TSTP Solution File: DAT085_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT085_1 : TPTP v8.1.0. Released v6.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n025.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 14:36:40 EDT 2022
% Result : Theorem 0.11s 0.39s
% Output : Proof 0.17s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 96
% Syntax : Number of formulae : 235 ( 97 unt; 6 typ; 0 def)
% Number of atoms : 553 ( 467 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 453 ( 159 ~; 181 |; 0 &)
% ( 98 <=>; 15 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of FOOLs : 30 ( 30 fml; 0 var)
% Number arithmetic : 1980 ( 48 atm; 664 fun;1066 num; 202 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 8 ( 5 >; 3 *; 0 +; 0 <<)
% Number of predicates : 10 ( 5 usr; 3 prp; 0-3 aty)
% Number of functors : 11 ( 6 usr; 4 con; 0-2 aty)
% Number of variables : 386 ( 342 !; 0 ?; 386 :)
% Comments :
%------------------------------------------------------------------------------
tff(count_type,type,
count: ( $int * list ) > $int ).
tff(cons_type,type,
cons: ( $int * list ) > list ).
tff(nil_type,type,
nil: list ).
tff(head_type,type,
head: list > $int ).
tff(length_type,type,
length: list > $int ).
tff(append_type,type,
append: ( list * list ) > list ).
tff(1,plain,
( ( 0 = 1 )
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(2,plain,
^ [H: $int,T: list] :
refl(
( ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
<=> ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ) )),
inference(bind,[status(th)],]) ).
tff(3,plain,
( ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
<=> ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ) ),
inference(quant_intro,[status(thm)],[2]) ).
tff(4,plain,
^ [H: $int,T: list] :
rewrite(
( ( length(cons(H,T)) = $sum(1,length(T)) )
<=> ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ) )),
inference(bind,[status(th)],]) ).
tff(5,plain,
( ! [H: $int,T: list] : ( length(cons(H,T)) = $sum(1,length(T)) )
<=> ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [H: $int,T: list] : ( length(cons(H,T)) = $sum(1,length(T)) )
<=> ! [H: $int,T: list] : ( length(cons(H,T)) = $sum(1,length(T)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(7,axiom,
! [H: $int,T: list] : ( length(cons(H,T)) = $sum(1,length(T)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l_1) ).
tff(8,plain,
! [H: $int,T: list] : ( length(cons(H,T)) = $sum(1,length(T)) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ),
inference(modus_ponens,[status(thm)],[8,5]) ).
tff(10,plain,
! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ),
inference(skolemize,[status(sab)],[9]) ).
tff(11,plain,
! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 ),
inference(modus_ponens,[status(thm)],[10,3]) ).
tff(12,plain,
( ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) )
<=> ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(13,plain,
( ( $sum($product(-1,length(nil)),length(cons(0,nil))) = 1 )
<=> ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(14,plain,
$sum(length(cons(0,nil)),$product(-1,length(nil))) = $sum($product(-1,length(nil)),length(cons(0,nil))),
inference(rewrite,[status(thm)],]) ).
tff(15,plain,
( ( $sum(length(cons(0,nil)),$product(-1,length(nil))) = 1 )
<=> ( $sum($product(-1,length(nil)),length(cons(0,nil))) = 1 ) ),
inference(monotonicity,[status(thm)],[14]) ).
tff(16,plain,
( ( $sum(length(cons(0,nil)),$product(-1,length(nil))) = 1 )
<=> ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[15,13]) ).
tff(17,plain,
( ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(0,nil)),$product(-1,length(nil))) = 1 ) )
<=> ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[16]) ).
tff(18,plain,
( ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(0,nil)),$product(-1,length(nil))) = 1 ) )
<=> ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) ) ),
inference(transitivity,[status(thm)],[17,12]) ).
tff(19,plain,
( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(0,nil)),$product(-1,length(nil))) = 1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(20,plain,
( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(0,nil)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
$sum(length(nil),$product(-1,length(cons(0,nil)))) = -1,
inference(unit_resolution,[status(thm)],[20,11]) ).
tff(22,plain,
( ( $sum(length(nil),$product(-1,length(cons(0,nil)))) != -1 )
| $lesseq($sum(length(nil),$product(-1,length(cons(0,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(23,plain,
$lesseq($sum(length(nil),$product(-1,length(cons(0,nil)))),-1),
inference(unit_resolution,[status(thm)],[22,21]) ).
tff(24,plain,
( ( $sum(length(nil),$product(-1,length(cons(0,nil)))) != -1 )
| $greatereq($sum(length(nil),$product(-1,length(cons(0,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(25,plain,
$greatereq($sum(length(nil),$product(-1,length(cons(0,nil)))),-1),
inference(unit_resolution,[status(thm)],[24,21]) ).
tff(26,plain,
( ( length(nil) = 0 )
<=> ( length(nil) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(27,axiom,
length(nil) = 0,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l) ).
tff(28,plain,
length(nil) = 0,
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
( ( length(nil) != 0 )
| $lesseq(length(nil),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(30,plain,
$lesseq(length(nil),0),
inference(unit_resolution,[status(thm)],[29,28]) ).
tff(31,plain,
( ( length(nil) != 0 )
| $greatereq(length(nil),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(32,plain,
$greatereq(length(nil),0),
inference(unit_resolution,[status(thm)],[31,28]) ).
tff(33,plain,
^ [K: $int,L: list] :
refl(
( ( $sum(K,$product(-1,head(cons(K,L)))) = 0 )
<=> ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(34,plain,
( ! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 )
<=> ! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,plain,
^ [K: $int,L: list] :
trans(
monotonicity(rewrite($sum(head(cons(K,L)),$product(-1,K)) = $sum($product(-1,K),head(cons(K,L)))),
( ( $sum(head(cons(K,L)),$product(-1,K)) = 0 )
<=> ( $sum($product(-1,K),head(cons(K,L))) = 0 ) )),
rewrite(
( ( $sum($product(-1,K),head(cons(K,L))) = 0 )
<=> ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ) )),
( ( $sum(head(cons(K,L)),$product(-1,K)) = 0 )
<=> ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(36,plain,
( ! [K: $int,L: list] : ( $sum(head(cons(K,L)),$product(-1,K)) = 0 )
<=> ! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ) ),
inference(quant_intro,[status(thm)],[35]) ).
tff(37,plain,
^ [K: $int,L: list] :
rewrite(
( ( head(cons(K,L)) = K )
<=> ( $sum(head(cons(K,L)),$product(-1,K)) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(38,plain,
( ! [K: $int,L: list] : ( head(cons(K,L)) = K )
<=> ! [K: $int,L: list] : ( $sum(head(cons(K,L)),$product(-1,K)) = 0 ) ),
inference(quant_intro,[status(thm)],[37]) ).
tff(39,plain,
( ! [K: $int,L: list] : ( head(cons(K,L)) = K )
<=> ! [K: $int,L: list] : ( head(cons(K,L)) = K ) ),
inference(rewrite,[status(thm)],]) ).
tff(40,axiom,
! [K: $int,L: list] : ( head(cons(K,L)) = K ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l1) ).
tff(41,plain,
! [K: $int,L: list] : ( head(cons(K,L)) = K ),
inference(modus_ponens,[status(thm)],[40,39]) ).
tff(42,plain,
! [K: $int,L: list] : ( $sum(head(cons(K,L)),$product(-1,K)) = 0 ),
inference(modus_ponens,[status(thm)],[41,38]) ).
tff(43,plain,
! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ),
inference(modus_ponens,[status(thm)],[42,36]) ).
tff(44,plain,
! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 ),
inference(modus_ponens,[status(thm)],[44,34]) ).
tff(46,plain,
( ~ ! [K: $int,L: list] : ( $sum(K,$product(-1,head(cons(K,L)))) = 0 )
| ( $sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(47,plain,
$sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))) = 0,
inference(unit_resolution,[status(thm)],[46,45]) ).
tff(48,plain,
( ( $sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))) != 0 )
| $lesseq($sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(49,plain,
$lesseq($sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))),0),
inference(unit_resolution,[status(thm)],[48,47]) ).
tff(50,plain,
( ( $sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))) != 0 )
| $greatereq($sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(51,plain,
$greatereq($sum(length(cons(0,nil)),$product(-1,head(cons(length(cons(0,nil)),nil)))),0),
inference(unit_resolution,[status(thm)],[50,47]) ).
tff(52,plain,
head(cons(length(cons(0,nil)),nil)) = 1,
inference(theory_lemma,[status(thm)],[51,49,32,30,25,23]) ).
tff(53,plain,
length(cons(0,nil)) = head(cons(length(cons(0,nil)),nil)),
inference(theory_lemma,[status(thm)],[51,49,32,30,25,23,32,30,25,23]) ).
tff(54,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(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 ) )
<=> ! [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)],[54]) ).
tff(56,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(57,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)],[56]) ).
tff(58,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(59,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)],[58]) ).
tff(60,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(61,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(62,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(63,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)],[62]) ).
tff(64,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)],[63,61]) ).
tff(65,axiom,
! [K: $int,H: $int,T: list,N: $int] :
( ( K = H )
=> ( count(K,cons(H,T)) = $sum(count(K,T),1) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_4) ).
tff(66,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,plain,
! [K: $int,H: $int,T: list] :
( ( K != H )
| ( count(K,cons(H,T)) = $sum(1,count(K,T)) ) ),
inference(modus_ponens,[status(thm)],[66,60]) ).
tff(68,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)],[67,59]) ).
tff(69,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)],[68,57]) ).
tff(70,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)],[69]) ).
tff(71,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)],[70,55]) ).
tff(72,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(73,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(74,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(75,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(76,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)],[75]) ).
tff(77,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)],[76,74]) ).
tff(78,plain,
( ~ $true
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(79,plain,
( ( 0 = 0 )
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(80,plain,
$sum(0,0) = 0,
inference(rewrite,[status(thm)],]) ).
tff(81,plain,
$product(-1,0) = 0,
inference(rewrite,[status(thm)],]) ).
tff(82,plain,
$sum(0,$product(-1,0)) = $sum(0,0),
inference(monotonicity,[status(thm)],[81]) ).
tff(83,plain,
$sum(0,$product(-1,0)) = 0,
inference(transitivity,[status(thm)],[82,80]) ).
tff(84,plain,
( ( $sum(0,$product(-1,0)) = 0 )
<=> ( 0 = 0 ) ),
inference(monotonicity,[status(thm)],[83]) ).
tff(85,plain,
( ( $sum(0,$product(-1,0)) = 0 )
<=> $true ),
inference(transitivity,[status(thm)],[84,79]) ).
tff(86,plain,
( ( $sum(0,$product(-1,0)) != 0 )
<=> ~ $true ),
inference(monotonicity,[status(thm)],[85]) ).
tff(87,plain,
( ( $sum(0,$product(-1,0)) != 0 )
<=> $false ),
inference(transitivity,[status(thm)],[86,78]) ).
tff(88,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)],[87,77]) ).
tff(89,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)],[88,73]) ).
tff(90,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)],[89]) ).
tff(91,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)],[90,72]) ).
tff(92,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(93,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)],[92,91]) ).
tff(94,plain,
$sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) = -1,
inference(unit_resolution,[status(thm)],[93,71]) ).
tff(95,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(96,plain,
$lesseq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1),
inference(unit_resolution,[status(thm)],[95,94]) ).
tff(97,plain,
( ( $sum(count(0,nil),$product(-1,count(0,cons(0,nil)))) != -1 )
| $greatereq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(98,plain,
$greatereq($sum(count(0,nil),$product(-1,count(0,cons(0,nil)))),-1),
inference(unit_resolution,[status(thm)],[97,94]) ).
tff(99,plain,
^ [K: $int] :
refl(
( ( count(K,nil) = 0 )
<=> ( count(K,nil) = 0 ) )),
inference(bind,[status(th)],]) ).
tff(100,plain,
( ! [K: $int] : ( count(K,nil) = 0 )
<=> ! [K: $int] : ( count(K,nil) = 0 ) ),
inference(quant_intro,[status(thm)],[99]) ).
tff(101,plain,
( ! [K: $int] : ( count(K,nil) = 0 )
<=> ! [K: $int] : ( count(K,nil) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(102,axiom,
! [K: $int] : ( count(K,nil) = 0 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a) ).
tff(103,plain,
! [K: $int] : ( count(K,nil) = 0 ),
inference(modus_ponens,[status(thm)],[102,101]) ).
tff(104,plain,
! [K: $int] : ( count(K,nil) = 0 ),
inference(skolemize,[status(sab)],[103]) ).
tff(105,plain,
! [K: $int] : ( count(K,nil) = 0 ),
inference(modus_ponens,[status(thm)],[104,100]) ).
tff(106,plain,
( ~ ! [K: $int] : ( count(K,nil) = 0 )
| ( count(0,nil) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(107,plain,
count(0,nil) = 0,
inference(unit_resolution,[status(thm)],[106,105]) ).
tff(108,plain,
( ( count(0,nil) != 0 )
| $lesseq(count(0,nil),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(109,plain,
$lesseq(count(0,nil),0),
inference(unit_resolution,[status(thm)],[108,107]) ).
tff(110,plain,
( ( count(0,nil) != 0 )
| $greatereq(count(0,nil),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(111,plain,
$greatereq(count(0,nil),0),
inference(unit_resolution,[status(thm)],[110,107]) ).
tff(112,plain,
length(cons(0,nil)) = count(0,cons(0,nil)),
inference(theory_lemma,[status(thm)],[111,109,98,96,32,30,25,23]) ).
tff(113,plain,
count(0,cons(0,nil)) = length(cons(0,nil)),
inference(symmetry,[status(thm)],[112]) ).
tff(114,plain,
count(0,cons(0,nil)) = 1,
inference(transitivity,[status(thm)],[113,53,52]) ).
tff(115,plain,
^ [L: list] :
refl(
( ( append(nil,L) = L )
<=> ( append(nil,L) = L ) )),
inference(bind,[status(th)],]) ).
tff(116,plain,
( ! [L: list] : ( append(nil,L) = L )
<=> ! [L: list] : ( append(nil,L) = L ) ),
inference(quant_intro,[status(thm)],[115]) ).
tff(117,plain,
( ! [L: list] : ( append(nil,L) = L )
<=> ! [L: list] : ( append(nil,L) = L ) ),
inference(rewrite,[status(thm)],]) ).
tff(118,axiom,
! [L: list] : ( append(nil,L) = L ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',l_6) ).
tff(119,plain,
! [L: list] : ( append(nil,L) = L ),
inference(modus_ponens,[status(thm)],[118,117]) ).
tff(120,plain,
! [L: list] : ( append(nil,L) = L ),
inference(skolemize,[status(sab)],[119]) ).
tff(121,plain,
! [L: list] : ( append(nil,L) = L ),
inference(modus_ponens,[status(thm)],[120,116]) ).
tff(122,plain,
( ~ ! [L: list] : ( append(nil,L) = L )
| ( append(nil,nil) = nil ) ),
inference(quant_inst,[status(thm)],]) ).
tff(123,plain,
append(nil,nil) = nil,
inference(unit_resolution,[status(thm)],[122,121]) ).
tff(124,plain,
nil = append(nil,nil),
inference(symmetry,[status(thm)],[123]) ).
tff(125,plain,
cons(0,nil) = cons(0,append(nil,nil)),
inference(monotonicity,[status(thm)],[124]) ).
tff(126,plain,
cons(0,append(nil,nil)) = cons(0,nil),
inference(symmetry,[status(thm)],[125]) ).
tff(127,plain,
( ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) )
<=> ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(128,plain,
( ( $sum($product(-1,length(nil)),length(cons(-1,nil))) = 1 )
<=> ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) ),
inference(rewrite,[status(thm)],]) ).
tff(129,plain,
$sum(length(cons(-1,nil)),$product(-1,length(nil))) = $sum($product(-1,length(nil)),length(cons(-1,nil))),
inference(rewrite,[status(thm)],]) ).
tff(130,plain,
( ( $sum(length(cons(-1,nil)),$product(-1,length(nil))) = 1 )
<=> ( $sum($product(-1,length(nil)),length(cons(-1,nil))) = 1 ) ),
inference(monotonicity,[status(thm)],[129]) ).
tff(131,plain,
( ( $sum(length(cons(-1,nil)),$product(-1,length(nil))) = 1 )
<=> ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) ),
inference(transitivity,[status(thm)],[130,128]) ).
tff(132,plain,
( ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(-1,nil)),$product(-1,length(nil))) = 1 ) )
<=> ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) ) ),
inference(monotonicity,[status(thm)],[131]) ).
tff(133,plain,
( ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(-1,nil)),$product(-1,length(nil))) = 1 ) )
<=> ( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) ) ),
inference(transitivity,[status(thm)],[132,127]) ).
tff(134,plain,
( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(-1,nil)),$product(-1,length(nil))) = 1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(135,plain,
( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[134,133]) ).
tff(136,plain,
$sum(length(nil),$product(-1,length(cons(-1,nil)))) = -1,
inference(unit_resolution,[status(thm)],[135,11]) ).
tff(137,plain,
( ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) != -1 )
| $lesseq($sum(length(nil),$product(-1,length(cons(-1,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(138,plain,
$lesseq($sum(length(nil),$product(-1,length(cons(-1,nil)))),-1),
inference(unit_resolution,[status(thm)],[137,136]) ).
tff(139,plain,
length(append(nil,nil)) = length(nil),
inference(monotonicity,[status(thm)],[123]) ).
tff(140,plain,
length(nil) = length(append(nil,nil)),
inference(symmetry,[status(thm)],[139]) ).
tff(141,plain,
( ( length(nil) != length(append(nil,nil)) )
| $greatereq($sum(length(nil),$product(-1,length(append(nil,nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(142,plain,
$greatereq($sum(length(nil),$product(-1,length(append(nil,nil)))),0),
inference(unit_resolution,[status(thm)],[141,140]) ).
tff(143,plain,
( ~ ! [H: $int,T: list] : ( $sum(length(cons(H,T)),$product(-1,length(T))) = 1 )
| ( $sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))) = 1 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(144,plain,
$sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))) = 1,
inference(unit_resolution,[status(thm)],[143,11]) ).
tff(145,plain,
( ( $sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))) != 1 )
| $lesseq($sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(146,plain,
$lesseq($sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))),1),
inference(unit_resolution,[status(thm)],[145,144]) ).
tff(147,plain,
( $greatereq($sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))),0)
| ~ $lesseq($sum(length(nil),$product(-1,length(cons(-1,nil)))),-1)
| ~ $greatereq($sum(length(nil),$product(-1,length(append(nil,nil)))),0)
| ~ $lesseq($sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(148,plain,
$greatereq($sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))),0),
inference(unit_resolution,[status(thm)],[147,146,142,138]) ).
tff(149,plain,
( ( $sum(length(nil),$product(-1,length(cons(-1,nil)))) != -1 )
| $greatereq($sum(length(nil),$product(-1,length(cons(-1,nil)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(150,plain,
$greatereq($sum(length(nil),$product(-1,length(cons(-1,nil)))),-1),
inference(unit_resolution,[status(thm)],[149,136]) ).
tff(151,plain,
( ( length(nil) != length(append(nil,nil)) )
| $lesseq($sum(length(nil),$product(-1,length(append(nil,nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(152,plain,
$lesseq($sum(length(nil),$product(-1,length(append(nil,nil)))),0),
inference(unit_resolution,[status(thm)],[151,140]) ).
tff(153,plain,
( ( $sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))) != 1 )
| $greatereq($sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(154,plain,
$greatereq($sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))),1),
inference(unit_resolution,[status(thm)],[153,144]) ).
tff(155,plain,
( $lesseq($sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))),0)
| ~ $greatereq($sum(length(nil),$product(-1,length(cons(-1,nil)))),-1)
| ~ $lesseq($sum(length(nil),$product(-1,length(append(nil,nil)))),0)
| ~ $greatereq($sum(length(cons(0,append(nil,nil))),$product(-1,length(append(nil,nil)))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(156,plain,
$lesseq($sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))),0),
inference(unit_resolution,[status(thm)],[155,154,152,150]) ).
tff(157,plain,
( ( $sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) = 0 )
| ~ $lesseq($sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))),0)
| ~ $greatereq($sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(158,plain,
$sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) = 0,
inference(unit_resolution,[status(thm)],[157,156,148]) ).
tff(159,plain,
^ [L1: list,L2: list] :
refl(
( ( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) )
<=> ( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ) )),
inference(bind,[status(th)],]) ).
tff(160,plain,
( ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) )
<=> ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ) ),
inference(quant_intro,[status(thm)],[159]) ).
tff(161,plain,
^ [L1: list,L2: list] :
rewrite(
( ( ( $sum(length(L1),$product(-1,length(L2))) != 0 )
| ( L1 = L2 ) )
<=> ( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ) )),
inference(bind,[status(th)],]) ).
tff(162,plain,
( ! [L1: list,L2: list] :
( ( $sum(length(L1),$product(-1,length(L2))) != 0 )
| ( L1 = L2 ) )
<=> ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ) ),
inference(quant_intro,[status(thm)],[161]) ).
tff(163,plain,
^ [L1: list,L2: list] :
rewrite(
( ( ( length(L1) != length(L2) )
| ( L1 = L2 ) )
<=> ( ( $sum(length(L1),$product(-1,length(L2))) != 0 )
| ( L1 = L2 ) ) )),
inference(bind,[status(th)],]) ).
tff(164,plain,
( ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) )
<=> ! [L1: list,L2: list] :
( ( $sum(length(L1),$product(-1,length(L2))) != 0 )
| ( L1 = L2 ) ) ),
inference(quant_intro,[status(thm)],[163]) ).
tff(165,plain,
( ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) )
<=> ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(166,plain,
( ~ ~ ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) )
<=> ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(167,plain,
( ~ ! [L1: list,L2: list] :
( ( length(L1) = length(L2) )
=> ( L1 = L2 ) )
<=> ~ ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(168,plain,
( ~ ~ ! [L1: list,L2: list] :
( ( length(L1) = length(L2) )
=> ( L1 = L2 ) )
<=> ~ ~ ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ) ),
inference(monotonicity,[status(thm)],[167]) ).
tff(169,plain,
( ~ ~ ! [L1: list,L2: list] :
( ( length(L1) = length(L2) )
=> ( L1 = L2 ) )
<=> ! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ) ),
inference(transitivity,[status(thm)],[168,166]) ).
tff(170,axiom,
~ ~ ! [L1: list,L2: list] :
( ( length(L1) = length(L2) )
=> ( L1 = L2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',c) ).
tff(171,plain,
! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ),
inference(modus_ponens,[status(thm)],[170,169]) ).
tff(172,plain,
! [L1: list,L2: list] :
( ( length(L1) != length(L2) )
| ( L1 = L2 ) ),
inference(modus_ponens,[status(thm)],[171,165]) ).
tff(173,plain,
! [L1: list,L2: list] :
( ( $sum(length(L1),$product(-1,length(L2))) != 0 )
| ( L1 = L2 ) ),
inference(modus_ponens,[status(thm)],[172,164]) ).
tff(174,plain,
! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ),
inference(modus_ponens,[status(thm)],[173,162]) ).
tff(175,plain,
! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ),
inference(skolemize,[status(sab)],[174]) ).
tff(176,plain,
! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) ),
inference(modus_ponens,[status(thm)],[175,160]) ).
tff(177,plain,
( ( ~ ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) )
| ( $sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) != 0 )
| ( cons(0,append(nil,nil)) = cons(-1,nil) ) )
<=> ( ~ ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) )
| ( $sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) != 0 )
| ( cons(0,append(nil,nil)) = cons(-1,nil) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(178,plain,
( ~ ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) )
| ( $sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) != 0 )
| ( cons(0,append(nil,nil)) = cons(-1,nil) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(179,plain,
( ~ ! [L1: list,L2: list] :
( ( $sum(length(L2),$product(-1,length(L1))) != 0 )
| ( L1 = L2 ) )
| ( $sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) != 0 )
| ( cons(0,append(nil,nil)) = cons(-1,nil) ) ),
inference(modus_ponens,[status(thm)],[178,177]) ).
tff(180,plain,
( ( $sum(length(cons(-1,nil)),$product(-1,length(cons(0,append(nil,nil))))) != 0 )
| ( cons(0,append(nil,nil)) = cons(-1,nil) ) ),
inference(unit_resolution,[status(thm)],[179,176]) ).
tff(181,plain,
cons(0,append(nil,nil)) = cons(-1,nil),
inference(unit_resolution,[status(thm)],[180,158]) ).
tff(182,plain,
cons(-1,nil) = cons(0,append(nil,nil)),
inference(symmetry,[status(thm)],[181]) ).
tff(183,plain,
cons(-1,nil) = cons(0,nil),
inference(transitivity,[status(thm)],[182,126]) ).
tff(184,plain,
count(0,cons(-1,nil)) = count(0,cons(0,nil)),
inference(monotonicity,[status(thm)],[183]) ).
tff(185,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))) = 0 ) )
<=> ( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(186,plain,
( ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[185]) ).
tff(187,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))) = 0 ) )
<=> ( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(188,plain,
( ! [K: $int,H: $int,T: list] :
( ( $sum(K,$product(-1,H)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
<=> ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[187]) ).
tff(189,plain,
^ [K: $int,H: $int,T: list] :
rewrite(
( ( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) )
<=> ( ( $sum(K,$product(-1,H)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(190,plain,
( ! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = 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))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[189]) ).
tff(191,plain,
( ! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(192,plain,
( ! [K: $int,H: $int,T: list,N: $int] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ) ),
inference(elim_unused_vars,[status(thm)],]) ).
tff(193,plain,
^ [K: $int,H: $int,T: list,N: $int] :
rewrite(
( ( ( K != H )
=> ( count(K,cons(H,T)) = count(K,T) ) )
<=> ( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ) )),
inference(bind,[status(th)],]) ).
tff(194,plain,
( ! [K: $int,H: $int,T: list,N: $int] :
( ( K != H )
=> ( count(K,cons(H,T)) = count(K,T) ) )
<=> ! [K: $int,H: $int,T: list,N: $int] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ) ),
inference(quant_intro,[status(thm)],[193]) ).
tff(195,plain,
( ! [K: $int,H: $int,T: list,N: $int] :
( ( K != H )
=> ( count(K,cons(H,T)) = count(K,T) ) )
<=> ! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ) ),
inference(transitivity,[status(thm)],[194,192]) ).
tff(196,axiom,
! [K: $int,H: $int,T: list,N: $int] :
( ( K != H )
=> ( count(K,cons(H,T)) = count(K,T) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_3) ).
tff(197,plain,
! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ),
inference(modus_ponens,[status(thm)],[196,195]) ).
tff(198,plain,
! [K: $int,H: $int,T: list] :
( ( K = H )
| ( count(K,cons(H,T)) = count(K,T) ) ),
inference(modus_ponens,[status(thm)],[197,191]) ).
tff(199,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(K,$product(-1,H)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ),
inference(modus_ponens,[status(thm)],[198,190]) ).
tff(200,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ),
inference(modus_ponens,[status(thm)],[199,188]) ).
tff(201,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ),
inference(skolemize,[status(sab)],[200]) ).
tff(202,plain,
! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) ),
inference(modus_ponens,[status(thm)],[201,186]) ).
tff(203,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(204,plain,
( ( $false
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(205,plain,
( ( $sum($product(-1,count(0,nil)),count(0,cons(-1,nil))) = 0 )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(206,plain,
$sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = $sum($product(-1,count(0,nil)),count(0,cons(-1,nil))),
inference(rewrite,[status(thm)],]) ).
tff(207,plain,
( ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 )
<=> ( $sum($product(-1,count(0,nil)),count(0,cons(-1,nil))) = 0 ) ),
inference(monotonicity,[status(thm)],[206]) ).
tff(208,plain,
( ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ),
inference(transitivity,[status(thm)],[207,205]) ).
tff(209,plain,
( ( -1 = 0 )
<=> $false ),
inference(rewrite,[status(thm)],]) ).
tff(210,plain,
$sum(-1,0) = -1,
inference(rewrite,[status(thm)],]) ).
tff(211,plain,
$sum(-1,$product(-1,0)) = $sum(-1,0),
inference(monotonicity,[status(thm)],[81]) ).
tff(212,plain,
$sum(-1,$product(-1,0)) = -1,
inference(transitivity,[status(thm)],[211,210]) ).
tff(213,plain,
( ( $sum(-1,$product(-1,0)) = 0 )
<=> ( -1 = 0 ) ),
inference(monotonicity,[status(thm)],[212]) ).
tff(214,plain,
( ( $sum(-1,$product(-1,0)) = 0 )
<=> $false ),
inference(transitivity,[status(thm)],[213,209]) ).
tff(215,plain,
( ( ( $sum(-1,$product(-1,0)) = 0 )
| ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 ) )
<=> ( $false
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ) ),
inference(monotonicity,[status(thm)],[214,208]) ).
tff(216,plain,
( ( ( $sum(-1,$product(-1,0)) = 0 )
| ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 ) )
<=> ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ),
inference(transitivity,[status(thm)],[215,204]) ).
tff(217,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(-1,$product(-1,0)) = 0 )
| ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ) ),
inference(monotonicity,[status(thm)],[216]) ).
tff(218,plain,
( ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(-1,$product(-1,0)) = 0 )
| ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 ) )
<=> ( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ) ),
inference(transitivity,[status(thm)],[217,203]) ).
tff(219,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(-1,$product(-1,0)) = 0 )
| ( $sum(count(0,cons(-1,nil)),$product(-1,count(0,nil))) = 0 ) ),
inference(quant_inst,[status(thm)],]) ).
tff(220,plain,
( ~ ! [K: $int,H: $int,T: list] :
( ( $sum(H,$product(-1,K)) = 0 )
| ( $sum(count(K,cons(H,T)),$product(-1,count(K,T))) = 0 ) )
| ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[219,218]) ).
tff(221,plain,
$sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) = 0,
inference(unit_resolution,[status(thm)],[220,202]) ).
tff(222,plain,
( ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) != 0 )
| $lesseq($sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(223,plain,
$lesseq($sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))),0),
inference(unit_resolution,[status(thm)],[222,221]) ).
tff(224,plain,
( ( $sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))) != 0 )
| $greatereq($sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(225,plain,
$greatereq($sum(count(0,nil),$product(-1,count(0,cons(-1,nil)))),0),
inference(unit_resolution,[status(thm)],[224,221]) ).
tff(226,plain,
count(0,cons(-1,nil)) = 0,
inference(theory_lemma,[status(thm)],[111,109,225,223]) ).
tff(227,plain,
0 = count(0,cons(-1,nil)),
inference(symmetry,[status(thm)],[226]) ).
tff(228,plain,
0 = 1,
inference(transitivity,[status(thm)],[227,184,114]) ).
tff(229,plain,
$false,
inference(modus_ponens,[status(thm)],[228,1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : DAT085_1 : TPTP v8.1.0. Released v6.1.0.
% 0.10/0.11 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.11/0.32 % Computer : n025.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 : Wed Aug 31 01:59:30 EDT 2022
% 0.11/0.32 % CPUTime :
% 0.11/0.32 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.11/0.32 Usage: tptp [options] [-file:]file
% 0.11/0.32 -h, -? prints this message.
% 0.11/0.32 -smt2 print SMT-LIB2 benchmark.
% 0.11/0.32 -m, -model generate model.
% 0.11/0.32 -p, -proof generate proof.
% 0.11/0.32 -c, -core generate unsat core of named formulas.
% 0.11/0.32 -st, -statistics display statistics.
% 0.11/0.32 -t:timeout set timeout (in second).
% 0.11/0.32 -smt2status display status in smt2 format instead of SZS.
% 0.11/0.32 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.11/0.32 -<param>:<value> configuration parameter and value.
% 0.11/0.32 -o:<output-file> file to place output in.
% 0.11/0.39 % SZS status Theorem
% 0.11/0.39 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------