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Datei: finite_cyclic_groups.prf Sprache: Lisp

Original von: PVS^©

(finite_cyclic_groups
(IMP_finite_groups_TCC1 0
  (IMP_finite_groups_TCC1-1 nil 3408118121
   ("" (rewrite "fullset_is_group") nil nil)
   ((fullset_is_group formula-decl nil finite_cyclic_groups nil)) nil))
(prime_order_cycle 0
  (prime_order_cycle-2 nil 3407853657
   ("" (skosimp*)
    (("" (expand "cyclic?")
      (("" (expand "order")
        (("" (case "card(G!1) > 1")
          (("1" (case "(EXISTS (a: (G!1)): a /= one)")
            (("1" (skosimp*)
              (("1" (name "HH" "generated_by(a!1)")
                (("1" (inst + "a!1")
                  (("1" (assert)
                    (("1" (replace -1)
                      (("1" (case "subgroup?(HH,G!1)")
                        (("1" (lemma "Lagrange")
                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (expand "prime?")
                                (("1"
                                  (inst - "order(HH)")
                                  (("1"
                                    (expand "order")
                                    (("1"
                                      (split -5)
                                      (("1" (propax) nil nil)
                                       ("2"
                                        (assert)
                                        (("2"
                                          (flatten)
                                          (("2"
                                            (lemma
                                             "generated_by_card_1")
                                            (("2"
                                              (inst?)
                                              (("2"
                                                (inst - "G!1")
                                                (("2"
                                                  (expand "member")
                                                  (("2"
                                                    (propax)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil)
                                       ("3"
                                        (flatten)
                                        (("3"
                                          (assert)
                                          (("3"
                                            (hide -2 -4)
                                            (("3"
                                              (expand "subgroup?")
                                              (("3"
                                                (lemma
                                                 "same_card_subset[T]")
                                                (("3"
                                                  (inst - "HH" "G!1")
                                                  (("3"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (typepred "G!1")
                              (("2"
                                (lemma "finite_subgroups")
                                (("2"
                                  (inst?)
                                  (("2" (assert) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (lemma "generated_is_subgroup")
                          (("2" (inst?)
                            (("2" (inst - "G!1")
                              (("2" (assert) nil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide -2 2)
              (("2" (lemma "card_2_has_2[T]")
                (("2" (inst - "G!1")
                  (("2" (assert)
                    (("2" (skosimp*)
                      (("2" (inst-cp + "x!1")
                        (("2" (inst + "y!1") (("2" (assert) nil nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (expand "prime?") (("2" (flatten) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((cyclic? const-decl "boolean" group nil)
    (finite_group nonempty-type-eq-decl nil group nil)
    (finite_group? const-decl "bool" group_def nil)
    (one formal-const-decl "T" finite_cyclic_groups nil)
    (* formal-const-decl "[T, T -> T]" finite_cyclic_groups nil)
    (card const-decl "{n: nat | n = Card(S)}" finite_sets nil)
    (Card const-decl "nat" finite_sets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (nat nonempty-type-eq-decl nil naturalnumbers nil)
    (>= const-decl "bool" reals nil)
    (int nonempty-type-eq-decl nil integers nil)
    (integer_pred const-decl "[rational -> boolean]" integers nil)
    (rational nonempty-type-from-decl nil rationals nil)
    (rational_pred const-decl "[real -> boolean]" rationals nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (is_finite const-decl "bool" finite_sets nil)
    (set type-eq-decl nil sets nil)
    (T formal-nonempty-type-decl nil finite_cyclic_groups 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)
    (y!1 skolem-const-decl "T" finite_cyclic_groups nil)
    (x!1 skolem-const-decl "T" finite_cyclic_groups nil)
    (G!1 skolem-const-decl "finite_group[T, *, one]"
     finite_cyclic_groups nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (card_2_has_2 formula-decl nil finite_sets nil)
    (generated_is_subgroup formula-decl nil cyclic_group nil)
    (Lagrange formula-decl nil lagrange nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (finite_subgroups formula-decl nil group nil)
    (finite_monad nonempty-type-eq-decl nil monad nil)
    (finite_monad? const-decl "bool" monad_def nil)
    (posnat nonempty-type-eq-decl nil integers nil)
    (nonneg_int nonempty-type-eq-decl nil integers nil)
    (member const-decl "bool" sets nil)
    (generated_by_card_1 formula-decl nil finite_groups nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (same_card_subset formula-decl nil finite_sets nil)
    (prime? const-decl "bool" primes "ints/")
    (HH skolem-const-decl "group[T, *, one]" finite_cyclic_groups nil)
    (subgroup? const-decl "bool" group_def nil)
    (real_gt_is_strict_total_order name-judgement
     "(strict_total_order?[real])" real_props nil)
    (generated_by const-decl "group" group nil)
    (group nonempty-type-eq-decl nil group nil)
    (group? const-decl "bool" group_def nil)
    (/= const-decl "boolean" notequal nil)
    (order const-decl "posnat" monad nil))
   nil)
  (prime_order_cycle-1 nil 3407082460
   (";;; Proof prime_order_cycle-3 for formula finite_cyclic_groups.prime_order_cycle"
    (skosimp*)
    ((";;; Proof prime_order_cycle-3 for formula finite_cyclic_groups.prime_order_cycle"
      (expand "cyclic?")
      ((";;; Proof prime_order_cycle-3 for formula finite_cyclic_groups.prime_order_cycle"
        (expand "order")
        ((";;; Proof prime_order_cycle-3 for formula finite_cyclic_groups.prime_order_cycle"
          (case "card(G!1) > 1")
          (("1" (case "(EXISTS (a: (G!1)): a /= one)")
            (("1" (skosimp*)
              (("1" (name "HH" "generated_by(a!1)")
                (("1" (inst + "a!1")
                  (("1" (assert)
                    (("1" (replace -1)
                      (("1" (case "subgroup?(HH,G!1)")
                        (("1" (lemma "Lagrange")
                          (("1" (inst?)
                            (("1" (assert)
                              (("1"
                                (expand "prime?")
                                (("1"
                                  (inst - "order(HH)")
                                  (("1"
                                    (expand "order")
                                    (("1"
                                      (assert)
                                      (("1"
                                        (split -5)
                                        (("1"
                                          (assert)
                                          (("1"
                                            (expand "divides")
                                            (("1" (propax) nil)))))
                                         ("2"
                                          (flatten)
                                          (("2"
                                            (lemma
                                             "generated_by_card_1")
                                            (("2"
                                              (inst?)
                                              (("2"
                                                (inst - "G!1")
                                                (("2"
                                                  (expand "member")
                                                  (("2"
                                                    (assert)
                                                    nil)))))))))))
                                         ("3"
                                          (flatten)
                                          (("3"
                                            (assert)
                                            (("3"
                                              (hide -2 -4)
                                              (("3"
                                                (expand "subgroup?")
                                                (("3"
                                                  (lemma
                                                   "same_card_subset[T]")
                                                  (("3"
                                                    (inst - "HH" "G!1")
                                                    (("3"
                                                      (assert)
                                                      nil)))))))))))))))))))))))))
                             ("2" (typepred "G!1")
                              (("2"
                                (lemma "finite_subgroups")
                                (("2"
                                  (inst?)
                                  (("2" (assert) nil)))))))))))
                         ("2" (lemma "generated_is_subgroup")
                          (("2" (inst?)
                            (("2" (inst - "G!1")
                              (("2" (assert) nil)))))))))))))))))))
             ("2" (hide -2 2)
              (("2" (lemma "card_2_has_2[T]")
                (("2" (inst - "G!1")
                  (("2" (assert)
                    (("2" (skosimp*)
                      (("2" (inst-cp + "x!1")
                        (("2" (inst + "y!1")
                          (("2" (assert) nil)))))))))))))))))
           ("2" (expand "prime?") (("2" (flatten) nil))))))))))
    ";;; developed with shostak decision procedures")
   ((cyclic? const-decl "boolean" group nil)
    (finite_group nonempty-type-eq-decl nil group nil)
    (finite_group? const-decl "bool" group_def nil)
    (generated_is_subgroup formula-decl nil group nil)
    (Lagrange formula-decl nil lagrange nil)
    (finite_subgroups formula-decl nil group nil)
    (finite_monad nonempty-type-eq-decl nil monad nil)
    (finite_monad? const-decl "bool" monad_def nil)
    (subgroup? const-decl "bool" group_def nil)
    (generated_by const-decl "group" group nil)
    (group nonempty-type-eq-decl nil group nil)
    (group? const-decl "bool" group_def nil)
    (order const-decl "posnat" monad nil))
   nil)))

¤ Dauer der Verarbeitung: 0.19 Sekunden (vorverarbeitet) ¤

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