SET007 Axioms: SET007+516.ax
%------------------------------------------------------------------------------
% File : SET007+516 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 329 ( 7 equ)
% Maximal formula atoms : 43 ( 21 avg)
% Number of connectives : 346 ( 32 ~; 0 |; 169 &)
% ( 4 <=>; 141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 16 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 42 ( 41 usr; 0 prp; 1-4 aty)
% Number of functors : 55 ( 55 usr; 38 con; 0-6 aty)
% Number of variables : 116 ( 109 !; 7 ?)
% 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)) ) ).
%------------------------------------------------------------------------------