%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP183-2 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:07:23 AM UTC 2025
% Result : Unsatisfiable 63.73s 63.93s
% Output : Proof 63.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 24
% Number of leaves : 99
% Syntax : Number of formulae : 345 ( 263 unt; 0 typ; 0 def)
% Number of atoms : 455 ( 440 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 150 ( 54 ~; 50 |; 0 &)
% ( 46 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 2 avg)
% Maximal term depth : 7 ( 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 : 6 ( 6 usr; 2 con; 0-2 aty)
% Number of variables : 417 ( 384 !; 0 ?; 417 :)
% Comments :
%------------------------------------------------------------------------------
tff(identity_type,type,
identity: $i ).
tff(greatest_lower_bound_type,type,
greatest_lower_bound: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(least_upper_bound_type,type,
least_upper_bound: ( $i * $i ) > $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(1,plain,
^ [X: $i] :
refl(( ( multiply(inverse(X),X) = identity )
<=> ( multiply(inverse(X),X) = identity ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-0.ax',left_inverse) ).
tff(5,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(identity),identity) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
multiply(inverse(identity),identity) = identity,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( ( inverse(identity) = identity )
<=> ( inverse(identity) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(11,axiom,
inverse(identity) = identity,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',p20_1) ).
tff(12,plain,
inverse(identity) = identity,
inference(modus_ponens,[status(thm)],[11,10]) ).
tff(13,plain,
identity = inverse(identity),
inference(symmetry,[status(thm)],[12]) ).
tff(14,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(15,plain,
multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)) = identity,
inference(unit_resolution,[status(thm)],[14,7]) ).
tff(16,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(17,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)],[16]) ).
tff(18,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(19,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/sandbox/benchmark/Axioms/GRP004-2.ax',monotony_glb1) ).
tff(20,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)],[19,18]) ).
tff(21,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)],[20]) ).
tff(22,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)],[21,17]) ).
tff(23,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,identity)),greatest_lower_bound(a,identity)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)),
inference(unit_resolution,[status(thm)],[23,22]) ).
tff(25,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)) = multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[24]) ).
tff(26,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)) = inverse(identity),
inference(transitivity,[status(thm)],[25,15,13]) ).
tff(27,plain,
multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)),identity) = multiply(inverse(identity),identity),
inference(monotonicity,[status(thm)],[26]) ).
tff(28,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(29,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)],[28]) ).
tff(30,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(31,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/sandbox/benchmark/Axioms/GRP004-2.ax',monotony_glb2) ).
tff(32,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)],[31,30]) ).
tff(33,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)],[32]) ).
tff(34,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)],[33,29]) ).
tff(35,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)),identity) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(36,plain,
multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)),identity) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity)),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity)) = multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(inverse(greatest_lower_bound(a,identity)),identity)),identity),
inference(symmetry,[status(thm)],[36]) ).
tff(38,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(39,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)],[38]) ).
tff(40,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(41,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-0.ax',associativity) ).
tff(42,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(identity,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(46,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[45,44]) ).
tff(47,plain,
multiply(inverse(greatest_lower_bound(a,identity)),multiply(identity,identity)) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity),
inference(symmetry,[status(thm)],[46]) ).
tff(48,plain,
multiply(identity,identity) = multiply(inverse(identity),identity),
inference(monotonicity,[status(thm)],[13]) ).
tff(49,plain,
multiply(identity,identity) = identity,
inference(transitivity,[status(thm)],[48,9]) ).
tff(50,plain,
multiply(inverse(greatest_lower_bound(a,identity)),multiply(identity,identity)) = multiply(inverse(greatest_lower_bound(a,identity)),identity),
inference(monotonicity,[status(thm)],[49]) ).
tff(51,plain,
multiply(inverse(greatest_lower_bound(a,identity)),identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(identity,identity)),
inference(symmetry,[status(thm)],[50]) ).
tff(52,plain,
multiply(inverse(greatest_lower_bound(a,identity)),identity) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity),
inference(transitivity,[status(thm)],[51,47]) ).
tff(53,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(54,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,identity)),
inference(unit_resolution,[status(thm)],[53,44]) ).
tff(55,plain,
multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,identity)) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),
inference(symmetry,[status(thm)],[54]) ).
tff(56,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),identity)) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(multiply(inverse(greatest_lower_bound(a,identity)),identity),identity)),
inference(monotonicity,[status(thm)],[55,52]) ).
tff(57,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,plain,
( ! [X: $i] : ( inverse(inverse(X)) = X )
<=> ! [X: $i] : ( inverse(inverse(X)) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(59,axiom,
! [X: $i] : ( inverse(inverse(X)) = X ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',p20_2) ).
tff(60,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(skolemize,[status(sab)],[60]) ).
tff(62,plain,
! [X: $i] : ( inverse(inverse(X)) = X ),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(64,plain,
inverse(inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(a,identity),
inference(unit_resolution,[status(thm)],[63,62]) ).
tff(65,plain,
greatest_lower_bound(a,identity) = inverse(inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[64]) ).
tff(66,plain,
multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))) = multiply(inverse(inverse(greatest_lower_bound(a,identity))),inverse(greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[65]) ).
tff(67,plain,
multiply(inverse(inverse(greatest_lower_bound(a,identity))),inverse(greatest_lower_bound(a,identity))) = multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[66]) ).
tff(68,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(inverse(greatest_lower_bound(a,identity))),inverse(greatest_lower_bound(a,identity))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(69,plain,
multiply(inverse(inverse(greatest_lower_bound(a,identity))),inverse(greatest_lower_bound(a,identity))) = identity,
inference(unit_resolution,[status(thm)],[68,7]) ).
tff(70,plain,
identity = multiply(inverse(inverse(greatest_lower_bound(a,identity))),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[69]) ).
tff(71,plain,
identity = multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))),
inference(transitivity,[status(thm)],[70,67]) ).
tff(72,plain,
multiply(inverse(greatest_lower_bound(a,identity)),identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity)))),
inference(monotonicity,[status(thm)],[71]) ).
tff(73,plain,
multiply(inverse(greatest_lower_bound(a,identity)),multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity)))) = multiply(inverse(greatest_lower_bound(a,identity)),identity),
inference(symmetry,[status(thm)],[72]) ).
tff(74,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),inverse(greatest_lower_bound(a,identity))) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(75,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),inverse(greatest_lower_bound(a,identity))) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[74,44]) ).
tff(76,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),inverse(greatest_lower_bound(a,identity))) = multiply(identity,inverse(greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[15]) ).
tff(77,plain,
multiply(identity,inverse(greatest_lower_bound(a,identity))) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[76]) ).
tff(78,plain,
^ [X: $i] :
refl(( ( multiply(identity,X) = X )
<=> ( multiply(identity,X) = X ) )),
inference(bind,[status(th)],[]) ).
tff(79,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(quant_intro,[status(thm)],[78]) ).
tff(80,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,axiom,
! [X: $i] : ( multiply(identity,X) = X ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-0.ax',left_identity) ).
tff(82,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(skolemize,[status(sab)],[82]) ).
tff(84,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[83,79]) ).
tff(85,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,inverse(greatest_lower_bound(a,identity))) = inverse(greatest_lower_bound(a,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(86,plain,
multiply(identity,inverse(greatest_lower_bound(a,identity))) = inverse(greatest_lower_bound(a,identity)),
inference(unit_resolution,[status(thm)],[85,84]) ).
tff(87,plain,
inverse(greatest_lower_bound(a,identity)) = multiply(identity,inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[86]) ).
tff(88,plain,
inverse(greatest_lower_bound(a,identity)) = multiply(inverse(greatest_lower_bound(a,identity)),identity),
inference(transitivity,[status(thm)],[87,77,75,73]) ).
tff(89,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(multiply(inverse(greatest_lower_bound(a,identity)),a))) = multiply(inverse(greatest_lower_bound(a,identity)),a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(90,plain,
inverse(inverse(multiply(inverse(greatest_lower_bound(a,identity)),a))) = multiply(inverse(greatest_lower_bound(a,identity)),a),
inference(unit_resolution,[status(thm)],[89,62]) ).
tff(91,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(92,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)],[91]) ).
tff(93,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(94,axiom,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',p20_3) ).
tff(95,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[94,93]) ).
tff(96,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(skolemize,[status(sab)],[95]) ).
tff(97,plain,
! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) ),
inference(modus_ponens,[status(thm)],[96,92]) ).
tff(98,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(99,plain,
inverse(multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[98,97]) ).
tff(100,plain,
multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))) = inverse(multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(symmetry,[status(thm)],[99]) ).
tff(101,plain,
inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))) = inverse(inverse(multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(monotonicity,[status(thm)],[100]) ).
tff(102,plain,
inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))) = multiply(inverse(greatest_lower_bound(a,identity)),a),
inference(transitivity,[status(thm)],[101,90]) ).
tff(103,plain,
multiply(inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),inverse(identity)) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),
inference(monotonicity,[status(thm)],[102,12]) ).
tff(104,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))))) = multiply(inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(105,plain,
inverse(multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))))) = multiply(inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),inverse(identity)),
inference(unit_resolution,[status(thm)],[104,97]) ).
tff(106,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(107,plain,
multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))) = multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[106,84]) ).
tff(108,plain,
multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))) = multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[107]) ).
tff(109,plain,
inverse(multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),
inference(transitivity,[status(thm)],[99,108]) ).
tff(110,plain,
inverse(inverse(multiply(inverse(greatest_lower_bound(a,identity)),a))) = inverse(multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))))),
inference(monotonicity,[status(thm)],[109]) ).
tff(111,plain,
multiply(inverse(greatest_lower_bound(a,identity)),a) = inverse(inverse(multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(symmetry,[status(thm)],[90]) ).
tff(112,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(113,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)],[112]) ).
tff(114,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(115,axiom,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-2.ax',lub_absorbtion) ).
tff(116,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[116]) ).
tff(118,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[117,113]) ).
tff(119,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(a,greatest_lower_bound(a,identity)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(120,plain,
least_upper_bound(a,greatest_lower_bound(a,identity)) = a,
inference(unit_resolution,[status(thm)],[119,118]) ).
tff(121,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(122,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)],[121]) ).
tff(123,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(124,axiom,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-2.ax',symmetry_of_lub) ).
tff(125,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[124,123]) ).
tff(126,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(skolemize,[status(sab)],[125]) ).
tff(127,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[126,122]) ).
tff(128,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(a,greatest_lower_bound(a,identity)) = least_upper_bound(greatest_lower_bound(a,identity),a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(129,plain,
least_upper_bound(a,greatest_lower_bound(a,identity)) = least_upper_bound(greatest_lower_bound(a,identity),a),
inference(unit_resolution,[status(thm)],[128,127]) ).
tff(130,plain,
least_upper_bound(greatest_lower_bound(a,identity),a) = least_upper_bound(a,greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[129]) ).
tff(131,plain,
least_upper_bound(greatest_lower_bound(a,identity),a) = a,
inference(transitivity,[status(thm)],[130,120]) ).
tff(132,plain,
multiply(inverse(greatest_lower_bound(a,identity)),least_upper_bound(greatest_lower_bound(a,identity),a)) = multiply(inverse(greatest_lower_bound(a,identity)),a),
inference(monotonicity,[status(thm)],[131]) ).
tff(133,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(134,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)],[133]) ).
tff(135,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(136,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/sandbox/benchmark/Axioms/GRP004-2.ax',monotony_lub1) ).
tff(137,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)],[136,135]) ).
tff(138,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)],[137]) ).
tff(139,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)],[138,134]) ).
tff(140,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,identity)),least_upper_bound(greatest_lower_bound(a,identity),a)) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(141,plain,
multiply(inverse(greatest_lower_bound(a,identity)),least_upper_bound(greatest_lower_bound(a,identity),a)) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(unit_resolution,[status(thm)],[140,139]) ).
tff(142,plain,
least_upper_bound(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(inverse(greatest_lower_bound(a,identity)),least_upper_bound(greatest_lower_bound(a,identity),a)),
inference(symmetry,[status(thm)],[141]) ).
tff(143,plain,
multiply(multiply(identity,inverse(greatest_lower_bound(a,identity))),a) = multiply(inverse(greatest_lower_bound(a,identity)),a),
inference(monotonicity,[status(thm)],[86]) ).
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,identity))),a) = multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(145,plain,
multiply(multiply(identity,inverse(greatest_lower_bound(a,identity))),a) = multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(unit_resolution,[status(thm)],[144,44]) ).
tff(146,plain,
multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(multiply(identity,inverse(greatest_lower_bound(a,identity))),a),
inference(symmetry,[status(thm)],[145]) ).
tff(147,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(identity,greatest_lower_bound(identity,a)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(148,plain,
least_upper_bound(identity,greatest_lower_bound(identity,a)) = identity,
inference(unit_resolution,[status(thm)],[147,118]) ).
tff(149,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(150,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)],[149]) ).
tff(151,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(152,axiom,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-2.ax',symmetry_of_glb) ).
tff(153,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[152,151]) ).
tff(154,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(skolemize,[status(sab)],[153]) ).
tff(155,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[154,150]) ).
tff(156,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(a,identity) = greatest_lower_bound(identity,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(157,plain,
greatest_lower_bound(a,identity) = greatest_lower_bound(identity,a),
inference(unit_resolution,[status(thm)],[156,155]) ).
tff(158,plain,
least_upper_bound(identity,greatest_lower_bound(a,identity)) = least_upper_bound(identity,greatest_lower_bound(identity,a)),
inference(monotonicity,[status(thm)],[157]) ).
tff(159,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(identity,greatest_lower_bound(a,identity)) = least_upper_bound(greatest_lower_bound(a,identity),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(160,plain,
least_upper_bound(identity,greatest_lower_bound(a,identity)) = least_upper_bound(greatest_lower_bound(a,identity),identity),
inference(unit_resolution,[status(thm)],[159,127]) ).
tff(161,plain,
least_upper_bound(greatest_lower_bound(a,identity),identity) = least_upper_bound(identity,greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[160]) ).
tff(162,plain,
least_upper_bound(greatest_lower_bound(a,identity),identity) = identity,
inference(transitivity,[status(thm)],[161,158,148]) ).
tff(163,plain,
multiply(least_upper_bound(greatest_lower_bound(a,identity),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(monotonicity,[status(thm)],[162]) ).
tff(164,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(165,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)],[164]) ).
tff(166,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(167,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/sandbox/benchmark/Axioms/GRP004-2.ax',monotony_lub2) ).
tff(168,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)],[167,166]) ).
tff(169,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)],[168]) ).
tff(170,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)],[169,165]) ).
tff(171,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(least_upper_bound(greatest_lower_bound(a,identity),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(172,plain,
multiply(least_upper_bound(greatest_lower_bound(a,identity),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(unit_resolution,[status(thm)],[171,170]) ).
tff(173,plain,
least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a))) = multiply(least_upper_bound(greatest_lower_bound(a,identity),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(symmetry,[status(thm)],[172]) ).
tff(174,plain,
multiply(inverse(greatest_lower_bound(a,identity)),a) = multiply(multiply(identity,inverse(greatest_lower_bound(a,identity))),a),
inference(symmetry,[status(thm)],[143]) ).
tff(175,plain,
multiply(inverse(greatest_lower_bound(a,identity)),a) = multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(transitivity,[status(thm)],[174,145]) ).
tff(176,plain,
least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(inverse(greatest_lower_bound(a,identity)),a)) = least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(identity,multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(monotonicity,[status(thm)],[175]) ).
tff(177,plain,
least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(inverse(greatest_lower_bound(a,identity)),a),
inference(transitivity,[status(thm)],[176,173,163,146,143]) ).
tff(178,plain,
identity = multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[15]) ).
tff(179,plain,
least_upper_bound(identity,least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(inverse(greatest_lower_bound(a,identity)),a))) = least_upper_bound(multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(monotonicity,[status(thm)],[178,177]) ).
tff(180,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(181,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)],[180]) ).
tff(182,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(183,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/sandbox/benchmark/Axioms/GRP004-2.ax',associativity_of_lub) ).
tff(184,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)],[183,182]) ).
tff(185,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)],[184]) ).
tff(186,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)],[185,181]) ).
tff(187,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(identity,least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(inverse(greatest_lower_bound(a,identity)),a))) = least_upper_bound(least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),multiply(inverse(greatest_lower_bound(a,identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(188,plain,
least_upper_bound(identity,least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(inverse(greatest_lower_bound(a,identity)),a))) = least_upper_bound(least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(unit_resolution,[status(thm)],[187,186]) ).
tff(189,plain,
least_upper_bound(least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),multiply(inverse(greatest_lower_bound(a,identity)),a)) = least_upper_bound(identity,least_upper_bound(multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(symmetry,[status(thm)],[188]) ).
tff(190,plain,
multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))) = identity,
inference(transitivity,[status(thm)],[66,69]) ).
tff(191,plain,
multiply(multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))),a) = multiply(identity,a),
inference(monotonicity,[status(thm)],[190]) ).
tff(192,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))),a) = multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(193,plain,
multiply(multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))),a) = multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(unit_resolution,[status(thm)],[192,44]) ).
tff(194,plain,
multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(multiply(greatest_lower_bound(a,identity),inverse(greatest_lower_bound(a,identity))),a),
inference(symmetry,[status(thm)],[193]) ).
tff(195,plain,
multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = multiply(identity,a),
inference(transitivity,[status(thm)],[194,191]) ).
tff(196,plain,
multiply(inverse(identity),identity) = multiply(identity,identity),
inference(symmetry,[status(thm)],[48]) ).
tff(197,plain,
identity = multiply(inverse(identity),identity),
inference(symmetry,[status(thm)],[9]) ).
tff(198,plain,
identity = multiply(identity,identity),
inference(transitivity,[status(thm)],[197,196]) ).
tff(199,plain,
least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))) = least_upper_bound(multiply(identity,identity),multiply(identity,a)),
inference(monotonicity,[status(thm)],[198,195]) ).
tff(200,plain,
least_upper_bound(multiply(identity,identity),multiply(identity,a)) = least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(symmetry,[status(thm)],[199]) ).
tff(201,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(identity,least_upper_bound(identity,a)) = least_upper_bound(multiply(identity,identity),multiply(identity,a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(202,plain,
multiply(identity,least_upper_bound(identity,a)) = least_upper_bound(multiply(identity,identity),multiply(identity,a)),
inference(unit_resolution,[status(thm)],[201,139]) ).
tff(203,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(a,identity) = least_upper_bound(identity,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(204,plain,
least_upper_bound(a,identity) = least_upper_bound(identity,a),
inference(unit_resolution,[status(thm)],[203,127]) ).
tff(205,plain,
least_upper_bound(identity,a) = least_upper_bound(a,identity),
inference(symmetry,[status(thm)],[204]) ).
tff(206,plain,
multiply(identity,least_upper_bound(identity,a)) = multiply(identity,least_upper_bound(a,identity)),
inference(monotonicity,[status(thm)],[205]) ).
tff(207,plain,
multiply(identity,least_upper_bound(a,identity)) = multiply(identity,least_upper_bound(identity,a)),
inference(symmetry,[status(thm)],[206]) ).
tff(208,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,least_upper_bound(a,identity)) = least_upper_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(209,plain,
multiply(identity,least_upper_bound(a,identity)) = least_upper_bound(a,identity),
inference(unit_resolution,[status(thm)],[208,84]) ).
tff(210,plain,
least_upper_bound(a,identity) = multiply(identity,least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[209]) ).
tff(211,plain,
least_upper_bound(a,identity) = least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(transitivity,[status(thm)],[210,207,202,200]) ).
tff(212,plain,
least_upper_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = least_upper_bound(least_upper_bound(identity,multiply(greatest_lower_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(monotonicity,[status(thm)],[211]) ).
tff(213,plain,
inverse(multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))))) = inverse(inverse(multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(symmetry,[status(thm)],[110]) ).
tff(214,plain,
multiply(inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),inverse(identity)) = inverse(multiply(identity,multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity)))))),
inference(symmetry,[status(thm)],[105]) ).
tff(215,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity) = multiply(inverse(multiply(inverse(a),inverse(inverse(greatest_lower_bound(a,identity))))),inverse(identity)),
inference(symmetry,[status(thm)],[103]) ).
tff(216,plain,
greatest_lower_bound(identity,a) = greatest_lower_bound(a,identity),
inference(symmetry,[status(thm)],[157]) ).
tff(217,plain,
greatest_lower_bound(identity,a) = inverse(inverse(greatest_lower_bound(a,identity))),
inference(transitivity,[status(thm)],[216,65]) ).
tff(218,plain,
multiply(greatest_lower_bound(identity,a),inverse(greatest_lower_bound(a,identity))) = multiply(inverse(inverse(greatest_lower_bound(a,identity))),inverse(greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[217]) ).
tff(219,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,a),inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(multiply(identity,inverse(greatest_lower_bound(a,identity))),multiply(a,inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(220,plain,
multiply(greatest_lower_bound(identity,a),inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(multiply(identity,inverse(greatest_lower_bound(a,identity))),multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[219,34]) ).
tff(221,plain,
greatest_lower_bound(multiply(identity,inverse(greatest_lower_bound(a,identity))),multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(identity,a),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[220]) ).
tff(222,plain,
greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = greatest_lower_bound(multiply(identity,inverse(greatest_lower_bound(a,identity))),multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(monotonicity,[status(thm)],[87]) ).
tff(223,plain,
greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = identity,
inference(transitivity,[status(thm)],[222,221,218,69]) ).
tff(224,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),
inference(monotonicity,[status(thm)],[223]) ).
tff(225,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(226,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(unit_resolution,[status(thm)],[225,22]) ).
tff(227,plain,
greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[226]) ).
tff(228,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(229,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[228,44]) ).
tff(230,plain,
multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[229]) ).
tff(231,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(monotonicity,[status(thm)],[230]) ).
tff(232,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(233,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(unit_resolution,[status(thm)],[232,34]) ).
tff(234,plain,
greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(monotonicity,[status(thm)],[88]) ).
tff(235,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(symmetry,[status(thm)],[234]) ).
tff(236,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,identity)),greatest_lower_bound(identity,a)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(237,plain,
multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(identity,a)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(unit_resolution,[status(thm)],[236,22]) ).
tff(238,plain,
multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(identity,a)) = multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)),
inference(monotonicity,[status(thm)],[216]) ).
tff(239,plain,
multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(a,identity)) = multiply(inverse(greatest_lower_bound(a,identity)),greatest_lower_bound(identity,a)),
inference(symmetry,[status(thm)],[238]) ).
tff(240,plain,
identity = greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(transitivity,[status(thm)],[178,239,237,235]) ).
tff(241,plain,
multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(monotonicity,[status(thm)],[240]) ).
tff(242,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(243,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)],[242]) ).
tff(244,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(245,axiom,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-2.ax',glb_absorbtion) ).
tff(246,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[245,244]) ).
tff(247,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[246]) ).
tff(248,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[247,243]) ).
tff(249,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(identity,least_upper_bound(identity,a)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(250,plain,
greatest_lower_bound(identity,least_upper_bound(identity,a)) = identity,
inference(unit_resolution,[status(thm)],[249,248]) ).
tff(251,plain,
greatest_lower_bound(identity,least_upper_bound(a,identity)) = greatest_lower_bound(identity,least_upper_bound(identity,a)),
inference(monotonicity,[status(thm)],[204]) ).
tff(252,plain,
greatest_lower_bound(identity,least_upper_bound(a,identity)) = identity,
inference(transitivity,[status(thm)],[251,250]) ).
tff(253,plain,
multiply(greatest_lower_bound(identity,least_upper_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(monotonicity,[status(thm)],[252]) ).
tff(254,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,least_upper_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = greatest_lower_bound(multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(255,plain,
multiply(greatest_lower_bound(identity,least_upper_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = greatest_lower_bound(multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(unit_resolution,[status(thm)],[254,34]) ).
tff(256,plain,
greatest_lower_bound(multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) = multiply(greatest_lower_bound(identity,least_upper_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[255]) ).
tff(257,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(a,inverse(greatest_lower_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(258,plain,
multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(a,inverse(greatest_lower_bound(a,identity))),
inference(unit_resolution,[status(thm)],[257,84]) ).
tff(259,plain,
multiply(a,inverse(greatest_lower_bound(a,identity))) = multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[258]) ).
tff(260,plain,
greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(monotonicity,[status(thm)],[259]) ).
tff(261,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(a,least_upper_bound(a,identity)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(262,plain,
greatest_lower_bound(a,least_upper_bound(a,identity)) = a,
inference(unit_resolution,[status(thm)],[261,248]) ).
tff(263,plain,
multiply(greatest_lower_bound(a,least_upper_bound(a,identity)),inverse(greatest_lower_bound(a,identity))) = multiply(a,inverse(greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[262]) ).
tff(264,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(a,identity)),inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(265,plain,
multiply(greatest_lower_bound(a,least_upper_bound(a,identity)),inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[264,34]) ).
tff(266,plain,
greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(a,least_upper_bound(a,identity)),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[265]) ).
tff(267,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,identity),inverse(greatest_lower_bound(a,identity))) = least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(268,plain,
multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) = least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),
inference(unit_resolution,[status(thm)],[267,170]) ).
tff(269,plain,
least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))) = multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[268]) ).
tff(270,plain,
greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity)))),
inference(monotonicity,[status(thm)],[269]) ).
tff(271,plain,
greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity))))) = multiply(a,inverse(greatest_lower_bound(a,identity))),
inference(transitivity,[status(thm)],[270,266,263]) ).
tff(272,plain,
greatest_lower_bound(greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity))))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(monotonicity,[status(thm)],[271]) ).
tff(273,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(274,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)],[273]) ).
tff(275,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(276,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/sandbox/benchmark/Axioms/GRP004-2.ax',associativity_of_glb) ).
tff(277,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)],[276,275]) ).
tff(278,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)],[277]) ).
tff(279,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)],[278,274]) ).
tff(280,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(a,inverse(greatest_lower_bound(a,identity))),greatest_lower_bound(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity)))))) = greatest_lower_bound(greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity))))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(281,plain,
greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),greatest_lower_bound(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity)))))) = greatest_lower_bound(greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity))))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(unit_resolution,[status(thm)],[280,279]) ).
tff(282,plain,
greatest_lower_bound(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(monotonicity,[status(thm)],[269]) ).
tff(283,plain,
greatest_lower_bound(multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[282]) ).
tff(284,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(least_upper_bound(a,identity),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(285,plain,
multiply(least_upper_bound(a,identity),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(unit_resolution,[status(thm)],[284,22]) ).
tff(286,plain,
multiply(least_upper_bound(a,identity),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))) = multiply(least_upper_bound(a,identity),identity),
inference(monotonicity,[status(thm)],[223]) ).
tff(287,plain,
multiply(least_upper_bound(a,identity),identity) = multiply(least_upper_bound(a,identity),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[286]) ).
tff(288,plain,
multiply(least_upper_bound(a,identity),identity) = greatest_lower_bound(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(transitivity,[status(thm)],[287,285,283]) ).
tff(289,plain,
greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),identity)) = greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),greatest_lower_bound(least_upper_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(identity,inverse(greatest_lower_bound(a,identity)))),multiply(least_upper_bound(a,identity),multiply(a,inverse(greatest_lower_bound(a,identity)))))),
inference(monotonicity,[status(thm)],[288]) ).
tff(290,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(a,inverse(greatest_lower_bound(a,identity)))) = multiply(identity,multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[241]) ).
tff(291,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),a)),multiply(a,inverse(greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[233]) ).
tff(292,plain,
greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),inverse(greatest_lower_bound(a,identity))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity)))),multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[231]) ).
tff(293,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity) = multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),multiply(a,inverse(greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[224]) ).
tff(294,plain,
multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity) = multiply(a,inverse(greatest_lower_bound(a,identity))),
inference(transitivity,[status(thm)],[293,226,292,291,290,258]) ).
tff(295,plain,
greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(least_upper_bound(a,identity),identity)) = greatest_lower_bound(multiply(a,inverse(greatest_lower_bound(a,identity))),multiply(least_upper_bound(a,identity),identity)),
inference(monotonicity,[status(thm)],[294]) ).
tff(296,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(greatest_lower_bound(Y,Z),X) = greatest_lower_bound(multiply(Y,X),multiply(Z,X)) )
| ( multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),least_upper_bound(a,identity)),identity) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(least_upper_bound(a,identity),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(297,plain,
multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),least_upper_bound(a,identity)),identity) = greatest_lower_bound(multiply(multiply(inverse(greatest_lower_bound(a,identity)),a),identity),multiply(least_upper_bound(a,identity),identity)),
inference(unit_resolution,[status(thm)],[296,34]) ).
tff(298,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(299,plain,
greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(unit_resolution,[status(thm)],[298,155]) ).
tff(300,plain,
greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[299]) ).
tff(301,plain,
multiply(greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),identity) = multiply(greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),a),least_upper_bound(a,identity)),identity),
inference(monotonicity,[status(thm)],[300]) ).
tff(302,plain,
multiply(greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),identity) = multiply(inverse(greatest_lower_bound(a,identity)),a),
inference(transitivity,[status(thm)],[301,297,295,289,281,272,260,256,253,241,233,231,227,224,215,214,213,90]) ).
tff(303,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(least_upper_bound(a,identity))) = least_upper_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(304,plain,
inverse(inverse(least_upper_bound(a,identity))) = least_upper_bound(a,identity),
inference(unit_resolution,[status(thm)],[303,62]) ).
tff(305,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(identity,least_upper_bound(a,identity))) = multiply(inverse(least_upper_bound(a,identity)),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(306,plain,
inverse(multiply(identity,least_upper_bound(a,identity))) = multiply(inverse(least_upper_bound(a,identity)),inverse(identity)),
inference(unit_resolution,[status(thm)],[305,97]) ).
tff(307,plain,
inverse(least_upper_bound(a,identity)) = inverse(multiply(identity,least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[210]) ).
tff(308,plain,
inverse(least_upper_bound(a,identity)) = multiply(inverse(least_upper_bound(a,identity)),inverse(identity)),
inference(transitivity,[status(thm)],[307,306]) ).
tff(309,plain,
inverse(inverse(least_upper_bound(a,identity))) = inverse(multiply(inverse(least_upper_bound(a,identity)),inverse(identity))),
inference(monotonicity,[status(thm)],[308]) ).
tff(310,plain,
inverse(multiply(inverse(least_upper_bound(a,identity)),inverse(identity))) = inverse(inverse(least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[309]) ).
tff(311,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,identity),identity) = least_upper_bound(multiply(a,identity),multiply(identity,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(312,plain,
multiply(least_upper_bound(a,identity),identity) = least_upper_bound(multiply(a,identity),multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[311,170]) ).
tff(313,plain,
least_upper_bound(multiply(a,identity),multiply(identity,identity)) = multiply(least_upper_bound(a,identity),identity),
inference(symmetry,[status(thm)],[312]) ).
tff(314,plain,
inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity))) = inverse(multiply(least_upper_bound(a,identity),identity)),
inference(monotonicity,[status(thm)],[313]) ).
tff(315,plain,
inverse(multiply(least_upper_bound(a,identity),identity)) = inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity))),
inference(symmetry,[status(thm)],[314]) ).
tff(316,plain,
( ~ ! [Y: $i,X: $i] : ( inverse(multiply(X,Y)) = multiply(inverse(Y),inverse(X)) )
| ( inverse(multiply(least_upper_bound(a,identity),identity)) = multiply(inverse(identity),inverse(least_upper_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(317,plain,
inverse(multiply(least_upper_bound(a,identity),identity)) = multiply(inverse(identity),inverse(least_upper_bound(a,identity))),
inference(unit_resolution,[status(thm)],[316,97]) ).
tff(318,plain,
multiply(inverse(identity),inverse(least_upper_bound(a,identity))) = inverse(multiply(least_upper_bound(a,identity),identity)),
inference(symmetry,[status(thm)],[317]) ).
tff(319,plain,
multiply(inverse(identity),inverse(least_upper_bound(a,identity))) = multiply(identity,multiply(inverse(least_upper_bound(a,identity)),inverse(identity))),
inference(monotonicity,[status(thm)],[12,308]) ).
tff(320,plain,
multiply(identity,multiply(inverse(least_upper_bound(a,identity)),inverse(identity))) = multiply(inverse(identity),inverse(least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[319]) ).
tff(321,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,multiply(inverse(least_upper_bound(a,identity)),inverse(identity))) = multiply(inverse(least_upper_bound(a,identity)),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(322,plain,
multiply(identity,multiply(inverse(least_upper_bound(a,identity)),inverse(identity))) = multiply(inverse(least_upper_bound(a,identity)),inverse(identity)),
inference(unit_resolution,[status(thm)],[321,84]) ).
tff(323,plain,
multiply(inverse(least_upper_bound(a,identity)),inverse(identity)) = multiply(identity,multiply(inverse(least_upper_bound(a,identity)),inverse(identity))),
inference(symmetry,[status(thm)],[322]) ).
tff(324,plain,
multiply(inverse(least_upper_bound(a,identity)),inverse(identity)) = inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity))),
inference(transitivity,[status(thm)],[323,320,318,315]) ).
tff(325,plain,
inverse(multiply(inverse(least_upper_bound(a,identity)),inverse(identity))) = inverse(inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity)))),
inference(monotonicity,[status(thm)],[324]) ).
tff(326,plain,
inverse(inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity)))) = inverse(multiply(inverse(least_upper_bound(a,identity)),inverse(identity))),
inference(symmetry,[status(thm)],[325]) ).
tff(327,plain,
( ~ ! [X: $i] : ( inverse(inverse(X)) = X )
| ( inverse(inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity)))) = least_upper_bound(multiply(a,identity),multiply(identity,identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(328,plain,
inverse(inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity)))) = least_upper_bound(multiply(a,identity),multiply(identity,identity)),
inference(unit_resolution,[status(thm)],[327,62]) ).
tff(329,plain,
least_upper_bound(multiply(a,identity),multiply(identity,identity)) = inverse(inverse(least_upper_bound(multiply(a,identity),multiply(identity,identity)))),
inference(symmetry,[status(thm)],[328]) ).
tff(330,plain,
multiply(least_upper_bound(a,identity),identity) = least_upper_bound(a,identity),
inference(transitivity,[status(thm)],[312,329,326,310,304]) ).
tff(331,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),identity)) = least_upper_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),
inference(monotonicity,[status(thm)],[330,302]) ).
tff(332,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(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),identity) = least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(333,plain,
multiply(least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),identity) = least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a)),identity)),
inference(unit_resolution,[status(thm)],[332,170]) ).
tff(334,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))) = least_upper_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(335,plain,
least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))) = least_upper_bound(a,identity),
inference(unit_resolution,[status(thm)],[334,118]) ).
tff(336,plain,
least_upper_bound(a,identity) = least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),
inference(symmetry,[status(thm)],[335]) ).
tff(337,plain,
multiply(least_upper_bound(a,identity),identity) = multiply(least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),multiply(inverse(greatest_lower_bound(a,identity)),a))),identity),
inference(monotonicity,[status(thm)],[336]) ).
tff(338,plain,
least_upper_bound(a,identity) = inverse(inverse(least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[304]) ).
tff(339,plain,
least_upper_bound(a,identity) = multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,identity)),
inference(transitivity,[status(thm)],[338,309,325,328,313,337,333,331,212,189,179,142,132,111,110,105,103,54]) ).
tff(340,plain,
greatest_lower_bound(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(multiply(inverse(greatest_lower_bound(a,identity)),multiply(a,identity)),multiply(inverse(greatest_lower_bound(a,identity)),identity)),
inference(monotonicity,[status(thm)],[339,88]) ).
tff(341,plain,
greatest_lower_bound(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) = identity,
inference(transitivity,[status(thm)],[340,56,37,27,9]) ).
tff(342,plain,
( ( greatest_lower_bound(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) != identity )
<=> ( greatest_lower_bound(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) != identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(343,axiom,
greatest_lower_bound(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) != identity,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_p20) ).
tff(344,plain,
greatest_lower_bound(least_upper_bound(a,identity),inverse(greatest_lower_bound(a,identity))) != identity,
inference(modus_ponens,[status(thm)],[343,342]) ).
tff(345,plain,
$false,
inference(unit_resolution,[status(thm)],[344,341]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP183-2 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n004.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 15:45:08 EDT 2025
% 0.12/0.34 % CPUTime :
% 63.73/63.93 % SZS status Unsatisfiable
% 63.73/63.93 % SZS output start Proof
% See solution above
% 63.73/63.97 % E exiting
%------------------------------------------------------------------------------