%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP022-1 : 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 05:06:52 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.47s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 40
% Syntax : Number of formulae : 104 ( 44 unt; 0 typ; 0 def)
% Number of atoms : 424 ( 79 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 573 ( 267 ~; 261 |; 0 &)
% ( 45 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of FOOLs : 14 ( 14 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-2 aty)
% Number of variables : 304 ( 279 !; 0 ?; 304 :)
% Comments :
%------------------------------------------------------------------------------
tff(a_type,type,
a: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(identity_type,type,
identity: $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(1,plain,
^ [Y: $i,X: $i] :
refl(( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
<=> ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
<=> ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',total_function1) ).
tff(5,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(identity,inverse(a),multiply(identity,inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(identity,inverse(a),multiply(identity,inverse(a))),
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,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
product(identity,inverse(a),inverse(a)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,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(23,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)],[22]) ).
tff(24,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(25,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[26]) ).
tff(28,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[27,20]) ).
tff(29,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ( inverse(a) = multiply(identity,inverse(a)) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ( inverse(a) = multiply(identity,inverse(a)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
( ( ( inverse(a) = multiply(identity,inverse(a)) )
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ( inverse(a) = multiply(identity,inverse(a)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(31,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(identity,inverse(a)) )
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ( inverse(a) = multiply(identity,inverse(a)) ) ) ),
inference(monotonicity,[status(thm)],[30]) ).
tff(32,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(identity,inverse(a)) )
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ( inverse(a) = multiply(identity,inverse(a)) ) ) ),
inference(transitivity,[status(thm)],[31,29]) ).
tff(33,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(identity,inverse(a)) )
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(34,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),multiply(identity,inverse(a)))
| ( inverse(a) = multiply(identity,inverse(a)) ) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
inverse(a) = multiply(identity,inverse(a)),
inference(unit_resolution,[status(thm)],[34,28,18,9]) ).
tff(36,plain,
multiply(identity,inverse(a)) = inverse(a),
inference(symmetry,[status(thm)],[35]) ).
tff(37,plain,
( product(a,multiply(identity,inverse(a)),identity)
<=> product(a,inverse(a),identity) ),
inference(monotonicity,[status(thm)],[36]) ).
tff(38,plain,
( product(a,inverse(a),identity)
<=> product(a,multiply(identity,inverse(a)),identity) ),
inference(symmetry,[status(thm)],[37]) ).
tff(39,plain,
^ [X: $i] :
refl(( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(43,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(a,inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
product(a,inverse(a),identity),
inference(unit_resolution,[status(thm)],[46,45]) ).
tff(48,plain,
product(a,multiply(identity,inverse(a)),identity),
inference(modus_ponens,[status(thm)],[47,38]) ).
tff(49,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))),
inference(unit_resolution,[status(thm)],[49,7]) ).
tff(51,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(inverse(a)),inverse(inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(52,plain,
product(identity,inverse(inverse(a)),inverse(inverse(a))),
inference(unit_resolution,[status(thm)],[51,16]) ).
tff(53,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a)))
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a)))
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(54,plain,
( ( ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a))))
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a))) )
<=> ( ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a)))
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a))))
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a)))
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) ) ),
inference(monotonicity,[status(thm)],[54]) ).
tff(56,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a))))
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a)))
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) ) ),
inference(transitivity,[status(thm)],[55,53]) ).
tff(57,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a))))
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(58,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(inverse(a)) = multiply(identity,inverse(inverse(a))) )
| ~ product(identity,inverse(inverse(a)),inverse(inverse(a)))
| ~ product(identity,inverse(inverse(a)),multiply(identity,inverse(inverse(a)))) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
inverse(inverse(a)) = multiply(identity,inverse(inverse(a))),
inference(unit_resolution,[status(thm)],[58,28,52,50]) ).
tff(60,plain,
multiply(identity,inverse(inverse(a))) = inverse(inverse(a)),
inference(symmetry,[status(thm)],[59]) ).
tff(61,plain,
( product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity)
<=> product(inverse(a),inverse(inverse(a)),identity) ),
inference(monotonicity,[status(thm)],[36,60]) ).
tff(62,plain,
( product(inverse(a),inverse(inverse(a)),identity)
<=> product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity) ),
inference(symmetry,[status(thm)],[61]) ).
tff(63,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(inverse(a),inverse(inverse(a)),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(64,plain,
product(inverse(a),inverse(inverse(a)),identity),
inference(unit_resolution,[status(thm)],[63,45]) ).
tff(65,plain,
product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity),
inference(modus_ponens,[status(thm)],[64,62]) ).
tff(66,plain,
( product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a))))
<=> product(identity,inverse(inverse(a)),inverse(inverse(a))) ),
inference(monotonicity,[status(thm)],[60,60]) ).
tff(67,plain,
( product(identity,inverse(inverse(a)),inverse(inverse(a)))
<=> product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a)))) ),
inference(symmetry,[status(thm)],[66]) ).
tff(68,plain,
product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a)))),
inference(modus_ponens,[status(thm)],[52,67]) ).
tff(69,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(( ( product(X,V,W)
| ~ 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) ) )),
inference(bind,[status(th)],[]) ).
tff(70,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(72,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(U,Z,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
rewrite(( ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
<=> ( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) )
<=> ( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
inference(bind,[status(th)],[]) ).
tff(73,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',associativity1) ).
tff(75,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(modus_ponens,[status(thm)],[74,73]) ).
tff(76,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(modus_ponens,[status(thm)],[75,71]) ).
tff(77,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(modus_ponens,[status(thm)],[77,70]) ).
tff(79,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a))))
| ~ product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity)
| ~ product(a,multiply(identity,inverse(a)),identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a))))
| ~ product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity)
| ~ product(a,multiply(identity,inverse(a)),identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a))))
| ~ product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity)
| ~ product(a,multiply(identity,inverse(a)),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(81,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a))))
| ~ product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity)
| ~ product(a,multiply(identity,inverse(a)),identity) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
( product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(identity,multiply(identity,inverse(inverse(a))),multiply(identity,inverse(inverse(a))))
| ~ product(multiply(identity,inverse(a)),multiply(identity,inverse(inverse(a))),identity)
| ~ product(a,multiply(identity,inverse(a)),identity) ),
inference(unit_resolution,[status(thm)],[81,78]) ).
tff(83,plain,
product(a,identity,multiply(identity,inverse(inverse(a)))),
inference(unit_resolution,[status(thm)],[82,68,65,48]) ).
tff(84,plain,
^ [X: $i] :
refl(( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],[]) ).
tff(85,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[84]) ).
tff(86,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(88,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[87,86]) ).
tff(89,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[88]) ).
tff(90,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[89,85]) ).
tff(91,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(a,identity,a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(92,plain,
product(a,identity,a),
inference(unit_resolution,[status(thm)],[91,90]) ).
tff(93,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(a,identity,a) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(a,identity,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(94,plain,
( ( ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,a)
| ~ product(a,identity,multiply(identity,inverse(inverse(a)))) )
<=> ( ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(a,identity,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(95,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,a)
| ~ product(a,identity,multiply(identity,inverse(inverse(a)))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(a,identity,a) ) ),
inference(monotonicity,[status(thm)],[94]) ).
tff(96,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,a)
| ~ product(a,identity,multiply(identity,inverse(inverse(a)))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(a,identity,a) ) ),
inference(transitivity,[status(thm)],[95,93]) ).
tff(97,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,a)
| ~ product(a,identity,multiply(identity,inverse(inverse(a)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(98,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(identity,inverse(inverse(a))) = a )
| ~ product(a,identity,multiply(identity,inverse(inverse(a))))
| ~ product(a,identity,a) ),
inference(modus_ponens,[status(thm)],[97,96]) ).
tff(99,plain,
multiply(identity,inverse(inverse(a))) = a,
inference(unit_resolution,[status(thm)],[98,28,92,83]) ).
tff(100,plain,
inverse(inverse(a)) = a,
inference(transitivity,[status(thm)],[59,99]) ).
tff(101,plain,
( ( inverse(inverse(a)) != a )
<=> ( inverse(inverse(a)) != a ) ),
inference(rewrite,[status(thm)],[]) ).
tff(102,axiom,
inverse(inverse(a)) != a,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_inverse_of_inverse_is_original) ).
tff(103,plain,
inverse(inverse(a)) != a,
inference(modus_ponens,[status(thm)],[102,101]) ).
tff(104,plain,
$false,
inference(unit_resolution,[status(thm)],[103,100]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP022-1 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 15:29:31 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.47 % SZS status Unsatisfiable
% 0.19/0.47 % SZS output start Proof
% See solution above
% 0.19/0.48 % E exiting
%------------------------------------------------------------------------------