TPTP Problem File: GRP627+1.p
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%------------------------------------------------------------------------------
% File : GRP627+1 : TPTP v8.2.0. Released v3.4.0.
% Domain : Group Theory
% Problem : On the Group of Inner Automorphisms T18
% 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 : t18_autgroup [Urb08]
% Status : Theorem
% Rating : 1.00 v3.4.0
% Syntax : Number of formulae : 93 ( 27 unt; 0 def)
% Number of atoms : 323 ( 22 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 279 ( 49 ~; 2 |; 146 &)
% ( 6 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-4 aty)
% Number of functors : 21 ( 21 usr; 1 con; 0-6 aty)
% Number of variables : 190 ( 168 !; 22 ?)
% 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(t18_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] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k4_autgroup(A))
=> m2_fraenkel(k7_funct_2(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),C,B),u1_struct_0(A),u1_struct_0(A),k4_autgroup(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_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_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(commutativity_k2_tarski,axiom,
! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ).
fof(d1_binop_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ).
fof(d1_group_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_group_1(A),B,C) ) ) ) ).
fof(d4_autgroup,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_group_1(A)
& v3_group_1(A)
& v4_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_fraenkel(B,u1_struct_0(A),u1_struct_0(A))
=> ( B = k4_autgroup(A)
<=> ! [C] :
( m2_fraenkel(C,u1_struct_0(A),u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(A)))
=> ( r2_hidden(C,B)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,E) = k2_group_3(A,E,D) ) ) ) ) ) ) ) ).
fof(d4_group_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ( v4_group_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ) ).
fof(d5_tarski,axiom,
! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(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_binop_1,axiom,
$true ).
fof(dt_k1_fraenkel,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> m1_fraenkel(k1_fraenkel(A,B),A,B) ) ).
fof(dt_k1_funct_1,axiom,
$true ).
fof(dt_k1_funct_2,axiom,
$true ).
fof(dt_k1_group_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k1_tarski,axiom,
$true ).
fof(dt_k1_xboole_0,axiom,
$true ).
fof(dt_k1_zfmisc_1,axiom,
$true ).
fof(dt_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C)
& m1_subset_1(E,A)
& m1_subset_1(F,B) )
=> m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ).
fof(dt_k2_group_3,axiom,
! [A,B,C] :
( ( ~ 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(C,u1_struct_0(A)) )
=> m1_subset_1(k2_group_3(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k2_tarski,axiom,
$true ).
fof(dt_k2_zfmisc_1,axiom,
$true ).
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_k4_tarski,axiom,
$true ).
fof(dt_k5_relat_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_relat_1(B) )
=> v1_relat_1(k5_relat_1(A,B)) ) ).
fof(dt_k7_funct_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B)
& v1_funct_1(E)
& v1_funct_2(E,B,C)
& m1_relset_1(E,B,C) )
=> ( v1_funct_1(k7_funct_2(A,B,C,D,E))
& v1_funct_2(k7_funct_2(A,B,C,D,E),A,C)
& m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ).
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_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_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_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_fraenkel,axiom,
! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k5_relat_1(A,B))
& v1_funct_1(k5_relat_1(A,B)) ) ) ).
fof(fc1_funct_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ~ v1_xboole_0(k1_funct_2(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(fc2_funct_2,axiom,
! [A] : ~ v1_xboole_0(k1_funct_2(A,A)) ).
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_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_k1_fraenkel,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> k1_fraenkel(A,B) = k1_funct_2(A,B) ) ).
fof(redefinition_k2_binop_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C)
& m1_subset_1(E,A)
& m1_subset_1(F,B) )
=> k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ).
fof(redefinition_k7_funct_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B)
& v1_funct_1(E)
& v1_funct_2(E,B,C)
& m1_relset_1(E,B,C) )
=> k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ).
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_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(t12_funct_2,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> r2_hidden(B,k1_funct_2(A,A)) ) ).
fof(t1_subset,axiom,
! [A,B] :
( r2_hidden(A,B)
=> m1_subset_1(A,B) ) ).
fof(t20_group_3,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_subset_1(C,u1_struct_0(A))
=> ( k2_group_3(A,B,C) = k1_group_1(A,k1_group_1(A,k3_group_1(A,C),B),C)
& k2_group_3(A,B,C) = k1_group_1(A,k3_group_1(A,C),k1_group_1(A,B,C)) ) ) ) ) ).
fof(t21_funct_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( r2_hidden(C,A)
=> ( B = k1_xboole_0
| k1_funct_1(k5_relat_1(D,E),C) = k1_funct_1(E,k1_funct_1(D,C)) ) ) ) ) ).
fof(t25_group_1,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_subset_1(C,u1_struct_0(A))
=> k3_group_1(A,k1_group_1(A,B,C)) = k1_group_1(A,k3_group_1(A,C),k3_group_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) ) ).
%------------------------------------------------------------------------------