                                                                                                                           -1)
                                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                                                                                            nil
                                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                                                                      (assert)
                                                                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                                                      (mult-by
                                                                                                                                                                                                                                                       4
                                                                                                                                                                                                                                                       "pvc")
                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                        (hide-all-but
                                                                                                                                                                                                                                                         (1
                                                                                                                                                                                                                                                          4))
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (typepred
                                                                                                                                                                                                                                                           "sqv(c)")
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (expand
                                                                                                                                                                                                                                                             "pvc")
                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                              (expand
                                                                                                                                                                                                                                                               "pv")
                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                (grind
                                                                                                                                                                                                                                                                 :exclude
                                                                                                                                                                                                                                                                 ("inter_circle_max_time"))
                                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                                                        (case
                                                                                                                                                                                                                                                         "pvc = 0")
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (replace
                                                                                                                                                                                                                                                           -1)
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                              (replace
                                                                                                                                                                                                                                                               -19
                                                                                                                                                                                                                                                               :dir
                                                                                                                                                                                                                                                               rl)
                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                (assert)
                                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil)
                                                                                                                                                                                                                                                         ("2"
                                                                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                                     ("3"
                                                                                                                                                                                                                                                      (mult-by
                                                                                                                                                                                                                                                       5
                                                                                                                                                                                                                                                       "puc")
                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                        (hide-all-but
                                                                                                                                                                                                                                                         (1
                                                                                                                                                                                                                                                          5))
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (typepred
                                                                                                                                                                                                                                                           "sqv(c)")
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (expand
                                                                                                                                                                                                                                                             "pu")
                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                              (expand
                                                                                                                                                                                                                                                               "puc")
                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                (grind
                                                                                                                                                                                                                                                                 :exclude
                                                                                                                                                                                                                                                                 ("inter_circle_max_time"))
                                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                                                        (case
                                                                                                                                                                                                                                                         "puc = 0")
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (replace
                                                                                                                                                                                                                                                           -1)
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                              (replace
                                                                                                                                                                                                                                                               -20
                                                                                                                                                                                                                                                               :dir
                                                                                                                                                                                                                                                               rl)
                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                (assert)
                                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil)
                                                                                                                                                                                                                                                         ("2"
                                                                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil)
                                                                                                                                                                                                                                         ("2"
                                                                                                                                                                                                                                          (lemma
                                                                                                                                                                                                                                           "sq_le")
                                                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                                                            (inst?)
                                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                                        (case
                                                                                                                                                                                                                                         "FORALL (a1,a2,b1,b2:nnreal): a1<=a2 AND b1<=b2 IMPLIES a1*b1<=a2*b2")
                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                          (inst
                                                                                                                                                                                                                                           -
                                                                                                                                                                                                                                           "puc"
                                                                                                                                                                                                                                           "pvc"
                                                                                                                                                                                                                                           "-(c*u)"
                                                                                                                                                                                                                                           "-(c*v)")
                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                              (split
                                                                                                                                                                                                                                               -)
                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                                 (-1
                                                                                                                                                                                                                                                  6))
                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                                   "pv")
                                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                                    (expand
                                                                                                                                                                                                                                                     "pu")
                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                      (expand
                                                                                                                                                                                                                                                       "pvc")
                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                        (expand
                                                                                                                                                                                                                                                         "puc")
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (grind)
                                                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                nil)
                                                                                                                                                                                                                                               ("2"
                                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                                 (-18
                                                                                                                                                                                                                                                  1))
                                                                                                                                                                                                                                                (("2"
                                                                                                                                                                                                                                                  (grind)
                                                                                                                                                                                                                                                  nil
                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil)
                                                                                                                                                                                                                                         ("2"
                                                                                                                                                                                                                                          (hide-all-but
                                                                                                                                                                                                                                           1)
                                                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                                                            (skeep)
                                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                                              (mult-ineq
                                                                                                                                                                                                                                               -1
                                                                                                                                                                                                                                               -2)
                                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                            (flatten)
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (lemma
                                                                                                                                                                                                                               "sq_lt")
                                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                                (inst
                                                                                                                                                                                                                                 -
                                                                                                                                                                                                                                 "-(v*c)"
                                                                                                                                                                                                                                 "-(u*c)")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (rewrite
                                                                                                                                                                                                                                   "sq_neg")
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (rewrite
                                                                                                                                                                                                                                     "sq_neg")
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (assert)
                                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                                                   (1
                                                                                                                                                                                                                                    5))
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                 ("3"
                                                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                                                   (1
                                                                                                                                                                                                                                    4))
                                                                                                                                                                                                                                  (("3"
                                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                  (split
                                                                                                                                                                                                                   +)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (hide-all-but
                                                                                                                                                                                                                     1)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (grind)
                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil)
                                                                                                                                                                                                                   ("2"
                                                                                                                                                                                                                    (reveal
                                                                                                                                                                                                                     "sqvv")
                                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                                      (reveal
                                                                                                                                                                                                                       "sqvu")
                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil)
                                                                                                                                                                                                                   ("3"
                                                                                                                                                                                                                    (lemma
                                                                                                                                                                                                                     "inter_circle_max_time_def")
                                                                                                                                                                                                                    (("3"
                                                                                                                                                                                                                      (inst
                                                                                                                                                                                                                       -
                                                                                                                                                                                                                       "R"
                                                                                                                                                                                                                       "c"
                                                                                                                                                                                                                       _)
                                                                                                                                                                                                                      (("3"
                                                                                                                                                                                                                        (inst-cp
                                                                                                                                                                                                                         -
                                                                                                                                                                                                                         "v")
                                                                                                                                                                                                                        (("3"
                                                                                                                                                                                                                          (inst
                                                                                                                                                                                                                           -
                                                                                                                                                                                                                           "u")
                                                                                                                                                                                                                          (("3"
                                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                                            (("3"
                                                                                                                                                                                                                              (flatten)
                                                                                                                                                                                                                              (("3"
                                                                                                                                                                                                                                (hide
                                                                                                                                                                                                                                 (-2
                                                                                                                                                                                                                                  -4
                                                                                                                                                                                                                                  -6
                                                                                                                                                                                                                                  -7))
                                                                                                                                                                                                                                (("3"
                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                   "circle?")
                                                                                                                                                                                                                                  (("3"
                                                                                                                                                                                                                                    (case
                                                                                                                                                                                                                                     "FORALL (vv1,vv2:Vect2): norm(vv1) = R AND norm(vv2) = R IMPLIES sqv(vv1) = sqv(vv2)")
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                                      (hide-all-but
                                                                                                                                                                                                                                       1)
                                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                                        (case
                                                                                                                                                                                                                                         "FORALL (vvv1:Vect2): norm(vvv1) = R IMPLIES sqv(vvv1) = sq(R)")
                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                          (skeep)
                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                            (inst-cp
                                                                                                                                                                                                                                             -
                                                                                                                                                                                                                                             "vv1")
                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                              (inst
                                                                                                                                                                                                                                               -
                                                                                                                                                                                                                                               "vv2")
                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                (assert)
                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil)
                                                                                                                                                                                                                                         ("2"
                                                                                                                                                                                                                                          (hide
                                                                                                                                                                                                                                           2)
                                                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                                                            (skeep)
                                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                                                               "sqrt_eq"
                                                                                                                                                                                                                                               1
                                                                                                                                                                                                                                               :dir
                                                                                                                                                                                                                                               rl)
                                                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                                                (rewrite
                                                                                                                                                                                                                                                 "sqrt_sqv_norm")
                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil)
                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                            (hide
                                                                                                                                                                                                             2)
                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                              (hide
                                                                                                                                                                                                               (-1
                                                                                                                                                                                                                -2))
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (split
                                                                                                                                                                                                                 +)
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (flatten)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (case
                                                                                                                                                                                                                     "c = zero")
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (replace
                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                      (hide-all-but
                                                                                                                                                                                                                       (-1
                                                                                                                                                                                                                        1))
                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                        (expand
                                                                                                                                                                                                                         "perpR")
                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                          (expand
                                                                                                                                                                                                                           "zero")
                                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                                            (flatten)
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (decompose-equality
                                                                                                                                                                                                                               +)
                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                  (hide-all-but
                                                                                                                                                                                                                   1)
                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                    (grind)
                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil)
                                                                                                                                                                                               ("2"
                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                 (-1
                                                                                                                                                                                                  1))
                                                                                                                                                                                                (("2"
                                                                                                                                                                                                  (skeep)
                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                    (inst
                                                                                                                                                                                                     -
                                                                                                                                                                                                     "e1"
                                                                                                                                                                                                     "e2"
                                                                                                                                                                                                     _)
                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                      (assert)
                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                        (case
                                                                                                                                                                                                         "FORALL (w: Vect2):
                                                                                                                                                                                      sqv(w) = sq((w * e1) / sqv(e1))*sqv(e1) + sq((w * e2) / sqv(e2))*sqv(e2)")
                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                          (hide
                                                                                                                                                                                                           -5)
                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                            (inst-cp
                                                                                                                                                                                                             -
                                                                                                                                                                                                             "w1")
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (inst
                                                                                                                                                                                                               -
                                                                                                                                                                                                               "w2")
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (case
                                                                                                                                                                                                                 "sqv(w1) = sqv(w2)")
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                   -1
                                                                                                                                                                                                                   :dir
                                                                                                                                                                                                                   rl)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (hide
                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (replace
                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (hide
                                                                                                                                                                                                                         -1)
                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                          (case
                                                                                                                                                                                                                           "sqv(e1) = sqv(e2)")
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (replace
                                                                                                                                                                                                                             -1
                                                                                                                                                                                                                             :dir
                                                                                                                                                                                                                             rl)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (hide
                                                                                                                                                                                                                               -1)
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (rewrite
                                                                                                                                                                                                                                 "sq_div")
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (rewrite
                                                                                                                                                                                                                                   "sq_div")
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (rewrite
                                                                                                                                                                                                                                     "sq_div")
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                       "sq_div")
                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                        (mult-by
                                                                                                                                                                                                                                         -1
                                                                                                                                                                                                                                         "sqv(e1)")
                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                          (expand
                                                                                                                                                                                                                                           "sq")
                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                            (ground)
                                                                                                                                                                                                                                            nil
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                        nil)
                                                                                                                                                                                                                                       ("2"
                                                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                                                         "sqv_eq_0"
                                                                                                                                                                                                                                         :dir
                                                                                                                                                                                                                                         rl)
                                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                       "sqv_eq_0"
                                                                                                                                                                                                                                       :dir
                                                                                                                                                                                                                                       rl)
                                                                                                                                                                                                                                      nil
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil)
                                                                                                                                                                                                                                   ("2"
                                                                                                                                                                                                                                    (rewrite
                                                                                                                                                                                                                                     "sqv_eq_0"
                                                                                                                                                                                                                                     :dir
                                                                                                                                                                                                                                     rl)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                  (rewrite
                                                                                                                                                                                                                                   "sqrt_eq"
                                                                                                                                                                                                                                   :dir
                                                                                                                                                                                                                                   rl)
                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                    (rewrite
                                                                                                                                                                                                                                     "sqv_eq_0"
                                                                                                                                                                                                                                     :dir
                                                                                                                                                                                                                                     rl)
                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                            (rewrite
                                                                                                                                                                                                                             "sqrt_eq"
                                                                                                                                                                                                                             :dir
                                                                                                                                                                                                                             rl)
                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                                               "sqrt_sqv_norm")
                                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                                (rewrite
                                                                                                                                                                                                                                 "sqrt_sqv_norm")
                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                  (propax)
                                                                                                                                                                                                                  nil
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil)
                                                                                                                                                                                                         ("2"
                                                                                                                                                                                                          (skeep)
                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                            (inst
                                                                                                                                                                                                             -
                                                                                                                                                                                                             "w")
                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                              (name
                                                                                                                                                                                                               "cv1"
                                                                                                                                                                                                               "sq((w * e1) / sqv(e1)) * sqv(e1) + sq((w * e2) / sqv(e2)) * sqv(e2)")
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (replace
                                                                                                                                                                                                                 -1)
                                                                                                                                                                                                                (("2"
                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                   -5
                                                                                                                                                                                                                   +)
                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                    (case
                                                                                                                                                                                                                     "sqv(e1) = sqv(e2)")
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (replace
                                                                                                                                                                                                                       -1
                                                                                                                                                                                                                       :dir
                                                                                                                                                                                                                       rl)
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                                         "sqv_add")
                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                          (expand
                                                                                                                                                                                                                           "orthogonal?")
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (replace
                                                                                                                                                                                                                             -3)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (hide-all-but
                                                                                                                                                                                                                                 (1
                                                                                                                                                                                                                                  2
                                                                                                                                                                                                                                  3))
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                   "cv1")
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (rewrite
                                                                                                                                                                                                                                     "sqv_scal")
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                       "sqv_scal")
                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                       "sqrt_eq"
                                                                                                                                                                                                                       :dir
                                                                                                                                                                                                                       rl)
                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                                         "sqrt_sqv_norm")
                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                          (rewrite
                                                                                                                                                                                                                           "sqrt_sqv_norm")
                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                nil))
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil)
                                                                                                                                                                                                         ("3"
                                                                                                                                                                                                          (rewrite
                                                                                                                                                                                                           "sqv_eq_0"
                                                                                                                                                                                                           :dir
                                                                                                                                                                                                           rl)
                                                                                                                                                                                                          (("3"
                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                            (("3"
                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                               "sqv_eq_0"
                                                                                                                                                                                                               :dir
                                                                                                                                                                                                               rl)
                                                                                                                                                                                                              nil
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil)
                                                                                                                                                                                                         ("4"
                                                                                                                                                                                                          (rewrite
                                                                                                                                                                                                           "sqv_eq_0"
                                                                                                                                                                                                           :dir
                                                                                                                                                                                                           rl)
                                                                                                                                                                                                          (("4"
                                                                                                                                                                                                            (rewrite
                                                                                                                                                                                                             "sqv_eq_0"
                                                                                                                                                                                                             :dir
                                                                                                                                                                                                             rl)
                                                                                                                                                                                                            (("4"
                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                              nil
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil))
                                                                                                                                                                                                          nil))
                                                                                                                                                                                                        nil))
                                                                                                                                                                                                      nil))
                                                                                                                                                                                                    nil))
                                                                                                                                                                                                  nil))
                                                                                                                                                                                                nil))
                                                                                                                                                                                              nil))
                                                                                                                                                                                            nil))
                                                                                                                                                                                          nil))
                                                                                                                                                                                        nil)
                                                                                                                                                                                       ("2"
                                                                                                                                                                                        (case
                                                                                                                                                                                         "sqv(pu-c) = sq(R) AND sqv(pv-c) = sq(R)")
                                                                                                                                                                                        (("1"
                                                                                                                                                                                          (flatten)
                                                                                                                                                                                          (("1"
                                                                                                                                                                                            (lemma
                                                                                                                                                                                             "orthogonal_basis")
                                                                                                                                                                                            (("1"
                                                                                                                                                                                              (inst
                                                                                                                                                                                               -
                                                                                                                                                                                               "c"
                                                                                                                                                                                               "perpR(c)"
                                                                                                                                                                                               _)
                                                                                                                                                                                              (("1"
                                                                                                                                                                                                (inst-cp
                                                                                                                                                                                                 -
                                                                                                                                                                                                 "pu-c")
                                                                                                                                                                                                (("1"
                                                                                                                                                                                                  (inst
                                                                                                                                                                                                   -
                                                                                                                                                                                                   "pv-c")
                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                    (assert)
                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                      (case
                                                                                                                                                                                                       "perpR(c) /= zero AND orthogonal?(c, perpR(c))")
                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                        (flatten)
                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                            (hide
                                                                                                                                                                                                             -1)
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (lemma
                                                                                                                                                                                                               "orthogonal_basis_sqv")
                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                (inst?)
                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                  (assert)
                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                    (replace
                                                                                                                                                                                                                     -5)
                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                      (hide
                                                                                                                                                                                                                       -2)
                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                        (lemma
                                                                                                                                                                                                                         "orthogonal_basis_sqv")
                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                          (inst?)
                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                            (assert)
                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                              (replace
                                                                                                                                                                                                                               -4)
                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                (replace
                                                                                                                                                                                                                                 -1
                                                                                                                                                                                                                                 -2)
                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                  (hide
                                                                                                                                                                                                                                   (-1
                                                                                                                                                                                                                                    -3))
                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                    (case
                                                                                                                                                                                                                                     "sqv(perpR(c)) = sqv(c)")
                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                      (replace
                                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                        (hide
                                                                                                                                                                                                                                         -1)
                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                          (rewrite
                                                                                                                                                                                                                                           "sq_div")
                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                            (rewrite
                                                                                                                                                                                                                                             "sq_div")
                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                                                               "sq_div")
                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                (rewrite
                                                                                                                                                                                                                                                 "sq_div")
                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                  (rewrite
                                                                                                                                                                                                                                                   "dot_sub_left")
                                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                                    (rewrite
                                                                                                                                                                                                                                                     "dot_sub_left")
                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                                       "dot_sub_left")
                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                                                                         "dot_sub_left")
                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                          (case
                                                                                                                                                                                                                                                           "c*perpR(c) = 0")
                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                            (replace
                                                                                                                                                                                                                                                             -1)
                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                (case
                                                                                                                                                                                                                                                                 "c*c = sqv(c)")
                                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                                  (replace
                                                                                                                                                                                                                                                                   -1)
                                                                                                                                                                                                                                                                  (("1"
                                                                                                                                                                                                                                                                    (case
                                                                                                                                                                                                                                                                     "FORALL (zpop:real): zpop^2 = sq(zpop)")
                                                                                                                                                                                                                                                                    (("1"
                                                                                                                                                                                                                                                                      (rewrite
                                                                                                                                                                                                                                                                       -1)
                                                                                                                                                                                                                                                                      (("1"
                                                                                                                                                                                                                                                                        (rewrite
                                                                                                                                                                                                                                                                         -1)
                                                                                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                                                                                          (rewrite
                                                                                                                                                                                                                                                                           -1)
                                                                                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                                                                                            (rewrite
                                                                                                                                                                                                                                                                             -1)
                                                                                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                                                                                               -1)
                                                                                                                                                                                                                                                                              (("1"
                                                                                                                                                                                                                                                                                (mult-by
                                                                                                                                                                                                                                                                                 2
                                                                                                                                                                                                                                                                                 "sqv(c)/sq(sqv(c))")
                                                                                                                                                                                                                                                                                (("1"
                                                                                                                                                                                                                                                                                  (assert)
                                                                                                                                                                                                                                                                                  nil
                                                                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                                                                  (flatten)
                                                                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                                                                    (cross-mult
                                                                                                                                                                                                                                                                                     -1)
                                                                                                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                                                                                                      (lemma
                                                                                                                                                                                                                                                                                       "sqv_eq_0")
                                                                                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                                                                                        (inst?)
                                                                                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                                                                                          (assert)
                                                                                                                                                                                                                                                                                          nil
                                                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                                                     ("2"
                                                                                                                                                                                                                                                                      (hide-all-but
                                                                                                                                                                                                                                                                       1)
                                                                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                                                                        (skeep)
                                                                                                                                                                                                                                                                        (("2"
                                                                                                                                                                                                                                                                          (expand
                                                                                                                                                                                                                                                                           "^")
                                                                                                                                                                                                                                                                          (("2"
                                                                                                                                                                                                                                                                            (expand
                                                                                                                                                                                                                                                                             "expt")
                                                                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                                                                              (expand
                                                                                                                                                                                                                                                                               "expt")
                                                                                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                                                                                (expand
                                                                                                                                                                                                                                                                                 "expt")
                                                                                                                                                                                                                                                                                (("2"
                                                                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                                                                   "sq")
                                                                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                                                                    (propax)
                                                                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                                                  (expand
                                                                                                                                                                                                                                                                   "sqv"
                                                                                                                                                                                                                                                                   1)
                                                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                                                    (propax)
                                                                                                                                                                                                                                                                    nil
                                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                                                            (hide-all-but
                                                                                                                                                                                                                                                             1)
                                                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                                                              (grind)
                                                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                  nil)
                                                                                                                                                                                                                                                 ("2"
                                                                                                                                                                                                                                                  (flatten)
                                                                                                                                                                                                                                                  (("2"
                                                                                                                                                                                                                                                    (lemma
                                                                                                                                                                                                                                                     "sqv_eq_0")
                                                                                                                                                                                                                                                    (("2"
                                                                                                                                                                                                                                                      (inst?)
                                                                                                                                                                                                                                                      (("2"
                                                                                                                                                                                                                                                        (assert)
                                                                                                                                                                                                                                                        nil
                                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                                      nil))
                                                                                                                                                                                                                                                    nil))
                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                nil)
                                                                                                                                                                                                                                               ("2"
                                                                                                                                                                                                                                                (flatten)
                                                                                                                                                                                                                                                (("2"
                                                                                                                                                                                                                                                  (rewrite
                                                                                                                                                                                                                                                   "sqv_eq_0")
                                                                                                                                                                                                                                                  nil
                                                                                                                                                                                                                                                  nil))
                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                              nil)
                                                                                                                                                                                                                                             ("2"
                                                                                                                                                                                                                                              (flatten)
                                                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                                                (rewrite
                                                                                                                                                                                                                                                 "sqv_eq_0")
                                                                                                                                                                                                                                                nil
                                                                                                                                                                                                                                                nil))
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil)
                                                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                                                            (flatten)
                                                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                                                              (rewrite
                                                                                                                                                                                                                                               "sqv_eq_0")
                                                                                                                                                                                                                                              nil
                                                                                                                                                                                                                                              nil))
                                                                                                                                                                                                                                            nil))
                                                                                                                                                                                                                                          nil))
                                                                                                                                                                                                                                        nil))
                                                                                                                                                                                                                                      nil)
                                                                                                                                                                                                                                     ("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"
                                                                                                                                                                                                        (split
                                                                                                                                                                                                         +)
                                                                                                                                                                                                        (("1"
                                                                                                                                                                                                          (case
                                                                                                                                                                                                           "c = zero")
                                                                                                                                                                                                          (("1"
                                                                                                                                                                                                            (replace
                                                                                                                                                                                                             -1)
                                                                                                                                                                                                            (("1"
                                                                                                                                                                                                              (assert)
                                                                                                                                                                                                              nil
                                                                                                                                                                                                              nil))
                                                                                                                                                                                                            nil)
                                                                                                                                                                                                           ("2"
                                                                                                                                                                                                            (hide-all-but
                                                                                                                                                                                                             (1
                                                                                                                                                                                                              2))
                                                                                                                                                                                                            (("2"
                                                                                                                                                                                                              (flatten)
                                                                                                                                                                                                              (("2"
                                                                                                                                                                                                                (expand
                                                                                                                                                                                                                 "zero")
                                                                                                                                                                                                                (("2"
--> --------------------

--> maximum size reached

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

Messung V0.5 in Prozent
C=100 H=100 G=100

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Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.