SET007 Axioms: SET007+844.ax
%------------------------------------------------------------------------------
% File : SET007+844 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Operation of Addition of Relational Structures
% 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 : latsum_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 30 ( 0 unt; 0 def)
% Number of atoms : 304 ( 12 equ)
% Maximal formula atoms : 25 ( 10 avg)
% Number of connectives : 311 ( 37 ~; 1 |; 162 &)
% ( 7 <=>; 104 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 11 ( 11 usr; 0 con; 1-6 aty)
% Number of variables : 107 ( 106 !; 1 ?)
% 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(fc1_latsum_1,axiom,
! [A,B] :
( ( l1_orders_2(A)
& ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ( ~ v3_struct_0(k1_latsum_1(A,B))
& v1_orders_2(k1_latsum_1(A,B)) ) ) ).
fof(fc2_latsum_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& l1_orders_2(B) )
=> ( ~ v3_struct_0(k1_latsum_1(A,B))
& v1_orders_2(k1_latsum_1(A,B)) ) ) ).
fof(t1_latsum_1,axiom,
! [A,B,C,D] :
~ ( r2_hidden(A,k2_xboole_0(C,D))
& r2_hidden(B,k2_xboole_0(C,D))
& ~ ( r2_hidden(A,k4_xboole_0(C,D))
& r2_hidden(B,k4_xboole_0(C,D)) )
& ~ ( r2_hidden(A,D)
& r2_hidden(B,D) )
& ~ ( r2_hidden(A,k4_xboole_0(C,D))
& r2_hidden(B,D) )
& ~ ( r2_hidden(A,D)
& r2_hidden(B,k4_xboole_0(C,D)) ) ) ).
fof(d1_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( r1_latsum_1(A,B)
<=> ! [C,D] :
( ( r2_hidden(C,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
& r2_hidden(D,k3_xboole_0(u1_struct_0(A),u1_struct_0(B))) )
=> ( r2_hidden(k4_tarski(C,D),u1_orders_2(A))
<=> r2_hidden(k4_tarski(C,D),u1_orders_2(B)) ) ) ) ) ) ).
fof(d2_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ( C = k1_latsum_1(A,B)
<=> ( u1_struct_0(C) = k2_xboole_0(u1_struct_0(A),u1_struct_0(B))
& u1_orders_2(C) = k2_xboole_0(k2_xboole_0(u1_orders_2(A),u1_orders_2(B)),k7_relset_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(B),u1_struct_0(B),u1_orders_2(A),u1_orders_2(B))) ) ) ) ) ) ).
fof(t2_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( r1_tarski(u1_orders_2(A),u1_orders_2(k1_latsum_1(A,B)))
& r1_tarski(u1_orders_2(B),u1_orders_2(k1_latsum_1(A,B))) ) ) ) ).
fof(t3_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( ( v2_orders_2(A)
& v2_orders_2(B) )
=> v2_orders_2(k1_latsum_1(A,B)) ) ) ) ).
fof(t4_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C,D] :
( ( r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
& r2_hidden(C,u1_struct_0(A))
& r2_hidden(D,u1_struct_0(A))
& r1_latsum_1(A,B)
& v3_orders_2(A) )
=> r2_hidden(k4_tarski(C,D),u1_orders_2(A)) ) ) ) ).
fof(t5_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C,D] :
( ( r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
& r2_hidden(C,u1_struct_0(B))
& r2_hidden(D,u1_struct_0(B))
& r1_latsum_1(A,B)
& v3_orders_2(B) )
=> r2_hidden(k4_tarski(C,D),u1_orders_2(B)) ) ) ) ).
fof(t6_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C,D] :
( ( r2_hidden(k4_tarski(C,D),u1_orders_2(A))
=> r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B))) )
& ( r2_hidden(k4_tarski(C,D),u1_orders_2(B))
=> r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B))) ) ) ) ) ).
fof(t7_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
=> ( r2_hidden(C,u1_struct_0(A))
| r2_hidden(C,k4_xboole_0(u1_struct_0(B),u1_struct_0(A))) ) ) ) ) ).
fof(t8_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k1_latsum_1(A,B)))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(k1_latsum_1(A,B)))
=> ( ( C = E
& D = F
& r1_latsum_1(A,B)
& v3_orders_2(A) )
=> ( r1_orders_2(A,C,D)
<=> r1_orders_2(k1_latsum_1(A,B),E,F) ) ) ) ) ) ) ) ) ).
fof(t9_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( C = E
& D = F
& r1_latsum_1(A,B)
& v3_orders_2(B) )
=> ( r1_orders_2(k1_latsum_1(A,B),C,D)
<=> r1_orders_2(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t10_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& l1_orders_2(B) )
=> ! [C] :
( r2_hidden(C,u1_struct_0(A))
=> m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B))) ) ) ) ).
fof(t11_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& l1_orders_2(B) )
=> ! [C] :
( r2_hidden(C,u1_struct_0(B))
=> m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B))) ) ) ) ).
fof(t12_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( r2_hidden(C,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
=> m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(t13_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( r2_hidden(C,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
=> m1_subset_1(C,u1_struct_0(B)) ) ) ) ).
fof(t14_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
=> ( ( r2_hidden(C,u1_struct_0(A))
& r2_hidden(D,u1_struct_0(B))
& r1_latsum_1(A,B) )
=> ( r1_orders_2(k1_latsum_1(A,B),C,D)
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k1_latsum_1(A,B)))
& r2_hidden(E,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
& r1_orders_2(k1_latsum_1(A,B),C,E)
& r1_orders_2(k1_latsum_1(A,B),E,D) ) ) ) ) ) ) ) ).
fof(t15_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( C = E
& D = F
& r1_latsum_1(A,B)
& v3_orders_2(A)
& v3_orders_2(B) )
=> ( r1_orders_2(A,C,D)
<=> r1_orders_2(B,E,F) ) ) ) ) ) ) ) ) ).
fof(t16_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( v1_waybel_0(B,A)
& v12_waybel_0(B,A)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k13_lattice3(A,C,D),B) ) ) ) ) ) ).
fof(t17_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C,D] :
( ( v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
& r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
& r2_hidden(C,u1_struct_0(B)) )
=> r2_hidden(D,u1_struct_0(B)) ) ) ) ).
fof(t18_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
=> ( ( v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
& r1_orders_2(k1_latsum_1(A,B),C,D)
& r2_hidden(C,u1_struct_0(B)) )
=> r2_hidden(D,u1_struct_0(B)) ) ) ) ) ) ).
fof(t19_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& l1_orders_2(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( v1_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
& v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
& r1_latsum_1(A,B)
& C = E
& D = F )
=> k13_lattice3(A,C,D) = k13_lattice3(B,E,F) ) ) ) ) ) ) ) ).
fof(t20_latsum_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_yellow_0(A)
& v1_lattice3(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_yellow_0(B)
& v1_lattice3(B)
& l1_orders_2(B) )
=> ( ( ~ v1_xboole_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
& v1_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B))) )
=> r2_hidden(k3_yellow_0(B),u1_struct_0(A)) ) ) ) ).
fof(t21_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C,D] :
( ( v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
& r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
& r2_hidden(D,u1_struct_0(A)) )
=> r2_hidden(C,u1_struct_0(A)) ) ) ) ).
fof(t22_latsum_1,axiom,
! [A,B,C] :
( l1_orders_2(C)
=> ! [D] :
( l1_orders_2(D)
=> ~ ( r2_hidden(k4_tarski(A,B),u1_orders_2(k1_latsum_1(C,D)))
& v13_waybel_0(k3_xboole_0(u1_struct_0(C),u1_struct_0(D)),C)
& m1_subset_1(k3_xboole_0(u1_struct_0(C),u1_struct_0(D)),k1_zfmisc_1(u1_struct_0(C)))
& ~ ( r2_hidden(A,u1_struct_0(C))
& r2_hidden(B,u1_struct_0(C)) )
& ~ ( r2_hidden(A,u1_struct_0(D))
& r2_hidden(B,u1_struct_0(D)) )
& ~ ( r2_hidden(A,k4_xboole_0(u1_struct_0(C),u1_struct_0(D)))
& r2_hidden(B,k4_xboole_0(u1_struct_0(D),u1_struct_0(C))) ) ) ) ) ).
fof(t23_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
=> ( ( v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
& r1_orders_2(k1_latsum_1(A,B),C,D)
& r2_hidden(D,u1_struct_0(A)) )
=> r2_hidden(C,u1_struct_0(A)) ) ) ) ) ) ).
fof(t24_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( ( r1_latsum_1(A,B)
& v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
& v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
& v3_orders_2(A)
& v4_orders_2(A)
& v3_orders_2(B)
& v4_orders_2(B) )
=> v4_orders_2(k1_latsum_1(A,B)) ) ) ) ).
fof(t25_latsum_1,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( ( v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
& v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
& m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
& r1_latsum_1(A,B)
& v3_orders_2(A)
& v3_orders_2(B) )
=> v3_orders_2(k1_latsum_1(A,B)) ) ) ) ).
fof(dt_k1_latsum_1,axiom,
! [A,B] :
( ( l1_orders_2(A)
& l1_orders_2(B) )
=> ( v1_orders_2(k1_latsum_1(A,B))
& l1_orders_2(k1_latsum_1(A,B)) ) ) ).
%------------------------------------------------------------------------------