TPTP Problem File: GRP644+2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : GRP644+2 : TPTP v8.2.0. Released v3.4.0.
% Domain : Group Theory
% Problem : On the Lattice of Subgroups of a Group T22
% 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 : t22_latsubgr [Urb08]
% Status : Theorem
% Rating : 0.69 v8.2.0, 0.75 v8.1.0, 0.81 v7.4.0, 0.77 v7.3.0, 0.83 v7.1.0, 0.78 v7.0.0, 0.80 v6.4.0, 0.77 v6.3.0, 0.75 v6.2.0, 0.84 v6.1.0, 0.87 v6.0.0, 0.83 v5.5.0, 0.85 v5.4.0, 0.86 v5.3.0, 0.89 v5.2.0, 0.80 v5.1.0, 0.81 v5.0.0, 0.92 v4.1.0, 0.91 v4.0.0, 0.96 v3.7.0, 1.00 v3.4.0
% Syntax : Number of formulae : 5661 (1554 unt; 0 def)
% Number of atoms : 31277 (3906 equ)
% Maximal formula atoms : 49 ( 5 avg)
% Number of connectives : 29490 (3874 ~; 212 |;15139 &)
% ( 951 <=>;9314 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 325 ( 323 usr; 1 prp; 0-4 aty)
% Number of functors : 710 ( 710 usr; 182 con; 0-9 aty)
% Number of variables : 13268 (12647 !; 621 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Bushy version: includes all articles that contribute axioms to the
% Normal version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+4.ax').
include('Axioms/SET007/SET007+5.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+8.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+13.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+22.ax').
include('Axioms/SET007/SET007+23.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+32.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+40.ax').
include('Axioms/SET007/SET007+48.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+59.ax').
include('Axioms/SET007/SET007+60.ax').
include('Axioms/SET007/SET007+71.ax').
include('Axioms/SET007/SET007+117.ax').
include('Axioms/SET007/SET007+200.ax').
include('Axioms/SET007/SET007+205.ax').
include('Axioms/SET007/SET007+210.ax').
include('Axioms/SET007/SET007+212.ax').
include('Axioms/SET007/SET007+213.ax').
include('Axioms/SET007/SET007+218.ax').
include('Axioms/SET007/SET007+242.ax').
include('Axioms/SET007/SET007+246.ax').
include('Axioms/SET007/SET007+247.ax').
include('Axioms/SET007/SET007+248.ax').
include('Axioms/SET007/SET007+252.ax').
include('Axioms/SET007/SET007+255.ax').
include('Axioms/SET007/SET007+295.ax').
include('Axioms/SET007/SET007+312.ax').
include('Axioms/SET007/SET007+335.ax').
include('Axioms/SET007/SET007+375.ax').
include('Axioms/SET007/SET007+432.ax').
%------------------------------------------------------------------------------
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_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_k3_latsubgr,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B)
& v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_funct_1(k3_latsubgr(A,B,C))
& v1_funct_2(k3_latsubgr(A,B,C),u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B)))
& m2_relset_1(k3_latsubgr(A,B,C),u1_struct_0(k11_group_4(A)),u1_struct_0(k11_group_4(B))) ) ) ).
fof(t1_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> u1_struct_0(k9_group_2(A,B,C)) = k3_xboole_0(u1_struct_0(B),u1_struct_0(C)) ) ) ) ).
fof(t2_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( r2_hidden(B,k1_group_3(A))
<=> ? [C] :
( v1_group_1(C)
& m1_group_2(C,A)
& B = C ) ) ) ).
fof(t3_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( ( v1_group_1(C)
& m1_group_2(C,A) )
=> ( B = u1_struct_0(C)
=> r1_group_2(A,k5_group_4(A,B),C) ) ) ) ) ).
fof(t4_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( D = k2_xboole_0(u1_struct_0(B),u1_struct_0(C))
=> r1_group_2(A,k8_group_4(A,B,C),k5_group_4(A,D)) ) ) ) ) ) ).
fof(t5_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r1_rlvect_1(B,D)
| r1_rlvect_1(C,D) )
=> r1_rlvect_1(k8_group_4(A,B,C),D) ) ) ) ) ) ).
fof(t6_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> ? [E] :
( v1_group_1(E)
& m1_group_2(E,B)
& u1_struct_0(E) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D)) ) ) ) ) ) ).
fof(t7_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,B)
=> ? [E] :
( v1_group_1(E)
& m1_group_2(E,A)
& u1_struct_0(E) = k3_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D)) ) ) ) ) ) ).
fof(t8_latsubgr,axiom,
$true ).
fof(t9_latsubgr,axiom,
$true ).
fof(t10_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,A)
=> ! [E] :
( m1_group_2(E,A)
=> ! [F] :
( m1_group_2(F,B)
=> ! [G] :
( m1_group_2(G,B)
=> ( ( u1_struct_0(F) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D))
& u1_struct_0(G) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(E))
& m1_group_6(D,A,E) )
=> m1_group_6(F,B,G) ) ) ) ) ) ) ) ) ).
fof(t11_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_group_6(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_group_2(D,B)
=> ! [E] :
( m1_group_2(E,B)
=> ! [F] :
( m1_group_2(F,A)
=> ! [G] :
( m1_group_2(G,A)
=> ( ( u1_struct_0(F) = k3_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(D))
& u1_struct_0(G) = k3_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(E))
& m1_group_6(D,B,E) )
=> m1_group_6(F,A,G) ) ) ) ) ) ) ) ) ).
fof(t12_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> r1_tarski(k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,u1_struct_0(k5_group_4(A,D)))) ) ) ) ) ).
fof(t13_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_group_1(B)
& v4_group_1(B)
& l1_group_1(B) )
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_group_2(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
=> ( F = k2_xboole_0(u1_struct_0(C),u1_struct_0(D))
=> k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,u1_struct_0(k8_group_4(A,C,D))) = k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,u1_struct_0(k5_group_4(A,F))) ) ) ) ) ) ) ) ).
fof(t14_latsubgr,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( B = k18_group_2(u1_struct_0(A),k2_group_1(A))
=> r1_group_2(A,k5_group_4(A,B),k5_group_2(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(t15_latsubgr,axiom,
$true ).
fof(t16_latsubgr,axiom,
$true ).
fof(t17_latsubgr,axiom,
$true ).
fof(t18_latsubgr,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) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k1_funct_1(k1_latsubgr(A),B))
<=> r1_rlvect_1(B,C) ) ) ) ) ).
fof(t19_latsubgr,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) )
=> r2_hidden(k2_group_1(A),k1_funct_1(k1_latsubgr(A),B)) ) ) ).
fof(t20_latsubgr,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) )
=> k1_funct_1(k1_latsubgr(A),B) != k1_xboole_0 ) ) ).
fof(t21_latsubgr,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) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,k1_funct_1(k1_latsubgr(A),B))
& r2_hidden(D,k1_funct_1(k1_latsubgr(A),B)) )
=> r2_hidden(k1_group_1(A,C,D),k1_funct_1(k1_latsubgr(A),B)) ) ) ) ) ) ).
fof(t22_latsubgr,conjecture,
! [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) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_hidden(C,k1_funct_1(k1_latsubgr(A),B))
=> r2_hidden(k3_group_1(A,C),k1_funct_1(k1_latsubgr(A),B)) ) ) ) ) ).
%------------------------------------------------------------------------------