%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP167-5 : TPTP v9.0.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n016.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 05:07:19 AM UTC 2025
% Result : Unsatisfiable 167.20s 167.51s
% Output : Proof 167.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 25
% Number of leaves : 78
% Syntax : Number of formulae : 244 ( 181 unt; 0 typ; 0 def)
% Number of atoms : 335 ( 320 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 115 ( 38 ~; 34 |; 0 &)
% ( 43 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 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 : 8 ( 8 usr; 2 con; 0-2 aty)
% Number of variables : 328 ( 299 !; 0 ?; 328 :)
% Comments :
%------------------------------------------------------------------------------
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(negative_part_type,type,
negative_part: $i > $i ).
tff(a_type,type,
a: $i ).
tff(positive_part_type,type,
positive_part: $i > $i ).
tff(greatest_lower_bound_type,type,
greatest_lower_bound: ( $i * $i ) > $i ).
tff(identity_type,type,
identity: $i ).
tff(least_upper_bound_type,type,
least_upper_bound: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(1,plain,
^ [X: $i] :
refl(( ( negative_part(X) = greatest_lower_bound(X,identity) )
<=> ( negative_part(X) = greatest_lower_bound(X,identity) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) )
<=> ! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) )
<=> ! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',lat4_2) ).
tff(5,plain,
! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( negative_part(X) = greatest_lower_bound(X,identity) )
| ( negative_part(a) = greatest_lower_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
negative_part(a) = greatest_lower_bound(a,identity),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( ( positive_part(X) = least_upper_bound(X,identity) )
<=> ( positive_part(X) = least_upper_bound(X,identity) ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) )
<=> ! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) )
<=> ! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',lat4_1) ).
tff(14,plain,
! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : ( positive_part(X) = least_upper_bound(X,identity) )
| ( positive_part(a) = least_upper_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
positive_part(a) = least_upper_bound(a,identity),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
multiply(positive_part(a),negative_part(a)) = multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity)),
inference(monotonicity,[status(thm)],[18,9]) ).
tff(20,plain,
multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity)) = multiply(positive_part(a),negative_part(a)),
inference(symmetry,[status(thm)],[19]) ).
tff(21,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(22,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)],[21]) ).
tff(23,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(24,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(25,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)],[24,23]) ).
tff(26,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)],[25]) ).
tff(27,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)],[26,22]) ).
tff(28,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),greatest_lower_bound(a,identity)) = least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(29,plain,
multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity)) = least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))),
inference(unit_resolution,[status(thm)],[28,27]) ).
tff(30,plain,
least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))) = multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[29]) ).
tff(31,plain,
^ [X: $i] :
refl(( ( multiply(identity,X) = X )
<=> ( multiply(identity,X) = X ) )),
inference(bind,[status(th)],[]) ).
tff(32,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(quant_intro,[status(thm)],[31]) ).
tff(33,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,axiom,
! [X: $i] : ( multiply(identity,X) = X ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-0.ax',left_identity) ).
tff(35,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(skolemize,[status(sab)],[35]) ).
tff(37,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
multiply(identity,least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))),
inference(unit_resolution,[status(thm)],[38,37]) ).
tff(40,plain,
^ [B: $i,A: $i] :
refl(( ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
<=> ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ) )),
inference(bind,[status(th)],[]) ).
tff(41,plain,
( ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
<=> ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ) ),
inference(quant_intro,[status(thm)],[40]) ).
tff(42,plain,
( ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
<=> ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(43,axiom,
! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',p10) ).
tff(44,plain,
! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ),
inference(skolemize,[status(sab)],[44]) ).
tff(46,plain,
! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) ),
inference(modus_ponens,[status(thm)],[45,41]) ).
tff(47,plain,
( ~ ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
| ( inverse(least_upper_bound(a,identity)) = greatest_lower_bound(inverse(a),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(48,plain,
inverse(least_upper_bound(a,identity)) = greatest_lower_bound(inverse(a),inverse(identity)),
inference(unit_resolution,[status(thm)],[47,46]) ).
tff(49,plain,
greatest_lower_bound(inverse(a),inverse(identity)) = inverse(least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[48]) ).
tff(50,plain,
greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity))) = greatest_lower_bound(inverse(identity),inverse(least_upper_bound(a,identity))),
inference(monotonicity,[status(thm)],[49]) ).
tff(51,plain,
greatest_lower_bound(inverse(identity),inverse(least_upper_bound(a,identity))) = greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity))),
inference(symmetry,[status(thm)],[50]) ).
tff(52,plain,
( ~ ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
| ( inverse(least_upper_bound(identity,least_upper_bound(a,identity))) = greatest_lower_bound(inverse(identity),inverse(least_upper_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(53,plain,
inverse(least_upper_bound(identity,least_upper_bound(a,identity))) = greatest_lower_bound(inverse(identity),inverse(least_upper_bound(a,identity))),
inference(unit_resolution,[status(thm)],[52,46]) ).
tff(54,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(55,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)],[54]) ).
tff(56,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(57,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(58,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[58]) ).
tff(60,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,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),identity)) = least_upper_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(62,plain,
least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),identity)) = least_upper_bound(a,identity),
inference(unit_resolution,[status(thm)],[61,60]) ).
tff(63,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(64,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)],[63]) ).
tff(65,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(66,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(67,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[66,65]) ).
tff(68,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(skolemize,[status(sab)],[67]) ).
tff(69,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[68,64]) ).
tff(70,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(identity,least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(a,identity),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(71,plain,
greatest_lower_bound(identity,least_upper_bound(a,identity)) = greatest_lower_bound(least_upper_bound(a,identity),identity),
inference(unit_resolution,[status(thm)],[70,69]) ).
tff(72,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(73,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)],[72]) ).
tff(74,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(75,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(76,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,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(80,plain,
least_upper_bound(a,identity) = least_upper_bound(identity,a),
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
greatest_lower_bound(identity,least_upper_bound(a,identity)) = greatest_lower_bound(identity,least_upper_bound(identity,a)),
inference(monotonicity,[status(thm)],[80]) ).
tff(82,plain,
greatest_lower_bound(identity,least_upper_bound(identity,a)) = greatest_lower_bound(identity,least_upper_bound(a,identity)),
inference(symmetry,[status(thm)],[81]) ).
tff(83,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(84,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)],[83]) ).
tff(85,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(86,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(87,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[87]) ).
tff(89,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[88,84]) ).
tff(90,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(91,plain,
greatest_lower_bound(identity,least_upper_bound(identity,a)) = identity,
inference(unit_resolution,[status(thm)],[90,89]) ).
tff(92,plain,
identity = greatest_lower_bound(identity,least_upper_bound(identity,a)),
inference(symmetry,[status(thm)],[91]) ).
tff(93,plain,
identity = greatest_lower_bound(least_upper_bound(a,identity),identity),
inference(transitivity,[status(thm)],[92,82,71]) ).
tff(94,plain,
least_upper_bound(least_upper_bound(a,identity),identity) = least_upper_bound(least_upper_bound(a,identity),greatest_lower_bound(least_upper_bound(a,identity),identity)),
inference(monotonicity,[status(thm)],[93]) ).
tff(95,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(identity,least_upper_bound(a,identity)) = least_upper_bound(least_upper_bound(a,identity),identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(96,plain,
least_upper_bound(identity,least_upper_bound(a,identity)) = least_upper_bound(least_upper_bound(a,identity),identity),
inference(unit_resolution,[status(thm)],[95,78]) ).
tff(97,plain,
least_upper_bound(identity,least_upper_bound(a,identity)) = least_upper_bound(a,identity),
inference(transitivity,[status(thm)],[96,94,62]) ).
tff(98,plain,
inverse(least_upper_bound(identity,least_upper_bound(a,identity))) = inverse(least_upper_bound(a,identity)),
inference(monotonicity,[status(thm)],[97]) ).
tff(99,plain,
inverse(least_upper_bound(a,identity)) = inverse(least_upper_bound(identity,least_upper_bound(a,identity))),
inference(symmetry,[status(thm)],[98]) ).
tff(100,plain,
greatest_lower_bound(inverse(a),inverse(identity)) = greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity))),
inference(transitivity,[status(thm)],[49,99,53,51]) ).
tff(101,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(inverse(a),inverse(identity)) = greatest_lower_bound(inverse(identity),inverse(a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(102,plain,
greatest_lower_bound(inverse(a),inverse(identity)) = greatest_lower_bound(inverse(identity),inverse(a)),
inference(unit_resolution,[status(thm)],[101,69]) ).
tff(103,plain,
least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity))) = least_upper_bound(greatest_lower_bound(inverse(identity),inverse(a)),greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity)))),
inference(monotonicity,[status(thm)],[102,100]) ).
tff(104,plain,
least_upper_bound(greatest_lower_bound(inverse(identity),inverse(a)),greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity)))) = least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity))),
inference(symmetry,[status(thm)],[103]) ).
tff(105,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) )
<=> ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ) )),
inference(bind,[status(th)],[]) ).
tff(106,plain,
( ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ) ),
inference(quant_intro,[status(thm)],[105]) ).
tff(107,plain,
( ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) )
<=> ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(108,axiom,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',lat4_4) ).
tff(109,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ),
inference(modus_ponens,[status(thm)],[108,107]) ).
tff(110,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ),
inference(skolemize,[status(sab)],[109]) ).
tff(111,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) ),
inference(modus_ponens,[status(thm)],[110,106]) ).
tff(112,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(greatest_lower_bound(X,Y),greatest_lower_bound(X,Z)) )
| ( greatest_lower_bound(inverse(identity),least_upper_bound(inverse(a),greatest_lower_bound(inverse(a),inverse(identity)))) = least_upper_bound(greatest_lower_bound(inverse(identity),inverse(a)),greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(113,plain,
greatest_lower_bound(inverse(identity),least_upper_bound(inverse(a),greatest_lower_bound(inverse(a),inverse(identity)))) = least_upper_bound(greatest_lower_bound(inverse(identity),inverse(a)),greatest_lower_bound(inverse(identity),greatest_lower_bound(inverse(a),inverse(identity)))),
inference(unit_resolution,[status(thm)],[112,111]) ).
tff(114,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(inverse(a),greatest_lower_bound(inverse(a),inverse(identity))) = inverse(a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(115,plain,
least_upper_bound(inverse(a),greatest_lower_bound(inverse(a),inverse(identity))) = inverse(a),
inference(unit_resolution,[status(thm)],[114,60]) ).
tff(116,plain,
greatest_lower_bound(inverse(identity),least_upper_bound(inverse(a),greatest_lower_bound(inverse(a),inverse(identity)))) = greatest_lower_bound(inverse(identity),inverse(a)),
inference(monotonicity,[status(thm)],[115]) ).
tff(117,plain,
greatest_lower_bound(inverse(identity),inverse(a)) = greatest_lower_bound(inverse(identity),least_upper_bound(inverse(a),greatest_lower_bound(inverse(a),inverse(identity)))),
inference(symmetry,[status(thm)],[116]) ).
tff(118,plain,
greatest_lower_bound(inverse(a),inverse(identity)) = least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity))),
inference(transitivity,[status(thm)],[102,117,113,104]) ).
tff(119,plain,
inverse(greatest_lower_bound(inverse(a),inverse(identity))) = inverse(least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity)))),
inference(monotonicity,[status(thm)],[118]) ).
tff(120,plain,
inverse(least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity)))) = inverse(greatest_lower_bound(inverse(a),inverse(identity))),
inference(symmetry,[status(thm)],[119]) ).
tff(121,plain,
( ~ ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
| ( inverse(least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity)))) = greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(122,plain,
inverse(least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity)))) = greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),
inference(unit_resolution,[status(thm)],[121,46]) ).
tff(123,plain,
greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))) = inverse(least_upper_bound(greatest_lower_bound(inverse(a),inverse(identity)),greatest_lower_bound(inverse(a),inverse(identity)))),
inference(symmetry,[status(thm)],[122]) ).
tff(124,plain,
greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))) = inverse(greatest_lower_bound(inverse(a),inverse(identity))),
inference(transitivity,[status(thm)],[123,120]) ).
tff(125,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))) = multiply(inverse(greatest_lower_bound(inverse(a),inverse(identity))),greatest_lower_bound(inverse(a),inverse(identity))),
inference(monotonicity,[status(thm)],[124]) ).
tff(126,plain,
multiply(inverse(greatest_lower_bound(inverse(a),inverse(identity))),greatest_lower_bound(inverse(a),inverse(identity))) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),
inference(symmetry,[status(thm)],[125]) ).
tff(127,plain,
^ [X: $i] :
refl(( ( multiply(inverse(X),X) = identity )
<=> ( multiply(inverse(X),X) = identity ) )),
inference(bind,[status(th)],[]) ).
tff(128,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(quant_intro,[status(thm)],[127]) ).
tff(129,plain,
( ! [X: $i] : ( multiply(inverse(X),X) = identity )
<=> ! [X: $i] : ( multiply(inverse(X),X) = identity ) ),
inference(rewrite,[status(thm)],[]) ).
tff(130,axiom,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP004-0.ax',left_inverse) ).
tff(131,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(skolemize,[status(sab)],[131]) ).
tff(133,plain,
! [X: $i] : ( multiply(inverse(X),X) = identity ),
inference(modus_ponens,[status(thm)],[132,128]) ).
tff(134,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(greatest_lower_bound(inverse(a),inverse(identity))),greatest_lower_bound(inverse(a),inverse(identity))) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(135,plain,
multiply(inverse(greatest_lower_bound(inverse(a),inverse(identity))),greatest_lower_bound(inverse(a),inverse(identity))) = identity,
inference(unit_resolution,[status(thm)],[134,133]) ).
tff(136,plain,
identity = multiply(inverse(greatest_lower_bound(inverse(a),inverse(identity))),greatest_lower_bound(inverse(a),inverse(identity))),
inference(symmetry,[status(thm)],[135]) ).
tff(137,plain,
identity = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),
inference(transitivity,[status(thm)],[136,126]) ).
tff(138,plain,
multiply(identity,least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))),
inference(monotonicity,[status(thm)],[137]) ).
tff(139,plain,
multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = multiply(identity,least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[138]) ).
tff(140,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(141,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)],[140]) ).
tff(142,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(143,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(144,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[143,142]) ).
tff(145,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(skolemize,[status(sab)],[144]) ).
tff(146,plain,
! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) ),
inference(modus_ponens,[status(thm)],[145,141]) ).
tff(147,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(148,plain,
multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))))),
inference(unit_resolution,[status(thm)],[147,146]) ).
tff(149,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))))) = multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[148]) ).
tff(150,plain,
( ~ ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
| ( inverse(least_upper_bound(greatest_lower_bound(a,identity),identity)) = greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(151,plain,
inverse(least_upper_bound(greatest_lower_bound(a,identity),identity)) = greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity)),
inference(unit_resolution,[status(thm)],[150,46]) ).
tff(152,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(153,plain,
least_upper_bound(identity,greatest_lower_bound(a,identity)) = least_upper_bound(greatest_lower_bound(a,identity),identity),
inference(unit_resolution,[status(thm)],[152,78]) ).
tff(154,plain,
least_upper_bound(greatest_lower_bound(a,identity),identity) = least_upper_bound(identity,greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[153]) ).
tff(155,plain,
inverse(least_upper_bound(greatest_lower_bound(a,identity),identity)) = inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[154]) ).
tff(156,plain,
inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))) = inverse(least_upper_bound(greatest_lower_bound(a,identity),identity)),
inference(symmetry,[status(thm)],[155]) ).
tff(157,plain,
( ~ ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
| ( inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))) = greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(158,plain,
inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))) = greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),
inference(unit_resolution,[status(thm)],[157,46]) ).
tff(159,plain,
greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))) = inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[158]) ).
tff(160,plain,
greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))) = greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity)),
inference(transitivity,[status(thm)],[159,156,151]) ).
tff(161,plain,
greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))) = greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity))),
inference(monotonicity,[status(thm)],[160]) ).
tff(162,plain,
greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity))) = greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),
inference(symmetry,[status(thm)],[161]) ).
tff(163,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(164,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)],[163]) ).
tff(165,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(166,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(167,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
inference(modus_ponens,[status(thm)],[166,165]) ).
tff(168,plain,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
inference(skolemize,[status(sab)],[167]) ).
tff(169,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)],[168,164]) ).
tff(170,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) )
| ( greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity))) = greatest_lower_bound(greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))),inverse(identity)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(171,plain,
greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity))) = greatest_lower_bound(greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))),inverse(identity)),
inference(unit_resolution,[status(thm)],[170,169]) ).
tff(172,plain,
greatest_lower_bound(greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))),inverse(identity)) = greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(greatest_lower_bound(a,identity)),inverse(identity))),
inference(symmetry,[status(thm)],[171]) ).
tff(173,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(174,plain,
least_upper_bound(a,greatest_lower_bound(a,identity)) = a,
inference(unit_resolution,[status(thm)],[173,60]) ).
tff(175,plain,
inverse(least_upper_bound(a,greatest_lower_bound(a,identity))) = inverse(a),
inference(monotonicity,[status(thm)],[174]) ).
tff(176,plain,
( ~ ! [B: $i,A: $i] : ( inverse(least_upper_bound(A,B)) = greatest_lower_bound(inverse(A),inverse(B)) )
| ( inverse(least_upper_bound(a,greatest_lower_bound(a,identity))) = greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(177,plain,
inverse(least_upper_bound(a,greatest_lower_bound(a,identity))) = greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))),
inference(unit_resolution,[status(thm)],[176,46]) ).
tff(178,plain,
greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))) = inverse(least_upper_bound(a,greatest_lower_bound(a,identity))),
inference(symmetry,[status(thm)],[177]) ).
tff(179,plain,
greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))) = inverse(a),
inference(transitivity,[status(thm)],[178,175]) ).
tff(180,plain,
greatest_lower_bound(greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))),inverse(identity)) = greatest_lower_bound(inverse(a),inverse(identity)),
inference(monotonicity,[status(thm)],[179]) ).
tff(181,plain,
greatest_lower_bound(inverse(a),inverse(identity)) = greatest_lower_bound(greatest_lower_bound(inverse(a),inverse(greatest_lower_bound(a,identity))),inverse(identity)),
inference(symmetry,[status(thm)],[180]) ).
tff(182,plain,
greatest_lower_bound(inverse(a),inverse(identity)) = greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),
inference(transitivity,[status(thm)],[181,172,162]) ).
tff(183,plain,
multiply(greatest_lower_bound(inverse(a),inverse(identity)),a) = multiply(greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),a),
inference(monotonicity,[status(thm)],[182]) ).
tff(184,plain,
multiply(greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),a) = multiply(greatest_lower_bound(inverse(a),inverse(identity)),a),
inference(symmetry,[status(thm)],[183]) ).
tff(185,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(186,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)],[185]) ).
tff(187,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(188,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(189,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)],[188,187]) ).
tff(190,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)],[189]) ).
tff(191,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)],[190,186]) ).
tff(192,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(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),a) = greatest_lower_bound(multiply(inverse(a),a),multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(193,plain,
multiply(greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),a) = greatest_lower_bound(multiply(inverse(a),a),multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a)),
inference(unit_resolution,[status(thm)],[192,191]) ).
tff(194,plain,
greatest_lower_bound(multiply(inverse(a),a),multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a)) = multiply(greatest_lower_bound(inverse(a),greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity)))),a),
inference(symmetry,[status(thm)],[193]) ).
tff(195,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,a) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(196,plain,
multiply(identity,a) = a,
inference(unit_resolution,[status(thm)],[195,37]) ).
tff(197,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(identity),identity) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(198,plain,
multiply(inverse(identity),identity) = identity,
inference(unit_resolution,[status(thm)],[197,133]) ).
tff(199,plain,
multiply(multiply(inverse(identity),identity),a) = multiply(identity,a),
inference(monotonicity,[status(thm)],[198]) ).
tff(200,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(inverse(identity),identity),a) = multiply(inverse(identity),multiply(identity,a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(201,plain,
multiply(multiply(inverse(identity),identity),a) = multiply(inverse(identity),multiply(identity,a)),
inference(unit_resolution,[status(thm)],[200,146]) ).
tff(202,plain,
multiply(inverse(identity),multiply(identity,a)) = multiply(multiply(inverse(identity),identity),a),
inference(symmetry,[status(thm)],[201]) ).
tff(203,plain,
a = multiply(identity,a),
inference(symmetry,[status(thm)],[196]) ).
tff(204,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(205,plain,
greatest_lower_bound(a,identity) = greatest_lower_bound(identity,a),
inference(unit_resolution,[status(thm)],[204,69]) ).
tff(206,plain,
least_upper_bound(identity,greatest_lower_bound(a,identity)) = least_upper_bound(identity,greatest_lower_bound(identity,a)),
inference(monotonicity,[status(thm)],[205]) ).
tff(207,plain,
least_upper_bound(identity,greatest_lower_bound(identity,a)) = least_upper_bound(identity,greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[206]) ).
tff(208,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(209,plain,
least_upper_bound(identity,greatest_lower_bound(identity,a)) = identity,
inference(unit_resolution,[status(thm)],[208,60]) ).
tff(210,plain,
identity = least_upper_bound(identity,greatest_lower_bound(identity,a)),
inference(symmetry,[status(thm)],[209]) ).
tff(211,plain,
identity = least_upper_bound(identity,greatest_lower_bound(a,identity)),
inference(transitivity,[status(thm)],[210,207]) ).
tff(212,plain,
inverse(identity) = inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[211]) ).
tff(213,plain,
inverse(least_upper_bound(identity,greatest_lower_bound(a,identity))) = inverse(identity),
inference(symmetry,[status(thm)],[212]) ).
tff(214,plain,
greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))) = inverse(identity),
inference(transitivity,[status(thm)],[159,213]) ).
tff(215,plain,
multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a) = multiply(inverse(identity),multiply(identity,a)),
inference(monotonicity,[status(thm)],[214,203]) ).
tff(216,plain,
multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a) = a,
inference(transitivity,[status(thm)],[215,202,199,196]) ).
tff(217,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(a),a) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(218,plain,
multiply(inverse(a),a) = identity,
inference(unit_resolution,[status(thm)],[217,133]) ).
tff(219,plain,
greatest_lower_bound(multiply(inverse(a),a),multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a)) = greatest_lower_bound(identity,a),
inference(monotonicity,[status(thm)],[218,216]) ).
tff(220,plain,
greatest_lower_bound(identity,a) = greatest_lower_bound(multiply(inverse(a),a),multiply(greatest_lower_bound(inverse(identity),inverse(greatest_lower_bound(a,identity))),a)),
inference(symmetry,[status(thm)],[219]) ).
tff(221,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,greatest_lower_bound(a,identity)) = greatest_lower_bound(a,identity) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(222,plain,
multiply(identity,greatest_lower_bound(a,identity)) = greatest_lower_bound(a,identity),
inference(unit_resolution,[status(thm)],[221,37]) ).
tff(223,plain,
( ~ ! [X: $i] : ( multiply(inverse(X),X) = identity )
| ( multiply(inverse(least_upper_bound(a,identity)),least_upper_bound(a,identity)) = identity ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(224,plain,
multiply(inverse(least_upper_bound(a,identity)),least_upper_bound(a,identity)) = identity,
inference(unit_resolution,[status(thm)],[223,133]) ).
tff(225,plain,
multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(a,identity)) = multiply(inverse(least_upper_bound(a,identity)),least_upper_bound(a,identity)),
inference(monotonicity,[status(thm)],[49]) ).
tff(226,plain,
multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(a,identity)) = identity,
inference(transitivity,[status(thm)],[225,224]) ).
tff(227,plain,
multiply(multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(a,identity)),greatest_lower_bound(a,identity)) = multiply(identity,greatest_lower_bound(a,identity)),
inference(monotonicity,[status(thm)],[226]) ).
tff(228,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(a,identity)),greatest_lower_bound(a,identity)) = multiply(greatest_lower_bound(inverse(a),inverse(identity)),multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(229,plain,
multiply(multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(a,identity)),greatest_lower_bound(a,identity)) = multiply(greatest_lower_bound(inverse(a),inverse(identity)),multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity))),
inference(unit_resolution,[status(thm)],[228,146]) ).
tff(230,plain,
multiply(greatest_lower_bound(inverse(a),inverse(identity)),multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity))) = multiply(multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(a,identity)),greatest_lower_bound(a,identity)),
inference(symmetry,[status(thm)],[229]) ).
tff(231,plain,
multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(inverse(a),inverse(identity)),multiply(least_upper_bound(a,identity),greatest_lower_bound(a,identity))),
inference(monotonicity,[status(thm)],[30]) ).
tff(232,plain,
multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity)))) = multiply(greatest_lower_bound(inverse(a),inverse(identity)),a),
inference(transitivity,[status(thm)],[231,230,227,222,205,220,194,184]) ).
tff(233,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))))) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),a)),
inference(monotonicity,[status(thm)],[232]) ).
tff(234,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),a)) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),least_upper_bound(multiply(a,greatest_lower_bound(a,identity)),multiply(identity,greatest_lower_bound(a,identity))))),
inference(symmetry,[status(thm)],[233]) ).
tff(235,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(multiply(X,Y),Z) = multiply(X,multiply(Y,Z)) )
| ( multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),a) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),a)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(236,plain,
multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),a) = multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),multiply(greatest_lower_bound(inverse(a),inverse(identity)),a)),
inference(unit_resolution,[status(thm)],[235,146]) ).
tff(237,plain,
multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))) = identity,
inference(transitivity,[status(thm)],[125,135]) ).
tff(238,plain,
multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),a) = multiply(identity,a),
inference(monotonicity,[status(thm)],[237]) ).
tff(239,plain,
multiply(identity,a) = multiply(multiply(greatest_lower_bound(inverse(greatest_lower_bound(inverse(a),inverse(identity))),inverse(greatest_lower_bound(inverse(a),inverse(identity)))),greatest_lower_bound(inverse(a),inverse(identity))),a),
inference(symmetry,[status(thm)],[238]) ).
tff(240,plain,
a = multiply(positive_part(a),negative_part(a)),
inference(transitivity,[status(thm)],[203,239,236,234,149,139,39,30,20]) ).
tff(241,plain,
( ( a != multiply(positive_part(a),negative_part(a)) )
<=> ( a != multiply(positive_part(a),negative_part(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(242,axiom,
a != multiply(positive_part(a),negative_part(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_lat4) ).
tff(243,plain,
a != multiply(positive_part(a),negative_part(a)),
inference(modus_ponens,[status(thm)],[242,241]) ).
tff(244,plain,
$false,
inference(unit_resolution,[status(thm)],[243,240]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : GRP167-5 : TPTP v9.0.0. Bugfixed v1.2.1.
% 0.08/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n016.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 15:45:02 EDT 2025
% 0.13/0.34 % CPUTime :
% 167.20/167.51 % SZS status Unsatisfiable
% 167.20/167.51 % SZS output start Proof
% See solution above
% 167.20/167.54 % E exiting
%------------------------------------------------------------------------------