SET007 Axioms: SET007+352.ax
%------------------------------------------------------------------------------
% File : SET007+352 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Go-Board Theorem
% 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 : goboard4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 9 ( 2 unt; 0 def)
% Number of atoms : 113 ( 2 equ)
% Maximal formula atoms : 19 ( 12 avg)
% Number of connectives : 115 ( 11 ~; 0 |; 73 &)
% ( 1 <=>; 30 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 13 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 4 con; 0-4 aty)
% Number of variables : 27 ( 27 !; 0 ?)
% 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(t1_goboard4,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& v5_goboard1(A)
& v6_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__1,k3_finseq_1(B))
& r1_xreal_0(np__1,k3_finseq_1(C))
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_matrix_1(A))))
& r1_xboole_0(k2_relat_1(B),k2_relat_1(C)) ) ) ) ) ).
fof(t2_goboard4,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& v5_goboard1(A)
& v6_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& r1_xreal_0(np__2,k3_finseq_1(C))
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_matrix_1(A))))
& r1_xboole_0(k5_topreal1(np__2,B),k5_topreal1(np__2,C)) ) ) ) ) ).
fof(t3_goboard4,axiom,
! [A] :
( ( ~ v3_relat_1(A)
& v1_matrix_1(A)
& v3_goboard1(A)
& v4_goboard1(A)
& v5_goboard1(A)
& v6_goboard1(A)
& m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
& r1_xreal_0(np__2,k3_finseq_1(C))
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
& r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,np__1),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k3_finseq_1(B)),k2_relat_1(k7_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,np__1),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,np__1)))
& r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)),k2_relat_1(k8_matrix_1(u1_struct_0(k15_euclid(np__2)),A,k1_matrix_1(A))))
& r1_xboole_0(k5_topreal1(np__2,B),k5_topreal1(np__2,C)) ) ) ) ) ).
fof(d1_goboard4,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r1_goboard4(A,B,C)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(A))
=> ( r1_xreal_0(B,k1_goboard1(A,D))
& r1_xreal_0(k1_goboard1(A,D),C) ) ) ) ) ) ) ) ).
fof(t4_goboard4,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_xreal_0(np__2,k3_finseq_1(A))
& r1_xreal_0(np__2,k3_finseq_1(B))
& v1_topreal1(A)
& v1_topreal1(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(A))
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(A))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(C,np__1)) ) )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& r2_hidden(k1_nat_1(C,np__1),k4_finseq_1(B))
& k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,C) = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),B,k1_nat_1(C,np__1)) ) )
& r1_goboard4(k2_goboard1(A),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k2_goboard1(B),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k3_goboard1(A),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
& r1_goboard4(k3_goboard1(B),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
& r1_xboole_0(k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) ).
fof(t5_goboard4,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( v2_funct_1(A)
& v1_topreal1(A)
& r1_xreal_0(np__2,k3_finseq_1(A))
& v2_funct_1(B)
& v1_topreal1(B)
& r1_xreal_0(np__2,k3_finseq_1(B))
& r1_goboard4(k2_goboard1(A),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k2_goboard1(B),k1_goboard1(k2_goboard1(A),np__1),k1_goboard1(k2_goboard1(A),k3_finseq_1(A)))
& r1_goboard4(k3_goboard1(A),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
& r1_goboard4(k3_goboard1(B),k1_goboard1(k3_goboard1(B),np__1),k1_goboard1(k3_goboard1(B),k3_finseq_1(B)))
& r1_xboole_0(k5_topreal1(np__2,A),k5_topreal1(np__2,B)) ) ) ) ).
fof(t6_goboard4,axiom,
$true ).
fof(t7_goboard4,axiom,
$true ).
fof(t8_goboard4,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r1_topreal4(A,C,E)
& r1_topreal4(B,D,F)
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(G,k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,B))
=> ( r1_xreal_0(k21_euclid(C),k21_euclid(G))
& r1_xreal_0(k21_euclid(G),k21_euclid(E)) ) ) )
& ! [G] :
( m1_subset_1(G,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(G,k4_subset_1(u1_struct_0(k15_euclid(np__2)),A,B))
=> ( r1_xreal_0(k22_euclid(D),k22_euclid(G))
& r1_xreal_0(k22_euclid(G),k22_euclid(F)) ) ) )
& r1_xboole_0(A,B) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------