TPTP Problem File: SEU413+1.p
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%------------------------------------------------------------------------------
% File : SEU413+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Problem : The Operation of Addition of Relational Structures T23
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [RG04] Romanowicz & Grabowski (2004), The Operation of Additi
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t23_latsum_1 [Urb08]
% Status : Theorem
% Rating : 0.19 v8.1.0, 0.14 v7.5.0, 0.16 v7.4.0, 0.13 v7.3.0, 0.17 v7.2.0, 0.14 v7.1.0, 0.17 v7.0.0, 0.13 v6.4.0, 0.19 v6.3.0, 0.21 v6.2.0, 0.24 v6.1.0, 0.20 v6.0.0, 0.17 v5.5.0, 0.22 v5.4.0, 0.29 v5.3.0, 0.33 v5.2.0, 0.20 v5.1.0, 0.19 v5.0.0, 0.25 v4.1.0, 0.30 v4.0.1, 0.35 v4.0.0, 0.38 v3.7.0, 0.40 v3.5.0, 0.42 v3.4.0
% Syntax : Number of formulae : 42 ( 21 unt; 0 def)
% Number of atoms : 88 ( 9 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 52 ( 6 ~; 1 |; 16 &)
% ( 3 <=>; 26 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 1 con; 0-2 aty)
% Number of variables : 69 ( 62 !; 7 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Normal version: includes the axioms (which may be theorems from
% other articles) and background that are possibly necessary.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t23_latsum_1,conjecture,
! [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(abstractness_v1_orders_2,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_2(A)
=> A = g1_orders_2(u1_struct_0(A),u1_orders_2(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_relset_1,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> v1_relat_1(C) ) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ).
fof(d9_orders_2,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_orders_2(A,B,C)
<=> r2_hidden(k4_tarski(B,C),u1_orders_2(A)) ) ) ) ) ).
fof(dt_g1_orders_2,axiom,
! [A,B] :
( m1_relset_1(B,A,A)
=> ( v1_orders_2(g1_orders_2(A,B))
& l1_orders_2(g1_orders_2(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)) ) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_xboole_0,axiom,
$true ).
fof(dt_k4_tarski,axiom,
$true ).
fof(dt_l1_orders_2,axiom,
! [A] :
( l1_orders_2(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(dt_u1_orders_2,axiom,
! [A] :
( l1_orders_2(A)
=> m2_relset_1(u1_orders_2(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_orders_2,axiom,
? [A] : l1_orders_2(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_m1_relset_1,axiom,
! [A,B] :
? [C] : m1_relset_1(C,A,B) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : m1_subset_1(B,A) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(free_g1_orders_2,axiom,
! [A,B] :
( m1_relset_1(B,A,A)
=> ! [C,D] :
( g1_orders_2(A,B) = g1_orders_2(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : k3_xboole_0(A,A) = A ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
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(t2_boole,axiom,
! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ).
fof(t2_subset,axiom,
! [A,B] :
( m1_subset_1(A,B)
=> ( v1_xboole_0(B)
| r2_hidden(A,B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( m1_subset_1(A,k1_zfmisc_1(B))
<=> r1_tarski(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C)) )
=> m1_subset_1(A,C) ) ).
fof(t5_subset,axiom,
! [A,B,C] :
~ ( r2_hidden(A,B)
& m1_subset_1(B,k1_zfmisc_1(C))
& v1_xboole_0(C) ) ).
fof(t6_boole,axiom,
! [A] :
( v1_xboole_0(A)
=> A = k1_xboole_0 ) ).
fof(t7_boole,axiom,
! [A,B] :
~ ( r2_hidden(A,B)
& v1_xboole_0(B) ) ).
fof(t8_boole,axiom,
! [A,B] :
~ ( v1_xboole_0(A)
& A != B
& v1_xboole_0(B) ) ).
%------------------------------------------------------------------------------