TSTP Solution File: DAT088_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT088_1 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 14:36:41 EDT 2022
% Result : Theorem 0.18s 0.39s
% Output : Proof 0.18s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 46
% Syntax : Number of formulae : 115 ( 40 unt; 5 typ; 0 def)
% Number of atoms : 298 ( 269 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 270 ( 97 ~; 110 |; 0 &)
% ( 52 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of FOOLs : 15 ( 15 fml; 0 var)
% Number arithmetic : 859 ( 8 atm; 238 fun; 495 num; 118 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 6 ( 4 >; 2 *; 0 +; 0 <<)
% Number of predicates : 10 ( 5 usr; 3 prp; 0-3 aty)
% Number of functors : 10 ( 5 usr; 4 con; 0-2 aty)
% Number of variables : 210 ( 188 !; 0 ?; 210 :)
% Comments :
%------------------------------------------------------------------------------
tff(count_type,type,
count: ( $int * list ) > $int ).
tff(tail_type,type,
tail: list > list ).
tff(cons_type,type,
cons: ( $int * list ) > list ).
tff(nil_type,type,
nil: list ).
tff(head_type,type,
head: list > $int ).
tff(1,plain,
^ [K: $int,L: list] :
refl(
( ( tail(cons(K,L)) = L )
<=> ( tail(cons(K,L)) = L ) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [K: $int,L: list] : ( tail(cons(K,L)) = L )
<=> ! [K: $int,L: list] : ( tail(cons(K,L)) = L ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [K: $int,L: list] : ( tail(cons(K,L)) = L )
<=> ! [K: $int,L: list] : ( tail(cons(K,L)) = L ) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [K: $int,L: list] : ( tail(cons(K,L)) = L ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l2) ).
tff(5,plain,
! [K: $int,L: list] : ( tail(cons(K,L)) = L ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [K: $int,L: list] : ( tail(cons(K,L)) = L ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [K: $int,L: list] : ( tail(cons(K,L)) = L ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [K: $int,L: list] : ( tail(cons(K,L)) = L )
| ( tail(cons(0,nil)) = nil ) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
tail(cons(0,nil)) = nil,
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [K: $int,L: list] :
refl(
( ( cons(K,L) != nil )
<=> ( cons(K,L) != nil ) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [K: $int,L: list] : ( cons(K,L) != nil )
<=> ! [K: $int,L: list] : ( cons(K,L) != nil ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [K: $int,L: list] : ( cons(K,L) != nil )
<=> ! [K: $int,L: list] : ( cons(K,L) != nil ) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [K: $int,L: list] : ( cons(K,L) != nil ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l4) ).
tff(14,plain,
! [K: $int,L: list] : ( cons(K,L) != nil ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [K: $int,L: list] : ( cons(K,L) != nil ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [K: $int,L: list] : ( cons(K,L) != nil ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [K: $int,L: list] : ( cons(K,L) != nil )
| ( cons(0,nil) != nil ) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
cons(0,nil) != nil,
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [L: list] :
refl(
( ( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
<=> ( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
<=> ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
<=> ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(22,axiom,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',l3) ).
tff(23,plain,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ( ~ ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
| ( cons(0,nil) = nil )
| ( cons(0,nil) = cons(head(cons(0,nil)),tail(cons(0,nil))) ) )
<=> ( ~ ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
| ( cons(0,nil) = nil )
| ( cons(0,nil) = cons(head(cons(0,nil)),tail(cons(0,nil))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(27,plain,
( ~ ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
| ( cons(0,nil) = nil )
| ( cons(0,nil) = cons(head(cons(0,nil)),tail(cons(0,nil))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(28,plain,
( ~ ! [L: list] :
( ( L = nil )
| ( L = cons(head(L),tail(L)) ) )
| ( cons(0,nil) = nil )
| ( cons(0,nil) = cons(head(cons(0,nil)),tail(cons(0,nil))) ) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
cons(0,nil) = cons(head(cons(0,nil)),tail(cons(0,nil))),
inference(unit_resolution,[status(thm)],[28,25,18]) ).
tff(30,plain,
cons(head(cons(0,nil)),tail(cons(0,nil))) = cons(0,nil),
inference(symmetry,[status(thm)],[29]) ).
tff(31,plain,
tail(cons(head(cons(0,nil)),tail(cons(0,nil)))) = tail(cons(0,nil)),
inference(monotonicity,[status(thm)],[30]) ).
tff(32,plain,
tail(cons(head(cons(0,nil)),tail(cons(0,nil)))) = nil,
inference(transitivity,[status(thm)],[31,9]) ).
tff(33,plain,
count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))) = count(0,nil),
inference(monotonicity,[status(thm)],[32]) ).
tff(34,plain,
count(0,nil) = count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))),
inference(symmetry,[status(thm)],[33]) ).
tff(35,plain,
( ( count(0,nil) != count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))) )
| $greatereq($sum(count(0,nil),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(36,plain,
$greatereq($sum(count(0,nil),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))),0),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
^ [K: $int,H: $int,T: list] :
refl(
( ( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
<=> ( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ) )),
inference(bind,[status(th)],]) ).
tff(38,plain,
( ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ) ),
inference(quant_intro,[status(thm)],[37]) ).
tff(39,plain,
^ [K: $int,H: $int,T: list] :
rewrite(
( ( ( $sum(K,$product(-1,H)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
<=> ( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ) )),
inference(bind,[status(th)],]) ).
tff(40,plain,
( ! [K: $int,H: $int,T: list] :
( ( $sum(K,$product(-1,H)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
^ [K: $int,H: $int,T: list] :
rewrite(
( ( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) )
<=> ( ( $sum(K,$product(-1,H)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ) )),
inference(bind,[status(th)],]) ).
tff(42,plain,
( ! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(K,$product(-1,H)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,plain,
( ! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(44,plain,
( ! [K: $int,H: $int,T: list,N: $int] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) ),
inference(elim_unused_vars,[status(thm)],]) ).
tff(45,plain,
^ [K: $int,H: $int,T: list,N: $int] :
trans(
monotonicity(
rewrite(
( ( count(K,cons(H,T)) = $sum(count(K,T),1) )
<=> ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) )),
( ( ( K = H )
=> ( count(K,cons(H,T)) = $sum(count(K,T),1) ) )
<=> ( ( K = H )
=> ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) )),
rewrite(
( ( ( K = H )
=> ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) )
<=> ( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) )),
( ( ( K = H )
=> ( count(K,cons(H,T)) = $sum(count(K,T),1) ) )
<=> ( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) )),
inference(bind,[status(th)],]) ).
tff(46,plain,
( ! [K: $int,H: $int,T: list,N: $int] :
( ( K = H )
=> ( count(K,cons(H,T)) = $sum(count(K,T),1) ) )
<=> ! [K: $int,H: $int,T: list,N: $int] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) ),
inference(quant_intro,[status(thm)],[45]) ).
tff(47,plain,
( ! [K: $int,H: $int,T: list,N: $int] :
( ( K = H )
=> ( count(K,cons(H,T)) = $sum(count(K,T),1) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ) ),
inference(transitivity,[status(thm)],[46,44]) ).
tff(48,axiom,
! [K: $int,H: $int,T: list,N: $int] :
( ( K = H )
=> ( count(K,cons(H,T)) = $sum(count(K,T),1) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_4) ).
tff(49,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[49,43]) ).
tff(51,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(K,$product(-1,H)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ),
inference(modus_ponens,[status(thm)],[50,42]) ).
tff(52,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ),
inference(modus_ponens,[status(thm)],[51,40]) ).
tff(53,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ),
inference(skolemize,[status(sab)],[52]) ).
tff(54,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) ),
inference(modus_ponens,[status(thm)],[53,38]) ).
tff(55,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(56,plain,
( ( $false
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(57,plain,
( ( $sum($product(-1,count(0,nil)),count(0,cons(0,nil))) = 1 )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(58,plain,
$sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = $sum($product(-1,count(0,nil)),count(0,cons(0,nil))),
inference(rewrite,[status(thm)],]) ).
tff(59,plain,
( ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 )
<=> ( $sum($product(-1,count(0,nil)),count(0,cons(0,nil))) = 1 ) ),
inference(monotonicity,[status(thm)],[58]) ).
tff(60,plain,
( ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[59,57]) ).
tff(61,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(62,plain,
( ( 0 = 0 )
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(63,plain,
$sum(0,0) = 0,
inference(rewrite,[status(thm)],]) ).
tff(64,plain,
$product(-1,0) = 0,
inference(rewrite,[status(thm)],]) ).
tff(65,plain,
$sum(0,$product(-1,0)) = $sum(0,0),
inference(monotonicity,[status(thm)],[64]) ).
tff(66,plain,
$sum(0,$product(-1,0)) = 0,
inference(transitivity,[status(thm)],[65,63]) ).
tff(67,plain,
( ( $sum(0,$product(-1,0)) = 0 )
<=> ( 0 = 0 ) ),
inference(monotonicity,[status(thm)],[66]) ).
tff(68,plain,
( ( $sum(0,$product(-1,0)) = 0 )
<=> $true ),
inference(transitivity,[status(thm)],[67,62]) ).
tff(69,plain,
( ( $sum(0,$product(-1,0)) != 0 )
<=> ~ $true ),
inference(monotonicity,[status(thm)],[68]) ).
tff(70,plain,
( ( $sum(0,$product(-1,0)) != 0 )
<=> $false ),
inference(transitivity,[status(thm)],[69,61]) ).
tff(71,plain,
( ( ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 ) )
<=> ( $false
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[70,60]) ).
tff(72,plain,
( ( ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 ) )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[71,56]) ).
tff(73,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[72]) ).
tff(74,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ) ),
inference(transitivity,[status(thm)],[73,55]) ).
tff(75,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(0,$product(-1,0)) != 0 )
| ( $sum(count(0,cons(0,nil)),$product(-1,count(0,nil))) = 1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(76,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) != 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 1 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
$sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1,
inference(unit_resolution,[status(thm)],[76,54]) ).
tff(78,plain,
( ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) != -1 )
| $lesseq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(79,plain,
$lesseq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1),
inference(unit_resolution,[status(thm)],[78,77]) ).
tff(80,plain,
count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) = count(0,cons(0,nil)),
inference(monotonicity,[status(thm)],[30]) ).
tff(81,plain,
count(0,cons(0,nil)) = count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),
inference(symmetry,[status(thm)],[80]) ).
tff(82,plain,
( ( count(0,cons(0,nil)) != count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))) )
| $lesseq($sum(count(0,cons(0,nil)),$product(-1,count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(83,plain,
$lesseq($sum(count(0,cons(0,nil)),$product(-1,count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))))),0),
inference(unit_resolution,[status(thm)],[82,81]) ).
tff(84,plain,
( ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil )
<=> ( cons(0,nil) = nil ) ),
inference(monotonicity,[status(thm)],[30]) ).
tff(85,plain,
( ( cons(0,nil) = nil )
<=> ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil ) ),
inference(symmetry,[status(thm)],[84]) ).
tff(86,plain,
( ( cons(0,nil) != nil )
<=> ( cons(head(cons(0,nil)),tail(cons(0,nil))) != nil ) ),
inference(monotonicity,[status(thm)],[85]) ).
tff(87,plain,
cons(head(cons(0,nil)),tail(cons(0,nil))) != nil,
inference(modus_ponens,[status(thm)],[18,86]) ).
tff(88,plain,
^ [N: $int,L: list] :
refl(
( ( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) )
<=> ( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(89,plain,
( ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) )
<=> ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[88]) ).
tff(90,plain,
^ [N: $int,L: list] :
rewrite(
( ( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) )
<=> ( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(91,plain,
( ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) )
<=> ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[90]) ).
tff(92,plain,
( ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) )
<=> ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(93,plain,
( ~ ~ ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) )
<=> ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(94,plain,
( ~ ! [N: $int,L: list] :
( ( L != nil )
=> ( count(N,L) = count(N,tail(L)) ) )
<=> ~ ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(95,plain,
( ~ ~ ! [N: $int,L: list] :
( ( L != nil )
=> ( count(N,L) = count(N,tail(L)) ) )
<=> ~ ~ ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ) ),
inference(monotonicity,[status(thm)],[94]) ).
tff(96,plain,
( ~ ~ ! [N: $int,L: list] :
( ( L != nil )
=> ( count(N,L) = count(N,tail(L)) ) )
<=> ! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ) ),
inference(transitivity,[status(thm)],[95,93]) ).
tff(97,axiom,
~ ~ ! [N: $int,L: list] :
( ( L != nil )
=> ( count(N,L) = count(N,tail(L)) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',c) ).
tff(98,plain,
! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ),
inference(modus_ponens,[status(thm)],[97,96]) ).
tff(99,plain,
! [N: $int,L: list] :
( ( L = nil )
| ( count(N,L) = count(N,tail(L)) ) ),
inference(modus_ponens,[status(thm)],[98,92]) ).
tff(100,plain,
! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[99,91]) ).
tff(101,plain,
! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ),
inference(skolemize,[status(sab)],[100]) ).
tff(102,plain,
! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[101,89]) ).
tff(103,plain,
( ( ~ ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) )
| ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) = 0 ) )
<=> ( ~ ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) )
| ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(104,plain,
( ~ ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) )
| ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(105,plain,
( ~ ! [N: $int,L: list] :
( ( L = nil )
| ( $sum(count(N,L),$product(-1,count(N,tail(L)))) = 0 ) )
| ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) = 0 ) ),
inference(modus_ponens,[status(thm)],[104,103]) ).
tff(106,plain,
( ( cons(head(cons(0,nil)),tail(cons(0,nil))) = nil )
| ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) = 0 ) ),
inference(unit_resolution,[status(thm)],[105,102]) ).
tff(107,plain,
$sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) = 0,
inference(unit_resolution,[status(thm)],[106,87]) ).
tff(108,plain,
( ( $sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))) != 0 )
| $lesseq($sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(109,plain,
$lesseq($sum(count(0,cons(head(cons(0,nil)),tail(cons(0,nil)))),$product(-1,count(0,tail(cons(head(cons(0,nil)),tail(cons(0,nil))))))),0),
inference(unit_resolution,[status(thm)],[108,107]) ).
tff(110,plain,
$false,
inference(theory_lemma,[status(thm)],[109,83,79,36]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : DAT088_1 : TPTP v8.1.0. Released v6.1.0.
% 0.12/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n009.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 : Wed Aug 31 01:47:20 EDT 2022
% 0.18/0.33 % CPUTime :
% 0.18/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.18/0.34 Usage: tptp [options] [-file:]file
% 0.18/0.34 -h, -? prints this message.
% 0.18/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.18/0.34 -m, -model generate model.
% 0.18/0.34 -p, -proof generate proof.
% 0.18/0.34 -c, -core generate unsat core of named formulas.
% 0.18/0.34 -st, -statistics display statistics.
% 0.18/0.34 -t:timeout set timeout (in second).
% 0.18/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.18/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.18/0.34 -<param>:<value> configuration parameter and value.
% 0.18/0.34 -o:<output-file> file to place output in.
% 0.18/0.39 % SZS status Theorem
% 0.18/0.39 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------