SET007 Axioms: SET007+459.ax


%------------------------------------------------------------------------------
% File     : SET007+459 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Correspondence Between Signatures and Graphs. Part I
% Version  : [Urb08] axioms.
% English  : Correspondence Between Monotonic Many Sorted Signatures and
%            Well-Founded Graphs. Part I

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : msscyc_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   51 (   1 unit)
%            Number of atoms       :  399 (  30 equality)
%            Maximal formula depth :   23 (   8 average)
%            Number of connectives :  419 (  71 ~  ;   1  |; 208  &)
%                                         (   9 <=>; 130 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   49 (   1 propositional; 0-4 arity)
%            Number of functors    :   46 (   5 constant; 0-4 arity)
%            Number of variables   :  135 (   0 singleton; 121 !;  14 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_msscyc_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v2_graph_1(A)
      & v2_msscyc_1(A) ) )).

fof(rc2_msscyc_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v1_graph_1(A)
      & v2_graph_1(A)
      & v5_graph_1(A)
      & v6_graph_1(A)
      & v7_graph_1(A)
      & ~ v2_msscyc_1(A) ) )).

fof(fc1_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ~ v1_xboole_0(u2_graph_1(A)) ) )).

fof(rc3_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & ~ v1_xboole_0(B)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v8_graph_1(B,A) ) ) )).

fof(fc2_msscyc_1,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A)
        & ~ v1_xboole_0(B)
        & v8_graph_1(B,A)
        & m1_graph_1(B,A) )
     => ( ~ v1_xboole_0(k7_graph_2(A,B))
        & v1_relat_1(k7_graph_2(A,B))
        & v1_funct_1(k7_graph_2(A,B))
        & v1_finset_1(k7_graph_2(A,B))
        & v1_finseq_1(k7_graph_2(A,B)) ) ) )).

fof(cc1_msscyc_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( ( v2_graph_1(A)
          & v2_msscyc_1(A) )
       => ( v2_graph_1(A)
          & v3_msscyc_1(A) ) ) ) )).

fof(cc2_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_graph_1(B,A)
         => ( v1_xboole_0(B)
            & v1_relat_1(B)
            & v2_funct_1(B)
            & v1_finset_1(B)
            & v8_graph_1(B,A) ) ) ) )).

fof(cc3_msscyc_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( ( v2_graph_1(A)
          & v2_msscyc_1(A) )
       => ( v2_graph_1(A)
          & v4_msscyc_1(A) ) ) ) )).

fof(cc4_msscyc_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( ( v2_graph_1(A)
          & ~ v4_msscyc_1(A) )
       => ( v2_graph_1(A)
          & ~ v2_msscyc_1(A) ) ) ) )).

fof(rc4_msscyc_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v2_graph_1(A)
      & v4_msscyc_1(A) ) )).

fof(cc5_msscyc_1,axiom,(
    ! [A] :
      ( l1_graph_1(A)
     => ( ( v2_graph_1(A)
          & v4_msscyc_1(A) )
       => ( v2_graph_1(A)
          & v3_msscyc_1(A) ) ) ) )).

fof(rc5_msscyc_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v2_graph_1(A)
      & ~ v2_msscyc_1(A)
      & ~ v4_msscyc_1(A) ) )).

fof(rc6_msscyc_1,axiom,(
    ? [A] :
      ( l1_graph_1(A)
      & v2_graph_1(A)
      & v3_msscyc_1(A) ) )).

fof(rc7_msscyc_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & v3_msafree2(B,A)
        & l3_msualg_1(B,A) )
     => ? [C] :
          ( m1_msafree(C,A,B)
          & v1_relat_1(C)
          & v2_relat_1(C)
          & ~ v3_relat_1(C)
          & v1_funct_1(C)
          & v1_pre_circ(C,u1_struct_0(A)) ) ) )).

fof(fc3_msscyc_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & v1_pre_circ(B,u1_struct_0(A))
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k12_msafree(A,B,C))
        & v1_finset_1(k12_msafree(A,B,C)) ) ) )).

fof(t1_msscyc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) )
     => ( ! [B] :
            ( r2_hidden(B,k1_relat_1(A))
           => v1_finset_1(k1_funct_1(A,B)) )
       => v1_finset_1(k4_card_3(A)) ) ) )).

fof(d1_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( m1_graph_1(B,A)
          <=> ( m2_finseq_1(B,u2_graph_1(A))
              & ? [C] :
                  ( m2_finseq_1(C,u1_graph_1(A))
                  & r1_graph_2(A,C,B) ) ) ) ) ) )).

fof(t2_msscyc_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( r1_xreal_0(A,k3_finseq_1(B))
               => k1_graph_2(B,np__1,A) = k1_graph_2(k7_finseq_1(B,C),np__1,A) ) ) ) ) )).

fof(d2_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_graph_1(B,A)
         => ( v1_msscyc_1(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_graph_1(A))
                & r1_graph_2(A,C,B)
                & k1_funct_1(C,np__1) = k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) )).

fof(d3_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v2_msscyc_1(A)
      <=> v1_xboole_0(u2_graph_1(A)) ) ) )).

fof(t3_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => r1_tarski(k2_xboole_0(k2_relat_1(u3_graph_1(A)),k2_relat_1(u4_graph_1(A))),u1_graph_1(A)) ) )).

fof(t4_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B,C] :
          ( m1_subset_1(C,u1_graph_1(A))
         => ! [D] :
              ( m1_subset_1(D,u1_graph_1(A))
             => ( ( C = k1_funct_1(u3_graph_1(A),B)
                  & D = k1_funct_1(u4_graph_1(A),B) )
               => r1_graph_2(A,k4_lang1(u1_graph_1(A),C,D),k9_finseq_1(B)) ) ) ) ) )).

fof(t5_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( r2_hidden(B,u2_graph_1(A))
         => ( v8_graph_1(k9_finseq_1(B),A)
            & m1_graph_1(k9_finseq_1(B),A) ) ) ) )).

fof(t6_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_graph_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
             => ( ( v1_msscyc_1(B,A)
                  & r1_graph_2(A,C,B) )
               => k1_funct_1(C,np__1) = k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) )).

fof(t7_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( r2_hidden(B,u2_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m1_graph_1(C,A) )
             => ( C = k9_finseq_1(B)
               => k7_graph_2(A,C) = k10_finseq_1(k1_funct_1(u3_graph_1(A),B),k1_funct_1(u4_graph_1(A),B)) ) ) ) ) )).

fof(t8_msscyc_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => r1_xreal_0(k3_finseq_1(k1_graph_2(C,A,B)),k3_finseq_1(C)) ) ) ) )).

fof(t9_msscyc_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v2_graph_1(C)
                & ~ v2_msscyc_1(C)
                & l1_graph_1(C) )
             => ! [D] :
                  ( ( v8_graph_1(D,C)
                    & m1_graph_1(D,C) )
                 => ( ( r1_xreal_0(np__1,A)
                      & r1_xreal_0(A,B)
                      & r1_xreal_0(B,k3_finseq_1(D)) )
                   => ( v8_graph_1(k1_graph_2(D,A,B),C)
                      & m1_graph_1(k1_graph_2(D,A,B),C) ) ) ) ) ) ) )).

fof(t10_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => k3_finseq_1(k7_graph_2(A,B)) = k1_nat_1(k3_finseq_1(B),np__1) ) ) )).

fof(t11_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r1_xreal_0(C,k3_finseq_1(B)) )
               => ( k1_funct_1(k7_graph_2(A,B),C) = k1_funct_1(u3_graph_1(A),k1_funct_1(B,C))
                  & k1_funct_1(k7_graph_2(A,B),k1_nat_1(C,np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,C)) ) ) ) ) ) )).

fof(t12_msscyc_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ~ ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(A,B)
                  & r1_xreal_0(B,k3_finseq_1(C))
                  & v1_xboole_0(k1_graph_2(C,A,B)) ) ) ) ) )).

fof(t13_msscyc_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v2_graph_1(C)
                & ~ v2_msscyc_1(C)
                & l1_graph_1(C) )
             => ! [D] :
                  ( ( v8_graph_1(D,C)
                    & m1_graph_1(D,C) )
                 => ! [E] :
                      ( ( v8_graph_1(E,C)
                        & m1_graph_1(E,C) )
                     => ( ( r1_xreal_0(np__1,A)
                          & r1_xreal_0(A,B)
                          & r1_xreal_0(B,k3_finseq_1(D))
                          & E = k1_graph_2(D,A,B) )
                       => k7_graph_2(C,E) = k2_graph_2(u1_graph_1(C),k7_graph_2(C,D),A,k1_nat_1(B,np__1)) ) ) ) ) ) ) )).

fof(t14_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,k3_finseq_1(B))) ) ) )).

fof(t15_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v8_graph_1(C,A)
                & m1_graph_1(C,A) )
             => ( k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) = k1_funct_1(k7_graph_2(A,C),np__1)
              <=> ( ~ v1_xboole_0(k7_finseq_1(B,C))
                  & v8_graph_1(k7_finseq_1(B,C),A)
                  & m1_graph_1(k7_finseq_1(B,C),A) ) ) ) ) ) )).

fof(t16_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v8_graph_1(C,A)
                & m1_graph_1(C,A) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v8_graph_1(D,A)
                    & m1_graph_1(D,A) )
                 => ( B = k7_finseq_1(C,D)
                   => ( k1_funct_1(k7_graph_2(A,B),np__1) = k1_funct_1(k7_graph_2(A,C),np__1)
                      & k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) = k1_funct_1(k7_graph_2(A,D),k1_nat_1(k3_finseq_1(D),np__1)) ) ) ) ) ) ) )).

fof(t17_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => ( v1_msscyc_1(B,A)
           => k1_funct_1(k7_graph_2(A,B),np__1) = k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) ) ) ) )).

fof(t18_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & ~ v2_msscyc_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v8_graph_1(B,A)
            & m1_graph_1(B,A) )
         => ( v1_msscyc_1(B,A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ? [D] :
                    ( v8_graph_1(D,A)
                    & m1_graph_1(D,A)
                    & k3_finseq_1(D) = C
                    & ~ v1_xboole_0(k7_finseq_1(D,B))
                    & v8_graph_1(k7_finseq_1(D,B),A)
                    & m1_graph_1(k7_finseq_1(D,B),A) ) ) ) ) ) )).

fof(d4_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v3_msscyc_1(A)
      <=> ! [B] :
            ( ( v8_graph_1(B,A)
              & m1_graph_1(B,A) )
           => ~ ( ~ v1_xboole_0(B)
                & v1_msscyc_1(B,A) ) ) ) ) )).

fof(d5_msscyc_1,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( v4_msscyc_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_graph_1(A))
           => ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ! [D] :
                    ( ( v8_graph_1(D,A)
                      & m1_graph_1(D,A) )
                   => ( k1_funct_1(k7_graph_2(A,D),k1_nat_1(k3_finseq_1(D),np__1)) = B
                     => ( v1_xboole_0(D)
                        | r1_xreal_0(k3_finseq_1(D),C) ) ) ) ) ) ) ) )).

fof(t19_msscyc_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m1_trees_1(B,k1_relat_1(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => m1_trees_1(k16_finseq_1(k5_numbers,B,C),k1_relat_1(A)) ) ) ) )).

fof(t20_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_dtconstr(C,u1_struct_0(k5_msafree(A,B)),k5_trees_3(u1_struct_0(k5_msafree(A,B))),k1_msaterm(A,B))
             => ~ ( ~ v1_trees_9(C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_msualg_1(A))
                     => k1_funct_1(C,k1_xboole_0) != k4_tarski(D,u1_struct_0(A)) ) ) ) ) ) )).

fof(t21_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_msafree(C,A,B)
             => ! [D] :
                  ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
                 => ( r2_pboole(u1_struct_0(A),C,D)
                   => m1_msafree(D,A,B) ) ) ) ) ) )).

fof(t22_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v2_relat_1(C)
                & m1_msafree(C,A,B) )
             => ? [D] :
                  ( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B))
                  & r2_msualg_3(A,k11_msafree(A,C),B,D) ) ) ) ) )).

fof(t23_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v2_relat_1(C)
                & m1_msafree(C,A,B) )
             => ~ ( ~ v4_msafree2(B,A)
                  & v4_msafree2(k11_msafree(A,C),A) ) ) ) ) )).

fof(t24_msscyc_1,axiom,(
    $true )).

fof(t25_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_msualg_1(A))
             => ( k8_funct_2(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),C) = k1_xboole_0
               => k1_relat_1(k5_msualg_1(A,C,B)) = k1_tarski(k1_xboole_0) ) ) ) ) )).

fof(t26_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( v5_msscyc_1(A)
               => v1_finset_1(k1_msualg_2(A,B,C)) ) ) ) ) )).

fof(t28_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_msualg_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
                         => ! [G] :
                              ( m1_msaterm(G,A,B,k2_msaterm(A,B,E))
                             => ! [H] :
                                  ( m2_subset_1(H,k1_numbers,k5_numbers)
                                 => ! [I] :
                                      ( m1_subset_1(I,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),D))
                                     => ~ ( F = k4_trees_4(k4_tarski(E,u1_struct_0(A)),G)
                                          & r2_hidden(H,k4_finseq_1(G))
                                          & I = k1_funct_1(G,H)
                                          & r1_xreal_0(k9_msafree2(A,B,C,F),k9_msafree2(A,B,D,I)) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ( v5_msscyc_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => v1_finset_1(a_2_0_msscyc_1(A,B)) ) ) ) )).

fof(t27_msscyc_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => a_3_0_msscyc_1(A,B,C) = k2_xboole_0(k12_msafree(A,B,C),k1_msualg_2(A,k11_msafree(A,B),C)) ) ) ) )).

fof(fraenkel_a_2_0_msscyc_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_msscyc_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_msualg_1(B))
            & A = D
            & k2_msualg_1(B,D) = C ) ) ) )).

fof(fraenkel_a_3_0_msscyc_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & v2_relat_1(C)
        & m1_pboole(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_msscyc_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_funct_1(u4_msualg_1(B,k11_msafree(B,C)),D))
            & A = E
            & k9_msafree2(B,C,D,E) = np__0 ) ) ) )).
%------------------------------------------------------------------------------