%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP013-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n009.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 05:06:51 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.46s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 61
% Syntax : Number of formulae : 142 ( 52 unt; 0 typ; 0 def)
% Number of atoms : 671 ( 34 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 963 ( 457 ~; 445 |; 0 &)
% ( 61 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 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 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 6 ( 6 usr; 5 con; 0-1 aty)
% Number of variables : 502 ( 463 !; 0 ?; 502 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(a_type,type,
a: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(identity_type,type,
identity: $i ).
tff(c_type,type,
c: $i ).
tff(b_type,type,
b: $i ).
tff(d_type,type,
d: $i ).
tff(1,plain,
^ [X: $i] :
refl(( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(5,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(inverse(identity),inverse(inverse(identity)),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(inverse(identity),inverse(inverse(identity)),identity),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [B: $i,A: $i,C: $i] :
refl(( ( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
<=> ( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
<=> ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
<=> ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverses_have_property) ).
tff(14,plain,
! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(inverse(identity)),identity)
| product(identity,identity,inverse(identity)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(inverse(identity)),identity)
| product(identity,identity,inverse(identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(inverse(identity)),identity)
| product(identity,identity,inverse(identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(inverse(identity)),identity)
| product(identity,identity,inverse(identity)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
product(identity,identity,inverse(identity)),
inference(unit_resolution,[status(thm)],[19,16,9]) ).
tff(21,plain,
^ [X: $i] :
refl(( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],[]) ).
tff(22,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(25,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[26,22]) ).
tff(28,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,identity,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(29,plain,
product(identity,identity,identity),
inference(unit_resolution,[status(thm)],[28,27]) ).
tff(30,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(31,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)],[30]) ).
tff(32,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(33,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(34,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)],[33]) ).
tff(35,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(36,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[38,31]) ).
tff(40,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,identity,identity)
| ~ product(identity,identity,inverse(identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,identity,identity)
| ~ product(identity,identity,inverse(identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,identity,identity)
| ~ product(identity,identity,inverse(identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(42,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,identity,identity)
| ~ product(identity,identity,inverse(identity)) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
inverse(identity) = identity,
inference(unit_resolution,[status(thm)],[42,39,29,20]) ).
tff(44,plain,
( product(inverse(identity),inverse(a),inverse(a))
<=> product(identity,inverse(a),inverse(a)) ),
inference(monotonicity,[status(thm)],[43]) ).
tff(45,plain,
( product(identity,inverse(a),inverse(a))
<=> product(inverse(identity),inverse(a),inverse(a)) ),
inference(symmetry,[status(thm)],[44]) ).
tff(46,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
product(identity,inverse(a),inverse(a)),
inference(unit_resolution,[status(thm)],[46,27]) ).
tff(48,plain,
product(inverse(identity),inverse(a),inverse(a)),
inference(modus_ponens,[status(thm)],[47,45]) ).
tff(49,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(a),inverse(a))
| product(identity,inverse(a),a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(a),inverse(a))
| product(identity,inverse(a),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(a),inverse(a))
| product(identity,inverse(a),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(identity),inverse(a),inverse(a))
| product(identity,inverse(a),a) ),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,plain,
( ~ product(inverse(identity),inverse(a),inverse(a))
| product(identity,inverse(a),a) ),
inference(unit_resolution,[status(thm)],[51,16]) ).
tff(53,plain,
product(identity,inverse(a),a),
inference(unit_resolution,[status(thm)],[52,48]) ).
tff(54,plain,
^ [X: $i] :
refl(( product(inverse(X),X,identity)
<=> product(inverse(X),X,identity) )),
inference(bind,[status(th)],[]) ).
tff(55,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(quant_intro,[status(thm)],[54]) ).
tff(56,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(57,axiom,
! [X: $i] : product(inverse(X),X,identity),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',left_inverse) ).
tff(58,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(skolemize,[status(sab)],[58]) ).
tff(60,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(inverse(c)),inverse(c),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(62,plain,
product(inverse(inverse(c)),inverse(c),identity),
inference(unit_resolution,[status(thm)],[61,60]) ).
tff(63,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(inverse(c)),inverse(c),identity)
| product(inverse(c),identity,c) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(inverse(c)),inverse(c),identity)
| product(inverse(c),identity,c) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(64,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(inverse(c)),inverse(c),identity)
| product(inverse(c),identity,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(65,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(inverse(c)),inverse(c),identity)
| product(inverse(c),identity,c) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
product(inverse(c),identity,c),
inference(unit_resolution,[status(thm)],[65,16,62]) ).
tff(67,plain,
^ [X: $i] :
refl(( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],[]) ).
tff(68,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(70,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(71,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[70,69]) ).
tff(72,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[71]) ).
tff(73,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(c),identity,inverse(c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(75,plain,
product(inverse(c),identity,inverse(c)),
inference(unit_resolution,[status(thm)],[74,73]) ).
tff(76,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( c = inverse(c) )
| ~ product(inverse(c),identity,inverse(c))
| ~ product(inverse(c),identity,c) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( c = inverse(c) )
| ~ product(inverse(c),identity,inverse(c))
| ~ product(inverse(c),identity,c) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(77,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( c = inverse(c) )
| ~ product(inverse(c),identity,inverse(c))
| ~ product(inverse(c),identity,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(78,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( c = inverse(c) )
| ~ product(inverse(c),identity,inverse(c))
| ~ product(inverse(c),identity,c) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
c = inverse(c),
inference(unit_resolution,[status(thm)],[78,39,75,66]) ).
tff(80,plain,
inverse(c) = c,
inference(symmetry,[status(thm)],[79]) ).
tff(81,plain,
( product(a,b,inverse(c))
<=> product(a,b,c) ),
inference(monotonicity,[status(thm)],[80]) ).
tff(82,plain,
( product(a,b,c)
<=> product(a,b,inverse(c)) ),
inference(symmetry,[status(thm)],[81]) ).
tff(83,plain,
( product(a,b,c)
<=> product(a,b,c) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,axiom,
product(a,b,c),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_times_b_is_c) ).
tff(85,plain,
product(a,b,c),
inference(modus_ponens,[status(thm)],[84,83]) ).
tff(86,plain,
product(a,b,inverse(c)),
inference(modus_ponens,[status(thm)],[85,82]) ).
tff(87,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(b,inverse(b),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(88,plain,
product(b,inverse(b),identity),
inference(unit_resolution,[status(thm)],[87,7]) ).
tff(89,plain,
( ~ product(c,d,identity)
<=> ~ product(c,d,identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(90,axiom,
~ product(c,d,identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_c_times_d_is_identity) ).
tff(91,plain,
~ product(c,d,identity),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
( product(inverse(a),inverse(b),d)
<=> product(inverse(a),inverse(b),d) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,axiom,
product(inverse(a),inverse(b),d),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',inverse_a_times_inverse_b_is_d) ).
tff(94,plain,
product(inverse(a),inverse(b),d),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,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(96,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)],[95]) ).
tff(97,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(98,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(99,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)],[98]) ).
tff(100,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(101,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)],[100,99]) ).
tff(102,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)],[101,97]) ).
tff(103,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)],[102]) ).
tff(104,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)],[103,96]) ).
tff(105,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(c,d,identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(b,inverse(b),identity)
| ~ product(c,inverse(a),b) )
<=> ( ~ ! [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(c,d,identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(b,inverse(b),identity)
| ~ product(c,inverse(a),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(106,plain,
( ( product(c,d,identity)
| ~ product(b,inverse(b),identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(c,inverse(a),b) )
<=> ( product(c,d,identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(b,inverse(b),identity)
| ~ product(c,inverse(a),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(107,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(c,d,identity)
| ~ product(b,inverse(b),identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(c,inverse(a),b) )
<=> ( ~ ! [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(c,d,identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(b,inverse(b),identity)
| ~ product(c,inverse(a),b) ) ),
inference(monotonicity,[status(thm)],[106]) ).
tff(108,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(c,d,identity)
| ~ product(b,inverse(b),identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(c,inverse(a),b) )
<=> ( ~ ! [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(c,d,identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(b,inverse(b),identity)
| ~ product(c,inverse(a),b) ) ),
inference(transitivity,[status(thm)],[107,105]) ).
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(c,d,identity)
| ~ product(b,inverse(b),identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(c,inverse(a),b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(110,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(c,d,identity)
| ~ product(inverse(a),inverse(b),d)
| ~ product(b,inverse(b),identity)
| ~ product(c,inverse(a),b) ),
inference(modus_ponens,[status(thm)],[109,108]) ).
tff(111,plain,
~ product(c,inverse(a),b),
inference(unit_resolution,[status(thm)],[110,104,94,91,88]) ).
tff(112,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(c),inverse(b),inverse(a))
| product(c,inverse(a),b) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(c),inverse(b),inverse(a))
| product(c,inverse(a),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(113,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(c),inverse(b),inverse(a))
| product(c,inverse(a),b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(114,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( ~ product(inverse(A),inverse(B),C)
| product(A,C,B) )
| ~ product(inverse(c),inverse(b),inverse(a))
| product(c,inverse(a),b) ),
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
~ product(inverse(c),inverse(b),inverse(a)),
inference(unit_resolution,[status(thm)],[114,16,111]) ).
tff(116,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(117,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)],[116]) ).
tff(118,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(119,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(120,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)],[119]) ).
tff(121,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(122,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)],[121,120]) ).
tff(123,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)],[122,118]) ).
tff(124,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)],[123]) ).
tff(125,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)],[124,117]) ).
tff(126,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(b,inverse(b),identity)
| ~ product(a,identity,inverse(a))
| product(inverse(c),inverse(b),inverse(a))
| ~ product(a,b,inverse(c)) )
<=> ( ~ ! [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(b,inverse(b),identity)
| ~ product(a,identity,inverse(a))
| product(inverse(c),inverse(b),inverse(a))
| ~ product(a,b,inverse(c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(127,plain,
( ( product(inverse(c),inverse(b),inverse(a))
| ~ product(b,inverse(b),identity)
| ~ product(a,b,inverse(c))
| ~ product(a,identity,inverse(a)) )
<=> ( ~ product(b,inverse(b),identity)
| ~ product(a,identity,inverse(a))
| product(inverse(c),inverse(b),inverse(a))
| ~ product(a,b,inverse(c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(128,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(inverse(c),inverse(b),inverse(a))
| ~ product(b,inverse(b),identity)
| ~ product(a,b,inverse(c))
| ~ product(a,identity,inverse(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(b,inverse(b),identity)
| ~ product(a,identity,inverse(a))
| product(inverse(c),inverse(b),inverse(a))
| ~ product(a,b,inverse(c)) ) ),
inference(monotonicity,[status(thm)],[127]) ).
tff(129,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(inverse(c),inverse(b),inverse(a))
| ~ product(b,inverse(b),identity)
| ~ product(a,b,inverse(c))
| ~ product(a,identity,inverse(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(b,inverse(b),identity)
| ~ product(a,identity,inverse(a))
| product(inverse(c),inverse(b),inverse(a))
| ~ product(a,b,inverse(c)) ) ),
inference(transitivity,[status(thm)],[128,126]) ).
tff(130,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(inverse(c),inverse(b),inverse(a))
| ~ product(b,inverse(b),identity)
| ~ product(a,b,inverse(c))
| ~ product(a,identity,inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(131,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(b,inverse(b),identity)
| ~ product(a,identity,inverse(a))
| product(inverse(c),inverse(b),inverse(a))
| ~ product(a,b,inverse(c)) ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
( ~ product(a,identity,inverse(a))
| ~ product(a,b,inverse(c)) ),
inference(unit_resolution,[status(thm)],[131,125,88,115]) ).
tff(133,plain,
~ product(a,identity,inverse(a)),
inference(unit_resolution,[status(thm)],[132,86]) ).
tff(134,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(a),identity,inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(135,plain,
product(inverse(a),identity,inverse(a)),
inference(unit_resolution,[status(thm)],[134,73]) ).
tff(136,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,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(inverse(a),identity,inverse(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,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(inverse(a),identity,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(137,plain,
( ( product(a,identity,inverse(a))
| ~ product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( product(a,identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(inverse(a),identity,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(138,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,inverse(a))
| ~ product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(identity,inverse(a),inverse(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,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(inverse(a),identity,inverse(a)) ) ),
inference(monotonicity,[status(thm)],[137]) ).
tff(139,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,inverse(a))
| ~ product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(identity,inverse(a),inverse(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,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(inverse(a),identity,inverse(a)) ) ),
inference(transitivity,[status(thm)],[138,136]) ).
tff(140,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,inverse(a))
| ~ product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(141,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,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(identity,inverse(a),a)
| ~ product(inverse(a),identity,inverse(a)) ),
inference(modus_ponens,[status(thm)],[140,139]) ).
tff(142,plain,
$false,
inference(unit_resolution,[status(thm)],[141,125,47,135,133,53]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP013-1 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n009.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Thu Jun 19 15:27:48 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.46 % SZS status Unsatisfiable
% 0.19/0.46 % SZS output start Proof
% See solution above
% 0.19/0.48 % E exiting
%------------------------------------------------------------------------------