TPTP Problem File: GRP776+1.p
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% File : GRP776+1 : TPTP v9.0.0. Released v4.1.0.
% Domain : Group Theory
% Problem : A homomorphic mapping between two groups
% Version : Especial.
% English : A mapping between two groups that respects multiplication is a
% homomorphism.
% Refs : [Sta09] Stanovsky (2009), Email to Geoff Sutcliffe
% Source : [Sta09]
% Names : grp_hom [Sta09]
% Status : Theorem
% Rating : 0.36 v8.2.0, 0.39 v8.1.0, 0.33 v7.5.0, 0.44 v7.4.0, 0.30 v7.3.0, 0.31 v7.2.0, 0.34 v7.1.0, 0.30 v7.0.0, 0.37 v6.4.0, 0.42 v6.2.0, 0.52 v6.1.0, 0.67 v6.0.0, 0.61 v5.5.0, 0.70 v5.4.0, 0.75 v5.3.0, 0.81 v5.2.0, 0.00 v4.1.0
% Syntax : Number of formulae : 19 ( 3 unt; 0 def)
% Number of atoms : 42 ( 13 equ)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 24 ( 1 ~; 1 |; 7 &)
% ( 0 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 7 ( 7 usr; 2 con; 0-2 aty)
% Number of variables : 25 ( 25 !; 0 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
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%----G is closed under *, inv, eh
fof(sos01,axiom,
! [B,A] :
( ( g(A)
& g(B) )
=> g(product(A,B)) ) ).
fof(sos02,axiom,
! [A] :
( g(A)
=> g(inv(A)) ) ).
fof(sos03,axiom,
g(eh) ).
%----G is a group
fof(sos04,axiom,
! [C,B,A] :
( ( g(A)
& g(B)
& g(C) )
=> product(product(A,B),C) = product(A,product(B,C)) ) ).
fof(sos05,axiom,
! [A] :
( g(A)
=> product(eh,A) = A ) ).
fof(sos06,axiom,
! [A] :
( g(A)
=> product(A,eh) = A ) ).
fof(sos07,axiom,
! [A] :
( g(A)
=> product(A,inv(A)) = eh ) ).
fof(sos08,axiom,
! [A] :
( g(A)
=> product(inv(A),A) = eh ) ).
%----H is closed under *, inv, eh
fof(sos09,axiom,
! [B,A] :
( ( h(A)
& h(B) )
=> h(sum(A,B)) ) ).
fof(sos10,axiom,
! [B,A] :
( h(A)
=> h(opp(B)) ) ).
fof(sos11,axiom,
h(eg) ).
%----H is a group
fof(sos12,axiom,
! [C,B,A] :
( ( h(A)
& h(B)
& h(C) )
=> sum(sum(A,B),C) = sum(A,sum(B,C)) ) ).
fof(sos13,axiom,
! [A] :
( h(A)
=> sum(eg,A) = A ) ).
fof(sos14,axiom,
! [A] :
( h(A)
=> sum(A,eg) = A ) ).
fof(sos15,axiom,
! [A] :
( h(A)
=> sum(A,opp(A)) = eg ) ).
fof(sos16,axiom,
! [A] :
( h(A)
=> sum(opp(A),A) = eg ) ).
%----f: G -> H
fof(sos17,axiom,
! [A] :
( g(A)
=> h(f(A)) ) ).
fof(sos18,axiom,
! [B,A] : f(product(A,B)) = sum(f(A),f(B)) ).
%----f is a homomorphism
fof(goals,conjecture,
! [X0] :
( f(eh) = eg
& ( ~ g(X0)
| f(inv(X0)) = opp(f(X0)) ) ) ).
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