%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP180-2 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:07:22 AM UTC 2025
% Result : Unsatisfiable 27.39s 27.56s
% Output : Proof 27.39s
% Verified :
% SZS Type : Refutation
% Derivation depth : 28
% Number of leaves : 130
% Syntax : Number of formulae : 530 ( 417 unt; 0 typ; 0 def)
% Number of atoms : 671 ( 656 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 212 ( 85 ~; 81 |; 0 &)
% ( 46 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Maximal term depth : 9 ( 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 : 492 ( 459 !; 0 ?; 492 :)
% Comments :
%------------------------------------------------------------------------------
tff(least_upper_bound_type,type,
least_upper_bound: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
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(identity_type,type,
identity: $i ).
tff(1,plain,
^ [X: $i] :
refl(( ( multiply(identity,X) = X )
<=> ( multiply(identity,X) = X ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( multiply(identity,X) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_identity) ).
tff(5,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,b) = b ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
multiply(identity,b) = b,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( ( multiply(inverse(X),X) = identity )
<=> ( multiply(inverse(X),X) = identity ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_inverse) ).
tff(14,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(inverse(greatest_lower_bound(a,b))),inverse(greatest_lower_bound(a,b))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
multiply(inverse(inverse(greatest_lower_bound(a,b))),inverse(greatest_lower_bound(a,b))) = identity,
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(20,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(21,axiom,
! [X: $i] : ( inverse(inverse(X)) = X ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p11_2) ).
tff(22,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(skolemize,[status(sab)],[22]) ).
tff(24,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[23,19]) ).
tff(25,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(greatest_lower_bound(a,b))) = greatest_lower_bound(a,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(26,plain,
inverse(inverse(greatest_lower_bound(a,b))) = greatest_lower_bound(a,b),
inference(unit_resolution,[status(thm)],[25,24]) ).
tff(27,plain,
greatest_lower_bound(a,b) = inverse(inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[26]) ).
tff(28,plain,
multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))) = multiply(inverse(inverse(greatest_lower_bound(a,b))),inverse(greatest_lower_bound(a,b))),
inference(monotonicity,[status(thm)],[27]) ).
tff(29,plain,
multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))) = identity,
inference(transitivity,[status(thm)],[28,18]) ).
tff(30,plain,
multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),b) = multiply(identity,b),
inference(monotonicity,[status(thm)],[29]) ).
tff(31,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(32,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)],[31]) ).
tff(33,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(34,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(35,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[35]) ).
tff(37,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),b) = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),b) = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[38,37]) ).
tff(40,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),b),
inference(symmetry,[status(thm)],[39]) ).
tff(41,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) = b,
inference(transitivity,[status(thm)],[40,30,9]) ).
tff(42,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(a)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(43,plain,
inverse(inverse(a)) = a,
inference(unit_resolution,[status(thm)],[42,24]) ).
tff(44,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,inverse(a)) = inverse(a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(45,plain,
multiply(identity,inverse(a)) = inverse(a),
inference(unit_resolution,[status(thm)],[44,7]) ).
tff(46,plain,
inverse(a) = multiply(identity,inverse(a)),
inference(symmetry,[status(thm)],[45]) ).
tff(47,plain,
inverse(inverse(a)) = inverse(multiply(identity,inverse(a))),
inference(monotonicity,[status(thm)],[46]) ).
tff(48,plain,
inverse(multiply(identity,inverse(a))) = inverse(inverse(a)),
inference(symmetry,[status(thm)],[47]) ).
tff(49,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(50,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)],[49]) ).
tff(51,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(52,axiom,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p11_3) ).
tff(53,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(skolemize,[status(sab)],[53]) ).
tff(55,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[54,50]) ).
tff(56,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,inverse(a))) = multiply(inverse(inverse(a)),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(57,plain,
inverse(multiply(identity,inverse(a))) = multiply(inverse(inverse(a)),inverse(identity)),
inference(unit_resolution,[status(thm)],[56,55]) ).
tff(58,plain,
multiply(inverse(inverse(a)),inverse(identity)) = inverse(multiply(identity,inverse(a))),
inference(symmetry,[status(thm)],[57]) ).
tff(59,plain,
( ( inverse(identity) = identity )
<=> ( inverse(identity) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(60,axiom,
inverse(identity) = identity,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',p11_1) ).
tff(61,plain,
inverse(identity) = identity,
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
multiply(inverse(inverse(a)),inverse(identity)) = multiply(a,identity),
inference(monotonicity,[status(thm)],[43,61]) ).
tff(63,plain,
multiply(a,identity) = multiply(inverse(inverse(a)),inverse(identity)),
inference(symmetry,[status(thm)],[62]) ).
tff(64,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(65,plain,
multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b)) = identity,
inference(unit_resolution,[status(thm)],[64,16]) ).
tff(66,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b))) = multiply(a,identity),
inference(monotonicity,[status(thm)],[65]) ).
tff(67,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,b))),greatest_lower_bound(a,b)) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(68,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,b))),greatest_lower_bound(a,b)) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b))),
inference(unit_resolution,[status(thm)],[67,37]) ).
tff(69,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(70,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)],[69]) ).
tff(71,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(72,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(73,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[72,71]) ).
tff(74,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[73]) ).
tff(75,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[74,70]) ).
tff(76,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(a,greatest_lower_bound(a,b)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(77,plain,
least_upper_bound(a,greatest_lower_bound(a,b)) = a,
inference(unit_resolution,[status(thm)],[76,75]) ).
tff(78,plain,
a = least_upper_bound(a,greatest_lower_bound(a,b)),
inference(symmetry,[status(thm)],[77]) ).
tff(79,plain,
multiply(a,inverse(greatest_lower_bound(a,b))) = multiply(least_upper_bound(a,greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))),
inference(monotonicity,[status(thm)],[78]) ).
tff(80,plain,
multiply(least_upper_bound(a,greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))) = multiply(a,inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[79]) ).
tff(81,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(82,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)],[81]) ).
tff(83,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(84,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(85,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)],[84,83]) ).
tff(86,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)],[85]) ).
tff(87,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)],[86,82]) ).
tff(88,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(a,greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))) = least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(89,plain,
multiply(least_upper_bound(a,greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))) = least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),
inference(unit_resolution,[status(thm)],[88,87]) ).
tff(90,plain,
least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))) = multiply(least_upper_bound(a,greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[89]) ).
tff(91,plain,
least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))) = multiply(a,inverse(greatest_lower_bound(a,b))),
inference(transitivity,[status(thm)],[90,80]) ).
tff(92,plain,
multiply(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),greatest_lower_bound(a,b)) = multiply(multiply(a,inverse(greatest_lower_bound(a,b))),greatest_lower_bound(a,b)),
inference(monotonicity,[status(thm)],[91]) ).
tff(93,plain,
multiply(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),greatest_lower_bound(a,b)) = a,
inference(transitivity,[status(thm)],[92,68,66,63,58,48,43]) ).
tff(94,plain,
least_upper_bound(multiply(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),greatest_lower_bound(a,b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(a,b),
inference(monotonicity,[status(thm)],[93,41]) ).
tff(95,plain,
multiply(multiply(a,inverse(greatest_lower_bound(a,b))),greatest_lower_bound(a,b)) = multiply(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),greatest_lower_bound(a,b)),
inference(symmetry,[status(thm)],[92]) ).
tff(96,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b))) = multiply(multiply(a,inverse(greatest_lower_bound(a,b))),greatest_lower_bound(a,b)),
inference(symmetry,[status(thm)],[68]) ).
tff(97,plain,
multiply(a,identity) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[66]) ).
tff(98,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(a),a) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(99,plain,
multiply(inverse(a),a) = identity,
inference(unit_resolution,[status(thm)],[98,16]) ).
tff(100,plain,
multiply(a,multiply(inverse(a),a)) = multiply(a,identity),
inference(monotonicity,[status(thm)],[99]) ).
tff(101,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(a,inverse(a)),a) = multiply(a,multiply(inverse(a),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(102,plain,
multiply(multiply(a,inverse(a)),a) = multiply(a,multiply(inverse(a),a)),
inference(unit_resolution,[status(thm)],[101,37]) ).
tff(103,plain,
a = inverse(inverse(a)),
inference(symmetry,[status(thm)],[43]) ).
tff(104,plain,
multiply(a,inverse(a)) = multiply(inverse(inverse(a)),inverse(a)),
inference(monotonicity,[status(thm)],[103]) ).
tff(105,plain,
multiply(inverse(inverse(a)),inverse(a)) = multiply(a,inverse(a)),
inference(symmetry,[status(thm)],[104]) ).
tff(106,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(inverse(a)),inverse(a)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
multiply(inverse(inverse(a)),inverse(a)) = identity,
inference(unit_resolution,[status(thm)],[106,16]) ).
tff(108,plain,
identity = multiply(inverse(inverse(a)),inverse(a)),
inference(symmetry,[status(thm)],[107]) ).
tff(109,plain,
multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))) = multiply(a,inverse(a)),
inference(transitivity,[status(thm)],[28,18,108,105]) ).
tff(110,plain,
multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),a) = multiply(multiply(a,inverse(a)),a),
inference(monotonicity,[status(thm)],[109]) ).
tff(111,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),a) = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(112,plain,
multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),a) = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),
inference(unit_resolution,[status(thm)],[111,37]) ).
tff(113,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)) = multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),a),
inference(symmetry,[status(thm)],[112]) ).
tff(114,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)) = multiply(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),greatest_lower_bound(a,b)),
inference(transitivity,[status(thm)],[113,110,102,100,97,96,95]) ).
tff(115,plain,
least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,b))),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),greatest_lower_bound(a,b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[114]) ).
tff(116,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(117,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)],[116]) ).
tff(118,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(119,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(120,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)],[119,118]) ).
tff(121,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)],[120]) ).
tff(122,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)],[121,117]) ).
tff(123,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(a,b),least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(124,plain,
multiply(greatest_lower_bound(a,b),least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[123,122]) ).
tff(125,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(a,b)) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(126,plain,
multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(a,b)) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[125,122]) ).
tff(127,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(128,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)],[127]) ).
tff(129,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(130,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(131,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(skolemize,[status(sab)],[131]) ).
tff(133,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[132,128]) ).
tff(134,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(a,b) = least_upper_bound(b,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(135,plain,
least_upper_bound(a,b) = least_upper_bound(b,a),
inference(unit_resolution,[status(thm)],[134,133]) ).
tff(136,plain,
multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(a,b)) = multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(b,a)),
inference(monotonicity,[status(thm)],[135]) ).
tff(137,plain,
multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(b,a)) = multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(a,b)),
inference(symmetry,[status(thm)],[136]) ).
tff(138,plain,
multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(b,a)) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[137,126]) ).
tff(139,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(b,a))) = multiply(greatest_lower_bound(a,b),least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[138]) ).
tff(140,plain,
least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(a,b)),
inference(symmetry,[status(thm)],[126]) ).
tff(141,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,inverse(greatest_lower_bound(a,b))) = inverse(greatest_lower_bound(a,b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(142,plain,
multiply(identity,inverse(greatest_lower_bound(a,b))) = inverse(greatest_lower_bound(a,b)),
inference(unit_resolution,[status(thm)],[141,7]) ).
tff(143,plain,
multiply(multiply(identity,inverse(greatest_lower_bound(a,b))),b) = multiply(inverse(greatest_lower_bound(a,b)),b),
inference(monotonicity,[status(thm)],[142]) ).
tff(144,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(identity,inverse(greatest_lower_bound(a,b))),b) = multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(145,plain,
multiply(multiply(identity,inverse(greatest_lower_bound(a,b))),b) = multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[144,37]) ).
tff(146,plain,
multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(identity,inverse(greatest_lower_bound(a,b))),b),
inference(symmetry,[status(thm)],[145]) ).
tff(147,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(b),b) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(148,plain,
multiply(inverse(b),b) = identity,
inference(unit_resolution,[status(thm)],[147,16]) ).
tff(149,plain,
multiply(multiply(inverse(b),b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[148]) ).
tff(150,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(b),b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(151,plain,
multiply(multiply(inverse(b),b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[150,37]) ).
tff(152,plain,
multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(b),b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[151]) ).
tff(153,plain,
multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(greatest_lower_bound(a,b)),b),
inference(transitivity,[status(thm)],[152,149,146,143]) ).
tff(154,plain,
multiply(inverse(greatest_lower_bound(a,b)),inverse(inverse(a))) = multiply(inverse(greatest_lower_bound(a,b)),a),
inference(monotonicity,[status(thm)],[43]) ).
tff(155,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(inverse(a),greatest_lower_bound(a,b))) = multiply(inverse(greatest_lower_bound(a,b)),inverse(inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(156,plain,
inverse(multiply(inverse(a),greatest_lower_bound(a,b))) = multiply(inverse(greatest_lower_bound(a,b)),inverse(inverse(a))),
inference(unit_resolution,[status(thm)],[155,55]) ).
tff(157,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(158,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)],[157]) ).
tff(159,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(160,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(161,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)],[160,159]) ).
tff(162,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)],[161]) ).
tff(163,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)],[162,158]) ).
tff(164,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(a),greatest_lower_bound(a,b)) = greatest_lower_bound(multiply(inverse(a),a),multiply(inverse(a),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(165,plain,
multiply(inverse(a),greatest_lower_bound(a,b)) = greatest_lower_bound(multiply(inverse(a),a),multiply(inverse(a),b)),
inference(unit_resolution,[status(thm)],[164,163]) ).
tff(166,plain,
greatest_lower_bound(multiply(inverse(a),a),multiply(inverse(a),b)) = multiply(inverse(a),greatest_lower_bound(a,b)),
inference(symmetry,[status(thm)],[165]) ).
tff(167,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(b)) = b ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(168,plain,
inverse(inverse(b)) = b,
inference(unit_resolution,[status(thm)],[167,24]) ).
tff(169,plain,
b = inverse(inverse(b)),
inference(symmetry,[status(thm)],[168]) ).
tff(170,plain,
multiply(inverse(a),b) = multiply(inverse(a),inverse(inverse(b))),
inference(monotonicity,[status(thm)],[169]) ).
tff(171,plain,
multiply(inverse(a),inverse(inverse(b))) = multiply(inverse(a),b),
inference(symmetry,[status(thm)],[170]) ).
tff(172,plain,
multiply(identity,b) = multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),b),
inference(symmetry,[status(thm)],[30]) ).
tff(173,plain,
b = multiply(identity,b),
inference(symmetry,[status(thm)],[9]) ).
tff(174,plain,
inverse(inverse(b)) = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[168,173,172,39]) ).
tff(175,plain,
multiply(inverse(a),inverse(inverse(b))) = multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[46,174]) ).
tff(176,plain,
multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),inverse(inverse(b))),
inference(symmetry,[status(thm)],[175]) ).
tff(177,plain,
multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),b),
inference(transitivity,[status(thm)],[176,171]) ).
tff(178,plain,
identity = multiply(inverse(a),a),
inference(symmetry,[status(thm)],[99]) ).
tff(179,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(inverse(greatest_lower_bound(a,b)),b)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(180,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(inverse(greatest_lower_bound(a,b)),b)) = identity,
inference(unit_resolution,[status(thm)],[179,16]) ).
tff(181,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(182,plain,
inverse(multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),
inference(unit_resolution,[status(thm)],[181,55]) ).
tff(183,plain,
multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))) = inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[182]) ).
tff(184,plain,
multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[183]) ).
tff(185,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b) = multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(186,plain,
multiply(multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b) = multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[185,37]) ).
tff(187,plain,
multiply(inverse(b),greatest_lower_bound(a,b)) = multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),
inference(monotonicity,[status(thm)],[27]) ).
tff(188,plain,
multiply(multiply(inverse(b),greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))) = multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),
inference(monotonicity,[status(thm)],[187]) ).
tff(189,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(b),greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))) = multiply(inverse(b),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(190,plain,
multiply(multiply(inverse(b),greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))) = multiply(inverse(b),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),
inference(unit_resolution,[status(thm)],[189,37]) ).
tff(191,plain,
multiply(inverse(b),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))) = multiply(multiply(inverse(b),greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[190]) ).
tff(192,plain,
multiply(inverse(inverse(greatest_lower_bound(a,b))),inverse(greatest_lower_bound(a,b))) = multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[28]) ).
tff(193,plain,
identity = multiply(inverse(inverse(greatest_lower_bound(a,b))),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[18]) ).
tff(194,plain,
multiply(a,inverse(a)) = multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),
inference(transitivity,[status(thm)],[104,107,193,192]) ).
tff(195,plain,
multiply(inverse(b),multiply(a,inverse(a))) = multiply(inverse(b),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))),
inference(monotonicity,[status(thm)],[194]) ).
tff(196,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(b),a),inverse(a)) = multiply(inverse(b),multiply(a,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(197,plain,
multiply(multiply(inverse(b),a),inverse(a)) = multiply(inverse(b),multiply(a,inverse(a))),
inference(unit_resolution,[status(thm)],[196,37]) ).
tff(198,plain,
multiply(multiply(inverse(b),a),inverse(a)) = multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),
inference(transitivity,[status(thm)],[197,195,191,188]) ).
tff(199,plain,
multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b),
inference(monotonicity,[status(thm)],[198,41]) ).
tff(200,plain,
multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),a),
inference(transitivity,[status(thm)],[199,186,184,180,178]) ).
tff(201,plain,
greatest_lower_bound(multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(a),a),multiply(inverse(a),b)),
inference(monotonicity,[status(thm)],[200,177]) ).
tff(202,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(203,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)],[202]) ).
tff(204,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(205,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(206,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)],[205,204]) ).
tff(207,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)],[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)) ),
inference(modus_ponens,[status(thm)],[207,203]) ).
tff(209,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(multiply(inverse(b),a),inverse(a)),multiply(identity,inverse(a))),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = greatest_lower_bound(multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(210,plain,
multiply(greatest_lower_bound(multiply(multiply(inverse(b),a),inverse(a)),multiply(identity,inverse(a))),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = greatest_lower_bound(multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[209,208]) ).
tff(211,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),a),identity),inverse(a)) = greatest_lower_bound(multiply(multiply(inverse(b),a),inverse(a)),multiply(identity,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(212,plain,
multiply(greatest_lower_bound(multiply(inverse(b),a),identity),inverse(a)) = greatest_lower_bound(multiply(multiply(inverse(b),a),inverse(a)),multiply(identity,inverse(a))),
inference(unit_resolution,[status(thm)],[211,208]) ).
tff(213,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(b),greatest_lower_bound(a,b)) = greatest_lower_bound(multiply(inverse(b),a),multiply(inverse(b),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(214,plain,
multiply(inverse(b),greatest_lower_bound(a,b)) = greatest_lower_bound(multiply(inverse(b),a),multiply(inverse(b),b)),
inference(unit_resolution,[status(thm)],[213,163]) ).
tff(215,plain,
greatest_lower_bound(multiply(inverse(b),a),multiply(inverse(b),b)) = multiply(inverse(b),greatest_lower_bound(a,b)),
inference(symmetry,[status(thm)],[214]) ).
tff(216,plain,
identity = multiply(inverse(b),b),
inference(symmetry,[status(thm)],[148]) ).
tff(217,plain,
greatest_lower_bound(multiply(inverse(b),a),identity) = greatest_lower_bound(multiply(inverse(b),a),multiply(inverse(b),b)),
inference(monotonicity,[status(thm)],[216]) ).
tff(218,plain,
greatest_lower_bound(multiply(inverse(b),a),identity) = inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[217,215,187,183]) ).
tff(219,plain,
multiply(greatest_lower_bound(multiply(inverse(b),a),identity),inverse(a)) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),
inference(monotonicity,[status(thm)],[218]) ).
tff(220,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)) = multiply(greatest_lower_bound(multiply(inverse(b),a),identity),inverse(a)),
inference(symmetry,[status(thm)],[219]) ).
tff(221,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)) = greatest_lower_bound(multiply(multiply(inverse(b),a),inverse(a)),multiply(identity,inverse(a))),
inference(transitivity,[status(thm)],[220,212]) ).
tff(222,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(greatest_lower_bound(multiply(multiply(inverse(b),a),inverse(a)),multiply(identity,inverse(a))),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[221]) ).
tff(223,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),greatest_lower_bound(a,b)),
inference(transitivity,[status(thm)],[222,210,201,166]) ).
tff(224,plain,
inverse(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))) = inverse(multiply(inverse(a),greatest_lower_bound(a,b))),
inference(monotonicity,[status(thm)],[223]) ).
tff(225,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(226,plain,
inverse(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))),
inference(unit_resolution,[status(thm)],[225,55]) ).
tff(227,plain,
multiply(inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) = inverse(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[226]) ).
tff(228,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(229,plain,
inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),
inference(unit_resolution,[status(thm)],[228,55]) ).
tff(230,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)) = inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[229]) ).
tff(231,plain,
inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = inverse(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[230]) ).
tff(232,plain,
inverse(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),
inference(symmetry,[status(thm)],[231]) ).
tff(233,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(234,plain,
inverse(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[233,24]) ).
tff(235,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) = inverse(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[234]) ).
tff(236,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) = inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),
inference(transitivity,[status(thm)],[235,232]) ).
tff(237,plain,
inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = inverse(b),
inference(monotonicity,[status(thm)],[41]) ).
tff(238,plain,
inverse(b) = inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[237]) ).
tff(239,plain,
multiply(inverse(b),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))),
inference(monotonicity,[status(thm)],[238,236]) ).
tff(240,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(241,plain,
multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),
inference(unit_resolution,[status(thm)],[240,7]) ).
tff(242,plain,
multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[241,230]) ).
tff(243,plain,
inverse(multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) = inverse(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[242]) ).
tff(244,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) = multiply(inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(245,plain,
inverse(multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) = multiply(inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),inverse(identity)),
inference(unit_resolution,[status(thm)],[244,55]) ).
tff(246,plain,
multiply(inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),inverse(identity)) = inverse(multiply(identity,multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))),
inference(symmetry,[status(thm)],[245]) ).
tff(247,plain,
identity = inverse(identity),
inference(symmetry,[status(thm)],[61]) ).
tff(248,plain,
multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity) = multiply(inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),inverse(identity)),
inference(monotonicity,[status(thm)],[236,247]) ).
tff(249,plain,
multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[248,246,243,234]) ).
tff(250,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(greatest_lower_bound(a,b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(251,plain,
inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(greatest_lower_bound(a,b))),
inference(unit_resolution,[status(thm)],[250,55]) ).
tff(252,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(greatest_lower_bound(a,b))) = inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[251]) ).
tff(253,plain,
multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(greatest_lower_bound(a,b))),
inference(monotonicity,[status(thm)],[183]) ).
tff(254,plain,
multiply(multiply(inverse(b),a),inverse(a)) = inverse(b),
inference(transitivity,[status(thm)],[197,195,191,188,253,252,237]) ).
tff(255,plain,
multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity)) = multiply(inverse(b),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[254,249]) ).
tff(256,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity) = multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(257,plain,
multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity) = multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),
inference(unit_resolution,[status(thm)],[256,37]) ).
tff(258,plain,
multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)) = multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity),
inference(symmetry,[status(thm)],[257]) ).
tff(259,plain,
multiply(inverse(b),multiply(a,inverse(a))) = multiply(multiply(inverse(b),a),inverse(a)),
inference(symmetry,[status(thm)],[197]) ).
tff(260,plain,
multiply(inverse(b),multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b)))) = multiply(inverse(b),multiply(a,inverse(a))),
inference(symmetry,[status(thm)],[195]) ).
tff(261,plain,
multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))) = multiply(multiply(inverse(b),greatest_lower_bound(a,b)),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[188]) ).
tff(262,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(greatest_lower_bound(a,b))) = multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[253]) ).
tff(263,plain,
inverse(b) = multiply(multiply(inverse(b),a),inverse(a)),
inference(transitivity,[status(thm)],[238,251,262,261,190,260,259]) ).
tff(264,plain,
multiply(inverse(b),multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity))) = multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),identity)),
inference(monotonicity,[status(thm)],[263,258]) ).
tff(265,plain,
multiply(inverse(b),multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity))) = multiply(inverse(greatest_lower_bound(a,b)),a),
inference(transitivity,[status(thm)],[264,255,239,227,224,156,154]) ).
tff(266,plain,
least_upper_bound(multiply(inverse(b),multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity))),multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[265,153]) ).
tff(267,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(b),least_upper_bound(multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(inverse(b),multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity))),multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(268,plain,
multiply(inverse(b),least_upper_bound(multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(inverse(b),multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity))),multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[267,122]) ).
tff(269,plain,
multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[258,248,246,243,234]) ).
tff(270,plain,
least_upper_bound(multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[269]) ).
tff(271,plain,
least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[270]) ).
tff(272,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(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(273,plain,
multiply(least_upper_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[272,87]) ).
tff(274,plain,
multiply(least_upper_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[135]) ).
tff(275,plain,
multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(least_upper_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[274]) ).
tff(276,plain,
multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[275,273,271]) ).
tff(277,plain,
multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(b),least_upper_bound(multiply(a,multiply(multiply(inverse(greatest_lower_bound(a,b)),b),identity)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[276]) ).
tff(278,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(identity),identity) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(279,plain,
multiply(inverse(identity),identity) = identity,
inference(unit_resolution,[status(thm)],[278,16]) ).
tff(280,plain,
multiply(identity,identity) = multiply(inverse(identity),identity),
inference(monotonicity,[status(thm)],[247]) ).
tff(281,plain,
multiply(identity,identity) = identity,
inference(transitivity,[status(thm)],[280,279]) ).
tff(282,plain,
multiply(multiply(identity,identity),inverse(a)) = multiply(identity,inverse(a)),
inference(monotonicity,[status(thm)],[281]) ).
tff(283,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(identity,identity),inverse(a)) = multiply(identity,multiply(identity,inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(284,plain,
multiply(multiply(identity,identity),inverse(a)) = multiply(identity,multiply(identity,inverse(a))),
inference(unit_resolution,[status(thm)],[283,37]) ).
tff(285,plain,
multiply(identity,multiply(identity,inverse(a))) = multiply(multiply(identity,identity),inverse(a)),
inference(symmetry,[status(thm)],[284]) ).
tff(286,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(287,plain,
multiply(inverse(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))) = identity,
inference(unit_resolution,[status(thm)],[286,16]) ).
tff(288,plain,
inverse(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))) = inverse(inverse(multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[183]) ).
tff(289,plain,
inverse(inverse(multiply(inverse(greatest_lower_bound(a,b)),b))) = inverse(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),
inference(symmetry,[status(thm)],[288]) ).
tff(290,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(greatest_lower_bound(a,b)),b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(291,plain,
inverse(inverse(multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(greatest_lower_bound(a,b)),b),
inference(unit_resolution,[status(thm)],[290,24]) ).
tff(292,plain,
multiply(inverse(greatest_lower_bound(a,b)),b) = inverse(inverse(multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[291]) ).
tff(293,plain,
multiply(inverse(greatest_lower_bound(a,b)),b) = inverse(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),
inference(transitivity,[status(thm)],[292,289]) ).
tff(294,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))) = multiply(inverse(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),
inference(monotonicity,[status(thm)],[293]) ).
tff(295,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))) = identity,
inference(transitivity,[status(thm)],[294,287]) ).
tff(296,plain,
multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),inverse(a)) = multiply(identity,multiply(identity,inverse(a))),
inference(monotonicity,[status(thm)],[295,46]) ).
tff(297,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),inverse(a)) = multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(298,plain,
multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),inverse(a)) = multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(a))),
inference(unit_resolution,[status(thm)],[297,37]) ).
tff(299,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(a))) = multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b))))),inverse(a)),
inference(symmetry,[status(thm)],[298]) ).
tff(300,plain,
multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(a)) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),
inference(monotonicity,[status(thm)],[183]) ).
tff(301,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)) = multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(a)),
inference(symmetry,[status(thm)],[300]) ).
tff(302,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),inverse(a))),
inference(monotonicity,[status(thm)],[301]) ).
tff(303,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = inverse(a),
inference(transitivity,[status(thm)],[302,299,296,285,282,45]) ).
tff(304,plain,
multiply(multiply(inverse(b),a),multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) = multiply(multiply(inverse(b),a),inverse(a)),
inference(monotonicity,[status(thm)],[303]) ).
tff(305,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = multiply(multiply(inverse(b),a),multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(306,plain,
multiply(multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = multiply(multiply(inverse(b),a),multiply(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))),
inference(unit_resolution,[status(thm)],[305,37]) ).
tff(307,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(b),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(308,plain,
multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(b),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[307,37]) ).
tff(309,plain,
multiply(inverse(b),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[308]) ).
tff(310,plain,
multiply(inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),inverse(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)))) = multiply(inverse(b),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[239]) ).
tff(311,plain,
inverse(multiply(inverse(a),greatest_lower_bound(a,b))) = inverse(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[224]) ).
tff(312,plain,
multiply(inverse(greatest_lower_bound(a,b)),inverse(inverse(a))) = inverse(multiply(inverse(a),greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[156]) ).
tff(313,plain,
multiply(inverse(greatest_lower_bound(a,b)),a) = multiply(inverse(greatest_lower_bound(a,b)),inverse(inverse(a))),
inference(symmetry,[status(thm)],[154]) ).
tff(314,plain,
multiply(inverse(greatest_lower_bound(a,b)),a) = multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[313,312,311,226,310,309]) ).
tff(315,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = multiply(multiply(multiply(inverse(b),a),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),
inference(monotonicity,[status(thm)],[314]) ).
tff(316,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))) = inverse(b),
inference(transitivity,[status(thm)],[315,306,304,197,195,191,188,253,252,237]) ).
tff(317,plain,
multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[316]) ).
tff(318,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(319,plain,
multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[318,37]) ).
tff(320,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[319]) ).
tff(321,plain,
multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(b,a)),
inference(transitivity,[status(thm)],[320,317,277,268,266,140,136]) ).
tff(322,plain,
multiply(greatest_lower_bound(a,b),multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))))) = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),least_upper_bound(b,a))),
inference(monotonicity,[status(thm)],[321]) ).
tff(323,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(greatest_lower_bound(a,b),multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(324,plain,
multiply(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(greatest_lower_bound(a,b),multiply(multiply(inverse(greatest_lower_bound(a,b)),a),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))))),
inference(unit_resolution,[status(thm)],[323,37]) ).
tff(325,plain,
multiply(multiply(a,inverse(a)),a) = multiply(multiply(greatest_lower_bound(a,b),inverse(greatest_lower_bound(a,b))),a),
inference(symmetry,[status(thm)],[110]) ).
tff(326,plain,
multiply(a,multiply(inverse(a),a)) = multiply(multiply(a,inverse(a)),a),
inference(symmetry,[status(thm)],[102]) ).
tff(327,plain,
multiply(a,identity) = multiply(a,multiply(inverse(a),a)),
inference(symmetry,[status(thm)],[100]) ).
tff(328,plain,
a = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),
inference(transitivity,[status(thm)],[103,47,57,62,327,326,325,112]) ).
tff(329,plain,
multiply(a,multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[328]) ).
tff(330,plain,
multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[275,273]) ).
tff(331,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[330]) ).
tff(332,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[331]) ).
tff(333,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(334,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[333,122]) ).
tff(335,plain,
least_upper_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[334]) ).
tff(336,plain,
multiply(inverse(a),greatest_lower_bound(a,b)) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),
inference(monotonicity,[status(thm)],[27]) ).
tff(337,plain,
b = multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[173,172,39]) ).
tff(338,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),b) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[337]) ).
tff(339,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),b) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),
inference(transitivity,[status(thm)],[338,222,210,201,166,336]) ).
tff(340,plain,
multiply(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[339]) ).
tff(341,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(342,plain,
multiply(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),b),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[341,37]) ).
tff(343,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[342]) ).
tff(344,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[343,340]) ).
tff(345,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(346,plain,
multiply(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = identity,
inference(unit_resolution,[status(thm)],[345,16]) ).
tff(347,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[230]) ).
tff(348,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = identity,
inference(transitivity,[status(thm)],[347,346]) ).
tff(349,plain,
least_upper_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(identity,multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[348,344]) ).
tff(350,plain,
least_upper_bound(identity,multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[349]) ).
tff(351,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(inverse(greatest_lower_bound(a,b)),a)) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(352,plain,
inverse(multiply(inverse(greatest_lower_bound(a,b)),a)) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),
inference(unit_resolution,[status(thm)],[351,55]) ).
tff(353,plain,
multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))) = inverse(multiply(inverse(greatest_lower_bound(a,b)),a)),
inference(symmetry,[status(thm)],[352]) ).
tff(354,plain,
multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),a)),inverse(greatest_lower_bound(a,b))),
inference(monotonicity,[status(thm)],[353]) ).
tff(355,plain,
multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),a)),inverse(greatest_lower_bound(a,b))) = multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),
inference(symmetry,[status(thm)],[354]) ).
tff(356,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),a)),inverse(greatest_lower_bound(a,b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(357,plain,
inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),a)),inverse(greatest_lower_bound(a,b))),
inference(unit_resolution,[status(thm)],[356,55]) ).
tff(358,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a)) = multiply(a,identity),
inference(transitivity,[status(thm)],[113,110,102,100]) ).
tff(359,plain,
inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a))) = inverse(multiply(a,identity)),
inference(monotonicity,[status(thm)],[358]) ).
tff(360,plain,
inverse(multiply(a,identity)) = inverse(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),a))),
inference(symmetry,[status(thm)],[359]) ).
tff(361,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(a,identity)) = multiply(inverse(identity),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(362,plain,
inverse(multiply(a,identity)) = multiply(inverse(identity),inverse(a)),
inference(unit_resolution,[status(thm)],[361,55]) ).
tff(363,plain,
multiply(inverse(identity),inverse(a)) = inverse(multiply(a,identity)),
inference(symmetry,[status(thm)],[362]) ).
tff(364,plain,
multiply(inverse(identity),inverse(a)) = multiply(identity,inverse(a)),
inference(monotonicity,[status(thm)],[61]) ).
tff(365,plain,
multiply(identity,inverse(a)) = multiply(inverse(identity),inverse(a)),
inference(symmetry,[status(thm)],[364]) ).
tff(366,plain,
multiply(identity,inverse(a)) = multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),
inference(transitivity,[status(thm)],[365,363,360,357,355]) ).
tff(367,plain,
multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b),
inference(monotonicity,[status(thm)],[366,41]) ).
tff(368,plain,
multiply(multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b) = multiply(multiply(identity,inverse(a)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[367]) ).
tff(369,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b) = multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(370,plain,
multiply(multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b) = multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[369,37]) ).
tff(371,plain,
multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),inverse(greatest_lower_bound(a,b))),b),
inference(symmetry,[status(thm)],[370]) ).
tff(372,plain,
multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(a),b),
inference(transitivity,[status(thm)],[371,368,176,171]) ).
tff(373,plain,
least_upper_bound(identity,multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),b)),
inference(monotonicity,[status(thm)],[178,372]) ).
tff(374,plain,
least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),b)) = least_upper_bound(identity,multiply(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,b)))),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[373]) ).
tff(375,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(a),least_upper_bound(a,b)) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(376,plain,
multiply(inverse(a),least_upper_bound(a,b)) = least_upper_bound(multiply(inverse(a),a),multiply(inverse(a),b)),
inference(unit_resolution,[status(thm)],[375,122]) ).
tff(377,plain,
multiply(inverse(a),least_upper_bound(a,b)) = multiply(inverse(a),least_upper_bound(b,a)),
inference(monotonicity,[status(thm)],[135]) ).
tff(378,plain,
multiply(inverse(a),least_upper_bound(b,a)) = multiply(inverse(a),least_upper_bound(a,b)),
inference(symmetry,[status(thm)],[377]) ).
tff(379,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(multiply(inverse(a),least_upper_bound(b,a)),greatest_lower_bound(multiply(inverse(a),least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),least_upper_bound(b,a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(380,plain,
least_upper_bound(multiply(inverse(a),least_upper_bound(b,a)),greatest_lower_bound(multiply(inverse(a),least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),least_upper_bound(b,a)),
inference(unit_resolution,[status(thm)],[379,75]) ).
tff(381,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(382,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)],[381]) ).
tff(383,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(384,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(385,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[384,383]) ).
tff(386,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[385]) ).
tff(387,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[386,382]) ).
tff(388,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(b,least_upper_bound(b,a)) = b ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(389,plain,
greatest_lower_bound(b,least_upper_bound(b,a)) = b,
inference(unit_resolution,[status(thm)],[388,387]) ).
tff(390,plain,
multiply(greatest_lower_bound(b,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[389]) ).
tff(391,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(b,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(392,plain,
multiply(greatest_lower_bound(b,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[391,208]) ).
tff(393,plain,
greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(greatest_lower_bound(b,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[392]) ).
tff(394,plain,
greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[393,390]) ).
tff(395,plain,
multiply(inverse(b),greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[394]) ).
tff(396,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(b),greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(397,plain,
multiply(inverse(b),greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[396,163]) ).
tff(398,plain,
greatest_lower_bound(multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(b),greatest_lower_bound(multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[397]) ).
tff(399,plain,
multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(inverse(b),b),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[149]) ).
tff(400,plain,
multiply(inverse(greatest_lower_bound(a,b)),b) = multiply(multiply(identity,inverse(greatest_lower_bound(a,b))),b),
inference(symmetry,[status(thm)],[143]) ).
tff(401,plain,
multiply(inverse(greatest_lower_bound(a,b)),b) = multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[400,145,399,151]) ).
tff(402,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(b),multiply(b,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[401]) ).
tff(403,plain,
multiply(multiply(inverse(a),a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[99]) ).
tff(404,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(a),a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(405,plain,
multiply(multiply(inverse(a),a),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[404,37]) ).
tff(406,plain,
multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(a),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[405]) ).
tff(407,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(a,least_upper_bound(a,b)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(408,plain,
greatest_lower_bound(a,least_upper_bound(a,b)) = a,
inference(unit_resolution,[status(thm)],[407,387]) ).
tff(409,plain,
least_upper_bound(b,a) = least_upper_bound(a,b),
inference(symmetry,[status(thm)],[135]) ).
tff(410,plain,
greatest_lower_bound(a,least_upper_bound(b,a)) = greatest_lower_bound(a,least_upper_bound(a,b)),
inference(monotonicity,[status(thm)],[409]) ).
tff(411,plain,
greatest_lower_bound(a,least_upper_bound(b,a)) = a,
inference(transitivity,[status(thm)],[410,408]) ).
tff(412,plain,
multiply(greatest_lower_bound(a,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[411]) ).
tff(413,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,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(414,plain,
multiply(greatest_lower_bound(a,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[413,208]) ).
tff(415,plain,
greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(greatest_lower_bound(a,least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[414]) ).
tff(416,plain,
greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[415,412]) ).
tff(417,plain,
multiply(inverse(a),greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[416]) ).
tff(418,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(a),greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(419,plain,
multiply(inverse(a),greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[418,163]) ).
tff(420,plain,
greatest_lower_bound(multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(a),greatest_lower_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[419]) ).
tff(421,plain,
multiply(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(inverse(a),a),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[403]) ).
tff(422,plain,
multiply(inverse(greatest_lower_bound(a,b)),b) = multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[400,145,421,405]) ).
tff(423,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[422]) ).
tff(424,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(greatest_lower_bound(a,b)),b),
inference(transitivity,[status(thm)],[423,420,417,406,403,146,143]) ).
tff(425,plain,
greatest_lower_bound(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[424]) ).
tff(426,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(427,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)],[426]) ).
tff(428,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(429,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(430,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)],[429,428]) ).
tff(431,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)],[430]) ).
tff(432,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)],[431,427]) ).
tff(433,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(inverse(greatest_lower_bound(a,b)),b),greatest_lower_bound(multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))))) = greatest_lower_bound(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(434,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),greatest_lower_bound(multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))))) = greatest_lower_bound(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[433,432]) ).
tff(435,plain,
multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[254]) ).
tff(436,plain,
multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[435]) ).
tff(437,plain,
multiply(multiply(identity,inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[45]) ).
tff(438,plain,
multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(identity,inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[437]) ).
tff(439,plain,
greatest_lower_bound(multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(multiply(identity,inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[438,436]) ).
tff(440,plain,
greatest_lower_bound(multiply(multiply(identity,inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[439]) ).
tff(441,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(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = greatest_lower_bound(multiply(multiply(identity,inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(442,plain,
multiply(greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = greatest_lower_bound(multiply(multiply(identity,inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(multiply(multiply(inverse(b),a),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[441,208]) ).
tff(443,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(444,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)],[443]) ).
tff(445,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(446,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(447,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[446,445]) ).
tff(448,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(skolemize,[status(sab)],[447]) ).
tff(449,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[448,444]) ).
tff(450,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(a,b) = greatest_lower_bound(b,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(451,plain,
greatest_lower_bound(a,b) = greatest_lower_bound(b,a),
inference(unit_resolution,[status(thm)],[450,449]) ).
tff(452,plain,
greatest_lower_bound(b,a) = greatest_lower_bound(a,b),
inference(symmetry,[status(thm)],[451]) ).
tff(453,plain,
greatest_lower_bound(b,a) = inverse(inverse(greatest_lower_bound(a,b))),
inference(transitivity,[status(thm)],[452,27]) ).
tff(454,plain,
multiply(inverse(b),greatest_lower_bound(b,a)) = multiply(inverse(b),inverse(inverse(greatest_lower_bound(a,b)))),
inference(monotonicity,[status(thm)],[453]) ).
tff(455,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(b),greatest_lower_bound(b,a)) = greatest_lower_bound(multiply(inverse(b),b),multiply(inverse(b),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(456,plain,
multiply(inverse(b),greatest_lower_bound(b,a)) = greatest_lower_bound(multiply(inverse(b),b),multiply(inverse(b),a)),
inference(unit_resolution,[status(thm)],[455,163]) ).
tff(457,plain,
greatest_lower_bound(multiply(inverse(b),b),multiply(inverse(b),a)) = multiply(inverse(b),greatest_lower_bound(b,a)),
inference(symmetry,[status(thm)],[456]) ).
tff(458,plain,
greatest_lower_bound(identity,multiply(inverse(b),a)) = greatest_lower_bound(multiply(inverse(b),b),multiply(inverse(b),a)),
inference(monotonicity,[status(thm)],[216]) ).
tff(459,plain,
greatest_lower_bound(identity,multiply(inverse(b),a)) = inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[458,457,454,183]) ).
tff(460,plain,
multiply(greatest_lower_bound(identity,multiply(inverse(b),a)),inverse(a)) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),
inference(monotonicity,[status(thm)],[459]) ).
tff(461,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(identity,multiply(inverse(b),a)),inverse(a)) = greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(462,plain,
multiply(greatest_lower_bound(identity,multiply(inverse(b),a)),inverse(a)) = greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))),
inference(unit_resolution,[status(thm)],[461,208]) ).
tff(463,plain,
greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))) = multiply(greatest_lower_bound(identity,multiply(inverse(b),a)),inverse(a)),
inference(symmetry,[status(thm)],[462]) ).
tff(464,plain,
greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))) = multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),
inference(transitivity,[status(thm)],[463,460]) ).
tff(465,plain,
multiply(greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[464]) ).
tff(466,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(greatest_lower_bound(multiply(identity,inverse(a)),multiply(multiply(inverse(b),a),inverse(a))),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[465]) ).
tff(467,plain,
multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))) = greatest_lower_bound(multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(transitivity,[status(thm)],[466,442,440]) ).
tff(468,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),greatest_lower_bound(multiply(inverse(a),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(b),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))))),
inference(monotonicity,[status(thm)],[467]) ).
tff(469,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(470,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))) = greatest_lower_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[469,449]) ).
tff(471,plain,
greatest_lower_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(greatest_lower_bound(a,b)),b)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[470]) ).
tff(472,plain,
multiply(inverse(a),least_upper_bound(b,a)) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[378,376,374,350,335,332]) ).
tff(473,plain,
greatest_lower_bound(multiply(inverse(a),least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = greatest_lower_bound(multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[472]) ).
tff(474,plain,
greatest_lower_bound(multiply(inverse(a),least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(greatest_lower_bound(a,b)),b),
inference(transitivity,[status(thm)],[473,471,468,434,425,402,398,395,152,149,146,143]) ).
tff(475,plain,
least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity) = least_upper_bound(multiply(inverse(a),b),multiply(inverse(a),a)),
inference(monotonicity,[status(thm)],[171,178]) ).
tff(476,plain,
least_upper_bound(multiply(inverse(a),b),multiply(inverse(a),a)) = least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity),
inference(symmetry,[status(thm)],[475]) ).
tff(477,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(a),least_upper_bound(b,a)) = least_upper_bound(multiply(inverse(a),b),multiply(inverse(a),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(478,plain,
multiply(inverse(a),least_upper_bound(b,a)) = least_upper_bound(multiply(inverse(a),b),multiply(inverse(a),a)),
inference(unit_resolution,[status(thm)],[477,122]) ).
tff(479,plain,
multiply(inverse(a),least_upper_bound(b,a)) = least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity),
inference(transitivity,[status(thm)],[478,476]) ).
tff(480,plain,
least_upper_bound(multiply(inverse(a),least_upper_bound(b,a)),greatest_lower_bound(multiply(inverse(a),least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[479,474]) ).
tff(481,plain,
least_upper_bound(least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(inverse(a),least_upper_bound(b,a)),greatest_lower_bound(multiply(inverse(a),least_upper_bound(b,a)),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[480]) ).
tff(482,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(483,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)],[482]) ).
tff(484,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(485,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(486,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)],[485,484]) ).
tff(487,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)],[486]) ).
tff(488,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)],[487,483]) ).
tff(489,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(multiply(inverse(a),inverse(inverse(b))),least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity),multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(490,plain,
least_upper_bound(multiply(inverse(a),inverse(inverse(b))),least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(least_upper_bound(multiply(inverse(a),inverse(inverse(b))),identity),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[489,488]) ).
tff(491,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(greatest_lower_bound(a,b)),a))) = multiply(inverse(greatest_lower_bound(a,b)),b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(492,plain,
least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(greatest_lower_bound(a,b)),a))) = multiply(inverse(greatest_lower_bound(a,b)),b),
inference(unit_resolution,[status(thm)],[491,75]) ).
tff(493,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,b)),greatest_lower_bound(b,a)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(greatest_lower_bound(a,b)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(494,plain,
multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(b,a)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(greatest_lower_bound(a,b)),a)),
inference(unit_resolution,[status(thm)],[493,163]) ).
tff(495,plain,
multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(b,a)) = multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b)),
inference(monotonicity,[status(thm)],[452]) ).
tff(496,plain,
multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b)) = multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(b,a)),
inference(symmetry,[status(thm)],[495]) ).
tff(497,plain,
identity = multiply(inverse(greatest_lower_bound(a,b)),greatest_lower_bound(a,b)),
inference(symmetry,[status(thm)],[65]) ).
tff(498,plain,
identity = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(greatest_lower_bound(a,b)),a)),
inference(transitivity,[status(thm)],[497,496,494]) ).
tff(499,plain,
least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),identity) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,b)),b),multiply(inverse(greatest_lower_bound(a,b)),a))),
inference(monotonicity,[status(thm)],[498]) ).
tff(500,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),identity) = least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(501,plain,
least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),identity) = least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(unit_resolution,[status(thm)],[500,133]) ).
tff(502,plain,
least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,b)),b),identity),
inference(symmetry,[status(thm)],[501]) ).
tff(503,plain,
least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[502,499,492,400,145,421,405]) ).
tff(504,plain,
multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)) = inverse(inverse(b)),
inference(transitivity,[status(thm)],[40,30,9,169]) ).
tff(505,plain,
multiply(inverse(a),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),inverse(inverse(b))),
inference(monotonicity,[status(thm)],[504]) ).
tff(506,plain,
multiply(inverse(a),inverse(inverse(b))) = multiply(inverse(a),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[505]) ).
tff(507,plain,
least_upper_bound(multiply(inverse(a),inverse(inverse(b))),least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(inverse(a),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[506,503]) ).
tff(508,plain,
least_upper_bound(multiply(inverse(a),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(inverse(a),inverse(inverse(b))),least_upper_bound(identity,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[507]) ).
tff(509,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(inverse(a),least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(inverse(a),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(510,plain,
multiply(inverse(a),least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = least_upper_bound(multiply(inverse(a),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(unit_resolution,[status(thm)],[509,122]) ).
tff(511,plain,
multiply(least_upper_bound(a,greatest_lower_bound(a,b)),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[77]) ).
tff(512,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(a,greatest_lower_bound(a,b)),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(513,plain,
multiply(least_upper_bound(a,greatest_lower_bound(a,b)),multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[512,87]) ).
tff(514,plain,
least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(least_upper_bound(a,greatest_lower_bound(a,b)),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(symmetry,[status(thm)],[513]) ).
tff(515,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(516,plain,
least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(unit_resolution,[status(thm)],[515,133]) ).
tff(517,plain,
least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = least_upper_bound(multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(symmetry,[status(thm)],[516]) ).
tff(518,plain,
least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(transitivity,[status(thm)],[517,514,511]) ).
tff(519,plain,
multiply(inverse(a),least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))) = multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(monotonicity,[status(thm)],[518]) ).
tff(520,plain,
multiply(inverse(a),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b))) = multiply(inverse(a),least_upper_bound(multiply(greatest_lower_bound(a,b),multiply(inverse(greatest_lower_bound(a,b)),b)),multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(symmetry,[status(thm)],[519]) ).
tff(521,plain,
multiply(inverse(greatest_lower_bound(a,b)),b) = multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b))),
inference(transitivity,[status(thm)],[400,145,421,405,520,510,508,490,481,380,378,376,374,350,335,332]) ).
tff(522,plain,
multiply(a,identity) = a,
inference(transitivity,[status(thm)],[63,58,48,43]) ).
tff(523,plain,
multiply(multiply(a,identity),multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(a,multiply(multiply(inverse(multiply(inverse(greatest_lower_bound(a,b)),b)),inverse(a)),multiply(least_upper_bound(b,a),multiply(inverse(greatest_lower_bound(a,b)),b)))),
inference(monotonicity,[status(thm)],[522,521]) ).
tff(524,plain,
a = multiply(a,identity),
inference(transitivity,[status(thm)],[103,47,57,62]) ).
tff(525,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) = multiply(multiply(a,identity),multiply(inverse(greatest_lower_bound(a,b)),b)),
inference(monotonicity,[status(thm)],[524]) ).
tff(526,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) = least_upper_bound(a,b),
inference(transitivity,[status(thm)],[525,523,329,324,322,139,124,115,94]) ).
tff(527,plain,
( ( multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) != least_upper_bound(a,b) )
<=> ( multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) != least_upper_bound(a,b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(528,axiom,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) != least_upper_bound(a,b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_p11) ).
tff(529,plain,
multiply(a,multiply(inverse(greatest_lower_bound(a,b)),b)) != least_upper_bound(a,b),
inference(modus_ponens,[status(thm)],[528,527]) ).
tff(530,plain,
$false,
inference(unit_resolution,[status(thm)],[529,526]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP180-2 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n003.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Thu Jun 19 15:45:24 EDT 2025
% 0.11/0.33 % CPUTime :
% 27.39/27.56 % SZS status Unsatisfiable
% 27.39/27.56 % SZS output start Proof
% See solution above
% 27.47/27.61 % E exiting
%------------------------------------------------------------------------------