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Datei: sturmtarski.prf Sprache: Lisp

Original von: PVS^©

(vectors_3D
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    (* const-decl "Vector" vectors_3D nil)
    (jv const-decl "Vector" vectors_3D nil)
    (+ const-decl "Vector" vectors_3D nil)
    (kv const-decl "Vector" vectors_3D nil))
   shostak))
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      nil))
    nil)
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   nil))
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    (* const-decl "real" vectors_3D nil)
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          nil))
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                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
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   nil nil)
  (scal_cancel-1 nil 3255949401 ("" (postpone) nil nil) nil shostak))
(scal_div_mult_left 0
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    (Vector type-eq-decl nil vectors_3D nil)
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    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (* const-decl "Vector" vectors_3D nil))
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(scal_div_mult_right 0
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    (real_div_nzreal_is_real application-judgement "real" reals nil)
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(scal_eq_zero 0
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      nil))
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    (real nonempty-type-from-decl nil reals nil)
    (zero const-decl "Vector" vectors_3D nil)
    (* const-decl "Vector" vectors_3D nil))
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(dot_ge_dist 0
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    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_minus_real_is_real application-judgement "real" reals nil))
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(dot_gt_dist 0
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   ((real_gt_is_strict_total_order name-judgement
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    (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))
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(idem_right 0
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    (("" (prop)
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          nil)
         ("2" (replace -1) (("2" (propax) nil nil)) nil))
        nil)
       ("2" (replace -1)
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        nil)
       ("3" (replace -1) (("3" (grind) nil nil)) nil))
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    (* const-decl "Vector" vectors_3D nil)
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(sqv_neg 0
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    (minus_real_is_real application-judgement "real" reals nil))
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(sqv_add 0
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    (* const-decl "Vector" vectors_3D nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   nil))
(sqv_scal 0
  (sqv_scal-1 nil 3254160430 ("" (grind) nil nil)
   ((* const-decl "Vector" vectors_3D nil)
    (* const-decl "real" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_times_real_is_real application-judgement "real" reals nil))
   nil))
(sqv_sub 0
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    (* const-decl "real" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (* const-decl "Vector" vectors_3D nil)
    (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))
   nil))
(sqv_sub_scal 0
  (sqv_sub_scal-1 nil 3430826073
   ("" (skeep)
    (("" (rewrite "sqv_sub")
      (("" (rewrite "sqv_scal")
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        nil))
      nil))
    nil)
   ((sqv_sub formula-decl nil vectors_3D nil)
    (real nonempty-type-from-decl nil reals nil)
    (Vector type-eq-decl nil vectors_3D nil)
    (number nonempty-type-decl nil numbers nil)
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    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (* const-decl "Vector" vectors_3D nil)
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
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    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
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    (real_plus_real_is_real application-judgement "real" reals nil)
    (TRUE const-decl "bool" booleans nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (bijective? const-decl "bool" functions nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (* const-decl "real" vectors_3D nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal nil)
    (both_sides_times1 formula-decl nil real_props nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (neg_one_times formula-decl nil extra_tegies nil)
    (neg_neg formula-decl nil extra_tegies nil)
    (mult_neg formula-decl nil extra_tegies nil)
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (sqv_scal formula-decl nil vectors_3D nil))
   nil))
(sqv_sym 0
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   ((- const-decl "Vector" vectors_3D nil)
    (* const-decl "real" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (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))
   nil))
(sqrt_sqv_sq 0
  (sqrt_sqv_sq-1 nil 3255194049
   ("" (skosimp*)
    (("" (lemma "sqrt_square")
      (("" (inst -1 "sqv(v!1)") (("" (rewrite "sqrt_times") nil nil))
        nil))
      nil))
    nil)
   ((sqrt_square formula-decl nil sqrt "reals/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sqrt_times formula-decl nil sqrt "reals/")
    (sqv const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vector type-eq-decl nil vectors_3D 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
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    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   nil))
(norm_sym 0
  (norm_sym-1 nil 3254160430
   ("" (skosimp*)
    (("" (expand "norm")
      (("" (rewrite "sqrt_eq") (("" (rewrite "sqv_sym") nil nil)) nil))
      nil))
    nil)
   ((norm const-decl "nnreal" vectors_3D nil)
    (sqv_sym formula-decl nil vectors_3D nil)
    (- const-decl "Vector" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vector type-eq-decl nil vectors_3D 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)
    (sqrt_eq formula-decl nil sqrt "reals/"))
   nil))
(norm_neg 0
  (norm_neg-1 nil 3254160430 ("" (grind) nil nil)
   ((- const-decl "Vector" vectors_3D nil)
    (* const-decl "real" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (norm const-decl "nnreal" vectors_3D 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))
   nil))
(dot_sq_norm 0
  (dot_sq_norm-1 nil 3254160430
   ("" (skosimp*)
    (("" (expand "norm")
      (("" (rewrite "sq_sqrt") (("" (grind) nil nil)) nil)) nil))
    nil)
   ((norm const-decl "nnreal" vectors_3D nil)
    (* const-decl "real" vectors_3D nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (Vector type-eq-decl nil vectors_3D nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (sq_sqrt formula-decl nil sqrt "reals/"))
   nil))
(sq_norm 0
  (sq_norm-1 nil 3403524082 ("" (grind) nil nil)
   ((nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (* const-decl "real" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (norm const-decl "nnreal" vectors_3D nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil))
   nil))
(sqrt_sqv_norm 0
  (sqrt_sqv_norm-1 nil 3254160430
   ("" (skosimp*) (("" (expand "norm") (("" (propax) nil nil)) nil))
    nil)
   ((norm const-decl "nnreal" vectors_3D nil)) nil))
(norms_eq_sqv 0
  (norms_eq_sqv-1 nil 3427102244
   ("" (skeep)
    (("" (lemma "sq_eq")
      (("" (inst?)
        (("" (assert)
          (("" (replace -1 * rl)
            (("" (hide -1)
              (("" (rewrite "sq_norm")
                (("" (rewrite "sq_norm")
                  (("" (expand "sqv") (("" (propax) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((sq_eq formula-decl nil sq "reals/")
    (sqv const-decl "nnreal" vectors_3D nil)
    (sq_norm formula-decl nil vectors_3D nil)
    (norm const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vector type-eq-decl nil vectors_3D nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
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    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
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    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   shostak))
(norms_eq_sos 0
  (norms_eq_sos-1 nil 3602099225
   ("" (skeep)
    (("" (rewrite "norms_eq_sqv")
      (("" (rewrite "sqv_sos")
        (("" (rewrite "sqv_sos") (("" (assert) nil nil)) nil)) nil))
      nil))
    nil)
   ((norms_eq_sqv formula-decl nil vectors_3D nil)
    (real nonempty-type-from-decl nil reals nil)
    (Vector type-eq-decl nil vectors_3D nil)
    (sqv_sos formula-decl nil vectors_3D nil))
   nil))
(norm_le_sqv 0
  (norm_le_sqv-1 nil 3475937541
   ("" (skosimp*)
    (("" (expand "norm")
      (("" (lemma "sqrt_le") (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((norm const-decl "nnreal" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vector type-eq-decl nil vectors_3D 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)
    (sqrt_le formula-decl nil sqrt "reals/"))
   nil))
(norm_lt_sqv 0
  (norm_lt_sqv-1 nil 3475937556
   ("" (skosimp*)
    (("" (expand "norm")
      (("" (lemma "sqrt_lt") (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((norm const-decl "nnreal" vectors_3D nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (Vector type-eq-decl nil vectors_3D 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)
    (sqrt_lt formula-decl nil sqrt "reals/"))
   nil))
(norm_scal 0
  (norm_scal-1 nil 3254160430
   ("" (skosimp*)
    (("" (expand "norm")
      (("" (rewrite "sqv_scal")
        (("" (rewrite "sqrt_times")
          (("" (rewrite "sqrt_sq_abs") nil nil)) nil))
        nil))
      nil))
    nil)
   ((norm const-decl "nnreal" vectors_3D nil)
    (sqrt_times formula-decl nil sqrt "reals/")
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (nnreal type-eq-decl nil real_types nil)
    (sqv const-decl "nnreal" vectors_3D nil)
    (real_lt_is_strict_total_order name-judgement
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    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sqrt_sq_abs formula-decl nil sqrt "reals/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
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    (Vector type-eq-decl nil vectors_3D 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)
    (sqv_scal formula-decl nil vectors_3D nil))
   nil))
(caret_TCC1 0
  (caret_TCC1-2 nil 3430828042
   ("" (skeep)
    (("" (rewrite "norm_scal")
      (("" (expand "abs") (("" (grind-reals) nil nil)) nil)) nil))
    nil)
   ((nz_norm_gt_0 application-judgement "posreal" vectors_3D nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (norm_scal formula-decl nil vectors_3D 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)
    (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)
    (Vector type-eq-decl nil vectors_3D nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (norm const-decl "nnreal" vectors_3D nil)
    (zero const-decl "Vector" vectors_3D nil)
    (Nz_vector type-eq-decl nil vectors_3D nil)
    (nonzero_abs_is_pos application-judgement "{y: posreal | y >= x}"
     real_defs nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil))
   nil)
  (caret_TCC1-1 nil 3256053649 ("" (subtype-tcc) nil nil) nil shostak))
(norm_normalize 0
  (norm_normalize-1 nil 3560353164
   ("" (skeep) (("" (typepred "^(nzv)") (("" (propax) nil nil)) nil))
    nil)
   ((^ const-decl "Normalized" vectors_3D nil)
    (Normalized type-eq-decl nil vectors_3D nil)
    (Nz_vector type-eq-decl nil vectors_3D nil)
    (zero const-decl "Vector" vectors_3D nil)
    (/= const-decl "boolean" notequal nil)
    (norm const-decl "nnreal" vectors_3D nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" 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)
    (Vector type-eq-decl nil vectors_3D nil)
    (real nonempty-type-from-decl nil reals nil)
    (= const-decl "[T, T -> boolean]" equalities 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))
   shostak))
(dot_normalize 0
  (dot_normalize-2 nil 3256054056
   ("" (skosimp*)
    (("" (expand "^")
      (("" (assert)
        (("" (rewrite "dot_scal_right")
          (("" (assert)
            (("" (rewrite "dot_scal_left") (("" (assert) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((^ const-decl "Normalized" vectors_3D nil)
    (dot_scal_right formula-decl nil vectors_3D 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)
    (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)
    (Vector type-eq-decl nil vectors_3D nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nnreal type-eq-decl nil real_types nil)
    (norm const-decl "nnreal" vectors_3D nil)
    (zero const-decl "Vector" vectors_3D nil)
    (Nz_vector type-eq-decl nil vectors_3D nil)
    (* const-decl "Vector" vectors_3D nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (dot_scal_left formula-decl nil vectors_3D nil)
    (nz_nzv application-judgement "Nz_vector" vectors_3D nil)
    (posreal_div_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (nz_norm_gt_0 application-judgement "posreal" vectors_3D nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil))
   nil)
  (dot_normalize-1 nil 3255949259
   ("" (skosimp*)
    (("" (expand "normalize")
      (("" (assert)
        (("" (rewrite "dot_scal_right")
          (("" (assert)
            (("" (rewrite "dot_scal_left") (("" (assert) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   nil nil))
(normalize_normalize 0
  (normalize_normalize-1 nil 3268402058
   ("" (skosimp*)
--> --------------------

--> maximum size reached

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

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