TSTP Solution File: BOO004-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : BOO004-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% 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 : Tue Sep 6 17:18:35 EDT 2022
% Result : Unsatisfiable 0.44s 0.54s
% Output : Proof 0.48s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 43
% Syntax : Number of formulae : 89 ( 38 unt; 7 typ; 0 def)
% Number of atoms : 516 ( 0 equ)
% Maximal formula atoms : 20 ( 6 avg)
% Number of connectives : 776 ( 375 ~; 355 |; 0 &)
% ( 46 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 33 ( 33 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 9 ( 4 >; 5 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 354 ( 320 !; 0 ?; 354 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(x_type,type,
x: $i ).
tff(add_type,type,
add: ( $i * $i ) > $i ).
tff(multiplicative_identity_type,type,
multiplicative_identity: $i ).
tff(additive_identity_type,type,
additive_identity: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(1,plain,
^ [X: $i] :
refl(
( product(inverse(X),X,additive_identity)
<=> product(inverse(X),X,additive_identity) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : product(inverse(X),X,additive_identity)
<=> ! [X: $i] : product(inverse(X),X,additive_identity) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(inverse(X),X,additive_identity)
<=> ! [X: $i] : product(inverse(X),X,additive_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : product(inverse(X),X,additive_identity),
file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax',multiplicative_inverse1) ).
tff(5,plain,
! [X: $i] : product(inverse(X),X,additive_identity),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(inverse(X),X,additive_identity),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(inverse(X),X,additive_identity),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(inverse(X),X,additive_identity)
| product(inverse(x),x,additive_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
product(inverse(x),x,additive_identity),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(
( sum(X,inverse(X),multiplicative_identity)
<=> sum(X,inverse(X),multiplicative_identity) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [X: $i] : sum(X,inverse(X),multiplicative_identity)
<=> ! [X: $i] : sum(X,inverse(X),multiplicative_identity) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : sum(X,inverse(X),multiplicative_identity)
<=> ! [X: $i] : sum(X,inverse(X),multiplicative_identity) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [X: $i] : sum(X,inverse(X),multiplicative_identity),
file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax',additive_inverse2) ).
tff(14,plain,
! [X: $i] : sum(X,inverse(X),multiplicative_identity),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : sum(X,inverse(X),multiplicative_identity),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : sum(X,inverse(X),multiplicative_identity),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : sum(X,inverse(X),multiplicative_identity)
| sum(x,inverse(x),multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
sum(x,inverse(x),multiplicative_identity),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [X: $i] :
refl(
( sum(X,additive_identity,X)
<=> sum(X,additive_identity,X) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ! [X: $i] : sum(X,additive_identity,X)
<=> ! [X: $i] : sum(X,additive_identity,X) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [X: $i] : sum(X,additive_identity,X)
<=> ! [X: $i] : sum(X,additive_identity,X) ),
inference(rewrite,[status(thm)],]) ).
tff(22,axiom,
! [X: $i] : sum(X,additive_identity,X),
file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax',additive_identity2) ).
tff(23,plain,
! [X: $i] : sum(X,additive_identity,X),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [X: $i] : sum(X,additive_identity,X),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [X: $i] : sum(X,additive_identity,X),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ~ ! [X: $i] : sum(X,additive_identity,X)
| sum(x,additive_identity,x) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
sum(x,additive_identity,x),
inference(unit_resolution,[status(thm)],[26,25]) ).
tff(28,plain,
^ [Y: $i,X: $i] :
refl(
( sum(X,Y,add(X,Y))
<=> sum(X,Y,add(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(29,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)],[28]) ).
tff(30,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(31,axiom,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax',closure_of_addition) ).
tff(32,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(skolemize,[status(sab)],[32]) ).
tff(34,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(modus_ponens,[status(thm)],[33,29]) ).
tff(35,plain,
( ~ ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
| sum(x,x,add(x,x)) ),
inference(quant_inst,[status(thm)],]) ).
tff(36,plain,
sum(x,x,add(x,x)),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
refl(
( ( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
<=> ( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) )),
inference(bind,[status(th)],]) ).
tff(38,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) ),
inference(quant_intro,[status(thm)],[37]) ).
tff(39,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) ),
inference(rewrite,[status(thm)],]) ).
tff(40,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3) )
<=> ( ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) )),
( ( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4) )
<=> ( ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1)
| ~ sum(X,V3,V4) ) )),
rewrite(
( ( ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1)
| ~ sum(X,V3,V4) )
<=> ( ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) )),
( ( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4) )
<=> ( ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) )),
( ( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4)
| product(V1,V2,V4) )
<=> ( ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1)
| product(V1,V2,V4) ) )),
rewrite(
( ( ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1)
| product(V1,V2,V4) )
<=> ( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) )),
( ( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4)
| product(V1,V2,V4) )
<=> ( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) )),
inference(bind,[status(th)],]) ).
tff(41,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4)
| product(V1,V2,V4) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ) ),
inference(quant_intro,[status(thm)],[40]) ).
tff(42,axiom,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( ~ sum(X,Y,V1)
| ~ sum(X,Z,V2)
| ~ product(Y,Z,V3)
| ~ sum(X,V3,V4)
| product(V1,V2,V4) ),
file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax',distributivity5) ).
tff(43,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ),
inference(skolemize,[status(sab)],[44]) ).
tff(46,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) ),
inference(modus_ponens,[status(thm)],[45,38]) ).
tff(47,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| ~ sum(x,inverse(x),multiplicative_identity)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,additive_identity,x)
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x)) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| ~ sum(x,inverse(x),multiplicative_identity)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,additive_identity,x)
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(48,plain,
( ( product(multiplicative_identity,add(x,x),x)
| ~ sum(x,additive_identity,x)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,x,add(x,x))
| ~ sum(x,inverse(x),multiplicative_identity) )
<=> ( ~ sum(x,inverse(x),multiplicative_identity)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,additive_identity,x)
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(49,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,additive_identity,x)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,x,add(x,x))
| ~ sum(x,inverse(x),multiplicative_identity) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| ~ sum(x,inverse(x),multiplicative_identity)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,additive_identity,x)
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x)) ) ),
inference(monotonicity,[status(thm)],[48]) ).
tff(50,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,additive_identity,x)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,x,add(x,x))
| ~ sum(x,inverse(x),multiplicative_identity) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| ~ sum(x,inverse(x),multiplicative_identity)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,additive_identity,x)
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x)) ) ),
inference(transitivity,[status(thm)],[49,47]) ).
tff(51,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,additive_identity,x)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,x,add(x,x))
| ~ sum(x,inverse(x),multiplicative_identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(52,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( product(V1,V2,V4)
| ~ sum(X,V3,V4)
| ~ product(Y,Z,V3)
| ~ sum(X,Z,V2)
| ~ sum(X,Y,V1) )
| ~ sum(x,inverse(x),multiplicative_identity)
| ~ product(inverse(x),x,additive_identity)
| ~ sum(x,additive_identity,x)
| product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x)) ),
inference(modus_ponens,[status(thm)],[51,50]) ).
tff(53,plain,
product(multiplicative_identity,add(x,x),x),
inference(unit_resolution,[status(thm)],[52,46,36,27,18,9]) ).
tff(54,plain,
^ [X: $i] :
refl(
( product(multiplicative_identity,X,X)
<=> product(multiplicative_identity,X,X) )),
inference(bind,[status(th)],]) ).
tff(55,plain,
( ! [X: $i] : product(multiplicative_identity,X,X)
<=> ! [X: $i] : product(multiplicative_identity,X,X) ),
inference(quant_intro,[status(thm)],[54]) ).
tff(56,plain,
( ! [X: $i] : product(multiplicative_identity,X,X)
<=> ! [X: $i] : product(multiplicative_identity,X,X) ),
inference(rewrite,[status(thm)],]) ).
tff(57,axiom,
! [X: $i] : product(multiplicative_identity,X,X),
file('/export/starexec/sandbox2/benchmark/Axioms/BOO002-0.ax',multiplicative_identity1) ).
tff(58,plain,
! [X: $i] : product(multiplicative_identity,X,X),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [X: $i] : product(multiplicative_identity,X,X),
inference(skolemize,[status(sab)],[58]) ).
tff(60,plain,
! [X: $i] : product(multiplicative_identity,X,X),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
( ~ ! [X: $i] : product(multiplicative_identity,X,X)
| product(multiplicative_identity,x,x) ),
inference(quant_inst,[status(thm)],]) ).
tff(62,plain,
product(multiplicative_identity,x,x),
inference(unit_resolution,[status(thm)],[61,60]) ).
tff(63,plain,
( ~ sum(x,x,x)
<=> ~ sum(x,x,x) ),
inference(rewrite,[status(thm)],]) ).
tff(64,axiom,
~ sum(x,x,x),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_both_equalities) ).
tff(65,plain,
~ sum(x,x,x),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,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(67,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)],[66]) ).
tff(68,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(69,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(70,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)],[69]) ).
tff(71,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/sandbox2/benchmark/Axioms/BOO002-0.ax',distributivity1) ).
tff(72,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)],[71,70]) ).
tff(73,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)],[72,68]) ).
tff(74,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)],[73]) ).
tff(75,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)],[74,67]) ).
tff(76,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(x,x,x)
| ~ product(multiplicative_identity,x,x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,add(x,x),x) )
<=> ( ~ ! [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(x,x,x)
| ~ product(multiplicative_identity,x,x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,add(x,x),x) ) ),
inference(rewrite,[status(thm)],]) ).
tff(77,plain,
( ( sum(x,x,x)
| ~ product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,x,x)
| ~ product(multiplicative_identity,x,x) )
<=> ( sum(x,x,x)
| ~ product(multiplicative_identity,x,x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,add(x,x),x) ) ),
inference(rewrite,[status(thm)],]) ).
tff(78,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(x,x,x)
| ~ product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,x,x)
| ~ product(multiplicative_identity,x,x) )
<=> ( ~ ! [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(x,x,x)
| ~ product(multiplicative_identity,x,x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,add(x,x),x) ) ),
inference(monotonicity,[status(thm)],[77]) ).
tff(79,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(x,x,x)
| ~ product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,x,x)
| ~ product(multiplicative_identity,x,x) )
<=> ( ~ ! [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(x,x,x)
| ~ product(multiplicative_identity,x,x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,add(x,x),x) ) ),
inference(transitivity,[status(thm)],[78,76]) ).
tff(80,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(x,x,x)
| ~ product(multiplicative_identity,add(x,x),x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,x,x)
| ~ product(multiplicative_identity,x,x) ),
inference(quant_inst,[status(thm)],]) ).
tff(81,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(x,x,x)
| ~ product(multiplicative_identity,x,x)
| ~ sum(x,x,add(x,x))
| ~ product(multiplicative_identity,add(x,x),x) ),
inference(modus_ponens,[status(thm)],[80,79]) ).
tff(82,plain,
$false,
inference(unit_resolution,[status(thm)],[81,75,65,36,62,53]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : BOO004-1 : TPTP v8.1.0. Released v1.0.0.
% 0.08/0.14 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.14/0.35 % Computer : n003.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 30 02:31:07 EDT 2022
% 0.14/0.35 % CPUTime :
% 0.14/0.36 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.14/0.36 Usage: tptp [options] [-file:]file
% 0.14/0.36 -h, -? prints this message.
% 0.14/0.36 -smt2 print SMT-LIB2 benchmark.
% 0.14/0.36 -m, -model generate model.
% 0.14/0.36 -p, -proof generate proof.
% 0.14/0.36 -c, -core generate unsat core of named formulas.
% 0.14/0.36 -st, -statistics display statistics.
% 0.14/0.36 -t:timeout set timeout (in second).
% 0.14/0.36 -smt2status display status in smt2 format instead of SZS.
% 0.14/0.36 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.14/0.36 -<param>:<value> configuration parameter and value.
% 0.14/0.36 -o:<output-file> file to place output in.
% 0.44/0.54 % SZS status Unsatisfiable
% 0.44/0.54 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------