%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP034-3 : TPTP v9.0.0. Released v1.0.0.
% 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 05:06:54 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 22
% Syntax : Number of formulae : 48 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 263 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 392 ( 187 ~; 181 |; 0 &)
% ( 24 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 10 ( 10 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 3 ( 3 usr; 2 con; 0-1 aty)
% Number of variables : 107 ( 99 !; 0 ?; 107 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(identity_type,type,
identity: $i ).
tff(subgroup_member_type,type,
subgroup_member: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(5,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(identity,inverse(a),inverse(a)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(14,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(a,inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
product(a,inverse(a),identity),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
( subgroup_member(a)
<=> subgroup_member(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(20,axiom,
subgroup_member(a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_is_in_subgroup) ).
tff(21,plain,
subgroup_member(a),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
^ [B: $i,A: $i,C: $i] :
refl(( ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
<=> ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
inference(bind,[status(th)],[]) ).
tff(23,plain,
( ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
<=> ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) ),
inference(quant_intro,[status(thm)],[22]) ).
tff(24,plain,
( ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
<=> ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(25,plain,
^ [B: $i,A: $i,C: $i] :
trans(monotonicity(rewrite(( ( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C) )
<=> ( ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
( ( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) )
<=> ( ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A)
| subgroup_member(C) ) )),
rewrite(( ( ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A)
| subgroup_member(C) )
<=> ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
( ( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) )
<=> ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
inference(bind,[status(th)],[]) ).
tff(26,plain,
( ! [B: $i,A: $i,C: $i] :
( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) )
<=> ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) ),
inference(quant_intro,[status(thm)],[25]) ).
tff(27,axiom,
! [B: $i,A: $i,C: $i] :
( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-2.ax',closure_of_product_and_inverse) ).
tff(28,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(modus_ponens,[status(thm)],[28,24]) ).
tff(30,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(skolemize,[status(sab)],[29]) ).
tff(31,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(modus_ponens,[status(thm)],[30,23]) ).
tff(32,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,plain,
( ( subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) )
<=> ( subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(monotonicity,[status(thm)],[33]) ).
tff(35,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(transitivity,[status(thm)],[34,32]) ).
tff(36,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(37,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
subgroup_member(identity),
inference(unit_resolution,[status(thm)],[37,31,21,18]) ).
tff(39,plain,
( ~ subgroup_member(inverse(a))
<=> ~ subgroup_member(inverse(a)) ),
inference(rewrite,[status(thm)],[]) ).
tff(40,axiom,
~ subgroup_member(inverse(a)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_a_inverse_is_in_subgroup) ).
tff(41,plain,
~ subgroup_member(inverse(a)),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity)
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity)
| ~ product(identity,inverse(a),inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(43,plain,
( ( subgroup_member(inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) )
<=> ( subgroup_member(inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity)
| ~ product(identity,inverse(a),inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity)
| ~ product(identity,inverse(a),inverse(a)) ) ),
inference(monotonicity,[status(thm)],[43]) ).
tff(45,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity)
| ~ product(identity,inverse(a),inverse(a)) ) ),
inference(transitivity,[status(thm)],[44,42]) ).
tff(46,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(a)
| ~ subgroup_member(identity)
| ~ product(identity,inverse(a),inverse(a)) ),
inference(modus_ponens,[status(thm)],[46,45]) ).
tff(48,plain,
$false,
inference(unit_resolution,[status(thm)],[47,31,21,41,38,9]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.11 % Problem : GRP034-3 : TPTP v9.0.0. Released v1.0.0.
% 0.02/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n002.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 15:29:46 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.20/0.39 % SZS status Unsatisfiable
% 0.20/0.39 % SZS output start Proof
% See solution above
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------