%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP183-4 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:07:23 AM UTC 2025
% Result : Unsatisfiable 2.36s 2.52s
% Output : Proof 2.36s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 70
% Syntax : Number of formulae : 212 ( 156 unt; 0 typ; 0 def)
% Number of atoms : 292 ( 279 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 101 ( 33 ~; 29 |; 0 &)
% ( 39 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of FOOLs : 12 ( 12 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 2 con; 0-2 aty)
% Number of variables : 312 ( 285 !; 0 ?; 312 :)
% Comments :
%------------------------------------------------------------------------------
tff(identity_type,type,
identity: $i ).
tff(greatest_lower_bound_type,type,
greatest_lower_bound: ( $i * $i ) > $i ).
tff(least_upper_bound_type,type,
least_upper_bound: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(1,plain,
^ [X: $i] :
refl(( ( multiply(inverse(X),X) = identity )
<=> ( multiply(inverse(X),X) = identity ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_inverse) ).
tff(5,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
multiply(inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))) = identity,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
<=> ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',monotony_lub1) ).
tff(14,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(a),least_upper_bound(a,identity)) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
multiply(inverse(a),least_upper_bound(a,identity)) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [Y: $i,X: $i] :
refl(( ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
<=> ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ) )),
inference(bind,[status(th)],[]) ).
tff(20,plain,
( ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
<=> ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
<=> ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,axiom,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',lub_absorbtion) ).
tff(23,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity))) = least_upper_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(27,plain,
least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity))) = least_upper_bound(a,identity),
inference(unit_resolution,[status(thm)],[26,25]) ).
tff(28,plain,
^ [Y: $i,X: $i] :
refl(( ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
<=> ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ) )),
inference(bind,[status(th)],[]) ).
tff(29,plain,
( ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
<=> ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ) ),
inference(quant_intro,[status(thm)],[28]) ).
tff(30,plain,
( ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
<=> ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(31,axiom,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',symmetry_of_glb) ).
tff(32,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(skolemize,[status(sab)],[32]) ).
tff(34,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[33,29]) ).
tff(35,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(36,plain,
greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)),
inference(symmetry,[status(thm)],[36]) ).
tff(38,plain,
least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity))),
inference(monotonicity,[status(thm)],[37]) ).
tff(39,plain,
^ [Y: $i,X: $i] :
refl(( ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
<=> ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
<=> ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
<=> ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',symmetry_of_lub) ).
tff(43,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity)),
inference(unit_resolution,[status(thm)],[46,45]) ).
tff(48,plain,
least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity)) = least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[47]) ).
tff(49,plain,
least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity)) = least_upper_bound(a,identity),
inference(transitivity,[status(thm)],[48,38,27]) ).
tff(50,plain,
multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity))) = multiply(inverse(a),least_upper_bound(a,identity)),
inference(monotonicity,[status(thm)],[49]) ).
tff(51,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity))) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),least_upper_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(52,plain,
multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity))) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),least_upper_bound(a,identity))),
inference(unit_resolution,[status(thm)],[51,16]) ).
tff(53,plain,
least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),least_upper_bound(a,identity))) = multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[52]) ).
tff(54,plain,
least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)) = multiply(inverse(a),least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[18]) ).
tff(55,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(56,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(57,axiom,
! [X: $i] : ( inverse(inverse(X)) = X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p20x_2) ).
tff(58,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(skolemize,[status(sab)],[58]) ).
tff(60,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(a)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(62,plain,
inverse(inverse(a)) = a,
inference(unit_resolution,[status(thm)],[61,60]) ).
tff(63,plain,
a = inverse(inverse(a)),
inference(symmetry,[status(thm)],[62]) ).
tff(64,plain,
multiply(a,inverse(a)) = multiply(inverse(inverse(a)),inverse(a)),
inference(monotonicity,[status(thm)],[63]) ).
tff(65,plain,
multiply(inverse(inverse(a)),inverse(a)) = multiply(a,inverse(a)),
inference(symmetry,[status(thm)],[64]) ).
tff(66,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(inverse(a)),inverse(a)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(67,plain,
multiply(inverse(inverse(a)),inverse(a)) = identity,
inference(unit_resolution,[status(thm)],[66,7]) ).
tff(68,plain,
identity = multiply(inverse(inverse(a)),inverse(a)),
inference(symmetry,[status(thm)],[67]) ).
tff(69,plain,
identity = multiply(a,inverse(a)),
inference(transitivity,[status(thm)],[68,65]) ).
tff(70,plain,
multiply(inverse(a),identity) = multiply(inverse(a),multiply(a,inverse(a))),
inference(monotonicity,[status(thm)],[69]) ).
tff(71,plain,
multiply(inverse(a),multiply(a,inverse(a))) = multiply(inverse(a),identity),
inference(symmetry,[status(thm)],[70]) ).
tff(72,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
<=> ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(73,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(75,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',associativity) ).
tff(76,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(a),a),inverse(a)) = multiply(inverse(a),multiply(a,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(80,plain,
multiply(multiply(inverse(a),a),inverse(a)) = multiply(inverse(a),multiply(a,inverse(a))),
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(a),a) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(82,plain,
multiply(inverse(a),a) = identity,
inference(unit_resolution,[status(thm)],[81,7]) ).
tff(83,plain,
multiply(multiply(inverse(a),a),inverse(a)) = multiply(identity,inverse(a)),
inference(monotonicity,[status(thm)],[82]) ).
tff(84,plain,
multiply(identity,inverse(a)) = multiply(multiply(inverse(a),a),inverse(a)),
inference(symmetry,[status(thm)],[83]) ).
tff(85,plain,
^ [X: $i] :
refl(( ( multiply(identity,X) = X )
<=> ( multiply(identity,X) = X ) )),
inference(bind,[status(th)],[]) ).
tff(86,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(quant_intro,[status(thm)],[85]) ).
tff(87,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,axiom,
! [X: $i] : ( multiply(identity,X) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_identity) ).
tff(89,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[88,87]) ).
tff(90,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(skolemize,[status(sab)],[89]) ).
tff(91,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[90,86]) ).
tff(92,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,inverse(a)) = inverse(a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(93,plain,
multiply(identity,inverse(a)) = inverse(a),
inference(unit_resolution,[status(thm)],[92,91]) ).
tff(94,plain,
inverse(a) = multiply(identity,inverse(a)),
inference(symmetry,[status(thm)],[93]) ).
tff(95,plain,
inverse(a) = multiply(inverse(a),identity),
inference(transitivity,[status(thm)],[94,84,80,71]) ).
tff(96,plain,
identity = multiply(inverse(a),a),
inference(symmetry,[status(thm)],[82]) ).
tff(97,plain,
least_upper_bound(identity,inverse(a)) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),
inference(monotonicity,[status(thm)],[96,95]) ).
tff(98,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(least_upper_bound(identity,inverse(a)),greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))) = least_upper_bound(identity,inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(99,plain,
least_upper_bound(least_upper_bound(identity,inverse(a)),greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))) = least_upper_bound(identity,inverse(a)),
inference(unit_resolution,[status(thm)],[98,25]) ).
tff(100,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(inverse(a),identity) = least_upper_bound(identity,inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(101,plain,
least_upper_bound(inverse(a),identity) = least_upper_bound(identity,inverse(a)),
inference(unit_resolution,[status(thm)],[100,45]) ).
tff(102,plain,
least_upper_bound(identity,inverse(a)) = least_upper_bound(inverse(a),identity),
inference(symmetry,[status(thm)],[101]) ).
tff(103,plain,
greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),
inference(monotonicity,[status(thm)],[102]) ).
tff(104,plain,
greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[103]) ).
tff(105,plain,
least_upper_bound(least_upper_bound(identity,inverse(a)),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = least_upper_bound(least_upper_bound(identity,inverse(a)),greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[104]) ).
tff(106,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
<=> ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ) )),
inference(bind,[status(th)],[]) ).
tff(107,plain,
( ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
<=> ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ) ),
inference(quant_intro,[status(thm)],[106]) ).
tff(108,plain,
( ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
<=> ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(109,axiom,
! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',associativity_of_lub) ).
tff(110,plain,
! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ),
inference(modus_ponens,[status(thm)],[109,108]) ).
tff(111,plain,
! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ),
inference(skolemize,[status(sab)],[110]) ).
tff(112,plain,
! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ),
inference(modus_ponens,[status(thm)],[111,107]) ).
tff(113,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(identity,least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)))) = least_upper_bound(least_upper_bound(identity,inverse(a)),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(114,plain,
least_upper_bound(identity,least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)))) = least_upper_bound(least_upper_bound(identity,inverse(a)),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),
inference(unit_resolution,[status(thm)],[113,112]) ).
tff(115,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(identity,least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)))) = least_upper_bound(least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(116,plain,
least_upper_bound(identity,least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)))) = least_upper_bound(least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),identity),
inference(unit_resolution,[status(thm)],[115,45]) ).
tff(117,plain,
least_upper_bound(least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),identity) = least_upper_bound(identity,least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)))),
inference(symmetry,[status(thm)],[116]) ).
tff(118,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(119,plain,
least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(least_upper_bound(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),identity),
inference(unit_resolution,[status(thm)],[118,112]) ).
tff(120,plain,
least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = multiply(inverse(a),least_upper_bound(a,identity)),
inference(transitivity,[status(thm)],[119,117,114,105,99,97,54]) ).
tff(121,plain,
least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[120]) ).
tff(122,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) = least_upper_bound(least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(123,plain,
least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) = least_upper_bound(least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),
inference(unit_resolution,[status(thm)],[122,112]) ).
tff(124,plain,
least_upper_bound(least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),least_upper_bound(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))),
inference(symmetry,[status(thm)],[123]) ).
tff(125,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(126,plain,
multiply(identity,least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity),
inference(unit_resolution,[status(thm)],[125,91]) ).
tff(127,plain,
least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),identity)),
inference(monotonicity,[status(thm)],[95]) ).
tff(128,plain,
least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),identity)) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)),
inference(symmetry,[status(thm)],[127]) ).
tff(129,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(130,plain,
multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),multiply(inverse(a),identity)),
inference(unit_resolution,[status(thm)],[129,16]) ).
tff(131,plain,
multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)),
inference(transitivity,[status(thm)],[130,128]) ).
tff(132,plain,
least_upper_bound(multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),multiply(identity,least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) = least_upper_bound(least_upper_bound(multiply(inverse(a),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),inverse(a)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),
inference(monotonicity,[status(thm)],[131,126]) ).
tff(133,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) )
<=> ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(134,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ) ),
inference(quant_intro,[status(thm)],[133]) ).
tff(135,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(136,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',monotony_lub2) ).
tff(137,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ),
inference(modus_ponens,[status(thm)],[136,135]) ).
tff(138,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ),
inference(skolemize,[status(sab)],[137]) ).
tff(139,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ),
inference(modus_ponens,[status(thm)],[138,134]) ).
tff(140,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(least_upper_bound(inverse(a),identity),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),multiply(identity,least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(141,plain,
multiply(least_upper_bound(inverse(a),identity),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),multiply(identity,least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))),
inference(unit_resolution,[status(thm)],[140,139]) ).
tff(142,plain,
least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)) = least_upper_bound(identity,inverse(a)),
inference(symmetry,[status(thm)],[97]) ).
tff(143,plain,
least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)) = least_upper_bound(inverse(a),identity),
inference(transitivity,[status(thm)],[142,102]) ).
tff(144,plain,
multiply(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = multiply(least_upper_bound(inverse(a),identity),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),
inference(monotonicity,[status(thm)],[143]) ).
tff(145,plain,
multiply(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),
inference(transitivity,[status(thm)],[144,141,132,124,121,53,50,18]) ).
tff(146,plain,
inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))) = inverse(multiply(inverse(a),least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[54]) ).
tff(147,plain,
inverse(multiply(inverse(a),least_upper_bound(a,identity))) = inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(symmetry,[status(thm)],[146]) ).
tff(148,plain,
^ [Y: $i,X: $i] :
refl(( ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
<=> ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(149,plain,
( ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
<=> ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ) ),
inference(quant_intro,[status(thm)],[148]) ).
tff(150,plain,
( ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
<=> ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(151,axiom,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p20x_3) ).
tff(152,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[151,150]) ).
tff(153,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(skolemize,[status(sab)],[152]) ).
tff(154,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[153,149]) ).
tff(155,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(inverse(a),least_upper_bound(a,identity))) = multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(156,plain,
inverse(multiply(inverse(a),least_upper_bound(a,identity))) = multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),
inference(unit_resolution,[status(thm)],[155,154]) ).
tff(157,plain,
multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))) = inverse(multiply(inverse(a),least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[156]) ).
tff(158,plain,
multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))) = inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(transitivity,[status(thm)],[157,147]) ).
tff(159,plain,
multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),multiply(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) = multiply(inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(monotonicity,[status(thm)],[158,145]) ).
tff(160,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),multiply(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(161,plain,
multiply(multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),multiply(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity)),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity))),
inference(unit_resolution,[status(thm)],[160,78]) ).
tff(162,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(163,plain,
greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity),
inference(unit_resolution,[status(thm)],[162,34]) ).
tff(164,plain,
greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[104]) ).
tff(165,plain,
greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))) = greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[164]) ).
tff(166,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) )
<=> ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ) )),
inference(bind,[status(th)],[]) ).
tff(167,plain,
( ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) )
<=> ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ) ),
inference(quant_intro,[status(thm)],[166]) ).
tff(168,plain,
( ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) )
<=> ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(169,axiom,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',associativity_of_glb) ).
tff(170,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
inference(modus_ponens,[status(thm)],[169,168]) ).
tff(171,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
inference(skolemize,[status(sab)],[170]) ).
tff(172,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
inference(modus_ponens,[status(thm)],[171,167]) ).
tff(173,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) )
| ( greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))) = greatest_lower_bound(greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))),least_upper_bound(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(174,plain,
greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))) = greatest_lower_bound(greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))),least_upper_bound(a,identity)),
inference(unit_resolution,[status(thm)],[173,172]) ).
tff(175,plain,
greatest_lower_bound(greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))),least_upper_bound(a,identity)) = greatest_lower_bound(identity,greatest_lower_bound(least_upper_bound(identity,inverse(a)),least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[174]) ).
tff(176,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(a,identity) = least_upper_bound(identity,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(177,plain,
least_upper_bound(a,identity) = least_upper_bound(identity,a),
inference(unit_resolution,[status(thm)],[176,45]) ).
tff(178,plain,
^ [Y: $i,X: $i] :
refl(( ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
<=> ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ) )),
inference(bind,[status(th)],[]) ).
tff(179,plain,
( ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
<=> ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ) ),
inference(quant_intro,[status(thm)],[178]) ).
tff(180,plain,
( ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
<=> ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(181,axiom,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',glb_absorbtion) ).
tff(182,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[181,180]) ).
tff(183,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[182]) ).
tff(184,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[183,179]) ).
tff(185,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(186,plain,
greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))) = identity,
inference(unit_resolution,[status(thm)],[185,184]) ).
tff(187,plain,
greatest_lower_bound(greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))),least_upper_bound(a,identity)) = greatest_lower_bound(identity,least_upper_bound(identity,a)),
inference(monotonicity,[status(thm)],[186,177]) ).
tff(188,plain,
greatest_lower_bound(identity,least_upper_bound(identity,a)) = greatest_lower_bound(greatest_lower_bound(identity,least_upper_bound(identity,inverse(a))),least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[187]) ).
tff(189,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(identity,least_upper_bound(identity,a)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(190,plain,
greatest_lower_bound(identity,least_upper_bound(identity,a)) = identity,
inference(unit_resolution,[status(thm)],[189,184]) ).
tff(191,plain,
identity = greatest_lower_bound(identity,least_upper_bound(identity,a)),
inference(symmetry,[status(thm)],[190]) ).
tff(192,plain,
identity = greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity),
inference(transitivity,[status(thm)],[191,188,175,165,163]) ).
tff(193,plain,
least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),
inference(monotonicity,[status(thm)],[192]) ).
tff(194,plain,
least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity),
inference(symmetry,[status(thm)],[193]) ).
tff(195,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(196,plain,
least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),
inference(unit_resolution,[status(thm)],[195,25]) ).
tff(197,plain,
greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),greatest_lower_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),
inference(symmetry,[status(thm)],[196]) ).
tff(198,plain,
greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) = least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity),
inference(transitivity,[status(thm)],[197,194]) ).
tff(199,plain,
multiply(multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = multiply(multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),identity)),
inference(monotonicity,[status(thm)],[198]) ).
tff(200,plain,
multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))) = multiply(inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(monotonicity,[status(thm)],[158]) ).
tff(201,plain,
multiply(inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))) = multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(symmetry,[status(thm)],[200]) ).
tff(202,plain,
identity = multiply(inverse(least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(symmetry,[status(thm)],[9]) ).
tff(203,plain,
identity = multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),
inference(transitivity,[status(thm)],[202,201]) ).
tff(204,plain,
multiply(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = multiply(multiply(multiply(inverse(least_upper_bound(a,identity)),inverse(inverse(a))),least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),identity))),greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[203]) ).
tff(205,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(206,plain,
multiply(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))) = greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)),
inference(unit_resolution,[status(thm)],[205,91]) ).
tff(207,plain,
greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity)) = multiply(identity,greatest_lower_bound(least_upper_bound(inverse(a),identity),least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[206]) ).
tff(208,plain,
greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) = identity,
inference(transitivity,[status(thm)],[36,207,204,199,161,159,9]) ).
tff(209,plain,
( ( greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) != identity )
<=> ( greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) != identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(210,axiom,
greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) != identity,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_20x) ).
tff(211,plain,
greatest_lower_bound(least_upper_bound(a,identity),least_upper_bound(inverse(a),identity)) != identity,
inference(modus_ponens,[status(thm)],[210,209]) ).
tff(212,plain,
$false,
inference(unit_resolution,[status(thm)],[211,208]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : GRP183-4 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n001.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
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 15:49:18 EDT 2025
% 0.12/0.34 % CPUTime :
% 2.36/2.52 % SZS status Unsatisfiable
% 2.36/2.52 % SZS output start Proof
% See solution above
% 2.36/2.53 % E exiting
%------------------------------------------------------------------------------