%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP003-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:06:48 AM UTC 2025
% Result : Unsatisfiable 2.84s 3.04s
% Output : Proof 2.84s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 50
% Syntax : Number of formulae : 115 ( 37 unt; 0 typ; 0 def)
% Number of atoms : 685 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 1047 ( 500 ~; 490 |; 0 &)
% ( 57 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 23 ( 23 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 : 4 ( 4 usr; 2 con; 0-2 aty)
% Number of variables : 536 ( 492 !; 0 ?; 536 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(a_type,type,
a: $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(identity_type,type,
identity: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(equalish_type,type,
equalish: ( $i * $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/sandbox/benchmark/Axioms/GRP005-0.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(inverse(a)),inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(inverse(inverse(a)),inverse(a),identity),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Y: $i,X: $i] :
refl(( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],[]) ).
tff(11,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)],[10]) ).
tff(12,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(13,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax',total_function1) ).
tff(14,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(a,identity,multiply(a,identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
product(a,identity,multiply(a,identity)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(identity,identity,multiply(identity,identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(20,plain,
product(identity,identity,multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[19,16]) ).
tff(21,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(22,plain,
product(inverse(a),a,identity),
inference(unit_resolution,[status(thm)],[21,7]) ).
tff(23,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(24,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)],[23]) ).
tff(25,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(26,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(27,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)],[26]) ).
tff(28,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/sandbox/benchmark/Axioms/GRP005-0.ax',associativity1) ).
tff(29,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)],[28,27]) ).
tff(30,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)],[29,25]) ).
tff(31,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)],[30]) ).
tff(32,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)],[31,24]) ).
tff(33,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(identity,identity,multiply(identity,identity))
| product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(a),a,identity)
| ~ product(a,identity,multiply(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(identity,identity,multiply(identity,identity))
| product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(a),a,identity)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,plain,
( ( product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(a,identity,multiply(a,identity))
| ~ product(inverse(a),a,identity) )
<=> ( ~ product(identity,identity,multiply(identity,identity))
| product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(a),a,identity)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(35,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(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(a,identity,multiply(a,identity))
| ~ product(inverse(a),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(identity,identity,multiply(identity,identity))
| product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(a),a,identity)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(monotonicity,[status(thm)],[34]) ).
tff(36,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(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(a,identity,multiply(a,identity))
| ~ product(inverse(a),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(identity,identity,multiply(identity,identity))
| product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(a),a,identity)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(transitivity,[status(thm)],[35,33]) ).
tff(37,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(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(a,identity,multiply(a,identity))
| ~ product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,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(identity,identity,multiply(identity,identity))
| product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(a),a,identity)
| ~ product(a,identity,multiply(a,identity)) ),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,plain,
product(inverse(a),multiply(a,identity),multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[38,32,22,20,18]) ).
tff(40,plain,
^ [X: $i] :
refl(( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],[]) ).
tff(41,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[40]) ).
tff(42,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(43,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax',left_identity) ).
tff(44,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[44]) ).
tff(46,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[45,41]) ).
tff(47,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,a,a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(48,plain,
product(identity,a,a),
inference(unit_resolution,[status(thm)],[47,46]) ).
tff(49,plain,
( ~ product(a,identity,a)
<=> ~ product(a,identity,a) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,axiom,
~ product(a,identity,a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_right_identity) ).
tff(51,plain,
~ product(a,identity,a),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,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(53,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)],[52]) ).
tff(54,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(55,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(56,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)],[55]) ).
tff(57,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/sandbox/benchmark/Axioms/GRP005-0.ax',associativity2) ).
tff(58,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)],[57,56]) ).
tff(59,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)],[58,54]) ).
tff(60,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)],[59]) ).
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)],[60,53]) ).
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) )
| product(a,identity,a)
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a)
| ~ product(a,identity,multiply(a,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(a,identity,a)
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(63,plain,
( ( product(a,identity,a)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a) )
<=> ( product(a,identity,a)
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
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) )
| product(a,identity,a)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a) )
<=> ( ~ ! [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(a,identity,a)
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(monotonicity,[status(thm)],[63]) ).
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(a,identity,a)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a) )
<=> ( ~ ! [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(a,identity,a)
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a)
| ~ product(a,identity,multiply(a,identity)) ) ),
inference(transitivity,[status(thm)],[64,62]) ).
tff(66,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(a,identity,a)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a) ),
inference(quant_inst,[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(a,identity,a)
| ~ product(identity,a,a)
| ~ product(identity,multiply(a,identity),a)
| ~ product(a,identity,multiply(a,identity)) ),
inference(modus_ponens,[status(thm)],[66,65]) ).
tff(68,plain,
~ product(identity,multiply(a,identity),a),
inference(unit_resolution,[status(thm)],[67,61,51,48,18]) ).
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,multiply(a,identity),a)
| ~ product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(inverse(a)),inverse(a),identity)
| ~ product(inverse(inverse(a)),multiply(identity,identity),a) )
<=> ( ~ ! [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,multiply(a,identity),a)
| ~ product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(inverse(a)),inverse(a),identity)
| ~ product(inverse(inverse(a)),multiply(identity,identity),a) ) ),
inference(rewrite,[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,multiply(a,identity),a)
| ~ product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(inverse(a)),inverse(a),identity)
| ~ product(inverse(inverse(a)),multiply(identity,identity),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(71,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,multiply(a,identity),a)
| ~ product(inverse(a),multiply(a,identity),multiply(identity,identity))
| ~ product(inverse(inverse(a)),inverse(a),identity)
| ~ product(inverse(inverse(a)),multiply(identity,identity),a) ),
inference(modus_ponens,[status(thm)],[70,69]) ).
tff(72,plain,
~ product(inverse(inverse(a)),multiply(identity,identity),a),
inference(unit_resolution,[status(thm)],[71,61,68,39,9]) ).
tff(73,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,identity,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(74,plain,
product(identity,identity,identity),
inference(unit_resolution,[status(thm)],[73,46]) ).
tff(75,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
refl(( ( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(76,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[75]) ).
tff(77,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(78,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
rewrite(( ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| equalish(Z,W) )
<=> ( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(79,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| equalish(Z,W) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[78]) ).
tff(80,axiom,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| equalish(Z,W) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax',total_function2) ).
tff(81,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[81,77]) ).
tff(83,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[82]) ).
tff(84,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[83,76]) ).
tff(85,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,identity,identity)
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,identity,identity)
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(86,plain,
( ( equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(identity,identity,identity) )
<=> ( ~ product(identity,identity,identity)
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(identity,identity,identity) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,identity,identity)
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity)) ) ),
inference(monotonicity,[status(thm)],[86]) ).
tff(88,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(identity,identity,identity) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,identity,identity)
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity)) ) ),
inference(transitivity,[status(thm)],[87,85]) ).
tff(89,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity))
| ~ product(identity,identity,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(90,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( equalish(Z,W)
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,identity,identity)
| equalish(identity,multiply(identity,identity))
| ~ product(identity,identity,multiply(identity,identity)) ),
inference(modus_ponens,[status(thm)],[89,88]) ).
tff(91,plain,
equalish(identity,multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[90,84,74,20]) ).
tff(92,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
<=> ( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(93,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ) ),
inference(quant_intro,[status(thm)],[92]) ).
tff(94,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(95,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
rewrite(( ( ~ equalish(X,Y)
| ~ product(W,Z,X)
| product(W,Z,Y) )
<=> ( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(96,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ equalish(X,Y)
| ~ product(W,Z,X)
| product(W,Z,Y) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ) ),
inference(quant_intro,[status(thm)],[95]) ).
tff(97,axiom,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ equalish(X,Y)
| ~ product(W,Z,X)
| product(W,Z,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP005-0.ax',product_substitution3) ).
tff(98,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ),
inference(modus_ponens,[status(thm)],[97,96]) ).
tff(99,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ),
inference(modus_ponens,[status(thm)],[98,94]) ).
tff(100,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ),
inference(skolemize,[status(sab)],[99]) ).
tff(101,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) ),
inference(modus_ponens,[status(thm)],[100,93]) ).
tff(102,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity))
| ~ product(inverse(a),a,identity) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity))
| ~ product(inverse(a),a,identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
( ( ~ product(inverse(a),a,identity)
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity)) )
<=> ( product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity))
| ~ product(inverse(a),a,identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(104,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| ~ product(inverse(a),a,identity)
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity))
| ~ product(inverse(a),a,identity) ) ),
inference(monotonicity,[status(thm)],[103]) ).
tff(105,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| ~ product(inverse(a),a,identity)
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity))
| ~ product(inverse(a),a,identity) ) ),
inference(transitivity,[status(thm)],[104,102]) ).
tff(106,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| ~ product(inverse(a),a,identity)
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(W,Z,X)
| product(W,Z,Y)
| ~ equalish(X,Y) )
| product(inverse(a),a,multiply(identity,identity))
| ~ equalish(identity,multiply(identity,identity))
| ~ product(inverse(a),a,identity) ),
inference(modus_ponens,[status(thm)],[106,105]) ).
tff(108,plain,
product(inverse(a),a,multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[107,101,22,91]) ).
tff(109,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(identity,a,a)
| ~ product(inverse(inverse(a)),inverse(a),identity)
| product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(inverse(a),a,multiply(identity,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(identity,a,a)
| ~ product(inverse(inverse(a)),inverse(a),identity)
| product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(inverse(a),a,multiply(identity,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(110,plain,
( ( product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(identity,a,a)
| ~ product(inverse(a),a,multiply(identity,identity))
| ~ product(inverse(inverse(a)),inverse(a),identity) )
<=> ( ~ product(identity,a,a)
| ~ product(inverse(inverse(a)),inverse(a),identity)
| product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(inverse(a),a,multiply(identity,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(111,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(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(identity,a,a)
| ~ product(inverse(a),a,multiply(identity,identity))
| ~ product(inverse(inverse(a)),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(identity,a,a)
| ~ product(inverse(inverse(a)),inverse(a),identity)
| product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(inverse(a),a,multiply(identity,identity)) ) ),
inference(monotonicity,[status(thm)],[110]) ).
tff(112,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(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(identity,a,a)
| ~ product(inverse(a),a,multiply(identity,identity))
| ~ product(inverse(inverse(a)),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(identity,a,a)
| ~ product(inverse(inverse(a)),inverse(a),identity)
| product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(inverse(a),a,multiply(identity,identity)) ) ),
inference(transitivity,[status(thm)],[111,109]) ).
tff(113,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(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(identity,a,a)
| ~ product(inverse(a),a,multiply(identity,identity))
| ~ product(inverse(inverse(a)),inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(114,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(identity,a,a)
| ~ product(inverse(inverse(a)),inverse(a),identity)
| product(inverse(inverse(a)),multiply(identity,identity),a)
| ~ product(inverse(a),a,multiply(identity,identity)) ),
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
$false,
inference(unit_resolution,[status(thm)],[114,32,48,108,9,72]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP003-2 : TPTP v9.0.0. Released v1.0.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n003.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 15:27:54 EDT 2025
% 0.12/0.33 % CPUTime :
% 2.84/3.04 % SZS status Unsatisfiable
% 2.84/3.04 % SZS output start Proof
% See solution above
% 2.84/3.06 % E exiting
%------------------------------------------------------------------------------