TPTP Problem File: LAT345+1.p
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%------------------------------------------------------------------------------
% File : LAT345+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Lattice Theory
% Problem : Dual Concept Lattices T20
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Sch01] Schwarzweller (2001), A Characterization of Concept La
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t20_conlat_2 [Urb08]
% Status : Theorem
% Rating : 0.56 v8.1.0, 0.47 v7.5.0, 0.56 v7.4.0, 0.47 v7.3.0, 0.48 v7.0.0, 0.53 v6.4.0, 0.58 v6.3.0, 0.54 v6.2.0, 0.64 v6.1.0, 0.70 v6.0.0, 0.65 v5.5.0, 0.70 v5.4.0, 0.71 v5.3.0, 0.74 v5.2.0, 0.70 v5.1.0, 0.76 v5.0.0, 0.79 v4.1.0, 0.83 v3.7.0, 0.85 v3.5.0, 0.84 v3.4.0
% Syntax : Number of formulae : 69 ( 19 unt; 0 def)
% Number of atoms : 195 ( 21 equ)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 166 ( 40 ~; 1 |; 72 &)
% ( 3 <=>; 50 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 15 ( 15 usr; 1 con; 0-3 aty)
% Number of variables : 122 ( 103 !; 19 ?)
% 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(t20_conlat_2,conjecture,
! [A] :
( ( ~ v3_conlat_1(A)
& v4_conlat_1(A)
& l2_conlat_1(A) )
=> ! [B] :
( ( v6_conlat_1(B,A)
& ~ v7_conlat_1(B,A)
& v9_conlat_1(B,A)
& l3_conlat_1(B,A) )
=> k9_conlat_2(k7_conlat_2(A),k9_conlat_2(A,B)) = B ) ) ).
fof(abstractness_v4_conlat_1,axiom,
! [A] :
( l2_conlat_1(A)
=> ( v4_conlat_1(A)
=> A = g2_conlat_1(u1_conlat_1(A),u2_conlat_1(A),u3_conlat_1(A)) ) ) ).
fof(abstractness_v6_conlat_1,axiom,
! [A,B] :
( ( l1_conlat_1(A)
& l3_conlat_1(B,A) )
=> ( v6_conlat_1(B,A)
=> B = g3_conlat_1(A,u4_conlat_1(A,B),u5_conlat_1(A,B)) ) ) ).
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(cc2_conlat_1,axiom,
! [A] :
( ( v3_conlat_1(A)
& l1_conlat_1(A) )
=> ! [B] :
( l3_conlat_1(B,A)
=> v8_conlat_1(B,A) ) ) ).
fof(d7_conlat_2,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A) )
=> k7_conlat_2(A) = g2_conlat_1(u2_conlat_1(A),u1_conlat_1(A),k6_relset_1(u1_conlat_1(A),u2_conlat_1(A),u3_conlat_1(A))) ) ).
fof(d8_conlat_2,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A) )
=> ! [B] :
( l3_conlat_1(B,A)
=> ! [C] :
( ( v6_conlat_1(C,k7_conlat_2(A))
& l3_conlat_1(C,k7_conlat_2(A)) )
=> ( C = k8_conlat_2(A,B)
<=> ( u4_conlat_1(k7_conlat_2(A),C) = u5_conlat_1(A,B)
& u5_conlat_1(k7_conlat_2(A),C) = u4_conlat_1(A,B) ) ) ) ) ) ).
fof(dt_g2_conlat_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> ( v4_conlat_1(g2_conlat_1(A,B,C))
& l2_conlat_1(g2_conlat_1(A,B,C)) ) ) ).
fof(dt_g3_conlat_1,axiom,
! [A,B,C] :
( ( l1_conlat_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_conlat_1(A)))
& m1_subset_1(C,k1_zfmisc_1(u2_conlat_1(A))) )
=> ( v6_conlat_1(g3_conlat_1(A,B,C),A)
& l3_conlat_1(g3_conlat_1(A,B,C),A) ) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k4_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> v1_relat_1(k4_relat_1(A)) ) ).
fof(dt_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> m2_relset_1(k6_relset_1(A,B,C),B,A) ) ).
fof(dt_k7_conlat_2,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A) )
=> ( ~ v3_conlat_1(k7_conlat_2(A))
& v4_conlat_1(k7_conlat_2(A))
& l2_conlat_1(k7_conlat_2(A)) ) ) ).
fof(dt_k8_conlat_2,axiom,
! [A,B] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A)
& l3_conlat_1(B,A) )
=> ( v6_conlat_1(k8_conlat_2(A,B),k7_conlat_2(A))
& l3_conlat_1(k8_conlat_2(A,B),k7_conlat_2(A)) ) ) ).
fof(dt_k9_conlat_2,axiom,
! [A,B] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A)
& ~ v7_conlat_1(B,A)
& v9_conlat_1(B,A)
& l3_conlat_1(B,A) )
=> ( v6_conlat_1(k9_conlat_2(A,B),k7_conlat_2(A))
& ~ v7_conlat_1(k9_conlat_2(A,B),k7_conlat_2(A))
& v9_conlat_1(k9_conlat_2(A,B),k7_conlat_2(A))
& l3_conlat_1(k9_conlat_2(A,B),k7_conlat_2(A)) ) ) ).
fof(dt_l1_conlat_1,axiom,
$true ).
fof(dt_l2_conlat_1,axiom,
! [A] :
( l2_conlat_1(A)
=> l1_conlat_1(A) ) ).
fof(dt_l3_conlat_1,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_conlat_1,axiom,
$true ).
fof(dt_u2_conlat_1,axiom,
$true ).
fof(dt_u3_conlat_1,axiom,
! [A] :
( l2_conlat_1(A)
=> m2_relset_1(u3_conlat_1(A),u1_conlat_1(A),u2_conlat_1(A)) ) ).
fof(dt_u4_conlat_1,axiom,
! [A,B] :
( ( l1_conlat_1(A)
& l3_conlat_1(B,A) )
=> m1_subset_1(u4_conlat_1(A,B),k1_zfmisc_1(u1_conlat_1(A))) ) ).
fof(dt_u5_conlat_1,axiom,
! [A,B] :
( ( l1_conlat_1(A)
& l3_conlat_1(B,A) )
=> m1_subset_1(u5_conlat_1(A,B),k1_zfmisc_1(u2_conlat_1(A))) ) ).
fof(existence_l1_conlat_1,axiom,
? [A] : l1_conlat_1(A) ).
fof(existence_l2_conlat_1,axiom,
? [A] : l2_conlat_1(A) ).
fof(existence_l3_conlat_1,axiom,
! [A] :
( l1_conlat_1(A)
=> ? [B] : l3_conlat_1(B,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_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l1_conlat_1(A) )
=> ~ v1_xboole_0(u2_conlat_1(A)) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
fof(fc2_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l1_conlat_1(A) )
=> ~ v1_xboole_0(u1_conlat_1(A)) ) ).
fof(fc4_subset_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ).
fof(free_g2_conlat_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> ! [D,E,F] :
( g2_conlat_1(A,B,C) = g2_conlat_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(free_g3_conlat_1,axiom,
! [A,B,C] :
( ( l1_conlat_1(A)
& m1_subset_1(B,k1_zfmisc_1(u1_conlat_1(A)))
& m1_subset_1(C,k1_zfmisc_1(u2_conlat_1(A))) )
=> ! [D,E,F] :
( g3_conlat_1(A,B,C) = g3_conlat_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(involutiveness_k4_relat_1,axiom,
! [A] :
( v1_relat_1(A)
=> k4_relat_1(k4_relat_1(A)) = A ) ).
fof(involutiveness_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k6_relset_1(A,B,k6_relset_1(A,B,C)) = C ) ).
fof(rc10_conlat_1,axiom,
! [A] :
( l1_conlat_1(A)
=> ? [B] :
( l3_conlat_1(B,A)
& v6_conlat_1(B,A) ) ) ).
fof(rc11_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l1_conlat_1(A) )
=> ? [B] :
( l3_conlat_1(B,A)
& v6_conlat_1(B,A)
& ~ v7_conlat_1(B,A) ) ) ).
fof(rc12_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l1_conlat_1(A) )
=> ? [B] :
( l3_conlat_1(B,A)
& v6_conlat_1(B,A)
& v8_conlat_1(B,A) ) ) ).
fof(rc13_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A) )
=> ? [B] :
( l3_conlat_1(B,A)
& ~ v7_conlat_1(B,A)
& v9_conlat_1(B,A) ) ) ).
fof(rc14_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A) )
=> ? [B] :
( l3_conlat_1(B,A)
& v6_conlat_1(B,A)
& ~ v7_conlat_1(B,A)
& v9_conlat_1(B,A) ) ) ).
fof(rc1_subset_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& ~ v1_xboole_0(B) ) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_subset_1,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
& v1_xboole_0(B) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc5_conlat_1,axiom,
? [A] :
( l2_conlat_1(A)
& v4_conlat_1(A) ) ).
fof(rc7_conlat_1,axiom,
? [A] :
( l2_conlat_1(A)
& ~ v3_conlat_1(A)
& v4_conlat_1(A) ) ).
fof(rc8_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l1_conlat_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_conlat_1(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(rc9_conlat_1,axiom,
! [A] :
( ( ~ v3_conlat_1(A)
& l1_conlat_1(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u2_conlat_1(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(redefinition_k6_relset_1,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> k6_relset_1(A,B,C) = k4_relat_1(C) ) ).
fof(redefinition_k9_conlat_2,axiom,
! [A,B] :
( ( ~ v3_conlat_1(A)
& l2_conlat_1(A)
& ~ v7_conlat_1(B,A)
& v9_conlat_1(B,A)
& l3_conlat_1(B,A) )
=> k9_conlat_2(A,B) = k8_conlat_2(A,B) ) ).
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(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) ) ).
%------------------------------------------------------------------------------