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© Kompilation durch diese Firma

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Datei: path_ops.prf   Sprache: Lisp

Original von: PVS©

(path_ops
 (walk?_del_vert 0
  (walk?_del_vert-1 nil 3363193931
   ("" (skosimp*)
    (("" (expand "walk?")
      (("" (flatten)
        (("" (prop)
          (("1" (hide -2)
            (("1" (expand "verts_in?")
              (("1" (skosimp*) (("1" (inst?) (("1" (grind) nil)))))))))
           ("2" (hide -1)
            (("2" (skosimp*)
              (("2" (inst?)
                (("2" (assert)
                  (("2" (lemma "edges_del_vert")
                    (("2" (expand "edge?")
                      (("2" (flatten)
                        (("2" (inst?)
                          (("1" (assertnil)
                           ("2" (inst?)
                            (("2" (assert)
                              nil))))))))))))))))))))))))))
    nil)
   ((walk? const-decl "bool" walks nil)
    (verts_in? const-decl "bool" walks nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (< const-decl "bool" reals nil)
    (below type-eq-decl nil nat_types nil)
    (T formal-type-decl nil path_ops nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (> const-decl "bool" reals nil)
    (prewalk type-eq-decl nil walks nil)
    (below type-eq-decl nil naturalnumbers nil)
    (finite_remove application-judgement "finite_set" finite_sets nil)
    (remove const-decl "set" sets nil)
    (member const-decl "bool" sets nil)
    (/= const-decl "boolean" notequal nil)
    (del_vert const-decl "graph[T]" graph_ops nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (nil application-judgement "finite_set[T]" path_ops nil)
    (edge? const-decl "bool" graphs nil)
    (set type-eq-decl nil sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (dbl const-decl "set[T]" doubletons nil)
    (doubleton type-eq-decl nil doubletons nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (pregraph type-eq-decl nil graphs nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (graph type-eq-decl nil graphs nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (edges_del_vert formula-decl nil graph_ops nil))
   nil))
 (walk?_del_vert_not 0
  (walk?_del_vert_not-1 nil 3363193931
   ("" (skosimp*)
    (("" (expand "walk?")
      (("" (flatten)
        (("" (prop)
          (("1" (hide -2)
            (("1" (expand "verts_in?")
              (("1" (skosimp*)
                (("1" (expand "verts_of")
                  (("1" (inst 2 "i!1")
                    (("1" (expand "finseq_appl")
                      (("1" (inst?)
                        (("1" (expand "del_vert")
                          (("1" (expand "remove")
                            (("1" (expand "member")
                              (("1" (assertnil)))))))))))))))))))))
           ("2" (skosimp*)
            (("2" (inst?)
              (("2" (hide -2)
                (("2" (expand "del_vert")
                  (("2" (expand "edge?")
                    (("2" (assert)
                      (("2" (flatten)
                        (("2" (assert)
                          (("2" (expand "verts_of")
                            (("2" (expand "dbl")
                              (("2"
                                (hide -3 1)
                                (("2"
                                  (split)
                                  (("1" (inst?) (("1" (assertnil)))
                                   ("2"
                                    (inst?)
                                    (("2"
                                      (assert)
                                      nil))))))))))))))))))))))))))))))))))
    nil)
   ((walk? const-decl "bool" walks nil)
    (verts_in? const-decl "bool" walks nil)
    (verts_of const-decl "finite_set[T]" walks nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (del_vert const-decl "graph[T]" graph_ops nil)
    (member const-decl "bool" sets nil)
    (remove const-decl "set" sets nil)
    (below type-eq-decl nil naturalnumbers nil)
    (prewalk type-eq-decl nil walks nil)
    (> const-decl "bool" reals nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (T formal-type-decl nil path_ops nil)
    (below type-eq-decl nil nat_types nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (dbl const-decl "set[T]" doubletons nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (edge? const-decl "bool" graphs nil)
    (nil application-judgement "finite_set[T]" path_ops nil))
   nil))
 (path?_del_vert 0
  (path?_del_vert-1 nil 3363193931
   ("" (skosimp*)
    (("" (expand "path?")
      (("" (flatten)
        (("" (lemma "walk?_del_vert")
          (("" (inst?) (("" (assertnil))))))))))
    nil)
   ((path? const-decl "bool" paths nil)
    (walk?_del_vert formula-decl nil path_ops nil)
    (prewalk type-eq-decl nil walks 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)
    (finseq type-eq-decl nil finite_sequences nil)
    (below type-eq-decl nil nat_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (graph type-eq-decl nil graphs nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pregraph type-eq-decl nil graphs nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (doubleton type-eq-decl nil doubletons nil)
    (dbl const-decl "set[T]" doubletons nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (/= const-decl "boolean" notequal nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil path_ops nil))
   nil))
 (path?_del_verts 0
  (path?_del_verts-1 nil 3363193931
   ("" (skosimp*)
    (("" (expand "path?")
      (("" (flatten)
        (("" (split +)
          (("1" (hide -2)
            (("1" (expand "walk?")
              (("1" (flatten)
                (("1" (split +)
                  (("1" (hide -2)
                    (("1" (expand "verts_in?")
                      (("1" (expand "del_verts")
                        (("1" (expand "difference")
                          (("1" (expand "member")
                            (("1" (skosimp*)
                              (("1"
                                (inst?)
                                (("1" (assertnil)))))))))))))))
                   ("2" (skosimp*)
                    (("2" (expand "finseq_appl")
                      (("2" (inst?)
                        (("2" (assert)
                          (("2" (expand "edge?")
                            (("2" (flatten)
                              (("2"
                                (assert)
                                (("2"
                                  (expand "del_verts")
                                  (("2"
                                    (propax)
                                    nil)))))))))))))))))))))))))
           ("2" (propax) nil))))))))
    nil)
   ((path? const-decl "bool" paths nil)
    (walk? const-decl "bool" walks nil)
    (verts_in? const-decl "bool" walks nil)
    (difference const-decl "set" sets nil)
    (below type-eq-decl nil naturalnumbers nil)
    (prewalk type-eq-decl nil walks nil)
    (> const-decl "bool" reals nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (T formal-type-decl nil path_ops nil)
    (below type-eq-decl nil nat_types nil)
    (< const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (member const-decl "bool" sets nil)
    (del_verts const-decl "graph[T]" sep_sets nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (edge? const-decl "bool" graphs nil)
    (nil application-judgement "finite_set[T]" path_ops nil))
   nil))
 (walk_to_path 0
  (walk_to_path-1 nil 3363193931
   ("" (skolem!)
    ((""
      (case "FORALL (w: prewalk): NOT walk_from?(G!1, w, s!1, t!1)
                  OR (EXISTS (p: prewalk): path?(G!1, p) AND walk_from?(G!1, p, s!1, t!1))")
      (("1" (flatten)
        (("1" (skosimp*)
          (("1" (inst -1 "w!1") (("1" (assertnil nil)) nil)) nil))
        nil)
       ("2" (hide 2)
        (("2" (induct "w" 1 "graph_induction_walk")
          (("2" (skosimp*)
            (("2"
              (case "(EXISTS (i,j: below(length(w!1))): i /= j AND w!1`seq(i) = w!1`seq(j))")
              (("1" (skosimp*)
                (("1" (case "i!1 < j!1")
                  (("1"
                    (inst -3 "w!1^(0,i!1) o w!1^(j!1+1,length(w!1)-1)")
                    (("1" (split -3)
                      (("1" (rewrite "walk?_cut"nil nil)
                       ("2" (propax) nil nil)
                       ("3" (hide -2 -3 3)
                        (("3" (expand "o ")
                          (("3" (expand"^" min empty_seq)
                            (("3" (lift-if) (("3" (ground) nil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (expand "o ")
                      (("2" (expand"^" min empty_seq)
                        (("2" (lift-if) (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil)
                   ("2" (assert)
                    (("2"
                      (inst -2
                       "w!1^(0,j!1) o w!1^(i!1+1,length(w!1)-1)")
                      (("1" (assert)
                        (("1" (split -2)
                          (("1" (hide 4)
                            (("1" (rewrite "walk?_cut"nil nil)) nil)
                           ("2" (propax) nil nil)
                           ("3" (expand "o ")
                            (("3" (expand"^" min empty_seq)
                              (("3"
                                (lift-if)
                                (("3" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (expand "o ")
                        (("2" (expand"^" min empty_seq)
                          (("2" (lift-if) (("2" (assertnil nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide -1)
                (("2" (inst 2 "w!1")
                  (("2" (assert)
                    (("2" (expand "path?")
                      (("2" (expand "walk_from?")
                        (("2" (expand "finseq_appl")
                          (("2" (flatten)
                            (("2" (assert)
                              (("2"
                                (skosimp*)
                                (("2"
                                  (inst?)
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((path? const-decl "bool" paths nil)
    (walk_from? const-decl "bool" walks nil)
    (graph type-eq-decl nil graphs nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pregraph type-eq-decl nil graphs nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (doubleton type-eq-decl nil doubletons nil)
    (dbl const-decl "set[T]" doubletons nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (/= const-decl "boolean" notequal nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (OR const-decl "[bool, bool -> bool]" booleans nil)
    (prewalk type-eq-decl nil walks 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)
    (finseq type-eq-decl nil finite_sequences nil)
    (T formal-type-decl nil path_ops nil)
    (below type-eq-decl nil nat_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types nil)
    (graph_induction_walk formula-decl nil walk_inductions 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 "bool" reals nil) (< const-decl "bool" reals nil)
    (below type-eq-decl nil naturalnumbers nil)
    (walk?_cut formula-decl nil walks nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (empty_seq const-decl "finseq" finite_sequences nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (j!1 skolem-const-decl "below(length(w!1))" path_ops nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (i!1 skolem-const-decl "below(length(w!1))" path_ops nil)
    (w!1 skolem-const-decl "prewalk[T]" path_ops nil)
    (^ const-decl "finseq" finite_sequences nil)
    (O const-decl "finseq" finite_sequences nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (even_minus_odd_is_odd application-judgement "odd_int" integers
     nil)
    (odd_plus_odd_is_even application-judgement "even_int" integers
     nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil))
   nil))
 (walk_to_path_less 0
  (walk_to_path_less-2 "underconstr" 3363278454
   ("" (skolem * ("G!1" "s!1" "t!1" "_"))
    (("" (induct "w" 1 "graph_induction_walk")
      (("" (skosimp*)
        ((""
          (case "(EXISTS (i,j: below(length(w!1))): i /= j AND seq(w!1)(i) = seq(w!1)(j))")
          (("1" (skosimp*)
            (("1" (case "i!1 < j!1")
              (("1" (inst -3 "w!1^(0,i!1) o w!1^(j!1+1,length(w!1)-1)")
                (("1" (split -3)
                  (("1" (skosimp*)
                    (("1"
                      (case " length(w!1 ^ (0, i!1) o w!1 ^ (j!1 + 1, length(w!1) - 1)) <= length(w!1)")
                      (("1" (inst 2 "p!1") (("1" (assertnil nil))
                        nil)
                       ("2" (hide -1 -2 -3 3)
                        (("2" (install-rewrites "walks[T]")
                          (("2" (assert)
                            (("2" (hide -3) (("2" (grind) nil nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide 3)
                    (("2" (install-rewrites "walks[T]")
                      (("2" (expand "walk_from?")
                        (("2" (bddsimp)
                          (("1" (lemma "walk?_cut")
                            (("1"
                              (inst -1 "G!1" "w!1" "s!1" "t!1" "i!1"
                               "j!1")
                              (("1"
                                (prop)
                                (("1"
                                  (expand "walk_from?")
                                  (("1" (prop) nil nil))
                                  nil)
                                 ("2"
                                  (expand "walk_from?")
                                  (("2" (prop) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (assert)
                            (("2" (hide -5)
                              (("2"
                                (lift-if 1)
                                (("2"
                                  (bddsimp)
                                  (("1"
                                    (expand"^" min empty_seq)
                                    (("1"
                                      (lift-if 1)
                                      (("1"
                                        (bddsimp)
                                        (("1"
                                          (lift-if)
                                          (("1"
                                            (prop)
                                            (("1" (assertnil nil)
                                             ("2" (assertnil nil))
                                            nil))
                                          nil)
                                         ("2" (assertnil nil)
                                         ("3"
                                          (lift-if -1)
                                          (("3"
                                            (bddsimp)
                                            (("3" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (assert)
                                    (("2"
                                      (expand"^" min empty_seq)
                                      (("2"
                                        (lift-if 2)
                                        (("2"
                                          (lift-if 1)
                                          (("2"
                                            (bddsimp)
                                            (("1" (assertnil nil)
                                             ("2" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("3" (assertnil nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("3" (hide -3 3)
                    (("3" (typepred "j!1") (("3" (grind) nil nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide -3 3)
                  (("2" (install-rewrites "walks[T]")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil)
               ("2" (inst -2 "w!1^(0,j!1) o w!1^(i!1+1,length(w!1)-1)")
                (("1" (assert)
                  (("1" (split -2)
                    (("1" (skosimp*)
                      (("1" (inst 3 "p!1")
                        (("1" (bddsimp)
                          (("1" (hide -1 -2 -5)
                            (("1" (install-rewrites "walks[T]")
                              (("1"
                                (assert)
                                (("1"
                                  (expand"^" min empty_seq)
                                  (("1"
                                    (lift-if -1)
                                    (("1"
                                      (bddsimp)
                                      (("1" (assertnil nil)
                                       ("2" (assertnil nil)
                                       ("3" (assertnil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide 4)
                      (("2" (lemma "walk?_cut")
                        (("2"
                          (inst -1 "G!1" "w!1" "s!1" "t!1" "j!1" "i!1")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil)
                     ("3" (hide -2 4)
                      (("3" (install-rewrites "walks[T]")
                        (("3" (assert)
                          (("3" (expand"^" min empty_seq)
                            (("3" (lift-if 1)
                              (("3"
                                (bddsimp)
                                (("1" (assertnil nil)
                                 ("2" (assertnil nil)
                                 ("3" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil)
                 ("2" (hide -2 4)
                  (("2" (install-rewrites "walks[T]")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (hide -1)
            (("2" (inst 2 "w!1")
              (("2" (assert)
                (("2" (install-rewrites "paths[T]")
                  (("2" (expand "path?")
                    (("2" (expand "walk_from?")
                      (("2" (bddsimp)
                        (("2" (skosimp*)
                          (("2" (expand "finseq_appl")
                            (("2" (inst 1 "i!1" "j!1")
                              (("2" (prop) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((nat nonempty-type-eq-decl nil naturalnumbers nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences 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)
    (prewalk type-eq-decl nil walks nil)
    (pred type-eq-decl nil defined_types nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (/= const-decl "boolean" notequal nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (dbl const-decl "set[T]" doubletons nil)
    (doubleton type-eq-decl nil doubletons nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (pregraph type-eq-decl nil graphs nil)
    (graph type-eq-decl nil graphs nil)
    (walk_from? const-decl "bool" walks nil)
    (path? const-decl "bool" paths nil)
    (<= const-decl "bool" reals nil)
    (T formal-type-decl nil path_ops nil)
    (graph_induction_walk formula-decl nil walk_inductions 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 "bool" reals nil) (< const-decl "bool" reals nil)
    (below type-eq-decl nil naturalnumbers nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (empty_seq const-decl "finseq" finite_sequences nil)
    (walk? const-decl "bool" walks nil)
    (edge? const-decl "bool" graphs nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (verts_in? const-decl "bool" walks nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (int_min application-judgement "{k: int | k <= i AND k <= j}"
     real_defs nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (nil application-judgement "finite_set[T]" path_ops nil)
    (walk?_cut formula-decl nil walks nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (j!1 skolem-const-decl "below(length(w!1))" path_ops nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (i!1 skolem-const-decl "below(length(w!1))" path_ops nil)
    (w!1 skolem-const-decl "prewalk[T]" path_ops nil)
    (^ const-decl "finseq" finite_sequences nil)
    (O const-decl "finseq" finite_sequences nil)
    (nnint_plus_posint_is_posint application-judgement "posint"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (odd_plus_even_is_odd application-judgement "odd_int" integers
     nil))
   shostak)
  (walk_to_path_less-1 nil 3363193979
   ("" (skolem!)
    ((""
      (case "FORALL (w: prewalk): NOT walk_from?(G!1, w, s!1, t!1)
                  OR (EXISTS (p: prewalk): path?(G!1, p) AND walk_from?(G!1, p, s!1, t!1))")
      (("1" (flatten)
        (("1" (skosimp*)
          (("1" (inst -1 "w!1") (("1" (assertnil)))))))
       ("2" (hide 2)
        (("2" (induct "w" 1 "graph_induction_walk")
          (("2" (skosimp*)
            (("2"
              (case "(EXISTS (i,j: below(length(w!1))): i /= j AND seq(w!1)(i) = seq(w!1)(j))")
              (("1" (skosimp*)
                (("1" (case "i!1 < j!1")
                  (("1"
                    (inst -3 "w!1^(0,i!1) o w!1^(j!1+1,length(w!1)-1)")
                    (("1" (split -3)
                      (("1" (rewrite "walk?_cut"nil)
                       ("2" (propax) nil)
                       ("3" (hide -2 -3 3)
                        (("3" (expand "o ")
                          (("3" (expand"^" min empty_seq)
                            (("3" (expand "empty_seq")
                              (("3"
                                (lift-if)
                                (("3" (ground) nil)))))))))))))
                     ("2" (expand "o ")
                      (("2" (expand"^" min empty_seq)
                        (("2" (lift-if)
                          (("2" (expand "empty_seq")
                            (("2" (propax) nil)))))))))))
                   ("2" (assert)
                    (("2"
                      (inst -2
                       "w!1^(0,j!1) o w!1^(i!1+1,length(w!1)-1)")
                      (("1" (assert)
                        (("1" (split -2)
                          (("1" (hide 4)
                            (("1" (rewrite "walk?_cut"nil)))
                           ("2" (propax) nil)
                           ("3" (expand "o ")
                            (("3" (expand"^" min empty_seq)
                              (("3"
                                (lift-if)
                                (("3"
                                  (expand "empty_seq")
                                  (("3" (propax) nil)))))))))))))
                       ("2" (expand "o ")
                        (("2" (expand"^" min empty_seq)
                          (("2" (expand "empty_seq")
                            (("2" (lift-if)
                              (("2" (ground) nil)))))))))))))))))
               ("2" (hide -1)
                (("2" (inst 2 "w!1")
                  (("2" (assert)
                    (("2" (expand "path?")
                      (("2" (expand "walk_from?")
                        (("2" (flatten)
                          (("2" (assert)
                            (("2" (skosimp*)
                              (("2"
                                (expand "finseq_appl")
                                (("2"
                                  (inst?)
                                  (("2"
                                    (assert)
                                    nil))))))))))))))))))))))))))))))))
    nil)
   nil nil))
 (walk_to_path_from 0
  (walk_to_path_from-1 nil 3363193931
   ("" (skosimp*)
    (("" (lemma "walk_to_path")
      (("" (inst?)
        (("" (inst -1 "s!1" "t!1")
          (("" (split -1)
            (("1" (skosimp*)
              (("1" (inst?)
                (("1" (expand "path_from?")
                  (("1" (assert)
                    (("1" (expand "walk_from?")
                      (("1" (flatten)
                        (("1" (expand "from?")
                          (("1" (assertnil)))))))))))))))
             ("2" (inst?) nil))))))))))
    nil)
   ((walk_to_path formula-decl nil path_ops nil)
    (path_from? const-decl "bool" paths nil)
    (walk_from? const-decl "bool" walks nil)
    (from? const-decl "bool" walks nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (> const-decl "bool" reals nil)
    (prewalk type-eq-decl nil walks nil)
    (graph type-eq-decl nil graphs nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pregraph type-eq-decl nil graphs nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (doubleton type-eq-decl nil doubletons nil)
    (dbl const-decl "set[T]" doubletons nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (/= const-decl "boolean" notequal nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil path_ops nil))
   nil))
 (walk_to_path_from_less 0
  (walk_to_path_from_less-1 nil 3363292497
   ("" (skosimp*)
    (("" (lemma "walk_to_path_less")
      (("" (inst?)
        (("" (prop)
          (("" (skosimp*)
            (("" (inst?)
              (("" (expand "path_from?")
                (("" (expand "walk_from?")
                  (("" (prop)
                    (("" (expand "from?") (("" (prop) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((walk_to_path_less formula-decl nil path_ops nil)
    (walk_from? const-decl "bool" walks nil)
    (from? const-decl "bool" walks nil)
    (path_from? const-decl "bool" paths nil)
    (prewalk type-eq-decl nil walks 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)
    (finseq type-eq-decl nil finite_sequences nil)
    (below type-eq-decl nil nat_types nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (graph type-eq-decl nil graphs nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (pregraph type-eq-decl nil graphs nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (doubleton type-eq-decl nil doubletons nil)
    (dbl const-decl "set[T]" doubletons nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (/= const-decl "boolean" notequal nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil path_ops nil))
   nil))
 (path_disjoint_TCC1 0
  (path_disjoint_TCC1-1 nil 3389448101 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (dbl const-decl "set[T]" doubletons nil)
    (doubleton type-eq-decl nil doubletons nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (pregraph type-eq-decl nil graphs nil)
    (graph type-eq-decl nil graphs nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (> const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (prewalk type-eq-decl nil walks nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (/= const-decl "boolean" notequal nil)
    (T formal-type-decl nil path_ops nil)
    (verts_in? const-decl "bool" walks nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (edge? const-decl "bool" graphs nil)
    (walk? const-decl "bool" walks nil)
    (path? const-decl "bool" paths nil)
    (from? const-decl "bool" walks nil)
    (path_from? const-decl "bool" paths nil)
    (nil application-judgement "finite_set[T]" path_ops nil))
   nil))
 (path_disjoint_TCC2 0
  (path_disjoint_TCC2-1 nil 3389448101 ("" (subtype-tcc) nil nil)
   ((boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set type-eq-decl nil sets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (dbl const-decl "set[T]" doubletons nil)
    (doubleton type-eq-decl nil doubletons nil)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (pregraph type-eq-decl nil graphs nil)
    (graph type-eq-decl nil graphs nil)
    (number nonempty-type-decl nil numbers nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (> const-decl "bool" reals nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (prewalk type-eq-decl nil walks nil)
    (min const-decl "{p: real | p <= m AND p <= n}" real_defs nil)
    (^ const-decl "finseq" finite_sequences nil)
    (int_plus_int_is_int application-judgement "int" integers nil)
    (int_min application-judgement "{k: int | k <= i AND k <= j}"
     real_defs nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (nnint_plus_nnint_is_nnint application-judgement "nonneg_int"
     integers nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (T formal-type-decl nil path_ops nil)
    (trunc1 const-decl "prewalk" walks nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (verts_of const-decl "finite_set[T]" walks nil)
    (member const-decl "bool" sets nil)
    (intersection const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil)
    (verts_in? const-decl "bool" walks nil)
    (/= const-decl "boolean" notequal nil)
    (edge? const-decl "bool" graphs nil)
    (walk? const-decl "bool" walks nil)
    (path? const-decl "bool" paths nil)
    (from? const-decl "bool" walks nil)
    (path_from? const-decl "bool" paths nil)
    (O const-decl "finseq" finite_sequences nil)
    (posint_plus_nnint_is_posint application-judgement "posint"
     integers nil)
    (finite_intersection1 application-judgement "finite_set"
     finite_sets nil)
    (nil application-judgement "finite_set[T]" path_ops nil))
   nil))
 (path_disjoint 0
  (path_disjoint-1 nil 3389448102
   ("" (skosimp*)
    (("" (case "walk_from?(G!1, p!1, u!1, yt!1)")
      (("1" (case " walk_from?(G!1, q!1, yt!1, v!1)")
        (("1" (lemma "walk_concat")
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            (("1" (inst -1 "G!1" "q!1")
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                  (inst -2 "G!1" "u!1" "v!1" "yt!1" "p!1" "rev(q!1)")
                  (("1" (lemma "rev_rev")
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                                          (hide -2 -3 -5 -11 -12)
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                                            (hide -10)
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                                                (lift-if -1)
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                                                         -5
                                                         -6
                                                         -11
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                                                        (("1"
                                                          (inst
                                                           -8
                                                           "i!1"
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                                                          (("1"
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                                                            (("1"
                                                              (hide
                                                               -5
                                                               -9)
                                                              (("1"
                                                                (expand
                                                                 "trunc1")
                                                                (("1"
                                                                  (expand
                                                                   "^")
                                                                  (("1"
                                                                    (assert)
                                                                    nil
                                                                    nil))
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                                                           ("2"
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                                                             "trunc1")
                                                            (("2"
                                                              (expand
                                                               "^")
                                                              (("2"
                                                                (expand
                                                                 "empty_seq")
                                                                (("2"
                                                                  (assert)
                                                                  nil
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                                                           ("3"
                                                            (expand
                                                             "trunc1")
                                                            (("3"
                                                              (expand
                                                               "^")
                                                              (("3"
                                                                (expand
                                                                 "empty_seq")
                                                                (("3"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (expand
                                                         "empty?")
                                                        (("2"
                                                          (expand
                                                           "member")
                                                          (("2"
                                                            (inst
                                                             -13
                                                             "seq(trunc1(p!1))(i!1)")
                                                            (("2"
                                                              (expand
                                                               "intersection")
                                                              (("2"
                                                                (expand
                                                                 "member")
                                                                (("2"
                                                                  (prop)
                                                                  (("1"
                                                                    (expand
                                                                     "verts_of")
                                                                    (("1"
                                                                      (inst
                                                                       1
                                                                       "i!1")
                                                                      (("1"
                                                                        (expand
                                                                         "finseq_appl")
                                                                        (("1"
                                                                          (propax)
                                                                          nil
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil)
                                                                   ("2"
                                                                    (replace
                                                                     -1
                                                                     1
                                                                     lr)
                                                                    (("2"
                                                                      (expand
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                                                                      (("2"
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                                                                        (("2"
                                                                          (inst
                                                                           1
                                                                           "j!1-length(trunc1(p!1))")
                                                                          (("2"
                                                                            (assert)
                                                                            nil
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
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                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (lift-if -1)
                                                    (("2"
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                                                      (("1"
                                                        (hide
                                                         -4
                                                         -5
                                                         -6
                                                         -7
                                                         -8
                                                         -9
                                                         -10
                                                         -11
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                                                        (("1"
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                                                          (("1"
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                                                            (("1"
                                                              (inst
                                                               -4
                                                               "seq(trunc1(p!1))(j!1)")
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                                                                (("1"
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                                                                          (inst
                                                                           1
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                                                                          nil
                                                                          nil))
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                                                                      nil)
                                                                     ("2"
                                                                      (replace
                                                                       -2
                                                                       1
                                                                       rl)
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                                                                        (expand
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                                                                        (("2"
                                                                          (expand
                                                                           "finseq_appl")
                                                                          (("2"
                                                                            (inst
                                                                             1
                                                                             "i!1-length(trunc1(p!1))")
                                                                            (("2"
                                                                              (assert)
                                                                              nil
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
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                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil)
                                                       ("2"
                                                        (inst
                                                         -10
                                                         "i!1-length(trunc1(p!1))"
                                                         "j!1-length(trunc1(p!1))")
                                                        (("1"
                                                          (assert)
                                                          nil
                                                          nil)
                                                         ("2"
                                                          (assert)
                                                          nil
                                                          nil)
                                                         ("3"
                                                          (assert)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
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                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide (-2 -4 -7 -10 -11 -13 -15))
                                  (("2"
                                    (install-rewrites "walks[T]")
                                    (("2"
                                      (assert)
                                      (("2"
                                        (expand"^" min empty_seq)
                                        nil
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (hide-all-but (-10 1))
                          (("2" (expand "path_from?")
                            (("2" (install-rewrites "paths[T]")
                              (("2"
                                (expand "walk_from?")
                                (("2"
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                                    (expand "from?")
                                    (("1" (prop) nil nil))
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                                   ("2"
                                    (expand "from?")
                                    (("2" (prop) nil nil))
                                    nil)
                                   ("3"
                                    (expand "path?")
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                                    nil))
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                                nil))
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                            nil))
                          nil)
                         ("3" (hide-all-but (-1 -4 -5 -6 1 2))
                          (("3" (install-rewrites "paths[T]")
                            (("3" (expand "walk_from?")
                              (("3"
                                (prop)
                                (("1"
                                  (expand "rev")
                                  (("1" (propax) nil nil))
                                  nil)
                                 ("2"
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                                  (("2" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (case "length(p!1)=1")
                        (("1" (replace -1 - lr)
                          (("1" (assert)
                            (("1" (prop)
                              (("1" (assertnil nil)
                               ("2" (assertnil nil))
                              nil))
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                          nil)
                         ("2" (assertnil nil))
                        nil))
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                    nil)
                   ("2" (expand "rev")
                    (("2" (typepred "q!1") (("2" (propax) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
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            nil))
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         ("2" (hide-all-but (-3 1))
          (("2" (install-rewrites "paths[T]")
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              (("2" (prop)
                (("2" (expand "walk_from?")
                  (("2" (prop)
                    (("1" (expand "from?") (("1" (prop) nil nil)) nil)
                     ("2" (expand "from?") (("2" (prop) nil nil)) nil)
                     ("3" (expand "path?") (("3" (prop) nil nil)) nil))
                    nil))
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       ("2" (hide-all-but (-1 1))
        (("2" (expand "path_from?")
          (("2" (expand "walk_from?")
            (("2" (prop)
              (("1" (expand "from?") (("1" (prop) nil nil)) nil)
               ("2" (expand "from?") (("2" (prop) nil nil)) nil)
               ("3" (expand "path?") (("3" (prop) nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((walk_from? const-decl "bool" walks nil)
    (prewalk type-eq-decl nil walks nil)
    (> const-decl "bool" reals nil)
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.83 Sekunden  (vorverarbeitet)  ¤





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