TSTP Solution File: GRP185-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP185-3 : TPTP v8.1.0. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n005.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 : Fri Sep 16 22:26:40 EDT 2022
% Result : Unsatisfiable 46.34s 29.56s
% Output : Proof 46.45s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 62
% Syntax : Number of formulae : 188 ( 136 unt; 6 typ; 0 def)
% Number of atoms : 246 ( 236 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 86 ( 31 ~; 27 |; 0 &)
% ( 28 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of FOOLs : 9 ( 9 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 6 ( 3 >; 3 *; 0 +; 0 <<)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 6 ( 6 usr; 3 con; 0-2 aty)
% Number of variables : 251 ( 230 !; 0 ?; 251 :)
% Comments :
%------------------------------------------------------------------------------
tff(least_upper_bound_type,type,
least_upper_bound: ( $i * $i ) > $i ).
tff(identity_type,type,
identity: $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(a_type,type,
a: $i ).
tff(greatest_lower_bound_type,type,
greatest_lower_bound: ( $i * $i ) > $i ).
tff(1,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(2,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)],[1]) ).
tff(3,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(4,axiom,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',lub_absorbtion) ).
tff(5,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)))) = least_upper_bound(multiply(a,b),identity) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)))) = least_upper_bound(multiply(a,b),identity),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,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(11,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)],[10]) ).
tff(12,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(13,axiom,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',symmetry_of_glb) ).
tff(14,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)) = greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))),
inference(symmetry,[status(thm)],[18]) ).
tff(20,plain,
least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)))),
inference(monotonicity,[status(thm)],[19]) ).
tff(21,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(22,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)],[21]) ).
tff(23,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(24,axiom,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',symmetry_of_lub) ).
tff(25,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) ),
inference(modus_ponens,[status(thm)],[26,22]) ).
tff(28,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(29,plain,
least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),identity)),
inference(unit_resolution,[status(thm)],[28,27]) ).
tff(30,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),identity)) = least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))),
inference(symmetry,[status(thm)],[29]) ).
tff(31,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(32,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)],[31]) ).
tff(33,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(34,axiom,
! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',associativity_of_lub) ).
tff(35,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)],[34,33]) ).
tff(36,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)],[35]) ).
tff(37,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)],[36,32]) ).
tff(38,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),identity)) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(39,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),identity)) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity),
inference(unit_resolution,[status(thm)],[38,37]) ).
tff(40,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),identity)),
inference(symmetry,[status(thm)],[39]) ).
tff(41,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(42,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)],[41]) ).
tff(43,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(44,axiom,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',glb_absorbtion) ).
tff(45,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(skolemize,[status(sab)],[45]) ).
tff(47,plain,
! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X ),
inference(modus_ponens,[status(thm)],[46,42]) ).
tff(48,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(identity,b),multiply(a,identity)))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(49,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(identity,b),multiply(a,identity)))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),
inference(unit_resolution,[status(thm)],[48,47]) ).
tff(50,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,greatest_lower_bound(X,Y)) = X )
| ( least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(51,plain,
least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),
inference(unit_resolution,[status(thm)],[50,7]) ).
tff(52,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(53,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)],[52]) ).
tff(54,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(55,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',monotony_lub1) ).
tff(56,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)],[55,54]) ).
tff(57,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)],[56]) ).
tff(58,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)],[57,53]) ).
tff(59,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(60,plain,
multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),
inference(unit_resolution,[status(thm)],[59,58]) ).
tff(61,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) = multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),
inference(symmetry,[status(thm)],[60]) ).
tff(62,plain,
greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity)) = greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),
inference(monotonicity,[status(thm)],[61]) ).
tff(63,plain,
greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)) = greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity)),
inference(symmetry,[status(thm)],[62]) ).
tff(64,plain,
least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity))),
inference(monotonicity,[status(thm)],[63]) ).
tff(65,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(66,plain,
least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(unit_resolution,[status(thm)],[65,27]) ).
tff(67,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))),
inference(symmetry,[status(thm)],[66]) ).
tff(68,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(69,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)],[68]) ).
tff(70,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(71,axiom,
! [Z: $i,Y: $i,X: $i] : ( multiply(least_upper_bound(Y,Z),X) = least_upper_bound(multiply(Y,X),multiply(Z,X)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',monotony_lub2) ).
tff(72,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)],[71,70]) ).
tff(73,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)],[72]) ).
tff(74,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)],[73,69]) ).
tff(75,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),least_upper_bound(b,identity)) = least_upper_bound(multiply(a,least_upper_bound(b,identity)),multiply(identity,least_upper_bound(b,identity))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(76,plain,
multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)) = least_upper_bound(multiply(a,least_upper_bound(b,identity)),multiply(identity,least_upper_bound(b,identity))),
inference(unit_resolution,[status(thm)],[75,74]) ).
tff(77,plain,
least_upper_bound(multiply(a,least_upper_bound(b,identity)),multiply(identity,least_upper_bound(b,identity))) = multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),
inference(symmetry,[status(thm)],[76]) ).
tff(78,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(b,identity) = least_upper_bound(identity,b) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(79,plain,
least_upper_bound(b,identity) = least_upper_bound(identity,b),
inference(unit_resolution,[status(thm)],[78,27]) ).
tff(80,plain,
multiply(identity,least_upper_bound(b,identity)) = multiply(identity,least_upper_bound(identity,b)),
inference(monotonicity,[status(thm)],[79]) ).
tff(81,plain,
multiply(identity,least_upper_bound(identity,b)) = multiply(identity,least_upper_bound(b,identity)),
inference(symmetry,[status(thm)],[80]) ).
tff(82,plain,
^ [X: $i] :
refl(
( ( multiply(identity,X) = X )
<=> ( multiply(identity,X) = X ) )),
inference(bind,[status(th)],]) ).
tff(83,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(quant_intro,[status(thm)],[82]) ).
tff(84,plain,
( ! [X: $i] : ( multiply(identity,X) = X )
<=> ! [X: $i] : ( multiply(identity,X) = X ) ),
inference(rewrite,[status(thm)],]) ).
tff(85,axiom,
! [X: $i] : ( multiply(identity,X) = X ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-0.ax',left_identity) ).
tff(86,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(skolemize,[status(sab)],[86]) ).
tff(88,plain,
! [X: $i] : ( multiply(identity,X) = X ),
inference(modus_ponens,[status(thm)],[87,83]) ).
tff(89,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,least_upper_bound(identity,b)) = least_upper_bound(identity,b) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(90,plain,
multiply(identity,least_upper_bound(identity,b)) = least_upper_bound(identity,b),
inference(unit_resolution,[status(thm)],[89,88]) ).
tff(91,plain,
least_upper_bound(identity,b) = multiply(identity,least_upper_bound(identity,b)),
inference(symmetry,[status(thm)],[90]) ).
tff(92,plain,
least_upper_bound(identity,b) = multiply(identity,least_upper_bound(b,identity)),
inference(transitivity,[status(thm)],[91,81]) ).
tff(93,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( multiply(X,least_upper_bound(Y,Z)) = least_upper_bound(multiply(X,Y),multiply(X,Z)) )
| ( multiply(a,least_upper_bound(b,identity)) = least_upper_bound(multiply(a,b),multiply(a,identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(94,plain,
multiply(a,least_upper_bound(b,identity)) = least_upper_bound(multiply(a,b),multiply(a,identity)),
inference(unit_resolution,[status(thm)],[93,58]) ).
tff(95,plain,
least_upper_bound(multiply(a,b),multiply(a,identity)) = multiply(a,least_upper_bound(b,identity)),
inference(symmetry,[status(thm)],[94]) ).
tff(96,plain,
least_upper_bound(least_upper_bound(multiply(a,b),multiply(a,identity)),least_upper_bound(identity,b)) = least_upper_bound(multiply(a,least_upper_bound(b,identity)),multiply(identity,least_upper_bound(b,identity))),
inference(monotonicity,[status(thm)],[95,92]) ).
tff(97,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))) = least_upper_bound(least_upper_bound(multiply(a,b),multiply(a,identity)),least_upper_bound(identity,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(98,plain,
least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))) = least_upper_bound(least_upper_bound(multiply(a,b),multiply(a,identity)),least_upper_bound(identity,b)),
inference(unit_resolution,[status(thm)],[97,37]) ).
tff(99,plain,
least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),
inference(transitivity,[status(thm)],[98,96,77,60]) ).
tff(100,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b)))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(monotonicity,[status(thm)],[99]) ).
tff(101,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b)))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(102,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b)))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))),
inference(unit_resolution,[status(thm)],[101,37]) ).
tff(103,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(a,b),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b)))),
inference(symmetry,[status(thm)],[102]) ).
tff(104,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(least_upper_bound(identity,b),multiply(a,identity)) = least_upper_bound(multiply(a,identity),least_upper_bound(identity,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(105,plain,
least_upper_bound(least_upper_bound(identity,b),multiply(a,identity)) = least_upper_bound(multiply(a,identity),least_upper_bound(identity,b)),
inference(unit_resolution,[status(thm)],[104,27]) ).
tff(106,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(identity,b),multiply(a,identity))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,identity),least_upper_bound(identity,b))),
inference(monotonicity,[status(thm)],[105]) ).
tff(107,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(identity,b),multiply(a,identity))) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),
inference(transitivity,[status(thm)],[106,103,100,67,64,51]) ).
tff(108,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(identity,b),multiply(a,identity)))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(monotonicity,[status(thm)],[107]) ).
tff(109,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(identity,b),multiply(a,identity)))),
inference(symmetry,[status(thm)],[108]) ).
tff(110,plain,
least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)))) = least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))),
inference(symmetry,[status(thm)],[20]) ).
tff(111,plain,
least_upper_bound(multiply(a,b),identity) = least_upper_bound(least_upper_bound(multiply(a,b),identity),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)))),
inference(symmetry,[status(thm)],[9]) ).
tff(112,plain,
least_upper_bound(multiply(a,b),identity) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity),
inference(transitivity,[status(thm)],[111,110,29,39]) ).
tff(113,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity)),
inference(monotonicity,[status(thm)],[112]) ).
tff(114,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity)) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)),
inference(symmetry,[status(thm)],[113]) ).
tff(115,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity)) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(116,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity)) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),
inference(unit_resolution,[status(thm)],[115,47]) ).
tff(117,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity)),
inference(symmetry,[status(thm)],[116]) ).
tff(118,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)),
inference(transitivity,[status(thm)],[117,114]) ).
tff(119,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = greatest_lower_bound(greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(monotonicity,[status(thm)],[118]) ).
tff(120,plain,
greatest_lower_bound(greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(symmetry,[status(thm)],[119]) ).
tff(121,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(122,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)],[121]) ).
tff(123,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(124,axiom,
! [Z: $i,Y: $i,X: $i] : ( greatest_lower_bound(X,greatest_lower_bound(Y,Z)) = greatest_lower_bound(greatest_lower_bound(X,Y),Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/GRP004-2.ax',associativity_of_glb) ).
tff(125,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)],[124,123]) ).
tff(126,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)],[125]) ).
tff(127,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)],[126,122]) ).
tff(128,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(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)))) = greatest_lower_bound(greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(129,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)))) = greatest_lower_bound(greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(unit_resolution,[status(thm)],[128,127]) ).
tff(130,plain,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))),
inference(monotonicity,[status(thm)],[61]) ).
tff(131,plain,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),
inference(transitivity,[status(thm)],[130,18]) ).
tff(132,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))),
inference(monotonicity,[status(thm)],[131]) ).
tff(133,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(least_upper_bound(multiply(a,b),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)))),
inference(symmetry,[status(thm)],[132]) ).
tff(134,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,Y) = greatest_lower_bound(Y,X) )
| ( greatest_lower_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(135,plain,
greatest_lower_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))),
inference(unit_resolution,[status(thm)],[134,16]) ).
tff(136,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(137,plain,
greatest_lower_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b))) = greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),
inference(unit_resolution,[status(thm)],[136,47]) ).
tff(138,plain,
greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)) = greatest_lower_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b))),
inference(symmetry,[status(thm)],[137]) ).
tff(139,plain,
greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),
inference(transitivity,[status(thm)],[138,135,133,129,120,109,49]) ).
tff(140,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity),
inference(monotonicity,[status(thm)],[139]) ).
tff(141,plain,
( ~ ! [Y: $i,X: $i] : ( greatest_lower_bound(X,least_upper_bound(X,Y)) = X )
| ( greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity)))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(142,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity)))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),
inference(unit_resolution,[status(thm)],[141,47]) ).
tff(143,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(144,plain,
least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity)),
inference(unit_resolution,[status(thm)],[143,27]) ).
tff(145,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity)) = least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)),
inference(symmetry,[status(thm)],[144]) ).
tff(146,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))),
inference(monotonicity,[status(thm)],[145]) ).
tff(147,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity))),
inference(symmetry,[status(thm)],[146]) ).
tff(148,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(identity,least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(149,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(identity,least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)))) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))),
inference(unit_resolution,[status(thm)],[148,37]) ).
tff(150,plain,
( ~ ! [Y: $i,X: $i] : ( least_upper_bound(X,Y) = least_upper_bound(Y,X) )
| ( least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) = least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(least_upper_bound(a,identity),b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(151,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) = least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(least_upper_bound(a,identity),b)),
inference(unit_resolution,[status(thm)],[150,27]) ).
tff(152,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(least_upper_bound(a,identity),b)) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),
inference(symmetry,[status(thm)],[151]) ).
tff(153,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(154,plain,
least_upper_bound(a,identity) = least_upper_bound(identity,a),
inference(unit_resolution,[status(thm)],[153,27]) ).
tff(155,plain,
least_upper_bound(identity,a) = least_upper_bound(a,identity),
inference(symmetry,[status(thm)],[154]) ).
tff(156,plain,
multiply(least_upper_bound(identity,a),identity) = multiply(least_upper_bound(a,identity),identity),
inference(monotonicity,[status(thm)],[155]) ).
tff(157,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(identity,a),identity) = least_upper_bound(multiply(identity,identity),multiply(a,identity)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(158,plain,
multiply(least_upper_bound(identity,a),identity) = least_upper_bound(multiply(identity,identity),multiply(a,identity)),
inference(unit_resolution,[status(thm)],[157,74]) ).
tff(159,plain,
least_upper_bound(multiply(identity,identity),multiply(a,identity)) = multiply(least_upper_bound(identity,a),identity),
inference(symmetry,[status(thm)],[158]) ).
tff(160,plain,
least_upper_bound(multiply(identity,identity),multiply(a,identity)) = multiply(least_upper_bound(a,identity),identity),
inference(transitivity,[status(thm)],[159,156]) ).
tff(161,plain,
least_upper_bound(least_upper_bound(multiply(identity,identity),multiply(a,identity)),multiply(least_upper_bound(a,identity),b)) = least_upper_bound(multiply(least_upper_bound(a,identity),identity),multiply(least_upper_bound(a,identity),b)),
inference(monotonicity,[status(thm)],[160]) ).
tff(162,plain,
( ~ ! [Z: $i,Y: $i,X: $i] : ( least_upper_bound(X,least_upper_bound(Y,Z)) = least_upper_bound(least_upper_bound(X,Y),Z) )
| ( least_upper_bound(multiply(identity,identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))) = least_upper_bound(least_upper_bound(multiply(identity,identity),multiply(a,identity)),multiply(least_upper_bound(a,identity),b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(163,plain,
least_upper_bound(multiply(identity,identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))) = least_upper_bound(least_upper_bound(multiply(identity,identity),multiply(a,identity)),multiply(least_upper_bound(a,identity),b)),
inference(unit_resolution,[status(thm)],[162,37]) ).
tff(164,plain,
( ~ ! [X: $i] : ( multiply(identity,X) = X )
| ( multiply(identity,identity) = identity ) ),
inference(quant_inst,[status(thm)],]) ).
tff(165,plain,
multiply(identity,identity) = identity,
inference(unit_resolution,[status(thm)],[164,88]) ).
tff(166,plain,
identity = multiply(identity,identity),
inference(symmetry,[status(thm)],[165]) ).
tff(167,plain,
least_upper_bound(identity,least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))) = least_upper_bound(multiply(identity,identity),least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))),
inference(monotonicity,[status(thm)],[166]) ).
tff(168,plain,
least_upper_bound(identity,least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b))) = least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),
inference(transitivity,[status(thm)],[167,163,161,152]) ).
tff(169,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(identity,least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(monotonicity,[status(thm)],[168]) ).
tff(170,plain,
least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),least_upper_bound(identity,least_upper_bound(multiply(a,identity),multiply(least_upper_bound(a,identity),b)))),
inference(symmetry,[status(thm)],[169]) ).
tff(171,plain,
least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity))) = least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity))),
inference(symmetry,[status(thm)],[64]) ).
tff(172,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) = least_upper_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),greatest_lower_bound(least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)),least_upper_bound(multiply(a,b),identity))),
inference(symmetry,[status(thm)],[51]) ).
tff(173,plain,
least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity)) = least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity))),
inference(transitivity,[status(thm)],[172,171,66,170,149,147]) ).
tff(174,plain,
greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(a,identity)))),
inference(monotonicity,[status(thm)],[173]) ).
tff(175,plain,
least_upper_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),multiply(a,b)),identity) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),
inference(symmetry,[status(thm)],[140]) ).
tff(176,plain,
least_upper_bound(multiply(a,b),identity) = least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),
inference(transitivity,[status(thm)],[111,110,29,39,175]) ).
tff(177,plain,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) = greatest_lower_bound(least_upper_bound(greatest_lower_bound(multiply(least_upper_bound(a,identity),least_upper_bound(b,identity)),least_upper_bound(multiply(a,b),identity)),identity),least_upper_bound(multiply(least_upper_bound(a,identity),b),multiply(least_upper_bound(a,identity),identity))),
inference(monotonicity,[status(thm)],[176,60]) ).
tff(178,plain,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) = least_upper_bound(multiply(a,b),identity),
inference(transitivity,[status(thm)],[177,174,142,140,40,30,20,9]) ).
tff(179,plain,
( ( greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) != least_upper_bound(multiply(a,b),identity) )
<=> ( greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) != least_upper_bound(multiply(a,b),identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(180,axiom,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) != least_upper_bound(multiply(a,b),identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_p22b) ).
tff(181,plain,
greatest_lower_bound(least_upper_bound(multiply(a,b),identity),multiply(least_upper_bound(a,identity),least_upper_bound(b,identity))) != least_upper_bound(multiply(a,b),identity),
inference(modus_ponens,[status(thm)],[180,179]) ).
tff(182,plain,
$false,
inference(unit_resolution,[status(thm)],[181,178]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : GRP185-3 : TPTP v8.1.0. Bugfixed v1.2.1.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n005.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Aug 31 15:35:03 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.35 Usage: tptp [options] [-file:]file
% 0.14/0.35 -h, -? prints this message.
% 0.14/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.35 -m, -model generate model.
% 0.14/0.35 -p, -proof generate proof.
% 0.14/0.35 -c, -core generate unsat core of named formulas.
% 0.14/0.35 -st, -statistics display statistics.
% 0.14/0.35 -t:timeout set timeout (in second).
% 0.14/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.35 -<param>:<value> configuration parameter and value.
% 0.14/0.35 -o:<output-file> file to place output in.
% 46.34/29.56 % SZS status Unsatisfiable
% 46.34/29.56 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------