TSTP Solution File: FLD041-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : FLD041-3 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n008.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 14:55:50 EDT 2022
% Result : Unsatisfiable 0.20s 0.48s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 110
% Syntax : Number of formulae : 229 ( 55 unt; 9 typ; 0 def)
% Number of atoms : 1355 ( 0 equ)
% Maximal formula atoms : 19 ( 6 avg)
% Number of connectives : 1966 ( 912 ~; 932 |; 0 &)
% ( 122 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 81 ( 81 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 9 ( 5 >; 4 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 6 ( 6 usr; 4 con; 0-1 aty)
% Number of variables : 808 ( 737 !; 0 ?; 808 :)
% Comments :
%------------------------------------------------------------------------------
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(additive_inverse_type,type,
additive_inverse: $i > $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(multiplicative_identity_type,type,
multiplicative_identity: $i ).
tff(defined_type,type,
defined: $i > $o ).
tff(a_type,type,
a: $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(multiplicative_inverse_type,type,
multiplicative_inverse: $i > $i ).
tff(b_type,type,
b: $i ).
tff(1,plain,
^ [X: $i] :
refl(
( ( defined(additive_inverse(X))
| ~ defined(X) )
<=> ( defined(additive_inverse(X))
| ~ defined(X) ) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
<=> ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
<=> ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',well_definedness_of_additive_inverse) ).
tff(5,plain,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( defined(additive_identity)
<=> defined(additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(9,axiom,
defined(additive_identity),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',well_definedness_of_additive_identity) ).
tff(10,plain,
defined(additive_identity),
inference(modus_ponens,[status(thm)],[9,8]) ).
tff(11,plain,
( ( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(additive_identity))
| ~ defined(additive_identity) )
<=> ( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(additive_identity))
| ~ defined(additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(12,plain,
( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(additive_identity))
| ~ defined(additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(13,plain,
( ~ ! [X: $i] :
( defined(additive_inverse(X))
| ~ defined(X) )
| defined(additive_inverse(additive_identity))
| ~ defined(additive_identity) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
defined(additive_inverse(additive_identity)),
inference(unit_resolution,[status(thm)],[13,10,7]) ).
tff(15,plain,
^ [X: $i] :
refl(
( ( ~ defined(X)
| sum(additive_identity,X,X) )
<=> ( ~ defined(X)
| sum(additive_identity,X,X) ) )),
inference(bind,[status(th)],]) ).
tff(16,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(18,plain,
^ [X: $i] :
rewrite(
( ( sum(additive_identity,X,X)
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(additive_identity,X,X) ) )),
inference(bind,[status(th)],]) ).
tff(19,plain,
( ! [X: $i] :
( sum(additive_identity,X,X)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ) ),
inference(quant_intro,[status(thm)],[18]) ).
tff(20,axiom,
! [X: $i] :
( sum(additive_identity,X,X)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',existence_of_identity_addition) ).
tff(21,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(modus_ponens,[status(thm)],[20,19]) ).
tff(22,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(skolemize,[status(sab)],[22]) ).
tff(24,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) ),
inference(modus_ponens,[status(thm)],[23,16]) ).
tff(25,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(additive_inverse(additive_identity))
| sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(additive_inverse(additive_identity))
| sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(26,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(additive_inverse(additive_identity))
| sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(additive_inverse(additive_identity))
| sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)) ),
inference(modus_ponens,[status(thm)],[26,25]) ).
tff(28,plain,
sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)),
inference(unit_resolution,[status(thm)],[27,24,14]) ).
tff(29,plain,
( defined(multiplicative_identity)
<=> defined(multiplicative_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(30,axiom,
defined(multiplicative_identity),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',well_definedness_of_multiplicative_identity) ).
tff(31,plain,
defined(multiplicative_identity),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(multiplicative_identity)
| sum(additive_identity,multiplicative_identity,multiplicative_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(multiplicative_identity)
| sum(additive_identity,multiplicative_identity,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(33,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(multiplicative_identity)
| sum(additive_identity,multiplicative_identity,multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(34,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(multiplicative_identity)
| sum(additive_identity,multiplicative_identity,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
sum(additive_identity,multiplicative_identity,multiplicative_identity),
inference(unit_resolution,[status(thm)],[34,24,31]) ).
tff(36,plain,
^ [X: $i] :
refl(
( ( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
<=> ( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) )),
inference(bind,[status(th)],]) ).
tff(37,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) ),
inference(quant_intro,[status(thm)],[36]) ).
tff(38,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(39,plain,
^ [X: $i] :
rewrite(
( ( sum(additive_inverse(X),X,additive_identity)
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) )),
inference(bind,[status(th)],]) ).
tff(40,plain,
( ! [X: $i] :
( sum(additive_inverse(X),X,additive_identity)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,axiom,
! [X: $i] :
( sum(additive_inverse(X),X,additive_identity)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',existence_of_inverse_addition) ).
tff(42,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[42,38]) ).
tff(44,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) ),
inference(modus_ponens,[status(thm)],[44,37]) ).
tff(46,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(additive_identity)
| sum(additive_inverse(additive_identity),additive_identity,additive_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(additive_identity)
| sum(additive_inverse(additive_identity),additive_identity,additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(47,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(additive_identity)
| sum(additive_inverse(additive_identity),additive_identity,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(48,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(additive_identity)
| sum(additive_inverse(additive_identity),additive_identity,additive_identity) ),
inference(modus_ponens,[status(thm)],[47,46]) ).
tff(49,plain,
sum(additive_inverse(additive_identity),additive_identity,additive_identity),
inference(unit_resolution,[status(thm)],[48,45,10]) ).
tff(50,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(
( ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(51,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[50]) ).
tff(52,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) ),
inference(rewrite,[status(thm)],]) ).
tff(53,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( sum(X,V,W)
| ~ sum(X,Y,U) )
<=> ( ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(Y,Z,V) ) )),
rewrite(
( ( ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(U,Z,W) ) )),
rewrite(
( ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W)
| ~ sum(U,Z,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
( ( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(54,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[53]) ).
tff(55,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',associativity_addition_1) ).
tff(56,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[56,52]) ).
tff(58,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(skolemize,[status(sab)],[57]) ).
tff(59,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[58,51]) ).
tff(60,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(61,plain,
( ( ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) )
<=> ( ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(62,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) ) ),
inference(monotonicity,[status(thm)],[61]) ).
tff(63,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) ) ),
inference(transitivity,[status(thm)],[62,60]) ).
tff(64,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(65,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,multiplicative_identity,multiplicative_identity)
| sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity),
inference(unit_resolution,[status(thm)],[65,59,49,35]) ).
tff(67,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) )),
inference(bind,[status(th)],]) ).
tff(68,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(rewrite,[status(thm)],]) ).
tff(70,plain,
^ [Z: $i,Y: $i,X: $i] :
rewrite(
( ( sum(Y,X,Z)
| ~ sum(X,Y,Z) )
<=> ( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) )),
inference(bind,[status(th)],]) ).
tff(71,plain,
( ! [Z: $i,Y: $i,X: $i] :
( sum(Y,X,Z)
| ~ sum(X,Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[70]) ).
tff(72,axiom,
! [Z: $i,Y: $i,X: $i] :
( sum(Y,X,Z)
| ~ sum(X,Y,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',commutativity_addition) ).
tff(73,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[72,71]) ).
tff(74,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[73,69]) ).
tff(75,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(skolemize,[status(sab)],[74]) ).
tff(76,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[75,68]) ).
tff(77,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity)
| sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity)
| sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(78,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity)
| sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(79,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_identity),multiplicative_identity,multiplicative_identity)
| sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[78,77]) ).
tff(80,plain,
sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity),
inference(unit_resolution,[status(thm)],[79,76,66]) ).
tff(81,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(multiplicative_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(multiplicative_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(82,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(multiplicative_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(83,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_inverse(X),X,additive_identity) )
| ~ defined(multiplicative_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) ),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,plain,
sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity),
inference(unit_resolution,[status(thm)],[83,45,31]) ).
tff(85,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ~ sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ~ sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(86,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ~ sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(87,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ~ sum(multiplicative_identity,additive_inverse(additive_identity),multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)),
inference(unit_resolution,[status(thm)],[87,59,84,80,28]) ).
tff(89,plain,
( product(a,b,additive_identity)
<=> product(a,b,additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(90,axiom,
product(a,b,additive_identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',product_5) ).
tff(91,plain,
product(a,b,additive_identity),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(
( ( ~ product(X,Y,Z)
| product(Y,X,Z) )
<=> ( ~ product(X,Y,Z)
| product(Y,X,Z) ) )),
inference(bind,[status(th)],]) ).
tff(93,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[92]) ).
tff(94,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ) ),
inference(rewrite,[status(thm)],]) ).
tff(95,plain,
^ [Z: $i,Y: $i,X: $i] :
rewrite(
( ( product(Y,X,Z)
| ~ product(X,Y,Z) )
<=> ( ~ product(X,Y,Z)
| product(Y,X,Z) ) )),
inference(bind,[status(th)],]) ).
tff(96,plain,
( ! [Z: $i,Y: $i,X: $i] :
( product(Y,X,Z)
| ~ product(X,Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[95]) ).
tff(97,axiom,
! [Z: $i,Y: $i,X: $i] :
( product(Y,X,Z)
| ~ product(X,Y,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',commutativity_multiplication) ).
tff(98,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[97,96]) ).
tff(99,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[98,94]) ).
tff(100,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(skolemize,[status(sab)],[99]) ).
tff(101,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[100,93]) ).
tff(102,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,b,additive_identity)
| product(b,a,additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,b,additive_identity)
| product(b,a,additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(103,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,b,additive_identity)
| product(b,a,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(104,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(a,b,additive_identity)
| product(b,a,additive_identity) ),
inference(modus_ponens,[status(thm)],[103,102]) ).
tff(105,plain,
product(b,a,additive_identity),
inference(unit_resolution,[status(thm)],[104,101,91]) ).
tff(106,plain,
( defined(a)
<=> defined(a) ),
inference(rewrite,[status(thm)],]) ).
tff(107,axiom,
defined(a),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_is_defined) ).
tff(108,plain,
defined(a),
inference(modus_ponens,[status(thm)],[107,106]) ).
tff(109,plain,
^ [X: $i] :
refl(
( ( ~ defined(X)
| product(multiplicative_identity,X,X) )
<=> ( ~ defined(X)
| product(multiplicative_identity,X,X) ) )),
inference(bind,[status(th)],]) ).
tff(110,plain,
( ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) )
<=> ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ) ),
inference(quant_intro,[status(thm)],[109]) ).
tff(111,plain,
( ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) )
<=> ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(112,plain,
^ [X: $i] :
rewrite(
( ( product(multiplicative_identity,X,X)
| ~ defined(X) )
<=> ( ~ defined(X)
| product(multiplicative_identity,X,X) ) )),
inference(bind,[status(th)],]) ).
tff(113,plain,
( ! [X: $i] :
( product(multiplicative_identity,X,X)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ) ),
inference(quant_intro,[status(thm)],[112]) ).
tff(114,axiom,
! [X: $i] :
( product(multiplicative_identity,X,X)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',existence_of_identity_multiplication) ).
tff(115,plain,
! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ),
inference(modus_ponens,[status(thm)],[115,111]) ).
tff(117,plain,
! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ),
inference(skolemize,[status(sab)],[116]) ).
tff(118,plain,
! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) ),
inference(modus_ponens,[status(thm)],[117,110]) ).
tff(119,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) )
| ~ defined(a)
| product(multiplicative_identity,a,a) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) )
| ~ defined(a)
| product(multiplicative_identity,a,a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(120,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) )
| ~ defined(a)
| product(multiplicative_identity,a,a) ),
inference(quant_inst,[status(thm)],]) ).
tff(121,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| product(multiplicative_identity,X,X) )
| ~ defined(a)
| product(multiplicative_identity,a,a) ),
inference(modus_ponens,[status(thm)],[120,119]) ).
tff(122,plain,
product(multiplicative_identity,a,a),
inference(unit_resolution,[status(thm)],[121,118,108]) ).
tff(123,plain,
( ~ sum(additive_identity,b,additive_identity)
<=> ~ sum(additive_identity,b,additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(124,axiom,
~ sum(additive_identity,b,additive_identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_sum_4) ).
tff(125,plain,
~ sum(additive_identity,b,additive_identity),
inference(modus_ponens,[status(thm)],[124,123]) ).
tff(126,plain,
( defined(b)
<=> defined(b) ),
inference(rewrite,[status(thm)],]) ).
tff(127,axiom,
defined(b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_is_defined) ).
tff(128,plain,
defined(b),
inference(modus_ponens,[status(thm)],[127,126]) ).
tff(129,plain,
^ [X: $i] :
refl(
( ( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
<=> ( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) )),
inference(bind,[status(th)],]) ).
tff(130,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) ),
inference(quant_intro,[status(thm)],[129]) ).
tff(131,plain,
( ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(132,plain,
^ [X: $i] :
trans(
monotonicity(
rewrite(
( ( product(multiplicative_inverse(X),X,multiplicative_identity)
| sum(additive_identity,X,additive_identity) )
<=> ( sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) )),
( ( product(multiplicative_inverse(X),X,multiplicative_identity)
| sum(additive_identity,X,additive_identity)
| ~ defined(X) )
<=> ( sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity)
| ~ defined(X) ) )),
rewrite(
( ( sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity)
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) )),
( ( product(multiplicative_inverse(X),X,multiplicative_identity)
| sum(additive_identity,X,additive_identity)
| ~ defined(X) )
<=> ( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) )),
inference(bind,[status(th)],]) ).
tff(133,plain,
( ! [X: $i] :
( product(multiplicative_inverse(X),X,multiplicative_identity)
| sum(additive_identity,X,additive_identity)
| ~ defined(X) )
<=> ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ) ),
inference(quant_intro,[status(thm)],[132]) ).
tff(134,axiom,
! [X: $i] :
( product(multiplicative_inverse(X),X,multiplicative_identity)
| sum(additive_identity,X,additive_identity)
| ~ defined(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',existence_of_inverse_multiplication) ).
tff(135,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[134,133]) ).
tff(136,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[135,131]) ).
tff(137,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ),
inference(skolemize,[status(sab)],[136]) ).
tff(138,plain,
! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[137,130]) ).
tff(139,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
| ~ defined(b)
| sum(additive_identity,b,additive_identity)
| product(multiplicative_inverse(b),b,multiplicative_identity) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
| ~ defined(b)
| sum(additive_identity,b,additive_identity)
| product(multiplicative_inverse(b),b,multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(140,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
| ~ defined(b)
| sum(additive_identity,b,additive_identity)
| product(multiplicative_inverse(b),b,multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(141,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,additive_identity)
| product(multiplicative_inverse(X),X,multiplicative_identity) )
| ~ defined(b)
| sum(additive_identity,b,additive_identity)
| product(multiplicative_inverse(b),b,multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[140,139]) ).
tff(142,plain,
product(multiplicative_inverse(b),b,multiplicative_identity),
inference(unit_resolution,[status(thm)],[141,138,128,125]) ).
tff(143,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(
( ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(144,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) ),
inference(quant_intro,[status(thm)],[143]) ).
tff(145,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) ),
inference(rewrite,[status(thm)],]) ).
tff(146,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( product(X,V,W)
| ~ product(X,Y,U) )
<=> ( ~ product(X,Y,U)
| product(X,V,W) ) )),
( ( product(X,V,W)
| ~ product(X,Y,U)
| ~ product(Y,Z,V) )
<=> ( ~ product(X,Y,U)
| product(X,V,W)
| ~ product(Y,Z,V) ) )),
rewrite(
( ( ~ product(X,Y,U)
| product(X,V,W)
| ~ product(Y,Z,V) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
( ( product(X,V,W)
| ~ product(X,Y,U)
| ~ product(Y,Z,V) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
( ( product(X,V,W)
| ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W)
| ~ product(U,Z,W) ) )),
rewrite(
( ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W)
| ~ product(U,Z,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
( ( product(X,V,W)
| ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(147,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) ),
inference(quant_intro,[status(thm)],[146]) ).
tff(148,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',associativity_multiplication_1) ).
tff(149,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ),
inference(modus_ponens,[status(thm)],[148,147]) ).
tff(150,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ),
inference(modus_ponens,[status(thm)],[149,145]) ).
tff(151,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ),
inference(skolemize,[status(sab)],[150]) ).
tff(152,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ),
inference(modus_ponens,[status(thm)],[151,144]) ).
tff(153,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
| ~ product(multiplicative_identity,a,a)
| ~ product(b,a,additive_identity)
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(multiplicative_inverse(b),additive_identity,a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
| ~ product(multiplicative_identity,a,a)
| ~ product(b,a,additive_identity)
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(multiplicative_inverse(b),additive_identity,a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(154,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
| ~ product(multiplicative_identity,a,a)
| ~ product(b,a,additive_identity)
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(multiplicative_inverse(b),additive_identity,a) ),
inference(quant_inst,[status(thm)],]) ).
tff(155,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
| ~ product(multiplicative_identity,a,a)
| ~ product(b,a,additive_identity)
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(multiplicative_inverse(b),additive_identity,a) ),
inference(modus_ponens,[status(thm)],[154,153]) ).
tff(156,plain,
product(multiplicative_inverse(b),additive_identity,a),
inference(unit_resolution,[status(thm)],[155,152,142,122,105]) ).
tff(157,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),additive_identity,a)
| product(additive_identity,multiplicative_inverse(b),a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),additive_identity,a)
| product(additive_identity,multiplicative_inverse(b),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(158,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),additive_identity,a)
| product(additive_identity,multiplicative_inverse(b),a) ),
inference(quant_inst,[status(thm)],]) ).
tff(159,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),additive_identity,a)
| product(additive_identity,multiplicative_inverse(b),a) ),
inference(modus_ponens,[status(thm)],[158,157]) ).
tff(160,plain,
product(additive_identity,multiplicative_inverse(b),a),
inference(unit_resolution,[status(thm)],[159,101,156]) ).
tff(161,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(b,multiplicative_inverse(b),multiplicative_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(b,multiplicative_inverse(b),multiplicative_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(162,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(b,multiplicative_inverse(b),multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(163,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| product(Y,X,Z) )
| ~ product(multiplicative_inverse(b),b,multiplicative_identity)
| product(b,multiplicative_inverse(b),multiplicative_identity) ),
inference(modus_ponens,[status(thm)],[162,161]) ).
tff(164,plain,
product(b,multiplicative_inverse(b),multiplicative_identity),
inference(unit_resolution,[status(thm)],[163,101,142]) ).
tff(165,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(b)
| sum(additive_identity,b,b) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(b)
| sum(additive_identity,b,b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(166,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(b)
| sum(additive_identity,b,b) ),
inference(quant_inst,[status(thm)],]) ).
tff(167,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(b)
| sum(additive_identity,b,b) ),
inference(modus_ponens,[status(thm)],[166,165]) ).
tff(168,plain,
sum(additive_identity,b,b),
inference(unit_resolution,[status(thm)],[167,24,128]) ).
tff(169,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,b,b)
| sum(b,additive_identity,b) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,b,b)
| sum(b,additive_identity,b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(170,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,b,b)
| sum(b,additive_identity,b) ),
inference(quant_inst,[status(thm)],]) ).
tff(171,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,b,b)
| sum(b,additive_identity,b) ),
inference(modus_ponens,[status(thm)],[170,169]) ).
tff(172,plain,
sum(b,additive_identity,b),
inference(unit_resolution,[status(thm)],[171,76,168]) ).
tff(173,plain,
^ [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
refl(
( ( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
<=> ( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
inference(bind,[status(th)],]) ).
tff(174,plain,
( ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
<=> ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) ),
inference(quant_intro,[status(thm)],[173]) ).
tff(175,plain,
( ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
<=> ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) ),
inference(rewrite,[status(thm)],]) ).
tff(176,plain,
^ [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
trans(
monotonicity(
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( sum(C,D,B)
| ~ sum(X,Y,A) )
<=> ( ~ sum(X,Y,A)
| sum(C,D,B) ) )),
( ( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B) )
<=> ( ~ sum(X,Y,A)
| sum(C,D,B)
| ~ product(A,Z,B) ) )),
rewrite(
( ( ~ sum(X,Y,A)
| sum(C,D,B)
| ~ product(A,Z,B) )
<=> ( ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
( ( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B) )
<=> ( ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
( ( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B)
| ~ product(X,Z,C) )
<=> ( ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B)
| ~ product(X,Z,C) ) )),
rewrite(
( ( ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B)
| ~ product(X,Z,C) )
<=> ( ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
( ( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B)
| ~ product(X,Z,C) )
<=> ( ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
( ( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B)
| ~ product(X,Z,C)
| ~ product(Y,Z,D) )
<=> ( ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B)
| ~ product(Y,Z,D) ) )),
rewrite(
( ( ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B)
| ~ product(Y,Z,D) )
<=> ( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
( ( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B)
| ~ product(X,Z,C)
| ~ product(Y,Z,D) )
<=> ( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) )),
inference(bind,[status(th)],]) ).
tff(177,plain,
( ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B)
| ~ product(X,Z,C)
| ~ product(Y,Z,D) )
<=> ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ) ),
inference(quant_intro,[status(thm)],[176]) ).
tff(178,axiom,
! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( sum(C,D,B)
| ~ sum(X,Y,A)
| ~ product(A,Z,B)
| ~ product(X,Z,C)
| ~ product(Y,Z,D) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',distributivity_1) ).
tff(179,plain,
! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ),
inference(modus_ponens,[status(thm)],[178,177]) ).
tff(180,plain,
! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ),
inference(modus_ponens,[status(thm)],[179,175]) ).
tff(181,plain,
! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ),
inference(skolemize,[status(sab)],[180]) ).
tff(182,plain,
! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) ),
inference(modus_ponens,[status(thm)],[181,174]) ).
tff(183,plain,
( ( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(b,additive_identity,b)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(additive_identity,multiplicative_inverse(b),a) )
<=> ( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(b,additive_identity,b)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(additive_identity,multiplicative_inverse(b),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(184,plain,
( ( ~ product(additive_identity,multiplicative_inverse(b),a)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ sum(b,additive_identity,b)
| sum(multiplicative_identity,a,multiplicative_identity) )
<=> ( sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(b,additive_identity,b)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(additive_identity,multiplicative_inverse(b),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(185,plain,
( ( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| ~ product(additive_identity,multiplicative_inverse(b),a)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ sum(b,additive_identity,b)
| sum(multiplicative_identity,a,multiplicative_identity) )
<=> ( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(b,additive_identity,b)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(additive_identity,multiplicative_inverse(b),a) ) ),
inference(monotonicity,[status(thm)],[184]) ).
tff(186,plain,
( ( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| ~ product(additive_identity,multiplicative_inverse(b),a)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ sum(b,additive_identity,b)
| sum(multiplicative_identity,a,multiplicative_identity) )
<=> ( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(b,additive_identity,b)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(additive_identity,multiplicative_inverse(b),a) ) ),
inference(transitivity,[status(thm)],[185,183]) ).
tff(187,plain,
( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| ~ product(additive_identity,multiplicative_inverse(b),a)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ sum(b,additive_identity,b)
| sum(multiplicative_identity,a,multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(188,plain,
( ~ ! [B: $i,D: $i,Z: $i,Y: $i,A: $i,X: $i,C: $i] :
( ~ product(Y,Z,D)
| ~ product(X,Z,C)
| ~ product(A,Z,B)
| ~ sum(X,Y,A)
| sum(C,D,B) )
| sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(b,additive_identity,b)
| ~ product(b,multiplicative_inverse(b),multiplicative_identity)
| ~ product(additive_identity,multiplicative_inverse(b),a) ),
inference(modus_ponens,[status(thm)],[187,186]) ).
tff(189,plain,
sum(multiplicative_identity,a,multiplicative_identity),
inference(unit_resolution,[status(thm)],[188,182,172,164,160]) ).
tff(190,plain,
( ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(a)
| sum(additive_identity,a,a) )
<=> ( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(a)
| sum(additive_identity,a,a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(191,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(a)
| sum(additive_identity,a,a) ),
inference(quant_inst,[status(thm)],]) ).
tff(192,plain,
( ~ ! [X: $i] :
( ~ defined(X)
| sum(additive_identity,X,X) )
| ~ defined(a)
| sum(additive_identity,a,a) ),
inference(modus_ponens,[status(thm)],[191,190]) ).
tff(193,plain,
sum(additive_identity,a,a),
inference(unit_resolution,[status(thm)],[192,24,108]) ).
tff(194,plain,
( ~ sum(additive_identity,a,additive_identity)
<=> ~ sum(additive_identity,a,additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(195,axiom,
~ sum(additive_identity,a,additive_identity),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',not_sum_3) ).
tff(196,plain,
~ sum(additive_identity,a,additive_identity),
inference(modus_ponens,[status(thm)],[195,194]) ).
tff(197,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(
( ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(198,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[197]) ).
tff(199,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) ),
inference(rewrite,[status(thm)],]) ).
tff(200,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( sum(U,Z,W)
| ~ sum(X,Y,U) )
<=> ( sum(U,Z,W)
| ~ sum(X,Y,U) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) ) )),
rewrite(
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
rewrite(
( ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
( ( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(201,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ) ),
inference(quant_intro,[status(thm)],[200]) ).
tff(202,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/FLD002-0.ax',associativity_addition_2) ).
tff(203,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[202,201]) ).
tff(204,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[203,199]) ).
tff(205,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(skolemize,[status(sab)],[204]) ).
tff(206,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) ),
inference(modus_ponens,[status(thm)],[205,198]) ).
tff(207,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(208,plain,
( ( sum(additive_identity,a,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) )
<=> ( sum(additive_identity,a,additive_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(209,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) ) ),
inference(monotonicity,[status(thm)],[208]) ).
tff(210,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) ) ),
inference(transitivity,[status(thm)],[209,207]) ).
tff(211,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(212,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_identity,a,additive_identity)
| ~ sum(additive_inverse(additive_identity),additive_identity,additive_identity)
| ~ sum(additive_identity,a,a)
| ~ sum(additive_inverse(additive_identity),a,additive_identity) ),
inference(modus_ponens,[status(thm)],[211,210]) ).
tff(213,plain,
~ sum(additive_inverse(additive_identity),a,additive_identity),
inference(unit_resolution,[status(thm)],[212,206,196,49,193]) ).
tff(214,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(215,plain,
( ( sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity))
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) )
<=> ( sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(216,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity))
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ) ),
inference(monotonicity,[status(thm)],[215]) ).
tff(217,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity))
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ) ),
inference(transitivity,[status(thm)],[216,214]) ).
tff(218,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity))
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(219,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,W) )
| sum(additive_inverse(additive_identity),a,additive_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_identity)
| ~ sum(multiplicative_identity,a,multiplicative_identity)
| ~ sum(additive_inverse(multiplicative_identity),multiplicative_identity,additive_inverse(additive_identity)) ),
inference(modus_ponens,[status(thm)],[218,217]) ).
tff(220,plain,
$false,
inference(unit_resolution,[status(thm)],[219,206,84,213,189,88]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : FLD041-3 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n008.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 31 02:50:40 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.20/0.48 % SZS status Unsatisfiable
% 0.20/0.48 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------