TPTP Problem File: GRP646+1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : GRP646+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Group Theory
% Problem : On the Lattice of Subgroups of a Group T25
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Gan96] Ganczarski (1996), On the Lattice of Subgroups of a Gr
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t25_latsubgr [Urb08]
% Status : Theorem
% Rating : 0.83 v8.1.0, 0.78 v7.5.0, 0.81 v7.4.0, 0.70 v7.3.0, 0.76 v7.1.0, 0.70 v7.0.0, 0.73 v6.3.0, 0.75 v6.2.0, 0.88 v6.1.0, 0.90 v6.0.0, 0.91 v5.5.0, 0.93 v5.3.0, 0.96 v5.2.0, 0.90 v5.0.0, 0.92 v4.1.0, 0.96 v3.7.0, 0.95 v3.5.0, 1.00 v3.4.0
% Syntax : Number of formulae : 86 ( 25 unt; 0 def)
% Number of atoms : 316 ( 23 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 285 ( 55 ~; 1 |; 146 &)
% ( 11 <=>; 72 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 22 ( 22 usr; 1 con; 0-4 aty)
% Number of variables : 162 ( 146 !; 16 ?)
% 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(t25_latsubgr,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_group_3(A))) )
=> ( r2_hidden(k5_group_2(A),B)
=> r1_group_2(A,k2_latsubgr(A,B),k5_group_2(A)) ) ) ) ).
fof(abstractness_v1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> ( v1_group_1(A)
=> A = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ).
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( r2_hidden(A,B)
=> ~ r2_hidden(B,A) ) ).
fof(cc1_funct_2,axiom,
! [A,B,C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_partfun1(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,A,B) ) ) ) ).
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(cc5_funct_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(cc6_funct_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B) )
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& ~ v1_xboole_0(C)
& v1_partfun1(C,A,B)
& v1_funct_2(C,A,B) ) ) ) ) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( r1_tarski(A,B)
& r1_tarski(B,A) ) ) ).
fof(d1_group_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( B = k1_group_3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_group_1(C)
& m1_group_2(C,A) ) ) ) ) ).
fof(d1_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_group_3(A),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(B,k1_group_3(A),k1_zfmisc_1(u1_struct_0(A))) )
=> ( B = k1_latsubgr(A)
<=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> k1_funct_1(B,C) = u1_struct_0(C) ) ) ) ) ).
fof(d2_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_group_3(A))) )
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ( C = k2_latsubgr(A,B)
<=> u1_struct_0(C) = k6_setfam_1(u1_struct_0(A),k2_funct_2(k1_group_3(A),k1_zfmisc_1(u1_struct_0(A)),k1_latsubgr(A),B)) ) ) ) ) ).
fof(d5_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& l1_group_1(B) )
=> ( m1_group_2(B,A)
<=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B)) ) ) ) ) ).
fof(d7_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( v1_group_1(B)
& m1_group_2(B,A) )
=> ( B = k5_group_2(A)
<=> u1_struct_0(B) = k6_domain_1(u1_struct_0(A),k2_group_1(A)) ) ) ) ).
fof(dt_g1_group_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_group_1(g1_group_1(A,B))
& l1_group_1(g1_group_1(A,B)) ) ) ).
fof(dt_k18_group_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k18_group_2(A,B),k5_finsub_1(A)) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_group_3,axiom,
$true ).
fof(dt_k1_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_funct_1(k1_latsubgr(A))
& v1_funct_2(k1_latsubgr(A),k1_group_3(A),k1_zfmisc_1(u1_struct_0(A)))
& m2_relset_1(k1_latsubgr(A),k1_group_3(A),k1_zfmisc_1(u1_struct_0(A))) ) ) ).
fof(dt_k1_realset1,axiom,
$true ).
fof(dt_k1_setfam_1,axiom,
$true ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ).
fof(dt_k2_group_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> m1_subset_1(k2_group_1(A),u1_struct_0(A)) ) ).
fof(dt_k2_latsubgr,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_group_3(A))) )
=> ( v1_group_1(k2_latsubgr(A,B))
& m1_group_2(k2_latsubgr(A,B),A) ) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k5_finsub_1,axiom,
! [A] : v4_finsub_1(k5_finsub_1(A)) ).
fof(dt_k5_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_group_1(k5_group_2(A))
& m1_group_2(k5_group_2(A),A) ) ) ).
fof(dt_k6_domain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k6_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> m1_subset_1(k6_setfam_1(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> m1_subset_1(k8_funct_2(A,B,C,D),B) ) ).
fof(dt_k9_relat_1,axiom,
$true ).
fof(dt_l1_group_1,axiom,
! [A] :
( l1_group_1(A)
=> l1_struct_0(A) ) ).
fof(dt_l1_struct_0,axiom,
$true ).
fof(dt_m1_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ( ~ v3_struct_0(B)
& v3_group_1(B)
& l1_group_1(B) ) ) ) ).
fof(dt_m1_group_6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_group_2(B,A) )
=> ! [C] :
( m1_group_6(C,A,B)
=> m1_group_2(C,A) ) ) ).
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_group_1,axiom,
! [A] :
( l1_group_1(A)
=> ( v1_funct_1(u1_group_1(A))
& v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_u1_struct_0,axiom,
$true ).
fof(existence_l1_group_1,axiom,
? [A] : l1_group_1(A) ).
fof(existence_l1_struct_0,axiom,
? [A] : l1_struct_0(A) ).
fof(existence_m1_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ? [B] : m1_group_2(B,A) ) ).
fof(existence_m1_group_6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_group_2(B,A) )
=> ? [C] : m1_group_6(C,A,B) ) ).
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_group_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ~ v1_xboole_0(k1_group_3(A)) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(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_subset_1,axiom,
! [A] : ~ v1_xboole_0(k1_tarski(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_g1_group_1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C,D] :
( g1_group_1(A,B) = g1_group_1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(rc1_funct_2,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B) ) ).
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_partfun1,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C) ) ).
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(rc3_struct_0,axiom,
? [A] :
( l1_struct_0(A)
& ~ v3_struct_0(A) ) ).
fof(rc5_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(redefinition_k18_group_2,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k18_group_2(A,B) = k1_tarski(B) ) ).
fof(redefinition_k2_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(redefinition_k6_domain_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k6_domain_1(A,B) = k1_tarski(B) ) ).
fof(redefinition_k6_setfam_1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> k6_setfam_1(A,B) = k1_setfam_1(B) ) ).
fof(redefinition_k8_funct_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(redefinition_m1_group_6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_group_2(B,A) )
=> ! [C] :
( m1_group_6(C,A,B)
<=> m1_group_2(C,B) ) ) ).
fof(redefinition_m2_relset_1,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
<=> m1_relset_1(C,A,B) ) ).
fof(redefinition_r1_group_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> ( r1_group_2(A,B,C)
<=> B = C ) ) ).
fof(reflexivity_r1_group_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> r1_group_2(A,B,B) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : r1_tarski(A,A) ).
fof(symmetry_r1_group_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& v1_group_1(B)
& m1_group_2(B,A)
& v1_group_1(C)
& m1_group_2(C,A) )
=> ( r1_group_2(A,B,C)
=> r1_group_2(A,C,B) ) ) ).
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(t43_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( B != k1_xboole_0
=> ! [E] :
( ? [F] :
( r2_hidden(F,A)
& r2_hidden(F,C)
& E = k1_funct_1(D,F) )
=> r2_hidden(E,k9_relat_1(D,C)) ) ) ) ).
fof(t4_setfam_1,axiom,
! [A,B] :
( r2_hidden(A,B)
=> r1_tarski(k1_setfam_1(B),A) ) ).
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(t77_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> m1_group_2(k5_group_2(A),B) ) ) ).
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) ) ).
%------------------------------------------------------------------------------