%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP194+1 : TPTP v9.0.0. Released v2.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:07:26 AM UTC 2025
% Result : Theorem 0.14s 0.39s
% Output : Proof 0.14s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 47
% Syntax : Number of formulae : 115 ( 31 unt; 0 typ; 0 def)
% Number of atoms : 775 ( 202 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 1301 ( 675 ~; 495 |; 58 &)
% ( 61 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of FOOLs : 34 ( 34 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-3 aty)
% Number of variables : 226 ( 194 !; 9 ?; 226 :)
% Comments :
%------------------------------------------------------------------------------
tff(phi_type,type,
phi: $i > $i ).
tff(f_left_zero_type,type,
f_left_zero: $i ).
tff(multiply_type,type,
multiply: ( $i * $i * $i ) > $i ).
tff(tptp_fun_Y_1_type,type,
tptp_fun_Y_1: ( $i * $i ) > $i ).
tff(h_type,type,
h: $i ).
tff(tptp_fun_Y_0_type,type,
tptp_fun_Y_0: $i > $i ).
tff(f_type,type,
f: $i ).
tff(group_member_type,type,
group_member: ( $i * $i ) > $o ).
tff(left_zero_type,type,
left_zero: ( $i * $i ) > $o ).
tff(1,plain,
^ [G: $i,X: $i] :
rewrite(( ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ~ ( ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) )
| ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ! [G: $i,X: $i] :
~ ( ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) )
| ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
^ [G: $i,X: $i] :
refl(( ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(4,plain,
( ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) ),
inference(quant_intro,[status(thm)],[3]) ).
tff(5,plain,
^ [G: $i,X: $i] :
rewrite(( ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(6,plain,
( ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) ),
inference(quant_intro,[status(thm)],[5]) ).
tff(7,plain,
( ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) ),
inference(transitivity,[status(thm)],[6,4]) ).
tff(8,plain,
^ [G: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
<=> ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) )),
rewrite(( ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) )
<=> ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )),
( ( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ( ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) )),
rewrite(( ( ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) )),
( ( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(9,plain,
( ! [G: $i,X: $i] :
( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) ),
inference(quant_intro,[status(thm)],[8]) ).
tff(10,plain,
^ [G: $i,X: $i] :
rewrite(( ( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [G: $i,X: $i] :
( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) )
<=> ! [G: $i,X: $i] :
( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
<=> ! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,plain,
^ [G: $i,X: $i] :
rewrite(( ( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( group_member(Y,G)
=> ( multiply(G,X,Y) = X ) ) ) )
<=> ( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( group_member(Y,G)
=> ( multiply(G,X,Y) = X ) ) ) )
<=> ! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,axiom,
! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( group_member(Y,G)
=> ( multiply(G,X,Y) = X ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',left_zero) ).
tff(16,plain,
! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
! [G: $i,X: $i] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ),
inference(modus_ponens,[status(thm)],[16,12]) ).
tff(18,plain,
! [G: $i,X: $i] :
( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ),
inference(skolemize,[status(sab)],[17]) ).
tff(19,plain,
! [G: $i,X: $i] :
( ( ~ left_zero(G,X)
| ( group_member(X,G)
& ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
& ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ),
inference(modus_ponens,[status(thm)],[18,11]) ).
tff(20,plain,
! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ),
inference(modus_ponens,[status(thm)],[19,9]) ).
tff(21,plain,
! [G: $i,X: $i] :
~ ( ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) )
| ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) ) ),
inference(modus_ponens,[status(thm)],[20,7]) ).
tff(22,plain,
! [G: $i,X: $i] :
~ ( ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) )
| ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) ),
inference(modus_ponens,[status(thm)],[21,2]) ).
tff(23,plain,
( ~ ! [G: $i,X: $i] :
~ ( ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) )
| ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) )
| ~ ( ~ ( left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) )
| ~ ( ~ left_zero(h,phi(f_left_zero))
| ~ ( ~ group_member(phi(f_left_zero),h)
| ~ ! [Y: $i] :
( ~ group_member(Y,h)
| ( multiply(h,phi(f_left_zero),Y) = phi(f_left_zero) ) ) ) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
~ ( ~ ( left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) )
| ~ ( ~ left_zero(h,phi(f_left_zero))
| ~ ( ~ group_member(phi(f_left_zero),h)
| ~ ! [Y: $i] :
( ~ group_member(Y,h)
| ( multiply(h,phi(f_left_zero),Y) = phi(f_left_zero) ) ) ) ) ),
inference(unit_resolution,[status(thm)],[23,22]) ).
tff(25,plain,
( ~ ( left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) )
| ~ ( ~ left_zero(h,phi(f_left_zero))
| ~ ( ~ group_member(phi(f_left_zero),h)
| ~ ! [Y: $i] :
( ~ group_member(Y,h)
| ( multiply(h,phi(f_left_zero),Y) = phi(f_left_zero) ) ) ) )
| left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) ),
inference(tautology,[status(thm)],[]) ).
tff(26,plain,
( left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) ),
inference(unit_resolution,[status(thm)],[25,24]) ).
tff(27,plain,
( ~ ! [G: $i,X: $i] :
~ ( ~ ( left_zero(G,X)
| ~ group_member(X,G)
| ~ ( ~ group_member(tptp_fun_Y_1(X,G),G)
| ( multiply(G,X,tptp_fun_Y_1(X,G)) = X ) ) )
| ~ ( ~ left_zero(G,X)
| ~ ( ~ group_member(X,G)
| ~ ! [Y: $i] :
( ~ group_member(Y,G)
| ( multiply(G,X,Y) = X ) ) ) ) )
| ~ ( ~ ( left_zero(f,f_left_zero)
| ~ group_member(f_left_zero,f)
| ~ ( ~ group_member(tptp_fun_Y_1(f_left_zero,f),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_1(f_left_zero,f)) = f_left_zero ) ) )
| ~ ( ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
~ ( ~ ( left_zero(f,f_left_zero)
| ~ group_member(f_left_zero,f)
| ~ ( ~ group_member(tptp_fun_Y_1(f_left_zero,f),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_1(f_left_zero,f)) = f_left_zero ) ) )
| ~ ( ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) ) ),
inference(unit_resolution,[status(thm)],[27,22]) ).
tff(29,plain,
( ~ ( left_zero(f,f_left_zero)
| ~ group_member(f_left_zero,f)
| ~ ( ~ group_member(tptp_fun_Y_1(f_left_zero,f),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_1(f_left_zero,f)) = f_left_zero ) ) )
| ~ ( ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) )
| ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) ),
inference(tautology,[status(thm)],[]) ).
tff(30,plain,
( ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) ),
inference(unit_resolution,[status(thm)],[29,28]) ).
tff(31,plain,
( left_zero(f,f_left_zero)
<=> left_zero(f,f_left_zero) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,axiom,
left_zero(f,f_left_zero),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',left_zero_for_f) ).
tff(33,plain,
left_zero(f,f_left_zero),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
( ~ ( ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) )
| ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) ),
inference(tautology,[status(thm)],[]) ).
tff(35,plain,
( ~ ( ~ left_zero(f,f_left_zero)
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) )
| ~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ) ),
inference(unit_resolution,[status(thm)],[34,33]) ).
tff(36,plain,
~ ( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ),
inference(unit_resolution,[status(thm)],[35,30]) ).
tff(37,plain,
( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) )
| group_member(f_left_zero,f) ),
inference(tautology,[status(thm)],[]) ).
tff(38,plain,
group_member(f_left_zero,f),
inference(unit_resolution,[status(thm)],[37,36]) ).
tff(39,plain,
^ [X: $i] :
refl(( ( ~ group_member(X,f)
| group_member(phi(X),h) )
<=> ( ~ group_member(X,f)
| group_member(phi(X),h) ) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) )
<=> ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) )
<=> ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,plain,
^ [X: $i] :
rewrite(( ( group_member(X,f)
=> group_member(phi(X),h) )
<=> ( ~ group_member(X,f)
| group_member(phi(X),h) ) )),
inference(bind,[status(th)],[]) ).
tff(43,plain,
( ! [X: $i] :
( group_member(X,f)
=> group_member(phi(X),h) )
<=> ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,axiom,
! [X: $i] :
( group_member(X,f)
=> group_member(phi(X),h) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism1) ).
tff(45,plain,
! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ),
inference(modus_ponens,[status(thm)],[45,41]) ).
tff(47,plain,
! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) ),
inference(modus_ponens,[status(thm)],[47,40]) ).
tff(49,plain,
( ( ~ ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) )
| ~ group_member(f_left_zero,f)
| group_member(phi(f_left_zero),h) )
<=> ( ~ ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) )
| ~ group_member(f_left_zero,f)
| group_member(phi(f_left_zero),h) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) )
| ~ group_member(f_left_zero,f)
| group_member(phi(f_left_zero),h) ),
inference(quant_inst,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [X: $i] :
( ~ group_member(X,f)
| group_member(phi(X),h) )
| ~ group_member(f_left_zero,f)
| group_member(phi(f_left_zero),h) ),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,plain,
( ~ group_member(f_left_zero,f)
| group_member(phi(f_left_zero),h) ),
inference(unit_resolution,[status(thm)],[51,48]) ).
tff(53,plain,
group_member(phi(f_left_zero),h),
inference(unit_resolution,[status(thm)],[52,38]) ).
tff(54,plain,
( ~ left_zero(h,phi(f_left_zero))
<=> ~ left_zero(h,phi(f_left_zero)) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,axiom,
~ left_zero(h,phi(f_left_zero)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_left_zero_h) ).
tff(56,plain,
~ left_zero(h,phi(f_left_zero)),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
( ~ ( left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) )
| left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) ),
inference(tautology,[status(thm)],[]) ).
tff(58,plain,
( ~ ( left_zero(h,phi(f_left_zero))
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) )
| ~ group_member(phi(f_left_zero),h)
| ~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ) ),
inference(unit_resolution,[status(thm)],[57,56]) ).
tff(59,plain,
~ ( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) ) ),
inference(unit_resolution,[status(thm)],[58,53,26]) ).
tff(60,plain,
( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) )
| group_member(tptp_fun_Y_1(phi(f_left_zero),h),h) ),
inference(tautology,[status(thm)],[]) ).
tff(61,plain,
group_member(tptp_fun_Y_1(phi(f_left_zero),h),h),
inference(unit_resolution,[status(thm)],[60,59]) ).
tff(62,plain,
^ [X: $i] :
refl(( ( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) )
<=> ( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) )
<=> ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
^ [X: $i] :
rewrite(( ( ~ group_member(X,h)
| ( group_member(tptp_fun_Y_0(X),f)
& ( phi(tptp_fun_Y_0(X)) = X ) ) )
<=> ( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(65,plain,
( ! [X: $i] :
( ~ group_member(X,h)
| ( group_member(tptp_fun_Y_0(X),f)
& ( phi(tptp_fun_Y_0(X)) = X ) ) )
<=> ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) ) ),
inference(quant_intro,[status(thm)],[64]) ).
tff(66,plain,
( ! [X: $i] :
( ~ group_member(X,h)
| ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) )
<=> ! [X: $i] :
( ~ group_member(X,h)
| ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(67,plain,
^ [X: $i] :
rewrite(( ( group_member(X,h)
=> ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) )
<=> ( ~ group_member(X,h)
| ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) ) )),
inference(bind,[status(th)],[]) ).
tff(68,plain,
( ! [X: $i] :
( group_member(X,h)
=> ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) )
<=> ! [X: $i] :
( ~ group_member(X,h)
| ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) ) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,axiom,
! [X: $i] :
( group_member(X,h)
=> ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',surjective) ).
tff(70,plain,
! [X: $i] :
( ~ group_member(X,h)
| ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
! [X: $i] :
( ~ group_member(X,h)
| ? [Y: $i] :
( group_member(Y,f)
& ( phi(Y) = X ) ) ),
inference(modus_ponens,[status(thm)],[70,66]) ).
tff(72,plain,
! [X: $i] :
( ~ group_member(X,h)
| ( group_member(tptp_fun_Y_0(X),f)
& ( phi(tptp_fun_Y_0(X)) = X ) ) ),
inference(skolemize,[status(sab)],[71]) ).
tff(73,plain,
! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) ),
inference(modus_ponens,[status(thm)],[72,65]) ).
tff(74,plain,
! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) ),
inference(modus_ponens,[status(thm)],[73,63]) ).
tff(75,plain,
( ( ~ ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) )
| ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ~ ( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) ) ) )
<=> ( ~ ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) )
| ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ~ ( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) ) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(76,plain,
( ~ ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) )
| ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ~ ( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) ) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(77,plain,
( ~ ! [X: $i] :
( ~ group_member(X,h)
| ~ ( ~ group_member(tptp_fun_Y_0(X),f)
| ( phi(tptp_fun_Y_0(X)) != X ) ) )
| ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ~ ( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) ) ) ),
inference(modus_ponens,[status(thm)],[76,75]) ).
tff(78,plain,
~ ( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) ) ),
inference(unit_resolution,[status(thm)],[77,74,61]) ).
tff(79,plain,
( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) )
| group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f) ),
inference(tautology,[status(thm)],[]) ).
tff(80,plain,
group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f),
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
( ~ group_member(f_left_zero,f)
| ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) )
| ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ) ),
inference(tautology,[status(thm)],[]) ).
tff(82,plain,
! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) ),
inference(unit_resolution,[status(thm)],[81,36]) ).
tff(83,plain,
( ( ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = f_left_zero ) )
<=> ( ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = f_left_zero ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,plain,
( ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = f_left_zero ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(85,plain,
( ~ ! [Y: $i] :
( ~ group_member(Y,f)
| ( multiply(f,f_left_zero,Y) = f_left_zero ) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = f_left_zero ) ),
inference(modus_ponens,[status(thm)],[84,83]) ).
tff(86,plain,
multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = f_left_zero,
inference(unit_resolution,[status(thm)],[85,82,80]) ).
tff(87,plain,
phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(f_left_zero),
inference(monotonicity,[status(thm)],[86]) ).
tff(88,plain,
^ [X: $i,Y: $i] :
refl(( ( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
<=> ( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
inference(bind,[status(th)],[]) ).
tff(89,plain,
( ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
<=> ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ) ),
inference(quant_intro,[status(thm)],[88]) ).
tff(90,plain,
^ [X: $i,Y: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( group_member(X,f)
& group_member(Y,f) )
<=> ~ ( ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
( ~ ( group_member(X,f)
& group_member(Y,f) )
<=> ~ ~ ( ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
rewrite(( ~ ~ ( ~ group_member(Y,f)
| ~ group_member(X,f) )
<=> ( ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
( ~ ( group_member(X,f)
& group_member(Y,f) )
<=> ( ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
( ( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ( ~ group_member(Y,f)
| ~ group_member(X,f)
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ) )),
rewrite(( ( ~ group_member(Y,f)
| ~ group_member(X,f)
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
( ( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ) )),
inference(bind,[status(th)],[]) ).
tff(91,plain,
( ! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ) ),
inference(quant_intro,[status(thm)],[90]) ).
tff(92,plain,
( ! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,plain,
^ [X: $i,Y: $i] :
rewrite(( ( ( group_member(X,f)
& group_member(Y,f) )
=> ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ) )),
inference(bind,[status(th)],[]) ).
tff(94,plain,
( ! [X: $i,Y: $i] :
( ( group_member(X,f)
& group_member(Y,f) )
=> ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )
<=> ! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ) ),
inference(quant_intro,[status(thm)],[93]) ).
tff(95,axiom,
! [X: $i,Y: $i] :
( ( group_member(X,f)
& group_member(Y,f) )
=> ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',homomorphism2) ).
tff(96,plain,
! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ),
inference(modus_ponens,[status(thm)],[95,94]) ).
tff(97,plain,
! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ),
inference(modus_ponens,[status(thm)],[96,92]) ).
tff(98,plain,
! [X: $i,Y: $i] :
( ~ ( group_member(X,f)
& group_member(Y,f) )
| ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) ),
inference(skolemize,[status(sab)],[97]) ).
tff(99,plain,
! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ),
inference(modus_ponens,[status(thm)],[98,91]) ).
tff(100,plain,
! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) ),
inference(modus_ponens,[status(thm)],[99,89]) ).
tff(101,plain,
( ( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ~ group_member(f_left_zero,f)
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) ) )
<=> ( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ~ group_member(f_left_zero,f)
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(102,plain,
( ( ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ~ group_member(f_left_zero,f) )
<=> ( ~ group_member(f_left_zero,f)
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
( ( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ~ group_member(f_left_zero,f) )
<=> ( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ~ group_member(f_left_zero,f)
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) ) ) ),
inference(monotonicity,[status(thm)],[102]) ).
tff(104,plain,
( ( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ~ group_member(f_left_zero,f) )
<=> ( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ~ group_member(f_left_zero,f)
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) ) ) ),
inference(transitivity,[status(thm)],[103,101]) ).
tff(105,plain,
( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) )
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ~ group_member(f_left_zero,f) ),
inference(quant_inst,[status(thm)],[]) ).
tff(106,plain,
( ~ ! [X: $i,Y: $i] :
( ( multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) )
| ~ group_member(Y,f)
| ~ group_member(X,f) )
| ~ group_member(f_left_zero,f)
| ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) ) ),
inference(modus_ponens,[status(thm)],[105,104]) ).
tff(107,plain,
multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))) = phi(multiply(f,f_left_zero,tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))),
inference(unit_resolution,[status(thm)],[106,100,38,80]) ).
tff(108,plain,
( ~ group_member(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)),f)
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) != tptp_fun_Y_1(phi(f_left_zero),h) )
| ( phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = tptp_fun_Y_1(phi(f_left_zero),h) ) ),
inference(tautology,[status(thm)],[]) ).
tff(109,plain,
phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))) = tptp_fun_Y_1(phi(f_left_zero),h),
inference(unit_resolution,[status(thm)],[108,78]) ).
tff(110,plain,
tptp_fun_Y_1(phi(f_left_zero),h) = phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h))),
inference(symmetry,[status(thm)],[109]) ).
tff(111,plain,
multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = multiply(h,phi(f_left_zero),phi(tptp_fun_Y_0(tptp_fun_Y_1(phi(f_left_zero),h)))),
inference(monotonicity,[status(thm)],[110]) ).
tff(112,plain,
multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero),
inference(transitivity,[status(thm)],[111,107,87]) ).
tff(113,plain,
( ~ group_member(tptp_fun_Y_1(phi(f_left_zero),h),h)
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) = phi(f_left_zero) )
| ( multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) != phi(f_left_zero) ) ),
inference(tautology,[status(thm)],[]) ).
tff(114,plain,
multiply(h,phi(f_left_zero),tptp_fun_Y_1(phi(f_left_zero),h)) != phi(f_left_zero),
inference(unit_resolution,[status(thm)],[113,59]) ).
tff(115,plain,
$false,
inference(unit_resolution,[status(thm)],[114,112]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.14 % Problem : GRP194+1 : TPTP v9.0.0. Released v2.0.0.
% 0.12/0.14 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n002.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu Jun 19 15:49:01 EDT 2025
% 0.14/0.36 % CPUTime :
% 0.14/0.39 % SZS status Theorem
% 0.14/0.39 % SZS output start Proof
% See solution above
% 0.14/0.42 % E exiting
%------------------------------------------------------------------------------