Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: integer_enumerations.pvs   Sprache: PVS

Untersuchung PVS©

(isomorphism_theorems
 (G_TCC1 0
  (G_TCC1-1 nil 3530909168 ("" (rewrite "T1_is_group"nil nil)
   ((T1_is_group formula-decl nil isomorphism_theorems nil)) nil))
 (GP_TCC1 0
  (GP_TCC1-1 nil 3530909168 ("" (rewrite "T2_is_group"nil nil)
   ((T2_is_group formula-decl nil isomorphism_theorems nil)) nil))
 (quotient_subgroup_TCC1 0
  (quotient_subgroup_TCC1-1 nil 3524781199
   ("" (skosimp*)
    (("" (lemma "normal_subgroup_tran[T1, *, one1]")
      (("1" (inst -1 "G!1" "M!1" "N!1")
        (("1" (assert)
          (("1" (typepred "M!1")
            (("1" (expand "normal_subgroup?") (("1" (propax) nil nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (rewrite "G_TCC1"nil nil))
      nil))
    nil)
   ((one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (fullset const-decl "set" sets nil)
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil))
   nil))
 (quotient_subgroup_TCC2 0
  (quotient_subgroup_TCC2-1 nil 3524781199
   ("" (skosimp*)
    (("" (prop)
      (("1" (skosimp)
        (("1" (inst 1 "a!1")
          (("1" (hide -)
            (("1" (typepred "a!1" "M!1")
              (("1" (expand"normal_subgroup?" "subgroup?")
                (("1" (flatten)
                  (("1" (expand"subset?" "member")
                    (("1" (inst?) (("1" (assertnil nil)) nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide -1)
        (("2" (inst 1 "lc_gen(M!1, N!1, x!1)") (("2" (assertnil nil))
          nil))
        nil))
      nil))
    nil)
   ((T1 formal-type-decl nil isomorphism_theorems nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (subgroup? const-decl "bool" group_def "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (subgroup type-eq-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (lc_gen const-decl "{a: T | G(a) AND lc = a * H}" cosets
     "algebra/"))
   nil))
 (quotient_subgroup_TCC3 0
  (quotient_subgroup_TCC3-1 nil 3524781199
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
       ("2" (typepred "G!1")
        (("2" (expand"group?" "monoid?" "monad?" "member")
          (("2" (flatten) nil nil)) nil))
        nil))
      nil))
    nil)
   ((G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil))
   nil))
 (quotient_subgroup_TCC4 0
  (quotient_subgroup_TCC4-1 nil 3524781199
   ("" (skosimp*)
    (("" (inst 1 "N!1")
      (("" (inst 1 "one1")
        (("1" (rewrite "left_coset_one[T1,*,one1]"nil nil)
         ("2" (typepred "G!1")
          (("2" (expand"group?" "monoid?" "monad?" "member")
            (("2" (flatten) nil nil)) nil))
          nil))
        nil))
      nil))
    nil)
   ((T1 formal-type-decl nil isomorphism_theorems nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (member const-decl "bool" sets nil)
    (monoid? const-decl "bool" monoid_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/"))
   nil))
 (quotient_subgroup 0
  (quotient_subgroup-1 nil 3524784283
   ("" (skosimp*)
    (("" (expand "subgroup?")
      (("" (prop)
        (("1" (hide -1)
          (("1" (expand"subset?" "member")
            (("1" (skosimp*)
              (("1" (expand "/")
                (("1" (expand "restrict")
                  (("1" (expand "left_cosets")
                    (("1" (skosimp)
                      (("1" (inst?)
                        (("1" (hide -1)
                          (("1" (typepred "a!1" "M!1")
                            (("1"
                              (expand"normal_subgroup?" "subgroup?"
                               "subset?" "member")
                              (("1"
                                (flatten)
                                (("1"
                                  (hide (-2 -4))
                                  (("1"
                                    (inst?)
                                    (("1" (assertnil nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (hide -1)
          (("2" (expand "group?")
            (("2" (prop)
              (("1" (expand "monoid?")
                (("1" (prop)
                  (("1" (expand "monad?")
                    (("1" (prop)
                      (("1" (expand "star_closed?")
                        (("1" (skosimp*)
                          (("1" (expand "member")
                            (("1" (expand "/")
                              (("1"
                                (expand "restrict")
                                (("1"
                                  (expand "left_cosets")
                                  (("1"
                                    (typepred "x!1" "y!1")
                                    (("1"
                                      (hide (-2 -4))
                                      (("1"
                                        (skosimp*)
                                        (("1"
                                          (inst 1 "a!1 * a!2")
                                          (("1"
                                            (replaces -1)
                                            (("1"
                                              (replaces -1)
                                              (("1"
                                                (lemma
                                                 "mult_lem[T1, *, one1]")
                                                (("1"
                                                  (inst?)
                                                  (("1"
                                                    (hide 2)
                                                    (("1"
                                                      (typepred
                                                       "a!2"
                                                       "M!1")
                                                      (("1"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("1"
                                                          (flatten)
                                                          (("1"
                                                            (hide
                                                             (-2 -4))
                                                            (("1"
                                                              (inst?)
                                                              (("1"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide 2)
                                                    (("2"
                                                      (typepred
                                                       "a!1"
                                                       "M!1")
                                                      (("2"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("2"
                                                          (flatten)
                                                          (("2"
                                                            (hide
                                                             (-2 -4))
                                                            (("2"
                                                              (inst?)
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil)
                                           ("2"
                                            (hide (-1 -2))
                                            (("2"
                                              (rewrite
                                               "product_in[T1, *, one1]")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (expand"member" "/" "left_cosets")
                        (("2" (expand "restrict")
                          (("2" (typepred "M!1")
                            (("2"
                              (expand"group?" "monoid?" "monad?"
                               "member")
                              (("2"
                                (flatten)
                                (("2"
                                  (hide-all-but (-2 1))
                                  (("2"
                                    (inst 1 "one1")
                                    (("2"
                                      (rewrite "left_coset_one")
                                      nil
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("3" (expand "identity?")
                        (("3" (skosimp*)
                          (("3" (expand "restrict")
                            (("3" (lemma "N_is_identity[T1, *, one1]")
                              (("3"
                                (inst?)
                                (("1"
                                  (flatten)
                                  (("1" (assertnil nil))
                                  nil)
                                 ("2"
                                  (hide 2)
                                  (("2"
                                    (typepred "x!1")
                                    (("2"
                                      (skosimp*)
                                      (("2"
                                        (inst?)
                                        (("2"
                                          (hide -)
                                          (("2"
                                            (typepred "a!1" "M!1")
                                            (("2"
                                              (expand*
                                               "normal_subgroup?"
                                               "subgroup?"
                                               "subset?"
                                               "member")
                                              (("2"
                                                (flatten)
                                                (("2"
                                                  (inst?)
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (expand "associative?")
                    (("2" (expand "restrict")
                      (("2" (skosimp*)
                        (("2" (typepred "z!1")
                          (("2" (typepred "y!1")
                            (("2" (typepred "x!1")
                              (("2"
                                (skosimp*)
                                (("2"
                                  (replace -1)
                                  (("2"
                                    (hide -1)
                                    (("2"
                                      (replace -2)
                                      (("2"
                                        (hide -2)
                                        (("2"
                                          (replace -3)
                                          (("2"
                                            (hide -3)
                                            (("2"
                                              (rewrite
                                               "mult_lem[T1, *, one1]")
                                              (("1"
                                                (rewrite
                                                 "mult_lem[T1, *, one1]")
                                                (("1"
                                                  (rewrite
                                                   "mult_lem[T1, *, one1]")
                                                  (("1"
                                                    (rewrite
                                                     "mult_lem[T1, *, one1]")
                                                    (("1"
                                                      (rewrite
                                                       "assoc[T1, *, one1]")
                                                      nil
                                                      nil)
                                                     ("2"
                                                      (hide-all-but 1)
                                                      (("2"
                                                        (typepred
                                                         "a!2"
                                                         "a!3"
                                                         "G!1"
                                                         "M!1")
                                                        (("2"
                                                          (expand
                                                           "normal_subgroup?")
                                                          (("2"
                                                            (flatten)
                                                            (("2"
                                                              (expand
                                                               "subgroup?")
                                                              (("2"
                                                                (expand
                                                                 "subset?")
                                                                (("2"
                                                                  (hide
                                                                   (-4
                                                                    -6))
                                                                  (("2"
                                                                    (copy
                                                                     -4)
                                                                    (("2"
                                                                      (inst
                                                                       -1
                                                                       "a!2")
                                                                      (("2"
                                                                        (inst
                                                                         -5
                                                                         "a!3")
                                                                        (("2"
                                                                          (expand
                                                                           "member")
                                                                          (("2"
                                                                            (expand*
                                                                             "group?"
                                                                             "monoid?"
                                                                             "monad?")
                                                                            (("2"
                                                                              (flatten)
                                                                              (("2"
                                                                                (hide-all-but
                                                                                 (-1
                                                                                  -4
                                                                                  -9
                                                                                  1))
                                                                                (("2"
                                                                                  (expand
                                                                                   "star_closed?")
                                                                                  (("2"
                                                                                    (inst?)
                                                                                    (("2"
                                                                                      (expand
                                                                                       "member")
                                                                                      (("2"
                                                                                        (propax)
                                                                                        nil
                                                                                        nil))
                                                                                      nil))
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil)
                                                     ("3"
                                                      (hide-all-but 1)
                                                      (("3"
                                                        (typepred
                                                         "a!1"
                                                         "M!1")
                                                        (("3"
                                                          (expand*
                                                           "normal_subgroup?"
                                                           "subgroup?"
                                                           "subset?"
                                                           "member")
                                                          (("3"
                                                            (flatten)
                                                            (("3"
                                                              (hide
                                                               (-2 -4))
                                                              (("3"
                                                                (inst?)
                                                                (("3"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("2"
                                                    (hide-all-but 1)
                                                    (("2"
                                                      (typepred
                                                       "a!3"
                                                       "M!1")
                                                      (("2"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("2"
                                                          (flatten)
                                                          (("2"
                                                            (hide
                                                             (-2 -4))
                                                            (("2"
                                                              (inst?)
                                                              (("2"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil)
                                                   ("3"
                                                    (hide-all-but 1)
                                                    (("3"
                                                      (typepred
                                                       "a!2"
                                                       "M!1")
                                                      (("3"
                                                        (expand*
                                                         "normal_subgroup?"
                                                         "subgroup?"
                                                         "subset?"
                                                         "member")
                                                        (("3"
                                                          (flatten)
                                                          (("3"
                                                            (hide
                                                             (-2 -4))
                                                            (("3"
                                                              (inst?)
                                                              (("3"
                                                                (assert)
                                                                nil
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("2"
                                                  (hide-all-but 1)
                                                  (("2"
                                                    (typepred
                                                     "a!3"
                                                     "M!1")
                                                    (("2"
                                                      (expand*
                                                       "normal_subgroup?"
                                                       "subgroup?"
                                                       "subset?"
                                                       "member")
                                                      (("2"
                                                        (flatten)
                                                        (("2"
                                                          (hide
                                                           (-2 -4))
                                                          (("2"
                                                            (inst?)
                                                            (("2"
                                                              (assert)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil)
                                                 ("3"
                                                  (hide-all-but 1)
                                                  (("3"
                                                    (typepred
                                                     "a!1"
                                                     "a!2"
                                                     "G!1"
                                                     "M!1")
                                                    (("3"
                                                      (expand
                                                       "normal_subgroup?")
                                                      (("3"
                                                        (flatten)
                                                        (("3"
                                                          (expand
                                                           "subgroup?")
                                                          (("3"
                                                            (expand
                                                             "subset?")
                                                            (("3"
                                                              (hide
                                                               (-4 -6))
                                                              (("3"
                                                                (copy
                                                                 -4)
                                                                (("3"
                                                                  (inst
                                                                   -1
                                                                   "a!1")
                                                                  (("3"
                                                                    (inst
                                                                     -5
                                                                     "a!2")
                                                                    (("3"
                                                                      (expand
                                                                       "member")
                                                                      (("3"
                                                                        (expand*
                                                                         "group?"
                                                                         "monoid?"
                                                                         "monad?")
                                                                        (("3"
                                                                          (flatten)
                                                                          (("3"
                                                                            (hide-all-but
                                                                             (-1
                                                                              -4
                                                                              -9
                                                                              1))
                                                                            (("3"
                                                                              (expand
                                                                               "star_closed?")
                                                                              (("3"
                                                                                (inst?)
                                                                                (("3"
                                                                                  (expand
                                                                                   "member")
                                                                                  (("3"
                                                                                    (propax)
                                                                                    nil
                                                                                    nil))
                                                                                  nil))
                                                                                nil))
                                                                              nil))
                                                                            nil))
                                                                          nil))
                                                                        nil))
                                                                      nil))
                                                                    nil))
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide-all-but 1)
                                                (("2"
                                                  (typepred
                                                   "a!2"
                                                   "M!1")
                                                  (("2"
                                                    (expand*
                                                     "normal_subgroup?"
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("2"
                                                      (flatten)
                                                      (("2"
                                                        (hide (-2 -4))
                                                        (("2"
                                                          (inst?)
                                                          (("2"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("3"
                                                (hide-all-but 1)
                                                (("3"
                                                  (typepred
                                                   "a!1"
                                                   "M!1")
                                                  (("3"
                                                    (expand*
                                                     "normal_subgroup?"
                                                     "subgroup?"
                                                     "subset?"
                                                     "member")
                                                    (("3"
                                                      (flatten)
                                                      (("3"
                                                        (hide (-2 -4))
                                                        (("3"
                                                          (inst?)
                                                          (("3"
                                                            (assert)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (expand "inv_exists?")
                (("2" (skosimp*)
                  (("2" (typepred "x!1")
                    (("2" (skosimp*)
                      (("2"
                        (inst 1
                         "*[T1, *, one1](inv[T1, *, one1](a!1), N!1)")
                        (("1" (replace -1)
                          (("1" (hide -1)
                            (("1" (rewrite "mult_lem[T1, *, one1]")
                              (("1"
                                (rewrite "mult_lem[T1, *, one1]")
                                (("1"
                                  (rewrite "inv_right[T1, *, one1]")
                                  (("1"
                                    (rewrite "inv_left[T1, *, one1]")
                                    (("1"
                                      (rewrite "left_coset_one")
                                      nil
                                      nil))
                                    nil))
                                  nil)
                                 ("2"
                                  (hide-all-but 1)
                                  (("2"
                                    (typepred "a!1" "M!1")
                                    (("2"
                                      (expand*
                                       "normal_subgroup?"
                                       "subgroup?"
                                       "subset?"
                                       "member")
                                      (("2"
                                        (flatten)
                                        (("2"
                                          (hide (-2 -4))
                                          (("2"
                                            (inst?)
                                            (("2" (assertnil nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil)
                                 ("3"
                                  (hide-all-but 1)
                                  (("3"
                                    (typepred "a!1" "M!1")
                                    (("3"
                                      (expand*
                                       "normal_subgroup?"
                                       "subgroup?"
                                       "subset?"
                                       "member")
                                      (("3"
                                        (flatten)
                                        (("3"
                                          (hide (-2 -4))
                                          (("3"
                                            (inst?)
                                            (("3"
                                              (assert)
                                              (("3"
                                                (rewrite
                                                 "inv_in[T1, *, one1]")
                                                nil
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (hide-all-but 1)
                                (("2"
                                  (typepred "a!1" "M!1")
                                  (("2"
                                    (expand*
                                     "normal_subgroup?"
                                     "subgroup?"
                                     "subset?"
                                     "member")
                                    (("2"
                                      (flatten)
                                      (("2"
                                        (hide (-2 -4))
                                        (("2"
                                          (inst?)
                                          (("2"
                                            (assert)
                                            (("2"
                                              (rewrite
                                               "inv_in[T1, *, one1]")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("3"
                                (hide-all-but 1)
                                (("3"
                                  (typepred "a!1" "M!1")
                                  (("3"
                                    (expand*
                                     "normal_subgroup?"
                                     "subgroup?"
                                     "subset?"
                                     "member")
                                    (("3"
                                      (flatten)
                                      (("3"
                                        (hide (-2 -4))
                                        (("3"
                                          (inst?)
                                          (("3" (assertnil nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (prop)
                          (("1" (expand "*")
                            (("1" (inst + "inv[T1, *, one1](a!1)")
                              (("1"
                                (rewrite "inv_in[T1, *, one1]")
                                nil
                                nil))
                              nil))
                            nil)
                           ("2" (expand "/")
                            (("2" (hide -2)
                              (("2"
                                (expand "restrict")
                                (("2"
                                  (expand "left_cosets")
                                  (("2"
                                    (inst 1 "inv[T1, *, one1](a!1)")
                                    (("2"
                                      (rewrite "inv_in[T1, *, one1]")
                                      nil
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((subgroup? const-decl "bool" group_def "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (a!2 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (mult_lem formula-decl nil factor_groups "algebra/")
    (product_in formula-decl nil group "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets type-eq-decl nil cosets "algebra/")
    (mult const-decl "left_cosets(G, H)" factor_groups "algebra/")
    (star_closed? const-decl "bool" groupoid_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (N_is_identity formula-decl nil factor_groups "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (x!1 skolem-const-decl "(M!1 / N!1)" isomorphism_theorems nil)
    (identity? const-decl "bool" operator_defs nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (a!3 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (a!2 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (assoc formula-decl nil group "algebra/")
    (associative? const-decl "bool" operator_defs nil)
    (monoid? const-decl "bool" monoid_def "algebra/")
    (inv_in formula-decl nil group "algebra/")
    (inv_right formula-decl nil group "algebra/")
    (inv_left formula-decl nil group "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (inv_exists? const-decl "bool" group_def "algebra/")
    (restrict const-decl "R" restrict nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (M!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (a!1 skolem-const-decl "(M!1)" isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one1 formal-const-decl "T1" isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (/ const-decl "group[left_cosets(G, N), mult(G, N), N]"
       right_left_cosets nil)
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil))
   shostak))
 (second_isomorphism_th_aux_TCC1 0
  (second_isomorphism_th_aux_TCC1-1 nil 3524841142
   ("" (skosimp*)
    (("" (replaces -1)
      (("" (lemma "HK_subgroup[T1, *, one1]")
        (("" (inst -1 "G!1" "H!1" "N!1")
          (("1" (assert)
            (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
              (("1" (inst?)
                (("1" (assert)
                  (("1" (expand "subgroup?") (("1" (propax) nil nil))
                    nil))
                  nil)
                 ("2" (hide (-1 2))
                  (("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (typepred "N!1")
            (("2" (expand "normal_subgroup?") (("2" (assertnil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_aux_TCC2 0
  (second_isomorphism_th_aux_TCC2-1 nil 3524841142
   ("" (skosimp*)
    (("" (replaces -2)
      (("" (replaces -1)
        (("" (lemma "HK_subgroup[T1, *, one1]")
          (("" (inst -1 "G!1" "H!1" "N!1")
            (("1" (assert)
              (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                (("1" (inst?)
                  (("1" (assert)
                    (("1" (replaces -1)
                      (("1" (lemma "H_K_are_subgroups[T1, *, one1]")
                        (("1" (inst?)
                          (("1" (prop)
                            (("1" (hide -2)
                              (("1"
                                (lemma
                                 "normal_subgroup_tran[T1, *, one1]")
                                (("1"
                                  (inst
                                   -1
                                   "G!1"
                                   "prod[T1, *, one1](N!1, H!1)"
                                   "N!1")
                                  (("1" (assertnil nil))
                                  nil)
                                 ("2" (rewrite "G_TCC1"nil nil))
                                nil))
                              nil)
                             ("2" (expand "subgroup?")
                              (("2" (assertnil nil)) nil))
                            nil)
                           ("2" (typepred "N!1")
                            (("2" (expand "normal_subgroup?" -2)
                              (("2" (assertnil nil)) nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil)
                   ("2" (hide 2)
                    (("2" (typepred "N!1")
                      (("2" (expand "normal_subgroup?")
                        (("2" (assertnil nil)) nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (hide 2)
              (("2" (typepred "N!1")
                (("2" (expand "normal_subgroup?")
                  (("2" (assertnil nil)) nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subgroup type-eq-decl nil group "algebra/")
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (H_K_are_subgroups formula-decl nil products_subgroups nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (normal_subgroup_tran formula-decl nil groups_scaf nil)
    (fullset const-decl "set" sets nil)
    (prod const-decl "set[T]" products_subgroups nil)
    (G_TCC1 assuming-tcc nil isomorphism_theorems nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (HK_subgroup formula-decl nil products_subgroups nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (one1 formal-const-decl "T1" isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_aux_TCC3 0
  (second_isomorphism_th_aux_TCC3-1 nil 3524841142
   ("" (skosimp*)
    (("" (inst 1 "one1")
      (("1" (rewrite "left_coset_one"nil nil)
       ("2" (replaces -1)
        (("2" (lemma "HK_subgroup[T1, *, one1]")
          (("2" (inst -1 "G!1" "H!1" "N!1")
            (("1" (assert)
              (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                (("1" (inst?)
                  (("1" (assert)
                    (("1" (replaces -1)
                      (("1"
                        (expand"subgroup?" "group?" "monoid?"
                         "monad?" "member")
                        nil nil))
                      nil))
                    nil)
                   ("2" (typepred "N!1")
                    (("2" (expand "normal_subgroup?")
                      (("2" (assertnil nil)) nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (typepred "N!1")
              (("2" (expand "normal_subgroup?")
                (("2" (assertnil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((one1 formal-const-decl "T1" isomorphism_theorems nil)
    (NH!1 skolem-const-decl "set[T1]" isomorphism_theorems nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T1 formal-type-decl nil isomorphism_theorems nil)
    (* formal-const-decl "[T1, T1 -> T1]" isomorphism_theorems nil)
    (normal_subgroup type-eq-decl nil normal_subgroups "algebra/")
    (normal_subgroup? const-decl "boolean" normal_subgroups "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/")
    (HK_subgroup formula-decl nil products_subgroups nil)
    (member const-decl "bool" sets nil)
    (monad? const-decl "bool" monad_def "algebra/")
    (monoid? const-decl "bool" monoid_def "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (HK_subgroup_permute formula-decl nil products_subgroups nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (G!1 skolem-const-decl "group[T1, *, one1]" isomorphism_theorems
     nil)
    (N!1 skolem-const-decl "normal_subgroup[T1, *, one1](G!1)"
     isomorphism_theorems nil))
   nil))
 (second_isomorphism_th_aux_TCC4 0
  (second_isomorphism_th_aux_TCC4-1 nil 3524841142
   ("" (skosimp*)
    (("" (rewrite "left_cosets_group[T1, *, one1]")
      (("1" (hide 2)
        (("1" (replaces -1)
          (("1" (lemma "HK_subgroup[T1, *, one1]")
            (("1" (inst -1 "G!1" "H!1" "N!1")
              (("1" (assert)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replaces -1)
                        (("1" (lemma "H_K_are_subgroups[T1, *, one1]")
                          (("1" (inst?)
                            (("1" (prop)
                              (("1"
                                (hide -2)
                                (("1"
                                  (lemma
                                   "normal_subgroup_tran[T1, *, one1]")
                                  (("1"
                                    (inst
                                     -1
                                     "G!1"
                                     "prod[T1, *, one1](N!1, H!1)"
                                     "N!1")
                                    (("1" (assertnil nil))
                                    nil)
                                   ("2" (rewrite "G_TCC1"nil nil))
                                  nil))
                                nil)
                               ("2"
                                (expand "subgroup?")
                                (("2" (assertnil nil))
                                nil))
                              nil)
                             ("2" (hide 2)
                              (("2"
                                (typepred "N!1")
                                (("2"
                                  (expand "normal_subgroup?")
                                  (("2" (assertnil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil)
                     ("2" (hide 2)
                      (("2" (typepred "N!1")
                        (("2" (expand "normal_subgroup?")
                          (("2" (assertnil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (hide 2)
                (("2" (typepred "N!1")
                  (("2" (expand "normal_subgroup?")
                    (("2" (assertnil nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (hide 2)
        (("2" (replaces -1)
          (("2" (lemma "HK_subgroup[T1, *, one1]")
            (("2" (inst -1 "G!1" "H!1" "N!1")
              (("1" (assert)
                (("1" (lemma "HK_subgroup_permute[T1, *, one1]")
                  (("1" (inst?)
                    (("1" (assert)
                      (("1" (replaces -1)
                        (("1"
                          (expand"subgroup?" "group?" "monoid?"
                           "monad?" "member" -1)
                          nil nil))
                        nil))
                      nil)
                     ("2" (hide-all-but 1)
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.169 Sekunden  (vorverarbeitet)  ¤





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