%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : BOO004-10 : TPTP v9.0.0. Released v7.5.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:00:05 AM UTC 2025
% Result : Unsatisfiable 0.18s 0.43s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 40
% Syntax : Number of formulae : 115 ( 84 unt; 0 typ; 0 def)
% Number of atoms : 162 ( 153 equ)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 57 ( 18 ~; 14 |; 0 &)
% ( 25 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of FOOLs : 8 ( 8 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 3 ( 1 usr; 1 prp; 0-2 aty)
% Number of functors : 9 ( 9 usr; 4 con; 0-4 aty)
% Number of variables : 248 ( 225 !; 0 ?; 248 :)
% Comments :
%------------------------------------------------------------------------------
tff(true_type,type,
true: $i ).
tff(sum_type,type,
sum: ( $i * $i * $i ) > $i ).
tff(x_type,type,
x: $i ).
tff(ifeq_type,type,
ifeq: ( $i * $i * $i * $i ) > $i ).
tff(add_type,type,
add: ( $i * $i ) > $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $i ).
tff(multiplicative_identity_type,type,
multiplicative_identity: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(1,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
refl(( ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true )
<=> ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',distributivity3) ).
tff(5,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(V3,X,V4),true,ifeq(product(Z,X,V2),true,ifeq(product(Y,X,V1),true,ifeq(sum(Y,Z,V3),true,sum(V1,V2,V4),true),true),true),true) = true )
| ( ifeq(product(add(x,x),multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true),true) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
ifeq(product(add(x,x),multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true),true) = true,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Y: $i,X: $i] :
refl(( ( sum(X,Y,add(X,Y)) = true )
<=> ( sum(X,Y,add(X,Y)) = true ) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true )
<=> ! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true )
<=> ! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_of_addition) ).
tff(14,plain,
! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Y: $i,X: $i] : ( sum(X,Y,add(X,Y)) = true )
| ( sum(x,x,add(x,x)) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
sum(x,x,add(x,x)) = true,
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
true = sum(x,x,add(x,x)),
inference(symmetry,[status(thm)],[18]) ).
tff(20,plain,
^ [X: $i] :
refl(( ( product(X,multiplicative_identity,X) = true )
<=> ( product(X,multiplicative_identity,X) = true ) )),
inference(bind,[status(th)],[]) ).
tff(21,plain,
( ! [X: $i] : ( product(X,multiplicative_identity,X) = true )
<=> ! [X: $i] : ( product(X,multiplicative_identity,X) = true ) ),
inference(quant_intro,[status(thm)],[20]) ).
tff(22,plain,
( ! [X: $i] : ( product(X,multiplicative_identity,X) = true )
<=> ! [X: $i] : ( product(X,multiplicative_identity,X) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,axiom,
! [X: $i] : ( product(X,multiplicative_identity,X) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',multiplicative_identity2) ).
tff(24,plain,
! [X: $i] : ( product(X,multiplicative_identity,X) = true ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [X: $i] : ( product(X,multiplicative_identity,X) = true ),
inference(skolemize,[status(sab)],[24]) ).
tff(26,plain,
! [X: $i] : ( product(X,multiplicative_identity,X) = true ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
( ~ ! [X: $i] : ( product(X,multiplicative_identity,X) = true )
| ( product(x,multiplicative_identity,x) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
product(x,multiplicative_identity,x) = true,
inference(unit_resolution,[status(thm)],[27,26]) ).
tff(29,plain,
product(x,multiplicative_identity,x) = sum(x,x,add(x,x)),
inference(transitivity,[status(thm)],[28,19]) ).
tff(30,plain,
ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),
inference(monotonicity,[status(thm)],[29]) ).
tff(31,plain,
ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),
inference(symmetry,[status(thm)],[30]) ).
tff(32,plain,
true = product(x,multiplicative_identity,x),
inference(symmetry,[status(thm)],[28]) ).
tff(33,plain,
ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true) = ifeq(product(x,multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true),
inference(monotonicity,[status(thm)],[32,31]) ).
tff(34,plain,
^ [B: $i,A: $i,C: $i] :
refl(( ( ifeq(A,A,B,C) = B )
<=> ( ifeq(A,A,B,C) = B ) )),
inference(bind,[status(th)],[]) ).
tff(35,plain,
( ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
<=> ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ) ),
inference(quant_intro,[status(thm)],[34]) ).
tff(36,plain,
( ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
<=> ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ) ),
inference(rewrite,[status(thm)],[]) ).
tff(37,axiom,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ifeq_axiom_001) ).
tff(38,plain,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,plain,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
inference(skolemize,[status(sab)],[38]) ).
tff(40,plain,
! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B ),
inference(modus_ponens,[status(thm)],[39,35]) ).
tff(41,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true) = ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(42,plain,
ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true) = ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),
inference(unit_resolution,[status(thm)],[41,40]) ).
tff(43,plain,
ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true),
inference(symmetry,[status(thm)],[42]) ).
tff(44,plain,
ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(product(x,multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true),
inference(transitivity,[status(thm)],[30,43,33]) ).
tff(45,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,product(add(x,x),multiplicative_identity,x),true) = product(add(x,x),multiplicative_identity,x) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(46,plain,
ifeq(true,true,product(add(x,x),multiplicative_identity,x),true) = product(add(x,x),multiplicative_identity,x),
inference(unit_resolution,[status(thm)],[45,40]) ).
tff(47,plain,
ifeq(true,true,product(add(x,x),multiplicative_identity,x),true) = ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),
inference(monotonicity,[status(thm)],[19]) ).
tff(48,plain,
ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true) = ifeq(true,true,product(add(x,x),multiplicative_identity,x),true),
inference(symmetry,[status(thm)],[47]) ).
tff(49,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true) = ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(50,plain,
ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true) = ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),
inference(unit_resolution,[status(thm)],[49,40]) ).
tff(51,plain,
^ [X: $i] :
refl(( ( sum(inverse(X),X,multiplicative_identity) = true )
<=> ( sum(inverse(X),X,multiplicative_identity) = true ) )),
inference(bind,[status(th)],[]) ).
tff(52,plain,
( ! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true )
<=> ! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true ) ),
inference(quant_intro,[status(thm)],[51]) ).
tff(53,plain,
( ! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true )
<=> ! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(54,axiom,
! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',additive_inverse1) ).
tff(55,plain,
! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true ),
inference(skolemize,[status(sab)],[55]) ).
tff(57,plain,
! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true ),
inference(modus_ponens,[status(thm)],[56,52]) ).
tff(58,plain,
( ~ ! [X: $i] : ( sum(inverse(X),X,multiplicative_identity) = true )
| ( sum(inverse(x),x,multiplicative_identity) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(59,plain,
sum(inverse(x),x,multiplicative_identity) = true,
inference(unit_resolution,[status(thm)],[58,57]) ).
tff(60,plain,
true = sum(inverse(x),x,multiplicative_identity),
inference(symmetry,[status(thm)],[59]) ).
tff(61,plain,
ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true) = ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),
inference(monotonicity,[status(thm)],[60]) ).
tff(62,plain,
ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true) = ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),
inference(symmetry,[status(thm)],[61]) ).
tff(63,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true) = ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(64,plain,
ifeq(true,true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true) = ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),
inference(unit_resolution,[status(thm)],[63,40]) ).
tff(65,plain,
^ [X: $i] :
refl(( ( sum(additive_identity,X,X) = true )
<=> ( sum(additive_identity,X,X) = true ) )),
inference(bind,[status(th)],[]) ).
tff(66,plain,
( ! [X: $i] : ( sum(additive_identity,X,X) = true )
<=> ! [X: $i] : ( sum(additive_identity,X,X) = true ) ),
inference(quant_intro,[status(thm)],[65]) ).
tff(67,plain,
( ! [X: $i] : ( sum(additive_identity,X,X) = true )
<=> ! [X: $i] : ( sum(additive_identity,X,X) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(68,axiom,
! [X: $i] : ( sum(additive_identity,X,X) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',additive_identity1) ).
tff(69,plain,
! [X: $i] : ( sum(additive_identity,X,X) = true ),
inference(modus_ponens,[status(thm)],[68,67]) ).
tff(70,plain,
! [X: $i] : ( sum(additive_identity,X,X) = true ),
inference(skolemize,[status(sab)],[69]) ).
tff(71,plain,
! [X: $i] : ( sum(additive_identity,X,X) = true ),
inference(modus_ponens,[status(thm)],[70,66]) ).
tff(72,plain,
( ~ ! [X: $i] : ( sum(additive_identity,X,X) = true )
| ( sum(additive_identity,x,x) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(73,plain,
sum(additive_identity,x,x) = true,
inference(unit_resolution,[status(thm)],[72,71]) ).
tff(74,plain,
true = sum(additive_identity,x,x),
inference(symmetry,[status(thm)],[73]) ).
tff(75,plain,
ifeq(true,true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true) = ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),
inference(monotonicity,[status(thm)],[74]) ).
tff(76,plain,
ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true) = ifeq(true,true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),
inference(symmetry,[status(thm)],[75]) ).
tff(77,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true) = ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(78,plain,
ifeq(true,true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true) = ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),
inference(unit_resolution,[status(thm)],[77,40]) ).
tff(79,plain,
^ [X: $i] :
refl(( ( product(X,inverse(X),additive_identity) = true )
<=> ( product(X,inverse(X),additive_identity) = true ) )),
inference(bind,[status(th)],[]) ).
tff(80,plain,
( ! [X: $i] : ( product(X,inverse(X),additive_identity) = true )
<=> ! [X: $i] : ( product(X,inverse(X),additive_identity) = true ) ),
inference(quant_intro,[status(thm)],[79]) ).
tff(81,plain,
( ! [X: $i] : ( product(X,inverse(X),additive_identity) = true )
<=> ! [X: $i] : ( product(X,inverse(X),additive_identity) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(82,axiom,
! [X: $i] : ( product(X,inverse(X),additive_identity) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',multiplicative_inverse2) ).
tff(83,plain,
! [X: $i] : ( product(X,inverse(X),additive_identity) = true ),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,plain,
! [X: $i] : ( product(X,inverse(X),additive_identity) = true ),
inference(skolemize,[status(sab)],[83]) ).
tff(85,plain,
! [X: $i] : ( product(X,inverse(X),additive_identity) = true ),
inference(modus_ponens,[status(thm)],[84,80]) ).
tff(86,plain,
( ~ ! [X: $i] : ( product(X,inverse(X),additive_identity) = true )
| ( product(x,inverse(x),additive_identity) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(87,plain,
product(x,inverse(x),additive_identity) = true,
inference(unit_resolution,[status(thm)],[86,85]) ).
tff(88,plain,
true = product(x,inverse(x),additive_identity),
inference(symmetry,[status(thm)],[87]) ).
tff(89,plain,
ifeq(true,true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true) = ifeq(product(x,inverse(x),additive_identity),true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true),
inference(monotonicity,[status(thm)],[88]) ).
tff(90,plain,
ifeq(product(x,inverse(x),additive_identity),true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true) = ifeq(true,true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true),
inference(symmetry,[status(thm)],[89]) ).
tff(91,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
refl(( ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true )
<=> ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ) )),
inference(bind,[status(th)],[]) ).
tff(92,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ) ),
inference(quant_intro,[status(thm)],[91]) ).
tff(93,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(94,axiom,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',distributivity7) ).
tff(95,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ),
inference(modus_ponens,[status(thm)],[94,93]) ).
tff(96,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ),
inference(skolemize,[status(sab)],[95]) ).
tff(97,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true ),
inference(modus_ponens,[status(thm)],[96,92]) ).
tff(98,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] : ( ifeq(product(Y,Z,V3),true,ifeq(sum(V3,X,V4),true,ifeq(sum(Z,X,V2),true,ifeq(sum(Y,X,V1),true,product(V1,V2,V4),true),true),true),true) = true )
| ( ifeq(product(x,inverse(x),additive_identity),true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true) = true ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(99,plain,
ifeq(product(x,inverse(x),additive_identity),true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true) = true,
inference(unit_resolution,[status(thm)],[98,97]) ).
tff(100,plain,
true = ifeq(product(x,inverse(x),additive_identity),true,ifeq(sum(additive_identity,x,x),true,ifeq(sum(inverse(x),x,multiplicative_identity),true,ifeq(sum(x,x,add(x,x)),true,product(add(x,x),multiplicative_identity,x),true),true),true),true),
inference(symmetry,[status(thm)],[99]) ).
tff(101,plain,
product(x,multiplicative_identity,x) = product(add(x,x),multiplicative_identity,x),
inference(transitivity,[status(thm)],[28,100,90,78,76,64,62,50,48,46]) ).
tff(102,plain,
ifeq(product(x,multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true) = ifeq(product(add(x,x),multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(product(x,multiplicative_identity,x),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),true),true),
inference(monotonicity,[status(thm)],[101,44]) ).
tff(103,plain,
ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(sum(x,x,add(x,x)),true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),
inference(monotonicity,[status(thm)],[19]) ).
tff(104,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(105,plain,
ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true) = ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),
inference(unit_resolution,[status(thm)],[104,40]) ).
tff(106,plain,
ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true) = ifeq(true,true,ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),true),
inference(symmetry,[status(thm)],[105]) ).
tff(107,plain,
ifeq(true,true,sum(x,x,x),true) = ifeq(sum(x,x,add(x,x)),true,sum(x,x,x),true),
inference(monotonicity,[status(thm)],[19]) ).
tff(108,plain,
( ~ ! [B: $i,A: $i,C: $i] : ( ifeq(A,A,B,C) = B )
| ( ifeq(true,true,sum(x,x,x),true) = sum(x,x,x) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(109,plain,
ifeq(true,true,sum(x,x,x),true) = sum(x,x,x),
inference(unit_resolution,[status(thm)],[108,40]) ).
tff(110,plain,
sum(x,x,x) = ifeq(true,true,sum(x,x,x),true),
inference(symmetry,[status(thm)],[109]) ).
tff(111,plain,
sum(x,x,x) = true,
inference(transitivity,[status(thm)],[110,107,106,103,43,33,102,9]) ).
tff(112,plain,
( ( sum(x,x,x) != true )
<=> ( sum(x,x,x) != true ) ),
inference(rewrite,[status(thm)],[]) ).
tff(113,axiom,
sum(x,x,x) != true,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_both_equalities) ).
tff(114,plain,
sum(x,x,x) != true,
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
$false,
inference(unit_resolution,[status(thm)],[114,111]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : BOO004-10 : TPTP v9.0.0. Released v7.5.0.
% 0.10/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n003.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Thu Jun 19 11:32:55 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.18/0.43 % SZS status Unsatisfiable
% 0.18/0.43 % SZS output start Proof
% See solution above
% 0.18/0.44 % E exiting
%------------------------------------------------------------------------------