%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP186-2 : 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 05:07:25 AM UTC 2025
% Result : Unsatisfiable 34.59s 34.76s
% Output : Proof 34.59s
% Verified :
% SZS Type : Refutation
% Derivation depth : 26
% Number of leaves : 105
% Syntax : Number of formulae : 400 ( 312 unt; 0 typ; 0 def)
% Number of atoms : 516 ( 501 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 162 ( 60 ~; 56 |; 0 &)
% ( 46 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of FOOLs : 14 ( 14 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 : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 421 ( 388 !; 0 ?; 421 :)
% Comments :
%------------------------------------------------------------------------------
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(greatest_lower_bound_type,type,
greatest_lower_bound: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(least_upper_bound_type,type,
least_upper_bound: ( $i * $i ) > $i ).
tff(identity_type,type,
identity: $i ).
tff(1,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(3,axiom,
! [X: $i] : ( inverse(inverse(X)) = X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p23_2) ).
tff(4,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[3,2]) ).
tff(5,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(skolemize,[status(sab)],[4]) ).
tff(6,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[5,1]) ).
tff(7,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(greatest_lower_bound(a,inverse(b)))) = greatest_lower_bound(a,inverse(b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(8,plain,
inverse(inverse(greatest_lower_bound(a,inverse(b)))) = greatest_lower_bound(a,inverse(b)),
inference(unit_resolution,[status(thm)],[7,6]) ).
tff(9,plain,
greatest_lower_bound(a,inverse(b)) = inverse(inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[8]) ).
tff(10,plain,
multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(greatest_lower_bound(a,inverse(b)))),
inference(monotonicity,[status(thm)],[9]) ).
tff(11,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[10]) ).
tff(12,plain,
^ [X: $i] :
refl(( ( multiply(inverse(X),X) = identity )
<=> ( multiply(inverse(X),X) = identity ) )),
inference(bind,[status(th)],[]) ).
tff(13,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(quant_intro,[status(thm)],[12]) ).
tff(14,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(15,axiom,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_inverse) ).
tff(16,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(skolemize,[status(sab)],[16]) ).
tff(18,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[17,13]) ).
tff(19,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(greatest_lower_bound(a,inverse(b)))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(20,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(greatest_lower_bound(a,inverse(b)))) = identity,
inference(unit_resolution,[status(thm)],[19,18]) ).
tff(21,plain,
identity = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[20]) ).
tff(22,plain,
identity = multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b)))),
inference(transitivity,[status(thm)],[21,11]) ).
tff(23,plain,
multiply(inverse(greatest_lower_bound(a,inverse(b))),identity) = multiply(inverse(greatest_lower_bound(a,inverse(b))),multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b))))),
inference(monotonicity,[status(thm)],[22]) ).
tff(24,plain,
multiply(inverse(greatest_lower_bound(a,inverse(b))),multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(greatest_lower_bound(a,inverse(b))),identity),
inference(symmetry,[status(thm)],[23]) ).
tff(25,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(26,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)],[25]) ).
tff(27,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(28,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(29,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[28,27]) ).
tff(30,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[29]) ).
tff(31,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[30,26]) ).
tff(32,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(inverse(greatest_lower_bound(a,inverse(b))),multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(33,plain,
multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(inverse(greatest_lower_bound(a,inverse(b))),multiply(greatest_lower_bound(a,inverse(b)),inverse(greatest_lower_bound(a,inverse(b))))),
inference(unit_resolution,[status(thm)],[32,31]) ).
tff(34,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(35,plain,
multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))) = identity,
inference(unit_resolution,[status(thm)],[34,18]) ).
tff(36,plain,
multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(identity,inverse(greatest_lower_bound(a,inverse(b)))),
inference(monotonicity,[status(thm)],[35]) ).
tff(37,plain,
multiply(identity,inverse(greatest_lower_bound(a,inverse(b)))) = multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))),inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[36]) ).
tff(38,plain,
^ [X: $i] :
refl(( ( multiply(identity,X) = X )
<=> ( multiply(identity,X) = X ) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,axiom,
! [X: $i] : ( multiply(identity,X) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_identity) ).
tff(42,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,inverse(greatest_lower_bound(a,inverse(b)))) = inverse(greatest_lower_bound(a,inverse(b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(46,plain,
multiply(identity,inverse(greatest_lower_bound(a,inverse(b)))) = inverse(greatest_lower_bound(a,inverse(b))),
inference(unit_resolution,[status(thm)],[45,44]) ).
tff(47,plain,
inverse(greatest_lower_bound(a,inverse(b))) = multiply(identity,inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[46]) ).
tff(48,plain,
inverse(greatest_lower_bound(a,inverse(b))) = multiply(inverse(greatest_lower_bound(a,inverse(b))),identity),
inference(transitivity,[status(thm)],[47,37,33,24]) ).
tff(49,plain,
multiply(a,inverse(greatest_lower_bound(a,inverse(b)))) = multiply(a,multiply(inverse(greatest_lower_bound(a,inverse(b))),identity)),
inference(monotonicity,[status(thm)],[48]) ).
tff(50,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,inverse(b))),identity)) = multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[49]) ).
tff(51,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),identity) = multiply(a,multiply(inverse(greatest_lower_bound(a,inverse(b))),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(52,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),identity) = multiply(a,multiply(inverse(greatest_lower_bound(a,inverse(b))),identity)),
inference(unit_resolution,[status(thm)],[51,31]) ).
tff(53,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(inverse(a)),inverse(a)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(54,plain,
multiply(inverse(inverse(a)),inverse(a)) = identity,
inference(unit_resolution,[status(thm)],[53,18]) ).
tff(55,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(a)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(56,plain,
inverse(inverse(a)) = a,
inference(unit_resolution,[status(thm)],[55,6]) ).
tff(57,plain,
a = inverse(inverse(a)),
inference(symmetry,[status(thm)],[56]) ).
tff(58,plain,
multiply(a,inverse(a)) = multiply(inverse(inverse(a)),inverse(a)),
inference(monotonicity,[status(thm)],[57]) ).
tff(59,plain,
multiply(a,inverse(a)) = identity,
inference(transitivity,[status(thm)],[58,54]) ).
tff(60,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,inverse(a))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),
inference(monotonicity,[status(thm)],[59]) ).
tff(61,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),inverse(a)) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(62,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),inverse(a)) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,inverse(a))),
inference(unit_resolution,[status(thm)],[61,31]) ).
tff(63,plain,
multiply(greatest_lower_bound(a,inverse(b)),inverse(a)) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),
inference(monotonicity,[status(thm)],[9]) ).
tff(64,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)) = multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),
inference(symmetry,[status(thm)],[63]) ).
tff(65,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a) = multiply(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),a),
inference(monotonicity,[status(thm)],[64]) ).
tff(66,plain,
multiply(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),a) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),
inference(symmetry,[status(thm)],[65]) ).
tff(67,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),a) = multiply(greatest_lower_bound(a,inverse(b)),multiply(inverse(a),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(68,plain,
multiply(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),a) = multiply(greatest_lower_bound(a,inverse(b)),multiply(inverse(a),a)),
inference(unit_resolution,[status(thm)],[67,31]) ).
tff(69,plain,
multiply(greatest_lower_bound(a,inverse(b)),multiply(inverse(a),a)) = multiply(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),a),
inference(symmetry,[status(thm)],[68]) ).
tff(70,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(a),a) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(71,plain,
multiply(inverse(a),a) = identity,
inference(unit_resolution,[status(thm)],[70,18]) ).
tff(72,plain,
multiply(greatest_lower_bound(a,inverse(b)),multiply(inverse(a),a)) = multiply(greatest_lower_bound(a,inverse(b)),identity),
inference(monotonicity,[status(thm)],[71]) ).
tff(73,plain,
multiply(greatest_lower_bound(a,inverse(b)),identity) = multiply(greatest_lower_bound(a,inverse(b)),multiply(inverse(a),a)),
inference(symmetry,[status(thm)],[72]) ).
tff(74,plain,
( ( inverse(identity) = identity )
<=> ( inverse(identity) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(75,axiom,
inverse(identity) = identity,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p23_1) ).
tff(76,plain,
inverse(identity) = identity,
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(identity)) = multiply(greatest_lower_bound(a,inverse(b)),identity),
inference(monotonicity,[status(thm)],[8,76]) ).
tff(78,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(79,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)],[78]) ).
tff(80,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(81,axiom,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p23_3) ).
tff(82,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(skolemize,[status(sab)],[82]) ).
tff(84,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[83,79]) ).
tff(85,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(86,plain,
inverse(multiply(identity,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(identity)),
inference(unit_resolution,[status(thm)],[85,84]) ).
tff(87,plain,
inverse(inverse(greatest_lower_bound(a,inverse(b)))) = inverse(multiply(identity,inverse(greatest_lower_bound(a,inverse(b))))),
inference(monotonicity,[status(thm)],[47]) ).
tff(88,plain,
inverse(inverse(greatest_lower_bound(a,inverse(b)))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),
inference(transitivity,[status(thm)],[87,86,77,73,69,66]) ).
tff(89,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),inverse(a)),
inference(monotonicity,[status(thm)],[88]) ).
tff(90,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),
inference(transitivity,[status(thm)],[89,62,60]) ).
tff(91,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))),
inference(monotonicity,[status(thm)],[90]) ).
tff(92,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[91]) ).
tff(93,plain,
multiply(inverse(inverse(a)),inverse(a)) = multiply(a,inverse(a)),
inference(symmetry,[status(thm)],[58]) ).
tff(94,plain,
identity = multiply(inverse(inverse(a)),inverse(a)),
inference(symmetry,[status(thm)],[54]) ).
tff(95,plain,
identity = multiply(a,inverse(a)),
inference(transitivity,[status(thm)],[94,93]) ).
tff(96,plain,
greatest_lower_bound(multiply(inverse(b),inverse(a)),identity) = greatest_lower_bound(multiply(inverse(b),inverse(a)),multiply(a,inverse(a))),
inference(monotonicity,[status(thm)],[95]) ).
tff(97,plain,
greatest_lower_bound(multiply(inverse(b),inverse(a)),multiply(a,inverse(a))) = greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),
inference(symmetry,[status(thm)],[96]) ).
tff(98,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
<=> ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ) )),
inference(bind,[status(th)],[]) ).
tff(99,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ) ),
inference(quant_intro,[status(thm)],[98]) ).
tff(100,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(101,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',monotony_glb2) ).
tff(102,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ),
inference(skolemize,[status(sab)],[102]) ).
tff(104,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) ),
inference(modus_ponens,[status(thm)],[103,99]) ).
tff(105,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(inverse(b),a),inverse(a)) = greatest_lower_bound(multiply(inverse(b),inverse(a)),multiply(a,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(106,plain,
multiply(greatest_lower_bound(inverse(b),a),inverse(a)) = greatest_lower_bound(multiply(inverse(b),inverse(a)),multiply(a,inverse(a))),
inference(unit_resolution,[status(thm)],[105,104]) ).
tff(107,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(108,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)],[107]) ).
tff(109,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(110,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(111,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[110,109]) ).
tff(112,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(skolemize,[status(sab)],[111]) ).
tff(113,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[112,108]) ).
tff(114,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(a,inverse(b)) = greatest_lower_bound(inverse(b),a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(115,plain,
greatest_lower_bound(a,inverse(b)) = greatest_lower_bound(inverse(b),a),
inference(unit_resolution,[status(thm)],[114,113]) ).
tff(116,plain,
greatest_lower_bound(inverse(b),a) = greatest_lower_bound(a,inverse(b)),
inference(symmetry,[status(thm)],[115]) ).
tff(117,plain,
greatest_lower_bound(inverse(b),a) = inverse(inverse(greatest_lower_bound(a,inverse(b)))),
inference(transitivity,[status(thm)],[116,9]) ).
tff(118,plain,
multiply(greatest_lower_bound(inverse(b),a),inverse(a)) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),
inference(monotonicity,[status(thm)],[117]) ).
tff(119,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)) = multiply(greatest_lower_bound(inverse(b),a),inverse(a)),
inference(symmetry,[status(thm)],[118]) ).
tff(120,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),inverse(a)) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),
inference(symmetry,[status(thm)],[89]) ).
tff(121,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,inverse(a))) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),a),inverse(a)),
inference(symmetry,[status(thm)],[62]) ).
tff(122,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,inverse(a))),
inference(symmetry,[status(thm)],[60]) ).
tff(123,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),
inference(transitivity,[status(thm)],[122,121,120,119,106,97]) ).
tff(124,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))) = multiply(greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))),
inference(monotonicity,[status(thm)],[123]) ).
tff(125,plain,
multiply(greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[124]) ).
tff(126,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(identity,least_upper_bound(identity,multiply(a,b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(127,plain,
multiply(greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(identity,least_upper_bound(identity,multiply(a,b)))),
inference(unit_resolution,[status(thm)],[126,104]) ).
tff(128,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(identity,least_upper_bound(identity,multiply(a,b)))) = multiply(greatest_lower_bound(multiply(inverse(b),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[127]) ).
tff(129,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,least_upper_bound(identity,multiply(a,b))) = least_upper_bound(identity,multiply(a,b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(130,plain,
multiply(identity,least_upper_bound(identity,multiply(a,b))) = least_upper_bound(identity,multiply(a,b)),
inference(unit_resolution,[status(thm)],[129,44]) ).
tff(131,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(identity,least_upper_bound(identity,multiply(a,b)))) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),
inference(monotonicity,[status(thm)],[130]) ).
tff(132,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(identity,least_upper_bound(identity,multiply(a,b)))),
inference(symmetry,[status(thm)],[131]) ).
tff(133,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(134,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)],[133]) ).
tff(135,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(136,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(137,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[136,135]) ).
tff(138,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[137]) ).
tff(139,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[138,134]) ).
tff(140,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity)) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(141,plain,
least_upper_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity)) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),
inference(unit_resolution,[status(thm)],[140,139]) ).
tff(142,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(multiply(a,b)),multiply(a,b)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(143,plain,
multiply(inverse(multiply(a,b)),multiply(a,b)) = identity,
inference(unit_resolution,[status(thm)],[142,18]) ).
tff(144,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(145,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)],[144]) ).
tff(146,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(147,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(148,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[147,146]) ).
tff(149,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[148]) ).
tff(150,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[149,145]) ).
tff(151,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(multiply(a,b),least_upper_bound(multiply(a,b),identity)) = multiply(a,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(152,plain,
greatest_lower_bound(multiply(a,b),least_upper_bound(multiply(a,b),identity)) = multiply(a,b),
inference(unit_resolution,[status(thm)],[151,150]) ).
tff(153,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(154,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)],[153]) ).
tff(155,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(156,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(157,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[156,155]) ).
tff(158,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(skolemize,[status(sab)],[157]) ).
tff(159,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[158,154]) ).
tff(160,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(multiply(a,b),identity) = least_upper_bound(identity,multiply(a,b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(161,plain,
least_upper_bound(multiply(a,b),identity) = least_upper_bound(identity,multiply(a,b)),
inference(unit_resolution,[status(thm)],[160,159]) ).
tff(162,plain,
least_upper_bound(identity,multiply(a,b)) = least_upper_bound(multiply(a,b),identity),
inference(symmetry,[status(thm)],[161]) ).
tff(163,plain,
greatest_lower_bound(multiply(a,b),least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(multiply(a,b),least_upper_bound(multiply(a,b),identity)),
inference(monotonicity,[status(thm)],[162]) ).
tff(164,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(multiply(a,b),least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(165,plain,
greatest_lower_bound(multiply(a,b),least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b)),
inference(unit_resolution,[status(thm)],[164,113]) ).
tff(166,plain,
greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b)) = greatest_lower_bound(multiply(a,b),least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[165]) ).
tff(167,plain,
greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b)) = multiply(a,b),
inference(transitivity,[status(thm)],[166,163,152]) ).
tff(168,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(169,plain,
inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)),
inference(unit_resolution,[status(thm)],[168,84]) ).
tff(170,plain,
multiply(inverse(b),inverse(a)) = inverse(multiply(a,b)),
inference(symmetry,[status(thm)],[169]) ).
tff(171,plain,
multiply(multiply(inverse(b),inverse(a)),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b))) = multiply(inverse(multiply(a,b)),multiply(a,b)),
inference(monotonicity,[status(thm)],[170,167]) ).
tff(172,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
<=> ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(173,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ) ),
inference(quant_intro,[status(thm)],[172]) ).
tff(174,plain,
( ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(175,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',monotony_glb1) ).
tff(176,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ),
inference(modus_ponens,[status(thm)],[175,174]) ).
tff(177,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ),
inference(skolemize,[status(sab)],[176]) ).
tff(178,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) ),
inference(modus_ponens,[status(thm)],[177,173]) ).
tff(179,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(multiply(inverse(b),inverse(a)),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b))) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(multiply(inverse(b),inverse(a)),multiply(a,b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(180,plain,
multiply(multiply(inverse(b),inverse(a)),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b))) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(multiply(inverse(b),inverse(a)),multiply(a,b))),
inference(unit_resolution,[status(thm)],[179,178]) ).
tff(181,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(multiply(inverse(b),inverse(a)),multiply(a,b))) = multiply(multiply(inverse(b),inverse(a)),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),multiply(a,b))),
inference(symmetry,[status(thm)],[180]) ).
tff(182,plain,
multiply(multiply(inverse(b),inverse(a)),multiply(a,b)) = multiply(inverse(multiply(a,b)),multiply(a,b)),
inference(monotonicity,[status(thm)],[170]) ).
tff(183,plain,
multiply(inverse(multiply(a,b)),multiply(a,b)) = multiply(multiply(inverse(b),inverse(a)),multiply(a,b)),
inference(symmetry,[status(thm)],[182]) ).
tff(184,plain,
identity = multiply(inverse(multiply(a,b)),multiply(a,b)),
inference(symmetry,[status(thm)],[143]) ).
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,multiply(a,b))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(186,plain,
greatest_lower_bound(identity,least_upper_bound(identity,multiply(a,b))) = identity,
inference(unit_resolution,[status(thm)],[185,150]) ).
tff(187,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(identity,least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(188,plain,
greatest_lower_bound(identity,least_upper_bound(identity,multiply(a,b))) = greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity),
inference(unit_resolution,[status(thm)],[187,113]) ).
tff(189,plain,
greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity) = greatest_lower_bound(identity,least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[188]) ).
tff(190,plain,
greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity) = multiply(multiply(inverse(b),inverse(a)),multiply(a,b)),
inference(transitivity,[status(thm)],[189,186,184,183]) ).
tff(191,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity)) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),multiply(multiply(inverse(b),inverse(a)),multiply(a,b))),
inference(monotonicity,[status(thm)],[190]) ).
tff(192,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(193,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)],[192]) ).
tff(194,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(195,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(196,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)],[195,194]) ).
tff(197,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)],[196]) ).
tff(198,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)],[197,193]) ).
tff(199,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(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity)) = greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(200,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity)) = greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity),
inference(unit_resolution,[status(thm)],[199,198]) ).
tff(201,plain,
greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity) = greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(least_upper_bound(identity,multiply(a,b)),identity)),
inference(symmetry,[status(thm)],[200]) ).
tff(202,plain,
greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity) = identity,
inference(transitivity,[status(thm)],[201,191,181,171,143]) ).
tff(203,plain,
greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),
inference(transitivity,[status(thm)],[132,128,125,92]) ).
tff(204,plain,
least_upper_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity)) = least_upper_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),identity),
inference(monotonicity,[status(thm)],[203,202]) ).
tff(205,plain,
least_upper_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),identity) = least_upper_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),greatest_lower_bound(greatest_lower_bound(multiply(multiply(inverse(b),inverse(a)),least_upper_bound(identity,multiply(a,b))),least_upper_bound(identity,multiply(a,b))),identity)),
inference(symmetry,[status(thm)],[204]) ).
tff(206,plain,
least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)),
inference(monotonicity,[status(thm)],[90]) ).
tff(207,plain,
least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),
inference(symmetry,[status(thm)],[206]) ).
tff(208,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))),identity) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(209,plain,
multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))),identity) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)),
inference(unit_resolution,[status(thm)],[208,104]) ).
tff(210,plain,
identity = inverse(identity),
inference(symmetry,[status(thm)],[76]) ).
tff(211,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(212,plain,
multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))),
inference(unit_resolution,[status(thm)],[211,178]) ).
tff(213,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))) = multiply(inverse(greatest_lower_bound(a,inverse(b))),greatest_lower_bound(a,inverse(b))),
inference(symmetry,[status(thm)],[212]) ).
tff(214,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))) = inverse(identity),
inference(transitivity,[status(thm)],[213,35,210]) ).
tff(215,plain,
multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))),identity) = multiply(inverse(identity),identity),
inference(monotonicity,[status(thm)],[214]) ).
tff(216,plain,
multiply(inverse(identity),identity) = multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b))),identity),
inference(symmetry,[status(thm)],[215]) ).
tff(217,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(identity),identity) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(218,plain,
multiply(inverse(identity),identity) = identity,
inference(unit_resolution,[status(thm)],[217,18]) ).
tff(219,plain,
identity = multiply(inverse(identity),identity),
inference(symmetry,[status(thm)],[218]) ).
tff(220,plain,
identity = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)),
inference(transitivity,[status(thm)],[219,216,209]) ).
tff(221,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),
inference(monotonicity,[status(thm)],[220]) ).
tff(222,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),
inference(symmetry,[status(thm)],[221]) ).
tff(223,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(224,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),
inference(unit_resolution,[status(thm)],[223,178]) ).
tff(225,plain,
greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),
inference(symmetry,[status(thm)],[224]) ).
tff(226,plain,
greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),
inference(transitivity,[status(thm)],[225,222]) ).
tff(227,plain,
multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),identity),
inference(monotonicity,[status(thm)],[226]) ).
tff(228,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),identity) = multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity),
inference(symmetry,[status(thm)],[227]) ).
tff(229,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),identity) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(identity,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(230,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),identity) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[229,31]) ).
tff(231,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(identity,identity)) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),identity),
inference(symmetry,[status(thm)],[230]) ).
tff(232,plain,
multiply(identity,identity) = multiply(inverse(identity),identity),
inference(monotonicity,[status(thm)],[210]) ).
tff(233,plain,
multiply(identity,identity) = identity,
inference(transitivity,[status(thm)],[232,218]) ).
tff(234,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(identity,identity)) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),
inference(monotonicity,[status(thm)],[233]) ).
tff(235,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(identity,identity)),
inference(symmetry,[status(thm)],[234]) ).
tff(236,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity),
inference(transitivity,[status(thm)],[235,231,228]) ).
tff(237,plain,
multiply(multiply(a,b),identity) = multiply(multiply(a,b),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),
inference(monotonicity,[status(thm)],[220]) ).
tff(238,plain,
multiply(multiply(a,b),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(multiply(a,b),identity),
inference(symmetry,[status(thm)],[237]) ).
tff(239,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(multiply(a,b),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(240,plain,
multiply(multiply(a,b),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),
inference(unit_resolution,[status(thm)],[239,178]) ).
tff(241,plain,
greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(multiply(a,b),greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),
inference(symmetry,[status(thm)],[240]) ).
tff(242,plain,
greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(multiply(a,b),identity),
inference(transitivity,[status(thm)],[241,238]) ).
tff(243,plain,
greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),
inference(transitivity,[status(thm)],[225,222,122,121,120]) ).
tff(244,plain,
multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(a,b),identity)),
inference(monotonicity,[status(thm)],[243,242]) ).
tff(245,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(a,b),identity)) = multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))),
inference(symmetry,[status(thm)],[244]) ).
tff(246,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(multiply(a,b))) = multiply(a,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(247,plain,
inverse(inverse(multiply(a,b))) = multiply(a,b),
inference(unit_resolution,[status(thm)],[246,6]) ).
tff(248,plain,
inverse(inverse(multiply(a,b))) = inverse(multiply(inverse(b),inverse(a))),
inference(monotonicity,[status(thm)],[169]) ).
tff(249,plain,
inverse(multiply(inverse(b),inverse(a))) = inverse(inverse(multiply(a,b))),
inference(symmetry,[status(thm)],[248]) ).
tff(250,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,multiply(inverse(b),inverse(a))) = multiply(inverse(b),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(251,plain,
multiply(identity,multiply(inverse(b),inverse(a))) = multiply(inverse(b),inverse(a)),
inference(unit_resolution,[status(thm)],[250,44]) ).
tff(252,plain,
multiply(inverse(b),inverse(a)) = multiply(identity,multiply(inverse(b),inverse(a))),
inference(symmetry,[status(thm)],[251]) ).
tff(253,plain,
inverse(multiply(inverse(b),inverse(a))) = inverse(multiply(identity,multiply(inverse(b),inverse(a)))),
inference(monotonicity,[status(thm)],[252]) ).
tff(254,plain,
inverse(multiply(identity,multiply(inverse(b),inverse(a)))) = inverse(multiply(inverse(b),inverse(a))),
inference(symmetry,[status(thm)],[253]) ).
tff(255,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,multiply(inverse(b),inverse(a)))) = multiply(inverse(multiply(inverse(b),inverse(a))),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(256,plain,
inverse(multiply(identity,multiply(inverse(b),inverse(a)))) = multiply(inverse(multiply(inverse(b),inverse(a))),inverse(identity)),
inference(unit_resolution,[status(thm)],[255,84]) ).
tff(257,plain,
multiply(inverse(multiply(inverse(b),inverse(a))),inverse(identity)) = inverse(multiply(identity,multiply(inverse(b),inverse(a)))),
inference(symmetry,[status(thm)],[256]) ).
tff(258,plain,
inverse(multiply(inverse(b),inverse(a))) = multiply(a,b),
inference(transitivity,[status(thm)],[249,247]) ).
tff(259,plain,
multiply(inverse(multiply(inverse(b),inverse(a))),inverse(identity)) = multiply(multiply(a,b),identity),
inference(monotonicity,[status(thm)],[258,76]) ).
tff(260,plain,
multiply(multiply(a,b),identity) = multiply(inverse(multiply(inverse(b),inverse(a))),inverse(identity)),
inference(symmetry,[status(thm)],[259]) ).
tff(261,plain,
multiply(multiply(a,b),identity) = multiply(a,b),
inference(transitivity,[status(thm)],[260,257,254,249,247]) ).
tff(262,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(a,b),identity)) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,b)),
inference(monotonicity,[status(thm)],[261]) ).
tff(263,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,b)) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(a,b),identity)),
inference(symmetry,[status(thm)],[262]) ).
tff(264,plain,
multiply(a,b) = inverse(inverse(multiply(a,b))),
inference(symmetry,[status(thm)],[247]) ).
tff(265,plain,
multiply(a,b) = inverse(multiply(inverse(b),inverse(a))),
inference(transitivity,[status(thm)],[264,248]) ).
tff(266,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(267,plain,
inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),
inference(unit_resolution,[status(thm)],[266,84]) ).
tff(268,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)) = inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))),
inference(symmetry,[status(thm)],[267]) ).
tff(269,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,b)) = multiply(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))),inverse(multiply(inverse(b),inverse(a)))),
inference(monotonicity,[status(thm)],[268,265]) ).
tff(270,plain,
multiply(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))),inverse(multiply(inverse(b),inverse(a)))) = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(a,b)),
inference(symmetry,[status(thm)],[269]) ).
tff(271,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = multiply(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))),inverse(multiply(inverse(b),inverse(a)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(272,plain,
inverse(multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = multiply(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b))))),inverse(multiply(inverse(b),inverse(a)))),
inference(unit_resolution,[status(thm)],[271,84]) ).
tff(273,plain,
multiply(multiply(inverse(a),a),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(identity,inverse(greatest_lower_bound(a,inverse(b)))),
inference(monotonicity,[status(thm)],[71]) ).
tff(274,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(a),a),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(275,plain,
multiply(multiply(inverse(a),a),inverse(greatest_lower_bound(a,inverse(b)))) = multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))),
inference(unit_resolution,[status(thm)],[274,31]) ).
tff(276,plain,
multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(multiply(inverse(a),a),inverse(greatest_lower_bound(a,inverse(b)))),
inference(symmetry,[status(thm)],[275]) ).
tff(277,plain,
multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = inverse(greatest_lower_bound(a,inverse(b))),
inference(transitivity,[status(thm)],[276,273,46]) ).
tff(278,plain,
multiply(inverse(b),multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b)))),
inference(monotonicity,[status(thm)],[277]) ).
tff(279,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(b),multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(280,plain,
multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(b),multiply(inverse(a),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))),
inference(unit_resolution,[status(thm)],[279,31]) ).
tff(281,plain,
multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b)))),
inference(transitivity,[status(thm)],[280,278]) ).
tff(282,plain,
inverse(multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = inverse(multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b))))),
inference(monotonicity,[status(thm)],[281]) ).
tff(283,plain,
inverse(multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b))))) = inverse(multiply(multiply(inverse(b),inverse(a)),multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))),
inference(symmetry,[status(thm)],[282]) ).
tff(284,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(inverse(b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(285,plain,
inverse(multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b))))) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(inverse(b))),
inference(unit_resolution,[status(thm)],[284,84]) ).
tff(286,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(inverse(b))) = inverse(multiply(inverse(b),inverse(greatest_lower_bound(a,inverse(b))))),
inference(symmetry,[status(thm)],[285]) ).
tff(287,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(b)) = b ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(288,plain,
inverse(inverse(b)) = b,
inference(unit_resolution,[status(thm)],[287,6]) ).
tff(289,plain,
b = inverse(inverse(b)),
inference(symmetry,[status(thm)],[288]) ).
tff(290,plain,
multiply(b,identity) = multiply(inverse(inverse(b)),inverse(identity)),
inference(monotonicity,[status(thm)],[289,210]) ).
tff(291,plain,
multiply(inverse(inverse(b)),inverse(identity)) = multiply(b,identity),
inference(symmetry,[status(thm)],[290]) ).
tff(292,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,inverse(b))) = multiply(inverse(inverse(b)),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(293,plain,
inverse(multiply(identity,inverse(b))) = multiply(inverse(inverse(b)),inverse(identity)),
inference(unit_resolution,[status(thm)],[292,84]) ).
tff(294,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,inverse(b)) = inverse(b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(295,plain,
multiply(identity,inverse(b)) = inverse(b),
inference(unit_resolution,[status(thm)],[294,44]) ).
tff(296,plain,
inverse(b) = multiply(identity,inverse(b)),
inference(symmetry,[status(thm)],[295]) ).
tff(297,plain,
inverse(inverse(b)) = inverse(multiply(identity,inverse(b))),
inference(monotonicity,[status(thm)],[296]) ).
tff(298,plain,
inverse(inverse(b)) = multiply(b,identity),
inference(transitivity,[status(thm)],[297,293,291]) ).
tff(299,plain,
multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(inverse(b))) = multiply(greatest_lower_bound(a,inverse(b)),multiply(b,identity)),
inference(monotonicity,[status(thm)],[8,298]) ).
tff(300,plain,
multiply(greatest_lower_bound(a,inverse(b)),multiply(b,identity)) = multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(inverse(b))),
inference(symmetry,[status(thm)],[299]) ).
tff(301,plain,
inverse(multiply(identity,inverse(b))) = inverse(inverse(b)),
inference(symmetry,[status(thm)],[297]) ).
tff(302,plain,
multiply(inverse(inverse(b)),inverse(identity)) = inverse(multiply(identity,inverse(b))),
inference(symmetry,[status(thm)],[293]) ).
tff(303,plain,
multiply(b,identity) = b,
inference(transitivity,[status(thm)],[290,302,301,288]) ).
tff(304,plain,
multiply(greatest_lower_bound(a,inverse(b)),multiply(b,identity)) = multiply(greatest_lower_bound(a,inverse(b)),b),
inference(monotonicity,[status(thm)],[303]) ).
tff(305,plain,
multiply(greatest_lower_bound(a,inverse(b)),b) = multiply(greatest_lower_bound(a,inverse(b)),multiply(b,identity)),
inference(symmetry,[status(thm)],[304]) ).
tff(306,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(a,inverse(b)),b) = greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(307,plain,
multiply(greatest_lower_bound(a,inverse(b)),b) = greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),
inference(unit_resolution,[status(thm)],[306,104]) ).
tff(308,plain,
greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)) = multiply(greatest_lower_bound(a,inverse(b)),b),
inference(symmetry,[status(thm)],[307]) ).
tff(309,plain,
greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)) = multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))),
inference(transitivity,[status(thm)],[308,305,300,286,283,272,270,263,245]) ).
tff(310,plain,
least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)) = least_upper_bound(multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))),multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)),
inference(monotonicity,[status(thm)],[309,236]) ).
tff(311,plain,
least_upper_bound(multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))),multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)),
inference(symmetry,[status(thm)],[310]) ).
tff(312,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(313,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)],[312]) ).
tff(314,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(315,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(316,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)],[315,314]) ).
tff(317,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)],[316]) ).
tff(318,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)],[317,313]) ).
tff(319,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),least_upper_bound(greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)) = least_upper_bound(multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))),multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(320,plain,
multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),least_upper_bound(greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)) = least_upper_bound(multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity)))),multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)),
inference(unit_resolution,[status(thm)],[319,318]) ).
tff(321,plain,
least_upper_bound(multiply(multiply(a,b),identity),multiply(identity,identity)) = least_upper_bound(multiply(a,b),identity),
inference(monotonicity,[status(thm)],[261,233]) ).
tff(322,plain,
multiply(inverse(identity),identity) = multiply(identity,identity),
inference(symmetry,[status(thm)],[232]) ).
tff(323,plain,
identity = multiply(identity,identity),
inference(transitivity,[status(thm)],[219,322]) ).
tff(324,plain,
least_upper_bound(greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity) = least_upper_bound(multiply(multiply(a,b),identity),multiply(identity,identity)),
inference(monotonicity,[status(thm)],[242,323]) ).
tff(325,plain,
least_upper_bound(greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity) = least_upper_bound(identity,multiply(a,b)),
inference(transitivity,[status(thm)],[324,321,161]) ).
tff(326,plain,
multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),least_upper_bound(greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)) = multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))),
inference(monotonicity,[status(thm)],[226,325]) ).
tff(327,plain,
multiply(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity),least_upper_bound(identity,multiply(a,b))) = multiply(greatest_lower_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),least_upper_bound(greatest_lower_bound(multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),a),identity)),multiply(multiply(a,b),multiply(multiply(inverse(greatest_lower_bound(a,inverse(b))),inverse(b)),identity))),identity)),
inference(symmetry,[status(thm)],[326]) ).
tff(328,plain,
multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),
inference(transitivity,[status(thm)],[91,327,320,311,207]) ).
tff(329,plain,
least_upper_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),identity) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),identity),
inference(monotonicity,[status(thm)],[328]) ).
tff(330,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),identity) = least_upper_bound(multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),identity),
inference(symmetry,[status(thm)],[329]) ).
tff(331,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(332,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)],[331]) ).
tff(333,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(334,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(335,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)],[334,333]) ).
tff(336,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)],[335]) ).
tff(337,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)],[336,332]) ).
tff(338,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(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),least_upper_bound(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(339,plain,
least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),least_upper_bound(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),identity),
inference(unit_resolution,[status(thm)],[338,337]) ).
tff(340,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(a,greatest_lower_bound(a,inverse(b))) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(341,plain,
least_upper_bound(a,greatest_lower_bound(a,inverse(b))) = a,
inference(unit_resolution,[status(thm)],[340,139]) ).
tff(342,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(a,greatest_lower_bound(a,inverse(b))) = least_upper_bound(greatest_lower_bound(a,inverse(b)),a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(343,plain,
least_upper_bound(a,greatest_lower_bound(a,inverse(b))) = least_upper_bound(greatest_lower_bound(a,inverse(b)),a),
inference(unit_resolution,[status(thm)],[342,159]) ).
tff(344,plain,
least_upper_bound(greatest_lower_bound(a,inverse(b)),a) = least_upper_bound(a,greatest_lower_bound(a,inverse(b))),
inference(symmetry,[status(thm)],[343]) ).
tff(345,plain,
least_upper_bound(greatest_lower_bound(a,inverse(b)),a) = a,
inference(transitivity,[status(thm)],[344,341]) ).
tff(346,plain,
multiply(least_upper_bound(greatest_lower_bound(a,inverse(b)),a),inverse(a)) = multiply(a,inverse(a)),
inference(monotonicity,[status(thm)],[345]) ).
tff(347,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(348,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)],[347]) ).
tff(349,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(350,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(351,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)],[350,349]) ).
tff(352,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)],[351]) ).
tff(353,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)],[352,348]) ).
tff(354,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(greatest_lower_bound(a,inverse(b)),a),inverse(a)) = least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),multiply(a,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(355,plain,
multiply(least_upper_bound(greatest_lower_bound(a,inverse(b)),a),inverse(a)) = least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),multiply(a,inverse(a))),
inference(unit_resolution,[status(thm)],[354,353]) ).
tff(356,plain,
least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),multiply(a,inverse(a))) = multiply(least_upper_bound(greatest_lower_bound(a,inverse(b)),a),inverse(a)),
inference(symmetry,[status(thm)],[355]) ).
tff(357,plain,
least_upper_bound(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),inverse(a)),multiply(a,inverse(a))),
inference(monotonicity,[status(thm)],[64,95]) ).
tff(358,plain,
least_upper_bound(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity) = identity,
inference(transitivity,[status(thm)],[357,356,346,58,54]) ).
tff(359,plain,
least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),least_upper_bound(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)) = least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),b),identity),
inference(monotonicity,[status(thm)],[308,358]) ).
tff(360,plain,
least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),b),identity) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),least_upper_bound(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),identity)),
inference(symmetry,[status(thm)],[359]) ).
tff(361,plain,
least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),b),identity) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),identity),
inference(monotonicity,[status(thm)],[307]) ).
tff(362,plain,
least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),identity) = least_upper_bound(multiply(greatest_lower_bound(a,inverse(b)),b),identity),
inference(symmetry,[status(thm)],[361]) ).
tff(363,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(identity,greatest_lower_bound(multiply(a,b),multiply(inverse(b),b))) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(364,plain,
least_upper_bound(identity,greatest_lower_bound(multiply(a,b),multiply(inverse(b),b))) = least_upper_bound(greatest_lower_bound(multiply(a,b),multiply(inverse(b),b)),identity),
inference(unit_resolution,[status(thm)],[363,159]) ).
tff(365,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(b),b) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(366,plain,
multiply(inverse(b),b) = identity,
inference(unit_resolution,[status(thm)],[365,18]) ).
tff(367,plain,
identity = multiply(inverse(b),b),
inference(symmetry,[status(thm)],[366]) ).
tff(368,plain,
least_upper_bound(identity,greatest_lower_bound(multiply(a,b),multiply(inverse(b),b))) = least_upper_bound(multiply(inverse(b),b),multiply(greatest_lower_bound(a,inverse(b)),b)),
inference(monotonicity,[status(thm)],[367,308]) ).
tff(369,plain,
least_upper_bound(multiply(inverse(b),b),multiply(greatest_lower_bound(a,inverse(b)),b)) = least_upper_bound(identity,greatest_lower_bound(multiply(a,b),multiply(inverse(b),b))),
inference(symmetry,[status(thm)],[368]) ).
tff(370,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(b),greatest_lower_bound(a,inverse(b))),b) = least_upper_bound(multiply(inverse(b),b),multiply(greatest_lower_bound(a,inverse(b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(371,plain,
multiply(least_upper_bound(inverse(b),greatest_lower_bound(a,inverse(b))),b) = least_upper_bound(multiply(inverse(b),b),multiply(greatest_lower_bound(a,inverse(b)),b)),
inference(unit_resolution,[status(thm)],[370,353]) ).
tff(372,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(inverse(b),greatest_lower_bound(inverse(b),a)) = inverse(b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(373,plain,
least_upper_bound(inverse(b),greatest_lower_bound(inverse(b),a)) = inverse(b),
inference(unit_resolution,[status(thm)],[372,139]) ).
tff(374,plain,
least_upper_bound(inverse(b),greatest_lower_bound(a,inverse(b))) = least_upper_bound(inverse(b),greatest_lower_bound(inverse(b),a)),
inference(monotonicity,[status(thm)],[115]) ).
tff(375,plain,
least_upper_bound(inverse(b),greatest_lower_bound(a,inverse(b))) = inverse(b),
inference(transitivity,[status(thm)],[374,373]) ).
tff(376,plain,
multiply(least_upper_bound(inverse(b),greatest_lower_bound(a,inverse(b))),b) = multiply(inverse(b),b),
inference(monotonicity,[status(thm)],[375]) ).
tff(377,plain,
multiply(inverse(b),b) = multiply(least_upper_bound(inverse(b),greatest_lower_bound(a,inverse(b))),b),
inference(symmetry,[status(thm)],[376]) ).
tff(378,plain,
identity = multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b))),
inference(transitivity,[status(thm)],[367,377,371,369,364,362,360,339,330,205,141,132,128,125,92]) ).
tff(379,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),identity) = multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b)))),
inference(monotonicity,[status(thm)],[378]) ).
tff(380,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b)))) = multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),identity),
inference(symmetry,[status(thm)],[379]) ).
tff(381,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),least_upper_bound(identity,multiply(a,b))) = multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(382,plain,
multiply(multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),least_upper_bound(identity,multiply(a,b))) = multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a)),least_upper_bound(identity,multiply(a,b)))),
inference(unit_resolution,[status(thm)],[381,31]) ).
tff(383,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(384,plain,
multiply(inverse(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))) = identity,
inference(unit_resolution,[status(thm)],[383,18]) ).
tff(385,plain,
inverse(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))) = inverse(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))),
inference(monotonicity,[status(thm)],[268]) ).
tff(386,plain,
inverse(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = inverse(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),
inference(symmetry,[status(thm)],[385]) ).
tff(387,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = multiply(a,inverse(greatest_lower_bound(a,inverse(b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(388,plain,
inverse(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))) = multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),
inference(unit_resolution,[status(thm)],[387,6]) ).
tff(389,plain,
multiply(a,inverse(greatest_lower_bound(a,inverse(b)))) = inverse(inverse(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))))),
inference(symmetry,[status(thm)],[388]) ).
tff(390,plain,
multiply(a,inverse(greatest_lower_bound(a,inverse(b)))) = inverse(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),
inference(transitivity,[status(thm)],[389,386]) ).
tff(391,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))) = multiply(inverse(multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),
inference(monotonicity,[status(thm)],[390]) ).
tff(392,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))) = identity,
inference(transitivity,[status(thm)],[391,384]) ).
tff(393,plain,
multiply(multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),least_upper_bound(identity,multiply(a,b))) = multiply(identity,least_upper_bound(identity,multiply(a,b))),
inference(monotonicity,[status(thm)],[392]) ).
tff(394,plain,
multiply(identity,least_upper_bound(identity,multiply(a,b))) = multiply(multiply(multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),multiply(inverse(inverse(greatest_lower_bound(a,inverse(b)))),inverse(a))),least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[393]) ).
tff(395,plain,
least_upper_bound(identity,multiply(a,b)) = multiply(identity,least_upper_bound(identity,multiply(a,b))),
inference(symmetry,[status(thm)],[130]) ).
tff(396,plain,
least_upper_bound(multiply(a,b),identity) = multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),
inference(transitivity,[status(thm)],[161,395,394,382,380,52,50]) ).
tff(397,plain,
( ( least_upper_bound(multiply(a,b),identity) != multiply(a,inverse(greatest_lower_bound(a,inverse(b)))) )
<=> ( least_upper_bound(multiply(a,b),identity) != multiply(a,inverse(greatest_lower_bound(a,inverse(b)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(398,axiom,
least_upper_bound(multiply(a,b),identity) != multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_p23) ).
tff(399,plain,
least_upper_bound(multiply(a,b),identity) != multiply(a,inverse(greatest_lower_bound(a,inverse(b)))),
inference(modus_ponens,[status(thm)],[398,397]) ).
tff(400,plain,
$false,
inference(unit_resolution,[status(thm)],[399,396]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.12 % Problem : GRP186-2 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.08/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n001.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 15:49:33 EDT 2025
% 0.12/0.34 % CPUTime :
% 34.59/34.76 % SZS status Unsatisfiable
% 34.59/34.76 % SZS output start Proof
% See solution above
% 34.59/34.81 % E exiting
%------------------------------------------------------------------------------