%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP036-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 05:06:55 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 31
% Syntax : Number of formulae : 71 ( 25 unt; 0 typ; 0 def)
% Number of atoms : 309 ( 34 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 428 ( 203 ~; 192 |; 0 &)
% ( 33 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 13 ( 13 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 3 ( 3 usr; 2 con; 0-1 aty)
% Number of variables : 227 ( 204 !; 0 ?; 227 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(identity_type,type,
identity: $i ).
tff(another_identity_type,type,
another_identity: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(subgroup_member_type,type,
subgroup_member: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(( product(inverse(X),X,identity)
<=> product(inverse(X),X,identity) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : product(inverse(X),X,identity),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',left_inverse) ).
tff(5,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(another_identity),another_identity,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(inverse(another_identity),another_identity,identity),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(14,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,another_identity,another_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
product(identity,another_identity,another_identity),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
( ( another_identity = identity )
<=> ( identity = another_identity ) ),
inference(commutativity,[status(thm)],[]) ).
tff(20,plain,
( ( identity = another_identity )
<=> ( another_identity = identity ) ),
inference(symmetry,[status(thm)],[19]) ).
tff(21,plain,
( ( identity != another_identity )
<=> ( another_identity != identity ) ),
inference(monotonicity,[status(thm)],[20]) ).
tff(22,plain,
( ( identity != another_identity )
<=> ( identity != another_identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,axiom,
identity != another_identity,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_identity_equals_another_identity) ).
tff(24,plain,
identity != another_identity,
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
another_identity != identity,
inference(modus_ponens,[status(thm)],[24,21]) ).
tff(26,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
refl(( ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(27,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[26]) ).
tff(28,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
rewrite(( ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(30,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[29]) ).
tff(31,axiom,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',total_function2) ).
tff(32,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[32,28]) ).
tff(34,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[33]) ).
tff(35,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[34,27]) ).
tff(36,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( another_identity = identity )
| ~ product(identity,another_identity,identity)
| ~ product(identity,another_identity,another_identity) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( another_identity = identity )
| ~ product(identity,another_identity,identity)
| ~ product(identity,another_identity,another_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(37,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( another_identity = identity )
| ~ product(identity,another_identity,identity)
| ~ product(identity,another_identity,another_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( another_identity = identity )
| ~ product(identity,another_identity,identity)
| ~ product(identity,another_identity,another_identity) ),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,plain,
( ~ product(identity,another_identity,identity)
| ~ product(identity,another_identity,another_identity) ),
inference(unit_resolution,[status(thm)],[38,35,25]) ).
tff(40,plain,
~ product(identity,another_identity,identity),
inference(unit_resolution,[status(thm)],[39,18]) ).
tff(41,plain,
( subgroup_member(another_identity)
<=> subgroup_member(another_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
subgroup_member(another_identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',another_identity_in_subgroup) ).
tff(43,plain,
subgroup_member(another_identity),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
^ [A: $i] :
refl(( ( ~ subgroup_member(A)
| product(another_identity,A,A) )
<=> ( ~ subgroup_member(A)
| product(another_identity,A,A) ) )),
inference(bind,[status(th)],[]) ).
tff(45,plain,
( ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) )
<=> ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) ) ),
inference(quant_intro,[status(thm)],[44]) ).
tff(46,plain,
( ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) )
<=> ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,axiom,
! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',another_left_identity) ).
tff(48,plain,
! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) ),
inference(modus_ponens,[status(thm)],[47,46]) ).
tff(49,plain,
! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) ),
inference(skolemize,[status(sab)],[48]) ).
tff(50,plain,
! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) ),
inference(modus_ponens,[status(thm)],[49,45]) ).
tff(51,plain,
( ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) )
| ~ subgroup_member(another_identity)
| product(another_identity,another_identity,another_identity) )
<=> ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) )
| ~ subgroup_member(another_identity)
| product(another_identity,another_identity,another_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(52,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) )
| ~ subgroup_member(another_identity)
| product(another_identity,another_identity,another_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(53,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| product(another_identity,A,A) )
| ~ subgroup_member(another_identity)
| product(another_identity,another_identity,another_identity) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
product(another_identity,another_identity,another_identity),
inference(unit_resolution,[status(thm)],[53,50,43]) ).
tff(55,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(( ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
<=> ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
inference(bind,[status(th)],[]) ).
tff(56,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) ),
inference(quant_intro,[status(thm)],[55]) ).
tff(57,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W)
| product(U,Z,W) ) )),
rewrite(( ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
inference(bind,[status(th)],[]) ).
tff(59,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) ),
inference(quant_intro,[status(thm)],[58]) ).
tff(60,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',associativity2) ).
tff(61,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(skolemize,[status(sab)],[62]) ).
tff(64,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(modus_ponens,[status(thm)],[63,56]) ).
tff(65,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(66,plain,
( ( product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity)
| ~ product(inverse(another_identity),another_identity,identity) )
<=> ( product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(67,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity)
| ~ product(inverse(another_identity),another_identity,identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity) ) ),
inference(monotonicity,[status(thm)],[66]) ).
tff(68,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity)
| ~ product(inverse(another_identity),another_identity,identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity) ) ),
inference(transitivity,[status(thm)],[67,65]) ).
tff(69,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity)
| ~ product(inverse(another_identity),another_identity,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(70,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,another_identity,identity)
| ~ product(another_identity,another_identity,another_identity)
| ~ product(inverse(another_identity),another_identity,identity) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
$false,
inference(unit_resolution,[status(thm)],[70,64,54,40,9]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP036-3 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n004.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 15:28:08 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.39 % SZS status Unsatisfiable
% 0.20/0.39 % SZS output start Proof
% See solution above
% 0.20/0.41 % E exiting
%------------------------------------------------------------------------------