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Datei: right_left_cosets.prf   Sprache: Lisp

Original von: PVS©

(right_left_cosets
 (IMP_lagrange_TCC1 0
  (IMP_lagrange_TCC1-1 nil 3530611400
   ("" (rewrite "fullset_is_group"nil nil)
   ((fullset_is_group formula-decl nil right_left_cosets nil)) nil))
 (nonempty_left_coset_TCC1 0
  (nonempty_left_coset_TCC1-1 nil 3528677081 ("" (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 right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (subgroup? const-decl "bool" group_def "algebra/"))
   nil))
 (nonempty_left_coset 0
  (nonempty_left_coset-1 nil 3528677207
   ("" (skosimp*)
    (("" (expand "left_coset")
      (("" (expand "member")
        (("" (expand "nonempty?")
          (("" (expand "empty?")
            (("" (expand "member")
              (("" (expand "*")
                (("" (inst - "a!1")
                  (("" (inst + "one")
                    (("1" (rewrite "one_right"nil nil)
                     ("2" (rewrite "one_in"nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((left_coset const-decl "{s: set[T] | subset?(s, G)}" cosets
     "algebra/")
    (nonempty? const-decl "bool" sets nil)
    (T formal-type-decl nil right_left_cosets nil)
    (monad nonempty-type-eq-decl nil monad "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (one_in formula-decl nil monad "algebra/")
    (one_right formula-decl nil group "algebra/")
    (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 "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (H!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (* const-decl "set[T]" cosets "algebra/")
    (empty? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil))
   shostak))
 (left_coset_finite_TCC1 0
  (left_coset_finite_TCC1-1 nil 3528677081 ("" (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 right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (subgroup? const-decl "bool" group_def "algebra/"))
   nil))
 (left_coset_finite 0
  (left_coset_finite-1 nil 3528677376
   ("" (skosimp)
    (("" (assert)
      (("" (typepred "G!1")
        (("" (expand "finite_group?")
          (("" (flatten)
            (("" (expand "left_coset")
              (("" (lemma "left_coset_subset")
                (("" (inst -1 "G!1" "H!1" "a!1")
                  (("" (lemma "finite_subset[T]")
                    (("" (inst?) (("" (assertnil nil)) nil)) nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((member const-decl "bool" sets nil)
    (left_coset const-decl "{s: set[T] | subset?(s, G)}" cosets
     "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (is_finite const-decl "bool" finite_sets nil)
    (G!1 skolem-const-decl "finite_group[T, *, one]" right_left_cosets
     nil)
    (* const-decl "set[T]" cosets "algebra/")
    (finite_set type-eq-decl nil finite_sets nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (finite_subset formula-decl nil finite_sets nil)
    (left_coset_subset formula-decl nil cosets "algebra/")
    (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 right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (finite_group? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/"))
   shostak))
 (left_coset_correspondence 0
  (left_coset_correspondence-1 nil 3528677400
   ("" (skosimp*)
    (("" (typepred "G!1")
      (("" (expand "subgroup?")
        (("" (assert)
          (("" (inst + "LAMBDA (x:(H!1)): a!1*x")
            (("1" (expand "bijective?")
              (("1" (split)
                (("1" (expand "injective?")
                  (("1" (skosimp*)
                    (("1" (rewrite "cancel_left"nil nil)) nil))
                  nil)
                 ("2" (expand "surjective?")
                  (("2" (skosimp*)
                    (("2" (typepred "y!1")
                      (("2" (replaces -5)
                        (("2" (expand "left_coset")
                          (("2" (expand "*")
                            (("2" (skosimp)
                              (("2"
                                (inst?)
                                (("2" (assertnil nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (skosimp)
              (("2" (replaces -4)
                (("2" (expand "left_coset")
                  (("2" (expand "*") (("2" (inst 1 "x!1"nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (member const-decl "bool" sets nil)
    (bijective? const-decl "bool" functions nil)
    (surjective? const-decl "bool" functions nil)
    (left_coset const-decl "{s: set[T] | subset?(s, G)}" cosets
     "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (injective? const-decl "bool" functions nil)
    (cancel_left formula-decl nil group "algebra/")
    (a!1 skolem-const-decl "T" right_left_cosets nil)
    (A!1 skolem-const-decl "set[T]" right_left_cosets nil)
    (H!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (subgroup? const-decl "bool" group_def "algebra/"))
   shostak))
 (left_coset_correspondence_inv 0
  (left_coset_correspondence_inv-1 nil 3530284707
   ("" (skosimp*)
    (("" (typepred "G!1")
      (("" (expand "subgroup?")
        (("" (assert)
          (("" (inst + "LAMBDA (x:(H!1)): a!1*x*inv(a!1)")
            (("1" (expand "bijective?")
              (("1" (split)
                (("1" (expand "injective?")
                  (("1" (skosimp*)
                    (("1" (rewrite "cancel_right")
                      (("1" (rewrite "cancel_left"nil nil)) nil))
                    nil))
                  nil)
                 ("2" (expand "surjective?")
                  (("2" (skosimp*)
                    (("2" (typepred "y!1")
                      (("2" (hide (-2 -3 -4))
                        (("2" (expand "*")
                          (("2" (skosimp)
                            (("2" (inst 1 "inv(a!1)*h!1")
                              (("1"
                                (replaces -1)
                                (("1" (rewrite "assoc"nil nil))
                                nil)
                               ("2"
                                (typepred "h!1")
                                (("2"
                                  (skosimp)
                                  (("2"
                                    (replaces -1)
                                    (("2"
                                      (typepred "h!2")
                                      (("2" (rewrite "assoc"nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (skosimp*)
              (("2" (expand "*" 1 1)
                (("2" (typepred "x!1")
                  (("2" (inst?)
                    (("2" (expand "*" 1 1) (("2" (inst?) nil nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (subset_is_partial_order name-judgement "(partial_order?[set[T]])"
     sets_lemmas nil)
    (member const-decl "bool" sets nil)
    (x!1 skolem-const-decl "(H!1)" right_left_cosets nil)
    (bijective? const-decl "bool" functions nil)
    (surjective? const-decl "bool" functions nil)
    (h!1 skolem-const-decl "({t: T | EXISTS (h: (H!1)): t = a!1 * h})"
     right_left_cosets nil)
    (one_left formula-decl nil group "algebra/")
    (inv_right formula-decl nil group "algebra/")
    (assoc formula-decl nil group "algebra/")
    (inv_left formula-decl nil group "algebra/")
    (injective? const-decl "bool" functions nil)
    (cancel_right formula-decl nil group "algebra/")
    (cancel_left formula-decl nil group "algebra/")
    (H!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (* const-decl "set[T]" cosets "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (a!1 skolem-const-decl "T" right_left_cosets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/"))
   shostak))
 (finite_left_coset_correspondence_TCC1 0
  (finite_left_coset_correspondence_TCC1-1 nil 3528677081
   ("" (skosimp)
    (("" (assert)
      (("" (lemma "left_coset_finite")
        (("" (inst?) (("" (assertnil nil)) nil)) nil))
      nil))
    nil)
   ((member const-decl "bool" sets nil)
    (T formal-type-decl nil right_left_cosets 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 "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (left_coset_finite formula-decl nil right_left_cosets nil))
   nil))
 (finite_left_coset_correspondence_TCC2 0
  (finite_left_coset_correspondence_TCC2-1 nil 3528677081
   ("" (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 right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (member const-decl "bool" sets nil)
    (subset? const-decl "bool" sets nil)
    (subgroup? const-decl "bool" group_def "algebra/"))
   nil))
 (finite_left_coset_correspondence_TCC3 0
  (finite_left_coset_correspondence_TCC3-1 nil 3528677081
   ("" (skosimp)
    (("" (lemma "left_coset_finite")
      (("" (inst?) (("" (assertnil nil)) nil)) nil))
    nil)
   ((left_coset_finite formula-decl nil right_left_cosets nil)
    (member const-decl "bool" sets nil)
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil))
   nil))
 (finite_left_coset_correspondence 0
  (finite_left_coset_correspondence-1 nil 3528677432
   ("" (skosimp)
    (("" (assert)
      (("" (name-replace "A" "left_coset(G!1, H!1)(a!1)" :hide? nil)
        (("" (name-replace "B" "left_coset(G!1, H!1)(b!1)" :hide? nil)
          (("" (lemma "card_eq_bij[T,T]")
            (("" (inst?)
              (("" (assert)
                (("" (hide (-4 -5 2))
                  (("" (lemma "left_coset_correspondence")
                    (("" (inst -1 "G!1" "H!1" "a!1" "A")
                      (("" (lemma "left_coset_correspondence")
                        (("" (inst -1 "G!1" "H!1" "b!1" "B")
                          (("" (assert)
                            (("" (skosimp*)
                              ((""
                                (lemma
                                 "bijective_inverse_exists[(H!1), (A)]")
                                ((""
                                  (inst -1 "f!2")
                                  ((""
                                    (expand "exists1")
                                    ((""
                                      (flatten)
                                      ((""
                                        (hide -2)
                                        ((""
                                          (skolem * "f!3")
                                          ((""
                                            (inst 1 "f!1 o f!3")
                                            ((""
                                              (lemma
                                               "bij_inv_is_bij_alt[(H!1), (A)]")
                                              ((""
                                                (inst -1 "f!2" "f!3")
                                                ((""
                                                  (hide-all-but
                                                   (-1 -3 1))
                                                  ((""
                                                    (lemma
                                                     "composition_bijective[(A), (H!1), (B)]")
                                                    ((""
                                                      (inst?)
                                                      ((""
                                                        (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))
          nil))
        nil))
      nil))
    nil)
   ((member const-decl "bool" sets nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (is_finite const-decl "bool" finite_sets nil)
    (H!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (G!1 skolem-const-decl "finite_group[T, *, one]" right_left_cosets
     nil)
    (A skolem-const-decl "{s: set[T] | subset?(s, G!1)}"
     right_left_cosets nil)
    (bijective? const-decl "bool" functions nil)
    (f!2 skolem-const-decl "[(H!1) -> (A)]" right_left_cosets nil)
    (bij_inv_is_bij_alt formula-decl nil function_inverse_def nil)
    (composition_bijective formula-decl nil func_composition
     "finite_sets/")
    (f!3 skolem-const-decl "[(A) -> (H!1)]" right_left_cosets nil)
    (inverse? const-decl "bool" function_inverse_def nil)
    (O const-decl "T3" function_props nil)
    (exists1 const-decl "bool" exists1 nil)
    (bijective_inverse_exists formula-decl nil function_inverse_def
     nil)
    (left_coset_correspondence formula-decl nil right_left_cosets nil)
    (card_eq_bij formula-decl nil finite_sets_card_eq "finite_sets/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (left_coset const-decl "{s: set[T] | subset?(s, G)}" cosets
     "algebra/")
    (subset? const-decl "bool" sets nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil))
   shostak))
 (set_left_cosets_full 0
  (set_left_cosets_full-1 nil 3528677586
   ("" (skosimp*)
    (("" (apply-extensionality 1 :hide? t)
      (("" (expand "Union")
        (("" (iff 1)
          (("" (prop)
            (("1" (skosimp*)
              (("1" (typepred "a!1")
                (("1" (expand "left_cosets")
                  (("1" (skosimp*)
                    (("1" (replace -1)
                      (("1" (hide -1)
                        (("1" (expand "*")
                          (("1" (skosimp*)
                            (("1" (replace -1)
                              (("1"
                                (hide -1)
                                (("1"
                                  (typepred "a!2")
                                  (("1"
                                    (typepred "h!1")
                                    (("1"
                                      (typepred "H!1")
                                      (("1"
                                        (expand "subgroup?")
                                        (("1"
                                          (expand "subset?")
                                          (("1"
                                            (inst?)
                                            (("1"
                                              (expand "member")
                                              (("1"
                                                (rewrite "star_closed")
                                                nil
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (inst + "x!1*H!1")
              (("1" (expand "*")
                (("1" (inst + "one")
                  (("1" (rewrite "one_right"nil nil)
                   ("2" (rewrite "one_in"nil nil))
                  nil))
                nil)
               ("2" (expand "left_cosets")
                (("2" (inst + "x!1"nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil right_left_cosets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (Union const-decl "set" sets nil) (set type-eq-decl nil sets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (x!1 skolem-const-decl "T" right_left_cosets nil)
    (H!1 skolem-const-decl "subgroup[T, *, one](G!1)" right_left_cosets
     nil)
    (G!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (one_right formula-decl nil group "algebra/")
    (one_in formula-decl nil monad "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (monad nonempty-type-eq-decl nil monad "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (star_closed formula-decl nil groupoid "algebra/")
    (star_closed? const-decl "bool" groupoid_def "algebra/")
    (groupoid nonempty-type-eq-decl nil groupoid "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil))
   shostak))
 (left_cosets_disjoint 0
  (left_cosets_disjoint-1 nil 3528677536
   ("" (skosimp*)
    (("" (expand "disjoint?")
      (("" (expand "empty?")
        (("" (assert)
          (("" (expand "intersection")
            (("" (assert)
              (("" (skosimp*)
                (("" (typepred "B!1")
                  (("" (typepred "A!1")
                    (("" (skosimp*)
                      (("" (replace -1)
                        (("" (hide -1)
                          (("" (replace -1)
                            (("" (hide -1)
                              ((""
                                (expand "*")
                                ((""
                                  (apply-extensionality 1 :hide? t)
                                  ((""
                                    (iff 1)
                                    ((""
                                      (skosimp*)
                                      ((""
                                        (prop)
                                        (("1"
                                          (skosimp*)
                                          (("1"
                                            (inst + "h!2*inv(h!1)*h!3")
                                            (("1"
                                              (case-replace
                                               "x!1*inv(h!1) = a!1")
                                              (("1"
                                                (assert)
                                                (("1"
                                                  (replace -4 -1)
                                                  (("1"
                                                    (rewrite
                                                     "assoc"
                                                     -1
                                                     :dir
                                                     rl)
                                                    (("1"
                                                      (rewrite
                                                       "associative"
                                                       1
                                                       :dir
                                                       rl)
                                                      (("1"
                                                        (replace -1)
                                                        (("1"
                                                          (propax)
                                                          nil
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil)
                                               ("2"
                                                (hide 2)
                                                (("2"
                                                  (hide -1 -3)
                                                  (("2"
                                                    (replaces -1)
                                                    (("2"
                                                      (rewrite
                                                       "assoc"
                                                       :dir
                                                       rl)
                                                      nil
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (rewrite "star_closed")
                                              (("2"
                                                (rewrite "star_closed")
                                                (("2"
                                                  (rewrite "inv_in")
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil)
                                         ("2"
                                          (skosimp*)
                                          (("2"
                                            (inst + "h!1*inv(h!2)*h!3")
                                            (("1"
                                              (replace -1)
                                              (("1"
                                                (hide -1)
                                                (("1"
                                                  (rewrite "assoc")
                                                  (("1"
                                                    (rewrite "assoc")
                                                    (("1"
                                                      (replace -1 1 rl)
                                                      (("1"
                                                        (case-replace
                                                         "x!1*inv(h!2) = a!2")
                                                        (("1"
                                                          (hide -1 2)
                                                          (("1"
                                                            (replaces
                                                             -1)
                                                            (("1"
                                                              (rewrite
                                                               "associative"
                                                               1)
                                                              nil
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("2"
                                              (rewrite "star_closed")
                                              (("2"
                                                (rewrite "star_closed")
                                                (("2"
                                                  (rewrite "inv_in")
                                                  nil
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((disjoint? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (left_cosets type-eq-decl nil cosets "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (h!3 skolem-const-decl "(H!1)" right_left_cosets nil)
    (groupoid nonempty-type-eq-decl nil groupoid "algebra/")
    (star_closed? const-decl "bool" groupoid_def "algebra/")
    (star_closed formula-decl nil groupoid "algebra/")
    (inv_in formula-decl nil group "algebra/")
    (associative formula-decl nil semigroup "algebra/")
    (assoc formula-decl nil group "algebra/")
    (one_right formula-decl nil group "algebra/")
    (inv_right formula-decl nil group "algebra/")
    (G!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (H!1 skolem-const-decl "subgroup[T, *, one](G!1)" right_left_cosets
     nil)
    (h!2 skolem-const-decl "(H!1)" right_left_cosets nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (h!1 skolem-const-decl "(H!1)" right_left_cosets nil)
    (h!3 skolem-const-decl "(H!1)" right_left_cosets nil)
    (intersection const-decl "set" sets nil)
    (empty? const-decl "bool" sets nil))
   shostak))
 (left_cosets_partition 0
  (left_cosets_partition-1 nil 3528677626
   ("" (skosimp*)
    (("" (expand "finite_partition?")
      (("" (lemma "set_left_cosets_full")
        (("" (inst - "G!1" "H!1")
          (("" (typepred "G!1")
            (("" (prop)
              (("1" (expand "partition?")
                (("1" (skosimp*)
                  (("1" (lemma "left_cosets_disjoint")
                    (("1" (inst - "G!1" "H!1" "a!1" "b!1")
                      (("1" (assertnil nil)
                       ("2" (typepred "b!1")
                        (("2" (expand "left_cosets")
                          (("2" (propax) nil nil)) nil))
                        nil)
                       ("3" (typepred "a!1")
                        (("3" (expand "left_cosets")
                          (("3" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("2" (lemma "Union_finite[T]")
                (("2" (inst?)
                  (("2" (assert)
                    (("2" (hide 2)
                      (("2" (replace -2)
                        (("2" (expand "finite_group?")
                          (("2" (propax) nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil)
               ("3" (expand "every")
                (("3" (skosimp*)
                  (("3" (typepred "x!1")
                    (("3" (expand "left_cosets")
                      (("3" (skosimp*)
                        (("3" (replace -1)
                          (("3" (lemma "left_coset_finite")
                            (("3" (inst -1 "H!1" "a!1" "G!1")
                              (("3"
                                (assert)
                                (("3"
                                  (expand "left_coset")
                                  (("3" (propax) nil nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((finite_partition? const-decl "bool" lagrange_scaf "algebra/")
    (T formal-type-decl nil right_left_cosets 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 "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (a!1 skolem-const-decl "(left_cosets(G!1, H!1))" right_left_cosets
     nil)
    (H!1 skolem-const-decl "subgroup[T, *, one](G!1)" right_left_cosets
     nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (G!1 skolem-const-decl "finite_group[T, *, one]" right_left_cosets
     nil)
    (b!1 skolem-const-decl "(left_cosets(G!1, H!1))" right_left_cosets
     nil)
    (left_cosets type-eq-decl nil cosets "algebra/")
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (left_cosets_disjoint formula-decl nil right_left_cosets nil)
    (partition? const-decl "bool" lagrange_scaf "algebra/")
    (Union_finite formula-decl nil finite_sets_of_sets nil)
    (left_coset const-decl "{s: set[T] | subset?(s, G)}" cosets
     "algebra/")
    (member const-decl "bool" sets nil)
    (left_coset_finite formula-decl nil right_left_cosets nil)
    (every const-decl "bool" sets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (set_left_cosets_full formula-decl nil right_left_cosets nil))
   shostak))
 (set_right_cosets_full_1 0
  (set_right_cosets_full_1-1 nil 3528677991
   ("" (skosimp*)
    (("" (apply-extensionality 1 :hide? t)
      (("" (expand "Union")
        (("" (iff 1)
          (("" (prop)
            (("1" (skosimp*)
              (("1" (typepred "a!1")
                (("1" (expand "right_cosets")
                  (("1" (skosimp*)
                    (("1" (replaces -1)
                      (("1" (expand "*")
                        (("1" (skosimp*)
                          (("1" (replaces -2)
                            (("1" (replaces -1)
                              (("1"
                                (typepred "a!2")
                                (("1"
                                  (typepred "h!1")
                                  (("1"
                                    (typepred "H!1")
                                    (("1"
                                      (expand "subgroup?")
                                      (("1"
                                        (expand "subset?")
                                        (("1"
                                          (inst?)
                                          (("1"
                                            (expand "member")
                                            (("1"
                                              (rewrite "star_closed")
                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil)
             ("2" (inst + "H!1*x!1")
              (("1" (expand "*")
                (("1" (inst + "one")
                  (("1" (rewrite "one_left"nil nil)
                   ("2" (rewrite "one_in"nil nil))
                  nil))
                nil)
               ("2" (expand "right_cosets")
                (("2" (inst + "x!1"nil nil)) nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil right_left_cosets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (right_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (Union const-decl "set" sets nil) (set type-eq-decl nil sets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (x!1 skolem-const-decl "T" right_left_cosets nil)
    (H!1 skolem-const-decl "subgroup[T, *, one](G!1)" right_left_cosets
     nil)
    (G!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (one_left formula-decl nil group "algebra/")
    (one_in formula-decl nil monad "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (monad nonempty-type-eq-decl nil monad "algebra/")
    (subset? const-decl "bool" sets nil)
    (member const-decl "bool" sets nil)
    (star_closed formula-decl nil groupoid "algebra/")
    (star_closed? const-decl "bool" groupoid_def "algebra/")
    (groupoid nonempty-type-eq-decl nil groupoid "algebra/")
    (* const-decl "set[T]" cosets "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil))
   shostak))
 (right_left_correspondence 0
  (right_left_correspondence-1 nil 3528678027
   ("" (skosimp*)
    ((""
      (inst 1
       "(LAMBDA (A: (right_cosets(G!1,H!1))): inv(rc_gen(G!1,H!1,A)) * H!1)")
      (("1" (expand "bijective?")
        (("1" (split)
          (("1" (expand "injective?")
            (("1" (skosimp*)
              (("1" (typepred "x1!1" "x2!1")
                (("1" (lemma "lc_eq")
                  (("1"
                    (inst -1 "G!1" "H!1" "inv(rc_gen(G!1, H!1, x1!1))"
                     "inv(rc_gen(G!1, H!1, x2!1))")
                    (("1" (assert)
                      (("1" (skosimp)
                        (("1" (rewrite "divby")
                          (("1"
                            (case "rc_gen(G!1, H!1, x2!1) = h!1 * rc_gen(G!1, H!1, x1!1)")
                            (("1" (hide (-2 -5))
                              (("1"
                                (lemma "rc_is_eq")
                                (("1"
                                  (inst
                                   -1
                                   "G!1"
                                   "H!1"
                                   "rc_gen(G!1, H!1, x2!1)"
                                   "rc_gen(G!1, H!1, x1!1)")
                                  (("1"
                                    (prop)
                                    (("1" (assertnil nil)
                                     ("2"
                                      (hide-all-but (-1 1))
                                      (("2" (inst?) nil nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil)
                             ("2" (hide-all-but (-1 1))
                              (("2"
                                (replace -1 1 rl)
                                (("2"
                                  (rewrite "assoc" :dir rl)
                                  nil
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil)
           ("2" (expand "surjective?")
            (("2" (skosimp)
              (("2" (typepred "y!1")
                (("2" (expand "left_cosets")
                  (("2" (skosimp)
                    (("2" (inst 1 "H!1 * inv(a!1)")
                      (("1" (replace -1 1)
                        (("1" (lemma "lc_is_eq")
                          (("1"
                            (inst -1 "G!1" "H!1"
                             "inv(rc_gen(G!1, H!1, H!1 * inv(a!1)))"
                             "a!1")
                            (("1" (assert)
                              (("1"
                                (hide 2)
                                (("1"
                                  (typepred
                                   "rc_gen(G!1, H!1, H!1 * inv(a!1))")
                                  (("1"
                                    (lemma "rc_eq")
                                    (("1"
                                      (inst
                                       -1
                                       "G!1"
                                       "H!1"
                                       "inv(a!1)"
                                       "rc_gen(G!1, H!1, H!1 * inv(a!1))")
                                      (("1"
                                        (assert)
                                        (("1"
                                          (skosimp)
                                          (("1"
                                            (hide-all-but (-1 1))
                                            (("1"
                                              (inst 1 "h!1")
                                              (("1"
                                                (lemma "inv_inv")
                                                (("1"
                                                  (inst?)
                                                  (("1"
                                                    (name-replace
                                                     "c"
                                                     "rc_gen(G!1, H!1, H!1 * inv(a!1))")
                                                    (("1"
                                                      (replaces -2)
                                                      (("1"
                                                        (rewrite
                                                         "inv_star")
                                                        (("1"
                                                          (replace
                                                           -1
                                                           1
                                                           rl)
                                                          (("1"
                                                            (rewrite
                                                             "assoc"
                                                             :dir
                                                             rl)
                                                            nil
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil)
                                   ("2"
                                    (hide (-1 -2 2))
                                    (("2"
                                      (inst?)
                                      (("2"
                                        (rewrite "inv_in")
                                        nil
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil))
                        nil)
                       ("2" (expand "right_cosets")
                        (("2" (inst?)
                          (("2" (rewrite "inv_in"nil nil)) nil))
                        nil))
                      nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("2" (skosimp)
        (("2" (expand "left_cosets")
          (("2" (inst?)
            (("1" (rewrite "inv_in"nil nil)
             ("2" (typepred "A!1")
              (("2" (expand "right_cosets") (("2" (propax) nil nil))
                nil))
              nil))
            nil))
          nil))
        nil)
       ("3" (skosimp)
        (("3" (typepred "A!1")
          (("3" (expand "right_cosets") (("3" (propax) nil nil)) nil))
          nil))
        nil)
       ("4" (skosimp) nil nil))
      nil))
    nil)
   ((H!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (G!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (right_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (setofsets type-eq-decl nil sets nil)
    (subgroup 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)
    (group nonempty-type-eq-decl nil group "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (setof type-eq-decl nil defined_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil)
    (* const-decl "set[T]" cosets "algebra/")
    (= const-decl "[T, T -> boolean]" equalities nil)
    (rc_gen const-decl "{a: T | G(a) AND rc = H * a}" cosets
     "algebra/")
    (right_cosets type-eq-decl nil cosets "algebra/")
    (inv const-decl "{y | x * y = one AND y * x = one}" group
     "algebra/")
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (* const-decl "set[T]" cosets "algebra/")
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (lc_eq formula-decl nil cosets "algebra/")
    (divby formula-decl nil group "algebra/")
    (inv_inv formula-decl nil group "algebra/")
    (inv_left formula-decl nil group "algebra/")
    (one_right formula-decl nil group "algebra/")
    (assoc formula-decl nil group "algebra/")
    (rc_is_eq formula-decl nil cosets "algebra/")
    (NOT const-decl "[bool -> bool]" booleans nil)
    (injective? const-decl "bool" functions nil)
    (a!1 skolem-const-decl "(G!1)" right_left_cosets nil)
    (lc_is_eq formula-decl nil cosets "algebra/")
    (inv_star formula-decl nil group "algebra/")
    (rc_eq formula-decl nil cosets "algebra/")
    (inv_in formula-decl nil group "algebra/")
    (surjective? const-decl "bool" functions nil)
    (bijective? const-decl "bool" functions nil)
    (A!1 skolem-const-decl "(right_cosets(G!1, H!1))" right_left_cosets
     nil))
   shostak))
 (finite_right_left_correspondence_TCC1 0
  (finite_right_left_correspondence_TCC1-1 nil 3528677081
   ("" (skosimp*)
    (("" (lemma "set_right_cosets_full_1")
      (("" (inst - "G!1" "H!1")
        (("" (typepred "G!1")
          (("" (lemma "Union_finite[T]")
            (("" (inst?)
              (("" (assert)
                (("" (hide 2)
                  (("" (replace -2)
                    (("" (expand "finite_group?")
                      (("" (propax) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((set_right_cosets_full_1 formula-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (right_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (Union_finite formula-decl nil finite_sets_of_sets nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil))
   nil))
 (finite_right_left_correspondence_TCC2 0
  (finite_right_left_correspondence_TCC2-1 nil 3528677081
   ("" (skosimp*)
    (("" (lemma "set_left_cosets_full")
      (("" (inst - "G!1" "H!1")
        (("" (typepred "G!1")
          (("" (lemma "Union_finite[T]")
            (("" (inst?)
              (("" (assert)
                (("" (hide 2)
                  (("" (replace -2)
                    (("" (expand "finite_group?")
                      (("" (propax) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((set_left_cosets_full formula-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (Union_finite formula-decl nil finite_sets_of_sets nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil))
   nil))
 (finite_right_left_correspondence 0
  (finite_right_left_correspondence-1 nil 3528678094
   ("" (skosimp)
    (("" (lemma "card_eq_bij[set[T], set[T]]")
      (("" (inst -1 "right_cosets(G!1, H!1)" "left_cosets(G!1, H!1)")
        (("" (assert)
          (("" (hide 2)
            (("" (lemma "right_left_correspondence")
              (("" (inst?) (("" (assertnil nil)) nil)) nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil)
    (card_eq_bij formula-decl nil finite_sets_card_eq "finite_sets/")
    (right_left_correspondence formula-decl nil right_left_cosets nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (right_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (setofsets type-eq-decl nil sets nil)
    (setof type-eq-decl nil defined_types nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (finite_set type-eq-decl nil finite_sets nil)
    (is_finite const-decl "bool" finite_sets nil))
   shostak))
 (index_TCC1 0
  (index_TCC1-1 nil 3528677081
   ("" (skosimp*)
    (("" (lemma "set_left_cosets_full")
      (("" (inst - "G!1" "H!1")
        (("" (typepred "G!1")
          (("" (lemma "Union_finite[T]")
            (("" (inst?)
              (("" (assert)
                (("" (hide 2)
                  (("" (replace -2)
                    (("" (expand "finite_group?")
                      (("" (propax) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((set_left_cosets_full formula-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (setofsets type-eq-decl nil sets nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (Union_surjective name-judgement
     "(surjective?[setofsets[T], set[T]])" sets_lemmas nil)
    (Union_finite formula-decl nil finite_sets_of_sets nil)
    (subgroup type-eq-decl nil group "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/")
    (finite_group? const-decl "bool" group_def "algebra/")
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets nil))
   nil))
 (index_gt1 0
  (index_gt1-1 nil 3530550237
   ("" (skosimp*)
    (("" (expand "index")
      (("" (case "nonempty?(left_cosets(G!1, H!1))")
        (("1" (rewrite "nonempty_card") (("1" (assertnil nil)) nil)
         ("2" (hide 2)
          (("2" (expand"nonempty?" "empty?" "member")
            (("2" (inst -1 "H!1")
              (("2" (expand "left_cosets")
                (("2" (inst 1 "one")
                  (("1" (rewrite "left_coset_one"nil nil)
                   ("2" (rewrite "one_in"nil nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((index const-decl "nat" right_left_cosets nil)
    (G!1 skolem-const-decl "finite_group[T, *, one]" right_left_cosets
     nil)
    (left_coset_one formula-decl nil cosets "algebra/")
    (one_in formula-decl nil monad "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (monad nonempty-type-eq-decl nil monad "algebra/")
    (member const-decl "bool" sets nil)
    (empty? const-decl "bool" sets nil)
    (nonempty_card formula-decl nil finite_sets nil)
    (is_finite const-decl "bool" finite_sets nil)
    (finite_set type-eq-decl nil finite_sets 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)
    (T formal-type-decl nil right_left_cosets nil)
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (setof type-eq-decl nil defined_types nil)
    (set type-eq-decl nil sets nil)
    (nonempty? const-decl "bool" sets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (subgroup? const-decl "bool" group_def "algebra/")
    (subgroup type-eq-decl nil group "algebra/")
    (setofsets type-eq-decl nil sets nil)
    (left_cosets const-decl "setofsets[T]" right_left_cosets nil)
    (finite_group? const-decl "bool" group_def "algebra/")
    (finite_group nonempty-type-eq-decl nil group "algebra/"))
   shostak))
 (divide_TCC1 0
  (divide_TCC1-1 nil 3530434541
   ("" (skosimp*)
    (("" (inst 1 "one")
      (("1" (rewrite "left_coset_one"nil nil)
       ("2" (rewrite "one_in"nil nil))
      nil))
    nil)
   ((G!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (group nonempty-type-eq-decl nil group "algebra/")
    (group? const-decl "bool" group_def "algebra/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (T formal-type-decl nil right_left_cosets 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/")
    (monad nonempty-type-eq-decl nil monad "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (one_in formula-decl nil monad "algebra/"))
   nil))
 (divide_TCC2 0
  (divide_TCC2-1 nil 3530434541
   ("" (skosimp*) (("" (rewrite "left_cosets_group[T,*,one]"nil nil))
    nil)
   ((left_cosets_group formula-decl nil factor_groups "algebra/")
    (boolean nonempty-type-decl nil booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (set type-eq-decl nil sets 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/")
    (T formal-type-decl nil right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets nil))
   nil))
 (divide_TCC3 0
  (divide_TCC3-1 nil 3530434541
   ("" (skosimp*)
    (("" (inst 1 "N!1")
      (("" (inst 1 "one")
        (("1" (rewrite "left_coset_one"nil nil)
         ("2" (rewrite "one_in"nil nil))
        nil))
      nil))
    nil)
   ((T formal-type-decl nil right_left_cosets 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 "[T, T -> T]" right_left_cosets nil)
    (one formal-const-decl "T" right_left_cosets 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[T, *, one](G!1)"
     right_left_cosets nil)
    (G!1 skolem-const-decl "group[T, *, one]" right_left_cosets nil)
    (monad nonempty-type-eq-decl nil monad "algebra/")
    (monad? const-decl "bool" monad_def "algebra/")
    (one_in formula-decl nil monad "algebra/")
    (left_coset_one formula-decl nil cosets "algebra/"))
   nil))
 (divide_TCC4 0
  (divide_TCC4-1 nil 3530434541
   ("" (skosimp*)
    (("" (typepred "N!1")
      (("" (expand "normal_subgroup?") (("" (assertnil nil)) nil))
      nil))
    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/")
    (one formal-const-decl "T" right_left_cosets nil)
    (* formal-const-decl "[T, T -> T]" right_left_cosets nil)
    (set type-eq-decl nil sets nil)
    (T formal-type-decl nil right_left_cosets nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   nil))
 (divide_TCC5 0
  (divide_TCC5-1 nil 3530434541
   ("" (skosimp*)
    (("" (expand "group?")
      (("" (prop)
        (("1" (expand "monoid?")
          (("1" (prop)
            (("1" (expand "monad?")
              (("1" (prop)
                (("1" (expand "star_closed?")
                  (("1" (skosimp*)
                    (("1" (assert)
                      (("1" (expand "restrict")
                        (("1" (expand "left_cosets")
--> --------------------

--> maximum size reached

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

¤ Dauer der Verarbeitung: 0.87 Sekunden  (vorverarbeitet)  ¤





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