TPTP Problem File: GRP628+1.p
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%------------------------------------------------------------------------------
% File : GRP628+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Group Theory
% Problem : On the Group of Inner Automorphisms T22
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Kor96] Kornilowicz (1996), On the Group of Inner Automorphism
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t22_autgroup [Urb08]
% Status : Theorem
% Rating : 0.33 v7.5.0, 0.38 v7.4.0, 0.23 v7.3.0, 0.34 v7.2.0, 0.31 v7.1.0, 0.30 v7.0.0, 0.23 v6.4.0, 0.31 v6.3.0, 0.33 v6.2.0, 0.40 v6.1.0, 0.47 v6.0.0, 0.48 v5.5.0, 0.52 v5.4.0, 0.54 v5.3.0, 0.59 v5.2.0, 0.45 v5.1.0, 0.48 v5.0.0, 0.54 v4.1.0, 0.52 v4.0.0, 0.54 v3.7.0, 0.55 v3.5.0, 0.58 v3.4.0
% Syntax : Number of formulae : 77 ( 17 unt; 0 def)
% Number of atoms : 298 ( 13 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 268 ( 47 ~; 1 |; 148 &)
% ( 3 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-4 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-2 aty)
% Number of variables : 137 ( 113 !; 24 ?)
% 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(t22_autgroup,conjecture,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_autgroup(A)))
=> ( B = C
=> k2_funct_1(B) = k3_group_1(k5_autgroup(A),C) ) ) ) ) ).
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_fraenkel,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ).
fof(cc1_funct_1,axiom,
! [A] :
( v1_xboole_0(A)
=> v1_funct_1(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_group_1,axiom,
! [A] :
( l1_group_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_group_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A) ) ) ) ).
fof(cc1_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)
=> v4_group_1(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(cc2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ) ).
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(d3_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> k3_autgroup(A) = g1_group_1(k1_autgroup(A),k2_autgroup(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_k1_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m1_fraenkel(k1_autgroup(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_funct_1(k2_autgroup(A))
& v1_funct_2(k2_autgroup(A),k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A))
& m2_relset_1(k2_autgroup(A),k2_zfmisc_1(k1_autgroup(A),k1_autgroup(A)),k1_autgroup(A)) ) ) ).
fof(dt_k2_funct_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_funct_1(A))
& v1_funct_1(k2_funct_1(A)) ) ) ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
fof(dt_k3_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( ~ v3_struct_0(k3_autgroup(A))
& v1_group_1(k3_autgroup(A))
& v3_group_1(k3_autgroup(A))
& v4_group_1(k3_autgroup(A))
& l1_group_1(k3_autgroup(A)) ) ) ).
fof(dt_k3_group_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k3_group_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k4_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> m1_fraenkel(k4_autgroup(A),u1_struct_0(A),u1_struct_0(A)) ) ).
fof(dt_k5_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ( v1_group_1(k5_autgroup(A))
& v1_group_3(k5_autgroup(A),k3_autgroup(A))
& m1_group_2(k5_autgroup(A),k3_autgroup(A)) ) ) ).
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_fraenkel,axiom,
! [A,B,C] :
( m1_fraenkel(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ).
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_relset_1,axiom,
$true ).
fof(dt_m1_subset_1,axiom,
$true ).
fof(dt_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ! [D] :
( m2_fraenkel(D,A,B,C)
=> ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) ) ) ) ).
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_fraenkel,axiom,
! [A,B] :
? [C] : m1_fraenkel(C,A,B) ).
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_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_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ? [D] : m2_fraenkel(D,A,B,C) ) ).
fof(existence_m2_relset_1,axiom,
! [A,B] :
? [C] : m2_relset_1(C,A,B) ).
fof(fc1_group_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( ~ v3_struct_0(g1_group_1(A,B))
& v1_group_1(g1_group_1(A,B)) ) ) ).
fof(fc1_struct_0,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ~ v1_xboole_0(u1_struct_0(A)) ) ).
fof(fc1_xboole_0,axiom,
v1_xboole_0(k1_xboole_0) ).
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_fraenkel,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_fraenkel(A) ) ).
fof(rc1_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A) ) ).
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_group_1,axiom,
? [A] :
( l1_group_1(A)
& v1_group_1(A) ) ).
fof(rc1_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)
& ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& v3_group_1(B)
& v4_group_1(B) ) ) ).
fof(rc1_group_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ? [B] :
( m1_group_2(B,A)
& ~ v3_struct_0(B)
& v1_group_1(B)
& v2_group_1(B)
& v3_group_1(B)
& v4_group_1(B)
& v1_group_3(B,A) ) ) ).
fof(rc1_partfun1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_xboole_0(A) ) ).
fof(rc1_xboole_0,axiom,
? [A] : v1_xboole_0(A) ).
fof(rc2_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_xboole_0(A)
& v1_funct_1(A) ) ).
fof(rc2_group_1,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A) ) ).
fof(rc2_partfun1,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C) ) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ v1_xboole_0(A) ).
fof(rc3_funct_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) ) ).
fof(rc3_group_1,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v1_group_1(A)
& v2_group_1(A)
& v3_group_1(A)
& v4_group_1(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_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ! [D] :
( m2_fraenkel(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
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(t11_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_autgroup(A)))
=> ( B = C
=> k2_funct_1(B) = k3_group_1(k3_autgroup(A),C) ) ) ) ) ).
fof(t13_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> m2_fraenkel(B,u1_struct_0(A),u1_struct_0(A),k1_autgroup(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(t51_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_group_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(t57_group_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_group_2(C,A)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ( D = B
=> k3_group_1(C,D) = k3_group_1(A,B) ) ) ) ) ) ).
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) ) ).
%------------------------------------------------------------------------------