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Datei: subtrees.prf   Sprache: Lisp

Original von: PVS©
(subtrees
 (walk_acr_TCC1 0
  (walk_acr_TCC1-1 nil 3251113233 ("" (subtype-tcc) nil nil) nil nil))
 (walk_acr_TCC2 0
  (walk_acr_TCC2-1 nil 3251113233 ("" (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)
    (T formal-type-decl nil subtrees 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)
    (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)
    (nonempty? const-decl "bool" sets 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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (> const-decl "bool" reals nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (prewalk type-eq-decl nil walks nil)
    (walk? const-decl "bool" walks nil)
    (Walk type-eq-decl nil walks 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)
    (nil application-judgement "finite_set[T]" subtrees nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (edge? const-decl "bool" graphs nil)
    (finseq_appl const-decl "[below[length(fs)] -> T]" finite_sequences
     nil)
    (verts_in? const-decl "bool" walks nil)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil))
   nil))
 (walk_acr_TCC3 0
  (walk_acr_TCC3-1 nil 3251113233 ("" (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)
    (T formal-type-decl nil subtrees 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)
    (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)
    (nonempty? const-decl "bool" sets 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)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (int nonempty-type-eq-decl nil integers nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (> const-decl "bool" reals nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (prewalk type-eq-decl nil walks nil)
    (walk? const-decl "bool" walks nil)
    (Walk type-eq-decl nil walks 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)
    (nil application-judgement "finite_set[T]" subtrees nil)
    (real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (int_minus_int_is_int application-judgement "int" integers nil)
    (real_le_is_total_order name-judgement "(total_order?[real])"
     real_props 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)
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil))
   nil))
 (walk_acr 0
  (walk_acr-1 nil 3251113233
   ("" (induct "n" 1 "NAT_induction")
    (("1" (skosimp*)
      (("1" (inst -1 "j!1-1")
        (("1" (assert)
          (("1" (inst?)
            (("1" (assert)
              (("1" (split -1)
                (("1" (skosimp*)
                  (("1" (inst 3 "j!2") (("1" (assert) nil nil)) nil))
                  nil)
                 ("2" (assert)
                  (("2" (inst 2 "j!1") (("2" (assert) nil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assert) nil nil))
        nil))
      nil)
     ("2" (skosimp*) (("2" (assert) nil nil)) nil)
     ("3" (skosimp*) (("3" (assert) nil nil)) nil))
    nil)
   ((real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_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)
    (j!1 skolem-const-decl "nat" subtrees nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (NAT_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types 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)
    (T formal-type-decl nil subtrees 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)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (graph type-eq-decl nil graphs nil)
    (nonempty? const-decl "bool" sets nil)
    (Graph type-eq-decl nil graphs nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (> const-decl "bool" reals nil)
    (prewalk type-eq-decl nil walks nil)
    (walk? const-decl "bool" walks nil)
    (Walk type-eq-decl nil walks nil) (< const-decl "bool" reals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (<= const-decl "bool" reals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
 (walk_acr2 0
  (walk_acr2-1 nil 3251113233
   ("" (induct "n" 1 "NAT_induction")
    (("1" (skosimp*)
      (("1" (inst -1 "j!1-1")
        (("1" (assert)
          (("1" (inst?)
            (("1" (assert)
              (("1" (split -1)
                (("1" (skosimp*)
                  (("1" (inst 3 "j!2") (("1" (assert) nil nil)) nil))
                  nil)
                 ("2" (inst 2 "j!1") (("2" (assert) nil nil)) nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (assert) nil nil))
        nil))
      nil)
     ("2" (skosimp*) (("2" (assert) nil nil)) nil)
     ("3" (skosimp*) (("3" (assert) nil nil)) nil))
    nil)
   ((real_lt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (real_gt_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)
    (j!1 skolem-const-decl "nat" subtrees nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (NAT_induction formula-decl nil naturalnumbers nil)
    (pred type-eq-decl nil defined_types 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)
    (T formal-type-decl nil subtrees 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)
    (IMPLIES const-decl "[bool, bool -> bool]" booleans nil)
    (graph type-eq-decl nil graphs nil)
    (nonempty? const-decl "bool" sets nil)
    (Graph type-eq-decl nil graphs nil)
    (below type-eq-decl nil nat_types nil)
    (finseq type-eq-decl nil finite_sequences nil)
    (> const-decl "bool" reals nil)
    (prewalk type-eq-decl nil walks nil)
    (walk? const-decl "bool" walks nil)
    (Walk type-eq-decl nil walks nil) (< const-decl "bool" reals nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (<= const-decl "bool" reals nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (int_minus_int_is_int application-judgement "int" integers nil))
   nil))
 (add_dbl_TCC1 0
  (add_dbl_TCC1-1 nil 3251113233 ("" (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)
    (T formal-type-decl nil subtrees 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)
    (nonempty? const-decl "bool" sets nil)
    (Graph type-eq-decl nil graphs nil)
    (member const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (nil application-judgement "finite_set[T]" subtrees nil)
    (G!1 skolem-const-decl "Graph[T]" subtrees nil)
    (/= const-decl "boolean" notequal nil))
   nil))
 (add_dbl 0
  (add_dbl-1 nil 3251113233
   ("" (skosimp*)
    (("" (expand "add")
      (("" (expand "member")
        (("" (split)
          (("1" (replace -1 * rl)
            (("1" (hide -1)
              (("1" (expand "dbl")
                (("1" (flatten) (("1" (ground) nil nil)) nil)) nil))
              nil))
            nil)
           ("2" (typepred "G!1")
            (("2" (inst?)
              (("2" (assert)
                (("2" (inst - "x!1") (("2" (assert) nil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((add const-decl "(nonempty?)" sets nil)
    (dbl const-decl "set[T]" doubletons nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (T formal-type-decl nil subtrees 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)
    (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)
    (nonempty? const-decl "bool" sets nil)
    (Graph type-eq-decl nil graphs nil)
    (member const-decl "bool" sets nil))
   nil))
 (max_tree_all_verts 0
  (max_tree_all_verts-1 nil 3251113233
   ("" (skosimp*)
    (("" (apply-extensionality 1 :hide? t)
      (("" (iff 1)
        (("" (prop)
          (("1" (hide -2)
            (("1" (lemma "max_subtree_subgraph")
              (("1" (inst?)
                (("1" (expand "subgraph?")
                  (("1" (flatten)
                    (("1" (hide -2)
                      (("1" (expand "subset?")
                        (("1" (inst?)
                          (("1" (expand "member")
                            (("1" (assert) nil nil)) nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (lemma "max_subtree_tree")
            (("2" (inst?)
              (("2" (name "TR" "max_subtree(G!1)")
                (("2" (case "(EXISTS (v: T): vert(TR)(v))")
                  (("1" (skosimp*)
                    (("1" (expand "path_connected?")
                      (("1" (flatten)
                        (("1" (inst -5 "v!1" "x!1")
                          (("1" (assert)
                            (("1" (skosimp*)
                              (("1"
                                (case
                                 "(EXISTS (j: posnat): j < length(w!1) AND vert(TR)(w!1(j-1)) AND
                                                           NOT vert(TR)(w!1(j)))")
                                (("1"
                                  (skosimp*)
                                  (("1"
                                    (typepred "w!1")
                                    (("1"
                                      (expand "walk?")
                                      (("1"
                                        (expand "verts_in?")
                                        (("1"
                                          (expand "finseq_appl")
                                          (("1"
                                            (flatten)
                                            (("1"
                                              (inst - "j!1-1")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (expand "edge?")
                                                  (("1"
                                                    (name
                                                     "ss"
                                                     "seq(w!1)(j!1 - 1)")
                                                    (("1"
                                                      (name
                                                       "tt"
                                                       "seq(w!1)(j!1)")
                                                      (("1"
                                                        (name
                                                         "TP"
                                                         "(# vert := add(tt,vert(TR)),
                                                      edges := add(dbl[T](ss,tt),edges(TR)) #)")
                                                        (("1"
                                                          (case
                                                           "tree?(TP)")
                                                          (("1"
                                                            (lemma
                                                             "max_subtree_max")
                                                            (("1"
                                                              (inst
                                                               -1
                                                               "G!1"
                                                               "TP")
                                                              (("1"
                                                                (replace
                                                                 -4)
                                                                (("1"
                                                                  (hide
                                                                   -2
                                                                   --4
                                                                   -5
                                                                   -6
                                                                   -7
                                                                   -8
                                                                   -9
                                                                   -10
                                                                   -11
                                                                   -13
                                                                   -14
                                                                   -15
                                                                   -16
                                                                   1
                                                                   3
                                                                   4)
                                                                  (("1"
                                                                    (replace
                                                                     -2
                                                                     *
                                                                     rl)
                                                                    (("1"
                                                                      (hide
                                                                       -2)
                                                                      (("1"
                                                                        (expand
                                                                         "size")
                                                                        (("1"
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--> --------------------

--> maximum size reached

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

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