%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : RNG037-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n004.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 05:23:58 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.45s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 26
% Syntax : Number of formulae : 55 ( 23 unt; 0 typ; 0 def)
% Number of atoms : 302 ( 0 equ)
% Maximal formula atoms : 20 ( 5 avg)
% Number of connectives : 444 ( 214 ~; 202 |; 0 &)
% ( 28 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 17 ( 17 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 6 ( 6 usr; 5 con; 0-1 aty)
% Number of variables : 206 ( 187 !; 0 ?; 206 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(a_type,type,
a: $i ).
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(b_type,type,
b: $i ).
tff(additive_inverse_type,type,
additive_inverse: $i > $i ).
tff(d_type,type,
d: $i ).
tff(c_type,type,
c: $i ).
tff(1,plain,
^ [X: $i] :
refl(( product(X,additive_identity,additive_identity)
<=> product(X,additive_identity,additive_identity) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : product(X,additive_identity,additive_identity)
<=> ! [X: $i] : product(X,additive_identity,additive_identity) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(X,additive_identity,additive_identity)
<=> ! [X: $i] : product(X,additive_identity,additive_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : product(X,additive_identity,additive_identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',multiplicative_identity2) ).
tff(5,plain,
! [X: $i] : product(X,additive_identity,additive_identity),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(X,additive_identity,additive_identity),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(X,additive_identity,additive_identity),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(X,additive_identity,additive_identity)
| product(a,additive_identity,additive_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(a,additive_identity,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/sandbox/benchmark/theBenchmark.p',additive_inverse2) ).
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(b,additive_inverse(b),additive_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
sum(b,additive_inverse(b),additive_identity),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',commutativity_of_addition) ).
tff(23,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(additive_inverse(b),b,additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(additive_inverse(b),b,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(additive_inverse(b),b,additive_identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(b,additive_inverse(b),additive_identity)
| sum(additive_inverse(b),b,additive_identity) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
sum(additive_inverse(b),b,additive_identity),
inference(unit_resolution,[status(thm)],[28,25,18]) ).
tff(30,plain,
( ~ sum(c,d,additive_identity)
<=> ~ sum(c,d,additive_identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(31,axiom,
~ sum(c,d,additive_identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_sum_is_additive_identity) ).
tff(32,plain,
~ sum(c,d,additive_identity),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
( product(a,additive_inverse(b),c)
<=> product(a,additive_inverse(b),c) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,axiom,
product(a,additive_inverse(b),c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_inverse_times_b) ).
tff(35,plain,
product(a,additive_inverse(b),c),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
( product(a,b,d)
<=> product(a,b,d) ),
inference(rewrite,[status(thm)],[]) ).
tff(37,axiom,
product(a,b,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_times_b) ).
tff(38,plain,
product(a,b,d),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
refl(( ( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
<=> ( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) )),
( ( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1)
| ~ product(X,V3,V4) ) )),
rewrite(( ( ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1)
| ~ product(X,V3,V4) )
<=> ( ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) )),
( ( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4) )
<=> ( ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) )),
( ( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4)
| sum(V1,V2,V4) )
<=> ( ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1)
| sum(V1,V2,V4) ) )),
rewrite(( ( ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1)
| sum(V1,V2,V4) )
<=> ( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) )),
( ( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4)
| sum(V1,V2,V4) )
<=> ( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) )),
inference(bind,[status(th)],[]) ).
tff(43,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4)
| sum(V1,V2,V4) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,axiom,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( ~ product(X,Y,V1)
| ~ product(X,Z,V2)
| ~ sum(Y,Z,V3)
| ~ product(X,V3,V4)
| sum(V1,V2,V4) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',distributivity1) ).
tff(45,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ),
inference(modus_ponens,[status(thm)],[45,41]) ).
tff(47,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) ),
inference(modus_ponens,[status(thm)],[47,40]) ).
tff(49,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,additive_identity,additive_identity) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,additive_identity,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ( sum(c,d,additive_identity)
| ~ product(a,additive_identity,additive_identity)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c) )
<=> ( sum(c,d,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,additive_identity,additive_identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(51,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,additive_identity,additive_identity)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,additive_identity,additive_identity) ) ),
inference(monotonicity,[status(thm)],[50]) ).
tff(52,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,additive_identity,additive_identity)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,additive_identity,additive_identity) ) ),
inference(transitivity,[status(thm)],[51,49]) ).
tff(53,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,additive_identity,additive_identity)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(54,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ product(X,V3,V4)
| ~ sum(Y,Z,V3)
| ~ product(X,Z,V2)
| ~ product(X,Y,V1) )
| sum(c,d,additive_identity)
| ~ product(a,b,d)
| ~ product(a,additive_inverse(b),c)
| ~ sum(additive_inverse(b),b,additive_identity)
| ~ product(a,additive_identity,additive_identity) ),
inference(modus_ponens,[status(thm)],[53,52]) ).
tff(55,plain,
$false,
inference(unit_resolution,[status(thm)],[54,48,38,35,32,29,9]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : RNG037-2 : TPTP v9.0.0. Released v1.0.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n004.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 03:14:53 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.45 % SZS status Unsatisfiable
% 0.20/0.45 % SZS output start Proof
% See solution above
% 0.20/0.46 % E exiting
%------------------------------------------------------------------------------