SET007 Axioms: SET007+516.ax


%------------------------------------------------------------------------------
% File     : SET007+516 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Birkhoff Theorem for Many Sorted Algebras
% Version  : [Urb08] axioms.
% English  :

% 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    : birkhoff [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   15 (   0 unit)
%            Number of atoms       :  329 (   7 equality)
%            Maximal formula depth :   25 (  16 average)
%            Number of connectives :  346 (  32 ~  ;   0  |; 169  &)
%                                         (   4 <=>; 141 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   42 (   0 propositional; 1-4 arity)
%            Number of functors    :   55 (  38 constant; 0-6 arity)
%            Number of variables   :  116 (   0 singleton; 109 !;   7 ?)
%            Maximal term depth    :    7 (   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(d1_birkhoff,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] :
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( m3_pboole(D,u1_struct_0(A),B,u4_msualg_1(A,C))
                 => ! [E] :
                      ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C))
                     => ( E = k1_birkhoff(A,B,C,D)
                      <=> ( r1_msualg_3(A,k11_msafree(A,B),C,E)
                          & r6_pboole(u1_struct_0(A),k1_msafree(u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C),k13_msafree(A,B),E),k3_msualg_3(u1_struct_0(A),k13_msafree(A,B),B,u4_msualg_1(A,C),k15_msafree(A,B),D)) ) ) ) ) ) ) ) )).

fof(t1_birkhoff,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_pboole(C,u1_struct_0(A)) )
             => ! [D] :
                  ( m3_pboole(D,u1_struct_0(A),C,u4_msualg_1(A,B))
                 => r2_pboole(u1_struct_0(A),k2_mssubfam(u1_struct_0(A),D),k2_mssubfam(u1_struct_0(A),k1_birkhoff(A,C,B,D))) ) ) ) ) )).

fof(s1_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s1_birkhoff)
            & l3_msualg_1(A,f1_s1_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s1_birkhoff)
                & l3_msualg_1(B,f1_s1_birkhoff) )
             => ( ( r6_msualg_3(f1_s1_birkhoff,A,B)
                  & p1_s1_birkhoff(A) )
               => p1_s1_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s1_birkhoff)
            & l3_msualg_1(A,f1_s1_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s1_birkhoff)
                & v5_msualg_1(B,f1_s1_birkhoff)
                & m1_msualg_2(B,f1_s1_birkhoff,A) )
             => ( p1_s1_birkhoff(A)
               => p1_s1_birkhoff(B) ) ) )
      & ! [A,B] :
          ( m2_pralg_2(B,A,f1_s1_birkhoff)
         => ( ! [C] :
                ~ ( r2_hidden(C,A)
                  & ! [D] :
                      ( l3_msualg_1(D,f1_s1_birkhoff)
                     => ~ ( D = k1_funct_1(B,C)
                          & p1_s1_birkhoff(D) ) ) )
           => p1_s1_birkhoff(k15_pralg_2(A,f1_s1_birkhoff,B)) ) ) )
   => ? [A] :
        ( v4_msualg_1(A,f1_s1_birkhoff)
        & v5_msualg_1(A,f1_s1_birkhoff)
        & l3_msualg_1(A,f1_s1_birkhoff)
        & ? [B] :
            ( m3_pboole(B,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A))
            & p1_s1_birkhoff(A)
            & r2_msualg_3(f1_s1_birkhoff,f2_s1_birkhoff,A,B)
            & ! [C] :
                ( ( v5_msualg_1(C,f1_s1_birkhoff)
                  & l3_msualg_1(C,f1_s1_birkhoff) )
               => ! [D] :
                    ( m3_pboole(D,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,C))
                   => ~ ( r1_msualg_3(f1_s1_birkhoff,f2_s1_birkhoff,C,D)
                        & p1_s1_birkhoff(C)
                        & ! [E] :
                            ( m3_pboole(E,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C))
                           => ~ ( r1_msualg_3(f1_s1_birkhoff,A,C,E)
                                & r6_pboole(u1_struct_0(f1_s1_birkhoff),k3_msualg_3(u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C),B,E),D)
                                & ! [F] :
                                    ( m3_pboole(F,u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C))
                                   => ( r6_pboole(u1_struct_0(f1_s1_birkhoff),k3_msualg_3(u1_struct_0(f1_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,f2_s1_birkhoff),u4_msualg_1(f1_s1_birkhoff,A),u4_msualg_1(f1_s1_birkhoff,C),B,F),D)
                                     => r6_pboole(u1_struct_0(f1_s1_birkhoff),E,F) ) ) ) ) ) ) ) ) ) )).

fof(s2_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s2_birkhoff)
            & l3_msualg_1(A,f1_s2_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s2_birkhoff)
                & l3_msualg_1(B,f1_s2_birkhoff) )
             => ( ( r6_msualg_3(f1_s2_birkhoff,A,B)
                  & p1_s2_birkhoff(A) )
               => p1_s2_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s2_birkhoff)
            & l3_msualg_1(A,f1_s2_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s2_birkhoff)
                & v5_msualg_1(B,f1_s2_birkhoff)
                & m1_msualg_2(B,f1_s2_birkhoff,A) )
             => ( p1_s2_birkhoff(A)
               => p1_s2_birkhoff(B) ) ) )
      & ! [A,B] :
          ( m2_pralg_2(B,A,f1_s2_birkhoff)
         => ( ! [C] :
                ~ ( r2_hidden(C,A)
                  & ! [D] :
                      ( l3_msualg_1(D,f1_s2_birkhoff)
                     => ~ ( D = k1_funct_1(B,C)
                          & p1_s2_birkhoff(D) ) ) )
           => p1_s2_birkhoff(k15_pralg_2(A,f1_s2_birkhoff,B)) ) ) )
   => ? [A] :
        ( v4_msualg_1(A,f1_s2_birkhoff)
        & v5_msualg_1(A,f1_s2_birkhoff)
        & l3_msualg_1(A,f1_s2_birkhoff)
        & ? [B] :
            ( m3_pboole(B,u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,A))
            & p1_s2_birkhoff(A)
            & ! [C] :
                ( ( v5_msualg_1(C,f1_s2_birkhoff)
                  & l3_msualg_1(C,f1_s2_birkhoff) )
               => ! [D] :
                    ( m3_pboole(D,u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,C))
                   => ~ ( p1_s2_birkhoff(C)
                        & ! [E] :
                            ( m3_pboole(E,u1_struct_0(f1_s2_birkhoff),u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C))
                           => ~ ( r1_msualg_3(f1_s2_birkhoff,A,C,E)
                                & r6_pboole(u1_struct_0(f1_s2_birkhoff),k3_msualg_3(u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C),B,E),D)
                                & ! [F] :
                                    ( m3_pboole(F,u1_struct_0(f1_s2_birkhoff),u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C))
                                   => ( ( r1_msualg_3(f1_s2_birkhoff,A,C,F)
                                        & r6_pboole(u1_struct_0(f1_s2_birkhoff),k3_msualg_3(u1_struct_0(f1_s2_birkhoff),f2_s2_birkhoff,u4_msualg_1(f1_s2_birkhoff,A),u4_msualg_1(f1_s2_birkhoff,C),B,F),D) )
                                     => r6_pboole(u1_struct_0(f1_s2_birkhoff),E,F) ) ) ) ) ) ) ) ) ) )).

fof(s3_birkhoff,axiom,
    ( ( p1_s3_birkhoff(f3_s3_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s3_birkhoff)
            & l3_msualg_1(A,f1_s3_birkhoff) )
         => ! [B] :
              ( m3_pboole(B,u1_struct_0(f1_s3_birkhoff),k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),u4_msualg_1(f1_s3_birkhoff,A))
             => ~ ( p1_s3_birkhoff(A)
                  & ! [C] :
                      ( m3_pboole(C,u1_struct_0(f1_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,A))
                     => ~ ( r1_msualg_3(f1_s3_birkhoff,f2_s3_birkhoff,A,C)
                          & r6_pboole(u1_struct_0(f1_s3_birkhoff),B,k3_msualg_3(u1_struct_0(f1_s3_birkhoff),k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,A),f4_s3_birkhoff,C)) ) ) ) ) ) )
   => ? [A] :
        ( m3_pboole(A,u1_struct_0(f1_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f3_s3_birkhoff))
        & r1_msualg_3(f1_s3_birkhoff,f2_s3_birkhoff,f3_s3_birkhoff,A)
        & r6_pboole(u1_struct_0(f1_s3_birkhoff),k1_birkhoff(f1_s3_birkhoff,k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),f3_s3_birkhoff,f5_s3_birkhoff),k3_msualg_3(u1_struct_0(f1_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,k11_msafree(f1_s3_birkhoff,k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers))),u4_msualg_1(f1_s3_birkhoff,f2_s3_birkhoff),u4_msualg_1(f1_s3_birkhoff,f3_s3_birkhoff),k1_birkhoff(f1_s3_birkhoff,k2_pre_circ(u1_struct_0(f1_s3_birkhoff),k5_numbers),f2_s3_birkhoff,f4_s3_birkhoff),A)) ) )).

fof(s4_birkhoff,axiom,
    ( ( k1_funct_1(k1_msualg_3(u1_struct_0(f1_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,k11_msafree(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers))),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),k1_birkhoff(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),f2_s4_birkhoff,f3_s4_birkhoff),f4_s4_birkhoff),f5_s4_birkhoff) = k1_funct_1(k1_msualg_3(u1_struct_0(f1_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,k11_msafree(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers))),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),k1_birkhoff(f1_s4_birkhoff,k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),f2_s4_birkhoff,f3_s4_birkhoff),f4_s4_birkhoff),f6_s4_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s4_birkhoff)
            & l3_msualg_1(A,f1_s4_birkhoff) )
         => ! [B] :
              ( m3_pboole(B,u1_struct_0(f1_s4_birkhoff),k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),u4_msualg_1(f1_s4_birkhoff,A))
             => ~ ( p1_s4_birkhoff(A)
                  & ! [C] :
                      ( m3_pboole(C,u1_struct_0(f1_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,A))
                     => ~ ( r1_msualg_3(f1_s4_birkhoff,f2_s4_birkhoff,A,C)
                          & r6_pboole(u1_struct_0(f1_s4_birkhoff),B,k3_msualg_3(u1_struct_0(f1_s4_birkhoff),k2_pre_circ(u1_struct_0(f1_s4_birkhoff),k5_numbers),u4_msualg_1(f1_s4_birkhoff,f2_s4_birkhoff),u4_msualg_1(f1_s4_birkhoff,A),f3_s4_birkhoff,C)) ) ) ) ) ) )
   => ! [A] :
        ( ( v5_msualg_1(A,f1_s4_birkhoff)
          & l3_msualg_1(A,f1_s4_birkhoff) )
       => ( p1_s4_birkhoff(A)
         => r1_equation(f1_s4_birkhoff,A,f4_s4_birkhoff,k5_equation(f1_s4_birkhoff,f4_s4_birkhoff,f5_s4_birkhoff,f6_s4_birkhoff)) ) ) )).

fof(s5_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s5_birkhoff)
            & l3_msualg_1(A,f1_s5_birkhoff) )
         => ! [B] :
              ( m3_pboole(B,u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,u4_msualg_1(f1_s5_birkhoff,A))
             => ~ ( p1_s5_birkhoff(A)
                  & ! [C] :
                      ( m3_pboole(C,u1_struct_0(f1_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A))
                     => ~ ( r1_msualg_3(f1_s5_birkhoff,f3_s5_birkhoff,A,C)
                          & r6_pboole(u1_struct_0(f1_s5_birkhoff),k3_msualg_3(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A),f4_s5_birkhoff,C),B)
                          & ! [D] :
                              ( m3_pboole(D,u1_struct_0(f1_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A))
                             => ( ( r1_msualg_3(f1_s5_birkhoff,f3_s5_birkhoff,A,D)
                                  & r6_pboole(u1_struct_0(f1_s5_birkhoff),k3_msualg_3(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,u4_msualg_1(f1_s5_birkhoff,f3_s5_birkhoff),u4_msualg_1(f1_s5_birkhoff,A),f4_s5_birkhoff,D),B) )
                               => r6_pboole(u1_struct_0(f1_s5_birkhoff),C,D) ) ) ) ) ) ) )
      & p1_s5_birkhoff(f3_s5_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s5_birkhoff)
            & l3_msualg_1(A,f1_s5_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s5_birkhoff)
                & v5_msualg_1(B,f1_s5_birkhoff)
                & m1_msualg_2(B,f1_s5_birkhoff,A) )
             => ( p1_s5_birkhoff(A)
               => p1_s5_birkhoff(B) ) ) ) )
   => ( v2_relat_1(k14_pboole(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,f4_s5_birkhoff))
      & m1_msafree(k14_pboole(u1_struct_0(f1_s5_birkhoff),f2_s5_birkhoff,f4_s5_birkhoff),f1_s5_birkhoff,f3_s5_birkhoff) ) )).

fof(s6_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s6_birkhoff)
            & l3_msualg_1(A,f1_s6_birkhoff) )
         => ! [B] :
              ( m3_pboole(B,u1_struct_0(f1_s6_birkhoff),k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),u4_msualg_1(f1_s6_birkhoff,A))
             => ~ ( p1_s6_birkhoff(A)
                  & ! [C] :
                      ( m3_pboole(C,u1_struct_0(f1_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A))
                     => ~ ( r1_msualg_3(f1_s6_birkhoff,f2_s6_birkhoff,A,C)
                          & r6_pboole(u1_struct_0(f1_s6_birkhoff),k3_msualg_3(u1_struct_0(f1_s6_birkhoff),k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A),f3_s6_birkhoff,C),B)
                          & ! [D] :
                              ( m3_pboole(D,u1_struct_0(f1_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A))
                             => ( ( r1_msualg_3(f1_s6_birkhoff,f2_s6_birkhoff,A,D)
                                  & r6_pboole(u1_struct_0(f1_s6_birkhoff),k3_msualg_3(u1_struct_0(f1_s6_birkhoff),k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),u4_msualg_1(f1_s6_birkhoff,f2_s6_birkhoff),u4_msualg_1(f1_s6_birkhoff,A),f3_s6_birkhoff,D),B) )
                               => r6_pboole(u1_struct_0(f1_s6_birkhoff),C,D) ) ) ) ) ) ) )
      & p1_s6_birkhoff(f2_s6_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s6_birkhoff)
            & l3_msualg_1(A,f1_s6_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s6_birkhoff)
                & v5_msualg_1(B,f1_s6_birkhoff)
                & m1_msualg_2(B,f1_s6_birkhoff,A) )
             => ( p1_s6_birkhoff(A)
               => p1_s6_birkhoff(B) ) ) ) )
   => r2_msualg_3(f1_s6_birkhoff,k11_msafree(f1_s6_birkhoff,k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers)),f2_s6_birkhoff,k1_birkhoff(f1_s6_birkhoff,k2_pre_circ(u1_struct_0(f1_s6_birkhoff),k5_numbers),f2_s6_birkhoff,f3_s6_birkhoff)) )).

fof(s7_birkhoff,axiom,
    ( ( p2_s7_birkhoff(f2_s7_birkhoff)
      & p1_s7_birkhoff(f3_s7_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s7_birkhoff)
            & l3_msualg_1(A,f1_s7_birkhoff) )
         => ! [B] :
              ( m3_pboole(B,u1_struct_0(f1_s7_birkhoff),k2_pre_circ(u1_struct_0(f1_s7_birkhoff),k5_numbers),u4_msualg_1(f1_s7_birkhoff,A))
             => ~ ( p2_s7_birkhoff(A)
                  & ! [C] :
                      ( m3_pboole(C,u1_struct_0(f1_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,f3_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,A))
                     => ~ ( r1_msualg_3(f1_s7_birkhoff,f3_s7_birkhoff,A,C)
                          & r6_pboole(u1_struct_0(f1_s7_birkhoff),B,k3_msualg_3(u1_struct_0(f1_s7_birkhoff),k2_pre_circ(u1_struct_0(f1_s7_birkhoff),k5_numbers),u4_msualg_1(f1_s7_birkhoff,f3_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,A),f4_s7_birkhoff,C)) ) ) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s7_birkhoff)
            & l3_msualg_1(A,f1_s7_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s7_birkhoff)
                & l3_msualg_1(B,f1_s7_birkhoff) )
             => ( ( r6_msualg_3(f1_s7_birkhoff,A,B)
                  & p1_s7_birkhoff(A) )
               => p1_s7_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s7_birkhoff)
            & l3_msualg_1(A,f1_s7_birkhoff) )
         => ! [B] :
              ( ( v3_msualg_4(B,f1_s7_birkhoff,A)
                & v4_msualg_4(B,f1_s7_birkhoff,A)
                & m1_msualg_4(B,u1_struct_0(f1_s7_birkhoff),u4_msualg_1(f1_s7_birkhoff,A),u4_msualg_1(f1_s7_birkhoff,A)) )
             => ( p1_s7_birkhoff(A)
               => p1_s7_birkhoff(k14_msualg_4(f1_s7_birkhoff,A,B)) ) ) ) )
   => p1_s7_birkhoff(f2_s7_birkhoff) )).

fof(s8_birkhoff,axiom,
    ( ( ? [A] :
          ( m3_pboole(A,u1_struct_0(f1_s8_birkhoff),u4_msualg_1(f1_s8_birkhoff,f2_s8_birkhoff),u4_msualg_1(f1_s8_birkhoff,f3_s8_birkhoff))
          & r2_msualg_3(f1_s8_birkhoff,f2_s8_birkhoff,f3_s8_birkhoff,A) )
      & p1_s8_birkhoff(f2_s8_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s8_birkhoff)
            & l3_msualg_1(A,f1_s8_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s8_birkhoff)
                & l3_msualg_1(B,f1_s8_birkhoff) )
             => ( ( r6_msualg_3(f1_s8_birkhoff,A,B)
                  & p1_s8_birkhoff(A) )
               => p1_s8_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s8_birkhoff)
            & l3_msualg_1(A,f1_s8_birkhoff) )
         => ! [B] :
              ( ( v3_msualg_4(B,f1_s8_birkhoff,A)
                & v4_msualg_4(B,f1_s8_birkhoff,A)
                & m1_msualg_4(B,u1_struct_0(f1_s8_birkhoff),u4_msualg_1(f1_s8_birkhoff,A),u4_msualg_1(f1_s8_birkhoff,A)) )
             => ( p1_s8_birkhoff(A)
               => p1_s8_birkhoff(k14_msualg_4(f1_s8_birkhoff,A,B)) ) ) ) )
   => p1_s8_birkhoff(f3_s8_birkhoff) )).

fof(s9_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v4_msualg_1(A,f1_s9_birkhoff)
            & v5_msualg_1(A,f1_s9_birkhoff)
            & v3_msafree2(A,f1_s9_birkhoff)
            & m1_msualg_2(A,f1_s9_birkhoff,f2_s9_birkhoff) )
         => p1_s9_birkhoff(A) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s9_birkhoff)
            & l3_msualg_1(A,f1_s9_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s9_birkhoff)
                & l3_msualg_1(B,f1_s9_birkhoff) )
             => ( ( r6_msualg_3(f1_s9_birkhoff,A,B)
                  & p1_s9_birkhoff(A) )
               => p1_s9_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s9_birkhoff)
            & l3_msualg_1(A,f1_s9_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s9_birkhoff)
                & v5_msualg_1(B,f1_s9_birkhoff)
                & m1_msualg_2(B,f1_s9_birkhoff,A) )
             => ( p1_s9_birkhoff(A)
               => p1_s9_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s9_birkhoff)
            & l3_msualg_1(A,f1_s9_birkhoff) )
         => ! [B] :
              ( ( v3_msualg_4(B,f1_s9_birkhoff,A)
                & v4_msualg_4(B,f1_s9_birkhoff,A)
                & m1_msualg_4(B,u1_struct_0(f1_s9_birkhoff),u4_msualg_1(f1_s9_birkhoff,A),u4_msualg_1(f1_s9_birkhoff,A)) )
             => ( p1_s9_birkhoff(A)
               => p1_s9_birkhoff(k14_msualg_4(f1_s9_birkhoff,A,B)) ) ) )
      & ! [A,B] :
          ( m2_pralg_2(B,A,f1_s9_birkhoff)
         => ( ! [C] :
                ~ ( r2_hidden(C,A)
                  & ! [D] :
                      ( l3_msualg_1(D,f1_s9_birkhoff)
                     => ~ ( D = k1_funct_1(B,C)
                          & p1_s9_birkhoff(D) ) ) )
           => p1_s9_birkhoff(k15_pralg_2(A,f1_s9_birkhoff,B)) ) ) )
   => p1_s9_birkhoff(f2_s9_birkhoff) )).

fof(s10_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s10_birkhoff)
            & l3_msualg_1(A,f1_s10_birkhoff) )
         => ( p2_s10_birkhoff(A)
          <=> ! [B] :
                ( m1_subset_1(B,u1_struct_0(f1_s10_birkhoff))
               => ! [C] :
                    ( m1_subset_1(C,k1_funct_1(k4_equation(f1_s10_birkhoff),B))
                   => ( ! [D] :
                          ( ( v5_msualg_1(D,f1_s10_birkhoff)
                            & l3_msualg_1(D,f1_s10_birkhoff) )
                         => ( p1_s10_birkhoff(D)
                           => r1_equation(f1_s10_birkhoff,D,B,C) ) )
                     => r1_equation(f1_s10_birkhoff,A,B,C) ) ) ) ) )
      & p1_s10_birkhoff(f2_s10_birkhoff) )
   => p2_s10_birkhoff(f2_s10_birkhoff) )).

fof(s11_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s11_birkhoff)
            & l3_msualg_1(A,f1_s11_birkhoff) )
         => ( p2_s11_birkhoff(A)
          <=> ! [B] :
                ( m1_subset_1(B,u1_struct_0(f1_s11_birkhoff))
               => ! [C] :
                    ( m1_subset_1(C,k1_funct_1(k4_equation(f1_s11_birkhoff),B))
                   => ( ! [D] :
                          ( ( v5_msualg_1(D,f1_s11_birkhoff)
                            & l3_msualg_1(D,f1_s11_birkhoff) )
                         => ( p1_s11_birkhoff(D)
                           => r1_equation(f1_s11_birkhoff,D,B,C) ) )
                     => r1_equation(f1_s11_birkhoff,A,B,C) ) ) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s11_birkhoff)
            & l3_msualg_1(A,f1_s11_birkhoff) )
         => ! [B] :
              ( m3_pboole(B,u1_struct_0(f1_s11_birkhoff),k2_pre_circ(u1_struct_0(f1_s11_birkhoff),k5_numbers),u4_msualg_1(f1_s11_birkhoff,A))
             => ~ ( p2_s11_birkhoff(A)
                  & ! [C] :
                      ( m3_pboole(C,u1_struct_0(f1_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A))
                     => ~ ( r1_msualg_3(f1_s11_birkhoff,f2_s11_birkhoff,A,C)
                          & r6_pboole(u1_struct_0(f1_s11_birkhoff),k3_msualg_3(u1_struct_0(f1_s11_birkhoff),k2_pre_circ(u1_struct_0(f1_s11_birkhoff),k5_numbers),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A),f3_s11_birkhoff,C),B)
                          & ! [D] :
                              ( m3_pboole(D,u1_struct_0(f1_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A))
                             => ( ( r1_msualg_3(f1_s11_birkhoff,f2_s11_birkhoff,A,D)
                                  & r6_pboole(u1_struct_0(f1_s11_birkhoff),k3_msualg_3(u1_struct_0(f1_s11_birkhoff),k2_pre_circ(u1_struct_0(f1_s11_birkhoff),k5_numbers),u4_msualg_1(f1_s11_birkhoff,f2_s11_birkhoff),u4_msualg_1(f1_s11_birkhoff,A),f3_s11_birkhoff,D),B) )
                               => r6_pboole(u1_struct_0(f1_s11_birkhoff),C,D) ) ) ) ) ) ) )
      & p2_s11_birkhoff(f2_s11_birkhoff)
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s11_birkhoff)
            & l3_msualg_1(A,f1_s11_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s11_birkhoff)
                & l3_msualg_1(B,f1_s11_birkhoff) )
             => ( ( r6_msualg_3(f1_s11_birkhoff,A,B)
                  & p1_s11_birkhoff(A) )
               => p1_s11_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s11_birkhoff)
            & l3_msualg_1(A,f1_s11_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s11_birkhoff)
                & v5_msualg_1(B,f1_s11_birkhoff)
                & m1_msualg_2(B,f1_s11_birkhoff,A) )
             => ( p1_s11_birkhoff(A)
               => p1_s11_birkhoff(B) ) ) )
      & ! [A,B] :
          ( m2_pralg_2(B,A,f1_s11_birkhoff)
         => ( ! [C] :
                ~ ( r2_hidden(C,A)
                  & ! [D] :
                      ( l3_msualg_1(D,f1_s11_birkhoff)
                     => ~ ( D = k1_funct_1(B,C)
                          & p1_s11_birkhoff(D) ) ) )
           => p1_s11_birkhoff(k15_pralg_2(A,f1_s11_birkhoff,B)) ) ) )
   => p1_s11_birkhoff(f2_s11_birkhoff) )).

fof(s12_birkhoff,axiom,
    ( ( ! [A] :
          ( ( v5_msualg_1(A,f1_s12_birkhoff)
            & l3_msualg_1(A,f1_s12_birkhoff) )
         => ! [B] :
              ( ( v5_msualg_1(B,f1_s12_birkhoff)
                & l3_msualg_1(B,f1_s12_birkhoff) )
             => ( ( r6_msualg_3(f1_s12_birkhoff,A,B)
                  & p1_s12_birkhoff(A) )
               => p1_s12_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s12_birkhoff)
            & l3_msualg_1(A,f1_s12_birkhoff) )
         => ! [B] :
              ( ( v4_msualg_1(B,f1_s12_birkhoff)
                & v5_msualg_1(B,f1_s12_birkhoff)
                & m1_msualg_2(B,f1_s12_birkhoff,A) )
             => ( p1_s12_birkhoff(A)
               => p1_s12_birkhoff(B) ) ) )
      & ! [A] :
          ( ( v5_msualg_1(A,f1_s12_birkhoff)
            & l3_msualg_1(A,f1_s12_birkhoff) )
         => ! [B] :
              ( ( v3_msualg_4(B,f1_s12_birkhoff,A)
                & v4_msualg_4(B,f1_s12_birkhoff,A)
                & m1_msualg_4(B,u1_struct_0(f1_s12_birkhoff),u4_msualg_1(f1_s12_birkhoff,A),u4_msualg_1(f1_s12_birkhoff,A)) )
             => ( p1_s12_birkhoff(A)
               => p1_s12_birkhoff(k14_msualg_4(f1_s12_birkhoff,A,B)) ) ) )
      & ! [A,B] :
          ( m2_pralg_2(B,A,f1_s12_birkhoff)
         => ( ! [C] :
                ~ ( r2_hidden(C,A)
                  & ! [D] :
                      ( l3_msualg_1(D,f1_s12_birkhoff)
                     => ~ ( D = k1_funct_1(B,C)
                          & p1_s12_birkhoff(D) ) ) )
           => p1_s12_birkhoff(k15_pralg_2(A,f1_s12_birkhoff,B)) ) ) )
   => ? [A] :
        ( m4_pboole(A,u1_struct_0(f1_s12_birkhoff),k4_equation(f1_s12_birkhoff))
        & ! [B] :
            ( ( v5_msualg_1(B,f1_s12_birkhoff)
              & l3_msualg_1(B,f1_s12_birkhoff) )
           => ( p1_s12_birkhoff(B)
            <=> r2_equation(f1_s12_birkhoff,B,A) ) ) ) )).

fof(dt_k1_birkhoff,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & v5_msualg_1(C,A)
        & l3_msualg_1(C,A)
        & m3_pboole(D,u1_struct_0(A),B,u4_msualg_1(A,C)) )
     => m3_pboole(k1_birkhoff(A,B,C,D),u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,B)),u4_msualg_1(A,C)) ) )).
%------------------------------------------------------------------------------