TSTP Solution File: RNG003-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : RNG003-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n026.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 : Tue Sep 20 03:17:35 EDT 2022
% Result : Unsatisfiable 246.24s 158.42s
% Output : Proof 246.33s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 119
% Syntax : Number of formulae : 290 ( 93 unt; 8 typ; 0 def)
% Number of atoms : 1426 ( 96 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 2133 (1013 ~;1002 |; 0 &)
% ( 118 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of FOOLs : 24 ( 24 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 6 ( 3 >; 3 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 5 con; 0-2 aty)
% Number of variables : 969 ( 928 !; 0 ?; 969 :)
% Comments :
%------------------------------------------------------------------------------
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(a_type,type,
a: $i ).
tff(additive_inverse_type,type,
additive_inverse: $i > $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(b_type,type,
b: $i ).
tff(d_type,type,
d: $i ).
tff(c_type,type,
c: $i ).
tff(add_type,type,
add: ( $i * $i ) > $i ).
tff(1,plain,
^ [X: $i] :
refl(
( sum(additive_identity,X,X)
<=> sum(additive_identity,X,X) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : sum(additive_identity,X,X)
<=> ! [X: $i] : sum(additive_identity,X,X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : sum(additive_identity,X,X)
<=> ! [X: $i] : sum(additive_identity,X,X) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : sum(additive_identity,X,X),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',additive_identity1) ).
tff(5,plain,
! [X: $i] : sum(additive_identity,X,X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : sum(additive_identity,X,X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : sum(additive_identity,X,X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : sum(additive_identity,X,X)
| sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(
( sum(X,additive_inverse(X),additive_identity)
<=> sum(X,additive_inverse(X),additive_identity) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [X: $i] : sum(X,additive_inverse(X),additive_identity)
<=> ! [X: $i] : sum(X,additive_inverse(X),additive_identity) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : sum(X,additive_inverse(X),additive_identity)
<=> ! [X: $i] : sum(X,additive_inverse(X),additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [X: $i] : sum(X,additive_inverse(X),additive_identity),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',right_inverse) ).
tff(14,plain,
! [X: $i] : sum(X,additive_inverse(X),additive_identity),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : sum(X,additive_inverse(X),additive_identity),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : sum(X,additive_inverse(X),additive_identity),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : sum(X,additive_inverse(X),additive_identity)
| sum(additive_identity,additive_inverse(additive_identity),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
sum(additive_identity,additive_inverse(additive_identity),additive_identity),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(
( ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(22,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
rewrite(
( ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) )),
inference(bind,[status(th)],]) ).
tff(23,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) ),
inference(quant_intro,[status(thm)],[22]) ).
tff(24,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',addition_is_well_defined) ).
tff(25,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(skolemize,[status(sab)],[26]) ).
tff(28,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[27,20]) ).
tff(29,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_identity)
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ( additive_identity = additive_inverse(additive_identity) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_identity)
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ( additive_identity = additive_inverse(additive_identity) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(30,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_identity)
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ( additive_identity = additive_inverse(additive_identity) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(31,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_identity)
| ~ sum(additive_identity,additive_inverse(additive_identity),additive_inverse(additive_identity))
| ( additive_identity = additive_inverse(additive_identity) ) ),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
additive_identity = additive_inverse(additive_identity),
inference(unit_resolution,[status(thm)],[31,28,18,9]) ).
tff(33,plain,
additive_inverse(additive_identity) = additive_identity,
inference(symmetry,[status(thm)],[32]) ).
tff(34,plain,
( sum(a,additive_inverse(additive_identity),a)
<=> sum(a,additive_identity,a) ),
inference(monotonicity,[status(thm)],[33]) ).
tff(35,plain,
( sum(a,additive_identity,a)
<=> sum(a,additive_inverse(additive_identity),a) ),
inference(symmetry,[status(thm)],[34]) ).
tff(36,plain,
^ [X: $i] :
refl(
( sum(X,additive_identity,X)
<=> sum(X,additive_identity,X) )),
inference(bind,[status(th)],]) ).
tff(37,plain,
( ! [X: $i] : sum(X,additive_identity,X)
<=> ! [X: $i] : sum(X,additive_identity,X) ),
inference(quant_intro,[status(thm)],[36]) ).
tff(38,plain,
( ! [X: $i] : sum(X,additive_identity,X)
<=> ! [X: $i] : sum(X,additive_identity,X) ),
inference(rewrite,[status(thm)],]) ).
tff(39,axiom,
! [X: $i] : sum(X,additive_identity,X),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',additive_identity2) ).
tff(40,plain,
! [X: $i] : sum(X,additive_identity,X),
inference(modus_ponens,[status(thm)],[39,38]) ).
tff(41,plain,
! [X: $i] : sum(X,additive_identity,X),
inference(skolemize,[status(sab)],[40]) ).
tff(42,plain,
! [X: $i] : sum(X,additive_identity,X),
inference(modus_ponens,[status(thm)],[41,37]) ).
tff(43,plain,
( ~ ! [X: $i] : sum(X,additive_identity,X)
| sum(a,additive_identity,a) ),
inference(quant_inst,[status(thm)],]) ).
tff(44,plain,
sum(a,additive_identity,a),
inference(unit_resolution,[status(thm)],[43,42]) ).
tff(45,plain,
sum(a,additive_inverse(additive_identity),a),
inference(modus_ponens,[status(thm)],[44,35]) ).
tff(46,plain,
( ~ ! [X: $i] : sum(additive_identity,X,X)
| sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ),
inference(quant_inst,[status(thm)],]) ).
tff(47,plain,
sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))),
inference(unit_resolution,[status(thm)],[46,7]) ).
tff(48,plain,
^ [X: $i] :
refl(
( sum(additive_inverse(X),X,additive_identity)
<=> sum(additive_inverse(X),X,additive_identity) )),
inference(bind,[status(th)],]) ).
tff(49,plain,
( ! [X: $i] : sum(additive_inverse(X),X,additive_identity)
<=> ! [X: $i] : sum(additive_inverse(X),X,additive_identity) ),
inference(quant_intro,[status(thm)],[48]) ).
tff(50,plain,
( ! [X: $i] : sum(additive_inverse(X),X,additive_identity)
<=> ! [X: $i] : sum(additive_inverse(X),X,additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(51,axiom,
! [X: $i] : sum(additive_inverse(X),X,additive_identity),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',left_inverse) ).
tff(52,plain,
! [X: $i] : sum(additive_inverse(X),X,additive_identity),
inference(modus_ponens,[status(thm)],[51,50]) ).
tff(53,plain,
! [X: $i] : sum(additive_inverse(X),X,additive_identity),
inference(skolemize,[status(sab)],[52]) ).
tff(54,plain,
! [X: $i] : sum(additive_inverse(X),X,additive_identity),
inference(modus_ponens,[status(thm)],[53,49]) ).
tff(55,plain,
( ~ ! [X: $i] : sum(additive_inverse(X),X,additive_identity)
| sum(additive_inverse(additive_inverse(b)),additive_inverse(b),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(56,plain,
sum(additive_inverse(additive_inverse(b)),additive_inverse(b),additive_identity),
inference(unit_resolution,[status(thm)],[55,54]) ).
tff(57,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(58,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)],[57]) ).
tff(59,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(60,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',commutativity_of_addition) ).
tff(61,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(skolemize,[status(sab)],[61]) ).
tff(63,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[62,58]) ).
tff(64,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(b),additive_identity)
| sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(b),additive_identity)
| sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(65,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(b),additive_identity)
| sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(66,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(b),additive_identity)
| sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,plain,
sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity),
inference(unit_resolution,[status(thm)],[66,63,56]) ).
tff(68,plain,
( ~ ! [X: $i] : sum(X,additive_identity,X)
| sum(b,additive_identity,b) ),
inference(quant_inst,[status(thm)],]) ).
tff(69,plain,
sum(b,additive_identity,b),
inference(unit_resolution,[status(thm)],[68,42]) ).
tff(70,plain,
( ~ ! [X: $i] : sum(X,additive_inverse(X),additive_identity)
| sum(b,additive_inverse(b),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(71,plain,
sum(b,additive_inverse(b),additive_identity),
inference(unit_resolution,[status(thm)],[70,16]) ).
tff(72,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(73,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)],[72]) ).
tff(74,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(75,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W) )
<=> ( ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| ~ sum(X,V,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)
| 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,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W)
| 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(76,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W)
| 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)],[75]) ).
tff(77,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(X,V,W)
| sum(U,Z,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',associativity_of_addition2) ).
tff(78,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)],[77,76]) ).
tff(79,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)],[78,74]) ).
tff(80,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)],[79]) ).
tff(81,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)],[80,73]) ).
tff(82,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(b,additive_inverse(b),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(b,additive_identity,b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),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(b,additive_inverse(b),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(b,additive_identity,b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(83,plain,
( ( sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(b,additive_identity,b) )
<=> ( ~ sum(b,additive_inverse(b),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(b,additive_identity,b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(84,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_inverse(b)),b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(b,additive_identity,b) )
<=> ( ~ ! [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(b,additive_inverse(b),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(b,additive_identity,b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ) ),
inference(monotonicity,[status(thm)],[83]) ).
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_inverse(b)),b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(b,additive_identity,b) )
<=> ( ~ ! [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(b,additive_inverse(b),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(b,additive_identity,b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ) ),
inference(transitivity,[status(thm)],[84,82]) ).
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_inverse(b)),b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(b,additive_identity,b) ),
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(b,additive_inverse(b),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(b,additive_identity,b)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity) ),
inference(modus_ponens,[status(thm)],[86,85]) ).
tff(88,plain,
sum(additive_identity,additive_inverse(additive_inverse(b)),b),
inference(unit_resolution,[status(thm)],[87,81,71,69,67]) ).
tff(89,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ( b = additive_inverse(additive_inverse(b)) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ( b = additive_inverse(additive_inverse(b)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(90,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ( b = additive_inverse(additive_inverse(b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(91,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),b)
| ~ sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ( b = additive_inverse(additive_inverse(b)) ) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
b = additive_inverse(additive_inverse(b)),
inference(unit_resolution,[status(thm)],[91,28,88,47]) ).
tff(93,plain,
additive_inverse(additive_inverse(b)) = b,
inference(symmetry,[status(thm)],[92]) ).
tff(94,plain,
( ( a = additive_inverse(additive_inverse(b)) )
<=> ( a = b ) ),
inference(monotonicity,[status(thm)],[93]) ).
tff(95,plain,
( ( a = b )
<=> ( a = additive_inverse(additive_inverse(b)) ) ),
inference(symmetry,[status(thm)],[94]) ).
tff(96,plain,
( ( a != b )
<=> ( a != additive_inverse(additive_inverse(b)) ) ),
inference(monotonicity,[status(thm)],[95]) ).
tff(97,plain,
( ( a != b )
<=> ( a != b ) ),
inference(rewrite,[status(thm)],]) ).
tff(98,axiom,
a != b,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_a_equals_b) ).
tff(99,plain,
a != b,
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
a != additive_inverse(additive_inverse(b)),
inference(modus_ponens,[status(thm)],[99,96]) ).
tff(101,plain,
( sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
<=> sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ),
inference(monotonicity,[status(thm)],[33]) ).
tff(102,plain,
( sum(additive_identity,additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
<=> sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ),
inference(symmetry,[status(thm)],[101]) ).
tff(103,plain,
sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))),
inference(modus_ponens,[status(thm)],[47,102]) ).
tff(104,plain,
( sum(a,additive_inverse(a),additive_inverse(additive_identity))
<=> sum(a,additive_inverse(a),additive_identity) ),
inference(monotonicity,[status(thm)],[33]) ).
tff(105,plain,
( sum(a,additive_inverse(a),additive_identity)
<=> sum(a,additive_inverse(a),additive_inverse(additive_identity)) ),
inference(symmetry,[status(thm)],[104]) ).
tff(106,plain,
( ~ ! [X: $i] : sum(X,additive_inverse(X),additive_identity)
| sum(a,additive_inverse(a),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(107,plain,
sum(a,additive_inverse(a),additive_identity),
inference(unit_resolution,[status(thm)],[106,16]) ).
tff(108,plain,
sum(a,additive_inverse(a),additive_inverse(additive_identity)),
inference(modus_ponens,[status(thm)],[107,105]) ).
tff(109,plain,
( ~ ! [X: $i] : sum(X,additive_identity,X)
| sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(additive_inverse(additive_inverse(b)))) ),
inference(quant_inst,[status(thm)],]) ).
tff(110,plain,
sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(additive_inverse(additive_inverse(b)))),
inference(unit_resolution,[status(thm)],[109,42]) ).
tff(111,plain,
( ~ ! [X: $i] : sum(additive_inverse(X),X,additive_identity)
| sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(112,plain,
sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity),
inference(unit_resolution,[status(thm)],[111,54]) ).
tff(113,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(114,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(115,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity),
inference(unit_resolution,[status(thm)],[115,63,112]) ).
tff(117,plain,
( ~ ! [X: $i] : sum(X,additive_identity,X)
| sum(additive_inverse(b),additive_identity,additive_inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(118,plain,
sum(additive_inverse(b),additive_identity,additive_inverse(b)),
inference(unit_resolution,[status(thm)],[117,42]) ).
tff(119,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(b),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)) )
<=> ( ~ ! [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(b),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(120,plain,
( ( sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b)) )
<=> ( ~ sum(additive_inverse(b),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(121,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_inverse(additive_inverse(b))),additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b)) )
<=> ( ~ ! [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(b),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)) ) ),
inference(monotonicity,[status(thm)],[120]) ).
tff(122,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_inverse(additive_inverse(b))),additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b)) )
<=> ( ~ ! [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(b),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)) ) ),
inference(transitivity,[status(thm)],[121,119]) ).
tff(123,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_inverse(additive_inverse(b))),additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(124,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(b),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(additive_inverse(b))),additive_identity)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)) ),
inference(modus_ponens,[status(thm)],[123,122]) ).
tff(125,plain,
sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b)),
inference(unit_resolution,[status(thm)],[124,81,118,67,116]) ).
tff(126,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b))
| sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b))
| sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(127,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b))
| sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(128,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(b))
| sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b)) ),
inference(modus_ponens,[status(thm)],[127,126]) ).
tff(129,plain,
sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b)),
inference(unit_resolution,[status(thm)],[128,63,125]) ).
tff(130,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(additive_inverse(additive_inverse(b))))
| ( additive_inverse(b) = additive_inverse(additive_inverse(additive_inverse(b))) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(additive_inverse(additive_inverse(b))))
| ( additive_inverse(b) = additive_inverse(additive_inverse(additive_inverse(b))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(131,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(additive_inverse(additive_inverse(b))))
| ( additive_inverse(b) = additive_inverse(additive_inverse(additive_inverse(b))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(132,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(b))
| ~ sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_identity,additive_inverse(additive_inverse(additive_inverse(b))))
| ( additive_inverse(b) = additive_inverse(additive_inverse(additive_inverse(b))) ) ),
inference(modus_ponens,[status(thm)],[131,130]) ).
tff(133,plain,
additive_inverse(b) = additive_inverse(additive_inverse(additive_inverse(b))),
inference(unit_resolution,[status(thm)],[132,28,129,110]) ).
tff(134,plain,
additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(b),
inference(symmetry,[status(thm)],[133]) ).
tff(135,plain,
( sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b))))
<=> sum(c,additive_inverse(d),additive_inverse(b)) ),
inference(monotonicity,[status(thm)],[134]) ).
tff(136,plain,
( sum(c,additive_inverse(d),additive_inverse(b))
<=> sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ),
inference(symmetry,[status(thm)],[135]) ).
tff(137,plain,
( ~ ! [X: $i] : sum(X,additive_identity,X)
| sum(c,additive_identity,c) ),
inference(quant_inst,[status(thm)],]) ).
tff(138,plain,
sum(c,additive_identity,c),
inference(unit_resolution,[status(thm)],[137,42]) ).
tff(139,plain,
( sum(b,c,d)
<=> sum(b,c,d) ),
inference(rewrite,[status(thm)],]) ).
tff(140,axiom,
sum(b,c,d),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_plus_c_is_d) ).
tff(141,plain,
sum(b,c,d),
inference(modus_ponens,[status(thm)],[140,139]) ).
tff(142,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(b,c,d)
| ~ sum(c,additive_identity,c)
| sum(d,additive_identity,d) )
<=> ( ~ ! [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(b,c,d)
| ~ sum(c,additive_identity,c)
| sum(d,additive_identity,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(143,plain,
( ( sum(d,additive_identity,d)
| ~ sum(c,additive_identity,c)
| ~ sum(b,c,d)
| ~ sum(b,c,d) )
<=> ( ~ sum(b,c,d)
| ~ sum(c,additive_identity,c)
| sum(d,additive_identity,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(144,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(d,additive_identity,d)
| ~ sum(c,additive_identity,c)
| ~ sum(b,c,d)
| ~ sum(b,c,d) )
<=> ( ~ ! [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(b,c,d)
| ~ sum(c,additive_identity,c)
| sum(d,additive_identity,d) ) ),
inference(monotonicity,[status(thm)],[143]) ).
tff(145,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(d,additive_identity,d)
| ~ sum(c,additive_identity,c)
| ~ sum(b,c,d)
| ~ sum(b,c,d) )
<=> ( ~ ! [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(b,c,d)
| ~ sum(c,additive_identity,c)
| sum(d,additive_identity,d) ) ),
inference(transitivity,[status(thm)],[144,142]) ).
tff(146,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(d,additive_identity,d)
| ~ sum(c,additive_identity,c)
| ~ sum(b,c,d)
| ~ sum(b,c,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(147,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(b,c,d)
| ~ sum(c,additive_identity,c)
| sum(d,additive_identity,d) ),
inference(modus_ponens,[status(thm)],[146,145]) ).
tff(148,plain,
sum(d,additive_identity,d),
inference(unit_resolution,[status(thm)],[147,81,141,138]) ).
tff(149,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,c,d)
| sum(c,b,d) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,c,d)
| sum(c,b,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(150,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,c,d)
| sum(c,b,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(151,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,c,d)
| sum(c,b,d) ),
inference(modus_ponens,[status(thm)],[150,149]) ).
tff(152,plain,
sum(c,b,d),
inference(unit_resolution,[status(thm)],[151,63,141]) ).
tff(153,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(c,b,d)
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(d,additive_inverse(b),c)
| ~ sum(c,additive_identity,c) )
<=> ( ~ ! [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(c,b,d)
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(d,additive_inverse(b),c)
| ~ sum(c,additive_identity,c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(154,plain,
( ( sum(d,additive_inverse(b),c)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(c,b,d)
| ~ sum(c,additive_identity,c) )
<=> ( ~ sum(c,b,d)
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(d,additive_inverse(b),c)
| ~ sum(c,additive_identity,c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(155,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(d,additive_inverse(b),c)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(c,b,d)
| ~ sum(c,additive_identity,c) )
<=> ( ~ ! [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(c,b,d)
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(d,additive_inverse(b),c)
| ~ sum(c,additive_identity,c) ) ),
inference(monotonicity,[status(thm)],[154]) ).
tff(156,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(d,additive_inverse(b),c)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(c,b,d)
| ~ sum(c,additive_identity,c) )
<=> ( ~ ! [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(c,b,d)
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(d,additive_inverse(b),c)
| ~ sum(c,additive_identity,c) ) ),
inference(transitivity,[status(thm)],[155,153]) ).
tff(157,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(d,additive_inverse(b),c)
| ~ sum(b,additive_inverse(b),additive_identity)
| ~ sum(c,b,d)
| ~ sum(c,additive_identity,c) ),
inference(quant_inst,[status(thm)],]) ).
tff(158,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(c,b,d)
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(d,additive_inverse(b),c)
| ~ sum(c,additive_identity,c) ),
inference(modus_ponens,[status(thm)],[157,156]) ).
tff(159,plain,
sum(d,additive_inverse(b),c),
inference(unit_resolution,[status(thm)],[158,81,152,138,71]) ).
tff(160,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(d,additive_identity,d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(d,additive_inverse(b),c) )
<=> ( ~ ! [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(d,additive_identity,d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(d,additive_inverse(b),c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(161,plain,
( ( sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(d,additive_inverse(b),c)
| ~ sum(d,additive_identity,d) )
<=> ( ~ sum(d,additive_identity,d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(d,additive_inverse(b),c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(162,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(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(d,additive_inverse(b),c)
| ~ sum(d,additive_identity,d) )
<=> ( ~ ! [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(d,additive_identity,d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(d,additive_inverse(b),c) ) ),
inference(monotonicity,[status(thm)],[161]) ).
tff(163,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(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(d,additive_inverse(b),c)
| ~ sum(d,additive_identity,d) )
<=> ( ~ ! [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(d,additive_identity,d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(d,additive_inverse(b),c) ) ),
inference(transitivity,[status(thm)],[162,160]) ).
tff(164,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(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| ~ sum(d,additive_inverse(b),c)
| ~ sum(d,additive_identity,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(165,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(d,additive_identity,d)
| ~ sum(additive_inverse(b),additive_inverse(additive_inverse(b)),additive_identity)
| sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(d,additive_inverse(b),c) ),
inference(modus_ponens,[status(thm)],[164,163]) ).
tff(166,plain,
sum(c,additive_inverse(additive_inverse(b)),d),
inference(unit_resolution,[status(thm)],[165,81,159,148,67]) ).
tff(167,plain,
^ [Y: $i,X: $i] :
refl(
( sum(X,Y,add(X,Y))
<=> sum(X,Y,add(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(168,plain,
( ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
<=> ! [Y: $i,X: $i] : sum(X,Y,add(X,Y)) ),
inference(quant_intro,[status(thm)],[167]) ).
tff(169,plain,
( ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
<=> ! [Y: $i,X: $i] : sum(X,Y,add(X,Y)) ),
inference(rewrite,[status(thm)],]) ).
tff(170,axiom,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',closure_of_addition) ).
tff(171,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(modus_ponens,[status(thm)],[170,169]) ).
tff(172,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(skolemize,[status(sab)],[171]) ).
tff(173,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(modus_ponens,[status(thm)],[172,168]) ).
tff(174,plain,
( ~ ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
| sum(c,additive_inverse(additive_inverse(b)),add(c,additive_inverse(additive_inverse(b)))) ),
inference(quant_inst,[status(thm)],]) ).
tff(175,plain,
sum(c,additive_inverse(additive_inverse(b)),add(c,additive_inverse(additive_inverse(b)))),
inference(unit_resolution,[status(thm)],[174,173]) ).
tff(176,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(c,additive_inverse(additive_inverse(b)),add(c,additive_inverse(additive_inverse(b))))
| ( d = add(c,additive_inverse(additive_inverse(b))) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(c,additive_inverse(additive_inverse(b)),add(c,additive_inverse(additive_inverse(b))))
| ( d = add(c,additive_inverse(additive_inverse(b))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(177,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(c,additive_inverse(additive_inverse(b)),add(c,additive_inverse(additive_inverse(b))))
| ( d = add(c,additive_inverse(additive_inverse(b))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(178,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(additive_inverse(b)),d)
| ~ sum(c,additive_inverse(additive_inverse(b)),add(c,additive_inverse(additive_inverse(b))))
| ( d = add(c,additive_inverse(additive_inverse(b))) ) ),
inference(modus_ponens,[status(thm)],[177,176]) ).
tff(179,plain,
d = add(c,additive_inverse(additive_inverse(b))),
inference(unit_resolution,[status(thm)],[178,28,175,166]) ).
tff(180,plain,
add(c,additive_inverse(additive_inverse(b))) = d,
inference(symmetry,[status(thm)],[179]) ).
tff(181,plain,
( sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity)
<=> sum(additive_inverse(d),d,additive_identity) ),
inference(monotonicity,[status(thm)],[180]) ).
tff(182,plain,
( sum(additive_inverse(d),d,additive_identity)
<=> sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ),
inference(symmetry,[status(thm)],[181]) ).
tff(183,plain,
( ~ ! [X: $i] : sum(additive_inverse(X),X,additive_identity)
| sum(additive_inverse(d),d,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(184,plain,
sum(additive_inverse(d),d,additive_identity),
inference(unit_resolution,[status(thm)],[183,54]) ).
tff(185,plain,
sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity),
inference(modus_ponens,[status(thm)],[184,182]) ).
tff(186,plain,
( sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
<=> sum(d,additive_inverse(b),c) ),
inference(monotonicity,[status(thm)],[180]) ).
tff(187,plain,
( sum(d,additive_inverse(b),c)
<=> sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c) ),
inference(symmetry,[status(thm)],[186]) ).
tff(188,plain,
sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c),
inference(modus_ponens,[status(thm)],[159,187]) ).
tff(189,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b))
| sum(additive_identity,additive_inverse(b),additive_inverse(b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b))
| sum(additive_identity,additive_inverse(b),additive_inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(190,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b))
| sum(additive_identity,additive_inverse(b),additive_inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(191,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(b),additive_identity,additive_inverse(b))
| sum(additive_identity,additive_inverse(b),additive_inverse(b)) ),
inference(modus_ponens,[status(thm)],[190,189]) ).
tff(192,plain,
sum(additive_identity,additive_inverse(b),additive_inverse(b)),
inference(unit_resolution,[status(thm)],[191,63,118]) ).
tff(193,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(
( ( 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(Y,Z,V)
| ~ sum(X,Y,U) ) )),
inference(bind,[status(th)],]) ).
tff(194,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[193]) ).
tff(195,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) ),
inference(rewrite,[status(thm)],]) ).
tff(196,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ 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,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| 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(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) )
<=> ( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
inference(bind,[status(th)],]) ).
tff(197,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[196]) ).
tff(198,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',associativity_of_addition1) ).
tff(199,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(modus_ponens,[status(thm)],[198,197]) ).
tff(200,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(modus_ponens,[status(thm)],[199,195]) ).
tff(201,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(skolemize,[status(sab)],[200]) ).
tff(202,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(modus_ponens,[status(thm)],[201,194]) ).
tff(203,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(204,plain,
( ( sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) )
<=> ( ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(205,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ) ),
inference(monotonicity,[status(thm)],[204]) ).
tff(206,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ) ),
inference(transitivity,[status(thm)],[205,203]) ).
tff(207,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(208,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_identity,additive_inverse(b),additive_inverse(b))
| sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ),
inference(modus_ponens,[status(thm)],[207,206]) ).
tff(209,plain,
( sum(additive_inverse(d),c,additive_inverse(b))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(b),c)
| ~ sum(additive_inverse(d),add(c,additive_inverse(additive_inverse(b))),additive_identity) ),
inference(unit_resolution,[status(thm)],[208,202,192]) ).
tff(210,plain,
sum(additive_inverse(d),c,additive_inverse(b)),
inference(unit_resolution,[status(thm)],[209,188,185]) ).
tff(211,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),c,additive_inverse(b))
| sum(c,additive_inverse(d),additive_inverse(b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),c,additive_inverse(b))
| sum(c,additive_inverse(d),additive_inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(212,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),c,additive_inverse(b))
| sum(c,additive_inverse(d),additive_inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(213,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),c,additive_inverse(b))
| sum(c,additive_inverse(d),additive_inverse(b)) ),
inference(modus_ponens,[status(thm)],[212,211]) ).
tff(214,plain,
sum(c,additive_inverse(d),additive_inverse(b)),
inference(unit_resolution,[status(thm)],[213,63,210]) ).
tff(215,plain,
sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))),
inference(modus_ponens,[status(thm)],[214,136]) ).
tff(216,plain,
( sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
<=> sum(additive_inverse(a),d,c) ),
inference(monotonicity,[status(thm)],[180]) ).
tff(217,plain,
( sum(additive_inverse(a),d,c)
<=> sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c) ),
inference(symmetry,[status(thm)],[216]) ).
tff(218,plain,
( sum(a,c,d)
<=> sum(a,c,d) ),
inference(rewrite,[status(thm)],]) ).
tff(219,axiom,
sum(a,c,d),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_plus_c_is_d) ).
tff(220,plain,
sum(a,c,d),
inference(modus_ponens,[status(thm)],[219,218]) ).
tff(221,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(a,c,d)
| sum(c,a,d) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(a,c,d)
| sum(c,a,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(222,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(a,c,d)
| sum(c,a,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(223,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(a,c,d)
| sum(c,a,d) ),
inference(modus_ponens,[status(thm)],[222,221]) ).
tff(224,plain,
sum(c,a,d),
inference(unit_resolution,[status(thm)],[223,63,220]) ).
tff(225,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(c,additive_identity,c)
| sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d) )
<=> ( ~ ! [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(c,additive_identity,c)
| sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(226,plain,
( ( sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d)
| ~ sum(c,additive_identity,c) )
<=> ( ~ sum(c,additive_identity,c)
| sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(227,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(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d)
| ~ sum(c,additive_identity,c) )
<=> ( ~ ! [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(c,additive_identity,c)
| sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d) ) ),
inference(monotonicity,[status(thm)],[226]) ).
tff(228,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(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d)
| ~ sum(c,additive_identity,c) )
<=> ( ~ ! [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(c,additive_identity,c)
| sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d) ) ),
inference(transitivity,[status(thm)],[227,225]) ).
tff(229,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(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d)
| ~ sum(c,additive_identity,c) ),
inference(quant_inst,[status(thm)],]) ).
tff(230,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(c,additive_identity,c)
| sum(d,additive_inverse(a),c)
| ~ sum(a,additive_inverse(a),additive_identity)
| ~ sum(c,a,d) ),
inference(modus_ponens,[status(thm)],[229,228]) ).
tff(231,plain,
sum(d,additive_inverse(a),c),
inference(unit_resolution,[status(thm)],[230,81,224,138,107]) ).
tff(232,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(d,additive_inverse(a),c)
| sum(additive_inverse(a),d,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(d,additive_inverse(a),c)
| sum(additive_inverse(a),d,c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(233,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(d,additive_inverse(a),c)
| sum(additive_inverse(a),d,c) ),
inference(quant_inst,[status(thm)],]) ).
tff(234,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(d,additive_inverse(a),c)
| sum(additive_inverse(a),d,c) ),
inference(modus_ponens,[status(thm)],[233,232]) ).
tff(235,plain,
sum(additive_inverse(a),d,c),
inference(unit_resolution,[status(thm)],[234,63,231]) ).
tff(236,plain,
sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c),
inference(modus_ponens,[status(thm)],[235,217]) ).
tff(237,plain,
( sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
<=> sum(d,additive_inverse(d),additive_identity) ),
inference(monotonicity,[status(thm)],[180]) ).
tff(238,plain,
( sum(d,additive_inverse(d),additive_identity)
<=> sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity) ),
inference(symmetry,[status(thm)],[237]) ).
tff(239,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),d,additive_identity)
| sum(d,additive_inverse(d),additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),d,additive_identity)
| sum(d,additive_inverse(d),additive_identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(240,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),d,additive_identity)
| sum(d,additive_inverse(d),additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(241,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_inverse(d),d,additive_identity)
| sum(d,additive_inverse(d),additive_identity) ),
inference(modus_ponens,[status(thm)],[240,239]) ).
tff(242,plain,
sum(d,additive_inverse(d),additive_identity),
inference(unit_resolution,[status(thm)],[241,63,184]) ).
tff(243,plain,
sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity),
inference(modus_ponens,[status(thm)],[242,238]) ).
tff(244,plain,
( ~ ! [X: $i] : sum(additive_identity,X,X)
| sum(additive_identity,additive_inverse(a),additive_inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(245,plain,
sum(additive_identity,additive_inverse(a),additive_inverse(a)),
inference(unit_resolution,[status(thm)],[244,7]) ).
tff(246,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(a),additive_inverse(a))
| sum(additive_inverse(a),additive_identity,additive_inverse(a)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(a),additive_inverse(a))
| sum(additive_inverse(a),additive_identity,additive_inverse(a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(247,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(a),additive_inverse(a))
| sum(additive_inverse(a),additive_identity,additive_inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(248,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(additive_identity,additive_inverse(a),additive_inverse(a))
| sum(additive_inverse(a),additive_identity,additive_inverse(a)) ),
inference(modus_ponens,[status(thm)],[247,246]) ).
tff(249,plain,
sum(additive_inverse(a),additive_identity,additive_inverse(a)),
inference(unit_resolution,[status(thm)],[248,63,245]) ).
tff(250,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(a),additive_identity,additive_inverse(a))
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) )
<=> ( ~ ! [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(a),additive_identity,additive_inverse(a))
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(251,plain,
( ( sum(c,additive_inverse(d),additive_inverse(a))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(additive_inverse(a),additive_identity,additive_inverse(a)) )
<=> ( ~ sum(additive_inverse(a),additive_identity,additive_inverse(a))
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(252,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(c,additive_inverse(d),additive_inverse(a))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(additive_inverse(a),additive_identity,additive_inverse(a)) )
<=> ( ~ ! [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(a),additive_identity,additive_inverse(a))
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) ) ),
inference(monotonicity,[status(thm)],[251]) ).
tff(253,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(c,additive_inverse(d),additive_inverse(a))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(additive_inverse(a),additive_identity,additive_inverse(a)) )
<=> ( ~ ! [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(a),additive_identity,additive_inverse(a))
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) ) ),
inference(transitivity,[status(thm)],[252,250]) ).
tff(254,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(c,additive_inverse(d),additive_inverse(a))
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(additive_inverse(a),additive_identity,additive_inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(255,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(a),additive_identity,additive_inverse(a))
| ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) ),
inference(modus_ponens,[status(thm)],[254,253]) ).
tff(256,plain,
( ~ sum(additive_inverse(a),add(c,additive_inverse(additive_inverse(b))),c)
| ~ sum(add(c,additive_inverse(additive_inverse(b))),additive_inverse(d),additive_identity)
| sum(c,additive_inverse(d),additive_inverse(a)) ),
inference(unit_resolution,[status(thm)],[255,81,249]) ).
tff(257,plain,
sum(c,additive_inverse(d),additive_inverse(a)),
inference(unit_resolution,[status(thm)],[256,243,236]) ).
tff(258,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(259,plain,
( ( ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b))))
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) ) )
<=> ( ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(260,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b))))
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ) ),
inference(monotonicity,[status(thm)],[259]) ).
tff(261,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b))))
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ) ),
inference(transitivity,[status(thm)],[260,258]) ).
tff(262,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b))))
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(263,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(c,additive_inverse(d),additive_inverse(a))
| ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ),
inference(modus_ponens,[status(thm)],[262,261]) ).
tff(264,plain,
( ( additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a) )
| ~ sum(c,additive_inverse(d),additive_inverse(additive_inverse(additive_inverse(b)))) ),
inference(unit_resolution,[status(thm)],[263,28,257]) ).
tff(265,plain,
additive_inverse(additive_inverse(additive_inverse(b))) = additive_inverse(a),
inference(unit_resolution,[status(thm)],[264,215]) ).
tff(266,plain,
additive_inverse(a) = additive_inverse(additive_inverse(additive_inverse(b))),
inference(symmetry,[status(thm)],[265]) ).
tff(267,plain,
( sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
<=> sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity) ),
inference(monotonicity,[status(thm)],[266,33]) ).
tff(268,plain,
( sum(additive_inverse(additive_inverse(additive_inverse(b))),additive_inverse(additive_inverse(b)),additive_identity)
<=> sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity)) ),
inference(symmetry,[status(thm)],[267]) ).
tff(269,plain,
sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity)),
inference(modus_ponens,[status(thm)],[112,268]) ).
tff(270,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(271,plain,
( ( sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity)) )
<=> ( ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(272,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ) ),
inference(monotonicity,[status(thm)],[271]) ).
tff(273,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ) ),
inference(transitivity,[status(thm)],[272,270]) ).
tff(274,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b)))
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(275,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ),
inference(modus_ponens,[status(thm)],[274,273]) ).
tff(276,plain,
( ~ sum(additive_inverse(a),additive_inverse(additive_inverse(b)),additive_inverse(additive_identity))
| sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ~ sum(a,additive_inverse(a),additive_inverse(additive_identity))
| ~ sum(additive_inverse(additive_identity),additive_inverse(additive_inverse(b)),additive_inverse(additive_inverse(b))) ),
inference(unit_resolution,[status(thm)],[275,202]) ).
tff(277,plain,
sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b))),
inference(unit_resolution,[status(thm)],[276,269,108,103]) ).
tff(278,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,additive_inverse(additive_identity),a)
| ~ sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ( a = additive_inverse(additive_inverse(b)) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,additive_inverse(additive_identity),a)
| ~ sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ( a = additive_inverse(additive_inverse(b)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(279,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,additive_inverse(additive_identity),a)
| ~ sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ( a = additive_inverse(additive_inverse(b)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(280,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,additive_inverse(additive_identity),a)
| ~ sum(a,additive_inverse(additive_identity),additive_inverse(additive_inverse(b)))
| ( a = additive_inverse(additive_inverse(b)) ) ),
inference(modus_ponens,[status(thm)],[279,278]) ).
tff(281,plain,
( ~ sum(a,additive_inverse(additive_identity),a)
| ( a = additive_inverse(additive_inverse(b)) ) ),
inference(unit_resolution,[status(thm)],[280,28,277]) ).
tff(282,plain,
$false,
inference(unit_resolution,[status(thm)],[281,100,45]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : RNG003-1 : TPTP v8.1.0. Released v1.0.0.
% 0.11/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n026.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Sep 2 21:28:08 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 246.24/158.42 % SZS status Unsatisfiable
% 246.24/158.42 % SZS output start Proof
% See solution above
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