TPTP Problem File: GRP194+1.p
View Solutions
- Solve Problem
%--------------------------------------------------------------------------
% File : GRP194+1 : TPTP v9.0.0. Released v2.0.0.
% Domain : Group Theory (Semigroups)
% Problem : In semigroups, a surjective homomorphism maps the zero
% Version : [Gol93] axioms.
% English : If (F,*) and (H,+) are two semigroups, phi is a surjective
% homomorphism from F to H, and id is a left zero for F,
% then phi(id) is a left zero for H.
% Refs : [Gol93] Goller (1993), Anwendung des Theorembeweisers SETHEO a
% Source : [Gol93]
% Names :
% Status : Theorem
% Rating : 0.18 v9.0.0, 0.19 v8.1.0, 0.22 v7.5.0, 0.25 v7.4.0, 0.10 v7.3.0, 0.14 v7.2.0, 0.10 v7.1.0, 0.13 v7.0.0, 0.20 v6.4.0, 0.23 v6.3.0, 0.21 v6.2.0, 0.36 v6.1.0, 0.47 v6.0.0, 0.35 v5.5.0, 0.37 v5.4.0, 0.50 v5.3.0, 0.48 v5.2.0, 0.40 v5.1.0, 0.33 v4.1.0, 0.35 v4.0.1, 0.30 v4.0.0, 0.29 v3.7.0, 0.15 v3.5.0, 0.16 v3.4.0, 0.21 v3.3.0, 0.14 v3.2.0, 0.18 v3.1.0, 0.11 v2.7.0, 0.17 v2.6.0, 0.14 v2.5.0, 0.12 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1, 0.00 v2.1.0
% Syntax : Number of formulae : 8 ( 2 unt; 0 def)
% Number of atoms : 21 ( 4 equ)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 13 ( 0 ~; 0 |; 6 &)
% ( 1 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 2-2 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-3 aty)
% Number of variables : 15 ( 14 !; 1 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%--------------------------------------------------------------------------
%----Include Semigroup axioms
include('Axioms/GRP007+0.ax').
%--------------------------------------------------------------------------
%----Definition of a homomorphism
fof(homomorphism1,axiom,
! [X] :
( group_member(X,f)
=> group_member(phi(X),h) ) ).
fof(homomorphism2,axiom,
! [X,Y] :
( ( group_member(X,f)
& group_member(Y,f) )
=> multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ).
fof(surjective,axiom,
! [X] :
( group_member(X,h)
=> ? [Y] :
( group_member(Y,f)
& phi(Y) = X ) ) ).
%----Definition of left zero
fof(left_zero,axiom,
! [G,X] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y] :
( group_member(Y,G)
=> multiply(G,X,Y) = X ) ) ) ).
%----The conjecture
fof(left_zero_for_f,hypothesis,
left_zero(f,f_left_zero) ).
fof(prove_left_zero_h,conjecture,
left_zero(h,phi(f_left_zero)) ).
%--------------------------------------------------------------------------